OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

1988 Steele Prizes page 965

;I~ The AMS Centennial: Social and Mathematical Festivities page 970

SEPTEMBER 1988, VOLUME 35, NUMBER 7 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences

This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting. Abstracts should be sub­ and annual meetings are joint meetings of the Mathematical Associ­ mitted on special forms which are available in many departments of ation of America and the American Mathematical Society. The meet­ mathematics and from the headquarters office of the Society. Ab­ ing dates which fall rather far in the future are subject to change; this stracts of papers to be presented at the meeting must be received is particularly true of meetings to which no numbers have been as­ at the headquarters of the Society in Providence, Rhode Island, on signed. Programs of the meetings will appear in the issues indicated or before the deadline given below for the meeting. Note that the below. First and supplementary announcements of the meetings will deadline for abstracts for consideration for presentation at special have appeared in earlier issues. sessions is usually three weeks earlier than that specified below. For Abstracts of papers presented at a meeting of the Society are pub­ additional information, consult the meeting announcements and the lished in the journal Abstracts of papers presented to the American list of organizers of special sessions. Meetings

Abstract Program Meeting # Date Place Deadline Issue 845 * October 28-30, 1988 Lawrence, Kansas Expired October 846 * November 12-13, 1988 Claremont, California Expired October 847 * January 11-14, 1989 Phoenix, Arizonat October 12 ** December (95th Annual Meeting) * April 15-16, 1989 Worcester, Massachusetts January 25 March * May 19-20, 1989 Chicago, Illinois March 1 April August 7-10, 1989 Boulder, Colorado May 16 July I August (92nd Summer Meeting) October 21-22, 1989 Hoboken, New Jersey August 30 October October 27-28, 1989 Muncie, Indiana August 30 October January 17-20, 1990 Louisville, Kentucky (96th Annual Meeting) January 16-19, 1991 San Francisco, California (97th Annual Meeting) * Please refer to page 1052 for listing of special sessions t Preregistration/Housing deadline is November 10 ** MAA Contributed Paper deadline is September 30

Deadlines

October Issue November Issue December Issue January Issue Classified Ads* Aug 31, 1988 Oct 3, 1988 Oct 31, 1988 Nov 30, 1988 News Items Sept 6, 1988 Oct 7, 1988 Nov 4, 1988 Nov 25, 1988 Meeting Announcements** Aug 24, 1988 Sept 26, 1988 Oct 24, 1988 Nov 17, 1988 * Please contact AMS Advertising Department for an Advertising Rate Card for display advertising deadlines. ** For material to appear in the Mathematical Sciences Meetings and Conferences section. OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

ARTICLES DEPARTMENTS

965 1988 Steele Prizes Awarded 963 Letters to the Editor The 1988 Steele Prizes were awarded at the Society's ninety-first Summer 1002 News and Announcements Meeting and Centennial Celebration in Providence to Sigurdur Helgason for expository writing, to Gian-Carlo Rota for a fundamental paper, and to Deane 1007 NSF News and Reports Montgomery for his mathematical career. 1013 News from Washington 1018 Acknowledgement 970 The AMS Centennial: Social and Mathematical Festivities of Contributions An array of festivities, both mathematical and social, made this 1Oath birthday celebration a very special event. 1040 1988 AMS Elections 1041 Election Information 1043 Meetings and Conferences of the AMS (Listing) FEATURE COLUMNS 1055 Mathematical Sciences Meetings and Conferences 1061 New AMS Publications 974 Inside the AMS: Report on the Council Meeting, August 1988 1065 AMS Reports and In order to better acquaint members with the important role the Council plays Communications in the Society, this report on the most recent Council meeting is presented. Recent Appointments, 1065 976 Computers and Mathematicians Jon Barwise Statistics on Women Mathematicians, 1065 Six of the most popular mathematical computer programs on the market are Officers of the Society, 1067 reviewed by Barry Simon and Robert Wilson in their article, "Supercalculators on the PC." 1078 Miscellaneous Personal Items, 1078 Deaths, 1078 1080 Visiting Mathematicians (Supplement) 1082 Backlog of Mathematics Research Journals 1085 New Members of the AMS 1087 Classified Advertising 1107 Forms MR and CMP on CD-ROM AMERICAN MATHEMATICAL SOCIETY

One of the highlights of the Centennial Celebration in Providence was the AMS preview of a new information medium: a CD-ROM (Compact Disc­ Read Only Memory) which contains records from Mathematical Reviews EDITORIAL COMMITTEE Robert J. Blattner, Ralph P. Boas (MR) and Current Mathematical Publications ( CMP). The actual CD-ROM, Lucy J. Garnett, Mary Ellen Rudin called MathSci Disc, will be available to the public in January 1989 and Nancy K. Stanton, Steven H. Weintraub will contain all reviews and abstracts from MR 1985 through 1988 and over Everett Pitcher (Chairman) 50,000 entries from CMP. This first release in January will be followed by semi-annual discs that will incorporate the current six months of MR and MANAGING EDITOR CMP into material on the previous disc. Although a CD-ROM has tremen­ James A. Voytuk dous storage capacity, as more and more records are added the disc becomes ASSOCIATE EDITORS full, and eventually new discs will be started which include a portion of the Ronald L. Graham, Special Articles previous CD-ROM. Archival discs with MR records from years before 1985 Jeffrey C. Lagarlas, Spec/a/ Articles may also be produced in the future.

SUBSCRIPTION INFORMATION The CD-ROM technology is essentially the same as that used for audio com­ Subscription prices for Volume 35 (1988) are pact discs, where information is encoded on the disc in a digitized form $1051ist; $84 institutional member; $63 individual through a series of pits impressed into the disc. A "reader" retrieves the dig­ member. (The subscription price for members Is ital information by using the scattered light from a laser focused on the pits. Included In the annual dues.) A late charge of A CD-ROM is only 4 3/4 inches across, but it holds up to 500 megabytes 1Oo/o of the subscription price will be imposed upon orders received from nonmembers after of data, which is the equivalent of about 275,000 pages of printed material. January 1 of the subscription year. Add for post· It is this storage capacity that makes it possible to put four years, or almost age: Surface delivery outside the United States five linear feet, of MR on a single disc and still leave room for software and and lndia-$1 0; to lndia-$20; expedited deliv· indexes that will allow for quick and efficient searching of the information. ery to destinations in North America-$15; else­ A CD-ROM, like a magnetic disk, provides random access to the data, and where-$38. Subscriptions and orders for AMS publications should be addressed to the Amer­ the search software will allow an individual to find items by using a variety ican Mathematical Society, P.O. Box 1571, An· of descriptors that would identify a particular item. In the case of MR these nex Station, Providence, Rl 02901-9930. All or­ identifiers will include: author names, classification codes, words in the ti­ ders must be prepaid. tle of an article or book, words from the reviewer abstract, journal names, publishers, and other data elements. The only hardware necessary to use a ADVERTISING CD-ROM is a PC, such as an IBM-AT with a hard disk, and a CD-ROM Notices publishes situations wanted and classi­ fied advertising, and display advertising for pub­ reader. Shortly, software and hardware will be available for connecting a lishers and academic or scientific organizations. reader to a Macintosh. Copyright@ 1988 by the American Mathemat· The MathSci Disc will be produced for the AMS by SilverPlatter, a com­ lcal Society. All rights reserved. Printed in the United States of America. pany specializing in information products on CO-ROM's. SilverPlatter will The paper used in this journal is acid-free and provide search software and do the indexing of the MR and CMP files, and falls within the guidelines established to ensure they will carry the product through to the actual cutting of the master disc permanence and durability.@ and the duplication of copies for distribution. SilverPlatter will also provide [Notices of the American Mathematical Society is documentation and toll-free help services to the user. Records selected from published ten times a year (January, February, MathSci Disc can be downloaded to the microcomputer for editing or can be March, April, May/June, July/August, Septem­ processed with TEX software to convert the records into the typeset form in ber, October, November, December) by the Amer­ ican Mathematical Society at 201 Charles Street, which they appear in MR, with the actual mathematical expressions. Between Providence, Rl 02904. Second class postage paid the semi-annual updates of MathSci Disc, the online service of MathSci from at Providence, Rl and additional mailing offices. the vendors BRS, DIALOG, and ESA, can be used to search the most recent POSTMASTER: Send address change notices to monthly records which are concurrent with the printed journal. Notices of the American Mathematical Society, Membership and Sales Department, American The MathSci Disc will be distributed by the AMS, through an annual lease Mathematical Society, P. 0. Box 6248, Provi· arrangement, to both institutions and individuals, with substantial discounts dence, Rl 02940.] Publication here of the Soci· to individuals at subscribing institutions. More information about the pricing ety's street address, and the other information in structure for MathSci Disc can be found on page 1039. brackets above, Is a technical requirement of the U.S. Postal Service. All correspondence should be mailed to the Post Office Box, NOT the street address.

962 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Mathematics in the News Military Funding I've just taught Number Theory; it was a On the Referendum, etc. fortuitous time. I never before had "cur­ As a Naval officer still serving in the rent events" or "show and tell" in a class Navy Reserve, and as an industrial math­ in pure mathematics, but this Spring sev­ Letters ematician working principally on Navy eral of my students brought newspaper contracts, I feel a deep commitment to clippings on the day the Boston Globe told to the Editor the nation's defense. I put up with the of Miyaoka's proof of the Fermat conjec­ negative aspects of industrial mathemat­ ture. ics, such as long hours, lack of funding I'd had enough advance warning to for publication, and lack of representa­ prepare a lecture proving Miyaoka's The­ tion in the mathematical community be­ orem (for so I called it) in the special case cause I feel that the work I do is use­ n = 4, using Fermat's well known argu­ ful and important to the Navy. There­ ment by descent. fore I was disappointed-even dejected, I kept my class informed of the sta­ maybe-to read the triumphant letter re­ tus of the theorem as those entitled to prove theorems. Most mathematics ma­ garding the results of the referendum an opinion debated in wandering email. jors calculate well enough. Many can solve (Notices, p. 675). In time, as we now know, doubt over­ problems. Some can formulate conjectures Even though I voted for some of the whelmed belief, Miyaoka withdrew his if given appropriate experimental mate­ motions, it is very hard not to take the claim, and the Conjecture stands as be­ rial. But all too few can prove even things vote as a condemnation of what I do and fore. they know to be true. believe. Not only am I not represented, But there's more to the story. One of What I have always found most frus­ but the organization I once aspired to join the hardest things we teach students is to trating is not my students' inability to doesn't seem to want my kind around any­ prove a theorem-we all have our more. I thought our common bond was limits-but their inability to recognize mathematics, not politics. Policy on Letters to the Editor when they've written what I call nonsense. The adjacent letter from Professor Letters submitted for publication in Notices I'm irrationally offended by blatant faulty Gurevic also hit home. I, too, looked in are reviewed by the Editorial Committee, logic in student papers: theorems proved vain at the Atlanta meetings for people to whose task is to determine which ones are by example ("for all" read as "there ex­ talk mathematics with. The meetings are suitable for publication. The publication ists"), statements construed as their con­ the only time I even get to see other math­ schedule requires from two to four months verses, arguments which, if correct, would ematicians. I have uncovered some inter­ between receipt of the letter in Providence show that 1 = 0, .... We all know the esting network theory problems in my and publication of the earliest issue of No­ litany. tices in which it could appear. work in position-finding, command and But several weeks of this, following control networks, and software design, but Publication decisions are ultimately Miyaoka's dashed hopes, led me to apol­ made by majority vote of the Editorial Com­ I'll be darned if I could get anyone to talk ogize to my students, as I hope mittee, with ample provision for prior dis­ you will to me about them. I'm reduced to order­ cussion by committee members, by mail or to yours. It dawned on me, all too belat­ ing books through the mail, teaching my­ at meetings. Because of this discussion pe­ edly, that they struggle quite as hard at self, and slowly building a theory from riod, some letters may require as much as the edge of what they can comprehend scratch. This is not unlike the military seven months before a final decision is made. as we do when we do new mathematics. trying to find someone to help it with its Letters which have been, or may be, pub­ We know that believing something is the mathematical problems, and I would like lished elsewhere will be considered, but the first step toward proving it. While doubt to briefly discuss that analogy. Managing Editor of Notices should be in­ is a more reliable. ultimate testing mode, formed of this fact when the letter is sub­ From the point of view of the mili­ without hope nothing is possible. So they tary, each branch of the armed services mitted. and we and Miyaoka all hope. We're not The committee reserves the right to edit . has one or more agencies-which usually angry letters. at Miyaoka, nor ashamed of him, employ some mathematicians-whose job Notices does not ordinarily publish com­ nor do we consider him foolish or igno­ it is to analyze the requirements of the plaints about reviews of books or articles, rant or dumb or careless. Let's remember operational forces and provide hardware although rebuttals and correspondence con­ to treat our students' struggles with the and software to meet those requirements. cerning reviews in Bulletin of the Ameri­ same respect we accord his. These agencies let contracts to civilian can Mathematical Society will be consid­ firms, usually "defense contractors" who ered for publication. Ethan Bolker specialize in such work because of the Letters should be typed and in legible University of Massachusetts, Boston ex­ traordinary form or they will be returned to the sender, (Received May 25, 1988) demands of specification, test­ possibly resulting in a delay of publication. ing, etc. These firms also employ mathe­ Letters should be mailed to the Editor maticians. Employees of these firms, in­ of Notices, American Mathematical Soci­ cluding mathematicians, must account for ety, P.O. Box 6248, Providence, RI 02940, each hour spent by attributing it to a par­ and will be acknowledged on receipt. ticular task of a particular contract.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 963 Letters to the Editor

Failure to do so accurately can become • provide funding to outside mathe­ Politicization of the Society something that you read about in the news­ maticians to work with Navy personnel When I joined the AMS in 1965, I consid­ papers. These mathematicians are not sup­ or contractors in solving particular prob­ ered the Society to be a professional or­ posed to be doing "basic research"; in re­ lems; ganization devoted to the furtherance of ality, they do a great deal of it because • act as a mathematical "Board of In­ study and research in mathematics. Math­ they can't meet the contract requirements spection and Survey," visiting Navy Sys­ ematics is, in and of itself, devoid of po­ without it. Unfortunately, contracts are tems Commands, laboratories, and con­ litical content. I always felt that the AMS, rarely so well funded that they can sup­ tractors to see what research is being done; or any professional society for that mat­ port publication of the results-they pay exposing defective mathematics, if neces­ ter, should be similarly apolitical, to the only to get an "answer." sary, but generally fostering better math­ extent that I believe that the Society As one such mathematician, I rarely ematics at these sites; should not ever make political statements find in the literature the mathematics I • publish an informal journal-a of any kind, whether about the treatment need to solve problems. If my experience newsletter or electronic bulletin board, of Jews in Russia, Palestinians in Israel, is typical, it makes sense for the govern­ perhaps-which would allow defense en­ Greeks in Anatolia, blacks in South Africa, ment to fund efforts to create the math­ gineers and mathematicians to broach women anywhere, or whales by the ematics. The irony is that the research problems and make suggestions without Japanese. funding agencies, such as ONR, do not the intimidating requirements of formal With the recent passage of the refer­ know what the problems to be solved are. publication; endum issues by a large majority of the Their employees are civilians, and often • sponsor informal conferences on top­ membership of the Society, it is appar­ do not have security clearances allowing ics of broad interest, and otherwise put ent to me that not only has the Society them access to information about ongo­ mathematicians and engineers doing sim­ become extremely more politicized than ing projects. I have never heard of them ilar work in touch with each other. my worst fears, but that the ship of state visiting contractor facilities. They have a The mathematicians administering of the AMS has taken a sudden and sharp lot of employee turnover. They seem to such a program within ONR would them­ turn to the left and become a ship of fools. operate in response to proposals from the selves need to be gifted, broadly educated, I do not suffer fools gladly and have no academic community. and be granted access to classified infor­ wish to be associated with them, be they Because of this, I used to think that mation. The mathematics itself is usually mathematicians or politicians. Accord­ the reason for having such research fund­ unclassified, but as any applied mathema­ ingly, I shall not renew my membership ing agencies was to promote goodwill and tician knows, recognizing what the prob­ in the AMS in December, nor shall I seek to foster the growth of a talent pool-the lem is, or even that it is a mathematical to rejoin the Society at any time in the fu­ same reason the military might want to problem, is often the most difficult step ture unless I learn that it has cleansed it­ support the Boy Scouts or amateur radio, in the solution. It is also the step which self of this political folly and made struc­ for example. To accuse them of coercion usually requires a security clearance. tural changes which would prevent such of because they do not submit the proposals It was distressing to see the results from ever happening again. they receive to peer review seems absurd. the referendum, but perhaps some good It is with great sorrow that I find it In my opinion, the greatest criticism that will come of it. necessary to write this letter. I may hope can be made against them is that they are R. Peter DeLong that the Society will come to its senses making too little contribution to solving Hughes Aircraft Company, and correct this egregious error which it the military's mathematical problems. Fullerton, California has made, but I do not hold out much If the academic mathematical commu­ (Received June 27, 1988) hope for this. From the size of the vote, nity doesn't want military funding, and it appears that I am in a minority which military agencies are unable to attract at­ is too small to influence the course the tention to their mathematical problems, Society is taking. I can now only suggest both mathematics and the taxpayer would that in the interests of "truth in labelling," be better served by changing the way the the Society should now change its name military spends its research dollars. I to "The Left-wing American Mathemati­ would recommend enfranchising the cal Society." Even if I were a leftwinger, Mathematics Section of ONR, for exam­ which, as you may have guessed, I am not, ple, to I would not want to be a member of such • provide funding to enable defense a society. industry mathematicians to finish and publish their findings. I think money spent Randolph Constantine in this way would produce more useful Bayfield, Colorado mathematics than money spent anywhere (Received June 3, 1988) else;

964 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1988 STEELE PRIZES AWARDED AT CENTENNIAL CELEBRATION IN PROVIDENCE

Three Leroy P. Steele Prizes were awarded at the Society's (Academic Press, 1978), and Groups and Geometric Anal­ ninety-first Summer Meeting and Centennial Celebration ysis (Academic Press, 1984). in Providence, Rhode Island. In 1962 Sigurdur Helgason published a book which The Steele Prizes are made possible by a bequest to has become a classic. The subject matter included central the Society by Mr. Steele, a graduate of Harvard College, topics in geometry and Lie group theory, with important Class of 1923, in memory of George David Birkhoff, ramifications for harmonic analysis. More recently this William Fogg Osgood, and William Caspar Graustein. material has been revised and expanded into a two Three Steele Prizes are awarded each Summer: one volume treatment. for expository mathematical writing, one for a research Proceeding at a leisurely pace, the author first leads the paper of fundamental and lasting importance, and one reader through the basic theory of differential geometry, in recognition of cumulative influence extending over a emphasizing an invariant, coordinate-free development. career, including the education of doctoral students. The Next is a careful treatment of the foundations of the current award is $4,000 for each of these categories. theory of Lie groups, presented in a manner which The recipients of the Steele Prizes for 1988 are SIGUR­ since 1962 has served as a model for the treatment of DUR HELGASON for the expository award; GIAN-CARLO this subject by a number of subsequent authors. The RoTA for research work of fundamental importance; and central theme of symmetric spaces is related in a clear DEANE MONTGOMERY for the career award. fashion to the study of semisimple Lie groups and tools The Steele Prizes are awarded by the Council of are assembled for the classification of these objects, the Society, acting through a selection committee whose first into large classes, e.g., compact and noncompact members at the time of these selections were Frederick symmetric spaces, Hermitian symmetric spaces, then J. Almgren, Luis A. Caffarelli, Hermann Flaschka, John the fine classification. The last volume covers numerous P. Hempel, William S. Massey (chairman), Frank A. significant topics in harmonic analysis, from the Radon Raymond, Neil J. A. Sloane, Louis Solomon, Richard P. transform, to invariant differential operators, to Harish­ Stanley and Michael E. Taylor. Chandra's c-function, ending with a quick overview of The text that follows contains the Committee's ci­ harmonic analysis on compact symmetric spaces in terms tations for each award, the recipients' responses at the of the representation theory of compact Lie groups. prize session in Providence, and a brief biographical The exposition throughout is a model of clarity. sketch of each of the recipients. Professor Montgomery Arguments in proofs are very clean, the organization is was unable to attend the Summer Meeting to receive the superb, and the material ranges over a wide vista of prize in person. He did, however, send a written response important topics of interest to a broad segment of the to the award. mathematical community. Response Expository Writing I feel deeply grateful and honored to receive the Steele Sigurdur Helgason Prize at this Centennial Celebration. The first book in question, Differential Geometry Citation and Symmetric Spaces from 1962, represents my efforts (originating in 1955) at combining Elie Cartan's differ­ The 1988 Steele Prize for expository writing is awarded ential geometric work on symmetric spaces with some of to SIGURDUR HELGASON for his books Differential Ge­ Harish-Chandra's algebraic and analytic work on repre­ ometry and Symmetric Spaces (Academic Press, 1962), sentation theory of semisimple Lie groups. The ultimate Differential Geometry, Lie Groups, and Symmetric Spaces purpose, however, was to develop geometric analysis on

SEPTEMBER 1988, VOLUME 35, NUMBER 7 965 1988 Steele Prizes symmetric spaces in analogy with Fourier analysis and R. Hermann, A. Koninyi, B. Kostant, J. L. Koszul, Radon transforms on Rn and partial differential oper­ A. P. Mattuck, G. D. Mostow, K. Nomizu, R. Palais, ators with constant coefficients. My 1984 book, Groups J. Wolf. I remember this association with deep gratitude. and Geometric Analysis, treats the simplest examples and Harish-Chandra's papers offered an interesting con­ then deals with the first part of the general project.. trast to Cartan's work. While his papers reflected deep As this geometric analysis on symmetric spaces has originality and accumulated technical power, his proofs developed, some unexpected feedback in classical anal­ were careful in details so that motivation and patience ysis has materialized. For example, the familiar Poisson were sufficient for understanding, at least on the local integral formula level. It was a source of great satisfaction to me to integrate some of the works of these two great mathe­ u(x) P(x,b)F(b)db maticians in my 1962 book. =Is The original project, geometric analysis on Rieman­ for harmonic functions u in the unit disk D with nian symmetric spaces, is the subject of the 1984 volume boundary B becomes a special result in non-Euclidean and of a further volume in preparation. It is gratifying Fourier analysis on D considered as the hyperbolic plane. also to see analysis on nonRiemannian symmetric spaces This circumstance then suggested that each eigenfunction progressing vigorously in several quarters in recent years. u of the Laplace-Beltrami operator L on the hyperbolic plane, (say Lu = c(c- 1)u), should have the form

u(x) =Is P(x,b)cdT(b) with a certain functional T on the boundary B. A priori one would expect that the needed class of functionals T would depend on the eigenvalue c(c- 1), but to my surprise I found that the functionals needed were always exactly the hyperfunctions on B, independently of c. Thus hyperfunctions, which at that time ( 1970) had existed as rather isolated objects outside the mainstream of analysis, showed themselves to be firmly attached to basic analysis on symmetric spaces. This connection has been explored much further in the outstanding work of several Japanese mathematicians. During the fifties when I embarked on this work, differential geometry had not acquired the great popular­ ity which it enjoys today. Thus I felt compelled in my 1962 book to write an exposition of basic Riemannian geometry, particularly the Hadamard-Cartan's theory of manifolds of negative curvature, and Cartan's theory of symmetric spaces and semisimple Lie groups. It was an interesting experience trying to understand his work in these areas. While his thesis from 1894 was not too difficult to fathom, his papers during the late 1920's on symmetric spaces reflected his accumulated experience with Lie groups, combined with a remarkable geometric intuition; as a result some of his proofs were rather baffling in their informality. When I have taught this Sigurdur Helgason material on later occasions I have been embarassed by the clumsiness of some of my proofs. It seems that my exposition of these results was more intended to Biographical Sketch convince myself that the results were true rather than to Sigurdur Helgason was born on September 30, 1927 in explain them to others. In this pursuit I was helped by Akureyri, Iceland. He received his Ph.D. from Princeton many mathematicians through personal contact, seminar University in 1954. activity and written papers; here I would like to men­ During his academic career, Professor Helgason has tion A. Borel, S-S. Chern, J.l. Hano, Harish-Chandra, served as Moore Instructor of Mathematics at the

966 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1988 Steele Prizes

Massachusetts Institute of Technology (1954-1956) and discipline. Two of its most prominent features are its Louis Block Lecturer at the University of Chicago ( 1957- unifying techniques which bring together a host of 1959). At MIT, he moved from Assistant Professor previously disparate topics, and its deep connections of Mathematics to Associate Professor of Mathematics with other branches of mathematics, such as algebraic ( 1959-1965). He held visiting positions at Princeton Uni­ topology, algebraic geometry, commutative algebra, and versity (1956-1957) and at Columbia University (1959- representation theory. The single paper most responsible 1960). Since 1965, he has been Professor of Mathematics for bringing on this revolution is the paper of Rota cited at MIT. He has also been, on leave, at the Institute for above. It showed how the theory of Mobius functions of a Advanced Study {1964-1966, 1974-1975, and Fall1983), partially ordered set, as developed earlier by L. Weisner, and at the lnstitut Mittag-Leffier (1970-1971). P. Hall, and others, could be used to unify and generalize Professor Helgason has been a member of the Ameri­ a wide selection of combinatorial results. Moreover, can Mathematical Society for 35 years and has given the it hinted at connections with algebra, topology, and following addresses: Invited Address, Summer Meeting, geometry which were later to be extensively developed Boulder, August 1963; Summer Institute on Harmonic by Rota and his followers. Today the theory of Mobius Analysis on Homogeneous Spaces, Williamstown, July functions occupies a central position within algebraic 1972; Invited Address, Annual Meeting, Washington, combinatorics and has found many applications outside D.C., January 1975; Special Session on Representations combinatorics. Perhaps more importantly, Rota's paper of Lie groups, Washington, D.C., October 1979. He gave has inspired many mathematicians to develop systematic an Invited Address at the 1970 International Congress of techniques for solving combinatorial problems and to Mathematicians in Nice. He also served on the Organiz­ apply them to problems outside combinatorics. ing Committee for the 1972 Summer Research Institute and the 1984 AMS Summer Research Conference on Integral Geometry. Response Professor Helgason received the Gold Medal of the I feel deeply honored by the Steele Prize which the University of Copenhagen in 1951 and held a Guggen­ Society has voted to award me this year, and I am heim Fellowship at the Institute for Advanced Study in delighted to accept it. 1964-1965. He was awarded a Doctor Honoris Causa The generalization of the Mobius function of number from the University of Iceland in 1986 and from the theory to locally finite partially ordered sets is an idea University of Copenhagen in 1988. He is a member of whose time has come. The fact that I should have been the Icelandic Academy of Sciences, the Royal Danish the one to first point out the timeliness of this idea is a Academy of Sciences and Letters, and the American historical accident. Academy of Arts and Sciences. I am sure that some combinatorialists of the early Professor Helgason's research interests include Lie part of this century who leafed through Dickson'~ History groups and differential geometry, integral geometry, and of the Theory of Numbers had realized that many of the harmonic analysis and differential equations on Lie identities collected in that book relating to the number­ groups and coset spaces. theoretic Mobius function depended only on the divis­ ibility partial order on the integers. Hans Rademacher once told me that he had been struck by this fact, and Fundamental Paper admitted that he had not been able to carry through a Gian-Carlo Rota proper generalization. What he missed was an insight that came almost simultaneously to Louis Weisner and Citation to Philip Hall in the thirties. They realized that the The 1988 Steele Prize for a paper which has proved to generalization could be carried out using functions of be of fundamental or lasting importance in its field is two variables on a partially ordered set, rather than using awarded to GIAN-CARLO ROTA for his paper: analogs of the arithmetic functions of number theory. Functions of two variables on a partially ordered set On the foundations of combinatorial theory, (under certain restrictions) form an algebra, which in I .. Theory of Mobius functions. my paper I called the incidence algebra. This algebra can Zeitschrift fiir Wahrscheinlichkeitstheorie be viewed as a generalization of the algebra of upper und Verwandte Gebiete, 2 (1964), pages 340-368. triangular matrices. Only 25 years ago the subject of combinatorics was Applications of the Mobius inversion formula on a regarded with disdain by "mainstream" mathematicians, partially ordered set keep cropping up. We may recall T. who considered it as little more than a bag of ad P. Speed's theory of statistical cumulants, the generaliza­ hoc tricks. Now, however, the new subject of "algebraic tion to all finite group actions of the Moreau-Witt for­ combinatorics" is a highly active and universally accepted mula for the number of primitive necklaces, Zaslavsky's

SEPTEMBER 1988, VOLUME 35, NUMBER 7 967 1988 Steele Prizes theory of enumeration of regions in arrangements of hy­ of the same kind. This set of problems is known as perplanes in Rn, and the more recent flurry of activity on the critical problem. Although much work has been done the algebraic topology of finite topological spaces defined on the critical problem, it remains beyond the reach of by partially ordered sets, where the Mobius function today's mathematics, and we may at best wish we will computes some homology and homotopy invariants. live long enough to see it solved. Biographical Sketch Gian-Carlo Rota was born on April 27, 1932 in Italy. He came to the United States in 1950 and became an American citizen in 1961. He received his Ph.D. from Yale University in 1956 under the direction of Jacob T. Schwartz. Professor Rota began his academic career as a Fel­ low at the Courant Institute of Mathematical Sciences (1956-1957). At Harvard University, he served as a Ben­ jamin Peirce Instructor of Mathematics (1957-1959). At the Massachusetts Institute of Technology he progressed from Assistant Professor of Mathematics to Associate Professor of Mathematics (1959-1965). In 1965 Profes­ sor Rota transferred to Rockefeller University, where he was a Professor of Mathematics until 1967. Professor Rota returned to MIT in 1967, where he served as Pro­ fessor of Mathematics until 1974. Since 1974, he has been Professor of Applied Mathematics and Philosophy at this same institution. Professor Rota has been a member of the American Mathematical Society for 33 years. He was a Member­ at-large of the Council (1967-1968) and was Editor of the Bulletin of the American Mathematical Society Gian-Carlo Rota (1968-1973). Professor Rota gave Invited Addresses at the Interna­ More than fifty years ago, G. D. Birkhoff succeeded in tional Congresses of Mathematicians in Nice in 1970 and associating to every graph a polynomial in one variable in Helsinki in 1978. He was the Hardy Lecturer, London x, now called the chromatic polynomial. When evaluated Mathematical Society ( 1972); Professore Linceo, Scuola at x = n, the chromatic polynomial gives the number of Normale Superiore, Pisa ( 1979 and 1984); and gave the ways of coloring the graph in n colors. Garrett Birkhoff, in Hedrick Lectures, Mathematical Association of Amer­ the second edition of his "Lattice Theory", remarked that ica ( 1967). Professor Rota has also given the following the chromatic polynomial can be computed by Mobius AMS addresses: Symposium on Stochastic Processes, inversion on the lattice of contractions of the graph. New York, April 1963; Special Session on Combinatorial A similar, more general polynomial, the characteristic Mathematics, Annual Meeting, Chicago, January 1966; polynomial, can be defined on any finite partially ordered Symposium on Combinatorics, Los Angeles, March 1968; set by Mobius inversion. The values of the characteristic Invited Address, New York, March 1972; Special Session polynomial give combinatorial information on the partial on Combinatorial Algorithms, New York, April1974; In­ order. In the case of lattices of flats of matroids (for vited Address, Wellesley, October 1977; Special Session example, for arrangements of hyperplanes), the zeros on Combinatorics, Fairfield, October 1983. of the characteristic polynomial can be given explicit Professor Rota was an Alfred P. Sloan Fellow ( 1963- combinatorial interpretations in terms of the existence 1965). He is a member of the National Academy of or non-existence of certain extremal configurations, much Sciences, and a Corresponding Member of the Academia like Hadwiger conjectured in the case of graphs. Thanks Argentina de Ciencias. He is a Fellow of the American to the characteristic polynomial of a partially ordered Academy of Arts and Sciences, of the Institute of set, of which the chromatic polynomial of a graph is a Mathematical Statistics, of the American Association for special case, the problem of coloring a graph is seen to the Advancement of Science, and of the Los Alamos be only one instance (which, by chance, happened to be National Laboratory. In 1984, he received an honorary historically the first) of a wide class of combinatorial degree from the University of Strasbourg. Professor problems, old and new, all of them presenting difficulties Rota is the founder of the Journal of Combinatorial

968 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1988 Steele Prizes

Theory ( 1967), Advances in Mathematics ( 1967), and of 1978. He has been a member of the National Academy Advances in Applied Mathematics ( 1980). of Sciences since 1955. All these honors and obligations His areas of research interest include combinatorial testify to his standing in the international mathematical theory, probability, and phenomenology. community. Response Career Award It is gratifying to receive a Steele prize for my work in a Deane Montgomery profession which has given me so much pleasure. It has been my good fortune to have had the help of first rate Citation collaborators and congenial and eminent colleagues and The 1988 Steele Prize for cumulative influence is awarded friends. Mathematics has managed to remain a rather tO DEANE MONTGOMERY for his lasting impact on math­ unified subject; mathematicians don't always agree, but ematics, particularly mathematics in America. Mont­ they have usually come together in supporting the main gomery is one of the founders of the modern theory of goals of the subject in spite of its breadth and diversity. transformation groups. This subject has its roots in the 19th century with the work of Sophus Lie, Felix Klein, and Henri Poincare. The work by many renowned mathematicians on Hilbert's fifth problem during the first half of our century was a catalyst to the development of much of the theory of the structure of topological and Lie groups. Montgomery's contributions, which extended over fifteen years, to the solution of Hilbert's fifth problem are very well known. His book, Topological Transformation Groups, coauthored with Leo Zippin, provides a complete and accessible account of the problem and its solution. In the course of working on this and related problems, Montgomery and his collaborators provided much of the terminology, basic constructions, foundational ideas, and standard machinery of transformation groups. As the subject matured, Montgomery and his collab­ orators led the way with influential papers that incorpo­ rated the latest developments of topology. These seminal papers opened up entirely new areas for investigation. Today the subject has a symbiotic relationship with many parts of mathematics and often serves as a testing ground for the efficacy of new ideas in mathematics. In all its ramifications it is difficult to find pieces of the subject Deane Montgomery that do not bear Montgomery's imprint. Montgomery's influence is pervasive at the Insti­ Biographical Sketch tute for Advanced Study. He made a special effort to Deane Montgomery was born on September 2, 1909 search out promising young American mathematicians in Weaver, Minnesota. He received his Ph.D. from the and bring them to the Institute ~ He acquainted himself University of Iowa in 1933. with all the young visitors and cordially offered much Professor Montgomery began his professional career mathematical and moral support. He worked very hard as a National Research Council Fellow at Harvard to have the Institute provide the best environment for University ( 1933-1934) and at the Institute for Advanced the development of the young visitors' talents. Many will Study ( 1934-1935). He moved from Assistant Professor testify how this helped them to live up to their promise. of Mathematics to Associate Professor of Mathematics His legacy, on this score, is with the postdoctoral students at Smith College ( 1935-1946) and also, during this at the Institute. period, was a Guggenheim Fellow (1941-1942). While Montgomery has also been active and visible in pro­ teaching at Smith College, Professor Montgomery held fessional organizations for mathematicians. Two indica­ a concurrent position as a Visiting Associate Professor tions of this are his terms as President of the American of Mathematics at Princeton University ( 1943-1945). Mathematical Society in 1961 and 1962 and as President During 1945-1946 he worked for John von Neumann of the International Mathematical Union from 1975 to on a project concerning numerical analysis. He has

SEPTEMBER 1988, VOLUME 35, NUMBER 7 969 1988 Steele Prizes also served as an Associate Professor of Mathematics Executive Committee (1955-1956); Committee on the at Yale University ( 1946-1948). Since 1948 Professor Relationships Between Headquarters and Mathemati­ Montgomery has been at the Institute for Advanced cal Reviews (1957); Committee on Expository Books Study. He began as a permanent member and, in 1951, (1958, 1959); Committee to Consider Publishing Col­ he was named Professor of Mathematics. Since 1980, he lected Works of Mathematicians ( 1959); Committee to has been Professor Emeritus of Mathematics. Select Gibbs Lecturers ( 1961, 1962); Nominating Com­ Professor Montgomery has been a member of the mittee ( 1965). American Mathematical Society for 55 years and has Professor Montgomery has given the following ad­ served the Society as Vice President (1952-1953), as dresses: Topological Transformation Groups in Eu­ Trustee (1955-1961) and as President (1961-1962). He clidean Spaces, at a meeting of Section A, American As­ was president of the International Mathematical Union sociation for the Advancement of Science, Durham, June from 1975 to 1978. 1941; Invited Address, New York, October 1943; Collo­ Professor Montgomery has served on the following quium Lecture, Summer Meeting, Minneapolis, Septem­ AMS committees: Bulletin Editorial Committee ( 1946- ber 19 51; Invited Address, International Congress of 1949); Committee to Nominate Officers and Committees Mathematicians, September 1954; Presidential Address, for the International Congress of Mathematicians ( 1948); Annual Meeting, Miami, January 1964; Special Session Committee to Select Hour Speakers for Eastern Sectional on Semi-groups and Topological Algebras, Lexington, Meetings (1948-1949); Committee to Nominate Officers November 1965. and Members of the Council (1951, 1956); Commit­ Professor Montgomery is also a member of the tee to Select Hour Speakers for Annual and Summer National Academy of Science, the International Mathe­ Meetings (1951-1952); Committee to Nominate a Rep­ matical Union (President, 197 4-197 5), and the American resentative of the Society on the Policy Committee Philosophical Society. His areas of research interest in­ for Mathematics ( 1953); Colloquium Editorial Commit­ clude topology and topological groups. tee, 1953-1958; Committee on Publications (1954-1958);

THE AMS CENTENNIAL: SOCIAL AND MATHEMATICAL FESTMTIES

Almost 1700 people attended the AMS Centennial Cel­ the President on ceremonial occasions. Rhonda Hughes, ebration, held August 8-12, 1988, in Providence, Rhode President of the Association for Women in Mathematics, Island, home of the AMS headquarters. An array of fes­ presented the Society with a contribution to the AMS tivities, both mathematical and social, made this 1OOth Centennial Research Fellowship Fund. The audience was birthday party a very special event. also addressed by Charles W. Gear, President of the The Opening Ceremonies were held in the opulent Society for Industrial and Applied Mathematics. Providence Performing Arts Center, which originally Leonard Gillman, President of the Mathematical opened in 1928. In this grand setting embellished with Association of America (MAA), told the crowd that he brass, bronze, marble, and gilt, several hundred math­ could not bring the MAA's gift because it weighs about ematicians listened to a selection of songs chosen to 550 pounds. The gift is a sculpture in white Carrara showcase the Arts Center's Mighty Wurlitzer pipe organ. marble from the mountains of northern Italy. Entitled AMS President George Daniel Mostow, serving as the "Torus with Cross-cap and Vector Field," it was made master of ceremonies, introduced a succession of repre­ by Helaman Rolfe Pratt Ferguson, a topologist and sentatives from state and local government, Brown Uni­ sculptor at Brigham Young University. Ferguson says his versity, and other mathematical societies, who presented inspiration was a theorem saying that cqmpact surfaces their felicitations to the AMS. Christopher Zeeman, are determined by the number of holes and the number President of the London Mathematical Society, exuded of cross-caps. The sculpture was dedicated at a special British charm when he presented to the Society a gold ceremony the day before the Centennial Celebration medal to commemorate the Centennial and to be worn by began.

970 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial

The MAA graciously yielded the time it would ordi­ narily have had at a joint meeting in order to allow time for the Society to have a full program of lectures. The MAA was represented in the program only by its seven Minicourses, held the weekend before the Centennial. The AMS was not the only one to receive gifts at the Opening Ceremonies. Mostow presented the repre­ sentatives of the other societies with specially engraved Revere bowls commemorating their cooperation. In one of the most touching moments, Everett Pitcher received a special plaque and a standing ovation for his 21 years of service as AMS Secretary. Participants seemed to find the Opening Ceremonies enjoyable. "It was very sweet and charming," said Lance Small, AMS Associate Secretary. The MAA's gift to the AMS. Left to right: AMS President G. D. Mostow, sculptor Helaman R. P. Ferguson, and MAA President Leonard Gillman. A Star-studded Symposium The scientific program featured a special symposium which brought together some of the nation's brightest mathematical stars who are likely to significantly influ­ ence mathematical research into the year 2000. William P. Thurston, one of the symposium speakers, praised the "broad sweep" of mathematical areas covered in the symposium and the fact that, because of the histori­ cal nature of the meeting, the speakers seemed to be making a real effort to present the talks at a level that many could understand and appreciate. Frank Gilfeather Front and back views of the medal presented to the AMS by the said this kind of program should appear more often. London Mathematical Society. "It doesn't have to be the Centennial to get people to The keynote address was presented by Edward E. give good expository talks," he said. "This is just what David, Jr., President of EED Inc. and former Science mathematicians love to listen to and learn from." Adviser to the President. David, who was chairman In addition to the symposium, there were AMS-MAA of the committee that produced the influential report, Joint Invited Addresses by three seasoned mathemati­ "Renewing U.S. Mathematics" (usually called the David cians. Raoul Bott provided a warm, personal view of Report), surveyed its impact and made some recommen­ his days at Princeton from 1955 to 1957, with vivid dations for future action. In particular, he stressed the descriptions of the mathematics and the mathematical importance of education and of attracting young people personalities he found there. Peter Lax described some to the field. (The Managing Editor hopes to be able to of the major lines of development of applied mathemat­ publish the full text of David's speech soon in Notices). ics in the United States and touched on some science That evening, the Opening Re.ception was held at policy issues of importance to the mathematical commu­ the Rhode Island State House, which was built in 1900 nity. With a biting wit and an engaging style, Saunders of white Georgian marble and which boasts the third Mac Lane spoke on the history of mathematics research largest unsupported marble dome in the world. There departments in this country. were balloons and three birthday cakes-one decorated with "1888," one with "1988," and one with the special AMS Centennial logo. Adding flair to the reception were the color guard of the Kentish Guards of the Rhode .."'' ...... Ill 0 .Ji.,. . - .... Island militia. A high point came when Mostow cut the .,..,. .• ~~~~ . I cake with a saber borrowed from one of the guards. "It . ,._. . AMERICAN

touch," said AMS J_ ~ MATHEMATICAL was done with a very light, happy , · 9: SOCIETY President-elect William Browder, adding that "the whole ' I 1888-198::1 meeting is a fantastic organizational job, I just can't say enough about how well it's been done." AMS President G. D. Mostow presides over the Opening Ceremonies.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 971 AMS Centennial

AMS Short Course reception also honored the symposium speakers, each of whom received a specially engraved Revere bowl. The weekend before the meeting, the AMS sponsored Mostow and a representative of the Brown University an enormously successful short course on chaos and library addressed the crowd of about 100 people. fractals. The course attracted a record crowd of about 500. The audience heard presentations on such topics as the horseshoe map, chaotic attractors, Julia sets, and iterated function systems. In addition, they saw com­ puter generated illustrations and films representing the mathematical objects explored in the course. The course attracted an especially diverse crowd, with many gradu­ ate students, participants from industry and laboratories, and even some high school teachers.

AMS President G. D. Mostow cuts the cake at the Opening Reception. Social and Cultural Events Between 300 and 400 people took tours of the AMS headquarters office. According to the tour guides, the participants found the warehouse and print shop espe­ cially interesting. There were also tours of Providence's historic areas and of Newport, Rhode Island, with its dazzling turn-of-the-century mansions. The social high Everett Pitcher accepts a plaque commemorating his 21 years of point was an outdoor clambake, featuring live music service as AMS Secretary. and sports. The clams, fish, potatoes, corn, onions, and sausage were baked in the traditional manner with layers Semicentennial Reception of seaweed to add extra flavor. The tables were covered with white butcher paper, and Irving Segal took advan­ For those who attended the AMS Semicentennial in 1938, tage of it as he scribbled a little mathematics during his there was a special reception in the elegant Alderman's dinner conversation with Peter Lax. Chambers at the Providence City Hall. The convivial crowd numbered around 60 people, 35 of whom were Semicentennial attendees. John W. Green, Saunders Mac Lane, and William Ted Martin spoke to the crowd about the Semicentennial and the now-legendary "ungala" din­ ner, organized by about 20 mathematicians who refused to pay $3 for the "gala" Semicentennial dinner. Mac Lane said that the "ungala" group also sent a telegram to the official dinner, telling the diners that their meal was too expensive. Martin noted that about 30% of the Semicentennial attendees are still alive. "I'm no statisti­ cian," he said, "but I think there's a theorem there that says that attendance at the Semicentennial is good for longevity." There was also a special reception to mark the official transfer of the Society archives to Brown University. Centennial Coordinator Tricia Cross and AMS Director of Meetings Held at the Bell Gallery of Brown's List Art Center, the H. Hope Daly did a superb job organizing the Centennial.

972 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial

The Semicentennial attendees gather in the Providence City Hall for a commemorative photo.

There were several special exhibits at the Centennial, from the Society. William J. LeVeque, who is retiring including a display of AMS archival materials at Brown this month from his position as Executive Director, was University. Brown also commemorated the Centennial also presented with an engraved bowl to commemorate with an exhibit of rare mathematical books that included his eleven years of service to the Society. 16th century editions of Euclid's Elements of Geometry. The Business Meeting was the last event in the In addition, the Rhode Island School of Design sponsored Centennial. As the Celebration came to a close, Secretary a showing of drawings of magic squares and other designs Pitcher provided a perspective on the importance of by Royal Vale Heath (1883-1960). the Society's history when he said, "The officers of the The AMS archival exhibit was put together by AMS Society look to the accomplishments of the past as a staff member Tricia Cross, who, as the Centennial Coor­ foundation on which the Society may build its future dinator, was responsible for most of the special Centen­ service to research in mathematics." nial events. At the AMS Business Meeting, she received Allyn Jackson a special gift of an engraved Revere bowl as a thank-you Staff Writer

SEPTEMBER 1988. VOLUME 35, NUMBER 7 973 Inside the AMS

Report on the Council Meeting, August 1988

The AMS Council is the official policymaking body of Some Council members expressed concern that the the AMS. With around 45 members, it is composed of series would emphasize more established researchers members-at-large and a range of other AMS officials while removing opportunities for younger people, including the president, the secretary, the treasurer, who traditionally present the AMS Invited Addresses. and chairs of various AMS committees. Members of However, some pointed out that because the lecturers the staff, including the executive director and the would be chosen for excellence in exposition rather executive editor of Mathematical Reviews, attend to than for an established research reputation, younger assist the Council. The Council generally meets three mathematicians would not be excluded. times a year to discuss and make decisions about Many members particularly favored the idea of various Society functions. identifying the scientific area before choosing the . Although the Council plays an important role in speakers, and some went on to suggest that there could the Society, many members are unaware of how it be more coordination of the various sessions, with, for functions and the kinds of issues it discusses. In example, both a short course and the lecture series on order to better acquaint the membership with the the same mathematical theme. In addition, many felt workings of the Council, Notices prepared this report that, because the lectures would be expository, they on the most recent Council meeting, which took place would increase communication between subfields in on August 7, 1988, in conjunction with the AMS the mathematical sciences by informing a broad cross­ Centennial Celebration in Providence, Rhode Island. section of the community about new developments in At the meeting, about 30 Council members were in particular areas. The Council ended its discussion of attendance. the matter by giving its formal approval to the idea.

A New Lecture Series Structure of the JPBM President G. D. Mostow opened the discussion with The Council reviewed a report from the Ad Hoc the idea of establishing a new lecture series entitled Committee on the Proposed Structure of the JPBM Progress in Mathematics. To be held during the (Joint Policy Board for Mathematics). The JPBM summer meetings, the series would be similar to consists of the presidents and executive directors of the well-known Seminaire Bourbaki. The plan is to the AMS, the Mathematical Association of America identify a particular area the series would cover and to (MAA), and the Society of Industrial and Applied choose excellent expositors who would speak on work Mathematics (SIAM), together with an appointed not their own. Lecturers would prepare in advance representative of each body, presently the secretary of a detailed manuscript which would be distributed to the AMS, the treasurer of the MAA, and the chair attendees and which would later become part of a of the Board of Trustees of SIAM. The purpose of book series. To make room in the meeting schedule, the JPBM is to work on projects of common interest the number of regular AMS Invited Addresses would to the three societies. One of its main purposes is to be reduced by two or three. direct the Office of Governmental and Public Affairs, The idea for the lecture series first surfaced last headed in Washington, DC by Kenneth M. Hoffman. May, when the AMS Executive Committee and Board One of the ad hoc committee members, Marc of Trustees gave its approval to the idea. The Council's Rieffel, summarized the report by saying that the task was to give its opinion of the idea and possibly committee had found that the JPBM spent much of approve it formally. its time on budgetary and other managerial details of

974 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Inside the AMS

the "Washington presence" and was lacking a focus .on Reports to the Council science policy matters. Recently, the three executive Ronald G. Douglas, chair of the Science Policy Com­ directors of the societies formed a committee to mittee, presented handle these managerial details, and the report said a brief report to the Council. on various topics the committee had been that this arrangement seemed to be working well. The addressmg. The topics included: the update of the "Da~id ad hoc committee therefore recommended that the Re­ port " which the National Science FoundatiOn has managerial committee be institutionalized an.d that requested the AMS substitute its secretary's representation on fo~ally of the Board on Ma~hematical Sciences of the National Research Council the JPBM by the representation of a person elected (NRC); the NRC's MS2000 project, a comprehensive assess­ by the Council. The main role of the elected mem~er ment of ~he would be to provide a strong AMS-JPBM connectiOn collegiate and university mathematics; Collegiate on policy matters. Mathematics Education newsletter, which is jointly sponsored by the AMS and Some discussion of the recommendations followed. the MAA .and which will begin in 1989; and the panel the William J. LeVeque, executive director of the AMS, committee sponsors at the winter Joint Mathematical circulated a memorandum expressing some of his Meetings. The Council seemed to agree that views on and experience with the JPBM. In particular, regular reports of this kind were very informative and would help to he felt that the chair of the AMS Science Policy strengthen ties between the Council and the Committee and the secretary should be members of Science Policy Committee. the JPBM. Mostow suggested that the Council could Council member Carol Wood elect the chair of the Science Policy Committee, reported on the an­ nual International Science Fair, in which who would then become the elected representative on the AMS participated for the first time last May. the JPBM. However, president-elect William Browder Seven prizes totaling $3000 were given for outstanding math~­ said that he believed such a mechanism would lessen matical projects the policy influence of the president, who ordinarily by high school students. She said that recommendations for further appoints the chair of the Science Policy Committee. AMS involvement would be in an upcoming report, and she After some deliberation over when the first term encouraged the Council members of the proposed elected member should begin, the to become involved in such activities at the regional level. Wood Council approved the ad hoc committee's original also acknowl­ recommendation. edged the assistance and support of Council member William P. The report also recommended that the executive Thurston. The Council also heard a report directors of the three societies should be nonvoting on a recent sur­ vey conducted by associate executive director members of the JPBM. Jean Taylor, one of the ad James A. Voytuk at the request of a committee hoc committee members, explained that the executive headed by Thurston. The survey, directors should implement rather than set policy, so which sought opinions on AMS election procedures, a nonvoting status was appropriate. Some Council was sent to 150-17 5 people, in­ cluding officers members felt, however, that a nonvoting status would and ex-officers of the AMS. Voytuk reported that make the directors less involved and less active, about 40 responses were received, and the results or might make them feel "second class." LeVeque would be compiled in a report. One of the main purposes of the survey was to investigate pointed out that he was appointed to the JP~M b_Y attitudes about contested elections, and it was the president of the MAA, not by the Council, so It noted that the survey found that few other was unclear who was to decide on his voting status. societies have uncontested In addition, some pointed out that the voting status elections as the AMS does. of the JPBM members should be parallel for the three societies, but it was not clear how the MAA and SIAM Other Business felt about this question. The Council seemed to agree In addition to that they should take a "wait and see" attitude and these matters, the Council discussed a letter from Saunders assess the effects of having an elected member on Mac Lane that made several suggestions of actions the Council could take the JPBM before making any further changes in the and con­ sidered ways Board's. composition. However, a straw vote revealed to strengthen connections between the AMS and the that a majority was in favor of retaining the voting American Association for the Advance­ status of the executive directors. ment of Science. Allyn Jackson Staff Writer

SEPTEMBER 1988, VOLUME 35, NUMBER 7 975 Computers and Mathematics

Edited by Jon Barwise

Editorial Notes been modeled by mathematical and so escape from the founda­ objects of some sort, objects over tional problems that were trou­ Doing Mathematics on a Computer which the algorithm in question is bling mathematicians of the time. The computer is having a large defined. The results of Godel, Church, impact on mathematics. Many as­ This makes computer program­ Kleene and Turing show that pects of this impact are shared ming a kind of applied mathemat­ Hilbert's program is impossible. with other disciplines, like physics, ics. Every programmer is in the The irony rests in the fact that this biology or economics. In all these business of designing and imple­ work, while showing the limita­ fields it is changing the way peo­ menting mathematical algorithms. tions of algorithms in the domain ple do simulations and experi­ So good computer programmers of symbols, and so of Hilbert's ments, write, teach, and commu­ have to be good applied mathe­ idea, also showed the tremendous nicate with one another. But there maticians in ways that they do not power of that very idea, by way are some ways in which mathe­ have to be applied physicists or of the notion of Turing machine. matics is destined to have a special applied biologists, or whatever. Symbol manipulating machines, relationship to computers. One of However, the main focus of which is what digital computers these has to do with the use per­ this month's column is on the are, will never be able to replace vasive of mathematics in compu­ converse side of the mathemat­ humans in doing mathematics, but tation. ics/computation relationship. It is they can do one heck of a lot There are two sides to this about computer programs whose of mathematics, and so, through use of mathematics in computa­ task is to allow the user "to do mathematical modeling, one heck tion. There is a sense in which mathematics" explicitly. Here we of a lot besides, which is what whenever one wants to program have one of the grand ironies of makes them so important in to­ a computer to do anything, one mathematical history. For research day's world.* has to first come up with an al­ aimed at showing the impossibility Given this history, it is only gorithm, the algorithm that the of making the doing of mathemat­ fitting that computers return to computer program is going to em­ ics an algorithmic business, indeed their origin and give mathemati­ body. But of course algorithms research that succeeded in show­ cians tools for doing mathematics. are mathematical objects, defined ing just that, in fact led to the Not all mathematics, of course. over other mathematical objects. modem digital computer. We know their theoretical limita­ They presuppose that whatever David Hilbert formulated the tions. But they should provide us real-world task one is after has research program of showing that Do you have mathematical soft­ all of mathematics could, in prin­ cipal, be made algorithmic. His * This is surely one of the greatest exam­ ware that you are willing to ples of unexpected consequences of pure share with the mathematical idea was that by turning atten­ research in mathematics. Who could have community, either as freeware tion from the traditional domain dreamt of the profound practical conse­ of mathematical objects (numbers, quences of the efforts of Hilbert, GOdel, or shareware? If so, send a brief and Turing? It is amusing to imagine them description to Jon Barwise. His functions, sets, ... ) to the domain of the symbols we use in doing applying to NSF for money and the kind of address is on the next page of response their application would have got­ the Editorial Notes. calculations and giving proofs, we ten, especially as regards its role in overall could make the whole business structure of science and industry. finite, concrete, and algorithmic,

976 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

with useful tools for doing the rou­ Next month we will have a Fourier Analysis Applied to Trans­ tine computational parts of math­ review of the group theory pro­ mission Problems," Proc. Inst. Ra­ ematics in a much more efficient gram CAYLEY. As other pro­ dio Engrs., Vol 30, 1942, pp. way. grams come out, we intend to have 144-150). This citation is found There are many programs on them reviewed here as well. Please in Dr. Bracewell's textbook, The the market that aim at letting send suggestions for programs that Fourier Transform and Its Appli­ the user do mathematics on per­ should be reviewed, or potential cation, Second Edition, on the first sonal computers. Six of the most reviews, or other contributions, to page of Chapter 19, "The Discrete popular (EUREKA, TKSOLVER me. Hartley Transform". The trans­ PLUS, MATHCAD 2.0, POINT form discussed there is identical FIVE, PC-MATLAB and GAUSS) Reviewers Needed to the patented discrete Bracewell are reviewed this month by Barry The column has already started re­ Transform. Simon and Robert Wilson in their ceiving mathematical software for Sincerely, article "Supercalculators on the review. If you would be interested PC." James W. Fox in doing such a review, please Houston, Texas But Simon and Wilson do more write and tell me your areas of ex­ than just review and compare pertise, and what equipment you Editors Comment: these six existing programs. They have available. Be as specific as also formulate a list of features possible about the latter. And if There are very interesting issues as they think should be part of any you know of a program that you to when two algorithms are really program for doing mathematics would like to see reviewed, write different, just as there are ques­ by computer. In this way they not to the owner suggesting that a copy tions as to when the proofs are only help the would be user of be sent to this column. really different. We all know that existing programs, they also ini­ Professor Jon Barwise there can be many different al­ tiate a dialog with developers of Center for the Study of gorithms for computing the same future programs. If you have other Language and Information function. And there can be many features that you think should be Ventura Hall different programs that implement added to the list, write a letter to Stanford University a given algorithm. So neither easy the column. Stanford, CA 94305 answer is right. While there are Since I happen to be a Macin­ beginnings toward a theory of al­ tosh user, I want to mention that Email can be sent to: gorithms that might answer such while their article is called "Super­ [email protected]. questions (e.g. by Y. Moschovakis calculators on the PC," the pro­ at U.C.L.A.), there certainly is no grams reviewed are also available A Letter to the Column widely accepted theory today. So for machines other than the PC. how the patent office is going to de­ Just which machines a given pro­ The following letter was received cide the matter in particular cases, gram is available for is indicated in response to the article by Ed heaven only knows. in the review of the particular Zalta about the patenting of algo­ However, it seems that no such program. rithms in the previous issue. interesting issues are at stake in the In June, a powerful new pro­ case of Bracewell's patent. Rather, gram, Mathematica, was released DearAMS: it is simply a matter of historical by Wolfram Research. This prQ­ The situation surrounding the priority. As I understand it from gram is available for the ·Macin-· grant to R. N. Bracewell of a Zalta, the facts are as follows. The tosh and various SUN worksta­ patent for a "special purpose com­ Discrete Hartley Transform was, tions, but not for the IBM PC. puter" to perform "the Discrete in fact, discovered by Bracewell. IBM has plans to release this pro­ Bracewell Transform" is even more As often happens in mathematics, gram for the AIX/RT in the not interesting than you might surmise Bracewell modestly named it af­ too distant future. This program from "Are Algorithms Patentable?," ter someone whose work in the is billed as "A system for doing pp. 796-799 in the July I August continuous case was suggestive in mathematics by computer." It will Notices. his own discovery-namely after be reviewed in this column in a The continuous version of the Hartley. (Apparently the lawyer few months. transform was published by R. V. who drew up the patent switched L. Hartley ("A More Symmetrical the name to "Discrete Bracewell

SEPTEMBER 1988, VOLUME 35, NUMBER 7 977 Computers and Mathematics

Transform.") So, though the Dis­ I would be interested in let­ many of the aspects of a super­ crete Hartley transform is the Dis­ ters that address the question as calculator. But invariably, APL crete Bracewell Transform, it was to whether algorithms should be can handle larger problems with­ in fact discovered by Bracewell. patentable at all. If you write a out choking than any of the other To further confuse matters, the letter commenting on a previous packages so for comparison pur­ Fast Bracewell Transform is dis­ column, please indicate whether poses, we felt it appropriate to tinct from the Fast Hartley Trans­ you are willing to have some por­ add an APL package. The stan­ form, but both of these were de­ tion of your letter published in the dard for APL on the PC is clearly fined by Bracewell as well. column. STSC's APL*PLUS and we used Version 7.0 for this comparison (which we'll henceforth shorten to "APL"). And to really show the cost in time for any interpreter, we include for one of our bench­ marks, a comparison with a com­ piled Turbo Pascal program. Supercalculators on the PC In understanding which of these programs might be right for you, Barry Simon and Richard M. Wilson you must bear in mind that com­ paring certain of them is like com­ California Institute of Technology paring apples and oranges. MATH­ CAD 1.0 and EUREKA had es­ sentially no overlap of functional­ ity. While that has changed with MATHCAD 2.0, it is still true that they are basically intended to meet different needs. For that Introduction Scientific Software's SOLVEIT. We reason, we'll begin the article with Hand held supercalculators such examined six commercial programs a quick once over trying to high­ as the HP28S are wondrous beasts. that meet our perquisites: Bor­ light the distinctions. Next we'll But their system resources can't land's EUREKA, Universal Tech­ discuss some comparative issues compare to a microcomputer like nical System's TKSOLVER PLUS, like editing matrices. Then we'll the IBM-PC. In this review we Mathsoft's MATHCAD 2.0, Pa­ write about each program sepa­ want to survey the software avail­ cific Crest Software's POINT FIVE, rately. Next we'll present the re­ able for IBM compatible PC's in The Math Works' PC-MATLAT sults of a standard set of "bench­ the supercalculator category. For (the 'Mat' is short for matrix, marks" and then we'll try to give inclusion, we required programs not mathematics) and Aptech Sys­ the flavor of some of the packages to not only calculate and graph tem's GAUSS. by describing how each can be simple functions but to have some To some extent, any interpreted used to illustrate the Gibbs' phe­ kind of built in variables and language with mathematical func­ nomenon. We end with a section "real programmability". We did tions, e.g. BASIC, could be used on what's missing and the reader not look at symbolic manipulation as a kind of supercalculator, but may want to skip there to get some programs of which there are now the most suitable such possibility flavor of the state of the art. Infor­ a few available on the PC (we ex­ might well be APL which is struc­ mation on prices and publishers is pect that the additional resources tured in a way most congenial for at the end of the article. available under OS/2 will have mathematical manipulation. In a In a very rough sense, one a considerable effect on symbolic real sense, APL is different from can divide the programs into two manipulation) nor do we report the other programs reviewed here groups: TKSOLVER, POINT FIVE, on two programs which were more in that you have to program to MATLAB, GAUSS and APL are sophisticated than four function do things that are built into the all basically mathematical program­ calculators but which didn't meet alternatives although STSC has in­ ming languages. In the other group our criteria: Dalin Software's SCI­ cluded enough workspaces with are EUREKA, TKSOLVER and ENTIFIC WHEEL and Structured their version of APL that it has MATHCAD. These programs

978 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

allow you to do things off the shelf are available. EUREKA recasts the sigma with indices as displayed by without learning a new program­ set of equations as a minimization MATHCAD can't really be made ming language. This is not to say problem and uses a method of in text mode. While MATHCAD that there aren't rules of syntax steepest descent algorithm. is not a replacement for a techni­ to learn or that the other pack­ TKSOLVER PLUS is also an cal word processor for long papers, ages don't come with a number equation solver but more versatile it is usable as a kind of technical of sample programs doing basic and powerful than EUREKA. It word processor with "live" formu­ functions but rather that to get allows solution by direct substitu­ lae when writing brief reports and very much out of the package, tion if possible as well as using is unique among the programs in you really need to learn at least Newton's method. You can input this regard. the basics of their language. You'll and deal directly with matrices The remaining packages are note that TKSOLVER is on both and vectors, something that can all basically programming envi­ lists. You can use it to solve si­ only be done in EUREKA and in ronments (as is a part of TK­ multaneous equations without ac­ an ad hoc manner. And, unlike SOLVER). APL is an interpreter cessing its extensive programming EUREKA, it has a rather com­ for a language which is especially language which is there if you need plete programming language built tuned to arrays. It is unfortunate more. in and illustrated in a large li­ that its conditional and loop struc­ There is a second way of clas­ brary of functions and examples tures are so primitive. As a lan­ sifying these programs. Those who supplied with the program. Indeed guage, you can do almost anything insist on something relatively easy its programming language is in the in it. While we'd hate to write a to use should incline towards EU­ same league as that of MATLAB word processor with it, that could REKA, MATHCAD or POINT and GAUSS. The greater power be done. While it is possible to FIVE depending on whether they comes at the cost of a more com­ write both good or bad code in are interested in equation solv­ plicated user interface and of a any language, APL has a deserved ing, calculations/reports or matri­ much greater effort required to reputation for making cryptic code ces/ spreadsheets. learn the program. For simple possible. It also has its own charac­ We should mention that Simon occasional use, and probably for ter set which its users tend to love is a member of Borland's "exec­ classroom use, EUREKA is to be and dabblers tend to dislike. STSC utive advisory board" but since perferred but for "serious" use, APL is missing a lot of useful tools this is a non-paid position, we feel TKSOLVER is better. Another al­ present in some of the other pack­ there is no conflict in our reporting ternative for classroom use is the ages. Its numeric matrix editor is on a Borland product. free MiniTK program provided by limited, although you can write the publishers of TKSOLVER. If a better one; indeed some stu­ Once over lightly you mainly want the mathematical dents at Caltech did write one for programming language you may use in our linear algebra course. EUREKA, in most ways the most well prefer one of the other pro­ You could write your own New­ limited of the programs under gramming environments. ton's method routines but they are discussion, is the easiest to de­ MATHCAD's strong point is not built in. As a general purpose scribe. It is an equation solver al­ visual and typographical presen­ programming language, there are lowing several simultaneous equa­ tation. It runs in graphics mode third party books on the language tions. Some of the "equations" can with the virtues and vices inher­ some of which provide canned be inequalities and it understands ent in that, you need more expen­ workspaces, for example the book maxjmin. If you have ever used sive hardware to begin with and Abstract Algebra: A Computational one of Borland's compilers, you'll actions like scrolling a page are Approach by Charles Simms. STSC feel right at home with EUREKA: much slower than in text mode includes over 25 workspaces with you prepare a source file in a built programs like EUREKA. While all the basic package and sells an ad­ in editor following certain rules the other programs have graphics ditional 30 workspaces as part of of syntax and choose the Solve available, it runs separately from separate packages. command from the menu. If there their main screens-MATHCAD POINT FIVE can be thought of is a syntax error, you are thrown lets you integrate graphs and text. as APL with simplicity and built into the editor at the place the With one of the other programs, in conveniences at a trade off of error was noticed. Once you've you'll need some non-standard no­ much less raw power. You'll never solved the first time, many options tation for sum since a large greek be able to do in a single line of

SEPTEMBER 1988, VOLUME 35, NUMBER 7 979 Computers and Mathematics

POINT FIVE code what you can SOLVER, MATLAB, GAUSS and Once over lightly on matrices do with APL. Like APL, it is an in­ APL-are linearly ordered in terms and editing terpreter but the lines you type in of the pure programming power: You'd think that the one thing and the output in response to those for example APL gives you di­ that these programs would get lines are kept in· separate windows. rect access to memory · addresses right is matrices and, in partic­ There is a full fledged array edi­ and interrupts and you could only ular, editing matrices. As far as tor built in. MATHCAD 2.0 is a hope to write a word processoring editing matrices, POINT FIVE is rather recent upgrade which made program in GAUSS or APL. They a model but alas while it will it a more serious competitor to are ordered in roughly the opposite invert matrices and find determi­ the other programs under consid­ way on the issue of ease of use: nants, it won't even find eigenval­ eration. POINT FIVE 2.0 which the language component of TK­ ues directly. MATHCAD's editor is under development will be out SOLVER isn't any easier than in in general feels awkward to us soon with additional features that MATLAB or GAUSS but you have but it is no worse for matrices may make it more competitive. direct access to its solver mod­ than anything else and at least MATLAB is a set of matrix ule. In terms of supplied routines edits matrices in a fairly natural handling routines. It has any kind though, the structure is trapezoidal way. It goes down hill from there. of matrix handling you could imag­ with TKSOLVER, MATLAB and EUREKA has no matrix support ine and some you can't. While GAUSS providing much more in and TKSOLVER treats a matrix it is a matrix based language, it the way of built in mathematical as a vector of vectors. APL comes has support for many non-linear routines than the other two pro­ with a workspace that allows some functions. It has a long, illustri­ grams. matrix editing but it is hardly a ous history and, as a result, there If you are interested primar­ model of what one would like. We versions running on many other ily in one of these packages as find it remarkable that MATLAB machines including Macintoshes, a programming environment, you and GAUSS which are program­ VAXes and Suns. Its graphic rou­ will have to decide if it might ming powerhouses fail to provide tines, including 3D graphs are es­ not make more sense for you any serious matrix editing facili­ pecially impressive. It can be run to use a more conventional com­ ties: GAUSS' editor is amateurish as an interpreter in interactive puter language like FORTRAN, and doesn't let you see the matrix mode, or you can use an editor C or PASCAL with a library of structure while editing; MATLAB (not supplied-you'll need your mathematical routines. The IMSL doesn't have an editor-while you own) to write functions and rou­ library, Borland's TURBO PAS­ can import a matrix prepared in tines that are called from disk and CAL NUMERICAL METHODS your personal editor, such editors compiled the first time that you TOOLBOX and Quinn-Curtis' do not have any special facilities use them during a secession. SCIENCE AND ENGINEERING for handling the special structure GAUSS has some similarities TOOLS with versions of Pascal, of matrices. to MATLAB in that it is a pow­ Modula and C are among the On editing facilities in gen­ erful programming language built choices. Generally, the packages eral, these programs show a wide around manipulation of mathe­ considered in this article are "higher variability. APL's full screen edi­ matical objects with an interpreta­ order" languages-you are spared tor which will save pages of pre­ tive and a compiled mode. It has from issues like worrying about vious output and allow you to more programmability than MAT­ how the machine stores your data reissue commands in interactive LAB, especially more hooks to and the heap management that mode is a pleasure to use although the computer. It was consistently you often get into dealing with its program editor is only ade­ among the spediest programs in large structures under PASCAL­ quate. EUREKA's editor is the our tests and with its large num­ but there is no free lunch; in­ standard one in Borland compil­ ber of built in modules, it seemed variably, compiled languages are ers and is easy to use. POINT to us the most "extensive" of the faster an<~ able to cope with larger FIVE and TKSOLVER are line stand alone packages. But its user structures. oriented programs and their edi­ interface is rather clunky by the tors are similar-adequate but not standards we have become used to much better. MATHCAD's editor on the PC. is unique to take advantage of The five programming envi­ its graphics mode and we found ronments-POINT FIVE, TK-

980 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

it rather difficult to use under manipulative operations of "+.x" whopping 4 Gigabytes, 80386 spe­ various circumstances. For exam­ and "jot-dot". cific versions of these programs (as ple, if you get an error message, already available from MATLAB longer remove and announced by STSC for their (Backspace} will no Once over lightly on large MATLAB APL*PLUS and by GAUSS) will the offending material! data structures has no real editor although it keeps be able to handle very large arrays. a stack of previously issued com­ When PCMATLAB, GAUSS, and mands which you can edit. Its POINT FIVE, are asked to display Once over lightly on graphing philosophy is that you are best a large amount of data, the screen served by your own editor. Since scrolls and that which has scrolled You'll want to be able to graph you can invoke your editor from off the screen cannot be recovered functions and perhaps surfaces. within MATLAB rather easily (if (so you need to ask the program to There are several issues that you you have sufficient memory), this display a smaller amount). By de­ care about here. First, what kind is a viable point of view. GAUSS' fault, APL will save 75 lines other of monitors are supported. Next, editor is its weakest element. In than the 25 on the screen (and you what kind of printer support there interactive mode, you are limited can reconfigure it to save more); is and finally, exactly what you can to two screens with a fair amount moving with the arrow keys will graph. of clutter. In edit mode, file size allow one to see what has scrolled On monitor support, all work isn't limited but the editor seems off. GAUSS lets you recall your on the CGA and the original to be missing basic amenities like one previous screen before you Hercules monographics monitors. inserting an external file or mov­ ran the command whose display POINT FIVE shows graphs on the ing a set of lines. This editor will scrolled off the screen. MATH­ EGA in CGA resolution ( 640 x be improved in version 2.0 of CAD keeps a record of all that 200) only. The other programs all GAUSS due out by the time this has happened since the program support the 640 x 350 EGA high article appears. was loaded but as mentioned else­ resolution mode. MATHCAD and While on the issue of matri­ where, scrolling is very slow. The APL even have VGA support at ces and arrays, we should focus object editor in POINT FIVE will this time. on GAUSS, MATLAB and APL allow you to peruse a large matrix Printer support for graphs is in which arrays play a fundamen­ in a way where scrolling is possible varied. POINT FIVE and TK­ tal role. To a mathematician, ar­ and exceeds the other programs in SOLVER have no special support rays in APL are much more log­ convenience in this regard. and printing is limited to shift ically thought out than in MAT­ All the programs here other PrtSc if you have a suitable driver LAB which has a number of plain than APL respond to the 64K for your monitor/printer combi­ inconsistencies which require spe­ segment architecture of the Intel nation such as the GRAPHICS cific testing for special situations. 80xx family by limiting objects to program that comes with DOS For example the function sum (ar­ ones that fit 64K. Since reals typi­ and supports the CGA with the ray) gives the column sums of the cally take 8 bytes storage, that lim­ IBM Graphics Printer. The mak­ array, unless the array is a 1 by its arrays to 8192 elements which ers of TKSOLVER told us that n matrix in which case it gives will accommodate no larger than they plan on introducing printer the row sum. While GAUSS isn't a 90 by 90 square matrix. With drivers soon. Among the others, inconsistent, we find it unnatural APL, array size is limited only by APL, MATHCAD and MATLAB that the basic function in GAUSS available RAM and we have used have especially good printer sup­ which sums up the columns of 205 by 205 matrices there. With port. a matrix returns a column vec­ the right add-on tools, one can As to what you can graph, TK­ tor rather than a row vector. APL deal with even larger objects in SOLVER, MATLAB and GAUSS supports arrays of any dimension compiled languages. For example, all support three dimensional plots. so that in APL a vector is a one TURBO PROFESSIONAL from MATHCAD is unique in allowing dimensional object while in MAT­ Turbo Power Software will allow mixing of the graphs in the middle LAB it is a two dimensional object Turbo Pascal programs to access of text. which just happens to be one ele­ arrays limited only by available ment wide in one dimension. And disk space, EMS memory or AT no other language has the lovely Extended memory. Because the segment size on an 80386 is a

SEPTEMBER 1988, VOLUME 35, NUMBER 7 981 Computers and Mathematics

Capsule Reviews means. If there is at least the abil­ unknowns (it is limited to 6 equa­ ity to define complex polynomials tions if you have complex vari­ capsule reviews, In our individual and multiply complex matrices, ables and round off errors become to list the functions that we want but .special treatment of complex significant after about 10 equa­ each program supports. There is a quantities is required, we'll use the tions). It does this very well but set" we'll just abbreviate "standard word "limited". otherwise it is limited. No matri­ that comprises: Save Warning? indicates whe­ ces, no gamma or Bessel functions, set: four functions, standard ther the program warns you if you no log plots. And it seems to choke ln, sin, cos, power, exponential, try to exit without saving your on certain types of complex equa­ tan, cot, hyperbolic versions of the work. It is, of course, inexcus­ tion solving as we'll explain when four trig, inverse of trig and hyp, able for a program NOT to do describing benchmarks. factorial (integer only) that. A general request for confir­ EUREKA's strength is clearly We will also use the following mation of your intention to exit its user interface. With MATH­ abbreviations: without telling you that you have CAD's equation solver, you'll need stat means mean, standard de­ unsavedwork is not sufficient in to type in all the equations af­ viation, variance our opinions so we have indicated ter solving to verify the extent to gamma means the full Euler that situation with the answer "No which the equations actually hold gamma function (Exit confirmation only)." for the hypothetical solution. With Bessel means at least some sub­ Shell to DOS? This deals with EUREKA, a single key provides set of J and Y functions whether one can run a new shell of this information. Another key will FFf refers to the fast Fourier command.com ("shell only"), run graph the current function and one transform (discrete Ff on 2n points) programs ("yes") including a shell can put all this output easily into required indicates the RAM of DOS or do neither ("no"). a report. If you want, several of need free amount of memory you these separate modules can appear to load the program (to within in separate windows on the screen. about 5K). You'll at least need the EUREKA Especially given the fact that it is memory taken for the operating RAM required: 260K available at discount from mail or­ system in addition. In addition all Graph types: Line only of a der software houses, it is by far the use available of memory to store single variable least expensive of these programs so you'll typically need 50-lOOK Log scales in graphs: No for individual purchase. more than the minimum to do Graphics device support: CGA, But as our benchmarks also much work. Hercules, EGA, even primitive show, it consistently quits working Equation solving refers to the graphs in text mode with the smallest numbers of vari­ ability to solve simultaneous non­ Printer support: IBM graphics ables and had problems with com­ linear equations in several un­ printer compatible only plex polynomials. As we remarked knowns (more than one or two Math coprocessor: Used but already, this program could be use­ and more than linear equations not required, automatically de­ ful for classroom use and its easy which are discussed under matrix tected if present interface makes it suited for very support). Functions: Standard set, nu­ occasional use, but otherwise, we'd Complex numbers refer to the merical integration and differenti­ place it at the bottom of our list. ability to use complex numbers ation, simple financial like future There is a version of EUREKA freely as arguments, for example value for the Macintosh. to write sin(i). By "full"-support, Matrix support: None we mean the ability to effortlessly Equation solving: Yes enter a number like TKSOLVER PLUS Complex numbers: Full but X=1+3*i flawed RAM required: 290K Branching: No Graph types: Line, bar, pie and to multiply complex quanti­ Users defined functions: Yes built in, contour graphs via work ties with the same symbols used to External language support: No sheet multiply reals. It may be necessary Save Warning?: Yes Log scales in graphs: Yes (as it is in MATHCAD or EU­ Shell to DOS?: Shell only Graphics device support: EGA, REKA) to one time configure the EUREKA is intended to solve CGA, Hercules program to understand what "i" up to 20 equations in multiple

982 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Print Supporter: only via PrtSc ically adds the variables in the considerable effort to set up the and the DOS graphics command equation to the variable sheet. program so that the functions you Math coprocessor: Used but Keeping the variables separate and really want are easily available. not required; authomatically de­ ready for instant viewing is a nice With EUREKA or MATHCAD, tected, if present touch. The definitions of proce­ numerical differentiation is built Functions: Standard set, net dures and functions are kept in a in; with TKSOLVER, you'll need present value, gamma, Bessel; in separate sheet as is a list sheet to transfer it from the library into addition many functions provided for things like tables and ma­ your regular toolbox "model" if via work sheets including Simp­ trices. You can have up to two you want to use it regularly. The son's rule integration and numeric windows on a screen at a time bottom line is that if you have differentiation, fast Fourier trans­ and hitting"=" when at command need of this kind of program (i.e. form mode brings up a list of windows. equation solving) on a daily ba­ Matrix support: Via separately This separation is useful but it sis, it is probably well worth the loaded modules: multiplication, in­ does add considerable complexity effort to master this program and verse, determinant, eigenvalues to the program. Indeed, we con­ we feel, barring some special cir­ Equation solving: Yes, both by sistently found it difficult to use cumstances, this is the program of substitution and Newton's method TKSOLVER without frequent re­ choice for you. If your needs are Complex numbers: Limited course to the manual. TKSOLVER likely to be occasional, one of the (fairly complete but special nota­ has excellent context sensitive help other programs, probably MATH­ tion is needed to handle complex but while learning the program, we CAD would be a better choice. quantities) found that we needed to look in While we admire TKSOLVER, Branching: Yes, extensive op- the manual to get some idea how to we found it the most difficult pro­ tions attack problems. We suspect that gram "to get into". We spent as User defined funtions: Yes once one masters the program, the much time with the program as External language support: No online help would be enough. TK­ with any other but felt the least Save Warning?: No (exit con- SOLVER has a toll free number secure in using it. Indeed, most firmation only) for questions. of the methods we used in the Shell to DOS?: No TKSOLVER handles syntax and first draft of our benchmarks were This is the most powerful by equation inconsistency errors quite awkward and were replaced by far of the general purpose pack­ well. Equations with a problem superior suggestions made by the ages (EUREKA, it and MATH­ have a > placed in the margin and TKSOLVER staff. CAD). Compared to it, they seem moving the cursor to that > dis­ TKSOLVER has the unique rather limited. In addition to its plays an error message about the feature of combining its equation basic built-in equation solving via problem. solving with its other modules. If either direct substitution or New­ While this program is very you wanted to graph a parametric ton's method, it has an exten­ powerful, some of that power may equation where the relations of the sive language and comes with a be overkill: it's clearly nice to be graphed variables to the parameter wide library of worksheets with able to burst into your neighbor's were only implicit, TKSOLVER functions that you can use for office and boast that your super­ would do it rather easily and all many purposes: a fast Fourier calculator does Runge Kutta but of the other programs would do it transform, numerical differentia­ we suspect that if you really need only with difficulty, if at all. The tion, symbolic differentiation, for RK, you'd be best served to write cost for this power is that it be­ a very restricted class of functions, a compiled program to number comes a more involved process to even a Runge Kutta differential crunch rather than to use an inter­ set up parametric equations with equation solver! pretive language like TKSOLVER. only explicit equations than in any TKSOLVER uses the windowed The other downside of the power of the other programs. worksheet as its main meta- phor. involves a more complex user in­ There are two limited versions There are nine main windows but terface (although if you've been ofTKSOLVER availabe. McGraw­ each can have subsheets. Two are turned off by an earlier version Hill sells a student edition for most important: the rule sheet and of TKSOLVER, the precursor of $44.95-this is the same as the the variable sheet. If you type TKSOLVER PLUS, the interface main program but without the in an equation that you want to has improved considerably from extensive libraries of functions be one of the rules, it automat- those earlier versions) and some (which provide differential equa-

SEPTEMBER 1988, VOLUME 35, NUMBER 7 983 Computers and Mathematics

tion solving, numeric integration If it weren't for the issue of menu or from a command line. and other basic features), with speed, MATHCAD would be close While its initial RAM require­ only one of the three manuals and to an ideal supercalculator. Not ments are high, it supports EMS without support. There is also a that we haven't complaints but ex­ for its workspaces. And it has a free demo program available upon cept for symbolic manipulation, it toll free support number. request which allows solutions of has all the functions that you'd MATHCAD understands com­ up to 24 scalar equations in up to want: not only the standard ones plex numbers very well-you can 32 variables limited to the built in including statistics, but even linear ask it to compute sin(2i) if you functions without the programma­ correlations and cubic splines. It want. bility or the array handling. There has a Newton's method solver and We did have a few gripes: the are versions of TKSOLVER for FFT. It will integrate (using Simp­ editor is idiosyncratic-because of the Macintosh, Apple lie, TRS- son's rule) and do a numerical the structure of its equations, one 80 and several non-compatible MS differentiation. And it will allow has to expect some compromises DOS machines like the DEC Rain­ reports with real cap sigmas for but we found this the most dis­ bow. sums! Moreover, it does simple concerting part of the program: calculations with the least amount Del and Backspace mean the same of thinking and reading of manu­ thing and Ins toggles between two MATHCAD als. insert modes(!)-there really isn't RAM required: 340K Its Achilles' heel is the issue an overtype mode at all. It was Graph types: Line, dot, bar of speed. There are two different annoying that you couldn't config­ Log scales in graphs: Yes aspects of this. As our tests show, ure the program to use the keys Graphics device support: RE- it is relatively slow in its calcu­ you wanted: [means subscript­ QUIRES graphics monitor-CGA, lational ability. In addition, by too bad if you'd find something EGA high res, Hercules default it computes the formulae else more natural. And we never Printer- support: Together with scrolling into view as you scroll so did fully comprehend where the MATLAB and APL*PLUS, the you'll often have to sit there while cursor needed to be placed for best of all these programs includes WAIT flashes on the screen. On a certain operations. If you want to standard dot matrix, HP laser and 16 MHz '386, this wasn't too bad define a variable "n" to be equal plotters (but not PostScript sup­ but it was unpleasant on an 8 MHz to 2, you must have port) Toshiba 1100+-we suspect we'd n := 2 Math coprocessor: Not required find it unacceptable on an original but supported and recommended; PC or XT. You can tum off auto­ on the screen which you put in automatically detected if present matic calculation but scrolling is by typing "n : 2". If you start Functions: Standard set, stat, still slow because of the graphics typing "n ="you immediately get Bessel, fft, linear regression, linear mode and you need to remember an "undefined variable" error. We interpolation, cubic spline, gamma, to hit the Calc key regularly. Even never did figure out where to place erf, numerical integration and dif­ worse, you need to remember that the cursor to remove the = without ferentiation the Calc key isn't enough when also removing then! In particular, Matrix support: Yes, products, you do a printout. You must in­ after an error message, backspace inverses and determinants stead call a special command to does not undo the last keystroke. Equation solving: Yes recalc the entire document. For some reason, MATHCAD Complex numbers: Full Another weakness is the lim­ indexes vectors and matrices start­ Branching: VERY limited (two ited language capability but by ing at zero; that is, a vector of functions only "if (cond, v 1, v 2)" careful use of vector valued func­ length n has coordinates labeled by which returns v 1, resp v 2, if cond tions you can go quite far: we 0, 1, ... , n -1 instead of 1, 2, ... , n. is true, resp false and until (exp 1, figured out how to compute great­ We understand why this is done exp 2) which iterates exp 2 until est common divisors in spite of in assembly language and the C exp 1 is negative) the fact that their tech support language (where indices represent User defined functions: Yes said that it couldn't be done! "offsets") and is an option in APL. External language support: No Except for the speed issue, Despite the fact that it can be Save Warning?: Yes there is a lot to like about MATH­ changed, we find this an awkward Shell to DOS?: Yes CAD. You can issue commands and strange default in a pack­ through function keys, through age that otherwise makes a point

984 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

to closely approxi~ate the _most POINT FIVE some supercalculator features. It has among the most complete sta­ standard mathematical notation .. RAM required: 215K (8087 ver­ The online help isn't bad but It tistical analysis functions of t~e sion) is not context sensitive and it is~'t Graph types: Dot, line, bar, programs under di~cussio~ and It available when you really need It. has various built m routmes for stacked bar If you want to change the format­ computing interest payments and Log scales in graphs: No ting of an equation-you choose future values. Graphics device support: CGA, format from a menu and get a POINT FIVE has some of the EGA in CGA emulation only, Her­ cryptic list of the current values of feel of APL from which it bor­ cules the formatting parameters. You'd rows the idea of workspace. It Printer support: For graphs via think that it would be natural to is a friendlier environment than PrtSc command only call up help at that point and APL with the lovely idea of sepa­ Math coprocessor: Separate 8087 to see what the parameters mean and non-8087 versions rating output from input. _POI~T but help isn't available then! If FIVE's strength involves Its abil­ Functions: Standard set (except you need to check what the pa­ ity to deal with data st~ctures: It hyperbolic), stat plus _man~ addi­ rameters mean, you must accept has a built in array editor which tional statistical functions mclud­ the old formatting values, call up is a pleasure to use. Its ability_ to ing chi squared and t test, many help and then recall Format from manipulate these arrays makes It a financial functions the menu. One of the strengths tool that could reasonably replace Matrix support: Yes, products, of this program is the ability to a spreadsheet in many places that inverses and determinants provide screen and printer output one might want to use one. Equation solving: Linear only with characters which aren't in the POINT FIVE admits a data via matrix inverse text mode set. Then why limit me type called "tables". A table is a Complex numbers: No to only 17 Greek letters: what if matrix whose rows and columns Branching: For loops, if-then­ I happen to want a lower c~se can be labeled-and the column else allowing multiple statements gamma or a chi? And why not give and row vectors can then be re­ after then or else but no gotos and me access to italics and boldface­ ferred to by name. If your calculus no subroutines even an additional font for section class grades are kept in a table, you User defined functions: No headings? While most us~s of _a can ask POINT FIVE to calculate External language support: No supercalculator don't require se~­ sumUohndoe) or sd(exam2)-th_e Save Warning?: Yes ous programming constructs, theu latter computes the standard devi­ Shell to DOS?: No lack really limits this program to ation. POINT FIVE is a broader pro­ supercalculator functi?nality. . The program has a number. of But these complamts are mi­ gram in what it can. do tha~ the annoying features, most of which others under review m that It has nor. If your machine is fast en<~ug_h we've been told will be addressed many spreadsheet like properties or you are patient enough, this Is in version 2.0: if you want to and this will become more pro­ the program for you! multiply matrices you must type nounced when version 2.0 is avail­ One final remark. Addison­ out in full the keyword MATRIX­ able. In this sense, it is addressed Wesley sells a student edition of MULT (and similarly for _ot~er to a broader market than the other MATHCAD for not much more functions); its support for pnntmg programs under review including than 10% of the price of the re~­ graphs is limited to hitting PrtSc social scientists. But its mathemat­ ular program. This program is not and it doesn't support high resolu­ ical ability is more limited than only limited in the size of doc­ tion on the EGA. any other program under review uments allowed but is a limited Version 1.0 of this program is except perhaps for EUREKA. F~r edition of version 1.1 of MATH­ an interesting first step but it is pure supercalculator purposes, It CAD rather than the version 2.0 rather limited. We are excited by has many lacks: no user defined that we are reviewing (MATH­ what we've told will be in Ver­ functions, no equation solving, no CAD version 1.1 is missing the sion 2.0 but in a world with va­ integration or differentiation and matrix support and the ability to porware, one should exercise re­ no complex numbers. But it has solve sets of simultaneous equa­ straint in purchasing programs on a wide array of other functions tions). We have been told that a the basis of promised future ver­ that may make it attractive as a student edition ofMATHCAD 2.0 sions. Given its rather aggressive multi-purpose program that has is planned. sitewide licensing program though

SEPTEMBER 1988, VOLUME 35, NUMBER 7 985 Computers and Mathematics

and the functionality already built nipulate data either in immediate some programming experience to in, this is worth your considera­ mode or more often by writing catch on immediately. And once tion. scripts. Most of the functions that M -files are understood, matrices the program comes with are pre­ for analysis, or subroutines to be cisely in the form of these external incorporated into student work, PC-MATLAB script files called M -files. It does can be distributed easily on floppy RAM required: 270K (will run not come with its own editor so to disks or electronically. (The only with less but graphics not avail­ write M-files you'll need your own problem for some students might able) ASCII editor which you can call be the need for an external editor.) Graph types: Line, bar, polar, directly from the program. MAT­ If matrices are your stock-in­ 3-d surface, 3-d contour LAB will load the functions into trade or you need the portability Log scales in graphs: Yes memory the first time in a session to minicomputers, 386 or Weitek Graphics device support: CGA, it is called. For better or worse, support, then this is a serious pro­ EGA, Hercules you may not define more than one gram to consider. If not, GAUSS' Printer support: Together with function in a file, so our disk drive more powerful programming envi­ MATHCAD and APL*PLUS, the soon was full of dozens of these ronment may be a more sensible best printer support: Epson printer, M-files choice although it is harder to use. Postscript, HP LaserJet, various The script files can call a vast CGI array of matrix and vector func­ plotters, other devices with GAUSS support; the major lack is no sup­ tions-not merely the standard port for the Toshiba 24 pin dot ones but functions like Cholesky RAM required: 170K (graphics matrix printers factorization. The routines used requires much more; over 400K Math coprocessor: Required are based on the UNPACK and free) Functions: Standard set, gamma, EISPACK projects. There are also Graph types: Line, bar, polar, stat, fft, poly fit, many electrical fil­ logical functions and many pro­ 3-d surface, 3-d contour tering functions, splines gram control functions. Log scales in graphs: Yes Matrix support: Very extensive, MATLAB understands complex Graphics device support: CGA, including det, eigenvalues and numbers very well-you can ask it EGA, Hercules (VGA support eigenvectors, qr and Schur decom­ to compute sin(2i) if you want. promised in Version 2.0 which positions, others If you use ANSI.SYS to set may be out by the time this article Equation solving: Yes colors at the DOS prompt, MAT­ is published) Complex numbers: Yes LAB's text screens will be a mix­ Printer support: Various dot Branching: If, for, while, sub- ture of black and the ANSI colors matrix, HP LaserJet but only at routines, pause for keystroke for a rather awkward appearance. 150 dots/in resolution (300 dpi User defined functions: Yes Most impressive are MATLAB's and postscript support promised External language support: Mi­ graphics capabilities including in Version 2.0 which may be out crosoft C, Borland's Turbo C and some 3-d graphing. It has excellent by the time this article is pub­ Microsoft Fortran printer support; you need to save lished) Save Warning?: No the graphs in a special file then Math coprocessor: Required Shell to DOS?: Yes exit the basic MATLAB program Functions: Standard, stat, MATLAB is included here be­ to run another program to print, Gamma function, non-parametric cause it has functionality as a su­ but the graphs on a LaserJet are re­ statistics, FFT, time series func­ percalculator but that isn't its real ally impressive. Unlike the version tions (but no Bessel functions) purpose. It is intended as a se­ of GAUSS current at this writing, Matrix support: Determinants, rious language for any kind of MATLAB supports the LaserJet at products, eigenvalues linear mathematics and is used a full 300 dpi if your printer has Equation solving: Nonlinear for real number crunching. As dis­ sufficient memory. systems with up to 90 equations cussed earlier, it has support for We found MATLAB relatively claimed by a direct method with other computer systems (Macin­ easy to learn and use. We think several alternates, nonlinear maxi­ tosh, VAX, Suns) and is one of it would be quite practical to use mization also the first programs available with in, say, a linear algebra class. A Complex numbers: (Special support for 80386 based machines few examples provided by the in­ program files and symbols required) and Weitek coprocessors. You rna- structor would enable those with

986 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Branching: Do while, do until, directory on a separate line each missing or non-number entries in if . . . then . . . else time that you make an execution matrices. User defined functions: Exten­ so the screen quickly fills up. Since Aptech Systems, the publish­ sive including local variables you can only type on the current ers of GAUSS have announced External language support: Mi­ screen and access one saved screen GAUSS 2.0 which they expect to crosoft C and. Fortran, RM For­ of commands, this is most unfor­ ship before this article is pub­ tran, C86 (but not Turbo or Quick tunate. And like MATLAB, with lished. A version of GAUSS for C) ANSI installed, your screen will 386 machines is also expected out Save Warning?: No (Exit con­ be an awkward looking mixture soon. firm only) of black and color. In addition, Shell to DOS?: Yes since GAUSS uses DOS to get APL*PLUSPC In terms of what it can do, this keyboard input, if you've rede­ is an impressive product. It comes fined keys with ANSI.SYS, you'll RAM required: 180K (with graph­ with over 500 pages of documen­ have to undefine them before en­ ics) tation and 10 disks, mainly with tering GAUSS. Just as we were Graph types: Line and pie graphs extra modules allowing solutions finishing this review, we received Log scales in graphs: No unless of non-linear equations and so­ a beta test copy of Version 2.0 you program them yourself phisticated statistical analysis. It which is expected to be released Graphics device support: With is consistently among the faster before this review is printed. At VDI, support for CGA, Hercules, programs in our benchmarks and least some of our complaints about EGA, VGA the most powerful programming the interface have been answered Printer support: With VDI, very language of any of the programs although a quick look suggests it complete including Epson, Toshiba under discussion except APL. Its still has some serious lacks such as and HP printers/plotters but not only serious weak point from a a decent matrix editor. postscript mathematical point of view is that There is a possibility with Math coprocessor: Recom­ its support of complex numbers is GAUSS, you may no longer have mended but not required; auto­ limited. It doesn't directly under­ any need of Fortran, Pascal, C matically detected if present stand things like sin( i) although or whatever other programming Functions: Standard set, stat you could easily develop your own languages you may be using for built in-many others in workspaces library of such functions if you computations. (Even though APL Matrix support: Basic opera­ wanted. It has a number of func­ is also powerful and used con­ tions including inverses built in. tions to handle complex matrices stantly by one of us, there are Determinant, eigenvalue and eigen­ but you must always enter the tasks for which we find it awk­ vector workspace available complex matrix as a pair of ma­ ward and slow. And MATLAB and Equation solving: Only possi­ trices representing the real and the others that have programming ble by programming it yourself imaginary part. languages don't have quite enough (but a workspace which solves a Its graphic support is also ex­ control statements or access to the single complex equation of a single tensive and quite well done al­ system to satisfy us.) complex variable is provided) though it is unfortunate that even The basic data type in GAUSS Complex numbers: Limited on a LASER JET with sufficient is the matrix and there are hun­ (special complex workspace with memory it can only handle graphs dreds of functions built into special functions needed) at half density, a lack which is sup­ GAUSS for dealing with matri­ Branching: Yes posed to be corrected in Version ces. For example, given a matrix User defined functions: Yes 2.0. x, MININDC(x) will return a col­ External language support: As- In spite of all this, the inter­ umn vector containing the index sembly face to the program (in the version (i.e. row number) of the small­ Save Warning?: No current at the time we wrote this est element in each column of x. Shell to DOS?: Yes review) is extremely clunky. It has We didn't come close to learning A note on graphs: Tradition­ a direct mode where you can just them all; the Command Refer­ ally, APL graphics relied on a set type in an arithmetic string but ence section of the manual runs of routines called D G functions you must end each line with (F4} 300 pages. GAUSS also has so­ and they were far from transparent (F2} to execute. Moreover, it in­ phisticated ways for dealing with to the user. With the current ver­ sists on telling you that the current sion 7.0, VDI support from GSS

SEPTEMBER 1988, VOLUME 35, NUMBER 7 987 Computers and Mathematics

has been included with drivers for the CGA but they are far from An important aspect of APL is a wide array of printers and moni­ ideal and one losses considerable the idea of"workspace", which has tors. With maximum buffering for functionality in the user interface) been borrowed by many other pro­ bitmaps, and an EGAILASERJET or downloadable fonts on the EGA grams. Every function or variable combination, these drivers reduce or VGA. Moreover, by default created (and not erased) during a the memory available to APL by APL uses its own keyboard driver. session is saved in a single file by about 130K. If you use this driver, macro pro­ the command While MATHCAD does have grams, cut and paste utilities and some very limited programmabil­ other resident programs are not )SAVE wsname. ity, its basic philosophy is to pro­ available. You can choose to not vide you with canned routines that use the APL driver and have a key­ When the workspace is loaded at are relatively easy to use. A pro­ board macro program present but a later session, all functions and gram like MATLAB is basically a then you are forced into a strange variables can be freely used. programming language limited to keyboard remapping where, for ex­ STSC has announced APL* mathematical manipulation with ample a ")" is not entered with PLUS 111386, a version that runs lots of canned routines that you the key that has a ")" on it but on 80386 computers making use can invoke or modify by changing the one with a quote on it! So of the large address space of those the source code provided. APL is while a macro program will record computers. It allows nested arrays on the opposite end of the spec­ keystrokes properly for playback, and we are told that its handling trum from MATHCAD-it is a they will not appear in the macro of editing matrices and many of full fledged programming environ­ editor as they appear to APL! An­ the other issues we discuss is much ment with a limited number of other unfortunate consequence of improved. canned routines included with the the special character set is the im­ possibility of using one's favorite basic package. Benchmarks: Overview One could argue that we could editor to write programs. The full just as well discuss BASIC as APL, screen editor in STSC's implemen­ There are perhaps two types of but in fact APL was developed by tation of APL is limited but APL problems one might use as tests or a mathematician as a compact lan­ programs require special charac­ "benchmarks" for these programs. guage for mathematical expression ters not available in other editors. First, there are those problems and so stands out among program­ Among APL's strengths are its which are relatively simple and to ming languages that a mathema­ many array primitives. When we which one just wants a quick an­ tician might want to use and as said it didn't have many "canned swer with no fuss. Then there are time goes on, STSC includes more routines", we meant things like those problems which one might and more canned routines with L U -decomposition of matrices. wish to more intensely investigate their basic package. It seems to But you can quickly reverse a vec­ or research. For the latter, one us that the language is especially tor V with ¢ V, drop the first should be willing to spend hours well suited to problems in algebra five coordinates with 5 l V, or­ reading the manuals and learning and discrete mathematics. It takes der its elements with V[4VJ, find the programs. a little getting used to but once the value of the largest coordi­ We don't claim to have done you do, it becomes quite natural nate with r1 v, decide whether any research level problems, but to write something like some coordinate is 7 with 7 E V, we did choose the Gibbs Phe­ extract the positive coordinates nomenon as a topic to investigate +II 100 with (V> O)IV, and find the first in depth with several of the pro­ nonzero coordinate position with grams. In the other samples, we to sum the numbers from 1 to (V = 0)11. If M is a matrix, then limited the effort spent on our ap­ 100 (1 100 forms a vector with a M[L}.M;] reorders the rows lexico­ proach to the problem to what we 100 entries and +I sums the vec­ graphically. With a few keystrokes consider a reasonable amount so tor). Other aspects of the language and no programming, you can ro­ that we didn't do certain prob­ take more getting used to. It has tate the ith row of M i coordinates lems which would be possible if its own symbol set so you'll need to the right, add a column of ze­ we wrote a little program. to replace a character ROM on ros, delete the rows of all zeros, The sample problems were done a monochrome monitor or CGA etc. on an 8MH IBM PC-AT with (there are soft character fonts for 640K of memory (505888 bytes

988 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

free after the operating system Benchmarks: GCD and user's TSR's were loaded), an We first tried to define a func­ [0] A~ AGCDB 80827 mathematics coprocessor, tion GCD which could be used [I] --.. (B = O)pO and an EGA (enhanced graphics freely in subsequent work; only in [2] A~ BGCDBIA adapter). The time required for ex­ APL, PCMATLAB, TKSOLVER on other machines will be ecution and GAUSS were we able to do It took about I I 4 second to exe­ For example, a 4. 77MH different. this. But all programs except EU­ cute 8088-based IBM PC or compatible REKA did solve our isolated ex­ an 8087 coprocessor will without ample. 156562431911123 GCD require perhaps five times as long. POINT FIVE: We typed in the 442677773754356. Our list and a brief description four lines of the sample problems follows. GAUSS: We typed In describing the code we used 001 a= 156562431911123 to solve the problems, we have 002 b =442677773754356 » procgcd(a, b); putting chosen to save space by 003 for i = I to 50 do if b eq 0 if b = 0; retp(a); endif; several statements on one line. The then i = 51 else temp = b, rept(gcd( b, a%b)); endp « statements are separated by the b =mod(a, b), a= temp program's statement separators. 004 a and then pressed (F2) to define GCD: Compute the greatest gcd. In an instant, we had common divisor of two 15 digit It took about 1.5 seconds to supply 7 as an answer. integers. gcd(156562431911123, Lissajou: Graph x = sin(2t + 442677773 7 54356). 0.6), y = sin(5t) as t ranges over PCMATLAB: We created an [0, 2n]. M-file: EUREKA: Can't do it. Roots: Find the roots of x 3 - MATHCAD: It is not easy to x 2 - x- 1. function b = gcd(a, b) calculate GCDs in MATHCAD, Hilbert: What is the largest if a = 0, return, end but we did it. Here is an approxi­ = (11(i+j))7,J=I HilbertmatrixHn b = gcd(rem(b, a), a); mation of what our screen looked which can be successfully inverted? like after we finished: Hilbert matrices are chosen be­ (One of us is very fond of re­ cause they are very close to being cursive programs.) Then we asked v

SEPTEMBER 1988, VOLUME 35, NUMBER 7 989 Computers and Mathematics

loaded, one has the GCD function Typing is more involved than the previous available. If one has just followed programs to set up; we first had to the tutorial, it would be tempting fee GCD(156562431911123, edit DOS's CONFIG.SYS file to to set up a mle x = GCD(a, b) 442677773754356) include (to support an ega; a third and then go to the variable sheet file would have been needed if and put in the values of a and b caused an error message "Func­ we'd wanted to use VDI graphics but TK has an immediate evaluate tions nested too deep". While this on a supported printer) the lines command j EE (Evaluate Expres­ is better than MATLAB's dump to sion) after which typing in DOS, it doesn't seem much BET­ device= c : \apl\ibmega.sys TER but in fact one can recover device= c : \apl\gsscgi.sys GCD(156562431911123, completely. The values of a and 442677773754356) b at the time the nesting became and install an interrupt handler too deep are stored in the vari­ (APLCG12.EXE in our case), call provided the value 7 immediately. ables a and b, so that entering up APL and load the VDI.A WS Trying to roll our own, it took fee GCD(a, b) will complete the workspace. Then the procedure is us a while to figure out how to recursion yielding. 7. similar to the other programs: write procedures in TKSOLVER (the manual is not very good Benchmarks: Lissajou T +-- oO, (z200)-:- 100 on this; it emphasizes using the X +- 1 0 .6 + 2 X T provided applications rather than MATHCAD: We typed in writing your own), but in the end Y+-1o5xT it was simple. We created a proce­ f(x) := sin(2 * x + .6) DISPLAY '0 0 640 350 dure GCD which had a and b as g(x) := sin(5 * x) XYPLOT X, [1.5]Y' input variables and d as the output t := 0, .05 .. 6.3 variable, and which was defined by The graph displayed the 101 plot­ Then hit @ for a plot and graphed ted points (as dots) joined by line top: if b = 0 then goto done f(t) against g(t) with appropriate segments in EGA resolution with a call mod( a, b; rem) limits. Success. pair of axis (but without labelling a :=b POINT FIVE: We typed in or graduations). One of the obscurities of APL b :=rem t = index{120) * pij60 is apparent; instead of SIN(X), goto top x = sin(2 * t + .6) one must write 1 oX. If you com­ plain to an APL person about this, done: d :=a y=sin(5*t) he will respond that you can easily Then we typed linegraph(y, x) define SIN X in your workspace to be 1 "circle" X. Nevertheless, this fee GCD(156562431911123, Success. The graphing is done in (remembering the meaning of the CGA resolution, so was not as nice 442677773754356) symbols, not to mention where to as in MATHCAD. POINT FIVE find them on the keyboard) is a chooses the limits of the axes for found the answer 7 immediately. big problem for beginners-and if you, which was appreciated. you are a casual user and don't use We also tried a recursive func­ EUREKA: EUREKA can only tion following a suggestion from APL for a month, you will surely graph functions of a single vari­ forget. Similarly, the ",[1.5]" in­ the makers of TKSOLVER. One able; it cannot graph parametri­ makes a, b the input and d the dicates one of a large number cally. of different ways of concatinating output as above but lets the func­ APL: APL as a programming tion GCD be defined by the single matrices. language has quite decent graphing PCMATLAB: We typed in statement primitives. But it is not for casual use as a graphing program. t = 0: .05 : 6.3; if b > 0 Nevertheless, STSC has pro­ then d := GCD(b,mod(a,b)) vided a graphics support workspace x = sin(2 * t + .6); else d :=a which makes possible what might y = sin(5 * t); be called semi-casual graphing. It plot(x, y)

990 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

and PCMATLAB choose the limits plicitly, one would be hard pressed sufficient if you know some APL of the axes and numbered them to graph the function with MATH­ and, in the end, we liked the way for us. The program recognized CAD. In TKSOLVER, the struc­ it worked. However, we remark our EGA and gave us a red graph ture is such that setting up the that CXROOTS compared poorly with white axes. problem would be no harder than to some of the more specialized TKSOLVER: We went to the the simple parametric equation equation solvers when asked to Variable sheet and with a few that we discuss. find roots of the more complicated keystrokes specified t as an input GAUSS: First one types in functions that we tried. list and x, y as output lists. Then Here is what one must do after we had to highlight t and hit ">" » run graph2d.get « loading the workspace: We defined twice to "dive" into the subsheet a function CXfn (you must use for t. Then hitting "!" brought up followed by (F2) (F4). The double that name) by the single line a dialog to define t as a list from chevrons are not typed in (they are the GAUSS program delim­ 0 to 6.3 in units of 0.05. Next we Y +-(X ZSTAR 3) had to go to the Rule sheet to type iters). (F2) ends the program and -((X ZSTAR 2) +X+ 10) in the parametric equations and (F4) executes it and in this case 2 dimensional graphing routines are then hit (FlO) to evaluate the out­ Here ZSTAR is one of the func­ put lists. By default the program loaded into memory. Like so many other features, graphics must be tion sprovided in COMPLEX. Note insists on flashing answers as it that the workspace uses a vector of computes, so we had to wait a no­ loaded as an external program. Then one need only type in length two to represent a complex ticeable amount of time for what number (so that i is represented was only about 200 simple calcula­ » t = seqa(O, 0.05, 126); by 0 1) and a 2 by n matrix to tions (we later learned of a way of px = sin(2 * t + .6); represent a complex vector, etc. modifying this default which sped This means that the usual + and up the calculations considerably). py = sin(5 * t); - work as you would like with Then we had to go to the plot sheet call graph2d; complex arrays but you must use and specify that we wanted a line other provided functions for mul­ plot with x and y as the variables to get a graph with default axes tiplication, etc. on the x and y axes. Finally (F7) properly choosen. Before running Then we executed drew the graph. GAUSS, you run a configuration TKSOLVER recognized our program to pick colors and to tell CXROOTS123456 EGA and choose the axes for us. GAUSS that you have an EGA. The graph quality was good but Default axes, colors etc. can all be and were rewarded with the dis­ the number of keystrokes needed adjusted by setting certain global play and the involved interrelationships variables to the desired values. which one had to juggle made this far from the effortless task it -o.4196433776 -0.4196433776 was in POINTFIVE, MATHCAD, Benchmarks: Roots 0.6062907292 -o.6062907292 MATLAB or GAUSS. APL: STSC's APL*PLUS comes 1.839286755 We should report here the com­ with a workspace COMPLEX which -2. 90300813E-28 ment that the product manager for contains, in particular, a routine TKSOLVER made to this com­ CXROOTS which will attempt to plaint: for a simple contrived prob­ find the complex zeros of any func­ the columns of which represent the lem like this, TKSOLVER is more tion you can define in APL. Like three roots. This took 13.7 seconds complicated than need be but the all the provided workspaces, you when we supressed the display of complication is there to accom­ are not presented with what we the intermediate iterations and a modate more complex situations would call a polished user inter­ couple seconds more if we did not. where x and t might obey some face (use EUREKA if you want The numbers following CX­ involved implicit equation rather that). Also, as is typical, the writ­ ROOTS are "guesses" which you than an explicit one. There is ten documentation (a paragraph) provide to the function (and which some merit to this remark: while is limited and you are referred are interpreted as half as many MATHCAD shines in the trial to the workspace itself for addi­ complex numbers). You must pro­ problem, if x and t were given im- tional documentation, but this was vide at least as many pairs as you

SEPTEMBER 1988, VOLUME 35, NUMBER 7 991 Computers and Mathematics

want roots (although for some rea­ of a built-in variable called TOL GAUSS: One of the furnished son, you may not provide just one which specifies the allowed error "external functions" provided with pair). The function CXROOTS is and which defaults to 0.001 so GAUSS is a polynomial root finder. pretty clever in what it does with these answers are correct within It worked quite well; here's what the guesses; it never finds a root the allowed error set up initially. we had to do to get the answer: more than once and often any We went back to the initial value guesses sees to work. For example, x = 0 tried setting TOL to 0.0001 »run eig.arc « and were greeted with the message » run poly.set « CXROOTS 100101102103 "Does not converge" so we tried »LET c = 1- 1- 1- 1; 104105106107 x = 1.83 and got the same message BUT when we tried an initial guess polyroot( c) « produces as output of 1.9, we were rewarded with *No convergence for root num­ 1.83928676 and the complex root [Answer:] ber 4; value returned for that root was obtained accurately when we 1.83928676 0.00000000 is the computation after 6 itera­ took an initial value of i. The tions moral is that MATHCAD at least -0.41964338 0.60629073 seems quite sensitive to the initial -0.41964338 - 0.60629073 guess. -0.4196433776 -0.4196433776 POINT FIVE: POINT FIVE This procedure actually works by 0.6062907292 -0.6062907292 can not solve equations (even poly­ setting up a matrix whose charac­ nomials) primitively. We could teristic equation is the given poly­ 1.839286755 have failed it on this problem, nomial and finding the eigenval­ 6.058451752E-27 but we were a little more patient ues! 4.29896533E 18 and taught it Newton's Method EUREKA: We typed x~3 - x~2 - x - 1 = 0 into the Edit win­ -3.732554945E17 f(x;) Xi+! =X;- f'(X;) dow and EUREKA solved (for the real root) correctly in less than MATHCAD: We set the number for our particular polynomial. This a second. This was very easy. of digits displayed to 8 and typed may be more than many users wish We then choose Options from the in (the initial value of x is the to do in order to find roots. main menu, settings from the sub­ starting value of x) We typed in the lines menu, and changed Complex to YES. We added x := i to the Edit X=2 x:O window to tell EUREKA to start X= X- (X* X* X- X* X- 1)/ and in an instant it y : root(x~3- x~- x- 1, x) at that value (3*X*X-2*x-1),x had a wrong answer: which displayed as and executing the second line until Solution: the value of x no longer changed. x:=O Variables Values Thus we found the real root only. rex = .45791313 y := root[x3 - x 2 - x- l,x] PCMATLAB: This program has imx = -.63752431 a built in polynomial root finder. Maximum error is .11698753 When we typed in y =, MATH­ All we needed to type in was CAD replied 1.83930348. If we "»P = [1 - 1 - 1- 1] went back up to x := 0 and en­ Well, perhaps it is not wrong since tered x : 1i, since MATHCAD -:»format long it is within the maximum error. understands complex numbers, it -:»roots(p) When we changed the equation in replied the Edit window to X*X*X-X*X­ to which it responded x - 1, the complex root was found -0.41969715 + 0.60633699i 1.83928675521416 accurately, so the bug is in com­ -0.41964337760708 plex exponentiation. In addition, The observant reader may have EUREKA has an explicit "poly" +0.60629072920720i noticed, as we did, that these are function for finding all the roots not correct after the 3 digit! After -0.41964337760708 of a polynomial. If one types in consulting the manual, we learned -0.60629072920720i f(x) := poly(x, 1, -1, -1, -1) and

992 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

hits the Solve key, in an instant as they give some idea of how to a= max(abs(id­ one is greeted with use the programs. The inverting of matrixmult(h, h inv))) Roots to the polynomial the Hilbert matrix for n about 10 f b max(abs(id­ takes a noticable amount of time = a second or two, in all these pro-' matrixmult(h inv, h))) Real part Imaginary part grams. The double loop to form max( concat( transpose( a), -.41964338 0.60629073 the Hilbert matrix can take almost transpose( b))) -.41964338 .60629073 10 seconds (although it was in­ 1.8392868 .00000000 stantaneous in APL because of the built-in jot-dot operator). This was nicely done but only n=8 0.00000025331 partly makes up for the fact that In PCMATLAB, GAUSS, and APL, we had a choice of defining n=9 0.00000739097 doing it the way most users will n = 10 0.00016784667 try (if they haven't absorbed the functions (writing small programs) to compute the Hilbert matrix n = 11 0.01135253906 manual) doesn't work. n = 12 0.13476562500 TKSOLVER: We typed x~3- given n, or to compute ERR(n) from n, or doing everything from n = 13 1.18 7 50000000 x~2 - x - 1 = 0 into the Rule the command line. n = 14 360.00000000000 Sheet, went to the Variable Sheet n = 15 4.12500000000 and entered 2 as a "guess" for APL: A "DOMAIN ERROR" x and "G" in the status column was given when we attempted to which tells TKSOLVER that 2 is invert the Hilbert matrix when N 2:: 12. The code used follows. a guess and pressed (F9} to solve. PCMATLAB: For order 12, we Less than a second was required N +-- 10oH +-- ..;--1 got the message. for the real root as 1.8392868. Warning: Matrix is close to sin­ You enter complex numbers as + (zNt. + (zN) gular or badly scaled. Results may pairs (x, y). It does not understand H INV +-- ffi H be inaccurate. (x, y)~3 but it does understand ID +-- (zNt. = (zN) POWER{{x, y), 3) so we entered RCOND 3.572248e- 017 + ERR +-- r/, (ID - H + . = H POWER((X. Y), 3) x INV), and indeed this seems to be the -POWER((x, y), 2) (ID - H INV +. x H) case. Here is the code and the results. -(x,y)- (1,0) = (0,0) Here are the results: We created an M -file HILB.M {MATLAB comes with a very sim­ and made guesses for both x and 0. 000000 18348 N=8 ilar M -file of the same name) con­ y. It found the root -.4196434 + 0.00000898586 N=9 taining the lines .60629073i. TKSOLVER also has N = 10 0.00028113640 an explicit model dealing with cu­ N = 11 0.00697469388 function h = hilb( m) bic equations which we could have used. POINT FIVE: POINT FIVE and fori= 1: m MATHCAD seemed not to mind for j = 1: m being asked to invert near singular h(i,j) = 1/{i + j- 1); Benchmarks: Hilbert matrices; they just did their best end We attempted to calculate the in-· (and gave no warnings). Here is verse, H-I, of the Hilbert matri­ the code and the results with the end ces of orders n for n = 8, 9, 10, .... former: We defined ERR(n) to be the max­ Then we entered n = 10, h = shape(fill(O, n * n), n, n) imum of the absolute values of the n = 10; x = hilb(n); union of the elements of I -HH-1 for i = 1 to n do for j = 1 to n a= x * inv(x)- eye(n); and I - n-1 H (as computed by do h[i,j] = 1/(1 + j- 1) b inv(x) * x- eye(n); the programs). The methods we h inv = matrixinv(h) = had to use to construct the Hilbert max(max(abs([a, b]))) matrices and to calculate ERR( n) id = shape(fill(O, n * n), n, n), will also be of interest to the reader for i = 1 to n do id[i, i] = 1

SEPTEMBER 1988, VOLUME 35, NUMBER 7 993 Computers and Mathematics

at the command line. The answers: was executed (with various values The results were of n) to find ERR(n) as n=8 0. 00000069771 n=8 0.000000512 n=9 0.00004452642 n=8 0.00000017 n=9 0.000025083 n = 10 0.00166572630 n=9 0.00000481 n = 10 0.001103396 n = 11 0.03627222776 n = 10 0.00021564 n = 11 0.036533826 n = 12 1.68031120300 n = 11 0.01033556 n = 12 4.563780308 roughly comparable to MATLAB MATHCAD: It was easy to For n = 12, we got the message as GAUSS. form and invert the Hilbert ma­ "matrix singular". but not quite as good EUREKA: Since EUREKA only trix, but it was a challenge to com­ TKSOLVER: To TKSOLVER, deals with equations, we asked pute ERR( n )-we got tired and a matrix is a list of lists (the it to solve the eight systems of did it by eye. One types in (re­ rows). We loaded the model IN­ eight linear equations equivalent call that the origin of vectors and VERT.TK. (The documentation is H ;K H is the or­ matrices in MATHCAD is 0) provided only as comments in the to 8 = g where 8 der 8 Hilbert matrix and g ranged model.) We defined a "Procedure over the eight unit vectors. Each n := 9 i := O.. n j := O.. n Function", Hilbert in the Function system took between 5 and 4 7 sec­ Sheet as follows onds to solve (for some reason, the M;,j := 1/(i + j + 1) fori= 1 ton same system varied in the time and displays required when we tried to solve for j = 1 ton it again, and very slightly differ­ 'aO[i][j) := 1/(i + j- 1) M-1 • M- identity(n + 1) and ent solutions were found). The next j maximum error according to EU­ M · M- 1 - identity(n + 1) next i REKA varied from .00057587813 when g = (1,0,0,0,0,0,0,0)1 to and by eye looks for the entry of .61053876 which occurred when maximum absolute value (despite and a function worst defined by g = (0, 0, 0, 0, 0, 0, 1, 0) 1• We were the manual says, the max what tired of typing in the equations by function seems to only work on W:=O hand and did not carry this test vectors, not matrices). For 10 x 10 fori= 1 ton further. matrices this maximum was about 'w[i][i] := 0 x 0.0005, about 2.0 for 12 12, and W := max(W,max('w[i]), 188 for 14 x 14. Benchmarks: Inverse Squares GAUSS: We typed in - min('w[i])) The effective use of arrays in the next i programs that have them (mem­ ~proc hilb(n); those pro­ and finally the rule sheet ory permitting) allows local i, j, h; h = zeros(n, n); grams, interpretive as they are, to i = 1; call Mblank('aO, n); approach the speed of compiled do while i <= n; j = 1; initialization languages. Nevertheless, TURBO PASCAL was faster in actual exe­ do while j <= n; h[i, j) call Mblank('A, n) cution time compared to the fastest =1/(i+j-1); j = j + 1; call Mdelete(' a) call Hilbert(n); (GAUSS) of our array processing endo; i = i + 1; endo; the above procedure supercalculators. Loops in these retp(h); endp « call LUDMIN interpreters are terribly slow as (n, 'aO, 'a, 'A); our tests show below. and pressed (F2) to define the built in matrix inversion All programs used the math function hilb(n). Then, call Mproduct('aO, 'A, 'w) coprocessor and all those which W 1 = worst(n) did our problem gave the answer: n = 10; h = hilb(n); call Mproduct('A, 'aO, 'w) 1. 64480907 5. APL: We entered "+/+(z8000) max c(maxc(abs(h * inv(h) W2 = worst(n) ERR= max(W1, W2) x (z8000)" and got the answer in -eye(n)))); maxc(maxc 5.6, 4.8, and 4.8 seconds, respec­ (abs(irtv(h) * h- eye(n)))); tively, on three successive trials.

994 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

output variable and then typed in Then we pressed ";" to get to the the program Variable Sheet and put G's (for "guess") in the Status columns for z :=0 all three variables; then (F9) to [11 s..-O<>I.---1 fori= 1 ton solve. Within a second or so we [2] LOOP: S ..- S ++I x I z := z + 1/i'""l got the answer [3] --t (8000 2:= I.--- I+ 1)/LOOP next i Name Output PCMATLAB: We typed in x = X .59905376 Then we typed fee isqr (8000) and 1: 8000;sum((1)./(x. * x)) and 4 y 2.3959314 after 16 seconds we got the answer seconds later, we got z 2.0050148 1. 64480907 4660402. ans= GAUSS: We entered One unique feature ofTKSOLVER 1. 64480907 466040 is that it is not necessary to guess x = seqa( 1, 1, 8000); all three variables-it will mix di­ Programming an explicit loop with rect substitution and its indirect sumc(l./(x. * x)) method so that if you only guess s = 0; for i = 1 : 8000 two variables, it will use the third and received the answer in 1.92 equation to find the third vari­ s=s+1f(i*i); end; seconds; this was the fastest of able and then minimize the errors all our review programs. Even an s took 2 minutes, 14 seconds. from the other two equations. In a explicit loop simple problem like this, the time saving and potential increased ac­ POINT FIVE: We entered "sum i = 1; s = 0; curacy in this change from three do while i <= 8000; to two variables isn't important (1/(index(8000) * index(8000)))". s=s+1f(i*i); but it could be significant in larger problems. The fact that there are The answer was the same as APL's i = i + 1; endo; s only two variables does give one and PCMATLAB's, but took 17 took only 26 seconds. access to various graphical meth­ seconds. Five minutes, 15 seconds TURBO PASCAL: 1.26 sec­ ods and the TKSOLVER staff took were required for the statement onds with the 80287 coprocessor; this benchmark, used TK's con­ "fori= 1 to 8000 do s = s+ 1/{i* 3.90 seconds with Turbo's six byte tour graphing and located a second i);" to execute. reals and no coprocessor. We were MATHCAD: We entered solution at surprised that GAUSS did as well as it did relative to Turbo. i == 1..8000 Name Output X 5.1004127 Benchmarks: Simultaneous Equations y -2.6442372 and got our answer in 24 seconds. EUREKA: We typed the equations z 2.54382442 EUREKA: EUREKA accepted in the Edit Window, then pressed "x = sum(1/(n*n), n, l, 101)"when (Esc) and (S) to solve. In a little MATHCAD: One begins by asked to Solve and gave the correct over a second, we had an answer: typing in the desired accuracy and answer. When 101 was replaced by digits displayed if you want more 102 (or 8000), the error message Solution: than the default 3 places. Then "sum is too large" was displayed. Variables Values you put in guesses of the 3 num­ bers listed as equalities and follow TKSOLVER: We went to the X = .59905376 function sheet by typing "= F", y = 2.3959314 that by the key word "Given". One filled in the blanks to say we z = 2.0050148 then types in the equations using wanted to define a procedure called Alt- = to show equalities here. isqr and used ">" to dive into the Maximum error is End with "Find(x, y, z) ="and the correct values are displayed within program subsheet. On that sheet, 1. 7763568e - 15 we filled in blanks to indicate that a second. n was an input variable and z an TKSOLVER: We typed the three PCMATLAB: There is nothing equations into the Rule Sheet. advertised as an equation solver

SEPTEMBER 1988, VOLUME 35, NUMBER 7 995 Computers and Mathematics

but in the readme file on disk is the which loads the non-linear equa­ max(fk(x)) does not go to 7t/4 but description of a function added tion solver. Then one types in to a universal quantity. after the manual was printed: The goal of this benchmark is "nelder.m Nonlinear optimiza­ »let p=111; to study the Gibbs phenomenon tion" vf = zeros(rows(p), 1); graphically and analytically with which minimizes a function of proc f(w); special emphasis on / 20• We would several variables. By setting up an hope, at a minimum, to be able to local x, y, z; x = w[l, 1]; explicit sum of squares and min­ graph / 20 on the interval (0, 1.6), imizing, one solves the equations. y = w[2, 1]; z = w£3, 1]; say, and calculate its maximum First you've got to make an M-file, vf[l, 1] = sin(x) + y-"1 value in the interval. But we would call it xyz.m +ln(z)-7; vf[2,1]=3*X also like to compare, say / 8, /9, and / 10, perhaps by graphing all three + 2~y - z~3 + 1; on the same axes. In addition, we function q = xyz(p) vf£3, 1] = x + y + z- 5; have tried to calculate the values retp(vf); endp; of fsoo and !IOooo(x) for x = 1 and X= p(1); y = p(2); Z = p(3); 1.1. Another item was to compute x = nlsys(&f, p, 0, 1e- 6, 0, 1); q1 = sin(x) + y-"1 + log(z)- 7 the average of / 20 on the interval q 2 = 2 * x + 2~y - z~3 + 1 and within an instant, one gets the [1, 2], i.e. q3 =X + y + Z - 5; response 2 q = sum([q 1 q2 q3].'"1); ~ fzo(x) dx. Final estimates for x are: 0.599054 2.395931 2.005015 This should be close to 1t I 4 = Then, inside MATLAB, use the 0.7853981634. It is readily evalu­ In the above, the parameters for commands ated exactly as a sum of cosines the function nlsys include &f and has the value 0. 7860233539 which is a pointer to the func­ format long to ten places. tion f, p which is the initial guess pO = [111] 1e- 6 which is the tolerance. The p = nelder('xyz', pO, 1e- 8) fact that one cannot deal directly MATHCAD with vectors but must discuss 3 by To graph / , we had only to type 1 matrices and the arcane syntax 20 to obtain after 129 seconds (con­ in siderably slower than other pro­ are typical of GAUSS. grams!!): i:= 1,3.. 39 The Gibbs Phenomenon 0.59905375845510 f(x) := L sin(~· x) The Fourier series expansion for . l 2.39593140122909 the square wave s(x) defined by I 2.00501484103321 X:= 0, .01..1.6 1Cj4 if 0 < X < 1C s(x) = { Then, moving to a new area, type GAUSS: This program comes -1Cj4 if 1C

996 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

and fill in the limits, function, and (it was 0.9261319947); to do this, procedure Gibbs. In the fill-in-the­ variable. one enters blanks this time, we indicated n as Of all the programs we consid­ a parameter variable, x as input ered, MATHCAD did all this with f/20 GIBBS (llOO)-:- 1000 and z as output and typed in the the least manual reading. rules because for any vector V, f jV will return the maximum entry in z :=0 APL V similar to the way + j V will fork= 1 to 2 * n- 1 step 2 APL has no native graphing abil­ return the sum of the entries in z := z + sin(k * x)jk V. It took 17.7 and 21.5 seconds, ity. But its ability to calculate next k values of the partial sums fk(x) respectively, to compute greatly exceeds the abilities of the Getting graphical output is a lit­ other programs. fwooo(1) = 0.7853740028 tle tedious. One must go to a plot An experienced APL program­ sheet (with"= p") and specify lists mer would define a function GIBBS fwooo(l.l) = 0.785421729 to be plotted. Normally, one must something like what follows: The trapezoid rule · go to list subsheets to enter the lists but a recent TK.SOLVER newslet­ [0] V ..- K GIBBS X ( + /(20 GIBBS 1 2), ter (both TK.SOLVER and MATH­ [1] ODDS..- -1 + 2 x zK (2 x 20 GIBBS 1 + (z99) + 100)) CAD have informative newsletters [2] V ..- +/[1](1 o 0DDS0 • x X) with useful information for users) -:-200 -:-ODDS 0 • +ox x suggests setting up once and for all a function plot with input pa­ If one were satisfied to calculate gave 0. 7860150883 as the integral rameters fun xl, x2, m and the fk(x) only for scalars x, then one (*) in about 3.5 seconds. Simp­ definition would use son's rule call blank(' x) +/(1 oX x ODDS)-:- ODDS call blank(' y) (+/(20 GIBBS 1 2), dx := (x2- x1)jm as the line [2] of the definition. (2 x 20 GIBBS 1 + (z49) +50), But the above definition has the for i := 1 to m + 1 (4 x 20 GIBBS .99+ advantage that it accepts vector 'x[i] := x1 (z50)-:- 50))-:- 300 valued arguments X. For example, 'y[i] := apply(fun,x1) to calculate f 20(1 ), one enters x1:=x1+dx 20 GIBBS 1 gives 0. 7860234421 (and the true next i value is 0. 7860233539). and the answer appears immedi­ For graphing, we used the Here a leading ' refers to a list. ately below. To find f 500 (x) for VDI.AWS workspace (see the Lis­ The built in procedure "apply" al­ x = .001, .002, ... , .010, one enters sajou benchmark). An acceptable lows one to treat a function name plot was produced by the com­ as a variable. One would define mands plot one time and add it to ones 500GIBBS.001.002.003.004 repetoire of functions. After defin­ .005.006.007.008.009.010 VALUES ..- 20 GIBBS (llOO) ing Gibbs and plot, one must go to -:- 100 DISPLAY the list sheet and dive to a line plot and is (after 9 seconds of compu­ subsheet specifying 'x as the x axis tation) presented with '0 0 640 350 LINEPLOT VALUES' variable and 'y as the only y axis 0.4730415603 0.8027066335 variable. Next use"= V" to go to 0.9243265233 0.8791017243 the variable sheet and specify the TKSOLVER 0.7749654244 0.7123432723 value 20 for n. Then type "fee call 0. 7272979221 0. 7870935903 First we needed to define a func­ plot('Gibbs, 0, 1.6, 160)" which 0.8325207556 0.829174451 tion Gibbs depending on a pa­ took 13 seconds to evaluate. (F7} To find the maximum of the rameter, n. As usual we went to then produces a plot instantly. 100 terms f 2o(x) for x = the function sheet with "= f", Graphing the three functions .001, .002, ... , .100 took 7 seconds and said we wanted to define a requires you to change the value of

SEPTEMBER 1988, VOLUME 35, NUMBER 7 997 Computers and Mathematics

n, use the same" I ee call plot('Gibbs, yielding the answer 0. 78601508833. 0, 1.6, 160)" three times with the Evidently, POINT FIVE uses the use of use "I ee calllistcopy(' y ,' z?)" gibbs=acctotal( sin( matrixmult trapezoid rule rather than Simp­ with ?= 1 and then 2 between (n, tr(x)) )In) son's rule to do its integrals. calls to plot. In the plot sheet, Memory and time problems indicate that the series should be at which point, one gets a single do not make the calculation of y, z 1 and z2. Given that there graph with gibbs 10000 practical. is no stack of 1ee values, one x20 = tr(gibbs[20, must "type call plot('Gibbs, 0, 1.6, m EUREKA 160)" three times! Alternatively, linegraph(x, x20) one could write a single procedure Here is what we typed into the based on plot which defined these and multiple graphs with Edit window: three functions. By setting the value of n in the x8 = tr(gibbs[8, m y = sum(sin(x * (2 * n + 1)) variable sheet to 500 or 10000, x9 = tr(gibbs[9, m 1(2*n+ 1),n, 1,20) it was easy to compute !IOooo( 1) x10 = tr(gibbs[10, Ul) which took 31 seconds and the x>O linegraph(x, x8, x9, x10) other values. X< .1 One pays the price for the The CGA resolution and lack of $max(y) vast array of functions that TK­ easy print formatting were the SOLVER comes with that one has After asking EUREKA to main disappointment in the single to locate the particular functions "SOLVE" we got the following in graph. The markers used to distin­ that one needs in the right on disk the Solution Window: guish different graphs thoroughly model. On advice from the TK­ function SOLVER staff, we used a minor mucked up the three E:SOLUTION. Line 1 graph. The maximum of x20 is modification of the Golden mean Solution: computed as max(x20) and gives search section from OPTIM\ OP­ 0.925863837. Variables Values TIM 1C. TK for the maximum prob­ lem and defint3 (using a Romberg gibbs500 is evaluated fairly X = .0785398574 from DIFFINT\ y = .92613220 approximation) easily with DEFINTEG.TK to evaluate. We Confidence level= 95.6% then set up a Rule sheet saying: n = 2 * index(500)- 1 All constraints satisfied. gibbs500=sum(sin(n)ln) y = Gibbs(x) format(gibbs500, 20, 17) To plot, we replaced y by f(x) and call Golden(.05,.15, 'Gibbs;x,y) asked EUREKA to "plot" f with average = defint3('Gibbs, 1, 2) gibbs500 endpoints 0 and 3.14. After view­ ing the cute dots for a second, we set n = 20 in the Variable sheet and a similar formula for the pressed F5 to "Zoom" into graph­ and evaluated. The time to com­ value at x = 1.1. The answers are ics mode. About 16 seconds. Very pute both the maximum value 0.78506368425933900 nice. EUREKA chooses the y­ (. 9261322) and the integral 0. 78489105914780900. limits for you and does not require (. 78602335) was 72 seconds. The The integral is computed with you to specify the exact x values built in Romberg integration pro­ the program on which to evaluate the function. cedure is exact to the number of Evidently, it simply chooses the decimal places shown. x = index(101) * .01 + .99 values of x corresponding to hori­ n = index(20) * 2 - 1 zontal pixels on our EGA monitor. POINT FIVE gibbs=acctotal This is nice-all programs should As in APL, the Gibbs problem can (sin(matrixmult(n, tr(x)))ln) do this or have it as an option. There is no way to calculate be set up compactly. tr stands for x20 = tr(gibbs[20, m transpose, not trace. !IOooo(1). We got an error mes­ gibbsint=integrate( x20, .01) sage "sum is too large" when we x = index(160) * .01 format(gibbsint, 14, 11) tried even /102(1), but fiOt(1) was n = index(20) * 2 - 1 gibbsint calculated correctly.

998 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

PCMATLAB be easy to define an M-function time, one uses embodying it. As usual with MATLAB, to define a function, you call up your editor ~ u = seqa(O, .01, 160); ~X = 1 : .01 : 2; and define an M-file. We made a px =- u-u; ~Y = gibbs(20,x); file gibbs.m with the lines PY = gibbs(u, 8) ~(y(l) + y(IOI) + 2 *sum -gibbs(u, 9)­ function y = gibbs(k, x) (y(3 : 2 : 99)) + 4 * sum gibbs(u, 10); call graph2d « y = zeros(x); (y(2: 2: 100)))/300

for i = 1 : 2: (2 * k- 1) To find the maximum of 120, y = y + sin{i * x)ji; Not surprisingly, the answer agreed one need only type in with that found by APL. end GAUSS ~ y = seqa(O,.Ol, 160); Plotting F20 is easy; issue the maxc(gibbs(y, 20)) « commands: One first defines a function gibbs (x, n) as follows and in an instant, one gets the ~X= 0: .01 : 1.6; ~proc gibbs(x, n); answer (0.92586384). Typing in ~Y = gibbs(20,x); local i, I; "gibbs(seqa(l, .1, 2), 500)" resulted 6 seconds later in the answers ~plot(x, y) i = seqa(l, 2, n); There was noticable pause after I= (sumc((sin{i. * x')).ji)); 0. 78506368 0. 78489106 the line defining y, but the plot­ retp{f); ting was instantaneous. Finding endp « but trying gibbs( 1, 10000) gave an the maximum value was done with error message presumably because The "." before an operator indi­ a 10000 element sequence takes ~ max(y) cates outer product when the ob­ more than 64K, but, as in MAT­ LAB, the sum could be evaluated which returned the answer 0. 92586. jects are a row and column vector and element wise operations when by breaking it into two pieces. Comparing F8, F9 and FlO on one To compute the Simpson's rule set of axes is done with both are column vectors (and col­ umn by column operations when integral, one just types in ~ plot(x, gibbs(8, x), x, one is a matrix and the other ~ intsimp(&gibbs, gibbs(9, x), x, gibbs( 10, x)) a vector). By default vectors are column vectors. The ' in the for­ 1, 2, 20, le - 8) « while mula defining I is needed so that x can be a column vector as it and gets the answer 0. 78602335 must be to do graphing. Having which is particularly accurate and ~ y500 = gibbs(500, [11.1]) run graph2d.get at the start of the quick in comparison to the other session, one can graph ho quickly programs. returns 0. 78506368425934 and by just using 0.78489105914781. Because MAT­ What's missing LAB evaluates a sum by forming · ~ px = seqa(O,.Ol, 160); The short answer as to what's a vector (rather than by a direct py = gibbs(px, 20); call graph2d « do loop) and vectors are limited missing is "a lot". By taking the to 8092 elements, one cannot use Graphing multiple functions isn't best from each of these programs gibbs.m to evaluate F 10000 but difficult but it is a little awkward; one could get a very good supercal­ culator and some of the individual it is easy to break the sum into px and py must be matrices of programs are certainly functional two parts which one then sums the same shape so one uses the but we are struck by the fact that together. operator ~ which concatanates two the most powerful of these pro­ MATLAB has no built in inte­ matrices on the left-right basis. So, grams have flawed user interfaces gral function but it is easy to hand to graph 18, 19 and 110 at the same code Simpson's rule and it would and the one with the slickest in­ terface has so little power.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 999 Computers and Mathematics

The first thing that our dream bly handles the conflict between LAB, GAUSS and POINT FIVE program has is a decent editor memory usage and the number have adequate control structures and, in particular a decent ma­ of funtions best. If you call a and APL has real programmabil­ trix editor. While it is true that function not in memory, it looks ity without them. EUREKA and programers are often best served for an "M-file" on disk with that MATHCAD are just deficient in by their own favorite personal ed­ name and if it finds it, the file is this regard. As for an intelligent itor, the fact that one needs to compiled and stored in memory. compiler, only TKSOLVER and deal with specialized objects like The problem with this is that each EUREKA have that. matrices requires a separate ed­ function requires its own file; at We'd hope to see a snappy itor. But, recognizing that users a minimum file allocation size of user interface with drop down have to face too many editors, the 2K on most hard disks, this can menus, dialog boxes and mouse math editor in this dream program be very costly in disk space. support. The possibility of sepa­ is fully configurable (as only EU­ The actual list of functions rating the elements of the pro­ REKA's editor is). This is done should include lots of matrix con­ gram into separate windows is a in a configuration program which trol-eigenvalues, eigenvectors and good one-we especially like the means that it takes no memory various decompositions. We'd like separation of input and output from the main program. POINT a wide array of statistical func­ that is present in EUREKA, TK­ FIVE comes closest to a decent tions, including correlation and SOLVER and POINT FIVE. But matrix editor and APL and EU­ fitting procedures and functions separate windows/modules can be REKA to decent editors but none for solving a series of non-linear overdone; the seven separate win­ of the programs pass here and equations. Numerical integration dows in TKSOLVER are rather some do very badly indeed. and differentiation, a wide array duanting. The best user interfaces Next it should have a wide ar­ of mathematical functions (like are in EUREKA, MATHCAD and ray of basic mathematical func­ Bessel, gamma, erf and maybe POINT FIVE which are also the tions including statistical func­ hypergeometric) and fast Fourier least powerful packages. In our tions. Financial functions aren't transforms should be included. opinion, all but EUREKA need required although if it will sell As long as we are dreaming, we some snap added to their inter­ copies outside the scientific com­ can ask for symbolic differentia­ face. On the issue of mouse sup­ munity, there is no reason not tion and other symbolic manip­ port, we should mention that it is to include it. Rich sets of func­ ulation features. On this account unfortunate that none is present tions take a lot of memory and of a wide variety of functions in MATHCAD. A graphics mode thereby represent a problem. Our the programs fare pretty well. TK­ program like MATHCAD has a dream program will have two ways SOLVER, MATLAB and GAUSS problem with speed of just mov­ around this issue. First, it will all have a vast array of functions ing a cursor around the screen­ support the Lotus-Intel-Microsoft while MATHCAD has enough to in such circumstances, a mouse Expanded Memory Specification keep it acceptable. EUREKA and can be a valuable tool-it would (EMS). Users with an EMS board POINT FIVE are rather limited on be valuable with any program's will be able to load all the func­ these points while APL depends on graphics if functions like zoom tions into EMS memory. Other addons from STSC and/or third are implemented. users will be able to choose the party programs for some of these Finally, there is the issue of functions they want by making a things. graphics. Our dream program has simple ASCII list without having We'd like to have some real separate graphing modules for two to cut and paste together their per­ programmability built into this and three dimensions. The two sonalized workspace from a collec­ dream program with a coherent dimensional module allows easy tion of parts of canned workspaces. syntax. There should be all the graphing of one or more functions None of the programs other than standard control structures associ­ with different functions drawn in APL and MATHCAD support EMS ated with PASCAL or C, perhaps different colors or, optionally, for although all have serious memory even a syntax similar to theirs. monochrome setups in different limitations. TKSOLVER, APL, and If you try to run or compile a symbols (e.g. dots vs continuous GAUSS all have large parts of subprogram and there is an er­ line). The program should make their function library in separate ror, you should be thrown into the an intelligent guess of the axes files and manipulating them can editor with the cursor at the of­ to use but it should be easy to be a nuisance. MATLAB proba- fending place. TKSOLVER, MAT- reset them. Up to this point, sev-

1000 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

eral of the programs, most notably port to half density mode is un­ POINT FIVE is published by MATHCAD do a good job. In ad­ fortunate. Pacific Crest Software, 887 NW dition, there should be a crosshair Of course in discussing our Grant Ave., Corvallis, OR 97330, that can pop up and be moved to dream program, one has to bear in (503)754-1067. It lists for $295 give you the coordinates of any mind that the market for such pro­ ($195 for educational institutions). point. This could be used to read grams is much smaller and some­ Special sitewide licenses for aca­ off the intersection point of two what more price sensitive than demic institutions are available for graphs. And it should be possible for a genre like business presenta­ prices ranging from $500 to $5000 to popup an adjustable window tion graphics. Our dream program depending on size. which can be moved and reshaped would require a lot of develop­ PC-MATLAB is published by and then magnify what is in the ment resources and may be a while The MathWorks, 20 N. Main St., window to full screen. None of in coming. Suite 250, Sherborn, MA 01770, these programs offer these fea­ (617)653-1415. It lists for $695 tures, although we have seen stand for the regular version. Versions Price and Publishers alone graphing programs, includ­ are available that run in native ing an internal Caltech program, EUREKA: THE SOLVER is put­ mode on 386 machines with or that do. lished Borland International, 4585 without the Weitek coprocessor. The dream program will al­ Scotts Valley Drive, Scotts Val­ There is a special price of $395 low graphing of surfaces and also ley, CA 95066, ( 408)438-8400 and for universities and $200/each in contour maps of surfaces. TK­ lists for $167. Borland offers spe­ units of 10 or more. SOLVER, GAUSS and MATLAB cial educational discounts on its GAUSS is published by Aptech all allow such three dimensional entire product line. Contact Bor­ Systems, 1914 N. 34th St., Suite graphing but only MATLAB lets land educational sales at ( 408)438- 201, Seattle, WA 98103, (206)547- you easily rotate the perspective. 8400. 1733 and costs $395 (the 386 ver­ Finally, the dream program has TKSOLVER PLUS is published sion will cost $495). While the solid printer support. If you have by Universal Technical Systems, program itself comes with printed ever seen the output that a busi­ 1220 Rock St., Rockford, Ill., documentation, the extra module ness graphics program like Har­ (800)435-7887 or (815)963-2220 only comes with documentation vard Presentation Graphics can and lists for $395. Sitewide li­ on disk or printed documentation produce on even a 120 dots per censes are not available but there for an additional fee. Sitewide li­ inch 9 pin graphics printer, you'll is a 40% discount for academic censes are available and an edu­ know what high class, high resolu­ institutions. There is a reduced cational discount by $1 00 off list tion printer support can produce. cost/function student edition and price is available on single copy Some of these programs, e.g. TK­ a free demo with some functional­ purchases. SOLVER and POINT FIVE limit ity available. There are also addi­ The APL*PLUS SYSTEM FOR printer support to using Shift­ tional "SolverPacks" available for THE PC is published by STSC, PrtSc, an inexcusable way to brush $50-$70 each. 2115 E. Jefferson St., Rockville, off printer output. EUREKA, which MATHCAD is published by MD 20852, (301)984-5000 and limits the output to IBM Graph­ MathSoft, Inc., 1 Kendall Square, lists for $695 for an individ­ ics compatible printers, isn't much Cambridge, MA 02139, (800)628- ual copy. Academic discounts are better in the age of 24 pin dot 4223 or (617)577-1017 and lists available at about 50% off. Call matrix and laser printers. The oth­ for $349. Sitewide licenses are (301)984-5123 for information on ers are somewhat better although available. There is a reduced cost/ sitewide licenses. GAUSS' limitation of laser sup- function student edition available.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1001 News and Announcements

L. S. Pontryagin Golub Receives tion of minority students in mathe­ 1908-1988 Distinguished Service Award matics and science. Called the Charles A. Dana Cen­ Academician Lev Semenovich Pon­ Gene H. Golub, Professor of Com­ ter for Innovation in Mathematics tryagin died May 3, 1988, in his puter Science at Stanford Univer­ and Science Education, the center eightieth year. He was born in Moscow, sity, was honored with the SIAM will aid U.S. colleges and universi­ September 3, 1908. Award for Distinguished Service to ties in developing faculty efforts for Overcoming the handicap of blind­ the Profession from the Society for increasing the numbers of minori­ ness, suffered in an accident at the Industrial and Applied Mathematics ties in science and mathematics. The age of 14, Pontryagin entered Moscow (SIAM). The award was presented center will draw upon Treisman's University in 1925. He was influ­ to Golub at SIAM's Conference on research, which focused on the dif­ enced by P. S. Aleksandrov, and Applied Linear Algebra in Madison, ference in performance of Black and two years later his first results, on the Wisconsin in May. Asian students in calculus courses. Alexander duality theorem, were pro­ Golub received the award "for his Treisman, currently director of the duced. For approximately the next many contributions to applied math­ Professional Development Program quarter century his work was in ematics and to SIAM, especially for at U.C. Berkeley, created a workshop topology and algebra or, as he ex­ his strong and inspiring leadership that dramatically improved the per­ pressed it, "problems where these and for his dedication to his numer­ formance of B:ack students in these two domains of mathematics come ous students, to whom he has given courses. together." Then in 1952 he began mathematical maturity and profes­ Last year, Treisman was hon­ to work exclusively on applied prob­ sional responsibility." The citation ored by the same foundation with a lems, in particular differential equa­ goes on to say, "SIAM particularly $50,000 Dana Award for Pioneering tions and control theory. He pursued recognizes his formation of NAnet, Achievements in Health and Higher this activity for the remainder of his his founding of two SIAM journals Education for his research on mi­ life. (on scientific and statistical comput­ nority students' achievement and his After graduating from Moscow ing and on matrix analysis and ap­ educational innovations (see News University in 1929, Pontryagin be­ plications), and his successful efforts and Announcements, Notices, April came a member of the Mechanics­ to bring the world community of ap­ 1988, page 544). Mathematics Faculty there. Begin­ plied mathematicians together, cul­ ning in 1934, he was affiliated with minating in the First International Lehmann Receives the Steklov Institute, where he was Conference on Industrial and Ap­ R. A. Fisher Lectureship head of the Differential Equations plied Mathematics." and Award Department. In 1939 he was elected Corresponding Member of Academy Grant for Innovation The R. A. Fisher Award of the of Sciences of the USSR; full mem­ in Math and Committee of Presidents of Statis­ bership came in 1959. In 1970, he Science Education tical Societies was presented to Erich was elected Vice-President of the In­ Lehmann of the University of Cali­ ternational Mathematical Union. The Charles A. Dana Foundation fornia, Berkeley on August 24, 1988, has awarded $737,000 to Philip Uri at the Joint Statistical Meetings in Treisman of the University of Cali­ New Orleans. Lehmann was honored fornia, Berkeley to establish a center for his distinguished contributions to help colleges improve the educa- to mathematical statistics, especially

1002 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News and Announcements

in testing statistical hypotheses, non­ versity of Oxford for his work in student at Rhode Island College. This parametric ~et~ods, and the theory number theory. full roster of activities keeps Marcia of point esttmatiOn. tired, but she is looking forward to Lehmann received a Ph.D. in AMS Trustees Honor recharging her batteries during a 10- mathematics in 1946 from the Uni­ Twenty Year Employees day vacation in Hawaii early this versity of California, Berkeley, where fall. he has taught since the early 1940s. At its meeting in November 1987, LEE DAVOL was hired as a full­ He has received three Guggenheim the Board of Trustees of the Society time Varitypist in 1962-a time when Fellowships, and has been elected to adopted the following resolution: the AMS office was located at Butler the American Academy of Arts and "This year the Board of Trustees Hospital. She left in 1967 to take Sciences and the National Academy takes special note of the fact that care of her two children, Rod and of Sciences. In addition, he is an three more employees of the So­ Lori-Beth, returning in June of 1971 Honorary Fellow of the Royal Statis­ ciety have completed twenty years to work part-time nights in what was tical Society. With more than 90 pub­ of service. The Board expresses its then the South Main Street office. lications to his credit, Lehmann is profound gratitude to MARCIA C. Soon after, Lee switched to part-time well known to statistical researchers ALMEIDA, LEONORA T. DAVOL and days, and she has been working that and students for his concise and clear THERESA S. DRENNAN, who bring to schedule ever since. books. During his career, he super­ twenty-five the number of employees Lee has worked a total of 21 1/2 vised more than fifty Ph.D. students. of the Society, past and present, who years of service for the Composi­ At the Meetings, Lehmann also have devoted more than twenty years tion Department (or its equivalent). delivered the R. A. Fisher Lecture, as members of the AMS staff. The She has progressed, along with the entitled "Model Specification: Fisher's Trustees offer their special thanks AMS itself, from the Varitype ma­ Views and Some Later Strategies." and their best wishes to these three chine to the Compuwriter and IBM The tradition of the Fisher Lecture, long-term employees and wish them "Selectric" Composer, to TEX and begun in 1963, honors both the con­ well in the future." the computer terminal. Her present tributions of Sir Ronald A. Fisher MARCIA CHRISTINE ALMEIDA was position is as a Technical Typist do­ and the work of a current-day statis­ hired in 1967 as a full-time em­ ing TEX work along with a myriad tician in the advancement of statisti­ ployee. Her duties at that time were of other typing projects. Lee received cal theory and applications. miscellaneous clerical work: she had her 20-year award in December of -AMSTAT News come in to help out in the Editorial 1986. Department. TERRY DRENNAN joined the AMS LMS 1988 Prizes When the Committee on Employ­ in 1967 when the AMS was located ment and Educational Policy came on South Main Street. She became a The P6lya Prize is awarded to C. T. into being, Marcia worked with Dr. member of the Editorial Department C. Wall of the University of Liver­ Lincoln Durst in the preparation as an Editorial Assistant, and she pool for his significant contributions of the data gathering for the An­ continues her work in the Editorial to algebraic, geometric, and differen­ nual Survey. Since that time her du­ Department where her contribution tial topology. ties have expanded to include work­ is greatly valued. The Senior Berwick Prize is award­ ing on other miscellaneous publica­ Terry is well known for her artis­ ed to D. B. A. Epstein of the Uni­ tions including the Professional Di­ tic and creative talents. Her inter­ versity of Warwick for his papers en­ rectory, the Combined Membership ests include sewing, painting, flower titled "Notes on notes of Thurston" List, and Assistantships and Fellow­ arrangements, gardening, roller skat­ (with P. Green and R. D. Canary) ships. In addition, she manages the ing, all sorts of crafts, going to yard and "Convex hulls, a theorem of prepress production of the Contem­ sales and arts and crafts shows. Sullivan, and measured pleated sur­ porary Math series, and she is a She is married, and she and her faces" (with A. Marden). valued member of the Editorial De­ husband, Dave, reside in East Provi­ Junior Whitehead Prizes are award­ partment. dence. They have a daughter, Michelle, ed to S. M. Rees of the University Marcia has two children: Chris­ who lives in Rumford with her hus­ of Liverpool for her work on er­ tine, 15, and Monica, 8. She lives band, Robert; and Terry's grandson, godic theory and dynamical systems, with her husband, Ray, and their John Peter. to P. J. Webb of the University of children in Riverside. She is an active At its meeting in May 1988, the Manchester for his work on the rep­ member of the community where she Board of Trustees of the Society resentation theory and cohomology has participated as a Brownie Leader adopted the following resolution: of finite groups, and to A. Wiles of and as manager of the 30-Week Club "This year the Board of Trustees Princeton University and the Uni- for her church. She is also a part-time takes special note of the fact that two

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1003 News and Announcements more employees of the Society have of thousands of offprints each month (IMO), held July 15 and 16 in Can­ completed twenty years of service. to more than 1100 subscribers. Bar­ berra, Australia. Five of the team The Boa:t:d expresses its profound bara served as staff support for the members received silver medals. gratitude to SANDRA D'ALLESANDRO group of mathematicians that devel­ The Americans had a team score and BARBARA VEZNAIAN, who bring oped the AMS (MOS) Subject Clas­ of 153 out of 252. Ahead of them to twenty-seven the number of em­ sification Scheme ( 1970) for mathe­ were teams from the Soviet Union ployees of the Society, past and matics. (217), China and Romania (tied for present, who have devoted more than The Offprint Service, and its suc­ second place with scores of 201), twenty years as members of the AMS cessor, the Mathematical Title Ser­ West Germany ( 174), and Vietnam staff. The Trustees offer their special vice, were part of the Society's In­ (166). In all, 49 nations and 268 thanks and their best wishes to these formation Systems Development De­ students participated in the IMO. In three long-term employees and wish partment. When cost forced the end each of the two, 4 1/2 hour ses­ them well in the 'future." of MOS and MTS, the department's sions, the students worked on three When SANDY D'ALLESANDRO was director, Sam Whidden, invited Bar­ challenging mathematical problems. hired in March 1968 as a receptionist bara to join its programming staff. "The U.S. team turned in a strong for the American Mathematical So­ Barbara trained as a programmer, performance, earning five silver med­ ciety, she was operating a small PPX participating in the design and de­ als and a bronze on what was an un­ with 5 in and out trunks and approxi­ velopment of many of the data pro­ usually difficult set of problems," said mately 40 extensions. Her duties also cessing applications that remain in U.S. team coach Gerald Heuer in an included some typing and light cleri­ regular production today. In 1974, interview immediately after learn­ cal duties. She is presently operating Barbara became Manager of Pro­ ing the results. "While we are some­ a board with 21 trunks and 160 ex­ gramming and Systems Analysis. what disappointed at placing sixth, tensions, as well as continuing with In the late '70s, Barbara partici­ we have to admire the superb scores her clerical duties. Sandy is well liked pated in, and helped supervise, con­ of the top five countries and we look by staff, AMS members and outside version of all of the Society's applica­ forward to a stronger showing by the vendors, and it is hoped that she will tions from its original Univac com­ U.S. team in West Germany next stay with the AMS for many years to puter to its DECSYSTEM-20. Bar­ year." come. bara, along with her staff, played an The IMO judges awarded indi­ After graduating from the Uni­ important role in the design of the vidual first, second, and third prizes versity of Rhode Island as an English Mathematical Reviews Data Base, to deserving team members. Five major, and doing graduate work in implemented on the DECSYSTEM- U.S. team members received second mathematics at Boston University, 20 computers in 1983. prizes: Jordan Ellenberg of Potomac, BARBARA VEZNAIAN joined the AMS In 1984, the Information Sys­ Maryland; Tal Kubo of Brookline, in 1968 as an Editorial Assistant, tems Development Department be­ Massachusetts; Samuel Kutin of Old hired by Ellen Swanson, then the came the Computer Services Divi­ Westbury, New York; Eric Wepsic Society's Director of Editorial Ser­ sion, and Barbara became Head of of Boston, Massachusetts; and John vices. Barbara had served only a few its Programming and Analysis De­ Woo of Pepper Pike, Ohio. Hubert weeks in the Editorial Department partment. She manages a staff of Bray of Houston, Texas received a when she was tapped by Dr. Gor­ twelve systems and application pro­ bronze medal. don Walker, then Executive Direc­ grammers and analysts serving all The U.S. team was chosen on the tor, to head the Mathematical Off­ other departments of the Society's basis of performance in the U.S.A. print Service, a newly inaugurated Providence office. She and her de­ Mathematical Olympiad, held April information service for mathemati­ partment are responsible for design, 26, and on the basis of an evaluation cians. The Offprint Service (MOS) development, and implementation of of their work at a rigorous 4-week arranged to obtain offprints of arti­ all new Providence office computer training session this summer. cles to appear in mathematical jour­ applications, and are currently recre­ The Mathematical Olympiad ac­ nals, redistributing them to mathe­ ating all such applications on the tivities are sponsored by .seven na­ matician subscribers according to the Society's new VAX computers. tional associations in the mathemati­ subscribers' profiles of mathematical cal sciences, with arrangements made interest. Barbara managed arrange­ U.S. Team Places Sixth by the Mathematical Association of ments with participating journals, in Math Olympiad America. Financial support was pro­ subscribers, and subject classifiers, vided by International Business Ma­ and headed a staff that accomplished A team of six American high school chines, the Army Research Office, the the computer processing of offprints students placed sixth in the 29th In­ Office of Naval Research, Hewlett- against profiles, and the distribution ternational Mathematical Olympiad

1004 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News and Announcements packard, and the Matilda R. Wilson in the Black and Hispanic students' that parents in general believed math­ Fund. achievement. ematics to be important, and "par­ The two-year study, funded by ents of Black students were the most News from the the National Science Foundation, vocal in expressing this belief." The Institute for Mathematics involved some 28,000 students in study recommended greater efforts to and Its Applications the public schools of Montgomery promote understanding between the University of Minnesota County. For more than a decade school and the home. Montgomery County has sought to The study also gathered data on The Signal Processing six-week pro­ boost the performance of female, gender differences in mathematics gram this summer at IMA included Black, and Hispanic students in aca­ performance. While girls and boys a wide range of topics: image analy­ demic areas such as mathematics, !he performed equally well on standard­ sis and vision, radar, sonar, medical report says. Indeed, the study found ized tests in early grades, the study imaging, speech analysis and recog­ that, ten years ago, standardized tests found substantial differences in the nition, one-dimensional processing showed Black and Hispanic students junior and senior years of high school (operator theory methods, VLSI im­ in Montgomery County were per­ in their performance on the mathe­ plementation), and X-Ray crystallog­ forming below national averages in matics portion of the Scholastic Ap­ raphy. The program began with two their racial/ ethnic groups, but to­ titude Test. "This difference in SAT weeks of general survey talks fol­ day are performing better than the mathematics performance was ob­ lowed by periods of concentration. national average for students of all served regardless of the amount and One of the novel features was a joint races. complexity of mathematics and sci­ IMA-3M Symposium which was held Nonetheless, the perplexing ques­ ence courses taken by the students, on July 19 at the 3M Center in tion remains of why these students and despite the fact that female stu­ St. Paul, Minnesota. The symposium perform more poorly than do White dents received higher grades than included several talks by IMA vis­ and Asian students in Montgomery male students in all mathematics itors, presentation of problems by County. Starting off with lower per­ classes," the report stated. 3M scientists, and guided tours of formance in early grades is an impor­ The study found that female stu­ 3M laboratories. tant factor. "The evidence suggests dents tended to have less confidence The 1988-1989 program on Non­ that once a student falls below the in mathematics and tended to per­ linear Waves has just begun. An standard level of performance in the ceive the subject as less useful than opening workshop on Solitons in curriculum for hisjher grade level, did male students. There were also Physics and Mathematics, which is he/she is not likely to ever again differences in career aspirations, with partly tutorial, will include as speak­ catch up," the report says. males more likely than females to ers Alan Newell, Mark Ablowitz, As a result, the more advanced see themselves in occupations utiliz­ D. Kaup, Y. Yodama, H. Segur, D. mathematics classes are dominated ing mathematics or in management Sattinger, and M. Kruskal. Another by White and Asian students. Even positions. workshop on Solitons in Nonlinear when Black students take the more The report says that such atti­ Optics and Plasma Physics will take advanced classes, they do not per­ tudes are influenced by parents and, place in November. (For further de­ form on standardized tests at the to a lesser extent, by the school en­ tails see the News and Announce­ same level as students from· the other vironment. With the exceptions of ments section of the April 1988 issue racial/ethnic groups. The report says the top female mathematics students, of Notices.) that in such classes, Black students female students receive less encour­ tend to feel isolated and perceive agement in mathematics from the Survey Examines Students' an especially acute pressure to do school, home, and society, the report Math Performance well. "High-achieving Black students says. In questions put to parents, reported that they had to prove them­ the survey found that mothers do A recent study of students' mathe­ selves to the teacher each time the not see themselves as competent in matics achievement in Montgomery entered a new mathematics class," mathematics and that parents gen­ County, Maryland came to an all-to­ the report said. erally "still view mathematics and familiar conclusion: the performance School counselors and principals science careers as being primarily of Black and Hispanic students con­ interviewed for the study generally for men." A significant portion of sistently trails that of White and felt that the poor performance of school counselors and principals at­ Asian students. This gap in perfor­ Black and Hispanic students stemmed tributed differences in mathematics mance shows up in the early primary from problems in the home and cited performance of males and females grades and has persisted despite gains economic factors and fragmentation to such factors as a lack of inter­ of families. However, the study noted est on the part of females and the

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1005 News and Announcements females' perception that they do not tunity to derive benefits from the Due to an editing oversight, the need mathematics for their careers. traditional excellence of Hungarian article, "Research Mathematicians in The report offers many recom­ mathematics education. Mathematics Education, Part 1," mendations for addressing the prob­ • All courses are taught in English. which appeared in the July I August lems it uncovered: remediation and • Classes are held in small groups. issue, did not carry a byline. The enrichment programs, mechanisms • Emphasis is on creative problem article was written by Notices Staff to increase parental involvement and solving. Writer Allyn Jackson. support, and ways of training and • Credits are transferable to Amer­ The recipients of the Rollo David­ retraining teachers. The report also ican colleges and universities. son Prizes were announced in the suggests that "school systems lauch • Living costs are modest. May/June issue. The university af­ public relations campaigns to change Semesters start in the first week of filiation of one of the awardees, the image of mathematics" to make September and February each year, P. H. Baxendale, was incorrectly it seem more exciting. and are preceded by an optional two­ stated. Since 1973 he has been a In Montgomery County, a rela­ week intensive language course. The lecturer in the Department of Math­ tively affluent area near Washington, deadline for applications for the fall ematics, Aberdeen University. He is DC, students "receive educational semesters is April 30, and October currently on leave of absence in services that are equal to or bet­ 15 for the spring semesters. Early Florida. ter than those provided elsewhere in applications are encouraged and will A few errors have come to our the nation." The report says that the be processed promptly. Tuition is attention in the list of fifty-year persistence of such problems suggests $2150 (U.S.) per semester. AMS members, which appeared in that they are "deep-seated and diffi­ For further information and ap­ the July j August issue. The name of cult to address" and may in fact be plication forms, contact the Ameri­ member Haim Reingold was incor­ worse in other school districts across can Program Director: Professor Paul rectly listed. F. A. Ficken has been the nation. D. Humke, Department of Mathe­ deceased for several years. Thomas matics, Saint Olaf College, North­ W. Mullikin should not have been Budapest Semesters field, MN 55057, 507-663-3113. included in the list. In addition, in Mathematics there are persons who were elected Errata to membership in 1938 effective in Budapest Semesters in Mathematics 1939 who are now in their fiftieth are now administered at Saint Olaf The NSF News & Reports section of year of membership but who were College. Recommended by educators the May/June issue of Notices con­ not included in the list of fifty year and highly praised by former partici­ tained a news item with the headline, members. This omission was due to pants, this study abroad program will "National R and D Funds at Eleven the manner in which the list was soon complete its fourth successful Year Low." Because of an inadver­ compiled. At least one such individ­ year. During these years, undergrad­ tent deletion, this headline was incor­ ual has been identified, namely D. D. uates from over forty colleges and rect. The correct headline is, "Real Miller. universities from all over the U. S. Increase in National Rand D Funds and Canada have had the oppor- at Eleven Year Low."

1006 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY National Science Foundation News & Reports

Reflections of Departing year to Virginia Polytechnic Insti­ Indeed, some regard program di­ OMS Rotators tute, believes that increasing aware­ rectors as paper-shuffling bureaucrats, ness within the mathematical sci­ but in fact the personal commit­ Rotators are crucial to the quality of ences community about the breadth ment and scientific knowledge the the research programs at the NSF. and the political climate of the dis­ individual brings to the position is Because they come directly from a cipline would be beneficial to both essential to the welfare of the pro­ research environment, rotators bring the community and the NSF. "You gram. William Lakin, who has now in current technical knowledge as cannot live an isolated life when you returned to Old Dominion Univer­ well as fresh ideas and new per­ come to NSF," she says. "You get sity from a one-year stint as Pro­ spectives. In the Division of Mathe­ forced into knowing a lot of things, gram Director in Applied Mathemat­ matical Sciences (DMS), where only which is good, in the end. I enjoyed ics, notes that a program director has about one-quarter of the staff mem­ having a larger perspective, getting several ways of attracting additional bers are permanent employees of the involved in all aspects that concerned funding. "Proposed work is often of NSF, attracting highly qualified rota­ the discipline." interest to more than one program tors is essential. Despite the difficulty Many are unaware of the program at NSF," he says. "A program direc­ of finding such people, the DMS director's activities beyond sending tor can identify these other programs, has consistently maintained an excel­ out proposals for review and mak­ form cooperative links, and often ob­ lent staff. This summer, six rotators ing awards. For example, "a tremen­ tain split-funding from outside DMS are returning to the academic world dous amount of the activity goes into for quality proposals. Outreach to while six new rotators are joining the putting together the proposals of the other disciplines within the Founda­ OMS (see the DMS Staff List fol­ NSF budgets and justifying them, tion is sometimes difficult, but it's an lowing this article). Interviews with and that's a scientific activity," says essential part of the job." Program di­ the departing rotators show that they Flournoy of the annual DMS budget rectors can also identify elements of feel they have learned a great deal at formulation process. "You have to be proposals that fit funding priorities the NSF. able to articulate what is important set within the DMS. For example, "The experience is definitely a in your subfield and why, and also if the proposed research is a group broadening one and a very rewarding why it should be funded." activity, extra funds might be avail­ one," says Nancy Flournoy, who has "I came to understand how in­ able for visitors, graduate students, taken a position at American Uni­ tense the competition for funds is, or postdocs. versity after serving two years in the. how difficult it is to make the case One of the important roles of Statistics and Probability program. for increased funding of mathemat­ program directors is to act as a link "The experience is one that could ics, and how progress in this direction to the mathematical community on not be duplicated in terms of coming can be made," says Andre Manitius, science policy matters. "I think it's to an understanding of a broad vari­ who served as Program Director in good for the [mathematical] commu­ ety of current research activity that Applied Mathematics for two years nity to take an active part in how is going on, not only in the field in and has now left his one-year ap­ the science policy in mathematics which the rotator works, but across pointment as Deputy Division Di­ should go," says William Adams, a all of mathematics and extending to rector to accept a position at George Program Director for Algebra and the other sciences as well." Mason University. "I also saw how Number Theory who has returned to Yashaswini Mittal, who also many things depend on the personal the University of Maryland. "Lack served in Probability and Statistics dedication of many people working of awareness is a big problem," he for 2 years and will return this for NSF." says, but the program directors'

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1007 NSF News & Reports knowledge can help. "We can as­ to be done with a proposal and all the tor for the STC program, he oversaw sist the mathematics community by other things that a program director the entire review of the center pro­ explaining [science policy issues] so is supposed to do." posals having mathematical content, that they can take appropriate ac­ In addition to science policy ac­ a particularly difficult process, since tion." tivities, program directors also be­ the program is in its first year and In addition, Manitius notes that come involved in special activities since many of the proposals were for one of the roles of the DMS pro­ within the NSF. For example, Adams interdisciplinary research efforts. gram officers is to "help create a saw to it that the Research Experi­ Finding people to serve as pro­ better image for mathematics among ences for Undergraduates program gram directors is always difficult, people in other disciplines and other got off to a healthy start in its first and those who are rotating out point parts of NSF. This requires willing­ two years. Flournoy has been effec­ to a number of changes that would ness to talk to other people, beyond tive at establishing ties between the make the job more attractive, such your own discipline." Some of the DMS and other disciplinary divisions as decreasing the workload, provid­ other positive influences program di­ at the NSF, and Mittal worked hard ing time for research, and improving rectors can have is identifying lesser to consolidate the Probability and the financial compensation for senior known but promising researchers and Statistics program and stimulated in­ researchers. Often the rotators' de­ by identifying new directions where terest in the statistical community partments at their home institutions additional funds would make a large in the NSF's Science and Technol­ do not like the idea oflosing a faculty impact. ogy Centers (STC) program. Both member for one or two years, and A certain amount of bureaucracy effectively promoted the use of com­ sometimes the rotators are even de­ is inevitable at a federal agency, putation in their program. nied the standard salary raises. Part and rotators coming from a less Paul Goodey, back at the Uni­ of the reason people come to the structured research environment can versity of Oklahoma after two years DMS is that they believe they can make the change difficult. "Because as Program Director for Geomet­ contribute something that is impor­ of the presence of many rotators, rical Analysis, was instrumental in tant and that the community will rec­ there is some element of an aca­ formulating preliminary ideas for an ognize as valuable. Addressing such demic atmosphere in the DMS," says initative in geometry and focused the issues is one way the community can Manitius. "At the same time, there attention of the DMS Advisory Com­ show its recognition of and appreci­ are many constraints imposed by the mittee on this subject. Lakin ran the ation for rotators. government rules which frustrate the panel for the the Research Oppor­ professional personnel." "Still," Mit­ tunities for Women program, helped Mathematical Sciences Staff tal says, "NSF as a whole is a lot less to establish stronger links with the The Program Directors for 1988- bureaucratic than some other fed­ other NSF divisions, and was the 1989 are: eral agencies, but even within NSF DMS Coordinator for proposed Cos­ I think DMS is about as close to mology Initiative involving the DMS Classical Analysis an academic situation as a federal and the physics and astronomy divi­ 202-357-3455 agency can come." sions. John V. Ryff The DMS also seems to do a par­ In his three years at the DMS, Modem Analysis ticularly good job. "The DMS has, I Manitius has been actively involved 202-357-3697 believe, one of the best staffs among in several initiatives, most notably William L. Paschke all the NSF divisions," says Manitius. a program to promote new interac­ Geometric Analysis "The quality standards of the review tions between mathematics and the 202-357-3451 are quite high, and the peer review is biosciences and an initiative to fund Russell B. Walker involving U.S. reseach­ taken very seriously." DMS program joint projects Topology and Foundations INRIA directors also have one of the largest ers and those at France's 202-357-3457 Research in workloads in the NSF. "The large (National Institute for Ralph M. Krause workload and close competition cre­ Computer Science and Automation). ates some problems, mostly because He assisted the Office of Science Algebra and Number Theory it leads to delays in processing," he and Technology Policy in their ef­ 202-357-3695 says. "To an outside observer, the forts to develop a cross-agency pro­ Ann K. Boyle time it takes the DMS to decide gram which will combine the research Jonathan D. Lubin about a specific proposal may seem of mathematicians, statisticians, data Applied Mathematics quite long. Seen from the inside, this collectors, and epidemiologists in the 202-357-3686 time is not so long when one takes mathematical modeling of the AIDS Peter W. Bates into account all the things that need epidemic. In addition, as a Coordina- BartS. Ng

1008 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF News & Reports

Computational Mathematics Manitius was on leave from his po­ $2.52 million under its 1988 Visiting 202-357-3691 sition as Professor of Mathematical Professorships for Women program. Raymond C. Y. Chin Sciences at Rensselaer Polytechnic As an important part of the NSF's Statistics and Probability Institute. initiative to enhance the participa­ the U. S. science 202-357-3693 All of the DMS staff can be tion of women in Mary Ellen Bock re~ched via electronic mail. To form and engineering enterprise, the VPW Peter W. Arzberger an individual's address, take the first program provides support for some initial and last name, and append of the nation's most productive and Special Projects @note.nsf.gov for INTERNET (a talented scientists and engineers. 202-357-3453 collection of networks of which The program enables female sci­ Deborah F. Lockhart ARPANET is a subnetwork) or @nsf entists and engineers from indus­ Elbert A. Walker for BITNET. For example, to contact try, government and academic insti­ The administrative staff includes: John Ryff through INTERNET, use tutions to serve as visiting professors. Division Director the address [email protected]. The visiting professors will perform Judith S. Sunley research, and will conduct lecturing, 202-357-9669 Advisory Committee Meeting counseling, and other activities to in­ Deputy Division Director crease the visibility of female scien­ Bernard R. McDonald The NSF Advisory Committee for tists and engineers and to encourage 202-357-9669 the Mathematical Sciences will meet other women to pursue careers in Administrative Officer at NSF headquarters in Washington, science and engineering. Tyczer Henson DC on October 17-19, 1988. The The 25 award winners were se­ 202-357-3683 Committee's advice plays an impor­ lected on the basis of a rigorous two­ tant role in setting priorities and stage review process that focused on permanent staff consists of The developing long-range plans in the the excellence of the research and Krause, McDonald, Ryff, and Sun­ Division of Mathematical Sciences on the proposed plan for lecturing ley. The incoming rotators are Rus- (DMS). The last meeting included and serving as counselor and mentor - sell Walker, Montana State Univer­ informative presentations about the to women entering research careers. sity; Lubin, Brown University; Ng, Congressional budget approval pro­ The NSF awards range in size from Indiana University-Purdue Univer­ cess (see "Congress and the NSF $48,283 to $235,441, and in dura­ sity in Indianapolis; Bock, Purdue Budget," Notices, May/June 1988, tion from nine to 24 months. In University; Arzberger, University of page 707) as well as discussions with some cases host institutions are also Wisconsin at Madison; and Lockhart, DMS staff. At the October meeting, contributing to the awards. Michigan Technological University. the Committee will conduct its tri­ Two of the Visiting Professor­ Bernard R. McDonald became annual oversight reviews of the stan­ ships were awarded to mathematical Deputy Division Director in August. dard research projects program in the scientists. They are NANCY G. LEvE­ A permanent employee of the Foun­ DMS. To allow time for this addi­ soN, University of California, Irvine, dation, McDonald came to the DMS tional task, the meeting will last three California, to visit the Massachusetts in 1983 from his position as Profes­ days instead of the usual two. The Institute of Technology; JUDITH D. sor of Mathematics at the University reviews will be held in closed-door SALLY, Northwestern University, to of Oklahoma. He was Program Offi­ session; however, the mathematical visit Purdue University. cer in Algebra and Number Theory community is welcome to attend for three years before moving to the the rest of the meeting. Those who Visiting Professorships Special Projects program in 1986. wish to attend or to suggest topics for Women McDonald will succeed Andre for discussion should contact Trudy Manitius, who has taken a position Sensibaugh at the DMS at 202-357- The Visiting Professorships for as Professor in the School of Infor­ 9669; electronic mail address tsen­ Women program is designed to pro­ mation Technology and Engineering [email protected] (INTERNET) vide opportunities for women to ad­ at George Mason University. Man­ or tsensibaugh@nsf (BITNET). vance their careers in the disciplines itius came to the DMS in 1985 as of science and engineering and to Program Director of Applied Mathe­ NSF Awards for provide greater visibility of women matics and became Deputy Division Visiting Professorships scientists and engineers employed in the past year, Director in 1987. For industry, government, and academic he has also served as Coordinator for The National Science Foundation to­ institutions. the NSF's Science and Technology day announced 25 awards totalling The program allows a woman sci­ Centers program. While at the NSF, entist or engineer to pursue advanced

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1009 NSF News & Reports research at an academic institution. for domestic travel, and $2000 for back on its progress to other partici­ In addition to her research responsi­ foreign travel, will be applied. pants. bilities, the visiting professor under­ Applicants must be women hold­ This year, approximately 700 un­ takes lecturing, counseling, and other ing a doctorate in a field of research dergraduate faculty will take part activities to increase the visibility of supported by the Division of Mathe­ in the conferences and workshops. women scientists in the academic en­ matical Science of the NSF, or have The projects address both theoretical vironment of the host institution and equivalent experience. A woman may and experimental topics, and sev­ to provide encouragement for other not be awarded more than one grant eral of them are interdisciplinary. All women to pursue careers in science in any two-year period, and should are residential and utilize the facili­ and engineering. not have other available sources of ties of the host academic institutions The research must be in a field funding, except possibly partial insti­ to foster informal interaction among normally supported by the NSF. The tutional support. the participants in a collegial atmo­ instruction and other activities may There will be four award periods sphere. The awards cover the cost be at the undergraduate or graduate per year, with applications due as of instruction, facilities, and room levels, be directed to the community follows: November 1, 1988; February and board to all participants, and, in at large, or involve some combination 1, 1989; May 1, 1989; August 1, 1989. many cases, offer a modest stipend. of such activities. The usual award Applicants should send a discus­ The four awards in mathematics will be for 12 months for a full- or sion of how the proposed travel grant are described below. part-time professorship, but awards would benefit their research program, The Mathematics Association of for one academic semester or for 24 and a curriculum vita, to the Asso­ Two-Year Colleges of New Jersey, months will be considered. ciation for Women in Mathematics, through Union County College, or­ The deadline for applications is P. 0. Box 178, Wellesley College, ganized a workshop on applications November 15, 1988. For further in­ Wellesley, MA 02181. of discrete mathematics. In two ex­ formation about guidelines and eli­ tended weekends (one was held last gibility, contact the Program Direc­ Faculty Enhancement spring and another will be held this tor for Visiting Professorships for Program Awards fall) thirty mathematics faculty from Women, National Science Founda­ two-year colleges participated in work­ tion, 1800 G Street, N.W., Washing­ The NSF made 27 awards in the Un­ shops and lectures conducted by Fred ton DC 20550; telephone 202-357- dergraduate Faculty Enhancement S. Roberts of Rutgers University. 7734. Program, which is designed to help During the summer, they worked on revitalize the teaching of undergrad­ individual or small-group projects on NSF-AWM Travel uate science, mathematics, and en­ which they reported at the second Grants for Women gineering. Begun this year with a session. The principles of discrete in Mathematics budget of $3 million, the program mathematics and its applications in sponsors national or regional work­ such fields as genetics, engineering, The objective of the National Science shops and seminars in which faculty computer science, and social plan­ Foundation-Association for Women learn about new techniques and new ning were covered. Jean Lane of in Mathematics Travel Grants is to developments in their fields. Union County College is the prin­ enable women to attend research con­ The program stipulates that there cipal investigator on this $47,386 ferences in their field, thereby pro­ be sustained interaction among the grant. viding a valuable opportunity to ad­ participants and the project leaders, The University of Montana con­ vance women's research activities, as both during the course of a project vened an interdisciplinary group of well as to increase the awareness and on a continuing basis there­ 25 college teachers of biology and that women are actively involved after. Establishment of computer net­ applied mathematics in two sessions in research. If more women attend works, site visits by project leaders to to study the use of mathematical and meetings, then the size of the pool participants' home institutions, and computer models in biological re­ from which speakers at subsequent reunions at national meetings are source conservation. The first, four­ meetings are drawn is increased, and among the means of accomplishing week summer session consisted of thus the problem of the absence of this objective. In some instances, par­ lectures, seminars, workshops, and women speakers at many research ticipants may begin work on a project field trips into wilderness areas in conferences is addressed. during the session-often dealing Montana, culminating in a group The travel grants will support with incorporation of new material modeling project. During the com­ travel and subsistence to a meeting or into classroom use-and pursue it ing academic year, the faculty par­ conference in the applicant's field of at their home institutions, reporting ticipants will work on projects in specialization. A maximum of $1000 modeling at their home institutions

1010 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF News & Reports

and report on their wor~ a~d the~r The Undergraduate Faculty En­ The deadline for nominations for integration of the rnatenal m theu hancement program was established the 1989 award is December 31, 1988. undergraduate teaching at a followup last year as part of a major NSF The award committee requires that session the next summer. Robert W. effort to upgrade undergraduate sci­ the nominations be made on a spe­ McKelvey of the University of Mon­ ence, engineering, and mathematics. cial form, which may be requested tana is the principal investigator on The projects described above exem­ from the Alan T. Waterman Award this $97,917 grant. plify the kinds of activities the NSF Committee, National Science Foun­ Fort Lewis College in Durango, is funding in this area. dation, Washington, DC 20550. For Colorado held a short course de­ A new program announcement is additional information, contact the signed to provide an introduction to now available (Forms and Publica­ Executive Secretary, Lois J. Harnaty, mathematical modeling and to the tions, NSF, Washington DC 20550; 202-357-7512. topics underlying the construction of telephone 202-357-7861). During the mathematical models. The week-long corning academic year, there are two NSF Announces Awards course, held in July, also addressed deadlines for proposal: -Qctober 14, for Experimental Computer the issues that must be resolved in the 1988 and March 3, 1988. For more Research Facilities design of an undergraduate course in information on the program, call at Four Universities mathematical modeling. Thirty-five Duncan McBride, Program Director, college faculty participated in twelve at 202-357-7051. The National Science Foundation hours of class discussion on these (NSF) has made awards to estab­ topics and an additional six hours Call for Nominations lish four major research facilities for of instruction-discussion on specific for Waterman Award computer research at universities in models and the use of computers and California, Illinois, North Carolina, software to supplement the modeling In 1975 Congress established the and Rhode Island. The awards were process. Participants also had the op­ Alan T. Waterman Award to mark announced today after a merit review portunity to work on specific models the 25th anniversary of the NSF and of 23 proposals, and are the first-year and had access to computer facili­ to honor the first Director of the installments of five-year grants that ties. Gary W. Grefsrud of Fort Lewis NSF, Alan T. Waterman. This an­ are anticipated to total more than College is the principal investigator nual award recognizes an outstand­ $11,700,000 when the projects are on this $24,681 grant. ing young scientist in the forefront completed. Consortium for Mathematics and of science. In addition to a medal Brown University in Providence, Its Applications conducted two, two­ and other recognition, the awardee Rhode Island, has received a first­ week workshops this past summer, receives grants of up to $500,000 for year award of $745,000 through one in discrete mathematics at North­ a period of up to three years for NSF's Computer and Information eastern University and one in geom­ scientific research or advanced study Science and Engineering (CISE) In­ etry at Virginia Commonwealth Uni­ in the biological, mathematical, med­ stitutional Infrastructure program. versity. Involving a total of 25 college ical, engineering, physical, social, or The award will be used to begin teachers from across the country, the other sciences at the institution of developing a coordinated set of soft­ workshops focused on day-to-day in­ the recipient's choice. ware tools to simplify complex design teractions between the participants, The award committee annually problems, such as the development support faculty from the universities solicits nominations from a wide va­ of future computing systems. The involved, and experts in the par­ riety of sources. Candidates must be software tools will create an envi­ ticular fields. Each workshop culmi­ U.S. citizens or permanent residents ronment that should significantly in­ nated with the production of three and must be 35 years or younger, or crease the productivity of designers. or four sets of curriculum materials not more than 5 years beyond the re­ A primary goal of the project is to that the college teachers can use in ceipt of the Ph.D. degree by Decem­ provide tools to support a variety their own classrooms and share with ber 31 of the year in which the nom­ of problem-solving methods in a sin­ others throughout the country. These ination is made. Candidates should gle environment. The development materials were edited in the summer, have completed sufficient research to of this environment will be based on will be tested this fall, and will be have demonstrated outstanding ca­ a prototype computer-programming made available to all interested teach­ pability and exceptional promise for environment, called Garden, devel­ ers in the spring of 1989. Margaret B. significant future achievement. In oped previously at Brown. Initial ap­ Cozzens of Northeastern University addition, candidates should exhibit plications will be in the areas of is the principal investigator on this quality, innovation, and potential for software development and the de­ $141,387 grant. discovery in their research. sign of Very Large Scale Integrated (VLSI) circuits. Total NSF funding

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1011 NSF News & Reports for the project is anticipated to be bility for building prototypes of com­ and most are held during the summer $3,481,000. plete computing systems, including months. Scientists at the University of Cal­ computer chips, system architecture, REU Supplements augment on­ ifornia at Berkeley will use their operating systems, compilers, appli­ going NSF research grants in order first-year award of $934,327 to do cations software and user interfaces. to provide research experiences for a research in the development of com­ The development of this capability small number of undergraduate stu­ puter systems for solving problems will be done in such a way that it dents. Requests are made directly to that involve massive amounts of in­ can be transplanted to other loca­ the NSF program officer handling the formation. Researchers will develop tions, thereby creating the possibility original research award. a computing facility with a variety of of a decentralized national capability For both Sites and Supplements, storage media, including high-speed for experimental computer systems student stipends for summer projects memory and more economical, but research. Development of a portable are expected to be at least $2000, slower, magnetic disk and optical system that can produce prototypes with academic year stipends com­ disk memory. The facility will have of complete computer systems may parable on a pro rata basis. Total large amounts of each type of mem­ have an impact analogous to that costs are expected to average around ory, which will be used to investigate which the successful MOSIS sys­ $4000 per student. The amount of effective ways to structure massive tem has had on VLSI design. Total indirect costs allowed for REU is memory systems. A related emphasis NSF funding for the North Carolina limited to 25% of student stipends. will be on the development of soft­ project is projected to be $2,079,997. Other eligible costs include salaries ware to make use of the memory. Besides NSF, the four universities of involved faculty, relevant student The facility will be used for research have also made commitments to sup­ housing costs, indirect costs, and a in artificial intelligence, text process­ port researchers and their projects, modest allowance for supplies. In ad­ ing, programming systems, graphics, and these institutions will assume an dition, the student participants must computer vision and scientific pro­ increasing share of the project costs be citizens or permanent residents of cessing. NSF anticipates funding the each year throughout the grant pe­ the U.S. and its possessions. project for up to $3,782,372 over five riod. After five years, the facilities REU is run by the NSF research years. should be able to operate indepen­ divisions and coordinated through The University of Illinois at Ur­ dently of NSF support. the Office of Undergraduate Sci­ bana will receive $519,692 during -NSF News Release ence, Engineering, and Mathemat­ the first year and up to $2,448,657 ics Education in the Directorate of over five years to establish a parallel­ Research Experiences Science and Engineering Education. processing research laboratory, which for Undergraduates For descriptions of some of the will enable scientists to use more than REU projects funded by the Division one computer in tandem. Computers The Research Experiences for Un­ of Mathematical Sciences, see "Re­ based on parallel processing can per­ dergraduates (REU) program is de­ search Experiences for Undergradu­ form multiple tasks simultaneously signed to provide opportunities to ates," Notices, May I June 1988, page and quickly. Processors and memory undergraduates students to partici­ 686. are organized and connected differ­ pate in active research experiences Requests for REU Supplements ently in different parallel-processing in mathematics, science, and engi­ may be submitted at any time and computers, with different designs hav­ neering. REU was launched in 1987 require 2-3 months processing time. ing different strengths in application. as part of the NSF's expansion of The deadline for Sites is October The technology to be developed at activities directed at undergraduates. 10 annually. For more information, Illinois will enable the best aspects There are two kinds of awards contact the Office of Undergraduate of different designs to be used to­ made in the REU program: Sites and Science, Engineering and Mathemat­ gether for more powerful problem Supplements. REV Sites will conduct ics Education, Room 639, National solving. Users will access this com­ projects for a number of undergrad­ Science Foundation, 1800 G Street bined system, called Tapestry, from uate students. Because the REU Sites NW, Washington, DC 20550; tele­ work stations and existing facilities projects must have a well-defined fo­ phone 202-357-7051. For program via a high-speed network. Initial ap­ cus, most REU Sites are expected to announcements, request form 88-28 plications will be in the areas of be within the scope of a single disci­ from the Forms and Publications computer vision and graphics, and pline and/or single academic depart­ Unit, Room 232 at the street address differential equations. ment, although an interdisciplinary above, or call 202-357-7861. A first-year award of $549,999 to project with cohesively integrated ac­ the University of North Carolina will tivities is acceptable. The normal size support the development of a capa- for REU Sites is about 8 students,

1012 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News from Washington

National Medals tions to public and private scientific orated with Ellis Johnson in investi­ of Science Awarded enterprise." gating subadditive functions related Gomory was born May 7, 1929, to group problems that had played On July 15, 1988, President Reagan in Brooklyn Heights, New York. He a role in cutting-planes and comer awarded National Medals of Science graduated from Williams College in polyhedra. to nineteen individuals in recognition 1950, studied at Cambridge Univer­ Gomory's research has been rec­ of their achievements in mathemat­ sity, and received his Ph.D. in math­ ognized (in addition to the Lao­ ics, science, and engineering. Among ematics from Princeton University chester Prize) by the title of IBM the recipients were two mathematical in 1954. Gomory served in the Navy Fellow, IBM's highest technical rank, scientists, Ralph E. Gomory of Inter­ and was Higgins Lecturer in Math­ and by the Harry Goode Memorial national Business Machines (IBM) ematics at Princeton before joining Award of the American Federation Corporation and Joseph B. Keller of IBM's newly founded Research Di­ of Information Processing Societies Stanford University. vision in 1959 as a research mathe­ and the John von Neumann Theory The National Medal of Science matician. Prize given jointly by the Operations was established by Congress in 1959 In his student years (Williams, Research Society and the Institute of to provide special recognition to indi­ Cambridge, Princeton), Gomory did Management Science. He was elected viduals for their outstanding contri­ research on nonlinear differential to the National Academy of Sciertces butions to knowledge in the physical, equations, but voluntary active ser­ in 1972. biological, mathematical, engineer­ vice in the Navy ( 1954-57) turned Gomory served as chairman of ing, behavioral, or social sciences. his attention to the applied mathe­ IBM Research's Mathematical Sci­ Selection is based on the total im­ matics of operations research. Back ences Department from 1965-67 and pact and importance of an individ­ at Princeton he soon linked clas­ 1968-70 during an important period ual's work on the present state of sical Diophantine analysis with the of its growth and evolution. This his or her chosen field. In addition, new linear programming to obtain period saw the beginning of Shmuel achievements of an unusually signifi­ cutting-plane and all-integer algo­ Winograd's work on limits of algo­ cant nature are considered in relation rithms which established integer pro­ rithms and of Benoit Mandelbrot's to their potential effects on the de­ gramming as a rigorous and vital work on fractals. velopment of scientific thought. theory. It remains an active area of Gomory became director of re­ Ralph E. Gomory was awarded research today. search for IBM in 1970, with direct the Medal for his scientific and in­ At IBM Research in the early '60s, responsibility for IBM's Research Di­ dustrial contributions. IBM provided Gomory published imaginative pa­ vision comprising the research labo­ Notices with biographical informa­ pers with Paul Gilmore on the knap­ ratories at Yorktown Heights, NY tion and a description of Gomory's sack, traveling salesman and cutting­ (the Thomas J. Watson Research contributions: stock problems (the last of which Center); San Jose, CA (now the Al­ Ralph E. Gomory, IBM senior won the Lanchester Prize of the Op­ maden Research Center), and Zurich, vice president for science and tech­ erations Research Society in 1964), Switzerland. He has continued to be nology, was honored for "his sci­ and with T. C. Hu on flows in multi­ responsible for the Research Divi­ entific contributions to the mathe­ terminal networks and continua. In sion ever since, reporting variously matics of discrete optimization, for the late '60s he developed the asymp­ to the president or chief executive bringing to a leading position one of totic theory of integer programming officer of IBM. industry's most significant research and introduced the concept of comer Gomory was very much aware establishments, and for his contribu- polyhedra. In the early '70s he collab- of the need for practical as well as

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1013 News from Washington scientific results coming from the Re­ the company's Corporate Manage­ NRC. Later he served on the NRC search Division. He worked to estab­ ment Board in 1983, and elected Governing Board, and he is currently lish an atmosphere in which purely a senior vice president in 1985. In a member of the Committee on Sci­ scientific work could be done, and in 1986, he was named to the newly ence, Engineering and Public Policy which technology transfer into prod­ created position he now occupies, (COSEPUP). ucts could also occur. He made use which combines research with other In the last few years Gomory has of a number of new organizational technological functions. written papers on the nature of tech­ approaches to technology transfer, Gomory is a trustee of Prince­ nology development, on research in of which one, the so-called "joint ton University and of the Alfred industry, and on industrial compet­ programs," has been particularly suc­ P. Sloan Foundation. He was An­ itiveness. He is currently a member cessful. drew D. White (visiting) Professor­ of the White House Science Coun­ Members of the Research Divi­ at-Large at Cornell from 1970-76, cil and chairman of the Advisory sion have made major contributions and he has served on visiting com­ Committee to the President on High to IBM's products and to the com­ mittees of Stanford, MIT, Princeton, Temperature Superconductivity. puter industry. Yorktown Heights Chicago and Yale. He was chairman Joseph B. Keller was honored was the birthplace of what is now of the Advisory Council of the De­ "for his outstanding contribution to known as RISC architecture, and San partment of Mathematics at Prince­ the geometrical theory of diffraction. Jose was the birthplace of the con­ ton from 1984-85, and is chairman This is a major extension of geo­ cept, theory and first prototype of of the Harvard Visiting Committee metrical optics which succeeds, after relational data. Major contributions for the Division of Applied Sciences. many centuries, in adding the physics to semiconductor technology and to Gomory is a member of the Na­ of diffraction to the simple ray con­ magnetic disc technology, particu­ tional Academy of Engineering, the cepts of optics and of other wave larly the first thin film head, have American Academy of Arts and Sci­ motions." emerged. ences and the American Philosophi­ Joseph B. Keller was born on July On the scientific side, Zurich did cal Society as well as the NAS, and 31, 1923, in Paterson, New Jersey. the work that resulted in two suc­ a fellow of the Econometric Society. He was educated at New York Uni­ cessive Nobel Prizes in physics, one He has served on the Councils of the versity, where he received his B.A. in for the Scanning Tunneling Micro­ NAS, the NAE, the APS and the IRI. 1943, his M.S. in 1946, and his Ph.D. scope and one for the new high­ in 1948. In 1948, Professor Keller be­ temperature superconducting mate­ gan his teaching career at New York rials. These are only some highlights University as assistant professor of of a large body of both practical and mathematics, before he became as­ theoretical work that has included sociate research professor in 1952. major steps forward in lasers, solid He was promoted to full professor in state physics, algebraic complexity 1956, a position he held until 1979. theory, and the evolution and appli­ He was concurrently the chairman of cation of fractals. the mathematics department at NYU Gomory set clear goals for IBM's from 1967-1973. Since 1979, he has Research Division: to be famous for been a professor of mathematics at its science and technology, and vi­ Stanford University. tal to IBM. Through a sensitivity Professor Keller was elected to deriving from his own personal re­ the American Academy of Arts & search, he created an atmosphere in Sciences in 1969, to the National which outstanding work could flour­ Academy of Sciences in 1973, and to ish together with practical contribu­ Foreign Membership of the Royal So­ tions to the company. His technical ciety in 1986. He was vice president leadership has been recognized by of the Society for Industrial and Ap­ the award of the Medal of the In­ Ralph E. Gomory plied Mathematics ( 1978-1979) and dustrial Research Institute (IRI) in a Member at Large of the AMS Gomory was worked on many 1985, and by the Institute of Electri­ Council (1985-1987). studies for the National Research cal and Electronics Engineers (IEEE) Keller has received honorary doc­ Council. In 1978-79 he was chair­ Engineering Leadership Recognition torates from the Technical Univer­ man of the group that brought out Award earlier this year. sity of Copenhagen ( 1981) and from the first Five Year Outlook Report Gomory was elected a vice pres­ Northwestern University (1988). He (on science and technology) from the ident of IBM in 1973, named to has been awarded the Von Karman

1014 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News from Washington prize of the Society for Industrial cal Theory of Diffraction (GTD), cal quantization, he introduced an and Applied Mathematics (SIAM), which he originated for solving prob­ important measure corresponding to the Eringen Medal of the Society for lems of wave propagation. He be­ the number of times that a closed Engineering Science, and the Timo­ gan thinking about such problems curve passes through a caustic sur­ shenko Medal of the American So­ in his work, during WW II, on face. Later generalized to curves on ciety of Mechanical Engineers, and problems of sonar for the Colum­ Lagrangian manifolds by Maslov, twice received the Lester R. Ford bia University Division of War Re­ this measure is referred to as the Award of the Mathematical Associ­ search. GTD is an important exten­ Keller-Maslov index. This work too ation of America. He has been the sion of the Geometrical Theory of was subsequently extended by Keller Gibbs Lecturer of the AMS and the Optics (GTO), in which wave prop­ to eigenvalue problems in bounded von Neumann Lecturer of SIAM. agation is described by rays. The ex­ domains, not necessarily associated His areas of research interest in­ tension to GTD overcomes the diffi­ with quantum mechanics, but gov­ clude applied mathematics, acous­ culties of GTO which cannot account erned by general systems of partial tics, electromagnetic theory, fluid dy­ for phenomena such as diffraction, differential equations. namics, and geometrical optics. or the occurrence of signals where Bernard J. Matkowsky of North­ GTO predicts none. In his theory, western University was asked by the Keller developed a systematic way managing editor of Notices to com­ to treat high frequency wave prop­ ment on Keller's contributions. He agation problems and derived and responded: solved the equations determining the Joseph Bishop Keller is one of rays, or paths, along which signals the foremost contemporary creators propagate, as well as those equa­ of mathematical techniques to solve tions governing how energy propa­ problems in science and engineering. gates along those rays. These include He has earned this reputation by his predictions of what happens to the outstanding research contributions to rays as they encounter obstacles or both mathematical methodolgy and inhomogeneities of the medium in to a wide variety of areas of applica­ which they travel. Prior to Keller's tion. He combines creativity in the work, only a few isolated problems development of mathematical meth­ were solved and understood, and ods with very deep physical insight there was no general theory which and has an uncanny ability to de­ could be used by engineers and sci­ scribe real world problems by sim­ entists for the solution of more com­ ple yet realistic models, to solve the plex and technologically important Joseph B. Keller exist books de­ mathematical problem by sophisti­ problems. Now there Keller's work has stimulated a to Keller's theory, as engineers cated techniques, many of which he voted vast literature both in the U.S. and his systematic himself created, and then to explain and scientists employ abroad, not only in many areas of to this day. the results and their consequences and general theory science and engineering in which his shown that the in simple terms. He is a virtuoso Keller has also methods and results are routinely for wave prop­ in showing how to take ideas found methods he developed employed, but in the mathematics other useful in one area of science and agation could be extended to community as well. For example, his For adapt them for use in other areas. In classes of problems as well. work has been the impetus for a on addition, he has taught and trained example, his fundamental work number of developments in the the­ generalized generations of applied mathemati­ semiclassical mechanics ory of Fourier integral operators and of Planck, Bohr, cians who form what is referred to as the earlier work Lagrangian manifolds. the "Keller school of Applied Math­ Sommerfeld, Wilson, Einstein, and Keller also considered problems the correct quan­ ematics." Through his own work, as Brillouin to derive of wave propagation through hetero­ nonseparable sys­ well as that of his students and other tization rules for geneous, turbulent, or random me­ in scientists with whom he has inter­ tems, thus yielding results valid dia. In this work he originated two These results, acted, he has had a profound and any coordinate system. methods which are very widely used: lasting influence on the way that referred to as the Einstein-Brillouin­ the Smoothing Method for problems quantization rules, problems are formulated and solved Keller (E.B.K.) involving small amplitude variations, chemi­ mathematically. are currently employed by and the Multiple Scale Method for well as other sci­ One of Keller's most outstand­ cal physicists as problems corresponding to rapidly ing contributions is the Geometri- entists. In his work on semiclassi-

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1015 News from Washington varying coefficients. This theory has Council will hold its annual Depart­ • ICEMAP Agencies Roundtable, since been taken up by others and has ment Chairs' Colloquium on October organized by Judith S. Sunley, Na­ come to be known as the Theory of 14-15, 1988 at the Washington Mar­ tional Science Foundation. Presenta­ Homogenization, on which volumes riott Hotel in Washington, DC. An tion by agencies of Interagency Com­ have been written. His work was earlier announcement of the collo­ mittee for Extramural Mathematics characterized by a simple formula­ quium appeared in the July f August Programs (ICEMAP) on opportuni­ tion which overcame the nonunifor­ issue of Notices, on page 819. ties for funding mathematical and mities restricting earlier theories. The program features the follow­ statistical research, and a roundtable No stranger to national service, ing presentations: discussion of current issues. Keller has worked on many problems • Computer Graphics and Mathe­ • MS2000 and Report to the Na­ related to national security and has matical Sciences Research, organized tion, organized by Bernard Madison, served on various advisory boards, by Edward Wegman, George Ma­ University of Arkansas. Update on national panels, and committees. Af­ son University. Three presentations activities, progress and plans for the ter his work on sonar for the Columbia with high visual content highlight­ MS2000 project (Mathematical Sci­ University Division of War Research, ing uses of computer graphics in re­ ences in the Year 2000), including he worked on problems of underwa­ search in core mathematics, applied a presentation on the Report to the ter explosions. In the early 1950s, he mathematics and statistics, including Nation by Lynn Arthur Steen, St. served, with von Neumann, on the examples of interesting mathemati­ Olaf College. Committee on Underwater Explo­ cal and statistical problems arising in • BMS Update, organized by Law­ sion of Atomic Bombs of the Armed computer graphics. rence Cox, National Research Coun­ Forces Special Weapons Project to • Computers in the Classroom, cil. Update on programs and activ­ consider proposed tests of the ef­ organized by Donald Kreider, Dart­ ities of the Board on Mathematical fects of A-bomb explosions on ships mouth College. Three presentations Sciences, including planning for the and submarines. Among the scientific on uses of computers in teaching update of the David Report. consequences of his explosion work is post-secondary mathematics and sta­ • Washington Update, Kenneth the Keller-Kolodner theory of under­ tistics, followed by a brief panel dis­ M. Hoffman, Joint Policy Board for water explosion bubble oscillations, cussion on problems and opportuni­ Mathematics. Update on policy is­ similarity solutions of spherical gas ties. sues affecting the mathematical sci­ flows, and the Kranzer-Keller theory • Computation and Computation­ ences. of water waves produced by explo­ al Methods in Mathematical Sci­ • Beyond the Individual Investi­ sions. These theories are still being ences Research, organized by William gator Model: Workshop on Extraor­ used today. Eddy, Carnegie-Mellon University. dinary Methods for Organizing and In addition to his outstanding Three presentations on computational Funding Mathematical Sciences Re­ and wide-ranging research, Keller is a methods in research and research search, organized by Andre Manitius, teacher and expositor par excellence. on computational methods in core George Mason University. Four pre­ He has received awards from all the mathematics, applied mathematics sentations on methods for organiz­ three major U.S. mathematical so­ and statistics, followed by a brief ing and funding mathematical sci­ cieties, from various engineering so­ panel discussion on where the field is ences research beyond the individ­ cieties, as well as national societies going. ual model, including research and in the U.S. and abroad. The ap­ • Supercomputers and Mathemat­ technology centers, special years, and proximately fifty Ph.D. students and ical Sciences Research, organized by equipment and symposia grants. numerous additional postdoctoral as­ Donald M. Austin, Department of The registration fee is $175 and sociates whom he has trained, now Energy and Charles Holland, Air covers the cost of handouts and successful applied mathematicians in Force Office of Scientific Research. meals (including a conference re­ their own right, further attest to the Three presentations on uses of su­ ception and banquet). Space is lim­ impact that Keller has had. In short, percomputers in research, research ited and advance registration is re­ Keller is one of the most prolific and problems spawned by supercomput­ quired. For more information, con­ important investigators and educa­ ers, and opportunities for supercom­ tact: Board on Mathematical Sci­ tors of our time. puter access. ences, National Research Council, • Mathematics in Industrial Prob­ 2101 Constitution Avenue, NW, Annual Department Chairs' lems, Colloquium Banquet Address Room NAS 312, Washington, DC Colloquium by Avner Freedman, Director of In­ 20418; telephone 202-334-2421. stitute for Mathematics and its Ap­ The Board on Mathematical Sciences plications, University of Minnesota. (BMS) of the National Research

1016 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News from Washington

NRC Announces Minority Awards tuition and fees and which may Richard Miller, formerly Director extend up to three years. In addi­ of the Mathematical Optimization The National Research Council tion, 14 doctoral candidates working and Finite Mathematics programs at (NRC) has announced awards in its on dissertations will receive one-year AFOSR, has returned to Iowa State programs for minority fellowships stipends. MARINA L. KA.MAHELE will University, where he is professor of for postdoctoral research and for receive a predoctoral fellowship to mathematics. doctoral studies. Both programs are study computer science at the Uni­ sponsored by the Ford Foundation versity of Hawaii, Manoa. A predoc­ Staff Changes at ONR and administered by the NRC. toral fellowship also will go to MARIA According to Frank Press, chair­ MARGARET MORRILL, who will study After serving about 1 1/2 years as man of the National Research Coun­ mathematics at the University of Cal­ Director of the Mathematics Divi­ cil, the Postdoctoral Fellowships for ifornia, Los Angeles. sion at the Office of Naval Research, Minorities Program "gives outstand­ The deadline for the 1989 fel­ John R. Cannon has accepted a po­ ing teachers/scholars the freedom lowship competitions is November sition as Professor and Head of the to pursue research interests with­ 14, 1988. Information and applica­ mathematics department at Lamar out added teaching responsiblities." tions are available from the Fellow­ University in Beaumont, Texas. His Among the 36 scholars receiving the ship Office, National Research Coun­ successor has not yet been named, postdoctoral fellowships are two in cil, 2101 Constitution Avenue, N.W., but ONR will soon announce the the mathematical sciences. MARK Washington, DC 20418; telephone search for a new director. JosEPH GoTAY, whose home insti­ 202-334-2872. Julia Abrahams has accepted a tution is the United States Naval position as Scientific Officer in the Academy, will use his fellowship to Staff Changes at AFOSR Probability and Statistics program. pursue research in mathematics at Before coming to ONR in July, the Mathematical Sciences Research Charles J. Holland, formerly Direc­ Abrahams was Engineering Editor Institute, Berkeley, California. IsoM tor of the Computer Science Divi­ at Springer-Verlag. She received her HARRIS HERRON of Howard Univer­ sion of the Office of Naval Research, Ph.D. in electrical engineering from sity, will do research in the applica­ has been appointed Director of the Princeton University and taught that tions of mathematics at the Univer­ Mathematical and Information Sci­ subject at Rice University and Car­ sity of Maryland. ences Directorate at the Air Force Of­ negie-Mellon University. In 1983, she The fellowships for doctoral stud­ fice of Scientific Research (AFOSR). held a visiting position at ONR in the ies are designed to increase the rep­ Holland replaces James M. Crowley, Probability and Statistics program. resentation of minorities on the fac­ who will take a position in the Chief Her research interests include statis­ ulties of American colleges and uni­ Scientist's Office at Air Force Sys­ tical communications theory, applied versities. Forty predoctoral students tems Command. In addition, Eytan probability, and applied random pro­ will receive fellowships which in­ Barouch, professor of mathematics cesses. clude individual stipends and grants at Clarkson University, will become to graduate institutions in lieu of a program adviser at AFOSR.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1017 Acknowledgement of Contributions

The officers and the staff of the Society acknowledge with gratitude gifts and contributions received during the past year. The inside cover of each issue of Mathematical Reviews carries the names of the sponsoring societies which sup­ port that publication. Contributing members of the Society paid dues of $132 or more. In addition to contributions to the AMS Centennial Research Fellow­ ship Fund, there were a number of unrestricted general contributions. Some of the contributors have asked to remain anonymous. All of these gifts provide important support for the Society's programs. The names listed below include those whose contributions were received during the year ending March 31, 1988.

CONTRIBUTING MEMBERS

Akemann, Charles A. Glass, A. M. W. Killeen, John Otermat, Scott C. Spencer, Joel H. Amir-Moez, Ali R. Gould, Henry W. Knuth, Donald E. Palais, Richard S. Stakgold, lvar Andrews, George E. Grace, Edward E. Kopala, Conrad Palmer, Theodore W. Sternberg, David Assmus, Edward F., Jr. Graves, Robert L. Krause, Ralph M. Papanicolaou, George C. Strauss, Frederick B. Babcock, William W. Greicar, Richard K. Krueger, Charles G. Pearson, Robert W. Strichartz, Robert S. Bauer, Frances B. Greif, Stanley J. Kunen, Kenneth Perry, William L. Sudler, Culbreth, Jr. Baumslag, Gilbert Grimmer, Ronald C. Lebedin, Neal D. Petro, John W. Sullivan, Richard W. Beechler, Barbara J. Gromov, Mikhael Lemay, William H. Polking, John C. Swokowski, Earl W. Bjorklund, Peter B. Haddad, George F. B. Lenard, Andrew Pratt, Vaughan R. Taft, Margaret W. Booth, George W. Haddix, George F. Lewis, Hugh L. Ratliff, Louis J., Jr. Taylor, Laurence R. Bressoud, David M. Halberstam, Heini Lieb, Elliott H. Redheffer, Raymond M. Uhlenbeck, Karen Buianouckas, Francis R. Hassinger, Bill, Jr. Mamelak, Joseph S. Reed, David L. Una!, Aynur Carson, Robert C. Hedlund, Gustav A. Martin, Gary A. Reedy, Christopher L. Veech, William A. Clifford, Alfred H. Hemstead, Robert J. Mathsen, Ronald M. Reese, Matthias F. Warren, William E. Cohn, Leslie Hironaka, Heisuke Mattson, H. F., Jr. Riney, John S. Weintraub, Steven H. Cootz, Thomas A. Hochster, Melvin Meder, Albert E., Jr. Rosenblum, Marvin Wendroff, Burton Corrigan, Thomas Carney Horrigan, Timothy J. Miles, E. P., Jr. Ross, Kenneth A. White, George N., Jr. Daverman, Robert J. Howe, Roger E. Miller, W. F. Rovnyak, James L. Whitmore, William F. De Marr, Ralph E. Hufford, George A. Mislin, Guido Sally, Paul J., Jr. Wilkins, J. Ernest, Jr. DeFacio, Brian Hunt, Richard A. Moschovakis, Yiannis N. Samit, Jonathan Wilson, Robert Lee DeLeon, Morris Jack Hutchinson, George A. Muhly, Paul S. Sawyer, Stanley A. Woeppel, James J. Demana, Franklin D. Johnson, Roy A. Murphy, Donald P. Schori, Richard M. Woolf, William B. Dickerson, Charles E. Kang, Sebong Nishiura, Togo Seligman, George B. Xenos, Philippos John Dinneen, Gerald P. Kaplan, Wilfred Nummela, Eric C. Sexauer, Norman E. Yahya, S.M. Ecklund, Earl F., Jr. Kautzmann, Frank N., III Olum, Paul Shabazz, Abdul Alim Yohe, J. Michael Farrell, Roger H. Kelly, John B. Orlik, Peter P. Shaneson, Julius L. Anonymous (5) Galuten, Aaron Kiefer, James E. Osofsky, Barbara L. Singmaster, David B.

Corporate Members and Institutional Associates

The Society also acknowledges with gratitude the support rendered by the following corporations, who held either Corporate Memberships or Institutional Associateships in the Society during the past year.

Corporate Members Institutional Associtates Aerospace Corporation Princeton University Press AT&T Bell Laboratories D. Reidel Publishing Corporation General Motors Corporation Springer-Verlag New York Incorporated International Business Machines Corporation Supercomputer Research Center, Institute for Defense Analyses National Security Agency

1018 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY General Contributions

GENERAL CONTRIBUTIONS

Abi-Khuzam, Faruk Fuad Dai, Taen-Yu Hironaka, Heisuke Macwilliams, F. Jessie Quinn, Frank S., III Adams, William W. D'Angelo, John P. Hobbs, Arthur M. Madow, William G. Randolph, John F. Aissen, Michael I. David, Karl Hodgson, Jonathan P. E. Mamelak, Joseph S. Raney, George N. Alberts, George E. Davidoff, Giuliana P. Hoffman, William C. Mann, Benjamin M. Raskind, Wayne Mark Al-Medhadi, Nora Seif Day, Mahlon M. Holland, Samuel S., Jr. Marin, Irwin C. Reade, Maxwell 0. Alvis, Dean L. De Koning, Jan Holmes, Philip John Mayor, John R. Reed, David L. Antman, Stuart S. Dedecker, Paul Hom, Jean Macgregor McAdam, Stephen J. Reissner, Eric Aribaud, F. DeFacio, Brian Horrigan, Timothy J. McCaughan, Dennis J. Reitwiesner, George W. Arisaka, Nakaaki Defrancesco, Henry F. Hu,K. Y. McColm, Gregory Loren Renno, James G., Jr. Au, C. H. E. Delbaen, Freddy Hu, Shou-Jen McCormick, Ferris E. Reznick, Bruce A. Bachman, George Deprima, Charles R. Huerta, Carlos Cuevas McCready, Robert R. Richards, John F. Baik, Yong Bai DeTurck, Dennis Hughes, Joseph A. McDonald, Janet Richmond, Lawrence B. Baker, Edward D. Duren, William L., Jr. Hughes, Ruth L. McGuinness, Francis Oisin Rieger, Georg J. Baker, George A. Sr Dutton, Charles Everett Hulthen, Lamek McKenzie, Harvey C. Robinson, Paul Lee Ball, N. Hansen Ecklund, Earl F., Jr. Iha, Franklin T. Meadows, Lawrence F. Robinson, Raphael M. Balser, Arienne S. Edmonds, Allan L. Ioakimidis, N. I. Meder, Albert E., Jr. Rohrbach, Hans Bari, Ruth A. Edrei, Albert Ishimoto, Hiroyasu Meisner, Morris J. Roseman, Joseph J. Barnet, Frank Emery, Carol J. Reinhard Ito, Ryuichi Mellender, James W Rosenstein, George M., Jr. Bartle, Robert G. Ensey, Ronald J. Iwata, Koichi Melrose, Gordon Rutledge, Dorothy S. Baruch, Herbert M., Jr. Errico, Thomas G. Jackson, Stanley B. Metzen, Gerhard G. Rykken, Charles John Bateman, Lynn M. Evergreen State Coli Jacob, Niels Meyer, Herman Sahab, Salem M. Bauer, Frances B. Fan, Ky Janwa, Heera Lal Michel, Rene Salas, Satumino L. Baum, John D. Fass, Arnold L. Jeanquartier, Pierre Mielke, Paul T. Sarafyan, Diran Baur, Lynne A. Faulkner, Frank D. Jeroslow, R. G. Mizuno, Hirobumi Sarason, Donald E. Beckmann, William H. Ferrari, Carlo Johnson, Charles N. Mochizuki, Horace Y. Sarhan, Mahmoud Beebe, Alfred S. Finch, John V. Johnson, Joseph F. Moller, Raymond W. Sasaki, Shigeo Bengtson, Thomas E. Folk, Thomas F. Jones, Carl Hilton Mongelli, Dominic Sastri, Chelluri C. A. Bennett, Fredricka T. Forbes, Stephen H. Kabak, Bertram S. Moore, Richard A. Schlesinger, Ernest C. Berger, Alan E. Franzblau, Deborah S. Kahn, Donald W. Morawetz, Cathleen S. Schmetterer, Leopold K. Bergman, George M. Friedberg, Solomon Kakihara, Yuichiro Morris, Robert A. Schraut, K. C. Betancourt, Carlos M. Fritsche, Richard T. Karle, Jerome Mostow, George Daniel Sclove, Stanley L. Bianco, Anthony J. Frohlich, Jurg M. Kent, George A. Moyls, Benjamin N. Scott, Leonard L., Jr. Blair, William D. Fuchs, Wolfgang H. Kemast, John C. Mrowka, Stanislaw G. Scott-Thomas, John F. Blanc, M. L. Fujiwara, Daisuke Kiefer, James E. Murphy, Donald P. Sedgewick, Charles H. W. Boos, William Furutani, Kenro Kim, Wan Se Murray, Margaret A.M. Sehgal, Virindra M. Booth, George W. Gabriel, Richard F. King, Randall R. Myers, Earle F. Shabazz, Abdul Alim Brauer, Fred Galbraith, A. S. Kirby, Andrew R. Nashed, M. Zuhair Sharp, Henry, Jr. Brauer, George U. Gantos, Richard L. Kita, Hiroo Nelligan, John D. Shiga, Koji Brothers, John E. Glass, Michael S. Klein, Edwin M. Nelson, James, Jr. Shiraiwa, Kenichi Brown, Arthur B. Gleason, Andrew M. Kolodner, Ignace I. Nevai, Paul Shorter, William Ivan Brown, Robert E. Goddyn, Luis A. Kossack, C. R. Newman, Morris Silberger, Allan J. Bryn, Milo F. Goldberg, Samuel Kra, Irwin Nichols, Edward Sirao, Tunekiti Buianouckas, Francis R. Goldberg, Seymour Krause, Ralph M. Nishiura, Togo Slook, Thomas H. Bumby, Richard T. Goldman, William Mark Krumpholz, Gary R. Nordhaus, Edward A. Smith, George Rowan Burkholder, Donald L. Goss, Robert N. Kunoff, Sharon Norman, Francis Alexander Smith, Marianne Freundlich Burkill, John C. Gottlieb, Daniel H. Kurata, Yoshiki Nummela, Eric C. Smith, Roy C. T. Byrnes, James S. Grabiel, Federico Kurshan, Robert P. Oguchi, Kunio Smith, Wilbur L. Caldwell, Roderick P. C. Grace, Edward E. Kurtz, Douglas S. Okuyama, Akihiro Sommese, Andrew John Campbell, Gary A. Green, John W. Kurtzke, John F., Jr. Onishi, Hironori Sonnesso, Geraldine Carlson, Bengt G. Greville, Thomas N. E. Lackey, Jackson B. Orlik, Peter P. Spanier, Jerome Carmichael, Richard D. Griffiths, H. Brian Lai, Tony W. Orloff, Leo N. Spears, Otis S. Carson, Robert C. Gross, Louis J. Lander, Felix I. Orr, Martin Spielberg, Stephen E. Castore, Glen M. Grubb, Gerd Laning, J. H. Otermat, Scott C. Spragens, William H., Jr. Chang, Weita Guillot, J. C. Lanzano, Paolo Paciorek, Joseph W. Stamey, William L. Chihara, Theodore S. Gulden, S. L. Lapidus, Michel L. Paisner, Helen L. Stegeman, Jan D. Chinn, William G. Gurney, Margaret Lawson, Walter R. Palm, Mark A. Steinberg, Robert Chow, Tseng Yeh Hagis, Peter, Jr. Lebow, Arnold Palmer, Theodore W. Stevens, Richard S. Christensen, Chris Hamstrom, Mary-Elizabeth Lehrer, Gustav I. Pan, Liu-Hua Stinson, Kevin Clifford, Jane E. Harrington, Walter J. Lewan, Douglas Papageorgiou, Nikolaos S. Stortz, Clarence B. Cochran, James A. Harris, John C. Lind, David W. Parker, Francis D. Sturm, Teo Cohen, Daniel I. A. Hassinger, Bill, Jr. Liu, Jong-Diing Pascual, Felino Garcia Sun, Hugo Sui-Hwan Cole, Paul Dana Hecker, David A. Liu, Shih-Chao Passerini, Henry J. Sunouchi, Gen-ichiro Coleman, A. John· Hemasinha, Rohan Llave, Rafael de Ia Patton, Charles M. Sward, Marcia P. Comenetz, Daniel Herbst, Albert F. Lombardo, Charles J. Peck, N. Tenney Takagi, Izumi Comfort, Edwin H. Herrera, R. B. Louhivaara, Ilppo S. Pell, William H. Tam, Bit-Shun Cooper, E. D. Herrero, Domingo A. Lubin, Jonathan D. Perlman, Sanford Tanabe, Hiroki Cootz, Thomas A. Herzog, Emil R. Lukacs, Eugene Petrie, Emily R. Tatevossian, Leon H. Crane, George E. Hilbert, Stephen R. Luther, C. F. Phelps, Robert R. Taylor, Angus E. Cullen, Helen F. Hildebrandt, Theodore W. Machover, Maurice Pollak, H. 0. Tepping, Benjamin J. Dahlin, Rolf B. Hilpipre, Douglas H. Macphail, Moray S. Quade, Edward S. Thomas, Erik G. F.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1019 General Contributions

Thomas, George B., Jr. Tugue, Tosiyuki Voigt, PaulS. Werner, Willes L. Yajima, Kenji Thomas, Janet D. Tull, Jack P. Voytuk, James A. Willcox, Alfred B. Yasuhara, Ann Thomas, Lawrence E. Uchiyama, Akihito Wage, Michael L. Williams, Frederick C. Yasuhara, Mitsuru Thomas, P. Emery Unai, Yasushi Unai Wantland, Evelyn K. Williamson, Francis U. Yokoi, Hideo Thron, Wolfgang J. van der Vaart, H. Robert. Warren, Bette L. Wilson, Robert Lee Young, Paul M. Tits, Jacques L. Van Alstyne, John P. Warren, William E. Wong, Sherman K. Ziebur, Allen D. Toll, Kathryn B. Vance, James T., Jr. Warschawski, Stefan E. Woodruff, William M. Zilmer, Delbert E. Tolsted, Elmer Vasconcelos, Walmer V. Wass, Noel Christopher Wright, David J. Zorn, M.A. Traina, Charles R. Vilms, Jaak Weaver, John C. Wu, T. C. Anonymous (73) Tsuji, Kazo Vodola, Paul A. Weill, Lawrence R. Yachter, Morris

AMS CENTENNIAL RESEARCH FELLOWSHIP FUND

Aarts, Johannes M. Ahmad, Hamid K. Alliston, R. Michael Andreassian, Agnes Arzberger, Peter W. Abadie, Jean M. Ahmad, Kathleen Alltop, William 0. Andreoli, Dorothy W. Asadulla, Syed Abbott, H. L. Ahmad, Shair Al-Medhadi, Nora Seif Andrews, George E. Asano, Kouhei Abbott, James C. Ahmadi, Mohammad H. Almgren, Frederick J., Jr. Andrews, James J. Asano, Shigemoto Abbott, James H. Ahrens, James Dwight Alonso, Agustin Andriamanalimanana, Bruno Ascenzi, Maria-Grazia Abdel-Megied, Mohamed Aibeche, Aissa Alonso, Juan M. Andrushkiw, Roman I. Askey, Richard A. Abdullah, Masood B. Aidoo, Emmanuel B. Alperin, Roger C. Angad-Gaur, Hendrik W. K. Aspenleiter, Stephen W. Abe, Eiichi Aikele, M. Holbrook Alpern, Steven Anger, Frank D. Asprey, Winifred A. Abellanas, Pedro Aimar, Hugo Alsholm, Preben Angus, John Edward Assmann, Susan F. Abernathy, Robert 0. Airault, Helene Alsina, Claudi Anichini, Giuseppe Assmus, Edward F., Jr. Abi-Khuzam, Faruk Fuad Aissen, Michael I. Alspach, Dale E. Anick, David Jay Astbury, Kenneth A. Abrahamson, Brian Aizawa, Sachiko Althoen, Steven C. Anselone, Philip M. Atalla, Robert E. Abramovici, Flavian Akagi, Hatsue Al-Thukair, Fawzi Ahmed Anshel, Michael Atallah, Kamal Abramson, Lawrence Akin, Ethan J. Altomare, Francesco Ansorge, Rainer Athanassoulis, Gerassimos A'Campo, Norbert Akiyama, Yoshio Alvarez, Jairo Ivan Anton, Howard Atiyah, Michael F. Acar, Robert Akylas, Triantaphyllos R. Alvarez, Josefina Anton, Jose Manuel Atkin, Joy H. Achmatowicz, Richard L. Aladro, Gerardo Alvis, Dean L. Antoniadis, Anestis A. Atkinson, Frederick V. de Acosta, Alejandro Al-Amiri, Hassoon S. Ambos-Spies, Klaus Antoniadis, Jannis Atkinson, H. R. Adachi, Masahisa Alavi, Yousef Amburgey, Jay K. Apaloo, Joseph Atkinson, Stuart L. Adachi, Steven H. Al-Bassam, Mohammed Ames, Karen A. Appell, Jurgen Atsuji, Masahiko Adams, J. Frank Ali A. Amick, Charles James Applebaum, Joseph A. Au, C. H. E. Adams, Loyce Albert, John P. Amin, Ashok T. Appleson, Robert R. Auchmuty, Giles Adams, Malcolm R. Alberts, George E. Amir, Dan Appling, William D. L. Auer, Jan W. Adams, Rebecca S. Albrecht, Ernst Amiran, Edoh Y. Arad, Zvi Ault, J. C. Adams, William W. Albrecht, Felix R. Amir-Moez, Ali R. Aragon, Regina D. Auslander, Bernice L. Adamson, lain T. A. C. Albright, Hugh N. Amitsur, Shimshon A. Araki, Shoro Auslander, Maurice Adelberg, Arnold M. Alder, H. L. Ammar, Gregory S. Arapostathis, Aristotle Aust, Catherine C. Adem, Alejandro Aldous, David J. Amsbury, Wayne P. Arason, Jon Kr Austin, Charles W. Adem, Jose Alessandrini, Giovanni Anacker, Steven E. Arce, Pedro E. Au-Yeung, Yik-Hoi Adler, Alfred Alexander, John R., Jr. Anaman, Todd M. Archdeacon, Dan Avellaneda, Marco Marcelo Adler, George Alexander, Roger K. Anantharam, Venkatachalam Archer, Myla M. Aversa, Vincenzo L. Adler, Irving Alexander, Stephanie B. Anantharaman, Rajan Archibald, W. Thomas Avioli, John J. Adler, Robert J. Alexanderson, Gerald L. Anastasio, Salvatore Arena, Orazio Avrunin, GeorgeS. Adluri, lndrasena Alfeld, Peter Ancel, Fredric Davis Arenstorf, Richard F. Ax, James Aeppli, Alfred Alfonso, Fermin Jorge Andersen, E. Sparre Aribaud, F. Axelsson, Reynir Agard, Egbert Alfonso, Joseph A. Andersen, Kurt Munk Arima, Satoshi Ayers, Kathleen L. Agmon, Shmuel Ali, M. Kursheed Anderson, Allan G. Arisaka, Nakaaki Ayoub, Christine W. Agosto Torres, Francis Alif, Metod Anderson, Berti! Arkowitz, Martin A. Ayoub, Raymond G. Agoston, Max K. Alkhayat, Isa A. Anderson, Chris R. Armbrust, Manfred K. Azoff, Edward A. Aguiar, Glenn E. Allday, Christopher J. Anderson, Donald W. Armitage, John V. Azpeitia, Alfonso G. Aguilar, Marcelo Alberto Allegretto, Walter Anderson, Eugene R. Arms, Judith M. Azumaya, Goro Aharoni, Israel Allen, Howard Anderson, Glen D. Armstrong, Gerald M. Azzam, Ali Aharonov, D. Allen, Joseph N. Anderson, Joel H. Armstrong, Kenneth H. Baartmans, Alphonse H. Ahlbrandt, Gisela Allen, Margaret A. Anderson, John T. Armstrong, Mark Anthony Baas, Nils Andreas Ahlfors, L. V. Allen, Roger W., Jr. Anderson, Marlow E. Armstrong, Thomas E. Baayen, Pieter C. Anderson, Michael T. Arnberg, Robert L. Babai, Laszlo Anderson, Robert M. Arnold, David M. Babbitt, Albert E., Jr. Anderson, Ronald M. Arnold, Douglas N. Babcock, William W. AMS CENTENNIAL RESEARCH Anderson, Stuart N. Arnold, Leslie K. Bachelis, Gregory F. FELLOWSHIP CONTRIBUTION Anderson, Theodore W. Arnold, Timothy K. Bachman, George FROM Andersson, Lars Arnoux, Pierre Bachman, R. Andersson, Rolf L. Aroian, Leo A. Bachmuth, Seymour RICHARD R. GOLDBERG Andler, Martin J. Aronson, Donald G. Bacon, Harold M. This contribution was a gift from members of the Ando, Shiro Arthur, James G. Bade, William G. mathematics department of Vanderbilt University Ando, Tsuyoshi Artin, Michael Bader, Ralf to Professor Goldberg on the occasion of his retire- Andre, Peter P. Artola, Michel Baderian, Armen Robert Andreadakis, Stylianos Arveson, William B. Badrikian, Albert ment from the post of department chairman.

1020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Ethan D. Bae, Soon-Sook Baron, Simson Beekman, John A. Bemis, Francisco Bloch, Baernstein, Albert, II Barr, Betty Jane Beekmann, Wolfgang Bernstein, Herbert J. Block, Richard E. Bagby, Richard J. Barrar, Richard B. Beem, John Kelly Bernstein, Irwin S. Blomstrom, Carol A. Bagby, S. Thomas Barre, Raymond C. Beesley, Darin J. Berresford, Geoffrey C. Blonde!, Corinne Baginski, Frank E. . Barreca, J. R. Beesley, E. Maurice Berriozabal, Manuel P. Bloom, David S. Bahamonde, Antomo Barrett, David E. Beeson, Michael J. Bers, Lipman Bloom, Frederick Bahri, Anthony Peter Barrett, Lida K. Behnke, K. Berterman, John E. Bloom, Melvin Baica, Malvina Florica Barrett, Terence W. Behrens, Ernst-August Bertin, Emile M. Blum, Joseph Baigent, Edith Anna Barros-Neto, Jose Behzadnoori, Kourosh Berzsenyi, George Blum, Lenore Baik, Yong Bai Barsky, Daniel Beig, Robert Bessenrodt, Christine Blyth Rentzsch, Barbara J. Bailey, Evelyn C. Barsotti, Leo Belair, Luc Besson, Gerard Boardman, John M. Bailey, G. H. Bart, H. Belfi, Victor A. Betancourt, Carlos M. Boas, Ralph P. Bailey, Robert E. Barth, Karl F. Belgacem, Fethi Bethelmy, David G. Bobisud, Larry E. Baillieul, John B. Barth, Wendy S. Bell, Carol Jean Betsch, Gerhard Bobonis, Augusto Baily, Andrew M. Bartick, Philip R. Bell, Howard E. Betz, Ebon E. Bodnar, Istvan Baily, Walter L., Jr. Bartle, Robert G. Bell, James M. Beukers, F. Bodnarescu, M. V. Bailyn, Paul M. Bartnik, Robert A. Bell, Steven R. Bevis, Jean H. Boduch, David J. Bain, James A. de Bartolomeis, Paolo Bell, Stoughton Beylkin, Gregory Bodvarsson, Gunnar Bain, Osias Barton, Thomas J. Bellamy, David P. Bhaskaran, M. Boercker, Dale P. Bakelman, Ilya Baruch, Herbert M., Jr. Bellenot, Steven F. Bhattacharya, RabiN. Boffi, Giandomenico Bakenhus, Fred Baruch, Herbert Mark, III Belley, J. M. Bia, Ahmed Boisvert, Robert E. Baker, Alan Basor, Estelle Bellissard, Jean V. Bianco, Anthony J. Boivin, Andre Baker, Blanche J. Bass, Hyman Bellow, Alexandra Bianco, Lucio Bojanic, Ranko Baker, Edward D. Bassalynejad, Tousi Afsaneh Belpaire, Gerald Bickel, Marilyn S. Bojars, Juris A. Baker, Edward G. Bateman, Felice D. Benardete, Diego Bidwell, Leonard N. Boley, David B. Baker, Frances E. Bateman, Lynn M. Bendat, Julius S. Biedenharn, Lawrence C. Bollinger, Robert A. Baker, George A. Sr Bateman, Paul T. Benedetto, John J. Bienenfeld, Mel Bond, Bradford D. Baker, Kirby A. Bates, Grace E. Benes, Vaclav E. Bieri, Robert Bond, David B. Baker, Peter Fritz Batie Nicolau, Nadal Bengtson, Thomas E. Bierstedt, Klaus D. Bonet, Jose Saksa, Dennis John Bator, Elizabeth Mary Benitez, Carlos Bierstone, Edward Bonin, Joseph E. Bal, Kenan Y. Batt, Jurgen 0. Benkart, Georgia M. Biles, Daniel C. Bonner, Richard F. Balanis, George N. Batten, Lynn Margaret Benke, George Bill era, Louis J. Boo, Per-Anders Balbes, Raymond Batterson, Steven L. Benkhalti, Rachid Billigheimer, Claude E. Booker, Andrew J. Baldi, Paolo Bauer, Carole Ann Bennett, Andrew G. Billik, Martin de Boor, Carl Baldwin, John T. Bauer, Douglas Bennett, Curtis D. Bilodeau, Gerald G. Boos, William Bales, John W. Bauer, Frances B. Bennett, David G. Bindschadler, David E. Booth, George W. Balkanyi, Leslie R. Bauer, Heinz Bennett, Fredricka T. Bing, Kurt Boras, Michael Edward Ball, Joseph A. Bauldry, William C. Bennett, John Bruce Bingen, Franz Borchers, Hans-Jurgen Ball, N. Hansen Baum, John D. Benson, Dave Bingham, Stuart C. Borel, Armand Ball, Richard Neal Baum, Paul F. Benson, David Bernard Bini, Dario Borgers, Christoph Ballard, William R. Baumann, Volker H. Benson, F. Chal Binkley, Dean H. Borho, Walter Ballou, Donald H. Baumeister, Johann Benton, Eileen R. Birch, Bryan J. Borkowski, Mark S. Baloglou, George Baumgartner, Gerald B., Jr. Benyamini, Yoav Bircher, John J. Borrego, Joseph T. Balser, Arienne S. Baumslag, Gilbert Berbee, Henry C. P. Biriuk, George Borrelli, Robert L. Bandle, Catherine Baur, Lynne A. Bercovici, Hari Bishop, Alan A. Borucki, Leonard John Sandler, Wyllis Bautista, Maurino P. Berend, Daniel Bishop, Richard L. Borwein, David Banerjee, Utpal Baxendale, Peter H. Berestycki, Henri Bismut, Jean-Michel Bosch, Carlos Giral Bang, Seung Jin Baxter, Willard E. Beretta, E. F. Bisztriczky, Tibor Bosch, William W. Bang, Thoger S. V. Bayer-Fluckiger, Eva M. Berg, Gene A. Bivens, lrl C. Bose, Raj C. Bankston, Paul J. Bazinet, Jacques Berg, Gunnar Bix, Michael C. Bossavit, A. Bannai, Eiichi Beachy, John A. Berge, John Orrin Bjorklund, Peter B. Bossert, J. Michael Bantegnie, Robert Beadle, Allen Jay Berger, Alan E. Blackall, Clair J. Boswell, Rupert D., Jr. Bao, David D. W. Beale, J. Thomas Berger, Charles A. Blackman, Jerome Bott, Raoul H. Baouendi, M. Salah Beals, R. Michael Berger, Marcel Blackmore, Denis L. Bottema, Murk J. Baras, John S. Beals, Richard W. Berger, Marsha J. Blackwood, Philip D. Boultbee, Nan Barber, S. F. Bean, Ralph J. Berger, Robert W. Blad, Michael C. Bourgeat, Alain P. Barbosa, J. Lucas M. Beard, Jacob T. B., Jr. Berger, Ruth I. Blair, William D. Bourgeois, A. J. Bardaro, Carlo Beatrous, Fran~ H. Berglund, John F. Blais, J. A. R. Bourgeois, Brian A. Barendregt, Henk P. Beattie, Margaret Bergman, George M. Blanc, M. L. Bourgin, Richard D. Barety, Julio Edgardo Beaumont, Ross A. Bergstrand, Deborah J. Blanch, Gertrude Bourion, Georges Bari, Ruth A. Bechtel, Robert D. Bergum, Gerald E. Blanche, Ernest E. Bourn, Robert B. Barkauskas, Anthony E. Beck, William A. Beringer, Alfred Blanchet-Sadri, Francine Bousfield, Aldridge K. Barker, Sandra Lou Hogan Becker, David S. Berkovitz, Leonard D. Bland, John S. Boutin, Pierre Barker, William Hamblin, II Becker, Eberhard Berlekamp, Elwyn R. Blankinship, William A. Bouwsma, Ward D. Barker, William Henry Becker, Howard S. Berman, David R. Blanton, John D. Bove, Antonio Barnard, Robert D. Becker, Michael Berman, Gerald Blaskie, Marc Bovier, Anton Barnard, Roger W. Becker, Thomas Berman, J. Michael Blatter, Christian Bower, Julia W. Barner, Klaus Beckman, Frank S. Berman, Robert David Blatter, Jorg Boyack, Stephen W. Barner, Martin Beckmann, Sybilla K. Berman, Stephen Blattner, Robert J. Boyce, Elsa K. Barnes, David C. Beckmann, William H. Bermond, J.-C. Bleick, Willard E. Boyce, William E. Barnes, Ward C. Bedard, Robert Bermudez, Alfredo Bleiler, Steven A. Boyce, William M. Barnet, Frank Bedford, Eric D. Bernau, Simon J. Blij, Frederik Van Der Boyd, Geraldine Fuller Barnhart, Richard Dee Beebe, Alfred S. Berner, Paul David Blitch, Patricia M. Boyer, Steven Patrick Baron, John J. Beechler, Barbara J. Bernfeld, Stephen R. Bloch, Anthony Michael Boyke, Robert P.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1021 AMS Centennial Research Fellowship Fund

Boyle, Jeffrey Brown, Johnny E. Burns, Anne M. Carlson, Bille C. Chao, J. A. Boyle, Mike Brown, Kenneth S. Burns, Daniel M., Jr. Carlson, James A. Chao, Michael S. Bozeman, Robert E. Brown, Lawrence G. Burns, Keith H. Carlson, John R. Chaperon, Marc Bozonis, Petros Brown, Patrick J. Burr, Stefan A. Carlson, John W. Chapin, Steven Arthur Braaksma, B. L. J. Brown, Richard H. Burrill, Claude W. Carlson, Robert C. Chapman, Judith E. Braam, Peter J. Brown, Richard L. Burris, Stanley N. Carlson, Timothy J. Charney, Ruth M. Brachman, Malcolm· K. Brown, Robert E. Burstall, Francis E. Carmichael, Richard D. Chatterji, Srishti D. Brackx, Freddy F. Brown, Russell M. Burton, Leonard P. Carpenter, Gail Alexandra Chatzidakis, Zoe Bradford, Giles Edward, III Brown, Scott W. Buser, Peter J. Carr, Ralph W. Chaudhary, B. D. Bradford, James C. Brown, Sharan Inez Bush, George C. Carroll, Francis W. Chaudhury, Tulip Bradford, Phillip G. Brown, Thomas A. Bushnell, Colin J. Carroll, Michael P. Chauvin, Andre Bradley, John S. Brown, William G. Buss, Samuel R. Carroll, Robert W. Chavez, Joseph Dean Bradley, Madeline A. Browne, H. Nelson, Jr. Bustelo, Julia Ana Carruth, James H. Chee, Pak Soong Bradley, Richard C. Browne, Patrick J. Butchart, John H. Carson, Andrew B. Cheeger, Jeff Brady, Wray G. Browning, David S. Butcher, John C. Carstens, Allan M. Chein, Orin N. Brakke, Kenneth A. Brownstein, Alan B. Butcher, Ralph Stevens Cartan, Henri Chen, Concordia C. Bralver, Peter Jeffrey Bruck, Ronald E., Jr. Butler, James William Carter, David W. Chen, Daniel K. C. Bramble, James H. Bruckner, Andrew M. Butler, John B., Jr. Carter, Thomas J. Chen, Kwan-Wei Bramsen, John D. Brucks, Karen M. Butler, Lynne M. Cartier, Pierre Chen, Louis H. Y. Brandenburg, Harald Bruckstein, Alfred M. Butnariu, Dan Casanova, Toussaint Chen, So-Chin Brandli, Emil R. Bruggeman, Roelof W. Butson, Alton T. Casazza, Peter G. Chen, W. W. L. Brandstrom, Hugo F. K. Bruner, Robert R. Buttke, Thomas F. Casey, Robert Francis Chen, Y. W. Brandt, Dennis E. Brunet, Richard M. Butts, Thomas R. Casey, Stephen D. Chen, Yu-Chuan Samuel Branner, Bodil Brungs, Hans H. Byerly, Robert E. Casian, Luis Guillermo Chen, Zen-Yi Brant, Larry J. Bruni, Anthony J. Byrd, Kenneth A. Casler, Burtis G. Cheng, Kuo-Shung Brattstrom, Gudrun B. Bruning, Jochen Byrne, Charles L. Casner, Lisa Marie Cheon, Seok-Hyeon Braude, Eric J. Brunner, Hermann Byrnes, James S. Cassidy, Phyllis J. Cherlin, George Y. Brauer, Fred Bruno, Vincent J. Byrnes, Raymond Albert, Jr. Castagnoli, Erio A. Cherniavsky, John C. Brauer, George U. Bruns, Winfried Cable, Charles A. Castellini, Gabriele Chernoff, Herman Bravo, Raul Brunson, Barry W. Caccianotti, Luciano Castore, Glen M. Chernoff, Paul R. Breckenridge, John C. Brunswick, Natascha A. Cadogan, Charles C. Castro, Antonio de Cherny, Julius Breen, Stephen Bryan, Charles A. Caffarelli, Luis A. Castrucci, Benedito Cherowitzo, William E. Brender, Melvin Bryant, Billy F. Caflisch, Russel Catanese, Fabrizio M. E. Cheung, B. K. S. Brennan, Joseph P. Bryant, John G. Cagnac, Francis Cateforis, Vasily C. Cheung, Wing-Sum Brenneman, Franklin S. Bryce, Gerald M. Cain, Bryan E. Cater, Frank S. Chiarenza, Filippo Brenner, Philip Brylawski, Thomas H. Cain, George L., Jr. Cattabriga, L. Chicks, Charles H. Brent, Richard P. Brylinski, Jean-Luc Caines, Peter E. Causey, William M. Chicane, Carmen C. Brenti, Francesco Bryn, Milo F. Cairoli, R. Cauvi, Roberto Jose, Jr. Chidambaraswamy, Jayanthi Bressler, David W. Brzezinski, Juliusz Calderbank, Robert Cavender, James A. Chihara, Theodore S. Bressoud, David M. Bube, Kenneth P. Calderon, Alberto P. Caveny, James Chikwendu, Sunday C. Bressoud, Thomas C. Buchanan, Thomas Caldwell, Gordon R. Cawley, Robert Chillag, David Bretscher, Otto K. Buchholz, John W. Caldwell, Roderick P. C. Cayco, Bern Chillak, Edward W. Breuer, Shlomo Buchmann, Johannes A. Caldwell, Thomas E. Cazzaniga, Franco Chinburg, Ted C. Breves Filho, J. A. Buchner, Michael A. Caldwell, William V. Cebollero, Blanca S. Ching, Wai-Sin Brickell, Ernest F. Buchsbaum, David A. Callahan, James J. Cecil, Thomas E. Chinn, William G. Brickman, Jacob Buck, Marshall W. Callahan, Willie R. Cegrell, Urban Choe, Geon Ho Bridges, Douglas S. Buckley, Joseph T. Camacho, James, Jr. Cerda, Joan Choe, Young Han Briem, E. Buckley, Stephen M. Camarena, Vicente Cernuschi-Frias, Bruno Choi, Man-Duen Brieske, Thomas J. Bucy, RichardS. Cameron, Douglas E. Cerruti, Umberto Choi, Yun Sung Brieskorn, Egbert Budnik, Paul, Jr. Camina, A. R. Certain, Melinda W. Choksi, Jal R. Brigham, Robert C. Buechler, Steven Allen Campanino, Massimo Cesari, Lamberto Chong, Chi Tat Brin, Michael I. Buehler, Royce E. Campbell, Harold E. A. Chabert, J. L. Choudhury, Dejoy K. Brinkmann, Christoph Buekenhout, F. Campbell, James T. Chacon, Paul R. Chouinard, Leo G., II Britton, John L. Buhler, Joe P. Campbell, L. Andrew Chae, Dongho Chouteau, Charles C. Brobeck, Barbara Caldwell Buianouckas, Francis R. Campbell, L. Lome Chaffee, James W. Chover, Joshua Brock, Paul W. Bula, Witold Campbell, Paul J. Chakerian, Gulbank D. Chow, Tseng Yeh Brodziak, Jon K. T. Bulger, Leonard L. Campillo, Antonio Chamberlin, Richard E. Chow, Yuan S. Broise, Michel Jean Bulirsch, Roland Zdenek Campiti, Michele Chambers, Daniel Warren Chrestenson, Hubert E. Bromberg, Eleazer Bullen, Peter Campos, L. M. B. C. Champagne, Carol Z. Christ, F. Michael Bromley, Darwin P. Bulman-Fieming, Sydney D. Candy, Gregory T. Chan, Jack-Kang Christensen, Chris Brommundt, Eberhard W. Bumby, Richard T. Canfield, E. Rodney Chan, Jean B. Christy, Joe Brons, Hans Bump, Daniel Willis Cannon, James W. Chan, Y. H. Chu, Chinku Brooks, Burrow P. Bungart, Lutz Cantoni, Vittorio Chan, Yun K. Chu, Tienchen Brooks, James 0. Buraczewski, Adam Cantor, Robert M. Chang, Bomshik Chuang, Pi-Chun Brooks, Robert Burchard, Hermann G. Cantrell, James C. Chang, Chao P. Chung, Lung Ock Brosowski, Bruno H. Burchard, Paul Cap, Clemens Heinrich Chang, Chienming A. Church, Philip T. Brothers, John E. Burckel, Robert B. Capozzi, Alberto Chang, 1-Lok Churchill, Edmund Broue, Michel Burgess, C. Edmund Cappell, Sylvain E. Chang, Jen-Tseh Churchill, Richard C. Brown, Anne E. Burgess, Richard Dennis Caranti, A. E. Chang, Kun Soo Ciarlet, Philippe G. Brown, Arthur B. Burgess, Walter D. Carbery, Anthony Chang, Ping-Tung Cikanek, Dana L. Brown, Edgar H., Jr. Burillo, Pedro Cargal, James M. Chang, Shao-Chien Cima, Joseph A. Brown, Gerald D. Burkholder, Donald L. Carleson, Lennart A. E. Chang, Weita Cincotta, Dominick Anthony Brown, Howard H. Burkholder, Douglas G. Carlisle, Ronald L. Chang, Yu Claasen, H. L. Brown, Jack B. Burkill, John C. Carlson, Bengt G. Chanillo, Sagun Clapp, Michael H.

1022 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Clare, Loren Paul Conlon, Samuel B. Crauder, Bruce C. D'Antona, Ottavio Dekster, B. V. Clark, Curtis Conn, Jack Frederick Craven, Bruce D. Dantzig, George B. Del Riego de Del Castillo, L. Clark, Kenneth D. Connelly, Robert Crawford, James P. D'Aprile, Margherita Delahunty, Michael D. Clark, Robert A. Conner, Luther T., Jr. Crawford, John David Darafsheh, Mohammad-Reza Delange, Hubert Clarke, Francis Connett, William C. Creede, Geoffrey D. D'Archangelo, James Michael Delanoe, Philippe Clarke, Graham Thomas Connolly, Frank X. Creese, Thomas M. D'Aristotle, Anthony J. Delbaen, Freddy Clarkson, Peter A. Connolly, Mary V. Crew, Richard M. Dark, R. S. Delchamps, David F. Clary, Stuart Connors, Edward A. Criscenti, Jacqueline P. Darmody, Laurie M. Deleanu, Aristide Clausing, Achim Conrad, Bruce P. Crittenden, Richard J. Darnel, Michael R. Deii'Antonio, Gianfausto Claussen, David R. Conrey, J. Brian Cronin, Bernard Darwin, Sophia K. Delporte, Jean Cleaver, Charles E. Constantin, Peter S. Croom, Frederick H. D'Atri, Joseph E. Delver, Robert Clemens, Charles Herbert Constantinescu, Corneliu Cross, James J. Datskovsky, Boris A. Demana, Franklin D. Clifford, Alfred H. Contessa, Maria Cross, John S. Dauxois, J. Demers, David J. Clifford, Jane E. Conti, Roberto Cross, Ronald W. Daverman, Robert J. Deming, Robert W. Cline, Edward T. Cook, David Edwin Crossley, Sterling G. Davey, Brian A. Demko, Stephen Clivio, Andrea Giovanni Cook, Joseph M. Crossman, Howard L. David, Daniel Demmel, James W. Clover, William J., Jr. Cook, Larry T. Crothers, Derrick S. F. David, Karl DeMoss, Dinah L. Cobb, J. Daniel Cook, Thurlow A. Crow, Edwin L. David, Nancy A. Denbow, Carl H. Cobham, Ian P. Cooke, Kenneth L. Crow, John Anthony Davidoff, Giuliana P. Dencker, Nils Cochran, Douglas Cooke, Roger Lee Crowell, Richard H. Davidson, Kenneth R. Denny, Diane L. Cochran, James A. Cooley, Clifford R. Crowley, James M. Davies, Morton J. Denny, William F., II Cochran, Tim D. Coolsaet, Arnold A. Crumb, Cyndi A. Davis, Chandler Dent, Elliod Cockburn, Julio Bernardo Coonen, Jerome T. Cruse, Allan B. Davis, James F. D'Entremont, Edward Joseph Cockburn, Sally Patricia Cooper, E. D. Cruthirds, John E. Davis, Linda M. Deodhar, Vinay Vithal Coen, Salvatore Cooper, Hilda L. Cude, Joe E. Davis, Martin D. Deprima, Charles R. Coffee, Jane P. Cooper, J. B. Cudia, Dennis F. Davis, Richard L. Depue, Bill Earl Coffey, John Copeland, Arthur H., Jr. Culler, Marc Davis, Ruth M. Derdzinski, Andrzej J. Cogburn, Robert F. Copeland, Terry S. Cunkle, Charles H. Davis, Stephen F. Deriggi, Dennis F. Cogdell, James Wesley Coppin, Charles A. Cunningham, A. B. Davis, William J. Deruaz, Marcel A. F. Cohen, Aljeh M. Coppotelli, Fred Cunningham, Frederic, Jr. Davison, Walter F. Derwent, John E. Cohen, Daniel I. A. Coram, Donald S. Cunningham, Joel L. Davitt, Harold H. Desarmenien, Jacques Cohen, Elaine Corazza, Paul J. Cunningham, Richard J. Dawes, A.M. DeShalit, Ehud Cohen, Frederick R. Corbacho, Eusebio T. Cuntz, Joachim Dawson, Donald A. Desmond, James Eddy Cohen, Leon W. Cordero, Luis A. Cuoco, Albert A. Day, B. J. Desoer, Charles A. Cohen, Marshall M. Cordes, Craig M. Curran, Peter M. Day, Mahlon M. Desolneux-Moulis, Nicole Cohen, Martin J. Cordes, Heinz 0. Currier, Albert W. Day, Martin V. Dethlefsen, Douglas G. Cohen, Marvin N. Corduneanu, Constantin Currier, Robert J. Day, Wayne Detlefsen, Michael E. Cohen, Moses E. Cornelius, Thomas A. Curry, John D. Daybell, Dorothy A. Detofoli, Rino Cohen, Robert A. Cornell, Gary Curtis, Charles W. Dayton, Barry H. DeTurck, Dennis Cohn, Donald L. Correa, Nelson L. Curtis, Doug W. de Branges, Louis Deumens, Erik Cohn, Leslie Corrigan, Thomas Carney Curtis, Herbert J. De Bruin, Marcel G. Deutsch, Emeric Cohn, Stephen Edward Corsi Tani, Gabriella Curtis, Morton L. de Caen, D. Devaney, Robert L. Cohn, William S. Cosner, George C. Curtis, Philip C., Jr. De Canniere, Jean Deveney, James K. Coifman, Ronald R. Costa, David G. Curto, Raul E. De Figueiredo, Djairo G. Devinatz, Ethan S. Colbourn, Charles Joseph Costenoble, Steven R. Cushing, Jim M. De Iongh, Johan J. DeVore, Ronald A. Cole, Charles A. Cote, Louis J. Cusick, David A. De Jong, Frederik J. Dewallens, Jean H. Cole, Herman D. Cotter, Christopher S. Cuthill, Elizabeth H. De Koning, Jan Dhooghe, Paul F. J. Cole, Nancy Cotton, Robert M. Cuttle, Yvonne H. De La Penha, Guilherme M. Di Maio, Giuseppe Cole, Paul Dana Couch, W. Eugene Cwikel, Michael De La Rica, Mariano Di Paola, Jane W. Cole, Randal H. Coulter, Malcolm A. Daboussi, Hedi Hermida Diamond, Jack Coleman, A. John Countryman, William Mark Dabrowski, Romuald De La Sen, Manuel Dias Da Silva, J. A. Coleman, Courtney S. Covington, Judith L. Dacorso Netto, Cesar De Lange, Jan Dibner, Steve Coleman, Robert Cowan, Jack D. Dade, Everett C. De Marr, Ralph E. Dickenstein, Alicia M. Coleman, William Cowan, John C., III Dadok, Jiri De Meersman, Roger Dickerson, Charles E. Collier, Jarrell D. Cowan, Richard E. Dahl, Erlend De Rezende, Ketty Diederich, Klas Collier, William G., Jr. Cowens, J. Wayne Dahlin, Rolf B. De Snoo, HendrikS. V. Diener, Karl-Heinz Collins, Carol E. Cox, Gregory V. Dai, Taen-Yu De Villiers, Raoul A. Diestel, Joseph Collins, Dennis G. Cox, Raymond H. Daigle, Daniel De Vries, Hendrik Dietz, David Collins, Donald J. Cox, S. H., Jr. Dale, Knut T. Dean, Carolyn A. Dieudonne, Jean A. Collins, Karen L. Cox, Stephen E.· D'Alotto, Louis A. Dean, James R. Dijksma, Aalt Collinsworth, John Grady Coxson; Pamela G. Daly, John F. Dean, Richard A. Dilao, Rui Colonius, Fritz Coyne, Steven Thomas Daly, John T. DeBlassie, Richard Dante Dilcher, Karl H. Colquitt, Landon A. Cozzens, Margaret Barry D'Ambrosio, Ubiratan Debrunner, Hans E. Dillon, John F. Colson, Henry D. Crabtree, J. B. Damiano, David B. Deckert, Kenneth L. Dillon, RichardT. Comegys, Esther Craft, David L. Damkohler, Wilhelm L. Deeter, Charles R. Dimitracopoulos, Comellas, Francese Craft, George A. Damon, James Norman DeFacio, Brian Constantinos Comenetz, Daniel Craggs, Robert F. Dana, Martin P. Defrancesco, Henry F. Dimsdale, Bernard Comez, Dogan Craig, Jessica Marguerite Dancis, Jerome DeGroot, Jan Dinneen, Gerald P. Comfort, Edwin H. Crane, George E. Daney, Charles G. Dehesa, Jesus Sanchez Dinwoodie, Ian Comfort, Steven H. Crane, Jane P. Dangello, Frank Ralph Deift, Percy Alec Dion, Gloria S. Comfort, W. Wistar Crane, Louis D'Angelo, John P. Deiter, Daphne D. Dionne, Benoit Compton, Kevin J. Crank, Keith N. Daniel, Klaus H. Dekker, Jacob C. E. Dionne, Philippe A. Conlon, Joseph G. Cranston, Michael Craig Daniels, M. Peg M. Dekleine, Herbert A. DiPerna, Ronald J. Conlon, Lawrence W. Crapo, Henry H. Danset, Renaud DeKoninck, Jean Marie Dipper, Richard T.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1023 AMS Centennial Research Fellowship Fund

Divis-Poracka, Zita M. Dudley, Underwood Eigen, Stanley J. Ervynck, Gontran J. 0. H. Fellers, Norman V., Jr. Dixon, Crist D. Dudziak, James J. Eilenberg, Samuel Eschenbach, Dieter Fellman, Johan Dixon, Martyn Russell Duff, George F. D. Eirola, Timo J. Escobales, Richard H., Jr. Fellows, Michael Ralph Dobbins, Robert R. Duffus, Dwight A. Eiseman, Peter R. Eskew, Thomas Eugene Fenchel, Werner do Carmo, Manfredo P. Duggal, Krishan Lal Eisenberg, Robert S. Eskin, Gregory Fendrich, John W. Dodds, Peter G. Duke, William D. Eisenbud, David Eslinger, Robert C. Fenley, John C. Dodds, Theresa K. Dukelow, James S., Jr. Eisenstadt, B. J. Esparza, Sergio 0. Fenley, Sergio Roberto Dodson, Christopher T. J. Dulin, Bill J. Ekedahl, Torsten Essen, Matts R. Fennell, James G. Dodziuk, Jozef Duma, Andrei Ekeland, lvar Esser, Anne-Marie Simon Fenske, Christian Doerfler, Willibald Dumas, H. Scott Eklund, Hans G. Estrada, Luis Feofiloff, Paulo Doering, Boro Dunbar, William Dart, Jr. Ekman, Kenneth E. Etgen, Garret J. Ferguson, David J. Doerr, Thomas A. Duncan, John Eldridge, Klaus E. Etheridge, William L. Ferguson, Helaman Dokken, Douglas Paul Duncan, Richard D. Elhadad, Jimmy H. Evans, Martin J. Fernando, Feanandez Dolan, James M. Dunstan, Allyn W. EI-Hodiri, Mohamed Evens, Leonard Rodriguez Dolbeault, Pierre E. Duntsch, Ivo Elia, Michele Everitt, William N. Fernando, Thotage Sarath Dold, Albrecht E. Dunwoody, Martin John Eliahou, Shalom Every, Jennifer B. Ferrar, Joseph Dolgachev, Igor Dupre, Maurice J. Elias, Uri Evyatar, Asriel Ferrari, Carlo Dollard, John D. Durand, Loyal Eliason, Stanley B. Ewbank, Edye Marie Ferreira, Andrew P. Domar, Yngve Durbin, John R. El-Kareh, Ardith W. Ewing, Richard E. Ferri, Massimo Dombroski, Michael Duren, Peter L. Ellard, Cecil Andrew Ewy, Daniel J. Ferris, Ian M. Dombrowski, Joanne M. Duren, William L., Jr. Elliger, Sigurd Eyles, Joseph W. Ferro, Ruggero Domic, Antun Durfee, Alan H. Ellingham, Mark N. Fabec, Raymond C. Ferry, Steven C. Dominguez, J. M. Durfee, William H. Elliott, David Faber, Richard L. Feser, Victor G. Dominijanni, Roberto Durkin, Marilyn B. Elliott, David L. Faber, Vance Feuerbacher, Gary A. Domokos, Gabor Durst, Lincoln K. Elliott, George A. Fabes, Eugene B. Feyel, Denis F. Donaly, James P. Durumeric, Oguz C. Elliott, Herbert E., Jr. Fabrega, Enfedaque Albert Field, Michael J. Dong, Rui-Tao Duskin, John W., Jr. Elliott, Joanne Fack, T. Fieldhouse, D. Donoho, David Leigh Dutt, J. E. Elliott, Sheldon E. Fadell, Albert G. Fife, James H. Doob, Joseph L. Dutta, Sankar Prasad Ellis, Alan J. Faierman, Melvin Figueiredo, Ana Maria Doppel, Karl Dutton, Charles Everett Ellis, James W., Jr. Fair, Wyman G. Filippenko, Ivan Doran, Robert S. Dwork, Bernard M. Ellis, Richard S. Fairgrieve, Thomas F. Finch, David V. Dorfmeister, Josef F. Dydak, Jerzy Ellis, Robert Falbo, Oement E. Finch, John V. Dorman, David R. Dye, Judy E. Ellis, Sharon Theleman Falkner, Neil F. Finch, Steven Richard Dorroh, James R. Dym, Harry Ellison, Robert J. Fall, Thomas C. Fine, Benjamin Dorsett, Charles I. Dynin, Alexander Eloe, Paul W. Faltings, Kai Fine, Jonathan Dorsey, Ely A. Dyson, Verena H. El-Shobaky, Entisarat Fan, Ky Fingerman, Joel Dorsey, Peter N. Eachus, J. J. Elson, Constance McMillan Fang, Yi Finkelstein, Leib Dou, Alberto M. Eakin, Paul M., Jr. Elton, John H. Faran, James J. Finkelstein, Louis D. Dougherty, John P. Ealy, Clifton E., Jr. Elzanowski, Marek Farlow, Stanley J. Finlayson, Henry C. Dougherty, Randall Eames, William P. Emamirad, Hassan Farmer, William Michael Finol, Carlos E. Dougherty, Robert P. Earle, Clifford J., Jr. Emch, Gerard G. Farnsteiner, Rolf Fischer, Arthur E. Douglas, Craig C. Eaton, Timothy R. Emery, Carol J. Reinhard Farrell, Roger H. Fischer, Bernd Douglas, Jim, Jr. Ebel, David J. Emmanuele, Giovanni Farrer, Brian H. Fischer, David R. Douglas, Ronald G. Eberlein, Patrick Barry Emmer, Michele Farris, Frank A. Fischer, H. Dovermann, Karl Heinz Eberly, David H. Endler, Otto Farris, Mark K. Fischer, Hans R. Dow, Alan Ebiefung, Aniekan Asukwo Endo, Youichi Fass, Arnold L. Fishback, William T. Dowd, Robert E. Ecklund, Earl F., Jr. Endsley, Neil H. Faulkner, Frank D. Fisher, Stephen D. Dowds, Richard E. Eckmann, Beno Engeler, Erwin Faulkner, Gary Doyle Fishman, Louis Dowers, J. F. Eckmann, J.P. Engess,r, Hermann Faurot, Donald P. Fisk, Steve Dowling, Michael L. Edelson, Allan L. Engl, Heinz W. Faurre, Pierre L. Fitzgerald, Eugenia E. Dowling, Thomas A. Edelstein-Keshet, Leah Engle, Jessie Ann Fearn, Dean H. Fitz-Gerald, Gary F. Downey, Rodney G. Edet, Silas B. Engler, Hans P. Federbush, Paul Fitzgerald, Robert W. Doyen, Jean Edgar, Hugh Maxwell Wallace Engquist, Bjorn Feehan, Paul Matthew Niall Fitzpatrick, Ben G. Drager, Lance D. Edidin, Dan S. Engstrom, Philip G. Feeman, George F. Fitzpatrick, Ben, Jr. Dragt, Alex J. Edmonds, Allan L. Enomoto, Kazuyuki Feeman, Timothy G. Fitzsimmons, Patrick J. Dray, Tevian Edmonds, George F. Enomoto, Masatoshi Feferman, Solomon Fixman, Uri Drazin, Michael P. Edmondson, Don E. Enqvist, Arne Fegan, H. D. Flaherty, Francis J. Drbohlav, Karel Edrei, Albert Ensey, Ronald J. Fegreus, John F. Flahive, Mary Elizabeth Drelles, Paul G. Edwards, Barbara E. Epp, Helmut P. Feibleman, James K. Flanders, Harley Dresher, Melvin Edwards, C. M. Epp, Susanna S. Feichtinger, Hans Georg Flanigan, Francis J. Drew, Gerald C. Edwards, Charles H., Jr. Epstein, Benjamin Feigenbaum, Ruth I. Flapan, Erica L. Dribin, Daniel M. Edwards, David Albert Epstein, Bernard Feighn, Mark E. Flath, Dan Driscoll, Mark J. Edwards, Kathleen R. Erban, Daniel Fein, Burton I. Fleckinger, J. Driver, Bruce K. Edwards, Robert D. Erbe, Lynn H. Feingold, Alex J. Fleissner, William G. Drucker, Thomas L. Edwards, Stephen A. Erber, Friedrich M. Feingold, David Fleming, Charles G. Du Cloux, Fokko Edwards, Wallace B., Jr. Erbland, John P. Feinsilver, Philip Fleming, Richard J. Dubisch, Roy Eells, James Erdelyi, Ivan N. Feit, Walter Fleming, Wendell H. Dubois, Joseph J. Eenigenburg, Paul J. Erdman, John M. Feldman, Arnold D. Flensted-Jensen, Mogens Dubreil, Paul J. Ehlers, F. Edward Ericksen, Wilhelm S. Feldman, Chester Floer, Andreas Dubson, Alberto S. Ehrlich, Karel Erickson, Roger P. Feldman, Edgar A. Florence, John B. Due, Duong Minh Ehrlich, Paul Ewing Erkiletian, Dickran H., Jr. Feldman, Jacob Floret, Klaus Duchamp, Thomas E. Eiardi, Anthony J. Erlander, Sven Feldman, Marcus B. Flosser, H. 0. Duchon, Nicholas Eifrig, Bernd Erlandsson, Thomas Feldman, Norman Floyd, Denis R. Dudek, Zbigniew C. Eigel, Edwin G., Jr. Errico, Thomas G. Fell, James M. G. Floyd, Edwin E.

1024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Aynn, Lawrence E. Friend, Jay M. Garfinkle, Devra Gilmer, Gloria Gorenflo, Rudolf Aytzanis, E. Friesen, Donald K. Garland, Stephen J. Gilmer, Patrick M. Gorenstein, Daniel foehl, Henry Clay Frink, Orrin Gamer, Lynn E. Gilmour, Grant James Gorlich, Ernest Foglio Susana, f. L. de Fritsch, Rudolf Garnett, John B. Gilsdorf, Thomas E. Gormley, Michael J. Foley, Daniel M., Jr. Fritsche, Richard T. Garrett, James R. Gingrich, Ross B. Gorton, Robert Bruce Folk, Thomas f. Froese, Richard G. Garrett, John W. Gioia, Anthony A. Goss, Robert N. Folland, Gerald B. Frohardt, Daniel E. Garrison, Betty B. Girela, Daniel Goth, John A. Fong, Che-Kao Frohlich, Jurg M. Garsia, Adriano M. Glass, A. M. W. Goto, Hideo Fong, Kirby W. Frohliger, John A. Garst, Peter Glass, Michael S. Goto, Midori S. Fong, Paul Frohman, Charles D. Gartland, Eugene C., Jr. Glaz, Sarah Goto, Shiro Fontaine, Anne T. Fryer, S. Harvey Garzon, Maximiliano Glazebrook, James F. Gottlieb, Daniel H. Foran, James Fu, Lorraine Gasca, Mariano Gleason, Andrew M. Gould, Lawrence D. Ford, David Fu, Yen-Tzu Gaschutz, Wolfgang Glenn, Paul G. Gouvea, Fernando Quadros Ford, David A. Fuchs, Laszlo Gass, Michael D. Glimm, James G. Gover, Eugene H. ford, George P. Fuchs, Peter Rudolf Gass, Norbert Glover, David W. Ooze, Michel forelli, Frank Fuchs, Wolfgang H. Gatterdam, Ronald W. Glover, Henry H. Grabiner, Sandy Foreman, David L. Fuchssteiner, Benno Gatteschi, Luigi Gluck, Herman R. Gragg, Karen E. Forkeotes, Ann A. Fudenberg, David Melton Gauchman, Hillel Goblirsch, Richard P. Graham, C. Robin forman, John W. Fuglede, Bent Gaudefroy, Alain Godbole, Anant P. Grahaln, Ian Forman, Robin Fuglister, Frederick J. Gautschi, Walter Goddard, J. D. Graham, Timothy N. Forster, Otto F. Fuji-Hara, Ryoh Gebelt, Nicholas Watts Goddyn, Luis A. Graham, Victor W. Forsythe, Keith W. Fujimagari, Tetsuo Gehrig, Eugene J. Godfrey, Colin Grainger, Arthur D. Forte, Bruno Fujisaki, Rieko Gehring, Frederick W. Godin, Paul J. Gramsch, Bernhard Fossum, Robert M. Fujiwara, Daisuke Geissinger, Ladnor D. Goebel, K. Granado, Michael F. Foster, A. L. Fujiwara, Masahiko Gelbart, Abe Goethals, J. M. E. Grannell, Michael John Foster, Andrew Fukuda, Komei Gelbart, Stephen S. Golan, Jonathan S. Grant, David R. Foster, Lorraine L. T. Fukushima, Masatoshi Gelbaum, Bernard R. Golber, David L. Grantham, Stephen B. Fotiadis, Dimitrios I. Fukushima, Mitsuo Geller, Daryl Neil Gold, Steven Grasser, Horst-Siegfried Foulis, David J. Fuller, Kent R. Geman, Stuart Alan Goldberg, Richard R. Gratiaa, Marceline C. Foumeau, Rene Fuller, Leonard E. Genensky, Samuel M. Goldberg, Samuel Gratzer, George A. Foumelle, Thomas A. Fuller, Luther B. Genet, Jean Raymond Goldberg, Seymour Gravel, Pierre Fowler, Franklin H., Jr. Fuller, Mark E. Gentili, Graziano Golden, Kenneth M. Graver, Jack E. Fox, Jeffrey Stephen Fuller, William R. (Oregon) Geoghegan, Ross Goldfeld, Dorian Graves, Glenn W. Fox, John Michael Fuller, William R. (Indiana) George, John Clay Goldhaber, J. K. Graves, Larry K. Fox, Ralph Fulton, Curtis M. Georgiady, Michael S. Goldman, Jerry Graves, William H. Foxby, Hans-Bjom Fulton, William Georgiou, D. A. Goldman, William Mark Gravesen, Jens Fraga, Robert J. Fuqua, Jeffry B. Geramita, Anthony V. Goldstein, Gerald Gray, George A. Frame, Michael Lee Furutani, Kenro Geramita, Joan M. Goldstein, Jerome A. Gray, Jeremy John France!, Margaret Ann Futia, Carl A. Gerber, Marlies Goldstein, Larry J. Gray, John W. Francis, Bruce A. Fwu, Chihchy Geremia, Raymond S., Jr. Goldstein, Norman J. Gray, Robert E. Francis, Eugene A. Gaal, Lisl Novak Gerhard, J. Arthur Goldston, Daniel A. Gray, Samuel Hutchison Francisco, Gerson Gabay, Jacques Gerig, Stephen R. Goldstone, Richard J. Greaves, G. Frank, Ove Gabber, Ofer Gerlach, Eberhard G. P. Golightly, William L., Jr. Green, David, Jr. Franks, John M. Gabriel, Richard F. Germain, Paul Golitzin, George Green, James A. Fransen, Arne Gacs, Peter Gershenson, Hillel H. Gomez Pardo, Jose Luis Green, John W. Frantz, Henry W., III Gadella, Manuel Gersten, S. M. Gomez-Ruiz, Francisco Green, Mark L. Franzblau, Deborah S. Gadia, Shashi Kumar Gerstenhaber, Murray Gomi, Kunio Green, Scott Wesley Frappier, Clement Gage, Michael E. Gervais, Jean Jacques Gompf, Robert E. Green, Simon Frawley, William J. Gago, Felipe Gerver, Joseph L. Gondard-Cozette, Daniell J. Green, William L. Frazier, Michael W. Gaillard, Pierre Yves Gessaman, Margaret P. Gonzalez, Raul Emesto Greenbaun, Nicholas N. Frech, Bruce Gaines, Robert Earl Gessel, Ira M. Gonzalez-Guzman, Jorge Greenberg, Ralph Fredricksen, Harold M. Galbraith, A. S. Getu, Seyoum Gooch, B. Dean Greene, Peter H. Fredrickson, Elvy L. Gale, Carolyn Spencer Geue, Andrew S. Good, Irving John Greenfield, Gary R. Free, Norman S. Gallagher, Patrick X. Ghaffari, Abolghassem Goodearl, Kenneth R. Greenleaf, Allan T. Freed, Daniel S. Gallier, Jean H. Ghatage, Pratibha Goodman, A. W. Greenleaf, Frederick P. Freedman, Allen R. Galliher, Herbert P. Ghent, K. S. Goodman, Albert J. Greenspan, Bernard Freeland, MarkS. Galovich, Steve Ghisa, D. Goodman, Frederick M. Greenspan, Bernard D. Freeman, Edmund E. Gambi, Jose Maria Gil, Pedro Goodman, Jacob Eli Greenspan, Donald Freese, Ralph S. Gamelin, Theodore W. Gilbert, Freeman Goodman, James Thomas Gregory, David A. Freiberger, Walter F. Ganelius, Tord H. Gilbert, George Thomas Goodman, Jonathan B. Gregory, John Freimanis, Mary Louise Gani, Joseph M. Gilbert, Michael D. Goodman, Nicolas D. Gregory, Thomas Bradford Freire Nistal, J. L. Ganser, Carl C. Gilbert, Richard C. Goodman, Philip Greicar, Richard K. Freitag, Herta T. Ganster, Maximilian Gilchrist, Martin P. Goodman, Roe W. Greif, Anthony David Frey, Regina F. Gantner, Thomas E. Gildenhuys, Dion Goodman, Victor Greif, Stanley J. Freyd, Peter J. Gantos, Richard L. Gilewicz, Jacek Goodrich, Suzanne Dinga Greig-Smith, N. H. Frezza, Ruggero Garcia, Mariano, Jr. Gill, Tepper L. Goodrick, Richard E. Greiner, Peter C. Fridman, Burna L. Garcia-Loygorri, Cristobal Gillard, Roland D. Gopal, Mangalam R. Greller, Larry David Fried, David Garcia-Planas, Isabel Gillespie, James R., Jr. Gopalsamy, Kondalsamy Grenier, Douglas Friedberg, Stephen H. Gardiner, Christopher J. Gillespie, Robert G. Gordh, George R., Jr. Greville, Thomas N. E. Friedland, Shmuel Gardiner, Cyril F. Gilligan, Bruce C. Gordon, B. Brent Grey, Louis D. Friedman, Jane E. Gardner, Merritt P. Gillman, Daniel W. Gordon, Carolyn S. Griess, Robert Louis, Jr. Friedman, Joel Gardner, Robert A. Gillman, David S. Gordon, Michael W. Griesser, James W. Friedman, Merwyn M. Gardner, Robert B. Gilman, Jane P. Gordon, Robert Griffin, William Gerald Friedman, Sy D. Gardner, William Allen Gilman, Robert Gordon, Russell A. Griffing, Gary R.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1025 AMS Centennial Research Fellowship Fund

Griffiths, H. Brian Haddad, Labib S. Harker, Patrick T. Hecker, David A. Heuer, Gerald A. Griffiths, Phillip A. Haddix, George F. Harman, Jonathan W. Hedberg, Lars I. Heumos, Michael J. Griggs, T. S. Haddock, A. Glen Harmelin, Reuven Hedlund, Gustav A. Heuvers, Konrad John Grigis, Alain A. Hadid, Samir Bashir Harnish, Stephen Hefez, Abramo Hewett, Thomas James Grilliot, Thomas J. Haeberly, Jean-Pierre A. Harper, James D. Heijmans, Johannes G. Hewitt, P. R. Grimmer, Ronald C. Haemmerlin, Guenther Harper, Lawrence H. Heil, Christopher E. Heyer, Herbert K. Grimson, W. Eric L. Hafner, James L. Harrell, Evans M. Heiligman, Mark Isaac Hiai, Fumio Grinberg, Vladimir Hag, Per Harrington, Walter J. Heimes, Kenneth A. Hickernell, Fred J. Grinfeld, Michael Hagan, Robert M., Jr. Harris, Bruno Heineken, Hermann Hickey, Timothy J. Gripenberg, Gustaf Hagen, Walter Lowell Harris, David Carter Heinicke, Allan G. Hicks, Darrell Lee Gritzmann, Peter Hager, Anthony W. Harris, Gary A. Heinonen, Juha M. Hida, Takeyuki Grivel, Pierre P. Hagis, Peter, Jr. Harris, Joe Heins, Maurice H. Higgins, John A. Grobler, Jacobus J. Hagler, James N. Harris, John C. Heintze, Ernst Higgins, Philip J. Grognard, Rene J. M. Hahn, Gena Harris, Michael H. Hejtmanek, Johann Hans Higuchi, Yasunari Gromoll, Detlef Hahn, Kyong T. Harris, R. Thomas, Jr. Heldermann, Norbert Hilbert, Stephen R. Grone, Bob Hahn, Marjorie G. Harris, Steven Guy Helgason, Sigurdur Hildebrandt, Theodore W. Gropp, William D. Hahn, Neale M. Harris, Theodore E. Hellberg, H. Stefan Hilgers, John W. Gross, Fletcher I. Hahn, Susan G. Harrison, Carol Nicoline Hellen, Gregory R. Hill, David G. B. Gross, Herbert Hahne, Gerhard E. Harrison, Monica L. Heller, Alex Hill, James Gross, Leonard Haight, John A. Harrold, Orville G., Jr. Heller, Dorothy M. Hill, Joe B. Gross, Louis J. Haiman, Mark Harrop, Fred F. Heller, Robert Hill, Ned W., Jr. Grosser, Siegfried K. Haimo, Deborah Tepper Hart, Klaas Pieter Hellerman, Leo Hill, Robert N. Grossman, Marvin W. Hain, Richard Martin Hart, Susan 0. Hellerstein, Simon Hill, Shirley A. Grossman, Robert Haji-Ghassemi, Kamran Hartig, Donald Helmer, Gary A. Hill, Theodore P. Grosswald, Emil Hakim, Monique Hartl, Johann Helton, Burrell W. Hillman, Jonathan Arthur Grove, Edward A. Halberstam, Heini Hartman, James L. Helton, Frances J. Hilpipre, Douglas H. Grove, John W. Halchin, Judy D. Hartmanis, Juris Helton, J. William Hilt, Sandra N. Grove, Larry C. Hale, Douglas F. Hartshorne, Robin Helwig, Karl-Heinz Hilton, Peter J. Grow, David E. Hales, Alfred W. Haruki, Hiroshi Hemasinha, Rohan Himonas, Alexandrou A. von Grudzinski, Olaf Hales, R. Stanton, Jr. Haruki, Shigeru Hemenway, ClareT. Hinder, Rainer Gruenberg, Karl W. Hall, Glen Richard Harvey, F. Reese Hemmeter, Joseph S. Hindman, Neil B. Gruenhage, Gary F. Hall, Mark Edwin Harvey, George G. Hempel, John P. Hingston, Nancy Gualtierotti, A. F. Hall, Marshall, Jr. Hasenjaeger, Gisbert F. R. Hemstead, Robert J. Hinkkanen, Aimo Guenard, Francois Halpern, Herbert Haskell, Deirdre Henderson, David W. Hinman, Peter G. Guerin, Esther E. Hamana, Masamichi Haskell, Peter E. Henderson, Francis McVey Hinohara, Yukitoshi Guilbault, Craig R. Hamill, William A. Haskins, Scott Marshall Henderson, James P. Hinson, Edward K. Guillemin, Victor W. Hamilton, Howard B. Haslach, Henry W., Jr. Henkin, Leon A. Hintzman, William R. Guillen, Francisco Hamilton, Phillip M. Hass, Joel Hennequin, Paul-Louis Hirano, Tetsutaro Guillot, J. C. Hammer, Carl Hasselbach, Eric R. Henniart, Guy M. Hironaka, Heisuke Gulbenkian, Garabed Hammer, F. David Hassett, Judy J. Henniger, J.P. Hirsch, Michael D. Gulden, S. L. Hammer, Peter L. Hassinger, Bill, Jr. Henry, Myron S. Hirsch, Morris W. Gulick, Denny Hammond, Michael Eugene Hastad, Johan Hensel, Robin Ann Morgan Hirschfelder, Elizabeth S. Gulliver, Robert D., II Hammond, William F. Hastings, Harold M. Henstock, Ralph Hirzebruch, F. E. P. Gumas, Lora Hamrick, Gary C. Hastings, Stuart P. Hepp, Klaus Hirzebruch, Ulrich Gundy, Richard F. Hamstrom, Mary-Elizabeth Hasumi, Morisuke Herbst, Albert F. Hisao, Kato Gunn, Charles G. Han, Chong-Kyu Hatcher, Rhonda L. Herbst, Ben Hitotsuyanagi, Nobuo Gunnarsson, Thomas E. W. Han, Jongsook Hattori, Akio Herbst, Robert T. Hlavac, Paul P. Gunter, Elsa L. Hanche-Oisen, Harald Hausknecht, Adam O'Neill Herfort, Wolfgang Norbert Ho, Chi-Fai Guo, Kanghui Handel, David Hausmann, Jean-Claude Herman, Michael Ho, Chung-Wu Guo, Lih-Jia Handler, Ivan Arthur Haussmann, Werner Herman, Richard H. Hoare, A. Howard M. Gupta, Dharma P. Handwerker, A. M. Haver, William E. Hermann, Margaret A. Hobbs, Arthur M. Gupta, Rajiv Hanes, Kit Havermann, Hans Hermann, Robert Hochschild, Gerhard P. Gupta, S. N. Hanges, Nicholas William Haviland, E. K. Hernandez, Carlos Hochstadt, Harry Gurarie, David E. Hanisch, Herman Hawkins, William Grant Hernandez, Diego B. Hochster, Melvin Gurganus, Kenneth R. Hanlon, Philip J. Hay, George E. Hernandez, Eugenio Hochwald, Scott H. Gurney, David Robert Hanna, Charles C. Hayashi, Elmer K. Hernandez, Horacio Hockett, Kevin G. Gurney, Margaret Hanna, J. Ray Hayashi, Kazumichi Hernandez, Rafael Hodel, Richard E. Gustafson, Richard F. Hannan, Edward J. Hayashi, Mikihiro Hernandez Ruiperez, Daniel Hodges, Lucille M. Gustafson, Sven-Ake Hano, Jun-Ichi Haydon, Richard G. Herod, James V. Hodges, Wilfrid A. Gutek, Andrzej S. Hansen, R. 0. Hayek, Nacere Herrera, Ismael Hodgson, Jonathan P. E. Guterman, Martin M. Hansen, Sten Hayes, David F. Herrera, R. B. Hoefer, Edwin T. Gutknecht, Martin H. Hansen, Vagn Lundsgaard Hayes, David R. Herrero, Domingo A. Hoeffding, Wassily Ha, Chung-Wei Hansson, Kurt A. Hayne, Roger M. Herreshoff, James B. Hofer, Robert D. Ha, Kwang Chul Harada, Shigeharu Haynes, Nola A. Herring, Susan Kay Hoffman, Jerome William Haag, Vincent H. Haralampidou, M. Marina Hays, M. M. R. Herrmann, Joseph M. Hoffman, Kenneth R. Haario, Heikki Harborth, Heiko Head, Richard R. Herschorn, Michael Hoffman, Michael E. Habegger, Nathan B. Harbourne, Brian Head, Thomas J. Hersh, Reuben Hoffman, Michael J. Habetha, Klaus Hardell, William J. Healy, Dennis M. Herve, Michel Hoffman, William C. Habetler, George J. Hardt, Robert M. Heath, David C. Herwitz, Paul S. Hoffmann-Jorgensen, Jorgen Habibullah, G. M. Hardy, Dare! W. Heath, Lenwood S. Herzog, Emil R. Hofmann, Charles E., III Hackborn, William W. Hardy, John T., Jr. Heath, Robert W. Hess, Jacob J. Hofmann, Karl H. Hackenbroch, Wolfgang Harinath, K. S., II Heath-Brown, Roger Hestenes, Magnus R. Hofmann, Steve C. Hackler, Donald B. Harine, Katherine Jane Hechler, Stephen H. Hetherington, Richard G. Hoft, Hartmut Haddad, Hadi M. Haring-Smith, Robert H. Hecht, Henryk Hetzer, Georg Hoft, Margret H.

1026 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Hogan, Guy T. Hratz, Joseph A. lgodt, Paul G. Jaffe, Martha A. Johnstone, lain M. Hogbe-Nlend, Henri Hrbacek, Karel Iha, Franklin T. Jaffe, William J. Johnstone, P. T. Hogesteeger, F. W. Hsia, J. S. Iino, Riichi Jager, H. Jonah, David W. Hognas, Goran Hsieh, Simon C. lio, Tutomu Jagy, William C. Jones, Alfredo R. Hoksbergen, Rob R. A. M. Hsieh, Wan-Chen Ijjasz Vasquez, Ede J. Jahren, Bjorn Jones, Carl Hilton Holberton, John V. Hsu, Guan-Hsong Ikebe, Teruo Jain, Naresh C. Jones, Christopher K. R. T. Holden, Peter J. Hu,S. T. Ikeda, Kazumasa Jakel, David W. Jones, Dallas J., Jr. Holder, L. I. Hua, Yang Illan Gomez, Inmaculada James, Robert C. Jones, Eleanor Green Holenweg, Werner Huang, Xun-Cheng Illman, Soren A. Janicki, Ryszard Jones, F. Burton Holladay, Kenneth W. Huard, James G. Ilmanen, Tom Jankins, Mark Jones, G. A. Holland, Daniel L. Huber, Alfred 0. Im Hof, Hans Christoph Jans, James P. Jones, Gordon C. Holland, Finbarr Huber, Martin Imai, Hideo Jansen, Willem G. Jones, Harold T. Holland, SamuelS., Jr. Hudson, Anne L. Imam, Ibrahim N. Janson, Svante Jones, Jeffrey P. Holland, W. Charles Hudson, Steven M. Imayoshi, Yoichi Jantosciak, James S. Jones, Kathryn A. Holley, Frieda K. Huebschmann, Johannes Inaba, Hiroshi Janusz, Gerald J. Jones, Kim Holley, Richard Andrew Huerta, Carlos Cuevas Indlekofer, K. H. Janwa, Heera Lal Jones, Peter Wilcox Hollister, Lindsay Huet, Denise Infante, Ettore Jara Martinez, Pascual Jones, Phillip S. Holman, Wayne James, III Huff, Lawrence R. Infantozzi, Carlos Alberto Jarchow, Hans Jones, Theodore W. Holmann, Harald R. A. Huff, Melvyn E. Ingraham, Edward C. Jardine, John Frederick Jones, Vaughan F. R. Holmes, Charles S. Huff, W. N. Ingram, Nancy Jane Jarratt, Mary K. Jones, William R. Holmes, Dave R. Hufford, George A. Innami, Nobuhiro Jarvi, Pentti K. Jongen, Hubertus Th Holmes, Ernest J. Hugger, Jens Innes, Joan Jarvis, Trevor M. Jongmans, Francois Holmes, James Samuel, Jr. Hughart, Stanley P. Insall, Eugene M., Jr. Jea, Hae-Pyng Jonker, Leo B. Holmes, John P. Hughes, Anne Ioakimidis, N. I. Jea, Kang C. Jonsdottir, Kristin Halla Holmes, Philip John Hughes, Anthony Ionescu, Dan Jean, Michel Jonsson, Bjami Holmes, Randall Reed Hughes, C. Bruce Iorio, Rafael J., Jr. Jeanquartier, Pierre Jordan, G. Samuel Holmes, Richard B. Hughes, Carl C. Ipina, Lynne Kamstra Jeffrey, Alan Jordan, Kirk E. Holton, Derek A. Hughes, Cephas D. Isaacs, Godfrey L. Jekel, Solomon M. Jorge, Deborah A. Holvoet, Roger Hughes, Joseph A. Isaacs, I. Martin Jenkins, Emerson D. Jorgensen, Colleen K. Holzman, Robert E. Hughes, Kimball Isaacson, Eli Jenkins, Joe Wiley Jorgensen, P. E. T. Homer, Roger H. Hughes, Rhonda J. Isaacson, Eugene Jensen, Arne Jory, Virginia V. Homer, William D. Hughes, Richard P., Jr. Isaak, Garth T. Jenson, Richard Alan Joseph, George J. Honda, Kin-Ya Hughes, Ruth L. Isbell, John R. Jerison, David Josephson, William Allen Hong, Sungpyo Huh, Won Iseki, Kiyoshi Jerome, Carlos Joyner, David W. Hongo, Eishi Huibregtse, Mark E. Ishiguro, Kenshi Jerome, Joseph W. Juberg, Richard K. Honig, Chaim Samuel Huilgol, Raja R. Ishihara, Tadashige Jeroslow, R. G. Jubran, Isa S. Hoo, Cheong Seng Hulbert, Guy William Ishii, Jyun Jerrard, Richard P. Jun, Kit Woung Hoobler, Raymond T. Hulkower, Neal D. Ishikawa, Hirofumi Jessen, Borge del Junco, Andres Hood, Rodney T. Hull, Richard B. lshimoto, Hiroyasu Jha, V. Jungck, Gerald F. Hooker, John W. Hulsbergen, Wilfred W. J. Iskander, Awad A. Ji, Ronghui Junghenn, Hugo D. Hoover, D. N. Hulthen, Lamek Ito, Kiyosi Jian, Alan S. Jungnickel, Dieter Horadam, Kathryn J. Hume, Leigh R. Ito, Noboru Jimenez, Jose Alfredo Justice, James H. Hom, Alfred Hummel, James A. Ito, Ryuichi Jockusch, Carl G., Jr. Kaashoek, Marinus A. Hom, Jean MacGregor Hummel, Tamara J. Ives, Robert T. John, David J. Kabak, Bertram S. Home, J. G., Jr. Humphreys, James E. Iwamiya, Toshiyuki John, Fritz Kadlecek, David Homer, Mary Lynn Humphreys, John F. Iwamoto, Lynn Haremi Johnen, Hans Kahan, William M. Homer, T. S. Huneke, Craig L. Iwanaga, Yasuo Johnsen, Ronald L. Kahane, Charles S. Hornung, Ulrich Hung, Henry Hin-Lai Iwaniec, Tadeusz Johnson, Charles N. Kahane, Jean-Pierre Horowitz, Maurice Hunt, Richard A. Iwasaki, F. Shigeo Johnson, D. Randolph, Jr. Kahn, Donald W. Horrigan, Timothy J. Hunte, Beryl E. Iwasawa, Kenkichi Johnson, Daniel P. Kahn, Peter J. Horrobin, Elizabeth Ann Hunter, Colin B. Iwata, Koichi Johnson, David H. Kaiser, Gerald Horrocks, Geoffrey Hunter, Maxwell Norman Ize, Jorge A., II Johnson, David L. Kajiwara, Joji Horsfield, Christopher H. Hunter, Norma Gerkens Izmirlian, Grant, Jr. Johnson, Diane M. Kakihara, Yuichiro Horsley, Anthony Hunzeker, Hubert L. Izuchi, Keiji Johnson, Donald G. Kakiichi, Yoshiaki Horst, Ernst S. Hurder, Steven E. Izumi, Shin-Ichi Johnson, Eugene W. Kaleva, Osmo Horvath, Charles Daniel Hurley, Charles Rory Izumiya, Shyuichi Johnson, Gene Douglas Kalinde, Albert Kasangu Horvath, John M. Hurley, Donal J. Jablow, Eric Robert Johnson, Gerald W. Kalisch, Gerhard K. Hosack, John M. Hurley, James F. Jackins, Harvey Johnson, Guy, Jr. Kalish, Diana Hosseini, Naser Hurst, C. A. Jackson, Kenneth R. Johnson, James E. Kall, Peter Hoste, Jim E. Hurvicb, Oifford'M. Jackson, Stanley B. Johnson, Jon L. Kalliongis, John E. Hotta, Ryoshi Hurwitz, Solomon Jackson, William D. Johnson, Joseph F. Kallman, Ralph A. Hotze!, Eckehart Husain, Taqdir Jaco, William H. Johnson, Kenneth D. Kallstrom, Anders Houston, Raymond S. Husseiny, Daad Jacob, Henry G. Johnson, Kenneth Walter Kalton, N. J. Houze!, Christian Hustin, Deborah L. Jacob, Niels Johnson, Peter A. Kaman, Charles H. Howard, Fredric T. Hutches, David J. Jacob, William Burkley Johnson, R. Warren Kamejima, Kohji Howard, Paul E. Hutchinson, George A. Jacobs, Felix J. Johnson, Robert L. Kamin, Shoshana Howard, William A. Hutton, Edward L. Jacobs, Glenn K. Johnson, Robert Shepard Kaminker, Jerry Howe, Roger E. Hwang, Jang C. Jacobs, Jonathan M. Johnson, Rodney W. Kamishima, Yoshinobu Howell, Kenneth Hwang, John Jew-Chen Jacobs, Marc Q. Johnson, Roy A. Kammler, David W. Howes, Frederick A. Hyllengren, Anders R. Jacobsen, Martin Johnson, Russell A. Kamowitz, Herbert M. Howie, John M. Hyvonen, Jaakko P. Jacobson, Clas A. Johnson, Steven Darrel Kanda, Mamoru Hoyle, Hughes B., III Iarrobino, Anthony A., Jr. Jacobson, Florence D. Johnson, William B. Kandaswamy, P. Hoyle, Jonathan William Igarashi, Akira Jacobson, Nathan Johnston, Bernard L. Kaneda, Masaharu Hoyt, Kenneth Igari, Satoru Jaffa, Robert E. Johnston, Ernest R. Kaneko, Akira

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1027 AMS Centennial Research Fellowship Fund

Kanenobu, Taizo Kelly, G. Maxwell King, Larry Koestner, Alfred Kreith, Kurt Kang, Ming-Chang Kelly, John B. King, Randall R. Koethe, Gottfried M. Krener, Arthur J. Kangel, Jay Kelly, John F. King, Stephen C. Koether, Robb T. Kreyszig, Erwin 0. Kaniuth, Eberhard Kelly, Paul J. King, William Robert, Jr. Kogelman, Stanley Krieger, Henry A. Kanjin, Yuichi Kelly, Theresa D. Kingston, Albert W. Koh, Jee Heub Krieger, Michael M. Kann, Edgar D. Kemp, Daniel Campbell Kinukawa, Masakiti Koh, Kwangil Kriegsman, Helen F. Kanner, Richard A. Kempachiro, Ohashi Kioustelidis, John Kohler, Peter Krinik, Alan C. Kanold, Hans-Joachim Kemperman, J. H. B. Kipps, Thomas C. Kohn, Joseph J. Krishna, Bat Kaper, Hans G. Kendall, Wilfrid Stephen Kirby, Robion C. Kohn, Robert V. Kristiansen, GundorPh K. Kaplan, Aroldo Kengmana, Thongchai Kirch, Allan M. Koivisto, Pertti T. Krogstad, Harald E. Kaplan, Stanley Kenig, Carlos E. Kirchbraber, Urs Kojima, Sadayoshi Kronstein, Karl M. Kaplan, Wilfred Kennedy, Edward S. Kirchherr, Walter W. Kokoska, Stephen M. Krug, Don K. Kappe, Wolfgang P. Kenner, Paul M. Kirk, Joe E., Jr. Kolaitis, Phocion G. Krumpholz, Gary R. Karat, Frank C., Jr. Kennison, John F. Kirk, William A. Kolbig, Kurt Siegfried Krystock, Robert L. Karamatsu, Yoshikazu Kennison, Lawrence S. Kirkland, Steve J. Kolchin, E. R. Krzyz, Jan G. Karbaliotis, Peter Kenoyer, David B. Kirley, Paul 0. Kolesar, R. J. Kubert, Daniel S. Karch, Paul T. Kenschaft, Patricia Oark Kirmser, P. G. Kolitsch, Louis Worthy Kubo, Fumio Karel, Martin Lewis Kent, George A. Kirsch, Werner Kolk, Johan A. C. Kubo, Tadao Karle, Jerome Kent, Joseph F. Kirwan, Frances C. Kolodner, Ignace I. Kucera, Thomas Glen Karlin, Samuel Kent, Steven L. Kiselman, Christer 0. Komanska, Henryka Kudla, Stephen S. Karlovitz, Les A. Keown, Nancy A. Kiser, Terry Lee Krystyna Kudo, Tatsuji Karlsson, Per W. Kerman, R. A. Kishi, Masanori Komatsu, Hikosaburo Kuebler, Roy R., Jr. Karnik, Subhash Moreshwar Kernast, John C. Kishore, Masao Komatu, Yusaku Kuenzi, Norbert J. KarP, Richard M. Kersting, Gary Steven Kist, Joseph E. Komiya, Yurin Kugler, Lawrence D. KarPilovsky, G. Kerzman, Norberto L. Kister, Jane Komm, Horace Kuhlmann, Norbert KarPinski, Marek Kesten, Harry Kita, Hiroo Komura, Takako Kuhn, Nicholas J. Karr, Alan F. Kestenband, Barbu C. Kitada, Hitoshi Kon, Mark Andrew Kuhn, Robert M. Karrass, Abraham Kettler, Paul C. Kitano, Koh-Ichi Konig, Heinz J. Kuhn, William W. Karreman, Herman F. Key, Eric Stephen Kittaneh, Fuad A. Konvisser, Marc W. Kuhne!, Wolfgang Karrer, Guido Keys, Charles David Kittel, Phyllis M. Koo, Hyeng Keun Kuiken, Kathryn Karube, Takashi Kezdy, Andre Kivela, Simo K. Koo, Ja-Kyung Kuku,A. 0. Kasriel, Robert H. Khadr, Amr S. Kiyek, K. Koo, Reginald Kulenovic, Mustafa R. Kates, Louis K. Khaleelulla, S. M. Kiyohara, Mineo Koornwinder, Tom H. Kulesza, John S. Kato, Shin-ichi Khalifah, M. F. Klass, Michael J. Kopacek, Hal W. Kulisch, Ulrich Kato, Takao Khalimsky, Efim Davidovich Klatman, David S. Kopell, Nancy J. Kulp, Daniel Teibi Kato, Tosio Khalique, C. M. Klein, David M. Koppelberg, Sabine Kumagai, Donna J. Katok, Anatole Khan, Mohammad Ahmad Klein, Edwin M; Kopperman, Ralph D. Kumahara, Keisaku Katok, Svetlana R. ·Khoshkam, Mahmood Kleiner, Alexander F., Jr. Koranyi, Adam Kumasawa, Nin Katsura, Toshiyuki Kianos, Martin S. Kleiner, Bruce A. Korenblum, Boris Kumjian, Alexander A. Katz, Andrew Kibler, Robert E. Kleiner, Mark Koriyama, Akira Kunkle, Thomas Katz, Richard A. Kichenassamy, Satyanad Kleisli, H. Korman, Philip L. Kunle, Heinz Katz, Sheldon Kiefer, James E. Kleitman, Daniel J. Kortanek, Kenneth 0. Kunzi, Hans-Peter Katz, Victor J. Kierlanczyk, Marek Kletzing, Dennis R. Korzeniowski, Andrzej Kunzle, Hans Peter Katzen, Dahn Kiernan, Bryce M. Kliemann, Wolfgang H. Koslowski, Jurgen Kuo, Tzee Char Kauffman, Louis H. Kiessler, Peter C. Kline, Niall W. Kossack, C. R. Kupka, Ivan Kauffman, Robert M. Kieval, Harry S. Klingen, Helmut Kostenbauder, Adnah G. Kuranishi, Masatake Kaufman, William E. Kilgour, D. Marc Klingsberg, Paul R. Kotiuga, P. Robert Kurata, Yoshiki Kaup, Burchard Killeen, John Klopp, Howard D. Kotzen, Marshall J. Kuroda, S. T. Kaup, David J. Kilov, Haim Knaust, Helmut Kovarik, Zdislav V. Kurss, Herbert Kavanagh, John P. Kilpelainen, Tero Knight, Julia Kovesi-Domokos, S. Kurtz, David C. Kawai, Takahiro Kilpi, Ytjo 0. Knill, Ronald J. Kowalik, Virgil C. Kurtz, Douglas S. Kawamoto, Shunji Kim, Donggyun Knoerr, Alan P. Kozin, Frank Kurtz, Stuart A. Kawasaki, Tetsuro Kim, Hayon Knopp, Marvin I. Kozlowski, George Kurtz, Thomas G. Kazarinoff, Nicholas D. Kim, Hong Jong Knorr, K. R. K. Kra, Irwin Kurtzke, John F., Jr. Kazdan, Jerry L. Kim, Hyuk Knorrer, Horst Kracht, Manfred Kushner, Alberto Leon Keane, M.S. Kim, In Su Knowles, J. David Kraft, Jurgen Kushner, Harvey Keane, Michael K. Kim, Jong-Chul Knowlton, William B. Kraines, David P. Kusuda, Masaharu Kearton, Cherry Kim, Kyung Hwa Knudsen, John R. Krakowski, Fred Kusunoki, Yukio Keating, M. E. Kim, Moon W. Knupp, Patrick M. Kramarz, Luis Kuttler, Kenneth Latimer Keblesh, John W. Kim, Soon-Kyu Knuth, Donald E. Kramer, Earl S. Kutzko, Philip C. Keeler, Stephen P. Kim, Wan Se Knutson, Donald I. Kramer, Jurg Kwak, DoY. Keenan, Daniel Macrae Kim, Woo Jong Ko, Hai-Ping Kramer, Raymond F., Jr. Kwak, Jin Ho Keeping, Anthony J. Kim, Young-Kook Ko, Ki Hyoung Kramer, Robert J. Kwak,Nosup Kegel, Otto H. Kimber, John E., Jr. Kobayashi, Shoshichi Kramer, Thomas R. Kwok, Loong-Piu Keiding, Hans Kime, Katherine A. Kobayashi, Yuji Kranjc, Marko Kwon, KitH. Keigher, William F. Kimmerle, Wolfgang Otto Kober, Wolfgang Kranzer, Herbert C. Kydoniefs, Anastasios Keinert, Fritz Kimura, Naoki Kocan, Daniel Krasny, Robert Kyner, Walter T. Keisler, James E. Kimura, Tatsuo Koch, Lisa K. Krause, Ralph M. Kyrouz, Thomas J. Kelingos, John A. Kimura, Tosihusa Koch, Richard M. Kraut, Gertrud L. Kythe, Prem K. Kellaher, John F. Kincaid, Wilfred M. Kochman, Stanley Oscar Krauter, Arnold R. Kyuno, Shoji Keller, Hans H. Kindler, Philippe Kodama, Tetsuo Kravitsky, Naftali Laasonen, Pentti Keller, Wilfrid King, Donald R. Koditz, Helmut Kravvaritis, D. Labelle, Jacques Kellogg, Charles N. King, James R. Koehler, Anne B. Krebs, Neil E. Labesse, J.P. Kelly, David C. King, L. Richardson Koehler, T. L. Kree, Paul Labrousse, Jean-Philippe

1028 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

LaBudde, Christian D. Lauchli, Hans Leitzel, James R. C. Lien, Magnhild Long, David Darren Labute, John P. Laudenbach, Francois Leitzel, Joan P. Liffman, Kurt Long, Ding-Gwo Lacampagne, Carole B. Laurenceau, Fritz L. Lelek, Andrzej Light, Christina Long, John M. Lachlan, Alistair H. Lauria, Francesco E. Lelong, P. J. Lightbourne, James H., III Longobardi, Patrizia Lackey, Jackson B. Lauritzen, Steffen L. Lemaire, Luc Lighterman, Mark S. Longstaff, W. E. Lada, Thomas J. Lausch, Hans Lemei, Han Ligouras, Panagiotis K. Longyear, Judith Q. Ladas, Gerasimos Laush, George LeMesurier, Brenton John Lim, Chjan C. Loomis, Irene H. Ladd, Dorothy C. Lauwerier, Hendrik A. Lenard, Andrew Lim, Chong-Keang Loomis, L. H. Laderman, Julian David Lavalle, Irving H. Lenstra, H. W., Jr. Lima, Asvald Lopez, Amparo Laduke, Jeanne Laverell, W. David Lent, Arnold H. Lin, Chin-Yuan Lopez-Carmona, Antonio Ladwig, John A. Lavine, David Lentin, Andre Lin, Jindro Lopez-Escobar, Edgar G. K. Lafferty, John D. Lavine, Richard B. Leonard, Henry S., Jr. Lin, Lieh-San Lopp, Kenneth W. Laffey, Thomas J. Law, Peter R. Leonard, Philip A. Lin, Shen Lorch, Edgar R. Lagarias, Jeffrey C. Lawrence, John W. Leong, Yu Kiang Lin, Wen-Hsiung Lord, Nicholas J. Lagnese, John E. Laws, Zane John Lepisto, Timo Valter Lin, Zongzhu Lorentz, George G. Lahr, Charles Dwight Lawson, David F. Lepowsky, James I. Lind, David W. Lorentz, Rudolph A. Lai, TonyW. Lawson, H. Blaine, Jr. Leray, Jean Lind, Douglas A. Lorenz, Dan H. Laitinen, Erkki Lawson, Terry Curtis Lerche, Kenneth D. Lindberg, John A., Jr. Lorenz, Martin Lakshmibai, V. Lawson, Walter R. Lerman, Manuel Lindemann, Willi Lorimer, Joseph Wilson Lallement, Gerard J. Lawvere, F. William Leschen, John G. Lindenstrauss, Joram Loring, Terry A. Lalli, Bikkar S. Lax, Peter D. Leslie, Joshua A. Lindgren, Goerg Lotan, Moshe Lam, Kee Y. Lax, Robert F. Lesmes, Jaime Lindgren, John E. Lou, You-Shi Lam, Siu-Por Lay, David C. Lesniak, Linda Lindsay, Martin Loud, Warren S. Lam, Tsit-Yuen Lazard, Michel P. Leu, Hsiao-Tung Lindstrom, Ingrid Louhivaara, Ilppo S. Lambarth, David A. Lazarov, Connor Leung, Allen Yuklun Lindstrom, Mikael Loustau, John A. Lambert, John Patrick Lazarsfeld, Robert K. Leung, Anthony W. Lindstrom, Tom L. Loustaunau, Joaquin 0. Lambert, Richard W. Lazerson, Earl E. Leung, Denny Ho-Hon Lindstrom, Wendell D. Louton, Tom Lamoureux, Michael P. Le Cam, L. M. Leung, Tat-Wing Line, John P. Lovell, John R. Lampe, William A. Le, Dung Trang Levatin, JoAnne Lee Ling, William Halstead Loveys, James Lampert, David S. Lea, James D. Leveille, Michel Linnell, P. A. Lovric, Miroslav Lamphere, Robert L. Leach, Ernest B. Levenberg, Norman Lions, J. L. Low, Jonathan D. Lampone, Leo W. Leader, Jeffery J. Levi, Howard Lions, Pierre Louis Lowe, Peter G. Lancaster, Kirk E. Leader, Solomon Levi, Inessa Lipkin, Leonard J. Lowengrub, Morton Lancaster, Peter Leahy, John V. Levin, Jacob J. Liporace, Louis A. Loxton, John Harold Lance, Timothy L. Leary, Kevin J. Levin, Richard G. Lipschutz, Seymour Lozano, Maria Teresa Lander, Felix I. Leavitt, William G. Levine, Daniel S. Lipshutz, Robert J. Lu, Chin-Pi Lander, Julian Charles Lebow, Arnold Levine, Eugene Lipton, R. J. Lu, Kau-Un Landesman, Edward M. Lebowitz, Joel L. Levine, Harold I. Litchfield, Kay P. Lubarsky, Robert S. Landin, Joseph LeBrun, Claude R. Levine, Howard A. Little, Charles H. C. Lubin, Jonathan D. Lando, Clifton A. Lee, Chung N. Levine, Jack Little, John B. Luce, R. Duncan Landraitis, Charles K. Lee, Dong Hoon Levine, Jerome P. Littman, Walter Ludden, Gerald D. Landrock, Peter Lee, Do-Won Levitt, Norman J. Liu, Chyan-Jyue Ludington, Anne L. Landweber, Peter S. Lee, In-Ja B. Levy, Azriel Liu, Hsi-Ching Luecke, John Edwin Lanford, Oscar E., III Lee, Jang Woo Levy, Gary B. Liu, Jianguo Luehr, Charles P. Lang, Serge Lee, John M. Levy, Gene Liu, Jong-Diing Lueneburg, E. Lange, Ridgley Lee, June-Bok Levy, Joel Liu, Li-Qian Luh, Jiang Langebartel, Ray G. Lee, Ke-Seung Levy, Joshua B. Liu, Lishi Lukacs, Eugene Langenhop, Carl E. Lee, Kotik K. Lewan, Douglas Liu, Shih-Chao Lundell, Albert T. Langsam, Joseph A. Lee, Kyung Bai Lewis, Charles J. Liu, Shih-Hua Lundgren, J. R. Langston, Michael David Lee, Myrna Pike Lewis, David W. Liu, Tai-Ping Lundy, Elizabeth L. Laning, J. H. Lee, Peng-Yee Lewis, Donald C. Liu, Tsai-Sheng Lundy, Thomas James Lansdon, Robert Steven Lee, Ronnie Lewis, George M. Liulevicius, Arunas L. Lusztig, George Lanteri, Antonio Lee, Shyling Lewis, Houston I. Livingston, Albert E. Luther, C. F. Lantz, Vicky L. Lee, Tzong-Yow Lewis, Hugh L. Livingston, Charles Luther, Norman Y. Lanza de Cristoforis, M. Lee, WillY. Lewis, L. Gaunce, Jr. Llave, Rafael de Ia Lutoborski, Adam Lanzano, Paolo Lee, Yuh-Jia Lewis, Robert H. Llewellyn, James Lutterodt, Clement Henry Lapidus, Arnold Leeb, Klaus Lewis, Roger T. Llewellyn, Mark A. Lutwak, Erwin Lapidus, Michel L. Leemans, H. W. Lewis, William James Llibre Salo, Jaume Lutzer, David J. Lappan, Peter A., Jr. Leep, David B. Lhee, Byung Su Llosa, Joseph Luvisetto, Maria Luisa Larkin, Martha A. Lefkovitch, Leonard P. L'heureux, James E. Lloyd, Noel G. Luxemburg, Wilhelmus A. J. Larrazabal, J. M. Leger, George F. Li, Bingxi Lluis, Emilio R. Lyford, William C. Larsen, Kenneth M. Leggett, Robert N., Jr. Li, Li Lluis-Puebla, Emilio Lyndon, Roger C. Larson, David R. Lehner, Joseph Li, Li-Ge Lo, Libo Lyons, Russell D. Larson, Edward H. Lehr, Marguerite Li, Tien-Yien Lo, Tzee-Nan Kuo Lysko, Janusz M. Larson, Lee M. Lehrer, Gustav I. Liang, Jaung Lockhart, Jody Meyer Lytle, Raymond A. Larson, Richard G. Lehrer-Hamed, Y. Liang, Joseph J. Loday, Jean-Louis Lyubeznik, Gennady M. Larson, Roland E. Lehtinen, Matti T. Liardet, Pierre Loeb, Henry L. MacCamy, Richard C. LaSalle, Margaret M. Lehtonen, Ari Taavi Einari Libby, Richard Allan Loeb, Peter A. MacDonald, I. G. Lashof, Richard K. Leichter, Jerrold Liberman, Zvie Loeser, Francois MacDonald, Ian D. Laska, Eugene M. Leichtweiss, Kurt Lichtenbaum, Stephen Lok, Walter L. MacDonnell, John J. Laska, Michael Leighton, Frank Thomson Lichtenstein, Walter D. Lolli, Gabriele Macfarlane, Andrew Ian Latto, Andrew R. Leighty, Robert M. Lichtman, Alexander Lombardo, Charles J. MacGillivray, A. Dean Lau, William G. Leipold, Richard A. Lieb, Elliott H. Loncaric, Josip MacGregor, Thomas H. Laubenbacher, Reinhard Leith, Cecil E., Jr. Lieberman, Gary M. London, David Machover, Maurice

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1029 AMS Centennial Research Fellowship Fund

Mack, John Michael Marcus, Steven I. Matz, Carol Lang McGowan, Harry Meyer, Wolfgang T. MacKenzie, Robert E. Margerum, Eugene A. Mauceri, G. C. McGowan, Jeffrey K. Meyer, Yves F. Mackey, Michael C. Margosian, Zaven Maumary, Serge McGuinness, David J. Meyerhoff, Robert MacLaren, M. Donald Margulies, William Maurer, Christian E. G. McGuinness, Francis Oisin Meyers, Norman G. MacNaughton, John S. Marhuenda, Francisco Maurer, Donald E. Mcintosh, William D. Mez, Hans-Christian Macphail, Moray S. Maria, May H. Maurer, Stephen B. Mciver, Jeffrey L. Mezzino, Michael J., Jr. Madan, Manohar L. Marion, Jean Albert Mavrommatis, Kostas McKay, James H. Mherian, Hayrenik M. Madden, James Joseph Markanda, Raj K. Mawhin, Jean L. McKay, Thomas G. Miao, Lung-Chi Maddux, Roger D. Markel, Scott A. Mawyer, Farley McKean, Henry P., Jr. Michael, Ernest A. Mader, Adolf Markot, Robert P. Maxfield, John E. McKenzie, Lionel W. Michael, William A. Madison, Bernard L. Marie, Charles Michael Maxson, Carlton J. McKenzie, Ralph N. Michaelis, Walter J. Madow, William G. Marley, Anthony A. J. May, J. Peter McKibben, William P. Michalek, Gary E. Madsen, Ib H. Marotto, Frederick R. May, Michael K. McKiernan, M. A. Michel, Rene Maeda, Michie Marques, Carlos A. May, Warren L. McKinney, Richard L. Michiaki, Watanabe Maeder, Roman E. Marr, Robert B. May, William McLarnan, Timothy J. Michler, G. Maehara, Kazuhisa Marsden, Jerrold E. Mayer, Alan L. McLaughlin, Jack E. Michor, Peter Magenes, Enrico Marshall, Bernard P. Mayer, John Clyde McLaughlin, Joyce R. Mickens, Ronald E. Magid, Martin A. Marshall, David Imler Mayer, Raymond A., Jr. McLaughlin, Renate Mielke, Paul T. Magill, Kenneth D., Jr. Marshall, Murray Angus Maynard, James W. McLean, Jeffery Thomas Mientka, Walter E. Magnus, Arne Marsiliano, Judy Mayor, John R. McLeod, Edward B., Jr. Miesner, Walter Magnuson, Alan William Marstrand, John M. Maziarz, E. A. McLeod, Robert M. Mihalik, Michael L. Magruder, Richard B. Martens, Henrik H. Mazumdar, Tapas McMillan, Brockway Miklavcic, Milan Mahaney, Lou A. Marti, Jurg T. Mazzeo, Rafe R. McMillan, Daniel R., Jr. Mikolic-Torreira, Igor Mahar, T. Martin, Corindo I. McAdam, Stephen J. McMillan, Evelyn R. Miles, E. P., Jr. Mahdavi, Kazem Martin, Donald A. McAllister, Byron L. McMillen, William Robert Miles, Philip E. Mahrous, Mohamed-Amin A. Martin, E. Dale McAllister, Gregory T. McMinn, Trevor J. Miles, Thomas J. Mai, Liem Martin, George E. McArthur, C. W. McMullen, John R. Miletta, Peter D. Maiellaro, Michele Martin, John R. (Texas) McAsey, Michael J. McMurtrie, Kenneth A. Miliaras, George Maier, Franz W. Martin, John R. (Canada) McBrien, Vincent 0. McNerney, Gerald M. Milicic, Dragan Maier, Helmut Martin, Margaret A. McCall, Edward H. McNulty, George F. Millar, Robert F. Mairhuber, John C. Martin, Nigel McCallum, William G. McNulty, Nieves Austria Millar, T. S. Maj, Mercede Martin, Robert H., Jr. McCanna, Joseph E. McOwen, Robert C. Miller, Dale A. Majda, George J. Martin, Roland E., III McCarthy, Charles A. McRae, D. George Miller, David E. Makepeace, Coline M. Martindale, Wallace S., III McCarthy, Donald J. McShane, Edward J. Miller, Donald S. Makkai, Michael Martinez-Maurica, Javier McCarthy, John McWilliams, Ralph D. Miller, Forrest R. Makovoz, Yuly Martino, James R. McCarthy, John David Meadows, Douglas Shelby Miller, Haynes R. Makowsky, Johann Andreas Martinon, Antonio McCarthy, Paul J. Meadows, Lawrence F. Miller, Jack M. Malafronte, Thomas A. Martin-Reyes, Francisco McCarty, Carl P. Meagher, Jack R. Miller, James Curtis Malcolm, Wilfred Gordon Javier McCarty, Kevin P. Meakin, John C. Miller, John G. Malcolmson, Peter Martins, Joyce A. McCasland, Roy Lynn Meder, Albert E., Jr. Miller, Kenneth G. Malik, Davender D. S. Marty, Roger H. McCaughan, Dennis J. Medhin, Negash G. Miller, Kenneth S. Malkevitch, Joseph Marumoto, Yoshihiko McClanahan, Ray A. Meek, David W. Miller, Michael J. Malliaris, Tassos G. Maruyama, Toru McCleary, John H. Meeker, Loren David Miller, Nicholas S. Mallios, Anastasios Marvin, John W. McClenahan, Maurice D. Meilan, Gerard Miller, Richard R. Maim, Dennis R. Marzuq, Maher M. H. McColm, Gregory Loren Meisner, Morris J. Miller, Robert N. Malon, David M. Mascioni, Vania D. McComb, Peter J. Meister, Erhard V. Miller, Stephen S. Maloof, Giles W. Masha, Edward V. McConnell, John C. Meixner, Th Miller, Timothy L. Maltese, George J. Masih, Samuel McConnell, Terry R. Mejlbo, Lars C. Miller, Valerie A. Malyshev, Igor G. Maskit, Bernard McConnell, Thomas Melin, Anders E. Miller, Willard, Jr. Mamelak, Joseph S. Maslanka, David Joseph McCool, Kenneth Mellender, James W Miller, William David Manders, Kenneth L. Mason, Robert M. McCormick, Ferris E. Melrose, Richard B. Miller, William V. Manderscheid, David C. Massam, Helene McCoy, Robert A. Melvin, Paul M. Miller, Zevi Manfredi, Juan J. Massopust, Peter R. McCoy, Thomas L. Melzi, Giovanni Mills, Janet E. Manis, Merle E. Masterson, John T. McCracken, Marjorie Frost Menconi, Mario Mills, Stuart E. Manjarrez, Victor M. Masuda, Kyuya McCray, Patrick D. Mendelsohn, Eric Mills, William H. Mann, Benjamin M. Masumoto, Makoto McCrea, Michael M. Mendelssohn, Roy Milman, Mario Mann, Henry B. Matchett, Andrew J. McCready, Robert R. Mendoza, Charles Gerard Milman, Mark H. Mannur, Narasimha Reddy Mather, John Norman McCrimmon, Kevin M. Mendoza, Gerardo A. Milner, Eric C. Manogue, Joseph F. Matheson, John D. McCrory, Clint G. Mendoza-D'Paola, Leonardo Milnor, John W. Manoly, Bruce E. Mathews, John H. McCullough, Darryl Menezes, Lucia S. Milnor, Tilla Klotz Mansfield, Maynard J. Mathis, Frank H. McCurley, Kevin Snow Merindol, Jean-Yves Milojevic, P. S. Mantini, Lisa Mathsen, Ronald M. McDonald, Janet Merriell, David M. Milteer, Linda J. Mapa, Felina G. Matijevic, Jacob R. McDonough, J. M. Mertens, John R., Jr. Mimura, Mamoru Marathe, Kishore B. Matovsky, John-Anton McDonough, Joseph Michael Merzbach, Uta C. Minac, Jan Marble, Robert P. Charles McDowell, David G. Meserve, Bruce E. Minagawa, Takizo Marcati, Pierangelo A. Matsuki, Kenji McDuff, Dusa Mess, Geoffrey Minda, C. David Marchand, Margaret 0. Matsumoto, Yukio McGarvey, John R. Messer, Karen Minehart, Deborah Fairley Marchesin, Dan Matsumura, Hideyuki McGavran, Dennis R. Metas, Nick Minemura, Katsuhiro Marchetti, Federico Matsunaga, Teru McGee, Robert I. Metropolis, Nicholas C. Miner, Robert R. Marchiafava, Stefano Matsushita, Yasuo McGehee, 0. Carruth Meuser, Diane Mae Mines, Ray Marculescu, Doru Matti, Joseph T. McGehee, Richard P. Meyer, Burnett C. Ming, Ronald W. Marcus, David J. Mattingly, Robert Bruce McGibbon, Charles A. Meyer, Donald V. Minsker, Steven Marcus, Leo G. Mattson, Leroy T. McGill, Suzanne Meyer, Jean-Pierre G. Mintel, Ernest J. Marcus, Marvin Matumoto, Hisayosi McGough, Nancy Ann Meyer, Paul R . Minton, Roland B.

.1030 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Mirchandaney, Arjan Sobhraj Morawetz, Cathleen S. Myers, Dale W. Nelson, Larry D. Nottrot, Roel Mirelli, Vincent Morduchow, Morris Myers, Earle F. Nelson, Robert R. Novikoff, Albert B. Mirsky, Norman D. Moretz, Marjorie A. Myers, Russell W. Nelson, Roger B. Nulton, James D. Mischenko, Peter Andrew Morgan, Christopher L. Myerson, Gerald Nelson, Timothy L. Nummela, Eric C. Mishra, Sudhakara Morgan, Frank Mylvaganam, Kanaga Nemchenok, Jacob Nunke, Ronald J. Mislin, Guido Morimoto, Akihiko Sabapathy Nemethi, Csaba Nunley, Charles L. Misra, A. K. Moriyon, Roberto Nachman, Louis J. Nerode, Ani! Nusse, Helena E. Misra, Kailash C. Morley, Larry J. Nadell, Paul T. Nesbitt, Cecil J. Nutt, Michael D. Mital,J. N. Morrel, Bernard B. Naeem, Rana Khalid Ness, Linda A. Nyman, Philip D. Mitchell, A. Richard Morrill, John E. Nagahara, Takasi Neubauer, Gerhard J. Oates-Williams, Sheila Mitchell, Alan R. Morris, L. E. Nagai, Osamu Neugebauer, Christoph J. Obaid, Samih A. Mitchell, Emerson C. Morris, Peter C. Nagami, Keio Neumann, Bernhard H. Obata, Morio Mitchell, James C. Morris, Robert A. Nagano, Tadashi Neumann, Michael M. Oberg, Edwin N. Mitchell, Stephen Ames Morrison, David R. Nagao, Hirosi Neumann, Peter M. Oberguggenberger, Michael B. Mitchell, Stephen G. Morrison, John A. Nagarajan, K. R. Nevai, Paul Obermayer, Judith H. Mitchell, Theodore Morrison, John F. Nagasaka, Kenji Neveu, Jacques Oberschelp, Walter Mitchell, William J. Morrison, Kent E. Nagata, Jun-Iti Newberger, Stuart M. Obi, Wilson C. Mittenthal, Lothrop Morrison, Michael A. Nagel, Alexander J. Newhouse, Sheldon E. de las Obras, Carmen Mitter, Sanjoy K. Morrison, Sally D. Nagel, Bengt C. H. Newlander, August, Jr. Obrock, Arthur E. Miura, Robert M. Morse, Burt J. Naidoo, Logan Newman, Morris Ochanine, Serge Miyadera, Isao Morton, Blaise Naimpally, Somashekhar A. Newman, Rogers J. Ochoa, Juan C. Miyahara, Yoshio Morton, Keith W. Najarian, John P. Newsom, Carroll V. Ochs, Robert L., Jr. Miyazaki, Ken-Ichi Moschovakis, Joan Rand Nakagawa, Yoko Newton, Mervin E. Ocken, Stanley Mizumoto, Hisao Moschovakis, Yiannis N. Nakai, Mitsuru Newton, PaulK. Ocone, Daniel L. Mizuno, Hirobumi Moser, Joseph G. Nakamura, Masahiro Ney, Peter E. Oda, Tadao Mlitz, Rainer Moser, Jurgen K. Nakamura, Riichiro Ng, BartS. O'Dea, Paul J. Mochizuki, Horace Y. Moser, Lucy I. Nakanishi, Yasutaka Nguiffo, B. Michel Odlyzko, Andrew M. Mochizuki, Leslie Y. Moser, William 0. J. Nakano, Kazumi Nguyen, Bao Gia Odoni, R. W. K. Modica, Luciano Mosier, Ronald G. Nakashima, Martin M. Nguyen, Dung X. O'Farrell, Anthony G. Moen, Courtney H. Moss, Kenneth N. Nakata, Masaomi Nguyen, Hung Thac O'Farrell, John Michael Moghaddam, Mohammad Moss, Lawrence Stuart Nakata, Mie Nguyen, Hung Trung Ogawa, Hajimu Reza R. Moss, William F., III Nail, Van C. Nguyen-Huu-Bong Ogg, Andrew P. Mohamed, I. J. Mostow, George Daniel Nam, Soonkeon Nicholls, Peter J. Ogle, Crichton Mohamed, Saad Motegi, Masanori Namba, Kanji Nichols, Edward Oguchi, Kunio Mohler, Lee K. Motto, Michael H. Nambooripad, S. K. S. Nichols, Eugene D. Ogus, Arthur Moite, Sally M. de Mottoni, Piero L. Namioka, Isaac Nicholson, Victor A. Oh, Hi-Jun Moller, Raymond W. Moulden, Trevor H. Namkoong, Kyu-Oh Nickel, James A. Oharu, Shinnosuke Moog, Christopher Chsiu Mount, Kenneth R. Nance, Dana W. Nickerson, Helen K. Ohta, Haruto Mongelli, Dominic Moy, Allen Nandakumar, N. R. Nicol, Charles A. Ohwaki, Shin-Ichi Monheit, Lilly A. Moyls, Benjamin N. Nannicini, Antonella Nicol, Sherrie Jean Oikawa, Kotaro Monk, J. Donald Mrowka, Stanislaw G. Narang, Deborah Lynn Nicolaenko, Basil Okada, Masami Monleon-Pradas, Manuel Muder, Douglas J. Narang, Kamal Nicolas, Jean-Louis Okada, Susumu Monna, Antonie F. Mueller, Bruno J. W. Narayanaswami, P. P. Niederhausen, Heinrich Okamoto, Kiyosato Monroe, James Graham Mueller, James R. Narcowich, Francis J. Nievergelt, Yves Okayasu, Takateru Monson, Stephen R. Mugele, Raymond A. Narendra, K. S. Niino, Kiyoshi Okoh, Frank Montaldi, James A. Mulla, Fuad S. Narushima, Hiroshi Nilsson, Per Oksendal, Bernt K. Montesinos, Jose M. Muller, George M. Nashed, M. Zuhair Nirenberg, Louis Okumura, Masafumi Montgomery, Hugh L. Muller, Gert H. Nash-Williams, Crispin Nishida, Takaaki Okuyama, Akihiro Montgomery, M. Susan Mullin, Albert A. St J. A. Nishikawa, Seiki Oldham, Frank Ernest Montgomery, Peter L. Mullis, Robert Harvey, Jr. Nataro, Dean C. Nishiura, Togo O'Leary, Robbin Lerch Montgomery, Richard W. Munemasa, Akihiro Nath, Radha G. Nissim, Leonard J. Oler, Norman Montgomery, W. Duane Munkholm, Hans J. Nathanson, Melvyn B. Nitecki, Zbigniew H. Olesen, Dorte Marianne Monticone, Leone C. Munkres, James R. Nation, James B. Nitsche, Johannes C. C. Oliger, Joseph E. Moody, Robert V. Muiioz, Edgar Naugle, Norman Wakefield Nitzberg, Mark J. Oliker, Vladimir I. Moon, David S. Muiioz, Miguel-C Navarro, Francisco J. Njastad, Olav Olin, Robert F. Moon, Kyung-Ho Munoz-Porras, Jose M. Naveira, Antonio Martinez Nkashama, Mubenga Ngandu Olive, Gloria Moon, Myung-suk Murakami, Shingo Nayak, M. K. Nobauer, Wilfried H. Olivier, Reinhard M. Mooney, John J. Murase, Nobuyuki Nayar, Bhamini M. Nobusawa, Nobuo Olmstead, P. S. Moons, Theodoor M. Murata, Hiroshi Nduka, A. Noel, Guy G. J. Olmsted, John M. H. Moore, Alan L. Murdoch, David C. Neeb, Donna M. Noguchi, Hiroshi Olsen, Martin Moore, Charles Nelson Murnaghan, Fiona A. Neggers, Joseph Noguchi, Junjiro Olshen, A. C. Moore, Hal G. Murphy, Brian B. Negishi, Norio Nobel, John A. Olson, Loren D. Moore, James Andrew Murphy, Donald P. Negrepontis, Stylianos Nolan, John P. Olver, Frank W. J. Moore, Judy Hennessey Murphy, Grattan P. Negro, Angelo L. Nold, Annett Olver, Peter J. Moore, Polly Murphy, Katherine Adele Nehs, Robert M. Noll, Landon Curt Oman, John Arthur Moore, Richard A. Murphy, Paul F. Neidhardt, Arnold L. Noonburg, Virginia A. O'Meara, 0. Timothy Moore, Robert H. Murphy, Thomas N. Neidhardt, Wayne L. Nordgren, Eric A. Omladic, Matjaz Moore, Robert Laurens Murray, G. Graham Neidinger, Richard Dean Nordhaus, Edward A. Omori, Hideki Moore, Rufus A. Murray, Margaret A.M. Nel, Louis D. Norfleet, Jesse Peter Onat, Emin T. Moore, Vardeman G. Murre, Jacob P. Nelligan, John D. Norman, Francis Alexander O'Neil, Philip J. Moore, William E. Murty, Ram M. Nelson, Abraham Norman, Peter O'Neill, Anne F. Moori, Jamshid Muse, Archie Henry Nelson, David Norman, R. Daniel O'Neill, Bernard V., Jr. Moran, Charles W. Muto, Hideo Nelson, Edward Northcutt, Robert A. Onishi, Hironori Moran, William Mycielski, Jan Nelson, James, Jr. Norton, V. Alan Ono, Takashi

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1031 AMS Centennial Research Fellowship Fund

Onose, Hiroshi Palmer, Edgar M. Pedersen, Henrik Picardello, Massimo A. Porter, Gerald J. Oodaira, Hiroshi Palmer, Susan Mary Pedersen, Katherine L. Piccinni, Paolo Porter, Willis B. Oohashi, Tsunemichi Palmer, Theodore W. Peercy, David E. Pickel, Paul Frederick Poss, Richard L. Oono, Yoshitsugu Pan, Liu-Hua Peetre, Jaak Pickering, Douglas Arthur Post, Karel A. Opfer, Gerhard H. Pan, Ting K. Peglar, George W. Pickrell, Douglas M. Potts, Donald H. Oppelt, John A. Panchapagesan, T. V. Pego, Robert L. Picoult-Hart, Darri C. Poulsen, Ebbe T. Oppenheim, Joseph H. Pandit, Onkar Nath Peiffer, Barry L. Piegl, Leslie Pourciau, Bruce Hunter Oral, Haluk Pannullo, Jerome Edward Peixoto, Mauricio M. Piene, Ragni Pour-El, Marian B. Orellana, Chacin Mauricio Papadopol, Petru Peleg, Bezalel Pierce, John Festus Pouzet, Maurice Orey, Steven Papageorgiou, Nikolaos S. Peletier, Lambertus A. Pierce, Rebecca L. Powell, C. Spencer Orihuela, Jose Papangelou, Fredos Pell, William H. Pierce, Richard S. Powell, James H. Orlik, Peter P. Papanicolaou, George C. Pelletier, Donald H. Pignedoli, A. Powell, Jessica L. Orloff, Jeremy M. Papantonopoulou, Aigli Pelloni, Beatrice Pigno, Louis Powell, William C. Orloff, LeoN. Helen Pence, Dennis D. Pigozzi, Don L. Powers, Kathryn A. Ornstein, Avraham J. Papas, Theodore Thomas Pennell, Stephen A. Piirila, Olli-Pekka Powers, Morris Walton Orr, Kent Edward Pappas, Peter C. Penner, Robert Clark Pillay, Anand Powers, Robert T. Orr, Martin Pappas, Richard C. Penney, Carol W. Pilz, Gunter Powls, Keith W. Orr, Richard J. Pardee, Otway O'M Penney, David E. Pinch, Richard Gilmour Eric Pradines, Jean Orsillo, James E. Pareigis, Bodo Pennisten, John W. Pinchon, Didier Prakash, Jagdish Orsted, Bent Parikh, Rohit J. Pepper, Paul M. Pinero, Alfredo Prakash, Nirmala Ortner, Brian R. Park, Hong G. Pereira da Silva, C. Pinkham, Henry C. Pralhad, Rangarao N. Osborn, J. Marshall Park, Hong-Sub Perelli, Alberto Pinkus, Allan M. Prather, Carl Lindell Osborn, T. A. Park, Jong Geun Perez de Vargas, Alberto Pinney, Karen R. Prato, Elisa Osborne, Mason S. Park, Sehie Perez, German Augusto Pinsky, Mark A. Pratt, Vaughan R. Osborne, Michael Robert Park, Young Soo Perkins, Peter Pinsky, Ross George Preissendoerfer, Manfred W. Osgood, Brad G. Parker, Francis D. Perlman, Sanford Pinter, Joseph Kalman Preller, Anne O'Shea, Donal B. Parker, George D. Pernice, Michael Pippert, Raymond E. Prenowitz, Walter Oslon, Steve G. Parker, Thomas H. Perrin, D. Piranian, George Pressley, A. N. Osner, H. J. Parker, Willard A. Perrine, Serge Pisanski, Tomaz Prevot, Kenneth J. Osofsky, Barbara L. Parks, Harold R. Perry, Peter Anton Pisier, Gilles Price, G. Baley Osserman, Robert Parks, Teresa A. Perry, William J. Pisier, Gilles Price, Roderick A. Ostaszewski, Krzysztof M. Parr, James T. Perry, William L. Pitcairn, Joel Price, Thomas E., Jr. Osterburg, James Parrish, Herbert C. Persek, Stephen C. Pitcher, Everett Price, Thomas M. Osterby, Ole Parrott, David Pesotan, Hoshang Pitt, Loren D. Primer, Jeremy D. Osterwalder, Konrad Parry, Walter R. Pestana, Dinis Duarte Pittnauer, Franz Prince, Geoffrey Eamonn Ostrom, T. G. Parry, William Petermann, G. E. Pitts, Andrew M. Prindle, John C. Osundu, Kevin E. Parshall, Brian J. Peters, James Vincent Pitts, David R. Proctor, Robert A. Oswald, Urs Parsons, Charles D. Peters, John E. Pixton, Dennis G. Promel, Hans Jurgen Ota, Clem Z. Parsons, John D. Peters, Justin R., III Plachky, Detlef Protas, David S. Ota, Schoichi Parter, Seymour V. Peters, Klaus Plank, Donald L. Protopopescu, V. Otal, Javier Partheniadis, Evan C. Petersen, Bent E. Plano, Richard A. Protter, Murray H. Otal, Jean-Pierre J. P. Paschke, William L. Petersen, Karl E. Plante, Joseph F. Protter, Philip E. Otermat, Scott C. Pascual, Felino Garcia Petersen, Peter Platek, Richard A. Provencher, Normand Otero, Maria Carmen Pascual-Gainza, Pere Petersen, Troels Pless, VeraS. Pruitt, William E. Otsuki, Nobukazu Pascucci, Richard S. Petersen, Wesley P. Plotnick, Daniel H. Prussner, Laird Drew Oursler, C. C. Pasini, Antonio Peterson, Allan C. Plotnick, Steven P. Przybylski, Thomas L. Owa, Shigeyoshi Passerini, Henry J. Peterson, Charles L. Plumeri, Charles John Przytycki, Jozef H. Owen, Nicholas C. Passman, Donald S. Peterson, Gary L. Plymen, Roger J. Pucci, Carlo Owens, William H. Passty, Gregory B. Peterson, William C. Podber, David P. Pucci, Patrizia Owusu-Ansah, T. Pastro, Pier-Ivan Petricka, Russell J. Poel, Mannes Puerta, Fernando Oyabu, Takashi Pasztor, Ana Petro, John W. Poenaru, Valentin A. Puigbo, Jaume Vila Ozaki, !sao Patel, Alka Pettet, Martin R. Poggio, T. A. Pukanszky, Lajos Ozaydin, Murad Paterson, Alan L. T. Petty, Clinton M. Pogu, Marc Putnam, Alfred L. Ozimkoski, Raymond E. Pathak, Vijay Damodar Petulante, Nelson R. Pohjolainen, Seppo A. Putt, Harold L. Ozols, Vilnis Patil, Dattatraya J. Pew, Thomas Vincent Polachek, Harry Puttaswamy, T. K. Ozsvath, Istvan Patterson, Nick J. Peyrovian, Reza Polemarchakis, H. Pym, JohnS. Pacella, Filomena Patterson, Samuel E. Pfeffer, Washek F. Polimeni, Albert D. Quade, Edward S. Pacheco-Castelao, Jose M. Patton, Charles M. Pfeiffer, Paul E. Polking, John C. Quan, Loi T. Paciorek, Joseph W. Paulling, John M. B. Pflug, Peter Pollack, Richard Queen, Clifford S. Packer, Judith A. Paulowich, David G. Pfluger, Albert Pollak, H. 0. Quigley, Frank D. Pade, Offer Paulsen, Vern I. Pflugfelder, Hala 0. Pollatsek, Harriet S. Quinn, Declan P. F. Padma, Narasimhachari Paveri Fontana, S. L. Phadke, Bhalchandra Balvant Polonsky, Ivan P. Quinn, Frank S., III Page, Stanley S. Pavlopoulos, Haralabos Phan, Kok W. Ponce, Gustavo Alberto Quinn, Joseph E. Paige, Eugene C., Jr. Pawelke, Siegfried H. R. Phanord, Dieudonne D. Pons, Montserrat Quinn, Michael Paul Paige, Lowell J. Payne, Lawrence E. Pheidas, Athanasios C. Ponthieux, Delores Marie Quint, Richard A. Paisner, Helen L. Payne, Stanley E. Phelps, Robert R. Poon, Yat Sun Quinto, Eric Todd Pak, Jingyal Pazy, Amnon J. Phili, Christine Melle Poorman, Alan G. Quiring, Donald K. Palais, Eleanor G. Pears, Alan R. Phillips, Anthony V. van der Poorten, Alfred J. Rabinowitz, Paul H. Palais, Robert A. Pearson, Robert W. Phillips, Ralph S. Pop-Stojanovic, Zoran R. Rabinowitz, Stanley Palka, Bruce P. Pearson, Stephen C. Piacentini, Giulia Maria Poris, Michael J. Raddatz, William D. Pallaschke, Diethard Peccati, Lorenzo Piacenza, Robert John Poritz, Alan B. Radford, David E. Palled, Shivappa V. Peck, Paul S. Piatetski-Shapiro, I. Port, Sidney C. Radjavi, Heydar Palm, Andrew C. Pedersen, Edward A. Piazza, Barry L. Porta, Horacio A. Radkowski, Alfred F. Palm, Mark A. Pedersen, Flemming P. Picard, Rainer H. Porte, Jean Radunskaya, Ami

1032 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Ruan, Zhong-Jin Radzikowski, Pawel Rehmann, Ulf Riley, Geoffrey William Rooney, Paul G. Flores Espinoza Raggio, Guido A. Rehmer, Karl 0. Riley, Robert F. Roonguthai, Warut Ruben, Ragsdaie, Katherine B. Reichaw, Meir Rinaman, William C. Roosenraad, CrisT. Ruberman, Daniel Rahme, Elham Hanna Reid, James D. Riney, John S. Root, William L. Rubin, Karl Robert J. Raimi, Ralph A. Reid, K. Brooks Ringeisen, Richard D. Roozemond, Leen Rubin, Joachim H. Rajnak, Stanley L. Reid, Les Ringel, Emanuel I. Rosa, Alexander Rubinstein, Rubinstein, Robert Rakesh Reilly, Ivan L. Rio, Jose Rosati, Mario de Francia, Jose L. Rail, Louis B. Reilly, Norman Raymond Rios, Barbara Luersen Rose, David A. Rubio D. Ralph, William J. Reineck, James F. Risnes, Martin Rose, N.J. Rudin, Bernard Ramakrishnan, Dinakar Reiner, Irma M. Ritt, Robert K. Roselle, David P. Rudin, Mary E. Ramaley, James F. Reinhardt, Elsi A. Ritter, David L. Roseman, Dennis Martin Rudnick, Karl Jay Ramalho, Margarita Reinhardt, William N. Ritter, Eugene K. Roseman, Joseph J. Rudolph, Daniel Ramamurthi, V. S. Reinoehl, John H. Rno, Jung-Sik Rosen, Jay S. Ruelle, David P. Ramanathan, Jayakumar Reise!, Robert B. Roach, Michael D. Rosen, Kenneth H. Ruess, W. Ramanujan, Melapalayam S. Reissner, Eric Roan, Shi-Shyr Rosen, Lon M. Ruh, Ernst A. Ramaroson, Francois Rembis, Frederick C. Robacker, John T. Rosen, Michael I. Ruijsenaars, Simon Norbert Ramm, Alexander G. Rempel, Richard S. Robbin, Joel W. Rosen, Ned Ira Ruiz, Alberto Ramsay, Arlan B. Remus, Dieter Robbins, David P. Rosen, Ronald H. Ruiz, Ceferino Ramsay, Keith A. Renardy, Michael Robel, Gregory F. Rosenbaum, Robert A. Rulla, James L. Randol, Burton S. Renegar, James M. Robert, Alain Rosenberg, Ivo G. Rumely, Robert S. Randolph, John F. Renner, Lex E. Roberts, Dorothy K. Rosenberg, Jonathan M. Rumford, Fred K. Raney, George N. Renno, James G., Jr. Roberts, George A. Rosenberg, Milton Rumsey, Howard C., Jr. Range, R. Michael Renshaw, James H. Roberts, Howard M. Rosenberg, Steven Runck, Paul Otto Rankin, Samuel Murray Resch, Richard I. Roberts, Ian T. Rosenblatt, Joseph Max Rupp, Russell D. Rankin, Stuart A. Restrepo, Guillermo Roberts, Jean E. Rosenblum, Marvin Rush, David Eugene Ransom, Adelbert W. Retzloff, David G. Roberts, Joel L. Rosenfeld, Gerald Rushing, T. Benny Rao, Ranga R. Reyes, Francisco G. Roberts, Joseph B. Rosenfeld, Norman S. Rusin, David J. Raphael, Marc Reynolds, David W. Roberts, Lawrence Gordon Rosenfeld, Ronald L. Rusinak, William A. Rapp, P. E. Reynolds, Donald F. Roberts, Richard C. Rosenlicht, Maxwell A. Ruskai, M. Beth Rascle, Michel Reynolds, Robert J. Robertson, Alexander P. Rosenstein, George M., Jr. Rusodimos, Jason Raskind, Wayne Mark Reynolds, William F. Robertson, Mark M. Rosenthal, Eric S. Russell, David Martin Rassias, John Michael Rhai, Tong-Shieng Robertson, Neil Rosenthal, Erik J. Russell, Gary L. Rassias, Themistocles Rhee, Choon Jai Robidoux, Nicolas Rosenthal, Haskell P. Russell, Margaret T. Michael Rhie, Young H. Robinson, Clark Rosenthal, Jenny E. Russo, Bernard Ratcliff, Gail D. L. Rhodes, Frank Robinson, Daniel A. Rosenthal, William E. Russo, Dominick A. Ratiu, Tudor Stefan Riaza, Roman Robinson, Derek J. S. Rosenzweig, Samson M. Russo, Paula A. Ratliff, Louis J., Jr. Riazi-Kermani, Mohammad Robinson, E. Arthur, Jr. Rosoff, Jeffrey A. Rutland, Leon W., Jr. Ratner, David A. Ribeiro, Hugo B. Robinson, G. de B. Ross, Bertram Rutledge, Dorothy S. Ratz, Jurg J. Ribes, Luis Robinson, Helen Drummond Ross, Kenneth A. Ruttimann, Gottfried T. Raussen, Martin Rica Da Silva, Amaro J. Robinson, Margaret Maher Ross, Paul Ryan, Donald E. Rave, Werner J. Ricci, Fulvio Robinson, Paul Lee Ross, Shepley L., II Ryan, Kevin M. Ravenel, Douglas C. Ricci, Michael G. Robinson, Raphael M. Rossello, Francese A. Ryan, Michael D. Rawlings, Don Paul Ricci, Stephen J. Robinson, Sam L. Rossi, Guido A. Ryan, Patrick J. Rawnsley, John H. Rice, John R. Robinson, Stanley A. Rossi, Hugo Ryan, Peter M. Rawsthorne, Daniel A. Rice, John William Robkin, Eugene E. Rossler, Otto E. Rychlik, M. Ray, Phillip Paul Richards, Franklin B. Rocha, Carlos V. Rosson, John Rydeen, Clarence F. Rayburn, Marlon C. Richards, John F. Rocha, Luiz F. C. van Rossum-Wijsmuller, Ryeburn, David Ray-Chaudhuri, Richards, Pamela C. Roche, Claude Marijke Rykken, Charles John Dwijendra K. Richardson, Henry R. Rockwell, Richard D. Roth, Richard L. Saade, John Marshall Raymond, Frank A. Richardson, Roger W., Jr. Rod, David L. Roth, Rodney J. Saari, Donald G. Read, Dwight Richert, Hans-Egon Rodriguez, Gilmar Rothaus, Oscar S. Sabapathy, Thayaparan Reade, Maxwell 0. Richert, Norman J. Rodriguez, Sanjurjo Jose Rothberger, Fritz Sabbagh, Gabriel Reardon, Philip C. Richey, Matthew P. Roe, Robert P. Rothe, Erich H. Sabharwal, Ranjit S. Reber, Douglas C. Richman, Daniel J. Roettinger-Kaplan, Ida Rothenberg, Melvin G. Saccoman, John J. Rebnord, David A. Richmond, Glayvera Rofman, Edmundo Rothmaler, Peter A. Sachs, Robert L. Reddien, G. W., Jr. Richmond, Lawrence B. Roger, Claude Rothschild, Bruce L. Sacks, Paul E. Reddy, William L. Richter, Guenther E. Rogers, Jack W., Jr. Rothschild, Joseph G. Sadosky, Cora S. M. Redheffer, Raymond M. Richter, Horst P. Rogers, Robert C. Rothschild, Linda Preiss Saeed-ul-Islam, Redner, Richard A. Richter, Michael M. Roggenkamp, Klaus W. Rottenberg, Reuven R. Saeki, Sadahiro B. Reed, Coke Stevenson Richter, Stefan Rogowski, Paul B. Rotthaus, Christel Saff, Edward Reed, David L. Richter, Wayne H. Rohatgi, Vijay K. Rouche, Nicolas Sagan, Bruce E. Reed, Irving S. Richtmyer, Robert D. Rohl, Frank D. Roush, Fred Sagarman, Elma Rebecca Reed, James H. Rickart, Charles E. Rohlfs, Jurgen Rousseau, Cecil C. Sagle, Arthur A. Reed, John K. Rickman, B. Rohm, Dale M. Rousseau, Thomas H. Sah, Chih-Han Reed, Myra Jean Riddle, Malcolm Lee Rohrbach, Hans Roussos, Joannis Markos Sahab, Salem M. Reed, Robert E. Ridolfo, Anthony S. Rohrlich, David E. Roussou, Athina Saigusa, Youichi Reedy, Christopher L. Riedel, Herbert H. J. Roitman, Judith Rouys, Robert L. Saito, Akira Rees, Charles S. Rieger, Georg J. Romanelli, S. Rowan, William H. Saito, Kazuyuki Rees, Mina S. Riemenschneider, Oswald Romero-Fuster, Maria Rowe, Niles W. Saito, Kichi-Suke Reese, Matthias F. Riesel, Hans I. Carmen Roxin, Emilio 0. Saitoh, Saburou Reese, Sylvester Riess, Karlem Ronga, Felice L. L. Roy, Charles L. Sakai, Katsuro Reeves-Hicks, Ernestine Rigdon, Robert D. Rood, John J. Royden, Halsey L. Sakai, Makoto Regan, Francis Rigoli, Marco Roohan, James T. Royer, James Samuel Sakuma, Motoyoshi Rehm, Allan S. Rike, Thomas A. Rooney, Gerald E., Jr. Royster, Wimberly C. Salamin, Eugene

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1033 AMS Centennial Research Fellowship Fund

Salas, Saturnino L. Schafke, R. Schumacher, Georg B. Senechal, Lester J. Shimamoto, Don H. Saleem, Mohammad Schapiro, Philip J. Schumacher, Scott Senechal, Marjorie Shimizuike, Yuzi Salehi, Ebrahim Schaps, Maika Elisheva Schumaker, John A. Senft, J. R. Shimura, Goro Salinas, Luis C. Scharlau, Rudolf M. E. Schumaker, Larry Lee Senge, George Shin, Dong Sun Salinas, Norberto Scharlemann, Martin G. Schumitzky, Alan Sentilles, F. Dennis Shin, Yong-Tae Salisbury, Thomas S. Schatz, James R. Schunck, Brian G. Seppala-Holtzman, David Shinbrot, Marvin Salkind, William Schatzman, Michelle Schupp, Paul E. Serapioni, Raul Shinohara, Yaichi Salles, Maurice Schauer, Richard L. Schuster, Seymour Serbyn, William D. Shiohara, Hiroshi Sallman, Jeffrey E. Schaumberger, Norman Schuur, Susan E. Serfling, Robert J. Shiraiwa, Kenichi Saltiel, Leon Schay, Geza Schwartz, Ira B. Serre, Jean-Pierre Shiratani, Katsumi Salvadori, Anna Schechtman, Gideon Schwartz, Jacob T. Serrell, Robert Shochat, David D. Salzberg, Pablo M. Scheffer, Carel L. Schwartz, Laurent Serrin, James B. Sholander, Marlow C. Samelson, Christopher L. Scheffer, Vladimir Schwarz, Binyamin Servedio, Frank J. Shor, S. W. Williston Samelson, Hans Scheier, M. A. Schwarz, Gerald W. Sesay, Mohamed W. I. Shore, Terence R. Samolis, Ruth Anastasia Scheinberg, Stephen Schwarz, Thomas Sevensten, Arjen Shores, Thomas S. Samorodnitsky, Gennady Scheinerman, Edward R. Schweiger, Fritz Sever, Michael Short, Hamish B. Sampson, Gary Schell, Emil D. Schweizer, Berthold Sevilla, Alicia N. Shorter, William Ivan Samuel, Pierre Schellhorn, Jean-Pierre Schwietzer, Michael J. Sevrain, Christophe Jean-Paul Shreve, David C. Sanchez, David A. Schempp, Walter Scissors, Richard H. Sexauer, Norman E. Shreve, Warren E. Sanchez-Pedreno, Salvador Scherkoske, Kenneth D. Sclove, Stanley L. Sha, Xin Wei Shrikhande, Mohan Sharad Sanchez-Vazquez, Gonzalo Schiappacasse, Oscar Scoppola, Carlo M. Shabazz, Abdul Alim Shu, Chi-Wang Sanders, Georgia E. Schibell, Stephen T. Scott, Dana S. Shafer, John Shub, Michael Sanders, Jan A. Schirmeier, Horst Scott, Leonard L., Jr. Shaffer, Dorothy B. Shube, Beatrice Sanders, Robert W. Schlesinger, Ernest C. Scott, Paul James Shaftman, David H. Shucker, David S. Sands, Jonathan Winslow Schlichtkrull, Henrik Scott, Peter Shaharir Shulberg, Gary W. Sangwine-Yager, Jane Schlipf, John S. Scott, Ridgway Shahin, Mazen Mohammed Shult, Ernest E. Rosamund Schmeelk, John F. Scott, Warner Henry Shahshahani, Mehrdad M. Shultz, Harris S. Sant, Lino Schmerl, James H. Harvey, III Shaio, Jack Shutt, Elsie G. Santalo, L. A. Schmetterer, Leopold K. Scott, William R. Shalen, Peter B. Shwartz, Adam Santos Filho, Jose Ruidival Schmickler-Hirzebruch, Scott-Thomas, John F. Shallit, Jeffrey Outlaw Sibner, Lesley M. S.D. Ulrike Scourfield, Eira J. Shamee, Bishara Farid Sibner, Robert J. Sanz-Serna, J. M. Schmid, Josef Scovill¢, Richard Shanahan, Patrick Sichler, J. Saperstone, Stephen H. Schmid, Jurg Scriba, Christoph J. Shand, Sherman M. Siddiqi, Jamil A. Saphar, Pierre David Schmid, Rudolf Scull, Sidney C. Shaneson, Julius L. Siddiqi, Rafat N. Sapozhnikova, Vera D. Schmid, Wilfried Seal, Charles Edwin Shank, Herbert S. Sidek, Halim Sarafyan, Diran Schmidlin, Dean J. Searl, James E. Shapiro, Dawn M. Sidney, Stuart J. Sarapuka, Peter K. Schmidt, Dieter S. Sears, Michael Shapiro, George S. Sieg, Wilfried Sarason, Donald E. Schmidt .• Harvey J., Jr. Secunda, Jeanne Shapiro, Harold N. Siegel, Martha J. Sarason, Leonard Schmidt, Henry J., Jr. Sedaghat, Hassan Shapiro, Louis W. Siegrist, Kyle T. Sarian, Edward Schmidt, Kenneth C. Sedgewick, Charles H. W. Sharma, Ambikeshwar Siemon, Helmut Sario, Leo Schmidt, Klaus D. Sedory, Stephen A. Sharma, P. L. Siemons, Johannes Sarmah, Purandar Schmidt, Ralph 0. Sedwick, Jackson L. Sharma, Rakesh K. Sigurdsson, Ragnat Sarrat, Charles F. Schmidt, Roland Seebeck, Charles L., Ill Sharp, Henry, Jr. Silberger, Allan J. Sastri, Chelluri C. A. Schmidt, Wolfgang M. Seeley, Robert T. Sharpe, Michael J. Siler, Jack G. Satake, lchiro Schneebeli, H. R. Segers, Richard G. Sharpton, John T. Silva, Alessandro Sathaye, Avinash Schneider, David I. Segnari, T. James Shatz, Stephen S. Silva, Maria Welleda Sato, Hiroki Schneider, Hans Sehgal, Surinder Shaw, Guy B. Silvennoinen, Risto Pekka Sato, Ken-iti Schneider, Todd J. Sehgal, Virindra M. Shearer, Michael Silver, Daniel S. Sato, Masako Schneider, Victor P. Sehnert, James Ellis Sheffer, Albert D. Silverman, Herb Sattinger, David H. Schnitzer, Franz J. Seid, Howard A. Sheffer, Isador M. Silverman, Joseph H. Saut, Jean Claude Schnyder, Walter A. Seidel, Johan J. Sheil-Small, Terence Brian Silverstein, Martin L. Sauter, John K. Schochet, Claude L. Seiden, Esther Shekoury, Raymond Silvia, Evelyn Marie Savariraj, G. A. Schoenenberger-Deuel, Seidenberg, Abraham Shelly, Maynard W. Simkani, Mehrdad Sawka, John M. Johanna Seidman, Thomas I. Shelupsky, David I. Simmons, Charles W. Sawtell, Roy G., Jr. Schoenwaelder, Ulrich F. Seifert, Benedict Shemanske, Thomas R. Simo, Carles Sawyer, Emerson L. Scholl, Anthony J. Seifert, George Shen, C. Y. Simoes-Pereira, J. M.S. Sawyer, Eric T. Schon, Rolf Seifert, Herbert Shen, Mei-Chang Simons, Lloyd D. Sawyer, Stanley A. Schonbek, Maria Elena Seifter, Norbert Shepherd, J. C. Simons, Roger A. Sayeki, Hidemitsu Schoneborn, Heinz Seikkala, Seppo A. Shepherdson, John C. Sims, Charles C. Scanlon, Jane Cronin Schraegle, Horst Sein, Hla Hla Sherif, Nagwa Sinclair, Allan M. Scaramuzzi, Roberto Schrauf, Geza H. Seitz, Gary M. Sherk, F. A. Sinclair, Annette Scarpi, Giambattista Schraut, K. C. Sekino, Kaoru Sherlock, Philip Thomas Sinclair, George Edward Scarsini, Marco Schreiber, Bert M. Selby, Alan M. Sherman, Clayton C. Singer, Isadore M. Scattone, Francesco Schreiber, Morris Seldin, Jonathan P. Sherman, Thomas L. Singer, Phyllis E. Scedrov, Andre Schrijver, Alexander Selesnick, Stephen Allan Sherr, Joseph S. Singer, William M. Schaal, Werner G. H. Schroder, M. Seligman, George B. Shiba, Masakazu Singerman, David Schacher, Murray M. Schroeck, Franklin E., Jr. Seligman, Mark L. Shibata, Masanori K. Singh, Harpreet Schachermayer, W. Schroeder, Carolyn L. Seligson, Stuart A. Shibata, Yukio Singh, Kanhaya L. Schaefer, Paul T. Schryer, Norman L. Sellers, Peter H. Shibuya, Yhuji Singh, Mansa C. Schaeffer, John W. Schuett, Werner Selmer, Ernst S. Shiffman, Max Singh, Surjeet Schaettler, Heinz M. Schulte, Thomas R. Seltzer, Stanley E. Shiflett, Ray C. Singh, Udit N. Schafer, Alice T. Schultz, Martin Harvey Semmes, Stephen William Shiga, Koji Singleton, William Schafer, Richard D. Schulz, Friedmar J. Sempi, Carlo E. Shih, Weishu Eugene, Jr. Schaffer, Juan Jorge Schumacher, Carol S. Sen, R.N. Shimada, Nobuo Singman, David H.

1034 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Centennial Research Fellowship Fund

Singmaster, David B. Sobel, Bernard A. Stamey, William L. Stiller, Peter Frederick Sullivan, Richard W. Sinha, Shekhar Sohmer, Bernard Stammbach, Urs Stinson, Kevin Sullivan, Wayne G. Sink, Ann M. del Solar-Petit, Emilio Stampfli, Joseph G. Stob, Michael Summerhill, R. Richard Sinkala, Zachariah Soldate, Albert M., Jr. Stanaitis, Otonas E. Stock, John R. Summers, Timothy W. Sinkiewicz, Gerald P. Solin, Robert N. Stand, Donald L. Stocker, Harold Edward Summerville, Richard M. Sinnott, Warren Mann Solomon, Bruce Michael Stand, Mildred L. Stockmeyer, Paul K. Sun, Hugo Sui-Hwan Sirao, Tunekiti Solomon, Jimmy L. Stanek, Peter F. G. Stoddard, James H. Sun, Jiayong Sirisaengtaksin, Ongard Solomon, Louis Stanley, Lee James Stohs, Stephen M. Sun, Tien-Yu Sitaraman, Yegnaseshan Solomon, Ronald M. Stanley, Richard P. Stolarsky, Kenneth B. Sundaram, Sheila Sitia, Candido Solow, Donald D. Stanton, Dennis W. Stolberg, Harold J. Sundararaman, D. Siu, Man-Keung Sommese, Andrew John Stanton, Nancy K. Stoltenberg-Hansen, Viggo Sundaresan, Kondagunta Siu, Yum Tong Song, Sung-Yell Staples, Edmund Stoltzfus, Neal W. Sunderland, Benjamin B., Jr. Sivera, Rafael Song, Young-Seop Beauderc, III Stolz, Stephan A. Sundt, Mitchell E. Sizer, Walter S. Soni, Kusum K. Staples, John C. Stone, Alexander P. Sung, Li-yeng Skeath, James E. Soni, Raj Pal Starbird, Michael Stone, Charles J. Sunley, Judith S. Skiba, Sandra Ann Sonneborn, Lee M. Starbird, Thomas W. Stone, David A. Sunouchi, Gen-ichiro Skiffington, John Joseph, III Sonner, Johann Stark, Matthias J. Stone, David R. Sunseri, Mary V. Sklar, David Sonnesso, Geraldine Stark, Sheldon H., Jr. Stone, Eugene A. Suppes, Patrick Skoug, David L. Sons, Linda R. Starr, Edith N. Stone, Lawrence D. Suri, Manil Skupien, Zdzislaw Sontag, Alexia Henderson Stasheff, James D. Stone, Michael G. Suryanarayan, E. R. Slack, Michael D. Sontag, Eduardo Daniel State, Emile Y. Storey, Albert J. Sutner, Klaus Slaman, Ted Sonti, Babu Venkatesh Stavroudis, Orestes N. Storrer, Hans H. Sutton, Charles S. Slater, Michael Sontz, Stephen Bruce Steel, Thomas B., Jr. Stortz, Clarence B. Suwa, Tatsuo Sleeman, Brian D. Sorenson, John R. Steele, Alfred T. Story, Helen F. Suzuki, Haruo Slepian, Paul Sormani, Michael J. Steele, William F. Stotland, Stephanie Anne Suzuki, Michio Sloane, Neil J. A. Sorvali, Tuomas Steen brink, J. Stout, Edgar Lee Sverdlove, Ronald Slook, Thomas H. Sot, Richard Edward Stefani, Stefano T. Stout, Lawrence Neff Swamy, Srikanta M. N. Slowikowski, Wojciech Souganidis, Panagiotis E. Stefansson, Jon R. Stout, Quentin F. Swan, Richard G. Sluchin, Benny Sousa Ramos, Jose Steffani, Ronald R. Straflin, Philip D., Jr. Swanek, James J. Slye, John M. Spain, Philip G. Stegeman, Jan D. Stralka, Albert R. Swann, Howard S. G. Small, Lance W. Spake, Reuben M. Stegenga, David A. Strang, Gilbert Swanson, David L. Small, Lynne Barnes Sparr, Gunnar Steger, Arthur Strate, Gordon J. Sward, Marcia P. Smart, John R. Spears, Otis S. Steger, Tim J. Straube, Emil J. Swardson, Mary Anne Smiley, Leonard Specht, Edward J. Stegun, Irene A. Straubing, Howard Swart, Johan Smiley, Leonard Spector, Rene Stehle, S. P. Straughan, Brian Swartz, Joseph H. Smillie, John Speed, Terence Paul Stein, Alan H. Straume, Eldar J. Swearingen, Daniel B. Smith, Barry F. Speh, Birgit Stein, Michael C. Strauss, Hubertus M. H. Sweeney, Edward T. Smith, Bryan A. Speiser, Robert Steinback, Myriam Strauss, Leon Sweeney, Orval L. Smith, David A. Spencer, Domina E. Steinberg, Leon Strauss, Walter A. Sweeney, William J. Smith, Edgar C., Jr. Spencer, Guilford L., II Steinberg, Maria W. Streater, David Neil Sweet, Melvin M. Smith, F. R. Spencer, Thomas Crawford Steinberg, Robert Strebel, Kurt Swick, Kenneth E. Smith, Frank D., Jr. Sperb, R. P. Steiner, Anne K. Street, Ross H. Swift, Adrian Smith, George Rowan Sperry, Marvin G. Steiner, Donald D. Streitman, Max Swokowski, Earl W. Smith, Harold Robert, Jr. Spiegel, W. Steiner, Ray Phillip Stroeker, R. J. Sydora, Richard Dwayne Smith, Harvey A. Spielberg, Stephen E. Steinert, Leon A. Strogatz, Steven H. Sylvester, John Smith, James L. Spikes, Dolores Richard Steinhardt, Fritz Strohl, G. Ralph, Jr. Szabo, Manfred Egon Smith, John M. Spiro, Andrea Steinhorn, Charles I. Strom, Arne Szalajka, Walter S. Smith, Justin R. Spitzer, Frank L. Steketee, Jakob A. Stromberg, Marc R. Szczarba, Robert H. Smith, Lois E. B. Spivack, Mark Stelzer, Joerg M. Stromquist, Walter R. Szczyrba, Igor Smith, Marianne Freundlich Spivakovsky, Mark Stenger, Frank Stroock, Daniel W. Sze, Alexander Smith, Michael G. Spragens, William H., Jr. Stengle, Gilbert A. Stroth, Gernot H. W. Szekeres, George Smith, Mi-Soo Bae Sprague, Alan P. Stenson, David L. Stroud, Eric G. Szeto, George Smith, Perry B. Sprecher, David A. Stenson, Ellen M. Strouse, Elizabeth J. Szeto, Mabel Smith, Richard Allan Springer, Tonny A. Stenzel, Matthew B. Struik, Dirk J. Sznitman, Alain-Sol Smith, Robert Colman Spruck, Joel Sterling, Theodor D. Struppa, Daniele C. Szucs, Joseph M. Smith, Roland F. Squier, Craig C. Stern, Alan Spector Struwe, M. Szulkin, Andrzej Smith, Roy C. T. Srinivasan, Bhama Stern, Louis Gabriel Stuart, Donna J. Szwec, R. J. Smith, Stephen B. (Utah) Srinivasan, T. P. Stern, Mark A. Stueckle, Samuel K. Tabak, Barbara L. Smith, Stephen B. Srinivasan, V. K. Stern, Robert B. Stufken, John Tabib, Claudette (Massachusetts) Srivastav, Anupam Stern, Ronald J. Stukes, C. Randolph Tadic, Marko Smith, Stephen D. Srivastav, RamP. Sternberg, David Sturm, Teo Tadmor, Eitan Smith, Truett Lee Stacey, Peter John Sterrett, Andrew, Jr. Stynes, Martin Taft, Earl J. Smith, Wayne Stewart Stachniak, Zbigniew Stetkaer, Henrik Su, Yuh-Ling Tagawa, Masa-Yoshi Smith, Wilbur L. Stackelberg, Olaf P. Stevens, Damon R. Suarez, Pedro A. Taggart, Marcia D. Smith, William K. Stafney, James D. Stevens, Glenn Howard Suchanek, Ana M. Taiani, Geraldine Smolarski, Dennis C. Stahl, Saul Stevens, John G. Sueiro-Bal, Juan Taibleson, Mitchell H. Smoller, Joel A. Stahmer, Karl Pangburn Stevens, Randy S. Sueyoshi, Yutaka Taira, Kazuaki Smyth, William F. Staicu, Vasile Stevens, Richard S. Sugar, Alvin C. Tait, Robert James Snaith, Victor P. Stair, Lynda F. Stevens, T. Christine Sugawara, Masahiro Takagi, Izumi Sneddon, Ian N. Stakgold, Ivar Stewart, Cameron L. Sulanke, Robert A. Takahashi, Reiji Snell, J. Laurie Staknis, Victor R. Stewart, G. W. Sulis, William H. Takahashi, Shuichi Snover, James E. Stallard, M. Craig Stewart, Robert C. Sullivan, Dennis P. Takeda, Ziro Snyder, William Merritt, Jr. Stallings, Paul R. Stickel, Eberhard U. Sullivan, Francis J. Takenaka, Shigeo Soare, Robert I. Stallmann, Friedemann W. Stiles, Joseph L. Sullivan, Joseph A. Takesaki, Masamichi

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1035 AMS Centennial Research Fellowship Fund

Takeuchi, Masaru Thoele, Edward C. Torre, Nicolo Goodrich Turiel, F. J. Van Meter, Robert G. Takeuti, Gaisi Tholen, Walter P. Torrecillas, Bias Turisco, Joann Stephanie VanOsdol, Donovan H. Takizawa, Kiyoshi Thorn, Rene Torres de Squire, Maria L. Turkington, Bruce E. Van Rootselaar, Bob Takizawa, Seizi Thoma, Elmar H. Torres, Euclides Turner, Edward C. Van Rossum, Herman Talenti, Giorgio G. Thomas, Alan T. Torres, M. Turner, George W. Van Stone, Kathryn L. Talge, Gordon J., III Thomas, Charles B. Torres, Waldo A. Turner, Jim M. Van Tuyl, A. H. Talmadge, Richard B. Thomas, Clayton J. Totaro, Burt J. Turner, Lincoln H. Van Veldhuizen, M. Tam, Bit-Shun Thomas, Erik G. F. Tousley, Horace W. Turner, Peter R. Van Vleck, Fred S. Tamari, Dov Thomas, George B., Jr. Townsend, James T. Turner, Philip H. Van Wesenbeeck, Walter F. Tamraz, Abdullah J. Thomas, Janet D. Towsley, Gary W. Turquette, Atwell R. Vance, Elbridge P. Tan, Henry K. Thomas, Ken E. Toyoshima, Tetsunosuke Turrittin, Hugh L. Vance, James T., Jr. Tanabe, Hiroki Thomas, Lawrence E. Traina, Charles R. Turull, Alexandre Vanden Boss, Eugene L. Tanaka, Hisao Thomas, Marc Phillip Tranah, David A. Turyn, Richard J. Vanderbei, Robert Tanaka, Jun-Ichi Thomas, Mark A. Transue, William R. Tutte, W. T. Vanstone, James R. Tanaka, Katsumi Thomas, P. Emery Trautman, David A. Tuzar, Jaroslav Vaquie, Michel Tanasi, Corrado Thomas, Pascal J. Trebels, Walter Tuzhilin, Avgustin Varadhan, Srinivasa S. R. Tancredi, Alessandro Thome, Edward L. Tremain, Janet C. Tymchatyn, Edward D. Varadi, Ferenc Tancreto, Daniel Louis Thomee, Vidar Trench, William F. Tzavaras, Athanassios E. Varma, Badri N. Tang, Alfred S. Thompson, Harold B. Trenholme, Alice Ruth Tzeng, Chun-Hung Vasak, Janet Trzcinski Tang, Betty M. Thompson, Jon H. Treves, Francois Uchiyama, Akihito Vasavada, Mahavirendra H. Tang, Hwa Tsang Thompson, Layton 0. Treybig, L. Bruce Uchiyama, Saburo Vasconcelos, Wolmer V. Tanimoto, Taffee T. Thompson, Lee D. Tribbey, William P. Ueno, Kazushige Vaserstein, L. N. Tanovic, Adrian Benedict Thompson, Mary E. Tricerri, Franco Ueno, Sueo Vatsala, A. S. Tao, Yoko Thompson, Robert C. Trigiante, Donato Uesugi, Toshitane Vaughn, John B. Tariq, Nessim Thompson, Robert James Trimble, Selden Y. Uhlenbeck, Karen Vaught, Robert L. Tartakoff, David S. Thompson, Russell C. Tripp, John C. Uhlmann, Werner Vazquez, Juan L. Tashiro, Yoshihiro Thompson, William Rae Trivieri, Lawrence A. Ukegawa, Takasaburo Vazquez-Abal, Maria Elena Tatevossian, Leon H. Thomson, Brian S. Troesch, B. Andreas Ullery, William D. Vega de Prada, Jose Manuel Taussky, Olga Thorbergsson, Gudlaugur Troger, Hans Ullman, Joseph L. Vegas, Jose M. Tavantzis, John Thorne, John Farley, III Tromba, A. J. Ulrich, Bernd Veldman, Willem H. Tavares dos Santos, Geovan Thrall, Robert M. Trombi, Peter C. Ulrich, Klaus Vella, David C. Tavernini, Lucio Thrash, Joe B. Trone!, Gerard Umeda, Takato Velleman, Daniel Taylor, Angus E. Thron, Wolfgang J. Tropp, Henry S. Umegaki, Hisaharu Venzi, P. Taylor, B. A. Throop, T. Trotman, David J. A. Unai, Yasushi Unai Venzke, Paul B. Taylor, Donald E. Tichy, Robert F. Trotter, Hale F. Una!, Aynur Vera, Gabriel Taylor, Gladys Gillman Tickamyer, Cecil H., II Trotter, P. G. Una!, Aysenur Verde Star, Luis Taylor, Keith F. Tierney, Ann R. Troue, Jacques Underwood, Douglas H. Verdier, Jean-Louis Taylor, Lafayette K. Tierney, John A. Troy, Alan Ungar, Abraham Verma, Ghasi R. Taylor, Michael E. Tierney, Michael J. Troyanov, Marc Unger, Ernst R. Vermes, Robert Taylor, Richard M. Tijdeman, Robert Trucano, Timothy Guy Uno, Katsuhiro Veron, Laurent Taylor, S. James Tijs, Stef H. Trudinger, Neil S. Upatisringa, Visutdhi Verona, Maria Elena Taylor, Thomas J. Tilidetzke, Robert James Trueman, D. C. Upmeier, Harald Verrios, Katherine G. D. Taylor, Walter F. Tillmann, Heinz G. Trytten, George N. Upton, John A. W. Vervaat, Wim Taylor, William W. Timm, Mathew T. Tsai, Long-Yi Urda, Michael Vesentini, Edoardo Tee, Garry J. Timmerscheidt, Klaus Tsai, Tian-Yue Ursini, Aldo Vetter, Udo Teirlinck, Luc M. Tin, Saw Tsai, Wei-Ling Usmani, Riaz A. Vietsch, W. Karel Teissier, Bernard Tinsley, Frederick C. Tsai, Yen-Shung van der Vaart, H. Robert Villamil, Hilda P. Tekumalla, Kameswari V. Tippett, James M. Tsang, Kai-Man Vaillancourt, Jean Villar, Luzviminda Viray Teleman, Nicolae Tischler, David C. Tsangaris, Panayiotis Vainio, Reino Johan Vinacua, Alvaro Teleman, Silviu Titani, Satoko Tsatsomeros, Michael J. Vajravelu, Kuppalapalle Vincenzi, Antonio Temam, Roger Tits, Jacques L. Tsau, Chichen Michael Valenza, Robert J. Vinti, Calogero Tena, Juan Tjiok, Mouw-Ching Tschantz, Steven T. Valere Bouche, Liane Viola, Carlo Tenenblat, Keti Tobias, Richard J. Tsuchikura, Tamotsu Valeriote, Matthew Viswanathan, T. M. Tennant, Raymond Frederick Toby, Ellen H. Tsuchiya, Takuya Valero, Manuel Ovidio Vitale, Richard A. Tepedino A., Gaetano Toda, Nobushige Tsuji, Kazo Valle, Antonio Viterbo, Claude Teply, Mark L. Todd, John Tsukada, Makoto Vamanamurthy, Mavina K. Vitter, Albert L., III Tepping, Benjamin J. Todor, Fabian Tsukiyama, Kouzou Van Alstyne, John P. Viviente, Jose L. Terao, Hiroaki Todorov, Pavel Georgiev Tsukui, Yasuyuki Van Assche, Walter J. Vodola, Paul A. Terman, David Togo, Shigeaki Tsutsumi, Masayoshi Van Bendegem, Jean Paul Vogan, David A., Jr. Terng, Chuu-Lian Toledo, Domingo Tu, Cheng Van Casteren, Johannes A. Vogel, Thomas I. Terras, Riho Toll, Charles H. Tu, Yih-0 Van de Wetering, R. Lee Vogt, Andrew Terrier, Jean Marc Toll, Kathryn B. Tubaro, Luciano Van Den Ban, Erik P. Vogt, E. Tesei, Alberto Tolle, Louis F. Tucci, Ralph P. Vander Geer, Gerard Vogtmann, Karen Testerman, Donna M. Tolosa, Juan Tucker, Albert W. Van Der Hoek, John Voichick, Michael Tewfick, Ahmed Hisham Tolsted, Elmer Tucker, Charles T. Van Der Kallen, Wilberd Voiculescu, Dan Textorius, Bjorn 0. B. Tomaszewski, Boguslaw Tucker, Don H. Van Der Meer, Jan-Cees Voigt, PaulS. Thelen, Brian J. Tomczak-Jaegermann, N. Tucker, Howard G. VanDer Waerden, B. L. Voigt, Robert G. Theodorescu, Radu Tomer, Eugene R. Tucker, Thomas W. Van Eck, H. N. Vojta, Paul A. Theohari-Apostolidi, Tominaga, Hisao Tuckerman, Bryant Van Emde Boas, Peter Volkmann, Bodo Theodora Tomiyama, Jun Tugue, Tosiyuki Van Engelen, F. Voytuk, James A. Theusch, Colleen J. Tompson, R. N. Tull, Jack P. Van Geemen, Bert Vrbik, Jan Thibodeaux, Michael Tonga, Marcel Tuller, Annita Van Haagen, A. J. Vrscay, Edward R. Thickstun, Thomas L. Topsoe, Flemming Tung, Shih-Hsiung Van Keppel, Marilyn J. Vukomanovic, Djordje V. Thickstun, W. R., Jr. Torgerson, Norman Wesley Turgeon, Jean M. Van Lint, Jacobus H. Vyborny, Rudolf

1036 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY ~···------,------.------AMS Centennial Research Fellowship Fund

Wada, Hidekazu Warschawski, Stefan E. West, Dennis E. Williams, Joyce W. Wood, John W. Wada, Junzo Washburn, Robert B., Jr. West, Donald C. Williams, Kenneth S. Wood, Thomas E. Wada, Toshimasa Washington, Lawrence C. West, James E. Williams, Lawrence R. Woodring, Tom Wadsworth, Adrian R. Wasow, Wolfgang R. Westbrook, Edwin P. Williams, Mark Woodroofe, Michael B. Waelbroeck, Lucien Wasserman, Arthur G. Westman, Joel J. Williams, Richard R., Jr. Woodrow, Robert Edward Wage, Michael L. Watanabe, Shoji Westphal-Schmidt, U. Williams, Robert F. Woodruff, William M. Wagener, Raymond Watatani, Yasuo Westwood, Derek J. Williams, Ronald 0. Woods, Alan C. Wagner, Gretchen B. Waterhouse, William C. Wetherbee, Ted J. Williams, Ruth J. Woods, Dale Wagner, Richard C. Wathen, Judy A. Wetzel, Marion D. Williams, Stanley C. Woods, Jerry D. Wagoner, John B. Watkins, David S. Wexler, D. Williams, Susan Gayle Woods, R. Grant Wagreich, Philip D. Watkins, Murray S. Wheeler, Charles H., III Williams, Vincent C. Woolf, William B. Wagstaff, Samuel S., Jr. Watkinson, Anthony Wheeler, Robert F. Williams, William 0. Woolford, Thomas L. Wahba, Grace G. Watnick, Richard M. White, Alvin M. Williamson, Charles K. Woyczynski, Wojbor A. Wahl, Jonathan M. Watson, David P. White, Arthur T. Williamson, Frank, Jr. Wright, Charles R. B. Wahlbin, Lars B. Watson, Harry A., Jr. White, Benjamin S. Williamson, Jack Wright, David Wainwright, John Watson, James D. White, Brian Cabell Williamson, Susan Wright, David G. Waiveris, Charles F. Watson, Martha F. White, George N., Jr. Willis, Barton L. Wright, David J. Wajnryb, Bronislaw Watson, Rodwell B. White, Janis E. Wilson, David L. Wright, Jeffrey Allen Wakae, Masami Watson, Saleem White, John T. Wilson, John H. Wright, Jill D. Wakana, Kaoru Watson, Stephen White, Lee Wayne Wilson, Leslie Charles Wright, Marcus W. Wake, Bob Alan Wavrik, John J. White, Neil L. Wilson, Paul R. Wright, Mary H. Walczak, Hubert R. Wayman, Arthur K. White, Stevan Russell Wilson, Raj Wright, Thomas Perrin, Jr. Walde, Ralph E. Wayne, Alan White, Wm A. Wilson, Raymond B. Wrobleski, William J. Walden, George W. Wayne, David Scott Whitehart, F. E. Wilson, Robert Lee Wschebor, Mario Waldinger, Hermann V. Weaver, John C. Whitehead, George W. Wilson, Ted C. Wu, Ching-mu Wales, David B. Weaver, Robert Wooddell Whitfield, John H. M. Winarsky, Norman D. Wu, Hung-Hsi Walker, Carol L. Webb, Glenn F. Whitley, W. Thurmon Windham, Michael Parks Wu, Ling-Erl Eileen T. Walker, Homer F. Webster, Sidney M. Whitman, Andrew P. Wingate, John W. Wu, T. C. Walker, James Edward Wechsler, Martin T. Whitman, Philip M. Wingler, Eric J. Wulf, Leo M. Walker, James S. Wedel, Arnold M. Whitney, D. Ransom Winker, Steven K. Wylie, Clarence R., Jr. Wall, Charles T. C. Wee, In-Suk Whitney, James N. Winslow, Dennis N. Wyss, Walter Wall, Curtiss E. Wehausen, John V. Whitney, Ronald F. Winslow, Richard E. Xia, Jingbo Wall, G. E. Wehlen, Joseph A., Jr. Whitson, Kim J. Winston, Kenneth Yachter, Morris Wallace, David A. R. Weibel, John L. Whitt, Lee B. Winter, Eva P. Yadin, Micha Wallace, David Robert Weil, Clifford E. Whittaker, James V. Winters, Bobby N. Yahya, S.M. Wallach, Nolan R. Weiler, Jerome T. Whitten, Ruth Ann Wirszup, Izaak Yajima, Kenji Wallen, C. J. Weill, Lawrence R. Whitten, Wilbur Wise, Gary Lamar Yaku, Takeo Wallen, Lawrence J. Weiner, Jerome H. Whyburn, Kenneth G. Wise, William Albert Yale, Paul B. Walsh, John J. Weiner, Joel L. Whyburn, Lucille E. Wiskott, Bettina Yamada, Miyuki Walsh, John Thomas Weinert, Hanns Joachim Wick, Brian D. Wissner, Heinz-Wolfgang Yamada, Naoki Walsh, Mary L. Weinitschke, Hubertus J. Wicker, fletcher D. Witsenhausen, Hans S. Yamada, Shinichi Walster, G. William Weinkam, James J. Wickerhauser, Mladen Victor Witt, Donald M. Yamada, Toshihiko Walter, Charles N. Weinstein, Alan D. Wicks, Thomas M. Witte, David S. Yamaguchi, ltaru Walter, Gilbert G. Weinstein, Jack Widman, Kjell-Ove H. Witten, Louis Yamaguchi, Jinsei Walter, John H. Weinstein, Michael I. Wiegand, Roger A. Wittman, Richard H. Yamaguchi, Seiichi Walter, Wolfgang L. Weinstein, Stephen T. Wiegand, Sylvia Margaret Wittner, Ben Scott Yamakawa, Mineo Walters, Eleanor B. Weinzweig, Avrum I. Wiegmann, Klaus Werner Witzgall, Christoph Yamaki, Hiroyoshi Walters, Peter Weis, Lutz Wielandt, Helmut W. Woan, Wen-Jin Yamamoto, Koichi Waltman, Paul Weiser, Alan Wielenberg, Norbert J. Wochele, Mark D. Yamanoshita, Tsuneyo Walton, Jay R. Weisgram, Thomas P. Wiener, Howard W. Woess, Wolfgang Yamaoka, Kenya Walus, Herwig Weiss, David M. Wightman, Arthur S. Wolf, Edwin M. Yamasaki, Masayuki Wan, Frederic Y. Weiss, Guido L. Wihstutz, Volker Wolf, Herbert S. Yanagawa, Minoru Wanby, Goran Weiss, Norman J. Wiid, Frans G. Wolf, Thomas R. Yanagi, Kenjiro Wang, Cecilia Y. Weiss, Paul Wik, Anders Wolfe, Carvel S. Yang, Chung-Tao Wang, Derming Weiss, Richard M. Wik, Ingemar L. Wolff, Manfred P. H. Yang, Deane Wang, Edward T. H. Weitkamp, Galen L. Wilcox, Calvin H. Wolfson, Kenneth G. Yang, George Yanji Wang, Hsiao-Lan WelCh, Lloyd R. Wildenhain, Kenneth R. Wolk, Elliot S. Yang, Jae-Hyun Wang, Hwai-Chiuan Wellner, Jon A. Wilf, Herbert S. Wolkowicz, Gail S. K. Yang, Jeong Sheng Wang, Kwang-Shang Wells, Charles Michael Wilkins, J. Ernest, Jr. Wollman, Stephen Yang, Kung-Wei Wang, Stuart Sui-Sheng Wells, David M. Wilkinson, Steven V. Wolpert, Scott A. Yang, Wei-Shih Wang, Wei-Ping Wells, James H. Willard, Stephen Womble, David E. Yano, Kentaro Wang, Ying Hsiung Wells, R. 0., Jr. Willcox, Alfred B. Wong, James S. W. Yanowitch, Michael Wantland, Evelyn K. Welters, G. E. Williams, Bennie B. Wong, Maurice K. F. Yao, Andrew Chi-Chih Warchall, Henry A. Wendland, Wolfgang L. Williams, Charles W. Wong, Raymond Y. Yaqub, Fawzi M. Ward, Harold N. Wendroff, Burton Williams, Dana P. Wong, Roman Woon-Ching Yaqub, Jill S. Wardle, Robin W. Wendt, Dieter W. F. Williams, Daniel Arthur Wong, Sherman K. Yasue, Kunio Ware, Buck Wene, Gregory P. Williams, David A. Wong, Shiu-Chun Yasugi, Mariko Ware, Roger P. Wente, Henry C. Williams, David E. Wong, William W. Yasuhara, Ann Warfield, Robert B., Jr. Wenzl, Hans Williams, Dorothy J. Wong, Yung-Chow Yasuhara, Mitsuru Warhola, Gregory T. Wermer, John Williams, E. Bruce Woo, Sung-Sik Yeager, Dorian P. Warner, Frank W., III Werner, Peter Williams, Francis D. Wood, David H. Yebra, Jose Luis Andres Warnock, Robert L. Werner, Willes L. Williams, George K. Wood, Geoffrey V. Yen, David H. Y. Warren, Bette L. Wernick, Robert J. Williams, Hugh M. Wood, Jay A. Yhap, Ernesto Franklin Warren, William E. Wessel, W. Roy Williams, James G. Wood, John C. Yohe, J. Michael

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1037 AMS Centennial Research Fellowship Fund

Yokoi, Hideo Younglove, James N. Zaslove, Barry L. Zinn, J. Zuckerberg, Hyam L. Yoneda, Kaoru Ypma, Tjalling J. Zeheb, Ezra Zipse, Philip W. Zuckerman, Gregg J. Yoneguchi, Hajimu Yu, Lucille Chieh Zehnder, Eduard J. Zirilli, Francesco Zuckerman, Paul R. Yorke, James A. Yui, Noriko Zemmer, Joseph L., Jr. Zitron, Norman R. Zweibel, John A. Yoshiara, Satoshi Yung, Mordechai M. Zhao, Pei-Yi Zitzler, Siham Braidi Zweifel, Paul F. Yoshida, Norihiro Yung, Tin-Gun Zhou, Xian Zizler, Vaclav Zygmund, Antoni Yoshida, Zensho Zaballa, Ion Zhou, Zhiming Zlatev, Zahari Anonymous (88) Yoshimoto, Takeshi Zack, Matthew M. Zia, Lee L. Zo, Felipe J. Yoshino, Genji Zacks, Shelemyahu Ziegler, Zvi Zoch, Richmond T. Yoshino, Takashi Zadeh, Lotfi A. Ziemer, William P. Zoercher, C. Z. Yoshizawa, Taro Zagier, Don Bernard Zierau, Roger Craig Zomorrodian, Reza Young, Barry H. Zajac, Edward E. Zierler, Neal Zondek, Bernd Young, Donald F. Zalik, R. A. Ziller, Wolfgang Zoreda-Lozano, Juan J. Young, Eutiquio C. Zama, Nobuo Zilmer, Delbert E. Zorn, M.A. Young, Lael M. Zamfirescu, Christina M. Zimering, Shimshon Zorn, Paul Young, Paul M. Zanolin, Fabio Zimmerman, Donald W. Zoroa, P. Young, Sam Wayne Zarantonello, Sergio E. Zimmerman, Grenith J. Zorzitto, Frank A. Young, Wo-Sang Zaslavsky, Alan M. Zimmerman, Jay J. Zsido, Laszlo Younger, Daniel H. Zaslavsky, Thomas Zimmermann-Huisgen, Birge K. Zucker, Steven M.

MATHEMATICAL INTUITIONISM: INTRODUCTION TO PROOF THEORY A. G. Dragalin (Translations of Mathematical Monographs, Volume 67)

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. 03 logics are used in the theory of computations, 1980 Mathematics Subject Classification: Nonclassical ISBN 0-8218-4520-9, LC 87-32766 in information theory, and for the description of systems ISSN 0065-9282 of heuristic programming. lntuitionistic logic is a particularly 224 pages (hardcover), March 1988 important nonclassical logic. The aim of this book is to Individual member $45, List price $75, present the most important methods of proof theory in Institutional member $60 To order, please specify MMON0/67 NA intuitionistic logic and to acquaint the reader with the prir:~cipal axiomatic theories based on intuitionistic logic. The exposition, accessible to a wide audience, requires only an introductory course in classical mathematical logic.

Shipping/Handling: 1st book S2, each additional S1. maximum $25; by air, 1st book S5. each additional S3, maximum S100 Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station Providence, Rl 02901-9930, or call toll free 800-556-7774 to charge with VIsa or MasterCard

1038 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY ANNOUNCING...

Ill MathSci™on CD-ROM ill

Now you can access Mathematical Reviews (MR) and Current Mathematical Publications (CMP) on CD-ROM (Compact Disc-Read Only Memory). The CD, called MathSci Disc, will combine the search­ ing features of online MathSci with the browsing ease of printed MR. For a fixed annual fee, MathSci Disc can be used at leisure without access charges or telephone connections.

Semi-Annual Issue MathSci Disc will be produced by SilverPlatter® and will be issued semi-annually. The first MathSci Disc, available in January 1989, will contain all the reviews and abstracts from MR 1985 through 1988 and over 50,000 entries from CMP. The July disc will have all the information on the January disc plus the January-June updates. Access to current information between successive CD issues is available online from MathSci, which is updated monthly on DIALOG, BRS, and ESA. Eas:r.-to-Op.erate

MathSci Disc will be available for both the IBM PC and the Macintosh. SilverPlatter's user-friendly CD software with help screens and menus will make MathSci Disc easy to use. Words and phrases in the text of the reviews and abstracts will be searchable with an adjacency operator. Records can be downloaded from the CD to the hard disk for editing or for processing with TEX software into typeset form with mathematics.

Fixed Cost

The MathSci Disc annual lease fee will include the January and the July issues of MathSci Disc, the SilverPlatter search software for the IBM or the Macintosh, the SilverPlatter search manual with a MathSci Disc chapter, and a toll-free help line.

The 1989 MathSci Disc will be available at a low annual lease fee, equal to ~hat of the printed MR: Nonmem-bers: List price- $3,510* AMS members & MR subscribers: additional leases- $2,106* AMS Members: $2,808* MR subscribers: $2,106* AMS members with an MR or first MathSci Disc subscription: $1,685*

Individuals at institutions subscribing to MathSci Disc can order a copy for personal use at a 90% discount: $351 *. (*Plus shipping & handling for addresses outside the U.S. and Canada.)

For more information on MathSci Disc or to receive a copy of the lease agreement, please contact Taissa Kusma, Head, Database Services, American Mathematical Society, P. 0. Box 6248, Providence, RI 02940 or call (800) 556-7774 in the continental U.S. or (401) 272-9500. Internet: [email protected]; Telex: 797192; FAX: 401-331-3842.

™ - MathSci is a trademark of the American Mathematical Society ®- Silver Platter is are istered trademark of SilverPlatter International N.V. 1988 AMS Elections

Candidates Publication Committees*

OFFICERS American Journal of David Gieseker Mathematics Vice President (one to be elected) Sun-Yung Alice Chang Melvin Hochster Bulletin Murray H. Protter

Secretary* Colloquium Charles L. Fefferman Robert M. Fossum Mathematical Reviews Leonard D. Berkovitz Associate Secretaries* John L. Selfridge Joseph A. Cima W. Wistar Comfort (South East) (East) Mathematical Surveys David Kinderlehrer and Monographs Treasurer* Franklin P. Peterson Mathematics of Computation Andrew M. Odlyzko Treasurer* Associate Adams Steve Armentrout Proceedings William W. J. Marshall Ash Maurice Auslander Member-at-Large of the Council (five to be elected) Andreas R. Blass L. Michael C. Reed Jonathan Alperin Clifford J. Earle, Jr. S. Berger Hugo Rossi Melvyn Palle E. T. Jorgensen Fan R. K. Chung George R. Sell Lawrence J. Corwin William Yslas Velez Transactions and Eugene B. Fabes James H. Curry Robert J. Zimmer Memoirs

Board of Trustees (one to be elected) NOMINATING COMMITTEE FOR 1989 Edwin E. Floyd Paul J. Sally, Jr. (Preferential Ballot, four to be elected) Joan S. Birman Ray A. Kunze EDITORIAL AND COMMUNICATION John B. Garnett Andy Roy Magid COMMITTEES James E. Humphreys James D. Stasheff Committee to Monitor Problems in Communication* Victor Klee Alan D. Weinstein (two to be elected) Arthur M. Jaffe Paul Nevai *Uncontested offices

1040 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Election Information

The ballots for election of members of the Council and Board of SUGGESTIONS FOR 1989 NOMINATIONS Trustees of the Society for 1989 will be mailed on or shortly after Council and Board of Trustees September I 0, in order for members to receive their ballots well in advance of the November I 0 deadline. Prior to casting their ballots Vice President (2) members are urged to consult the following articles and sections of the Bylaws of the Society: article I, section I; article II, sections I, 2; article III, sections I, 2, 3; article IV, sections I, 2, 4; article VII, sections I, 2, 5. The complete text of the Bylaws appears on pages Associate Secretaries (2) 1155-1160 of the November 1987 issue of Notices. A list of the members of the Council and Board of Trustees serving terms during 1988 appears in the AMS Reports and Communications section of this issue. Member of the Bulletin Editorial Committee (I) SUGGESTIONS FOR 1989 NOMINATIONS Each year the members of the Society are given the opportunity to propose for nomination the names of those individuals they deem Member of the Colloquium Editorial Committee (I) both qualified and responsive to their views and needs as part of the mathematical community. Candidates will be nominated by the Council to fill positions on the Council and Board of Trustees to Member of the Mathematical Reviews Editorial Committee (I) replace those whose terms expire December 31, 1989. See the AMS Reports and Communications section of this issue for the list of cur­ rent members of the Council and Board of Trustees. Members are Member of the Mathematical Surveys Editorial Committee (I) requested to write their suggestions for such candidates in the ap­ propriate spaces on the form in the next column. REPLACEMENT BALLOTS Members of the Mathematics of Computation Editorial Committee (2) This year ballots for the AMS election will be mailed September I 0, 1988, or within a day or two thereafter. The deadline for receipt of ballots in Providence is November I 0, 1988. There has been a small but recurring and distressing problem concerning members who state that they have not received ballots Members of the Proceedings Editorial Committee (3) in the annual election. It occurs for several reasons, including fail­ ure of local delivery systems on university or corporate properties, failure of members to give timely notice of changes of address to the Providence office, failures of postal services, and other human errors. Member of the Transactions and Memoirs Editorial Committee (I) To help alleviate this problem, the following replacement proce­ dure has been devised: A member who has not received a ballot by October I 0, 1988, or who has received a ballot but has accidentally Members of the Committee to Monitor Problems in Communication spoiled it, may write after that date to the Secretary of the AMS, Post (2) Office Box 6248, Providence, RI 02940, asking for a second ballot. The request should include the individual's member code and the address to which the replacement ballot should be sent. Immediately upon receipt of the request in the Providence office, a second ballot, which will be indistinguishable from the original, will be sent by first Members-at-large of the Council ( 5) class or air mail. It must be returned in an. inner envelope, which will be supplied, on the outside of which is the following statement to be signed by the member: The ballot in this envelope is the only ballot that I am submitting in this election. I understand that if this statement is not correct then no ballot of mine will be counted.

Member of the Board of Trustees (I) signature

Although a second ballot will be supplied on request and will be sent The completed form should be addressed to AMS Nominating Com­ by first class or air mail, the deadline for receipt of ballots will not mittee, Post Office Box 6248, Providence, RI 02940, to arrive no be extended to accommodate these special cases. later than November 10, 1988.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1041 AMERICAN MATHEMATICAL SociETY Expands Publication Program

In its Centennial year, the American Mathematical Several new book series and journals are planned to Society looks forward to increased opportunity to begin publication in 1988 and 1989. Some of them-­ serve the mathematical community through an ex­ such as the History of Mathematics Series -- repre­ panded publications program. One of the Society's sent new directions for the Society. Others-- such central goals is to promote communication about as the translation of expository articles from the mathematical sciences research through a diverse Japanese mathematical journal SDgaku -- reflect the and dynamic publications program. As one of the need for mathematicians to communicate more world's largest mathematical publishers, the AMS is easily across international lines. All of them exem­ committed to providing the high quality, reasonably plify the Society's commitment to serving the mathe­ priced publications the mathematical community matical community in the next century and beyond. needs.

HISTORY OF MATHEMATICS SERIES ___ AMS REPRINTS------­ This series begins in 1988 with books about Ameri­ The Society receives many requests for help in can mathematics during the past century. The titles obtaining copies of a number of excellent books in this series will present historical perspectives on that have gone out of print. As a service to the individuals who have profoundly influenced the community, the AMS will publish the series AMS development of mathematics, as well as those who Reprints, consisting of important research mono­ have made great contributions to the mathematical graphs and graduate level textbooks that have community, or will trace the development of special been declared out of print by the original publishers. areas of research.

SUGAKU EXPOSITIONS------­ Sugaku, published by the Mathematical Society of UNIVERSITY LECTURE SERIES, ____ Japan, is the Japanese counterpart of the Bulletin Lecture series provide an excellent forum for in­ of the AMS. Each issue of Sugaku contains several depth, and sometimes inspired, presentation of expository articles which provide highly informative mathematical topics, but often benefit few beyond accounts of a variety of current areas of mathe­ those attending them. To address this limitation, matical research. The AMS will publish these the AMS will publish books in the new University articles in a new translation journal called Sugaku Lecture Series which will preserve important lecture Expositions. series given at various institutions by outstanding mathematicians.

To submit a manuscript, contact To begin a STANDING ORDER for titles In the Director of Publication History of Mathematics or University Lecture American Mathematical Society Series or to begin a subscription to Sugaku P.O. Box 6248 Expositions, contact the Membership and Sales Providence, Rl 02940 USA Department at the AMS. (401) 272-9500 or (800) 556-77741n the continental U.S. Meetings and Conferences of the AMS

FUTURE MEETINGS

Lawrence, Kansas October 28-30 1045

Claremont, California November 12-13 1049

Invited Speakers and Special Sessions 1052

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1043 AMERICAN MATHEMATICAL Celebrate SOCIETY 1888-1988 100 YEARS OF AMERICAN MATHEMAnCS theAMS Centennial!

Now you can have a remembrance of the American Mathematical Society's 1OOth anniversary for your own -- a beautiful commemorative poster for your home, office, or department.

This striking poster is a photographic reproduction of the sculpture 'Torus with Cross-cap and Vector Field" by Helaman Rolfe Pratt Ferguson of Brigham Young University. The sculpture is a gift from the Mathematical Association of America to the American Mathematical Society on the occasion of its Centennial.

The poster is printed on museum quality AMERICAN MA.THEMATJCAL SOCIETY glossy paper. The sculpture is white and CENTENNIAL CELEBRATION ·1888·1988 photographed on a rich blue background. Torus with Cross-cap and Vector Field To order, please specify POSTER/NA List $10 each, plus $2 shipping and handling per order

To order your AMS anniversary remembrance. simply send a check or money order for $10 each (plus $2 shipping and handling per order) to: American Mathematical Society Annex Station, P. 0. Box 1571 Providence, RI 02901-9930 USA

Send MasterCard or VISA orders to: American Mathematical Society P. 0. Box 6248 Providence, RI 02940 or callS00-556-7774 to order by phone in the continental U.S.

For your convenience, an order form is available at the back of this issue. Lawrence, Kansas University of Kansas October 28-29

Second Announcement

The eight-hundred-and-forty-fifth meeting of the Ameri­ Noah Rhee, Dan Sorensesn, Dan Szyld, and Robert van can Mathematical Society will be held at the University de Geijn. of Kansas in Lawrence, Kansas on Friday, October 28, Algebraic geometry, BRUCE CRAUDER and SHELDON and Saturday, October 29, 1988. KATz, Oklahoma State University. Tentative speakers are Alberto Albano, Donu Arapura, Jim Carlson, Susan Crauder, Lawrence Ein, Eileen Fritz, Addresses Colley, Brucer Invited William Fulton, Tony Geramita, Brent Gordon, Brian By invitation of the Committee to Select Hour Speakers Harboume, Bruce Hunt, Sheldon Katz, Gary Kennedy, for Central Sectional Meetings, there will be four invited William Lang, Pablo Lejarraga, Genneady Lyubeznik, one-hour addresses. The speakers, their affiliations, and Juan Migliore, Prabhakar Rao, Igor Reider, Ahmad titles are: Rhayyel, Nick Shepherd-Barron, Roy Smith, Peter Stiller, BJ0RN DAHLBERG, Washington University, Elliptic and Robert Varley. boundary value problems in non-smooth domains. Control theory, TYRONE DUNCAN, University of STEVEN E. HURDER, University of Illinois at Chicago, Kansas Geometry and the index theory offoliations. Applications of set theory, WILLIAM FLEISSNER, Uni­ PETER ScoTT, University of Michigan, Ann Arbor, versity of Kansas. Tentative speakers include Paul Least area surfaces in 3-manifolds. Bankston, Amer Berslagic, Paul Corazza, Peg Daniels, SIDNEY M. WEBSTER, University of Minnesota, Min­ Sheldon Davis, Alan Dow, Gary Gruenhage, Win Just, neapolis, The integrability problems of complex analysis. John Kulesza, Withold Marciszwski, Arnie Miller, Jack Porter, Judy Roitman, Mary Ellen Rudin, Charles and Frank Tall. Special Sessions Scindwein, Real analysis, JAMES FoRAN, University of Missouri By invitation of the same committee, there will be twelve at Kansas City. Tentative speakers are Edward Arnold, special sessions of selected twenty-minute papers. Topics Jack Brown, P. S. Bullen, Geraldo Soares De Souza, and the names and affiliations of the organizers and a Henry Fast, Richard Gibson, H. P. Heinig, Paul Humke, list of tentative speakers, when available, follow. Kenneth Kellem, Cheng-Ming Lee, Sandra Meinersha­ Partial differential equations- Geometric theory, AN­ gen, Krzysztof Ostaszewski, Darwin Peek, Zbigniew Pi­ DREW ACKER, Wichita State University otrowski, and Daniel Waterman. Geometry and mathematical physics, JOHN K. BEEM, Flat bundles and geometric structures, WILLIAM MARK University of Missouri and PHILLIP·E. PARKER, Wichita GoLDMAN, University of Maryland. Tentative speakers State University. Tentative speakerS include Ian M. are D. Burns, S. Y. Choi, K. Corlette, S. Frank, D. Anderson, Brian DeFacio, Tevian Dray, Gerard Emch, Fried, D. Gallo, R. Hain, C. Hodgson, Y. Kamishima, Paul Ehrlich, Francis J. Flaherty, Gregory J. Galloway, R. Kulkarni, S. Luttinger, W. L. Lok, J. Millson, W. Steven G. Harris, Justin Huang, David Lerner, Corrine Neumann, J. Scherk, C. Simpson, J. Smillie, S. Tan, W. Manogue, Bahram Mashhoon, Adrian Melott, Forest Thurston, D. Toledo and S. Zucker. Miller, David Retzloff, John R. Urani, Walter Wei, Steve Operator theory and applications to geometry, STEVEN Wilkinson, and Chi-Ming Yau. E. HURDER, University of Illinois at Chicago and NoR­ Numerical linear algebra, RALPH BYERs, University BERTO SALINAS, University of Kansas. Tentative speakers of Kansas. Tentative speakers include Greg Ammar, Jesse include Kevin Clancey, Jeff Fox, James Heitsch, Gary Barlow, Chris Beattie, Mike Berry, Chris Bischof, Ralph Jensen, Palle Jorgensen, Franz Kamber, Jerry Kaminker, Byers, Biswa Datta, AI Geist, Bill Gragg, Nick Higham, Steven Krantz, Mike Lamoureaux, Paul Muhly, Andrew Liz Jessup, George Miminis, Steven Nash, Alex Pothen,

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1045 Meetings

Map of Lawrence and the campus of the University of Kansas

§DENOTES TRAFFIC SIGNAL NORTH

WEST +EAST

SOUTH

1046 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Rich, Richard Rochberg, Walter Rudin, Philippe Ton­ Accommodations deur, Albert Sheu, Keren Yan and Kehe Zhu. Blocks of rooms are being held at the lodging establish­ Commutative algebra, DANIEL KATZ and JEFFERY ments listed below. Participants should make their own LANG, of Kansas University reservations directly with the hotel of their choice and be Potential theory and partial differential equations in sure to identify themselves with the AMS meeting at the nonsmooth domains, JILL PIPHER, University of Chicago university. Please make reservations by October 12, 1988. and GREGORY VERCHOTA, University of Illinois at After that date, reservations will be accepted on a space Chicago. Tentative speakers are Rodrigo Banuelos, Rus­ available basis only. Prices listed below are subject to sell Brown, Eugene Fabes, Robert Fefferman, Carlos change and do not include applicable taxes of 8 percent. Kenig, John Lewis, Margaret Murray, Jill Pipher, Zhong­ Wei Shen, Gregory Verchota, and Jang-Mei Wu. Westminister Inn 3-manifolds, PETER ScoTT, University of Michigan. W. 6th Street Tentative speakers are Colin Adams, Mark Baker, Telephone: 913-841-8410 Mladen Bestvina, Marshall Cohen, Mark Feign, Bill Single $24 Double $32 Floyd, Dave Gabai, Cameron Gordon, Joel Hass, John Hempel, Ravi Kulkarni, Darren Long, Darryl McCul­ Virginia Inn lough, Bill Meeks, Geoff Mess, Bobby Meyers, Walter W. 6th Street Neumann, Ulrich Oertrel, Alan Reid, Danny Ruberman, Telephone: 913-843-6611 Marty Scharlemann, and Abby Thompson. Single $24 Double $33 Contributed Papers Park Inn W. 6th Street There will also be sessions for contributed ten-minute Telephone: 913-842-7030 papers. Single $29 Double $37 Registration All Seasons Iowa Street The registration desk will be located in the Level 4 Lobby Telephone: 913-843-9100 of the Kansas Union and will be open on Friday, October 28, from 8:00 p.m. to 3:00 p.m., and Saturday, October Single $28 Double $33 29 from 8:00 a.m. to 11:00 a.m. The registration fees are Best Western $30 for members of the AMS, $45 for nonmembers, and Iowa Street $10 for students or unemployed mathematicians. Telephone: 913-841-6500 To reach the registration area from Parking Lot 91, enter and walk through the understreet tunnel located Single $28 Double $36 at the southeast corner of Lot 91. When you enter the Rooms have not been blocked at the following loca­ Kansas Union building, take the elevators on the south tions but are included for informational purposes only. side of the stairwell to the Level 4 Lobby area. Holiday Inn McDonald Drive Social Event Telephone: 913-841-7077 or Toll free 800-238-8000 A social hour with a cash bar will be held on Friday Single $44 Double $50 evening, October 28 from 5:30 p.in. to 7:00 p.m. at the Eldridge Hotel 13th and Adams Alumni Center located on the corner of Massachusetts Street Oread Streets. Telephone: 913-749-5011 Single $61 Double $68 Petition Table

A petition table will be set up in the registration area. Food Service Additional information about petition tables can be found in a box in the Atlanta meeting announcement on Most local lodging establishments have restaurants for page 68 of the January issue of Notices. breakfast, lunch, and dinner. The Kansas Union building houses several food service facilities ranging from salad bar to deli to traditional meals. Additionally, Lawrence has a variety of fine restaurants, such as the Eldridge

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1047 Meetings

Hotel; Arthur Porters, 1511 W. 23rd Street; and Costello's Parking and Local Travel Greenhouse restaurant, 3400 W. 6th Street. Fast food Courtesy ':isitor parking permits are available through the restaurants can be found throughout Lawrence. Informa­ mathe~at1cs department at the university. Please send tion on area eating establishments will be included in your wntten request to: The Department of Mathematics your registration packet and will also be available at the University of Kansas, Strong Hall, Lawrence, KS 6604S registration area. not later than October 1, 1988. . Me_tered ~isitor parking at the University of Kansas Travel lS avadable m Lot 91 located just behind the Kansas Union. Visitors must park at metered spaces. The cur­ The University of Kansas main campus is located in rent rate is 25 cents per hour to a maximum of ten Lawrence, about forty miles west of Kansas City Inter­ hours. Meters accommodate nickels, dimes and quarters. national Airport which is served by most major airlines. Visitors parking in nonmetered spaces will be ticketed. Lawrence is also accessible by Greyhound buslines. Lawrence has local taxicab service information avail­ For those driving or renting cars at the airport, able at local motels and hotels. Lawrence can be reached by taking 1-29 south (from the airport) to 1-635 south to 1-70 west. There are two Andy Roy Magid exits for Lawrence: the East exit leads to the downtown Associate Secretary area; the West exit leads to the campus and lodging Norman, Oklahoma establishments listed above. Public limousine service to and from Kansas City International Airport is available from Corporate Coach, 913-841-5466, four times daily. Advance reservations are required and must be made 24 hours in advance.

January 11-14, 1989 Phoenix, Arizona

1048 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Claremont, California Claremont McKenna College November 12 -13

Second Announcement

The eight-hundred-and-forty-sixth meeting of the Amer­ Carolyn Gordon, Peter Perry, Peter Sarnak, Paul Yang ican Mathematical Society will be held at Claremont and Steven Zelditch. McKenna College, Claremont, California, on Saturday Differential and difference equations, STAVROS N. and Sunday, November 12 and 13, 1988. BUSENBERG, Harvey Mudd College, and MARIO MARTELLI, California State University, Fullerton. Ten­ Invited Addresses tative speakers are: N. Ailikakos, S. N. Chow, Donald Cohen, J. Cronin, Jim Cushing, Hector Fattorini, William By invitation of the Committee to Select Hour Speak­ Fitzgibbon, Patrick Fitzpatric, Frederick Howes, Russell ers for Far Western Sectional Meetings, there will be Johnson, K. Kreith, G. Ladas, Edward Landesman, J. three invited one-hour addresses. The speakers, their Lorenz, Joseph Mahaffy, Kenneth Palmer, Gary Rosen, affiliations, and some of the titles follow: K. Scmitt, George Sell, Hal Smith, Horst Thieme, and WILLIAM JACOB, Oregon State University, Galois co­ Pauline van den Driessche. homology and K-theory: Applications to division algebras Division rings, WILLIAM JACOB and ADRIAN and quadratic forms. WADSWORTH, University of California, San Diego. Ten­ ROBERT BROOKS, University of Southern California, tative speakers include: A. Blanchet, T. Craven, F. The spectrum of the Laplacian in 1988. DeMeyer, D. Estes, B. Fein, L. Gerstein, T.-Y. Lam, FRANCIS BONAHON, University of Southern Califor­ J. Minac, P. Morandi, A. Rosenberg, D. Saltman, 0. nia, Riemann surfaces and measured laminations. Taussky-Todd, R. Ware, and S. Yuzvinsky.

Special Sessions Contributed Papers By invitation of the same committee, there will be five There will also be sessions for contributed ten-minute special sessions of selected twenty-minute papers. The papers. topics, names and affiliations of the organizers and a list of tentative speakers are: Activities Low dimensional geometry, FRANCIS BoNAHON and of Other Organizations DAVID GABAI, California Institute of Technology. Ten­ The Mathematical Association of America will meet on tative speakers include: David Austin, Mladen Bestvina, Saturday, November 12. LEONARD GILLMAN, University Peter Braam, Andrew Casson, Daryl Cooper, David of Texas and President of MAA, and MARVIN MARKUS, Gabai, Cameron Gordon, Matt Grayson, Joel Hass, University of California at Santa Barbara, will give Darren Long, Steve Kerckhoff, Eric Klassen, Darryl Mc­ AMS-MAA joint invited addresses. There will be a Cullough, Lee Mosher, Bob Penner, Marty Scharlemann, luncheon at noon in the Athaeneum; the cost is $8. and Ron Stern. SOLOMON GOLOMB, University of Southern California Computers and software in mathematical research, will speak on The numerical range. There will be two ROBERT BORRELLI, Harvey Mudd College, and COURT­ special sessions titled Mathematical notes and classroom NEY S. COLEMAN, Harvey Mudd College. Tentative capsules and Mathematics as a humanistic discipline. speakers are: Frederick Dashiell, Jr., David Fisher, Ned Freed, Charles Lawson, Gottfried Mayer-Kress, and Robert Valenza. Registration The spectrum of the Laplacian, RoBERT BROOKS and The meeting registration desk will be located in the lobby S.-Y. CHENG, University of California, Los Angeles. of Bauer Center. The desk will be open from 8:30 a.m. Tentative speakers include: Isaac Chavel, Peter Doyle,

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1049

:;-

f f

-

~ ~

l l

A A

N N

66 66

Airpon Airpon

lnlern:ttion:.tl lnlern:ttion:.tl

Ont:~riu Ont:~riu

BLVD./U.S. BLVD./U.S.

CLAREMONTMCKENNACOLLEGE CLAREMONTMCKENNACOLLEGE

CENTER CENTER

ATHENAEUM ATHENAEUM

BAUER BAUER

PARKING PARKING

STREET STREET

@ @

® ®

CD CD

TENTH TENTH

___) ___)

~ ~

~ ~ r r

0 0

'!: '!:

() () m m

'!: '!:

(/) (/)

~ ~

)> )>

0 0 m m

-1 -1 :I: :I:

z z '!: '!: z z :0 :0

m m

m m

)> )>

)> )>

0 0 ~ ~

:I: :I:

, , 0 0

......

(/) (/)

0 0

~ ~ 0 0 Meetings to 2:00 p.m. on both Saturday and Sunday, November Food Service for both days 12 and 13. The registration fees are $30 The hotels listed above have their own restaurants. In for members of the AMS, $45 for nonmembers, and $10 addition, there are many good restaurants in Claremont for students and unemployed mathematicians. There is a and the surrounding area. A list will be provided at the special one-day fee for MAA members on Saturday only meeting. of$15. Luncheon Petition Table There will be a joint luncheon for MAA and AMS registration area. A petition table will be set up in the participants on Saturday, November 12 at noon in the Additional information about petition tables can be Athenaeum on the college campus. found in a box in the Atlanta meeting announcement on page 68 of the January issue of Notices. Travel Accommodations Claremont is located 35 miles east of Los Angeles. Most major airlines serve Ontario International Air­ been blocked at the following hotels and Rooms have port (California). The drive from LAX is slow on Friday own reservations motels. Participants should make their afternoons, but not as slow on Saturdays and Sundays. them­ directly with the hotel of their choice, identifying To get to the Bauer Center of Claremont McKenna Society's selves as attending the American Mathematical College where registration and meetings will be held, Rates quoted meeting at Claremont McKenna College. you should head west on 9th Street from Claremont applicable taxes and are subject to change. do not include Boulevard. Please refer to the accompanying map. Shuttle service from and to Ontario International Airport is provided free of charge. The driving time is 15 to 20 minutes. Parking Griswold's Inn (walking distance) Parking is available in the lots on the south side of 9th 555 West Foothill Boulevard Street on the Claremont-McKenna College campus. (Comer of Indian Hill Boulevard) Claremont, CA 91 711 Lance W. Small Telephone: 800-854-5733 (except California), Associate Secretary 800-821-0341 (in California) or 714-626-2411 La Jolla, California Rooms (1-4 persons): $60 plus tax

Rooms must be reserved before October 21. Ramada Inn (2.5 miles) 840 South Indian Hill Boulevard (Next to San Bernadino Freeway-Interstate 10) Claremont, CA 91711 Telephone: 800-228-2828 or 714-621-4831 Rooms (1-2 guests): $49 pius tax Additional person: $6 per night

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1051 Invited Speakers and Special Sessions

Invited Speakers October 1988 Meeting in Lawrence at AMS Meetings Central Section Associate Secretary: Andy Roy Magid The individuals listed below have accepted invitations Deadline for organizers: Expired to address the Society at the times and places in­ Deadline for consideration: Expired dicated. For some meetings, the list of speakers is Andrew Acker, Partial differential equations-Geometric incomplete. equations John K. Beem and Phillip E. Parker, Geometry and mathematical physics Lawrence, October 1988 Ralph Byers, Numerical linear algebra Bj0rn Dahlberg Peter Scott Bruce Crauder and Sheldon Katz, Algebraic geometry Steven E. Hurder Sidney M. Webster Tyrone Duncan, Control theory Claremont, November 1988 William Fleissner, Applications of set theory William Jacob Francis Bonahon James Foran, Real analysis Robert Brooks William Mark Goldman, Flat bundles and geometric Phoenix, January 1989 structures E. Hurder and Noberto Salinas, Operatory Ralph P. Boas Peter Landweber Steven theory and applications to geometry John B. Conway Cathleen S. Morawetz Daniel Katz and Jeffery Lang, Commutative algebra Percy Alec Deift (AMS-MAA) and Gregory Vechota, Potential theory and David Fried Steve Smale Jill Pipher partial differential equations in nonsmooth domains Ronald L. Graham (AMS-MAA) Scott, 3-manifolds (AMS-MAA) Luc Tartar Peter Worcester, April1989 November 1988 Meeting in Claremont Igor Frankel Karl Rubin Far Western Section Thomas H. Parker Adrian Ocneanu Associate Secretary: Lance W. Small Deadline for organizers: Expired Chicago, May 1989 Deadline for consideration: Expired Henri Gillet Richard Rochberg Francis Bonahon and David Gabai, Low dimensional Nicholas Lerner Shmuel Weinberger geometry Muncie, October 1989 Robert Borrelli and Courtney S. Coleman, Computers Kenneth Meyer Steven Sperber and software in mathematical research Paul S. Muhly Robert Brooks and S. -Y. Cheng, The spectrum of the Laplacian Organizers and Topics Stavros N. Busenberg, Differential and difference equa­ of Special Sessions tions The list below contains all the information about William Jacob and Adrian Wadsworth, Division alge­ Special Sessions at meetings of the Society available bras at the time this issue of Notices went to the printer. The section below entitled Information for Organizers describes the timetable for announcing the existence of Special Sessions.

1052 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Invited Speakers and Special Sessions

January 1989 Meeting in Phoenix Cary Huffman and Neal Brand, Codes and designs Associate Secretary: Lance W. Small Ronnie Lee and Steven Weintraub, Algebraic topology Deadline for organizers: Expired of varieties Deadline for consideration: September 21, 1988 S. P. Singh, Nonlinear analysis and its applications Melvyn S. Berger, Mathematics of nonlinear science John B. Conway, Harry Gonshor, and Martin Kruskal, August 1989 Meeting in Boulder Surreal numbers Associate Secretary: Andy Roy Magid Percy Deift, Integrable systems Deadline for organizers: November 15, 1988 David Eisenbud and Craig Huneke, Commutative Deadline for consideration: April 25, 1989 algebra and algebraic geometry David Fried and Joseph Christie, Geometry of hyper­ October 1989 Meeting in Hoboken bolic dynamical systems Eastern Section Larry C. Grove and M. F. Newman, Computational Associate Secretary: W. Wistar Comfort group theory Deadline for organizers: January 15, 1989 Deadline for consideration: August 9, 1989 William A. Harris, Singular perturbation theory Victor J. Katz and Florence Fasanelli, History of October 1989 Meeting in Muncie Mathematics Central Section Albert Marden and Burton Rodin, Computational Associate Secretary: Andy Roy Magid aspects of complex analysis Deadline for organizers: January 15, 1989 Sidney Port, Stochastic processes Deadline for consideration: August 9, 1989 Marc A. Rieffell, Operator algebras and geometry Hal L. Smith and James Cushing, Mathematics in Information for Organizers population biology Special Sessions at Annual and Summer Meetings are held April1989 Meeting in Worcester under the supervision of the Program Committee for Na­ Eastern Section tional Meetings. They are administered by the Associate Associate Secretary: W. Wistar Comfort Secretary in charge of that meeting with staff assistance Deadline for organizers: Expired from the Meetings and Editorial Departments in the Society Deadline for consideration: January 4, 1989 office in Providence. Richard Herman and Adrian Ocneanu, Operator alge­ According to the "Rules for Special Sessions" of the bras, Galois theory and representations Society, Special Sessions are selected by the Program Com­ James Lepowsky, Infinite-dimentional symmetries in mittee from a list of proposed Special Sessions in essentially the same manner as Invited Speakers are selected. The mathematics and physics number of Special Sessions at a Summer or Annual Meeting Thomas H. Parker, Gauge theory and differential is limited. The algorithm that determines the number of geometry Special Sessions allowed at a given meeting, while simple, is Karl Rubin and Glenn Stevens, L-functions and arith­ not repeated here, but may be found in "Rules for Special metic Sessions" which can be found on page 614 in the Aprill988 Lee Rudolph, Knot theory and algebraic geometry in issue of Notices. the large Each Invited Speaker is invited to generate a Special Session, either by personally organizing one or by having a May 1989 Meeting in Chicago Special Session organized by others. Proposals to organize Central Section a Special Session are sometimes requested either by the Associate Secretary: Andy Roy Magid Program Committee or by the Associate Secretary. Other Deadline for organizers: Expired proposals to organize a Special Session may be submitted Deadline for consideration: February 8, 1989 to the Associate Secretary in charge of that meeting (who is Jeffery Bergen, Noncommutative ring theory an ex-officio member of the committee and whose address Martin Butinas and Billy Rhoades, Sequence spaces may be found below). These proposals must be in the and summability hands of the Program Committee well in advance of the Jonathan Cohen, Numerical methods in harmonic meeting and, in any case, at least nine (9) months prior analysis to the meeting at which the Special Session is to be held in order that the committee may consider all the proposals Vinay Deodhar, Kazhdan-Lusztig theory and related for Special Sessions simultaneously. Proposals that are sent topics to the Providence office of the Society, to the Notices, or Stephen Doty, Algebraic groups and related topics directed to anyone other than the Associate Secretary will Christine Haught, Recursion theory

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1053 Invited Speakers and Special Sessions

have to be forwarded and may not be received in time to be Eastern Section considered for acceptance. W. Wistar Comfort, Associate Secretary It should be noticed that Special Sessions must be Department of Mathematics announced in the Notices in such a timely fashion that Wesleyan University any member of the Society who so wishes may submit an Middletown, CT 06457 abstract for consideration for presentation in the Special (Telephone 203- 34 7-9411) Session before the deadline for such consideration. This Southeastern Section deadline is usually three (3) weeks before the Deadline for Frank T. Birtel, Associate Secretary Abstracts for the meeting in question. Department of Mathematics Special Sessions are very effective at Sectional Meetings Tulane University LA 70118 and can usually be accommodated. They are selected by the New Orleans, (Telephone 504-865-5646) Committee to Select Hour Speakers for the Section. The As a general rule, members who anticipate organizing Special processing of proposals for Special Sessions for Sectional Sessions at AMS meetings are advised to seek approval at least Meetings is handled by the Associate Secretary for the nine months prior to the scheduled date of the meeting. No Special Section, who then forwards the proposals to the Committee Sessions can be approved too late to provide adequate advance to Select which makes the final selection of the proposals. notice to members who wish to participate. Each Invited Speaker at a Sectional Meeting is invited to organize a Special Session. Just as for national meetings no Special Session at a Sectional Meeting may be approved Information for Speakers so late that its announcement appears past the deadline A great many of the papers presented in Special Sessions after which members can no longer send abstracts for at meetings of the Society are invited papers, but any consideration for presentation in that Special Session. member of the Society who wishes to do so may submit The Society reserves the right of first refusal for the an abstract for consideration for presentation in a Special publication of proceedings of any Special Session. These Session, provided it is received in Providence prior to proceedings appear in the book series Contemporary Math­ the special early deadline announced above and in the ematics. announcements of the meeting at which the Special Session More precise details concerning proposals for and orga­ has been scheduled. Contributors should know that there nizing of Special Sessions may be found in the "Rules for is a limitation in size of a single special session, so that Special Sessions" or may be obtained from any Associate it is sometimes true that all places are filled by invitation. Secretary. Papers not accepted for a Special Session are considered as ten-minute contributed papers. Send Proposals for Special Sessions to the Abstracts of papers submitted for consideration for pre­ Associate Secretaries sentation at a Special Session must be received by the The programs of sectional meetings are arranged by the Associate Providence office (Editorial Department, American Mathe­ Secretary for the section in question: matical Society, P. 0. Box 6248, Providence, RI 02940) by Far Western Section (Pacific and Mountain) the special deadline for Special Sessions, which is usually Lance W. Small, Associate Secretary three weeks earlier than the deadline for contributed papers Department of Mathematics for the same meeting. The Council has decreed that no University of California, San Diego paper, whether invited or contributed, may be listed in the La Jolla, CA 92093 program of a meeting of the Society unless an abstract of the (Telephone 619- 534-3590) paper has been received in Providence prior to the deadline. Central Section Andy Roy Magid, Associate Secretary Department of Mathematics University of Oklahoma 601 Elm PHSC 423 Norman, OK 73019 (Telephone 405- 325-2052)

1054 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY !987-1988. Academic Year Devoted to Mathematical Sciences Differential Geometry, University of North Carolina, Chapel Hill, North Car­ Meetings and Conferences olina. (November 1987, p. 1137) !987-!988. Academic Year Devoted to several Complex Variables, Mittag-Leffler Institute, Djursholm, Sweden. (January !98 7, p. 131) !988-1989. Academic Year Devoted to Operator Algebras, Mittag-Leffler Insti­ tute, Djursholm, Sweden. (February 1988, p. 307) March 6-December 24, 1988. Math­ ematisches Forschungsinstitut Oberwol­ of interest to fach (Weekly Conferences), Federal Re­ THIS SECTION contains announcements of meetings and conferences some segment of the mathematical public, including ad hoc, local, or regional meetings, of Germany. (March 1988, pp. public and meetings or symposia devoted to specialized topics, as well as announcements 457-458) of regularly scheduled meetings of national or international mathematical organizations. April 17-December 24, 1988. (Information on meetings of the Society, and on meetings sponsored by the Society, will Mathematisches Forschungsinstitut Ober­ be found inside the front cover.) wolfach (Weekly Conferences), Federal AN ANNOUNCEMENT will be published in Notices if it contains a call for papers, Republic of Germany. (October 1987, and specifies the place, date, subject (when applicable), and the speakers; a second p. 995) full announcement will be published only if there are changes or necessary additional information. Once an announcement has appeared, the event will be briefly noted in each October 1988-August 1989. Symposium issue until it has been held and a reference will be given in parentheses to the month, on Singularity Theory and Its Applica­ year, and page of the issue in which the complete information appeared. Asterisks (*) tions, University of Warwick, Coven­ mark those announcements containing new or revised information. try, United Kingdom. (May/June 1988, IN GENERAL, announcements of meetings and conferences held in North America carry p. 724) only date, title of meeting, place of meeting, names of speakers (or sometimes a general January !-December 23, 1989. Math­ statement on the program), deadlines for abstracts or contributed papers, and source the North American area may carry more ematisches Forschungsinstitut Oberwol­ of further information. Meetings held outside In any case, if there is any application deadline with respect to Federal Re­ detailed information. fach (Weekly Conferences), participation in the meeting, this fact should be noted. All communications on meetings public of Germany. (April 1988, p. 629) and conferences in the mathematical sciences should be sent to the Editor of Notices, care of the American Mathematical Society in Providence. DEADLINES for entries in this section are listed on the inside front cover of each issue. In September 1988 order to allow participants to arrange their travel plans, organizers of meetings are urged 18-21. IMACS International Sympo­ to submit information for these listings early enough to allow them to appear in more than sium on System Modelling and Simu­ one issue of Notices prior to the meeting in question. To achieve this, listings should be lation, University of Calabria, Province received in Providence SIX MONTHS prior to the scheduled date of the meeting. de Cosenza, Italy. (July/August 1988, p. 892) University; J. Douglas, Purdue 18-24. DMV-Jahrestagung 1988, Re­ * 24-25. Midwest Several Complex Vari­ Brown Fix, University of gensburg, Federal Republic of Germany. ables Meeting, Purdue University, West University; G. J. J. G. Glimm, New (March 1988, p. 464) Lafayette, Indiana. Texas-Arlington; York University; H. 0. Kreiss, Cali­ R. Department 19-23. Sixth International Colloquium INFORMA noN: S. Bell, fornia Institute of Technology; J. L. University, on Differential Geometry, Universidad of Mathematics, Purdue Lions, College de France; R. A. Nico­ 47907, 317- de Santiago de Compostela, Santiago West Lafayette, Indiana laides, Carnegie-Mellon University; S. de Compostela, Spain. (February 1988, 494-1956. J. Osher, University of California; D. p. 312) * 24-27. Conference on Numerical Solu­ Russell, University of Wisconsin; E. 19-23. Histoire de Ia Relativite Generale, tions of Partial Differential Equations, Tadmor, Tel Aviv University. Marseille, France. (March 1988, p. 464) Virginia Polytechnic Institute and State INFORMATION: N. D. Smith, ICAM, 23-25. Fall Foliage Combinatorial Group University, Blacksburg, Virginia. Virginia Tech, 620 North Main Street­ Theory Conference, Shaker Village, New B2, Blacksburg, Virginia 24061-0531, Hampshire. (May/June 1988, p. 729) SPONSORs: Interdisciplinary Center for 703-961-7667. and Depart­ 24-25. Last Fall Foliage Topology Sem­ Applied Mathematics Poly­ 25-0ctober 1. International Symposium inar, Shaker Village, New Hampshire. ment of Mathematics, Virginia in Honor of Rene Thorn, Paris, France. (MayjJune 1988, p. 729) technic Institute and State University. SPEAKERS: I. Babuska, Uni­ INVITED INFORMATION: V. Houllet, Colloque of Maryland; H. T. Banks, versity Rene Thorn I. H. E. S., 35, route

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1055 Meetings and Conferences

de Chartres, 91440 Bures-sur-Yvettte, 10-12. IEEE Workshop on Visual Lan­ ofNewYork, City College, New York, France. Telephone: 33.1.69.07.48.53, guages, University of Pennsylvania, Pitts­ New York 10031, 212-690-5346. poste 405. (Note changes from Feb­ burgh, Pennsylvania. (March 1988, p. 465) 4-5. Southeast Differential Equations ruary 1988, p. 312) '88: Second Sympo­ 10-12. Frontiers Conference, Athens, Georgia. (March 1988, sium on the Frontiers of Massively Par­ '88: Second International p. 465) * 26-28. CSCW allel Computation, George Mason Uni­ Coop­ Third Annual Pi Mu Epsilon Re­ Conference on Computer-supported versity, Fairfax, Virginia. (July I August * 4-5. Oregon. Conference, Saint Norbert College, erative Work, Portland, 1988, p. 893) gional De Pere, Wisconsin. SPONSORS: Association for Computing 10-14. Workshop on Mathematical Pro­ Machinery, with support from Lotus gramming, Catholic University of Rio De INVITED SPEAKER: P. Straffin, Beloit Development Corporation and Xerox Janeiro, Brazil. (May jJune 1988, p. 729) College. Corporation. 12-14. Sensor-Based Robots: Algorithms INFORMATION: R. Poss, Saint Norbert PROGRAM: This conference will bring and Architectures, Bonas, France. College, De Pere, Wisconsin 54115, together academic and industry rep­ (May /June 1988, p. 729) 414-337-3198. resentatives from many disciplines 12-15. Discrete Mathematics and Com­ including computer science, organi­ * 5. New York Graph Theory Day 16, The puter Science, University of Montreal, zation design, artificial intelligence, State University of New York, Purchase, Montreal, Quebec, Canada. (July/August cognitive science, the social sciences New York. 1988, p. 893) and engineering. [Note: Conference SPONSOR: The Mathematics Section 21-22. Tenth Midwest Probability Collo­ attendance is limited, so advance reg­ of the New York Academy of Sci­ quium, Northwestern University, Evanston, istration is strongly encouraged.] ences. Illinois. (May /June 1988, p. 730) INFORMATION: J. Kling, Lotus Devel­ INVITED SPEAKERS: F. Harary and A. opment Corporation, 617-577-8500, 24-26. Twenty-ninth Foundations of J. Hoffman. or S. Sylvia, 617-225-1860. Computer Science, White Plains, New ORGANIZING COMMITTEE: S. Auslan­ York. (July f August 1988, p. 893) der; F. Buckley; J. W. Kennedy; M. 26-30. Computational Intelligence '88, 28-29. Seventeenth Midwest Differential Lewinter; L. V. Quintas. University of Milano, Milano, Italy. Equations Conference, Iowa State Uni­ INFORMATION: M. Lewinter, Mathe­ (March 1988, p. 465) versity, Ames, Iowa. (May jJune 1988, matics Department, State University 26-0ctober 1. Fifth International Confer­ p. 730) of New York, Purchase, New York ence on Complex Analysis, Halle, German 28-30. Central Sectional Meeting, Law­ 10577, 914-253-5040. Democratic Republic. (October 1987, rence, Kansas. p. 1000) * 7-11. Workshop on Solitons in Nonlin­ 28-0ctober 9. Sixth International Sum­ INFORMATION: J. Balletto, American ear Optics and Plasma Physics, Institute Box mer School on Probability Theory and Mathematical Society, Post Office for Mathematics and Its Applications, 02940. Mathematical Statistics, Varna, Bulgaria. 6248, Providence, Rhode Island Minneapolis, Minnesota. (JulyI August 1988, p. 892) 30-November 4. Ninth International ORGANIZERS: D. Kaup and Y. Ko­ 30-0ctober 1. Sixteenth Annual Math­ Conference on Computer Communications, dama. ematics and Statistics Conference, Miami Tel Aviv, Israel. (July f August 1988, p. 893) INFORMATION: Institute for Mathe­ University, Oxford, Ohio. (April 1988, 31-November 18. Workshop in Math­ matics and Its Applications, Univer­ p. 637) ematical Ecology, Trieste, Italy. (March sity of Minnesota, Minneapolis, Min­ 1988, p. 465) nesota 55455, 612-624-6066. October 1988 * 11-13. Midwest Dynamical Systems l. Algebra Day, Carleton University, Ot­ November 1988 Conference, Emory University, Atlanta, Georgia. tawa, Canada. (July f August 1988, p. 892) * 3-4. Symposium on the Legacy of Emil 3-5. Knowledge-Based Robot Control, Post, City University of New York, City INFORMATION: S. Batterson (404-727- Bonas, France. (May j June 1988, p. 729) College, New York, New York. 7923) or J. Christy ( 404-727-7956), 4-6. Colloque Ergoia 88 Ergonomie et Emory University, Department of Math· PRINCIPAL SPEAKERS: G. Baumslag, Intelligence Artificelle, Biarritz, France. ematics, Atlanta, Georgia 30322. City University of New York, City (April 1988, p. 637) College; M. Davis, New York Univer­ 12-13. 846th Meeting of the AMS, 5-7. IFAC/IMACS/IFIP Symposium sity, Courant Institute of Mathemati­ Claremont, California. on Robot Control SYROCO '88, Karl­ cal Sciences; S. Ginzburg, University INFORMATION: J. Balletto, American sruhe, Federal Republic of Germany. of Southern California; J. Hartmanis, Mathematical Society, Post Office Box (July f August 1988, p. 893) Cornell University; M. Minsky, H. 6248, Providence, Rhode Island 02940. 9-13. International Conference on Com­ Rogers, and G. Sacks, Massachusetts puter Languages, Castle Premier, Miami Institute of Technology. 12-13. Twenty-fourth Midwest Partial Beach, Florida. (March 1988, p. 465) INFORMATION: J. Barshay, Depart­ Differntial Equations Seminar, University ment of Mathematics, City University

1056 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences of Notre Dame, Notre Dame, Indiana. ridge of Northern Illinois Universsity, CONFERENCE TOPICS: Arakelov the­ (July/August 1988, p. 893) as well as sessions for contributed pa­ ory; Tate-Shafarevich groups; Mordell's 13-17. French-Israeli Conference on pers, research ideas, and discussion. conjecture. combinatorics and Algorithms, Israel. Some support will be available for a INVITED SPEAKERS: R. Coleman; B. (February 1988, p. 313) limited number of participants. Mazur; J. Tate; K. Ribet; K. Rubin; 18-19. Quantitative Approaches to Di­ INFORMATION: R. J. Wisner, Fermat P. Vojta. S. Kamienny or W. abetes, Sydney, Australia. (April 1988, Symposium, Department of Mathe­ INFORMATION: matical Sciences, New Mexico State McCallum, Department of Mathe­ p. 638) University, Box 300001, Las Cruces, matics, University of Arizona, Tuc­ To­ 20-25. Eighth Annual Conference: New Mexico 88003-0001, 505-646- son, Arizona 85721. Scientific and wards a Whole Decade for 3901. (Note changes from May/June for Interna­ Technological Cooperation 1988, p. 730.) 9-13. Sixth Texas International Sympo­ tional Development, Jamaica, West In­ sium on Approximation Theory, College dies. (March 1988, p. 465) Station, Texas. (April 1988, p. 638) 21-December 16. College on Global January 1989 9-14. K-Theory and Dynamics, Uni­ Geometric and Topological Methods in 2-5. International Colloquium in Ring versity of Florida, Gainesville, Florida. Analysis, Trieste, Italy. (March 1988, Theory, Bar-Ilan University, Ramat-Gan, (April 1988, p. 638) p. 465) Israel. (MayjJune 1988, p. 730) * 10-11. AMS Short Course on Matrix * 2-5. Fifth Haifa Matrix Conference, Theory and Applications, Phoenix, Ari­ December 1988 Technicon City, Haifa, Israel. zona. 5-8. IMACS Conference on Expert ORGANIZING COMMITTEE: A. Berman; INFORMATION: M. Foulkes, Ameri­ Systems for Numerical Computing, Pur­ M. Goldberg; D. Hershkowitz; L. can Mathematical Society, Post Office due University, West Lafayette, Indiana. Lerer; R. Loewy; A. Zaks. Box 6248, Providence, Rhode Island (July/ August 1988, p. 893) INFORMATION: A. Berman, Depart­ 02940. 6-8. First International Conference on ment of Mathematics, Technicon-Israel 11-14. Joint Mathematics Meetings, Matter Elements Analysis, Guangzhou. Institute of Technology, Haifa 32000, Phoenix, Arizona. (April 1987, p. 553) Guangdong, People's Republic of China. Israel. (October 1987, p. 1000) INFORMATION: H. Daly, American * 3-10. Workshop on Two Phase Waves Mathematical Society, Meetings De­ 12-15. International Conference on Com­ in Fluidized Beds, Sedimentation, and partment, Post Office Box 6248, Prov­ Tarpon Springs, Florida. puter Vision, Granular Flows, Institute for Mathemat­ idence, Rhode Island 02940. (May/June 1988, p. 730) ics and its Applications, Minneapolis, 12-17. International Course on Com­ Minnesota. 15-20. American Association for the Ad­ putational Geometry, Dipartimento di vancement of Science Annual Meeting, B. Keyfitz, Matematica, Universita, Catania, Italy. ORGANIZERS: D. Joseph, San Francisco, California. (MayjJune (May/June 1988, p. 730) D. Schaeffer. 1988, p. 731) INFORMATION: Institute for Mathe­ 23-27. Workshop on Algorithms, Word 13-15. IMA/SIAM International Con­ matics and its Applications, Min­ Problems, and Classification in Combina­ ference on Mathematics of Signal Pro­ neapolis, Minnesota 55455, 612-624- (April 1988, torial Group Theory, Mathematical Sci­ cessing, Warwick, England. 6066. p. 638) ences Research Institute, Berkeley, Cali­ 13-17. KAC-Moody Lie Algebras and 4-6. American Statistical Association fornia. (February 1988, p. 313) Physics Conference, North Carolina State Winter Conference: Statistics in Soci­ Diego, California. (March 1988, University, Raleigh, North Carolina. ety, San February 1989 (July/August 1988, p. 894) p. 465) 14-16. Raj Chandra Bose Memorial 6-12. Nonlinear Wave Equations, George * 6-10. Minisymposium on Plasticity, In­ Conference on Combinatorial Mathemat­ Mason University, Fairfax, Virginia. stitute for Mathematics and its Applica­ ics and Applications, Calcutta, . India.· (April 1988, p. 638) tions, Minneapolis, Minnesota. (July/ August 1988, p. 894) 8-10. Symposium in Honor of the Seven­ ORGANIZER: D. Joseph. 27-31. Holiday Symposium on Fermat's tieth Birthday of Ted Harris, Los Angeles, INFORMATION: Institute for Mathe­ Last Theorem, New Mexico State Uni­ California. (May /June 1988, p. 730) matics and its Applications, Univer­ versity, Las Cruces, New Mexico. 8-11. First Caribbean Conference on sity of Minnesota, Minneapolis, Min­ Fluid Dynamics, Saint Augustine, Trini­ nesota 55455, 612-624-6066. PROGRAM: With partial support from dad, West Indies. (June 1987, p. 686) the National Science Foundation, the * 19-23. Analyse quantitative de Ia sensi­ 8-11. Conference on the Arithmetic of symposium will focus on Fermat's * bilite en optimisation, Centre de recherches Algebraic Curves, University of Arizona, Last Theorem and will consist of ten mathematiques, Universite de Montreal. lectures, two each day, by J. Tate, of Tucson, Arizona. Harvard University. There will be ad­ ORGANIZERS: R. T. Rockafeller, Wash­ ditional sessions organized by J. Self- ington; R. Wets, Davis.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1057 Meetings and Conferences

INFORMATION: F. H. Clarke, Director, between aero-, ocean- and seismo­ ORGANIZERS: J. Glimm, A. Majda. Centre de recherches mathematiques, acoustics; new solution techniques INFORMATION: Institute for Mathe­ Universite de Montreal, CP 6128-A, which have been made possible with matics and its Applications, Univer­ Montreal, Quebec H3C 3J7 Canada. the advent of new computer architec­ sity of Minnesota, Minneapolis, Min­ tures. nesota 55455, 612-624-6066. 21-23. Seventeenth Annual Computer INFORMATION: D. Lee, Code 3122, Science Conference, Commonwealth Con­ Naval Underwater Systems Center, 10-13. IEEE Artificial Neural Networks vention Center, Louisville, Kentucky. New London, Connecticut 06320, 203- Conference, Sheraton International Con­ (May /June 1988, p. 731) 440-4438. ference Center, Reston, Virginia. (Note date change, March 1988, p. 465) 19-22. ENAR Spring Meeting, Lexing­ * 13-15. Operators and Function Theory: March 1989 ton, Kentucky. (March 1988, p. 465) The Role of de Branges's Spaces, Univer­ * 6-10. Workshop on Ellipticity in Evolu­ * 22-25. Computer-Aided Proofs in Analy­ sity of Arkansas, Fayetteville, Arkansas. tion Equations, Institute for Mathematics sis, University of Cincinnati, Cincinnati, PRINCIPAL SPEAKER: D. Sarason, Uni­ and its Applications, Minneapolis, Min­ Ohio. versity of California, Berkeley. nesota. PROGRAM: The theme of the confer­ OTHER INVITED SPEAKERS: J. Agler; ORGANIZERS: D. Joseph, B. Keyfitz, ence is the use of the computer to help S. Axler; L. de Branges; C. Cowen; J. D. Schaeffer. establish rigorous proof of theorems Helton; T. Kriete; M. Rosenblum; J. INFORMATION: Institute for Mathe­ in various areas in analysis. The use Rovnyak; J. Shapiro; A. Shields. matics and its Applications, Univer­ of symbolic processors, interval arith­ CALL FOR PAPERS: Contributed pa­ sity of Minnesota, Minneapolis, Min­ metic, and careful error analysis will pers should be submitted before Feb­ nesota 55455, 612-624-6066. be discussed. There will be a plenary ruary 15, 1989. lecture and contributed papers. The INFORMATION: J. Duncan or I. Mon­ * 13-18. East European Category Seminar meeting has support from the Uni­ roe, Department of Mathematical Sci­ (EECS '89), Sofia, Bulgaria. versity of Cincinnati, the Institute for ences, SCEN 301, University of Arkan­ PuRPOSE: This seminar is a tradi­ Mathematics ans its Applications, and sas, Fayetteville, Arkansas 72701. tional annual meeting of scientists support from the NSF is anticipated. 15-16. Eastern Section Meeting, Col­ working on category theory and its CALL FOR PAPERS: Abstracts are due lege of the Holy Cross, Worcester, Mas­ applications in general. In the classi­ by February 15, 1989. sachusetts. (May/June 1988, p. 731) cal branches, topics covered will in­ INFORMATION: D. Schmidt, Depart­ clude logics, algebra, topology, geom­ ment of Computer Science, or K. * 16-20. NCGA '89, Philadelphia Civic etry, analysis and in some modern Meyer, Department of Mathematics, Center, Philadelphia, Pennsylvania. applied extensions: theoretical com­ University of Cincinnati, Cincinnati, INFORMATION: NCGA '89 Education puter science, system theory, pro­ Ohio 45221. Coordinator, 2722 Merrilee Drive, gramming, graph theory, theoretical 28-31. Annual Scientific Conference of Suite 200, Fairfax, Virginia 22031, physics, fuzzy sets theory and others. * the Society of Applied Mathematics and 1-800-225-NCGA or 703-698-9600. INFORMATION: K. G. Peeva, IEC Sofia, Mechanics (GAMM), University of Karl­ POB 384, Bulgaria. * 17-21. Minisymposium on Computa­ sruhe, West Germany. tional Issues for Nonlinear Hyperbolic * 15-1 7. Second IMACS International ORGANIZER: G. Alefeld. Waves, Institute for Mathematics and its Symposium on Computational Acoustics, PROGRAM: There will be approxi­ Applications, Minneapolis, Minnesota. Princeton University, Princeton, New mately twelve main lectures and short Jersey. ORGANIZERS: J. Glimm, P. Wood­ communications from all fields of ap­ ward. CONFERENCE COMMITTEE: D. Lee, plied mathematics and mechanics. INFORMATION: Institute for Mathe­ Naval Underwater Systems Center; INFORMATION: G. Alefeld, lnstitut matics and its Applications, Univer­ A. Cakmak, Princeton University; R. fur Angewandte Mathematik, Kaiser­ sity of Minnesota, Minneapolis, Min­ Vichnevetsky, Rutgers University. straBe 12, D-7500 Karlsruhe 1, West nesota 55455, 612-624-6066. SPONSORs: IMACS (International As­ Germany. sociation for Mathematics and Com­ puters in Simulation; NUSC (Naval May 1989 Apri11989 Underwater Systems Center); ONR * 4-5. Twentieth Annual Pittsburgh Con­ (Office of Naval Research); Princeton 3-6. IEEE International Conference on ference on Modeling and Simulation, Pitts­ University. Control and Applications, Jerusalem, Is­ burgh. SYMPOSIUM TOPICS: Computational rael. (April 1988, p. 638) methods to solve acoustics problems * 3-14. Workshop on Multidimensional CONFERENCE TOPICS: Emphasis will (including acroacoustics, seismoacous­ Hyperbolic Problems and Computations, be on computer science: artificial in­ tics, and ocean acoustics) and in gen­ Institute for Mathematics and its Appli­ telligence, expert systems, robotics, eral wave propagation problems; com­ cations, Minneapolis, Minnesota. microprocessors, and personal com­ putational aspects of the interaction puter applications and software.

1058 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

CALL FOR PAPERS: Papers on all as­ donald; E. A. Rahmanov; E. B. Saff; University; R. Murty, McGill Univer­ pects of control theory and applica­ H. Stahl; G. Viennot. sity. tions, as well as all of the traditional INFORMATION: OPsConf, cjo P. Nevai, INFORMATION: F. H. Clarke, Director, areas of modeling and simulation are Department of Mathematics, The Ohio Centre de recherches mathematiques, of interest and are welcomed. Two State University, 231 West Eighteenth Universite de Montreal, CP 6128-A, copies of titles, authors, all authors' Avenue, Columbus, Ohio 43210-1174. Montreal, Quebec H3C 317 Canada. addresses, abstracts and summaries should be submitted by January 31, 23-27. International Conference on Com­ 1989. Notification of acceptance for puting and Information, Toronto, On­ June 1989 tario, Canada. (April 1988, p. 638) presentation will be given by March * 5-7. An International Symposium on 8, 1989. Instructions and model pa­ * 28-June I. Sixteenth International Sym­ Asymptotic and Computational Analysis, per for the preparation of accepted posium on Computer Architecture, Winnipeg, Canada. papers will be mailed to each author. Jerusalem, Israel. The final typed manuscript will be CONFERENCE THEMES: Asymptotic SO­ SPONSORs: IEEE Computer Society lutions to differential equations; asymp­ due by May 5, 1989. and the Association for Computing INFORMATION: W. G. Vogt or M. totic approximations of integrals; sin­ Machinery. gular perturbation theory; applica­ H. Mickle, Modeling and Simulation CALL FOR PAPERS: Industry-oriented Conference, 348 Benedum Engineer­ tions of asymptotics; special func­ papers will be accepted for evaluation tions; computational analysis. ing Hall, University of Pittsburgh, until November II, 1988. Authors are Pittsburgh, Pennsylvania 15261. PROGRAM: There will be one-hour and asked to submit five copies of papers half-hour invited research or survey 8-10. Forty-Third Annual Quality Con­ in English and include a cover page, presentations. In addition, there will gress, Toronto, Canada. (March 1988, a I 00-word abstract, and a list of be a limited number of contributed p. 465) five key words. All authors from the talks ( 15 minutes each). United States should forward papers 8-12. Workshop on Arithmetic Groups INFORMATION AND PAPER SUBMIS­ to Arvind, Laboratory for Computer and Buildings, Mathematical Sciences SION: R. Wong, Department of Ap­ Science, Massachusetts Institute of Research Institute, Berkeley, California. plied Mathematics, University of Man­ Technology, 545 Technology Square, (March 1988, p. 465) itoba, Winnipeg, Manitoba R3T 2N2, Cambridge, Massachusetts 02139. Pa­ Canada, 204-4 74-816 7. 15-17. 1989 New Zealand Mathemat­ pers from Europe and Israel should ics Colloquium, Palmerston North, New be sent to J. Gurd, Computer Science 5-16. Workshop on the Geometry of Zealand. (May I June 1988, p. 731) Department, University of Manch­ Hamiltonian Systems, Mathematical Sci­ 19-20. Central Section Meeting, Loyola ester, Manchester Ml3 9PL, United ences Research Institute, Berkeley, Cali­ University, Chicago, Illinois. Kingdom. Authors from the Far East fornia. (April 1988, p. 638) INFORMATION: J. Balletto, American should send papers to M. Kitsure­ * 6-8. Fourth International Conference on Mathematical Society, Post Office Box gawa, Institute of Industrial Science, Boundary Element Technology, Windsor, 6248, Providence, Rhode Island 02940. University of Tokyo, Minato-ku, Ontario, Canada. Tokyo 106, Japan. *22-24. Workshop on Vortex Methods, INFORMATION: Sixteenth International INFORMATION: N. G. Zamani, Depart­ Mathematical Sciences Research Insti­ Symposium on Computer Architec­ ment of Mathematics and Statistics, tute, Berkeley, California. ture, 90A Hayarkon Street, Post Of­ University of Windsor, Windsor, On­ fice Box 3190, Tel Aviv 61031, Israel. tario, Canada N9B 3P4. ORGANIZING COMMITTEE: C. Ander­ Telephone: 972-3-246261. son; A. Chorin, Chairman; A. Majda; * 6-10. Analytic Number Theory, Centre P. Marcus; M. Pulvirenti. 28-June I 0. AMS-SIAM Summer Sem­ de recherches mathematiques, Universite INFORMATION: I. Kaplansky, Direc­ inar on the Mathematics of Random Me­ de Montreal. tor, Mathematical Sciences. Research dia, Virginia Polytechnic Institute and ORGANIZERS: R. Murty, McGill Uni­ Institute, I 000 Centennial Drive, State University, Blacksburg, Virginia. versity; J. Friedlander, Toronto Uni­ Berkeley, California 94720. (May/June 1988, p. 731) versity; H. lwaniec, Rutgers Univer­ sity. *22-June 3. NATO Advanced Study Insti­ 29-June I. Third International Confer­ F. H. Clarke, Director, tute on Orthogonal Polynomials and Their ence in Mathematics: Fractional Calculus INFORMATION: mathematiques, Applications, The Ohio State University, and Its Applications, Nihon University, Centre de recherches Columbus, Ohio. Tokyo, Japan. (May/June 1988, p. 731) Universite de Montreal, C. P. 6128-A, * 30-June 30. Analytic Number Theory, Montreal, Quebec H3C 317 Canada. ORGANIZING COMMITTEE: M. Ismail; Forms and Related Topics, Cen­ Modular * 7-9. Canadian Applied Mathematics So­ P. Nevai, Director; D. Stanton. de recherches mathematiques, Uni­ tre ciety Tenth Annual Meeting, Winnipeg, PRINCIPAL SPEAKERs: D. Bessis; W. versite de Montreal. Gautschi; Y. Genin; R. Haydock; T. Canada. J. Friedlander, Univer­ Koornwinder; D. S. Lubinsky; I. Mac- ORGANIZERS: MAJOR THEMES: Asymptotic and sity of Toronto; H. lwaniec, Rutgers Computational Analysis; Two-point

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1059 Meetings and Conferences

Boundary Value Problems and Ma­ 13-19. Fourth Conference on Differen­ trix Computation. tial Equations and Applications, Rousse, May 1990 INFORMATION AND PAPER SUBMIS­ Bulgaria. (May/June 1988, p. 731) SION: R. Wong, Department of Ap­ 20-September 6. Nineteenth Ecole d'ete 25-31. Tenth International Conference plied Mathematics, University of Man­ de Calcul des Probabilities, Saint-Flour, on Pattern Recognition, Resorts Hotel itoba, Winnipeg, Manitoba R3T 2N2 Cantal. (March 1988, p. 466) Atlantic City, New Jersey. (March 1988, ~ 466) , Canada 204-4 74-816 7. 28-September 1. IFIP 89: Eleventh San Fran­ 12-16. Computers and Mathematics, World Computer Conference, Massachusetts Institute of Technology, cisco, California. (April 1988, p. 639) June 1990 29-September 6. Forty-seventh Session Cambridge, Massachusetts. (July I August 6-12. 1990 Barcelona Conference on of the International Statistical Institute, * 1988, p. 894) Algebraic Topology, Centre de Recerca Paris, France. (April 1988, p. 639) Matematica, Barcelona, Spain. July 1989 INFORMATION: M. Castellet, Director, September 1989 3-7. Computational Ordinary Differen­ Centre de Recerca Matematica, Insti­ tial Equations, London, England. (April 16-0ctober 20. Sixth World Congress tut D'Estudis Catalans, Apartat 50 - 1988, p. 638) on Medical Information, Beijing, China. 08193 Bellaterra, Barcelona, Spain. 10-12. International Conference on Com­ (April 1988, p. 639) putational Techniques and Applications, 25-29. Third International Conference August 1990 Griffith University, Brisbane. (April 1988, on the Theory of Groups and Related p. 639) Topics, Australian National University, 6-9. 1990 Joint Statistical Meetings, Anaheim, California. (March 1988, p. 466) 10-21. Microprogram on Noncommu­ Canberra. (May/June 1988, p. 732) tative Rings, Mathematical Sciences Re­ search Institute, Berkeley, California. October 1989 January 1991 (May I June 1988, p. 731) 16-19. 97th Annual Meeting, San Fran­ 30-August 4. Sixteenth Annual Confer­ 21-22. Eastern Section Meeting, Stevens Hoboken, New cisco, California. ence and Exhibition on Computer Graph­ Institute of Technology, ics and Interactive Techniques (SIG­ Jersey. (May I June 1988, p. 732) INFORMATION: H. Daly, American GRAPH '89), Hynes Auditorium, Boston, 27-28. Central Section Meeting, Ball Mathematical Society, Post Office Box Massachusetts. (March 1988, p. 466) State University, Muncie, Indiana. 6248, Providence, Rhode Island 02940. 30-August 12. Harmonic Analysis on Re­ (May/June 1988, p. 732) ductive Groups, Bowdoin College, Bruns­ August 1991 wick, Maine. (January 1988, p. 160) January 1990 17-21. American Sociological Associa­ 17-20. Joint Mathematics Meetings, tion Annual Meeting, Cleveland, Ohio. August 1989 Louisville, Kentucky. (April1987, p. 553) (May /June 1988, p. 732) 1-9. Eighteenth International Congress INFORMATION: H. Daly, American 19-22. 1991 Joint Statistical Meetings, of the History of Science, Hamburg, 1- Mathematical Society, Meetings De­ Atlanta, Georgia. (March 1988, p. 466) 5; Munich, 6-9, Federal Republic of partment, Post Office Box 6248, Prov­ Germany. (April 1988, p. 639) idence, Rhode Island 02940. 6-10. 1989 Joint Statistical Meetings, Washington, District of Columbia. (March 1988, p. 466)

1060 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

Vladimirov, V. S. and Zharinov, V. V., Analytic methods in THEORETICAL AND MATHEMATICAL mathematical physics PHYSICS, A Collection of Survey Articles, Zubarev, D. N. and Tserkovnikov, Yu. A., The method of two-time temperature Green functions in equilibrium and Part Ill: On the Fiftieth Anniversary nonequilibrium statistical mechanics of the Institute Sedov, L. 1., Korobeinikov, V. P., and Markov, V. V., The v. S. Vladimirov (Editor-in-Chief), theory of propagation of blast waves E. F. Mishchenko, and A. K. Gushchin Ladyzhenskaya, 0. A., On some trends of investigations (Proceedings of the Steklov Institute, Volume 175) carried out in the laboratory of mathematical physics at the Leningrad Branch of the Mathematical Institute The Steklov Mathematical Institute was formed in 1934 by the 1980 Mathematics Subject Classifications: 32, 35, 40, 41 , 42, 45, 76, Soviet Academy of Sciences. In commemoration of the fiftieth 78, 81 , 82 and others anniversary of the institute, the Academic Council, acting ISBN 0-8218-3119-4, LC 88-22154 ISSN 0081-5438 on the initiative of the Director, I. M. Vinogradov, decided 265 pages (softcover), September 1988 to publish a cycle of "Trudy of the Steklov Institute of the Individual member $70, List price $117, Academy of Sciences:· The cycle consists of surveys of work Institutional member $94 on certain important trends and problems in mathematics that To order, please specify STEKL0/175N were pursued at the institute. The choice of the form and character of the surveys was left to the authors. This book, which represents the third issue in the cycle, COMBINED MEMBERSHIP LIST contains surveys of investigations in certain domains of theoretical and mathematical physics, in particular, statistical This CML is a comprehensive directory of the membership physics, mathematical diffraction, axiomatic quantum field of the AMS, the Mathematical Association of America, and theory, the theory of boundary value problems for the the Society for Industrial and Applied Mathematics. The list is equations of mathematical physics, and the gas dynamics of distributed as a privilege of membership to AMS members in explosion. Fundamental results on the microscopic theory even-numbered years and the MAA members in odd-numbered of superfluidity and superconductivity are presented, as well years. The CML is an invaluable reference for keeping in as a number of results on the theory of functions of several touch with colleagues and for making connections in the complex variables and multidimensional Tauberian theory mathematical sciences community in the U.S. and abroad. having important applications in mathematical physics. There are two lists of individual members. The first is Contents a complete alphabetical list of all members in all three Bogolyubov, N. N., Sr., Superfluidity and quasimeans in organizations. For each member, the CML provides his or her problems of statistical mechanics address, title, department, institution, and telephone number (if Babich, V. M., The mathematical theory of diffraction (A available), electronic address (if indicated), and also indicates survey of some investigations carried out in the laboratory of the mathematical organizations to which the individual belongs. mathematical problems of geophysics, Leningrad Branch of the The second lists individual members according to their Mathematical Institute) geographic locations. In addition, the CML lists all academic Bogoyavlenskii, 0. 1., Vladimirov, V. S., Volovich, 1. V., and institutional members and provides addresses and Gushchin, A. K., Drozhzhinov, Yu. N., Zharinov, V. V., and telephone numbers of mathematical sciences departments. Mikhailov, V. P., Boundary value problems of mathematical 1980 Mathematics Subject Classification: 00 physics ISBN 0-8218-0113-9 Vladimirov, V. S., Drozhzhinov, Yu. N., and Zav'yalov, B. 448 pages (softcover), September 1988 1., Tauberian theorems for generalized functions and their Individual member $20, List price $34, applications Institutional member $27 To order, please specify CML/88/89N

Use the order form in the back of this issue or call 800-556-7774 to use VISA or MasterCard. Shipping and handling charges will be added.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1061 New AMS Publications

The following book description is being reprinted with a development of mathematics. It will also make a fine addition corrected Table of Contents. to any library collection. Contents Thomas Scott Fiske, Mathematical progress in America; The beginnings of the American Mathematical Society, A CENTURY OF MATHEMATICS Reminiscences of Thomas Scott Fiske; IN AMERICA, Part I J. L. Synge, For the tOOth birthday of the American Peter L. Duren, Editor Mathematical Society; George E. Andrews, J. J. Sylvester,. Johns Hopkins, and with the assistance of Richard A. Askey Partitions; and Uta C. Merzbach Carolyn Eisele, Thomas S. Fiske and Charles S. Peirce; Solomon Lefschetz, Luther Pfahler Eisenhart; In the 100 years since the founding of the AMS, the D. V. Widder, Some mathematical reminiscences; American mathematical community has grown from a small Stephen C. Kleene, The role of logical investigations in group heavily dependent on European mathematicians mathematics since 1930; to a large and influential group that in many areas sets R. P. Boas, Memories of bygone meetings; the standard for the rest of the world. By the 1930s, Hassler Whitney, Moscow 1935: Topology moving toward there was a flourishing mathematical community to America; welcome the influx of mathematicians fleeing Europe. Deane Montgomery, Oswald Veblen; These refugees supplied additional strength and new P. R. Halmos, Some books of Auld Lang Syne; vigor to a field that increased dramatically as a result of Nathan Reingold, Refugee mathematicians in the United States World War II and the postwar recognition of mathematics. of America, 1933-1941: Reception and reaction; This volume, the first in the new History of Mathematics Solomon Lefschetz, Reminiscences of a mathematical series, brings together a variety of perspectives on the immigrant in the U.S.; Ivan Niven, The threadbare thirties; political, social, and mathematical forces that have shaped the American mathematical community in the past century. Lipman Bers, The European mathematicians' migration to Humorous, edifying, and poignant, this book presents the America; personal recollections of a number of mathematicians who Irving Kaplansky, Abraham Adrian Albert; D. H. Lehmer, A half century of reviewing; have influenced the development of mathematics in this country. G. Baley Price, American mathematicians in WWI; American mathematicians in War service; One of the highlights of the volume is Lipman Bers's Mina Rees, The mathematical sciences and World War service; paper which was presented as an AMS-MAA Joint Invited Peter Hilton, Reminiscences of Bletchley Park, 1942-45; Address in Atlanta in January 1988 and which gives a moving J. Barkley Rosser, Mathematics and mathematicians in WW/1; account of the reception that he and other European refugee Herman H. Goldstine, A brief history of the computer; mathematicians received in this country. Described here Saunders Mac Lane, Concepts and categories in perspective; are some of the success stories of this century-such as Marshall Hall, Jr., Mathematical biography; classification of finite simple groups, delineated by Daniel Shiing-Shen Chern, American differential geometry-some Gorenstein-as well as some of the problems-such as personal notes; the McCarthy period, chronicled by Chandler Davis. Paul R. G. Baley Price, The mathematical scene, 1940-1965; Halmos, one of the most influential textbook writers, tells of W. S. Massey, Reminiscences of forty years as a the textbooks he used when he was a student and young mathematician; professor and how they influenced him. Among the papers Chandler Davis, The purge; reprinted here are some that have appeared in journals not R. W. Hamming, The use of mathematics; ordinarily read by mathematicians, such as the article by Donald E. Knuth, Algorithmic themes; science historian Nathan Reingold, which appeared in The Daniel Gorenstein, The classification of the finite simple Annals of Science. groups, a personal journey: The early years.

Mathematicians, historians of science, and students alike will find this book illuminating and rewarding. That the lessons of the past can guide the resolution of present problems makes this book important reading for all who are concerned with the

Use the order form in the back of this issue or call 800-556-7774 to use VISA or MasterCard. Shipping and handling charges will be added.

1062 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

1g80 Mathematics Subject Classification: 01 Qualitative properties of solutions ISBN 0-8218-0124-4, LC 88-22155 Representations of solutions ISSN 0899-2428 Generalized solutions 86 pages (hardcover), August 1988 4 systems with constant coefficients Individual member $34, List price $57, Equations and Institutional member $46 General first-order equations and systems To order, please specify HMATH/1 N General higher-order equations and systems Hypoelliptic equations and systems Please note: Special combination prices have been Volume 2 established for the reviews volumes below. The book Elliptic equations and systems descriptions are being re-run from previous Notices issues. Parabolic equations and systems Reviews in Partial Differential Equations, 1980-86 and Volume 3 Reviews In Global Analysis, 1980-86. . Hyperbolic equations and systems special combined price. Individual member $283, L1st Equations and systems of mixed or composite type price $472, Institutional member $378, Reviewer $236 To order, please specify REVPGL0/86N Overdetermined systems Spectral theory and eigenvalue problems Reviews In Partial Differential Equations, 1980-86 and Special equations and problems Reviews in Numerical Analysis, 1980-86 Special combined price. Individual member $262, List Volume 4 price $436, Institutional member $349, Reviewer $218 Special equations and problems 32020 through 35099 To order, please specify REVPNAN/86N Miscellaneous topics Pseudodifferential operators Volume 5 REVIEWS IN PARTIAL DIFFERENTIAL Author and Key Indexes EQUATIONS, 1980-86 1980 Mathematics Subject Classifications: 35020; 35M05, 35J65 Introduction by Murray H. Protter ISBN 0-8218-0103-1, LC 88-6681 4040 pages, 5 volumes (softcover), July 1988 Comprising a significant portion of present-day research Individual member $177, List price $295, in analysis, the area of partial differential equations Institutional member $236, Reviewer $148 To order, please specify REVPDE/86N encompasses a broad spectrum of topics, from classical work in linear second-order equations to more recent work in a general nonlinear setting. In addition, the subject has deep and fundamental ties to a wide variety of scientific REVIEWS IN GLOBAL ANALYSIS, 1980-86 areas outside mathematics. This important reference work Introduction by Anthony J. Tromba makes the vast subject of partial differential equations much more accessible both to specialists working in this area and The term "global analysis" refers to the general area of to those interested in related areas of mathematics and its analysis on manifolds, in which the methods of modern applications. algebra, analysis, geometry, and topology are blended. Although the beginnings of these ideas can be traced to These five volumes contain the more than 19,200 reviews the 17th century, major contributions in this direction were that appeared in Mathematical Reviews from 1980 through made by Lie, Riemann, and Poincare toward the end of 1986 and have a primary or secondary classification in Partial the last century, followed by the work of G. D. Birkhoff, E. Differential Equations (classification number 35). Relevant Cartan, and Morse in the early part of this century. However, cross-references are provided with each review. The fifth it is only in recent years that the subject has attained its volume of this set contains author and key indexes which present central position in mathematics. The subject has make it very easy to locate· items written by a specific many rich applications to fields outside mathematics-such author or to get information about collections or conference as mechanics, quantum physics, and general relativity-as proceedings dealing with partial differential equations. well as within mathematics itself. Contents Today, this vital and active field is undergoing a virtual Volume 1 explosion of new and important results. Reviews in Introduction by Murray H. Protter Global Analysis makes information about the most recent Partial differential equations contributions to this rapidly growing field accessible both to General theory

Use the order form in the back of this issue or call 800-556-7774 to use VISA or MasterCard. Shipping and handling charges will be added.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1063 New AMS Publications

specialists working in global analysis, and to those in other areas of pure and applied mathematics. REVIEWS IN NUMERICAL ANALYSIS, These five volumes contain the more than 18,000 reviews 1980-86 that appeared in Mathematical Reviews from 1980 through Introduction by Gene H. Golub 1986 and have a primary or secondary classification in Global Analysis (classification number 58). Relevant cross-references These five volumes bring together a wealth of bibliographic are provided with each review. The fifth volume of this set information in the area of numerical analysis. Containing contains author and key indexes, making it very easy to locate over 17,600 reviews of articles, books, and conference items written by a specific author or to get information about proceedings, these volumes represent all the numerical collections or conference proceedings dealing with global analysis entries that appeared in Mathematical Reviews analysis. between 1980 and 1986. Classified according to the 1980 Mathematics Subject Classification scheme, the reviews are Contents listed in each subsection according to their MR classification Volume 1 number. Accompanying each entry is a list of all subsequent Global analysis, analysis on manifolds reviews that cite the entry as a reference. In addition, author General theory of differentiable manifolds and key indexes appear at the end of volume 5. Infinite-dimensional manifolds 1980 Mathematics Subject Classifications: 65XX, 76XX, 41 XX, 35XX Calculus on manifolds; nonlinear operators ISBN 0-8218-0102-3, LC 87-25478 Spaces and manifolds of mappings 3750 pages, 5 volumes (softcover), January 1988 Individual member $150, List price $250, Volume 2 Institutional member $200, Reviewer $125 Variational problems in infinite-dimensional spaces To order, please specify REVNAN/86N Ordinary differential equations on manifolds; dynamical systems Volume 3 Ordinary differential equations on manifolds; dynamical systems Volume 4 Partial differential equations on manifolds; differential operators Pseudogroups and general structures on manifolds Volume 5 Series contents Author index Key index 1980 Mathematics Subject Classification: 58 ISBN 0-8218-01 04-X, LC 88-1 0565 4060 pages, 5 volumes (softcover), August 1988 Individual member $177, List price $295, Institutional member $236, Reviewer $148 To order, please specify REVGL0/86N

Use the order form in the back of this issue or call 800-556-7774 to use VISA or MasterCard. Shipping and handling charges will be added.

1064 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Reports and Communications

Recent Appointments G. D. Mostow, AMS, and Leonard Female: names that were obvi­ Gillman, MAA, to the Committee ously female; on Arrangements for the Boulder Unknown: names that could Committee members' terms of Meeting August 7-10, 1989. Pro­ not be identified as clearly male office on standing committees fessor Ramsay will serve as chair­ or female (i.e., only initials given); expire on December 31 of the man. and year given in parentheses fol­ Richard A. Askey (Section L) Foreign: foreign names that lowing their names, unless oth­ and Jerry L. Bona (Section Q) could not be identified as clearly erwise specified. have been appointed as the Soci­ male or female. ety's representatives to the Amer­ The following is the second George E. Andrews, Kenneth Mil­ ican Association for the Advance­ reporting of this information. Up­ let, M. Susan Montgomery, and ment of Science by President G. dated reports will appear annually Paul J. Sally, Jr. have been ap­ D. Mostow. Terms expire on May in Notices. pointed by President G. D. Mostow 27, 1989. to the ad hoc Committee on Fel­ lowship Policy. Professor Mont­ Statistics on Women Members of the AMS gomery will serve as chairman. Mathematicians Compiled Residing in the U.S. Male: 12,167 75% Charles Fefferman ( 1991) and by the AMS Jun-ichi lgusa ( 1991) have been Female: 2, 196 14% At its August 1985 meeting, the Unknown: 752 5% appointed by President G. D. Council of the AMS approved a Foreign: 982 6% Mostow to the Committee to Se­ motion to regularly assemble and Total checked: 16,097 lect the Winner of the Steele Prize. report in Notices information on Continuing members of the com­ the relative numbers of men ver­ mittee are Frederick J. Almgren, Invited Hour Address Speakers sus women in at least the follow­ at AMS Meetings (1978-1987) Jr. ( 1989), Luis A. Caffarelli ( 1990), ing categories: membership in the William S. Massey ( 1989), chair­ Male: 366 92% AMS; invited hour addresses at Female: 25 6% man, Frank A. Raymond (1989), AMS meetings; speakers at special Unknown: 2 1% Neil J. A. Sloane (1990), Louis sessions at AMS meetings; and Foreign: 3 1% Solomon ( 1989), Richard P. Stan­ members of editorial boards of Total checked: 396 ley ( 1989), Michael E. Taylor AMS journals. (1990). It was subsequently decided Speakers at Special Sessions President G. D. Mostow has that this information would ·be at AMS Meetings (1983-1987) appointed Hyman Bass, James gathered by determining the sex Male: 2, 713 78% McKenna, Cathleen S. Morawetz, of the individuals in the above Female: 203 6% Warren Page, Gian-Carlo Rota and. categories based on name identi­ Unknown: 342 10% Alan D. Weinstein to the Liaison fication and that additional infor­ Foreign: 220 6% Committee with AAAS. mation on the number of Ph.D.'s Total checked: 3,478 Larry Baggett, William L. granted to women would also be Briggs, Richard Andrew Holley, collected using the AMS Annual Trustees and Council Members Frieda K. Holley, William H. Jaco Survey. Since name identification 1987 1986 ~ 1984 (ex officio), ·Andy Roy Magid (ex was used, the information for Total: 65 65 71 officio), Arlan Ramsay, William 70 some categories necessitated the Male: 52 80% 56 86% 61 86% 62 89% N. Reinhardt, Kenneth A. Ross use of four classifications: Female: 13 20% 9 14%10 14% 8 11% (ex officio), and Richard L. Roth Male: names that were obvi­ have been appointed by Presidents ously male;

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1065 AMS Reports and Communications

Members of Editorial Boards of AMS Journals 1987 1986 1985 1984 1983 1982 1981 1980 1979 1978 Total: 133 109 102 93 90 83 85 82 82 67 Male: 125 94% 104 95% 94 92% 85 91 o/o 84 93% 77 93% 79 93% 77 94% 78 95% 64 96% Female: 8 6% 5 5% 8 8% 8 9% 6 7% 6 7% 6 7% 5 6% 4 5% 3 4%

Ph.D.'s Granted to U.S. Citizens 1987 1986 1985 1984 1983 1982 1981 1980 1979 1978 Total: 362 386 396 433 455 519 567 578 596 634 Male: 289 80% 304 79% 315 80% 346 80% 366 80% 431 83% 465 82% 491 85% 503 84% 545 86% Female: 73 20% 82 21% 81 20% 87 20% 89 20% 88 17% 102 18% 87 15% 93 16% 89 14%

LATTICES WITH UNIQUE COMPLEMENTS INTRODUCTION TO ANALYTIC NUMBER THEORY V.N.Salii Volume 69) A. G. Postnikov (Translations of Mathematical Monographs, (Translations of Mathematical Monographs, Volume 68) The class of uniquely complemented lattices properly contains all Boolean lattices. However, no explicit example of a Aimed at a level between textbooks and the latest research non-Boolean lattice of this class has been found. In addition, monographs, this book is directed at researchers, teachers, the question of whether this class contains any complete and graduate students interested in number theory and its non-Boolean lattices remains unanswered. This book focuses connections with other branches of science. The author on these classical problems of lattice theory and the various has attempted to give as broad a picture as possible of attempts to solve them. Requiring no specialized knowledge, the problems of analytic number theory while avoiding the book is directed at researchers and students interested in specialization and those topics already sufficiently covered general algebra and mathematical logic. In particular, this book focuses on general in the literature. 1980 Mathematics Subject Classification: 06 additive number theory and the concept of a numerical ISBN 0-8218-4522-5 semigroup-and gives a systematic discussion of these ISSN 0065-9282 topics. 128 pages {hardcover), March 1988 1980 Mathematics Subject Classification: 11 Individual member $31, List price $51, ISBN 0-8218-4521-7 Institutional member $41 ISSN 0065-9282 To order, please specify MMON0/69NA 336 pages (hardcover), March 1988 Individual member $68, List price $114, Shipping/Handling: 1st book $2, each add'l $1, $25 max. By Institutional member $91 air, 1st book $5, each add'l $3, $100 max. To order, please specify MMON0/68 NA Prepayment required. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901-9930, or call 800-556-7774 to Shipping/Handling: 1st book S2, each add'l S1, S25 max. By use VISA or MasterCard. air, 1st book S5, each add'l S3, SlOO max. Prepayment required. Order from AMS, P.O. Box 1571, Annex Station, Providence, Rl 02901-9930, or call 800-556-7774 to use VISA or MasterCard. ------@------~ ]

1066 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Officers and Committee Members

Terms expire on December 31 of the year given unless otherwise specified.

Numbers to the left of headings are used as points of reference in Secretary Everett Pitcher 1988 a index to AMS committees which follows this listing. Primary and Associate secondary headings are: Secretaries* Frank T. Birtel 1988 W. Wistar Comfort 1988 Officers Andy Roy Magid 1989 1.1 Liaison Committee Lance W. Small 1989 2 Council Treasurer Franklin P. Peterson 1988 2.1 Executive Committee of the Council Associate 3 Board of Trustees Treasurer Steve Armentrout 1988 4 Committees 4.1 Editorial and Communications Committees 4.2 Committees of the Board of Trustees 2.0.2. Representatives of Committees 4.3 Internal Organization of the AMS American Journal of 4.4 Program and Meetings Mathematics Spencer Bloch 1988 4.5 Status of the Profession Bulletin Morris W. Hirsch 1989 4.6 Prizes and Awards Edgar Lee Stout 1988 4. 7 Institutes and Symposia Colloquium Raoul H. Bott 1990 4.8 Joint Committees Charles L. Fefferman 1988 5 Representatives Journal of the AMS Michael Artin 1991 6 Index Committee to Monitor Problems in Communication Marian B. Pour-EI 1989 1. Officers Mathematical Reviews Melvin Hochster 1989 Mathematical Surveys and President G. D. Mostow 1988 Monographs R. 0. Wells, Jr. 1988 President-Elect William Browder 1988 Mathematics of Vice-Presidents Barry Simon 1989 Computation Walter Gautschi 1989 William P. Thurston 1989 Hugh C. Williams 1988 Karen Uhlenbeck 1988 Proceedings William J. Davis 1988 Secretary Everett Pitcher 1988 Irwin Kra 1988 Associate Andrew M. Odlyzko 1989 Secretaries Frank T. Birtel 1988 Donald S. Passman 1988 W. Wistar Comfort 1988 Science Policy Committee Andy Roy Magid 1989 Ronald G. Douglas 1989 Lance W. Small 1989 Transactions and Memoirs Treasurer Franklin P. Peterson 1988 Ronald L. Graham 1988 Associate Treasurer Steve Armentrout 1988 2.0.3. Members-at-Large

1.1. Liaison Committee James G. Arthur 1988 Yiannis N. Moschovakis 1989 Jane P. Gilman 1988 Linda A. Ness 1989 All members of this committee serve ex officio. Richard K. Guy 1990 Marc A. Rieffel 1989 Chairman G. D. Mostow Rhonda J. Hughes 1990 Harold M. Stark 1990 Franklin P. Peterson Robion C. Kirby 1990 Jean E. Taylor 1988 Everett Piicher Irwin Kra 1990 Chuu-Lian Terng 1988 H. Blaine Lawson, Jr. 1989 William A. Veech 1988 Albert Marden 1990 Robert F. Williams 1988 Carol S. Wood 1989 2. Council------* Only one Associate Secretary at a time is a voting member of the 2.0.1. Officers of the AMS Council, namely the cognizant Associate Secretary for the scientific President G. D. Mostow 1988 sessions. President-Elect William Browder 1988 Vice-Presidents Barry Simon 1989 William P. Thurston 1989 Karen Uhlenbeck 1988

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1067 Officers and Committee Members

2.1. Executive Committee of the Council 4.1.4. Colloquium Chairman Raoul H. Bott 1990 William Browder ex officio Charles L. Fefferman 1988 Irwin Kra 1990 H. Jerome Keisler 1989 Chairman G. D. Mostow ex officio officio Everett Pitcher ex 4.1.5. Committee to Monitor Problems Jean E. Taylor 1988 in Communication William P. Thurston 1991 William A. Veech 1989 Sheldon Axler 1989 Jozef Dodziuk 1988 Consultant Nancy Gubman Arthur M. Jaffe 1988 3. Board of Trustees------Richard G. Larson 1990 Steve Armentrout ex officio William H. Jaco ex officio Chairman Ramesh A. Gangolli 1989 Richard S. Palais 1990 Frederick W. Gehring 1992 Chairman Marian B. Pour-El 1989 Secretary Ronald L. Graham 1991 4.1.6. Contemporary Mathematics M. Susan Montgomery 1990 G. D. Mostow ex officio M. Salah Baouendi 1988 Franklin P. Peterson ex officio Daniel M. Burns 1989 PaulJ. Sally, Jr. 1988 David Eisenbud 1989 Jonathan Goodman 1990 William H. Jaco 1988 Gerald J. Janusz 1989 4. Committees Chairman Irwin Kra 1988 Jan Mycielski 1990

4.1. Editorial and Communications Associate Editor: Thomas F. Banchoff Committees 4.1.7. Journal of the AMS 4.1.1. Abstracts Editorial Committee Chairman Michael Artin 1992 H. Blaine Lawson, Jr. 1991 All members of this committee serve ex officio. Richard B. Melrose 1990 Frank T. Birte1 Wilfried Schmid 1990 W. Wistar Comfort Robert E. Tarjan 1992 Andy Roy Magid Associate Editors Chairman Everett Pitcher Lance W. Small James G. Arthur 1992 Joe Harris 1990 Peter Bickel 1990 Hendrik W. Lenstra, Jr. 1991 4.1.2. American Journal of Mathematics, Gerd Faltings 1991 Andrew Majda 1992 Society's Representatives Charles L. Fefferman 1992 Hugh L. Montgomery 1990 M. Salah Baouendi 1989 Michael H. Freedman 1990 Paul H. Rabinowitz 1991 Chairman Spencer Bloch 1988 Daniel Friedan 1991 Karen Uhlenbeck 1992 Ronald L. Graham 1992 W. Hugh Woodin 1990 4.1.3. Bulletin (New Series) Chairman Morris W. Hirsch 1989 4.1.8. Mathematical Reviews Roger E. Howe 1990 Leonard Berkovitz 1988 Edgar Lee Stout 1988 Chairman Melvin Hochster 1989 Hans F. Weinberger 1990 Associate Editors for Research Announcements 4.1.9. Surveys and Monographs William B. Arveson 1990 Ronald L. Graham 1990 Mathematical Spencer Bloch 1990 Victor W. Guillemin 1990 Victor W. Guillemin 1989 Gergory L. Cherlin 1989 Frank S. Quinn III 1990 Irwin Kra 1988 Percy Alec Deift 1990 Peter B. Shalen 1990 M. Susan Montgomery 1990 Michael B. Fried 1990 Nolan R. Wallach 1990 Chairman R. 0. Wells, Jr. 1988

Associate Editors for Research - Expository Articles Associate Editor: Thomas F. Banchoff Lenore Blum 1989 David A. Vogan, Jr. 1989 4.uo. Mathematics of Computation Mazur 1989 Barry Alan D. Weinstein 1989 Chairman Walter Gautschi 1989 Richard S. Palais 1990 Guido L. Weiss 1990 Donald Goldfarb 1990 John E. Osborn 1989 Hugh C. Williams 1988

1068 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Officers and Committee Members

Associate Editors Ralph L. Cohen 1990 Eugene B. Fabes 1988 James Bramble 1989 Carl Pomerance 1989 Ronald Getoor 1990 Bille C. Carlson 1989 Rene Schoof 1989 Chairman Ronald L. Graham 1988 Eugene Isaacson 1989 Larry L. Schumaker 1988 Jerry L. Kazdan 1990 Heinz-Otto Kreiss 1990 Ridgway Scott 1989 Roger D. Nussbaum 1991 James N. Lyness 1989 Daniel Shanks 1990 Carl Pomerance 1991 Harald Niederreiter 1990 Frank Stenger 1989 David J. Saltman 1990 Syvert P. Nersett 1988 Hans J. Stetter 1988 Audrey A. Terras 1991 Andrew M. Odlyzko 1988 G. W. Stewart 1989 Robert J. Zimmer 1990 Frank W. J. Olver 1989 Vidar C. Thomee 1989 Stanley J. Osher 1989 Lars B. Wahlbin 1989 4.us. Translation from Chinese Beresford N. Parlett 1988 John W. Wrench, Jr. 1990 Sun-Yung Alice Chang S.-Y. Cheng 4.1.11. Notices Chairman Tsit-Yuen Lam Robert J. Blattner 1990 Tai-Ping Liu Ralph P. Boas 1988 Chung-Chun Yang Lucy J. Garnett 1990 Chairman Everett Pitcher ex officio 4.1.16. Translation from Japanese Mary Ellen Rudin 1988 Shoshichi Kobayashi Nancy K. Stanton 1990 Chairman Katsumi Nomizu James A. Voytuk ex officio Steven H. Weintraub 1988 Standing Committees Associate Editors 4.1.17. History of Mathematics Special Articles Ronald L. Graham Richard A. Askey J elfrey C. Lagarias Chairman Peter L. Duren Bruno Harris 4.1.12. Proceedings Uta C. Merzbach William Adams 1988 J. Marshall Ash 1988 4.us. Subcommittee on Russian Mathematical Thomas H. Brylawski 1989 History Dennis Burke 1989 Frederick R. Cohen 1991 V. I. Arnol'd John B. Conway 1988 Richard Askey Chairman William J. Davis 1991 S. G. Gindikin Thomas J. Jech 1988 N. K. Nikol'skii Barbara Keyfitz 1991 Chairman Allen Shields Irwin Kra 1988 R. Daniel Mauldin 1989 4.1.19. Reprinted Books Kenneth R. Meyer 1991 Eugenio Calabi Paul S. Muhly 1990 Charles W. Curtis Andrew M. Odlyzko 1989 Chairman Oscar S. Rothaus George C. Papanicolaou 1991 Donald S. Passman 1988 4.1.20. University Lecture Series Louis J. Ratliff, Jr. 1991 Theodore W. Gamelin Jonathan M. Rosenberg 1991 Donald S. Ornstein William D. Sudderth 1991 Chairman Hugo Rossi James E. West 1991 Leonard L. Scott Warren J. Wong 1991 4.2. Committees of the Board of Trustees Associate Editor Blass Andreas 4.2.1. Agenda and Budget officio. 4.1.13. Proceedings of Symposia in Applied All members of this committee serve ex Mathematics Steve Armentrout Stuart S. Antman 1988 Ramesh Gangolli Giles Auchmuty 1989 G. D. Mostow Chairman Jane Cronin Scanlon 1988 Franklin P. Peterson Everett Pitcher 4.1.14. Transactions and Memoirs Jean E. Taylor James E. Baumgartner 1991 James W. Cannon 1989

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1069 Officers and Committee Members

4.2.2. Appeals Committee on Discounted Subscriptions 4.2.12. The Publication Program Consultant Carol-Ann Blackwood Steve Armentrout 1988 Consultant Melvin Hochster Peter L. Duren 1989 William H. Jaco ex officio Ramesh Gangolli 1989 Chairman Morton Lowengrub Chairman Murray Gerstenhaber 1988 Franklin P. Peterson Consultant Mary C. Lane PaulJ. Sally, Jr. William H. Jaco ex officio Andrew M. Odlyzko 1989 4.2.3. Audit Everett Pitcher ex officio Frederick W. Gehring Hugo Rossi 1989 M. Susan Montgomery Paul J. Sally, Jr. 1989

4.2.4. Computer Operations and Facilities, 4.2.13. Salaries Visiting Committee on Steve Armentrout ex officio Jill P. Mesirov 1988 Ramesh A. Gangolli ex officio Chairman S. Tucker Taft 1989 Chairman Ronald L. Graham Peter J. Weinberger 1990 Franklin P. Peterson ex officio

4.2.5. Corporate Relations 4.2.14. Staff and Services Chairman Ramesh A. Gangolli Chairman Steve Armentrout ex officio Maria M. Klawe Franklin P. Peterson ex officio Oscar S. Rothaus Paul J. Sally, Jr.

4.2.6. Endowment Ad Hoc Committee Andrew M. Gleason Chairman W. Ted Martin 4.2.15. Institutional Membership Cathleen S. Morawetz Consultant Carol-Ann Blackwood 4.2.7. Investment Ramesh Gangolli Chairman Frederick W. Gehring Steve Armentrout ex officio William A. Veech Ramesh A. Gangolli James A. Voytuk ex officio Chairman Franklin P. Peterson ex officio

4.2.s. Legal Aid 4.3. Internal Organization of the Steve Armentrout American Mathematical Society Chairman Morton L. Curtis Todd Dupont Standing Committees Murray Gerstenhaber

4.2.9. Liaison Committee 4.3.t. Committee on Committees All members of this committee serve ex officio. James G. Glimm 1988 William H. Jaco 1988 Chairman G. D. Mostow Chairman Irwin K.ra 1988 Franklin P. Peterson G. D. Mostow ex officio Everett Pitcher Everett Pitcher ex officio 4.2.10. Long Range Planning Paul H. Rabinowitz 1988 Audrey A. Terras 1988 All members of this committee serve ex officio. William A. Veech 1988 Ramesh A. Gangolli Irwin Kra 4.3.2. Election Scheduling William H. Jaco ex officio Jane P. Gilman Franklin P. Peterson Irwin Kra Everett Pitcher Chairman William P. Thurston Chairman William A. Veech William A. Veech James A. Voytuk 4.2.11. Membership Chairman Frederick W. Gehring 1990 Melvin Henriksen 1990 Irwin Kra 1990 Jill P. Mesirov 1989 Hugo Rossi 1988

1070 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Officers and Committee Members

4.3.3. Nominating Committee 4.4. Program and Meetings M. Salah Baouendi 1988 Roger C. Alperin 1989 Standing Committees Ronald DeVore 1989 Paul C. Fife 1988 Chairman Jane P. Gilman 1989 4.4.1. Program Committee for National Meetings Carl Pomerance 1988 Peter B. Gilkey 1990 Leonard Scott 1989 George A. Hagedorn 1990 William P. Ziemer 1988 Linda Keen 1988 Hugh L. Montgomery 1989 Ad Hoc Committees Everett Pitcher ex officio Chairman Paul H. Rabinowitz 1988 Nolan R. Wallach 1988 4.3.4. Centennial Committee Felix E. Browder 4.4.2. Central Sectional Meetings Harold M. Edwards (Select Hour Speakers for) Andrew M. Gleason Chairman Donald G. Aronson 1988 G. D. Mostow David Drasin 1989 Chairman Everett Pitcher Jerry Kaminker 1988 4.3.S. Centennial Program Committee Andy Roy Magid ex officio Robert J. Zimmer 1989 Hyman Bass Chairman Felix E. Browder 4.4.3. Eastern Sectional Meetings Philip A. Griffiths (Select Hour Speakers for) John W. Milnor Ruth M. Charney 1989 Cathleen S. Morawetz W. Wistar Comfort ex officio Richard H. Herman 1988 4.3.6. Poster Committee Chairman Lesley M. Sibner 1988 Chairman Thomas F. Banchoff Thomas Crawford Spencer 1988 F. Alberto Grunbaum Nelson Lee Max 4.4.4. Far Western Sectional Meetings (Select Hour Speakers for) 4.3.7. Public Information Committee Ronald J. DiPerna 1989 John W. Addison, Jr. William M. Kantor 1989 Chairman Yousef Alavi Heinz-Otto Kreiss 1988 William G. Chinn Chairman Murray M. Schacher 1988 Ronald R. Coifman Lance W. Small ex officio Ronald L. Graham Peter J. Hilton 4.4.s. Southeastern Sectional Meetings Donald R. Lick (Select Hour Speakers for) Jean Pedersen Frank T. Birtel ex officio C. Taubes Jon F. Carlson 1988 Patrick B. Eberlein 1989 4.3.8. Travel Grants for the AMS Centennial Chairman Frank S. Quinn III 1988 Chairman Richard W. Beals 1989 John W. Bunce 4.4.6. Agenda for Business Meetings Anthony W .. Hager William F. Lucas M. Salah Baouendi 1988 P. Emery Thomas Chairman Everett Pitcher Frank Uhlig · Carol L. Walker 1988

4.3.9. Proposed Structure of JPBM 4.4.7. Gibbs Lecturers for 1989 and 1990, Robert M. Fossum Committee to Select Marc A. Rieffel Jane Cronin Scanlon Chairman Jean E. Taylor Chairman Thomas Crawford Spencer Shmuel Winograd 4.3.10. 1987 Election Tellers Krzysztof Galicki Alexander Smith

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1071 Officers and Committee Members

4.5. Status of the Profession 4.5.7. Service to Mathematicians in Developing Countries Standing Committees Chairman Raymond G. Ayoub James A. Donaldson 4.5.1. Academic Freedom, Tenure, and James Eells Employment Security Donald M. Hill Marshall H. Stone Jerome A. Goldstein 1989 Thomas G. Kurtz 1990 Ad Hoc Committees Barbara L. Osofsky 1990 Robert R. Phelps 1989 4.5.8. Applied Mathematics Chairman Halsey L. Royden 1988 Gail S. Young 1989 Constantine Dafermos David S. Kinderleher 4.5.2. Centennial Fellowships Eduardo Daniel Sontag Terms expire on June 30 Chairman Jean E. Taylor

Chairman Frederick J. Almgren, Jr. 1989 4.5.9. Cooperation with the Chinese David Eisenbud 1990 S.-Y. Cheng Lawrence Craig Evans 1990 Ronald L. Graham Dorian Goldfeld 1989 Chairman Richard S. Palais Victor L. Klee, Jr. 1990 John W. Morgan 1989 4.5.10. Fellowship Policy Karen Vogtmann 1990 George E. Andrews Kenneth Millet 4.5.3. Human Rights of Mathematicians Chairman M. Susan Montgomery Michael I. Brin 1990 Paul J. Sally Jr. Bettye Anne Case 1989 Patrick X. Gallagher 1989 4.6. Prizes and Awards Herman R. Gluck 1989 Leon A. Henkin 1988 Neil I. Koblitz 1988 Standing Committees Joel L. Lebowitz 1990 Chairman Alice T. Schafer 1990 4.6.1. National Awards and Public Representation 4.5.4. Professional Ethics William Browder ex officio Chairman G. D. Mostow ex officio C. Edmund Burgess 1990 David Mumford 1989 Frank L. Gilfeather 1990 Everett Pitcher ex officio Paul R. Halmos 1988 Chairman Linda Keen 1989 4.6.2. Steele Prizes Anneli Lax 1988 Terms expire on June 30 4.5.5. Recruitment of Young Mathematicians Frederick J. Almgren, Jr. 1989 Paul J. Sally, Jr. Luis A. Caffarelli 1990 Chairman James D. Stasheff Charles L. Fefferman 1991 Lynn A. Steen Jun-ichi lgusa 1991 Chairman William S. Massey 1989 4.5.6. Science Policy Frank A. Raymond 1989 Hyman Bass 1990 Neil J. A. Sloane 1990 Chairman Ronald G. Douglas 1989 Louis Solomon 1989 Frank L. Gilfeather 1990 Richard P. Stanley 1989 James G. Glimm 1989 Michael E. Taylor 1990 Ronald L. Lipsman 1989 James W. Maxwell ex officio Ad Hoc Committees G. D. Mostow ex officio Robert Osserman 1988 4.6.3. Automatic Theorem Proving, Committee John C. Polking 1990 to Recommend Winners of Prizes for Judith D. Sally 1988 David A. Sanchez 1989 Chairman David Mumford William P. Thurston 1989 Jacob T. Schwartz Guido L. Weiss 1988 John L. Selfridge

1072 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Officers and Committee Members

4.6.4. Bocher Prize for 1989, Committee to Select Janet Peterson (SIAM) 1988 the Winner of Linda Petzold (SIAM) 1989 Paul J. Cohen Alice T. Schafer (MAA) 1988 Chairman Richard B. Melrose Elizabeth L. Scott (IMS) Louis Nirenberg Bhama Srinivasan (AMS) 1989 4.8.3. AM8-ASL-IMS-SIAM Committee on 4.7. Institutes and Symposia Translations from Russian and Other Slavic Languages Standing Committee Chairman Courtney S. Coleman (AMS) 1989 AMS Subcommittee Members 4.7.L Summer Institutes and Special Symposia Terms expire on February 28 Consultant V. I. Arnol'd Joseph N. Bernstein 1990 Steven L. Kleiman 1990 Charles V. Coffman 1988 Haynes R. Miller 1991 Chairman Courtney S. Coleman 1989 Paul H. Rabinowitz 1989 Allen Devinatz 1988 Raghavan Narasimhan 1991 Consultant S. G. Gindikin Thomas Crawford Spencer 1990 Vladislav V. Goldberg 1988 Robert B. Warfield, Jr. 1989 John R. Isbell 1988 Anatole Katok 1990 4.7.2. Liaison Committee with AAAS L. G. Makar-Limanov 1989 Hyman Bass Paul G. Nevai 1989 James McKenna Consultant N. K. Nikol'skii Cathleen S. Morawetz Warren Page ASL Subcommittee Members Gian-Carlo Rota Vladimir Lifschitz 1990 Alan D. Weinstein Chairman Elliott Mendelson 1989 Gregory Mine 1990 4.8. Joint Committees B. F. Wells 1989 IMS Subcommittee Members 4.8.1. AMS-AAAS-MAA Committee on Opportunities in Mathematics Chairman Eugene B. Dynkin for Underrepresented Minorities B. Pittel A. Rukhin Manuel P. Berriozabal 1988 W. J. Studden Sylvia T. Bozeman 1989 James A. Donaldson 1988 4.8.4. AMS-IMS-SIAM Committee on Joint Chairman Gloria F. Gilmer 1988 Summer Research Conferences in the Shirley Malcom ex officio Mathematical Sciences Rogers J. Newman 1989 Clarence E. Stephens 1989 Terms expire on June 30 Consultant Argelia Velez-Rodriguez William B. Arveson (AMS) 1989 John A. Burns (SIAM) 1988 Martin Golubitsky (SIAM) 1988 Roger E. Howe (AMS) 1992 4.8.2. AMS-ASA-IMS-MAA-NCTM-SIAM Daniel J. Kleitman (AMS) 1989 Committee on Women in the Chairman Ingram Olkin (IMS) 1988 Mathematical Sciences Mary Ellen Rudin (AMS) 1989 NCTM members' terms expire April 1 of the year given. Lesley M. Sibner (AMS) 1990 Stephen G. Simpson (AMS) 1990 Grace M. Burton (NCTM) 1990 Susan J. Devlin (ASA) 4.8.s. AMS-MAA Ad Hoc Advisory Committee on a Marjorie M. Enneking (NCTM) 1989 Newletter on Collegiate Mathematics Education Susan Geller (AMS) 1987 Jerry L. Bona Marjorie G. Hahn (IMS) Chairman Joseph A. Gallian Jeanne W. Kerr (AMS) 1989 Frank Morgan Chairman Carole B. LaCampagne (AMS, MAA) 1987 Doris W. Schattschneider Jeanne LaDuke (MAA) 1989 Suzanne Marie Lenhart (AMS) 1989 Betty K. Lichtenberg (NCTM) 1989 Joyce R. McLaughlin (SIAM) 1990 Ingram Olkin (IMS)

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1073 Officers and Committee Members - 4.8.6. AMS-MAA Arrangements Committee for the 4.8.12. Employment Concerns Subcommittee Phoenix Meeting Morton Brown 1989 January 11-14, 1989 Audrey A. Terras 1989 William H. Jaco ex officio Robert J. Thompson 1988 Chairman John McDonald Chairman Barnet M. Weinstock 1989 Kenneth A. Ross ex officio Lance W. Small ex officio 4.8.13. Short Course Subcommittee Chairman Stefan A. Burr 1989 4.8.7. AMS-MAA Arrangements Committee for the R. Peter DeLong 1991 Boulder Meeting Lis! Novak Gaal 1989 August 7-10, 1989 Robert P. Kurshan 1990 Larry W. Baggett Barbara L. Osofsky 1988 William L. Briggs Marjorie L. Stein 1990 Richard Andrew Holley James J. Tattersall 1990 Frieda K. Holley William H. Jaco ex officio 4.8.14. AMS-MAA-SIAM Joint Committee on Andy Roy Magid ex officio Employment Opportunities Chairman Arlan Ramsay Ronald M. Davis (MAA) 1990 William N. Reinhardt Calvin T. Long (MAA) 1988 Kenneth A. Ross ex officio James W. Maxwell ex officio Richard L. Roth Chairman Brian J. McCartin (SIAM) 1989 John W. Petro (AMS) 1988 4.8.8. AMS-MAA Joint Program Committee Donald C. Rung (AMS) 1990 for the AMS Centennial Robert S. Stepleman (SIAM) 1988 Chairman Hugh L. Montgomery M. Susan Montgomery 4.8.15. AMS-MAA-SIAM Joint Administrative Ivan Niven Committee Richard S. Palais All members of this committee serve ex officio.

4.8.9. AMS-MAA Joint Meetings Committee I. Edward Block (SIAM) All members of this committee serve ex officio. James W. Daniel (SIAM) Shmuel Winograd (SIAM) Consultant H. Hope Daly Leonard Gillman (MAA) Chairman William H. Jaco William H. Jaco (AMS) Everett Pitcher Franklin P. Peterson (AMS) Kenneth A. Ross Chairman Everett Pitcher (AMS) Alfred B. Willcox Kenneth A. Ross (MAA) Alfred B. Willcox (MAA) 4.8.10. AMS-MAA Committee on Employment and Educational Policy 4.8.16. AMS-MAA-SIAM Joint Policy Board Morton Brown (MAA) 1989 for Mathematics Stefan A. Burr (AMS) 1989 I. Edward Block (SIAM) ex officio Chairman Edward A. Connors (AMS) 1988 Charles William Gear (SIAM) ex officio Philip C. Curtis, Jr. (MAA) 1990 Leonard Gillman (MAA) ex officio Don 0. Loftsgaarden (MAA) 1990 Donald L. Kreider (MAA) 1992 David J. Lutzer (MAA) 1989 William H. Jaco (AMS) ex officio James W. Maxwell ex officio G. D. Mostow (AMS) ex officio Audrey A. Terras (AMS) 1988 Everett Pitcher (AMS) 1988 Werner C. Rheinboldt (SIAM) 1988 4.8.11. Data Subcommittee Alfred B. Willcox (MAA) ex officio Chairman Edward A. Connors 1990 Consultant Lincoln K. Durst 4.8.17. Joint Policy Board for Mathematics John D. Fulton 1988 James Hurley 1988 Head, Office of Governmental and Public Affairs: Charlotte Lin 1989 Kenneth M. Hoffman Don 0. Loftsgaarden 1990 Public Information Director: Kathleen Holmay David J. Lutzer 1990 Telephone of Public Information Director: 301-588-6168 James W. Maxwell ex officio Donald E. McClure 1990 Donald C. Rung 1989

1074 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Officers and Committee Members

JPBM Office of Governmental and Public Affairs 5.o.4. Conference Board of the Mathematical Sciences 818 Connecticut Avenue, N.W., Suite 515 G. D. Mostow 1988 Washington, DC 20006 Telephone: 202-659-6444 5.0.5. Fulkerson Prize Committee Telephone of Public Information Director: Alan J. Hoffman 301-588-6168 5.0.6. American Association for the Advancement of 4.8.18. JPBM Joint Coordinating Committee on Public Science Understanding of Mathematics Terms expire on May 27 Ronald L. Graham (AMS) Section A Louise Hay 1989 Joseph B. Keller (SIAM) Section L Richard A. Askey 1989 Lynn A. Steen (MAA) Section Q Jerry L. Bona 1989

4.8.19. JPBM Committee for Mathematics Department 5.0.7. U.S. National Committee on Theoretical and Heads Applied Mechanics Terms expire in January Term expires on October 31 Donald F. Reynolds 1990 Stuart S. Antman 1988 Donna Szott 1990 Thomas W. Tucker 1990 6. Index------4.8.2o. AMS-SIAM Committee 1987 Election Tellers ...... 4.3.1 0 on Applied Mathematics ABC Committee ...... 4.2.1 Committee on Opportunities in Constantine M. Dafermos 1989 AMS-AAAS-MAA for Underrepresented Minorities . . . . 4.8.1 James M. Hyman 1990 Mathematics Committee on Women in Donald E. McClure 1988 AMS-ASA-IMS-MAA-NCTM-SIAM the Mathematical Sciences ...... 4.8.2 Chairman George C. Papanicolaou 1989 Committee on Translations from Francis Sullivan 1988 AMS-ASL-IMS-SIAM Other Slavic Languages ...... 4.8.3 Robert F. Warming 1990 Russian and AMS-IMS-SIAM Committee on Joint Summer Research 4.8.21. AMS-SIAM-SMB Committee on Conferences in the Mathematical Sciences ...... 4.8.4 Mathematics in the Life Sciences AMS-MAA Ad Hoc Advisory Committee on a Newletter on Collegiate Mathematics Education ...... 4.8.5 Jack D. Cowan 1991 AMS-MAA Arrangements Committee for the Boulder Michael C. Mackey 1989 Meeting August 7-10, 1989 ...... 4.8. 7 Hans G. Othmer 1988 AMS-MAA Arrangements Committee for the Phoenix Richard E. Plant 1988 Chairman Meeting January 11-14,1989 ...... 4.8.6 John M. Rinzel 1989 AMS-MAA Committee on Employment and Educational 4.8.1 0 4.8.22. AMS-SIAM Committee to Screen Applicants for Policy ...... Graduate Study from the People's Republic of AMS-MAA Joint Meetings Committee ...... 4.8.9 AMS-MAA Joint Program Committee for the AMS Centennial 4.8.8 China AMS-MAA-SIAM Joint Administrative Committee ...... 4.8.15 David Benney AMS-MAA-SIAM Joint Committee on Employment Robert Bryant Opportunities ...... 4.8.14 AMS-MAA-SIAM Joint Policy Board for Mathematics. . . . 4.8.16 AMS-SIAM Committee on Applied Mathematics ...... 4.8.20 s. Representatives------AMS-SIAM Committee to Screen Applicants for Graduate Study from the People's Republic of China ...... 4.8.22 5.o.J. Advisory Board of the National Translations AMS-SIAM-SMB Committee on Mathematics in the Life Center of the John Crerar Library Sciences ...... 4.8.21 Ralph P. Boas Abstracts Editorial Committee ...... 4.1.1 Academic Freedom, Tenure, and Employment Security . . . 4.5.1 5.0.2. Commission on Professionals in Science and Administrative Committee ...... 4.8.15 Technology Advisory Board of the National Translations Center of the Edward A. Connors John Crerar Library ...... 5.0.1 Agenda and Budget ...... 4.2.1 5.o.3. Committee on the American Mathematics Agenda for Business Meetings ...... 4.4.6 Competition American Association for the Advancement of Science . . . 5.0.6 American Journal of Mathematics, Society's Representatives 4.1.2 Term expires on June 30 American Mathematics Competition, Committee on . . . . . 5.0.3 Guido L. Weiss 1990 Appeals Committee on Discounted Subscriptions ...... 4.2.2 Applied Mathematics ...... 4.5.8

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1075 Officers and Committee Members

Applied Mathematics, AMS-SIAM Committee on 0 408020 JPBM Committee for Mathematics Department Heads 4o8ol9 Arrangements Committee for the Boulder Meeting 0 4o8o7 JPBM Joint Coordinating Committee on Public Arrangements Committee for the Phoenix Meeting 0 40806 Understanding of Mathematics 0 0 0 0 0 0 0 0 0 0 0 0 0 4o8ol8 Audit 0 0 0 0 0 0 0 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40203 Japanese, Translation from 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 401.!6 Automatic Theorem Proving, Committee to Recommend Joint Administrative Committee 0 0 0 0 0 0 0 0 0 0 4o8ol5 Winners of Prizes for 0 0 0 0 0 0 0 0 0 0 0 0 0 0· 0 0 0 0 0 40603 Joint Committees 0 0 0 0 0 0 0 0 0 0 0 0 o 0 0 0 0 0 4o8 Board of Trustees 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 Joint Meetings Committee 0 0 0 0 0 0 o 4o8o9 Bulletin (New Series) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 401.3 Joint Policy Board 0 0 0 0 0 0 0 0 0 0 0 0 4o8ol6 Bocher Prize for 1989, Committee to Select the Winner of 4o6.4 Joint Policy Board for Mathematics: CAFfES o 0 0 0 0 o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40501 AMS-MAA-SIAM Joint Policy Board for Mathematics 4o8ol6 CBMS o o o o o o o o o o o o o o o o o o o o o 5o0.4 Joint Policy Board of Mathematics, Office of CEEP o o o o 0 o 0 o o 0 0 o 0 0 0 0 0 0 0 0 4o8ol0 Governmental and Public Affairs 0 0 0 0 0 0 0 o 0 0 0 0 4o8ol7 Centennial Committee 0 0 0 0 0 0 0 0 0 0 0 0 0 4o3.4 JPBM Committee for Mathematics Department Heads 4o8ol9 Centennial Fellowships 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o5o2 JPBM Joint Coordinating Committee on Public Centennial Program Committee 0 0 0 0 0 0 0 0 0 0 4o3o5 Understanding of Mathematics 0 0 0 0 0 0 0 0 0 0 0 0 0 4o8ol8 Central Sectional Meetings (Select Hour Speakers for) 4.4o2 Joint Program Committee for the AMS Centennial 0 0 0 0 0 0 4o8o8 Chinese, Cooperation with 0 0 0 0 0 0 0 0 0 4o5o9 Joint Summer Research Conferences 0 0 0 0 0 0 0 0 408.4 Chinese, Translation from 0 0 0 0 0 0 0 0 0 0 4ol.l5 Journal of the AMS 0 0 0 0 0 0 0 0 0 0 0 0 401.7 Colloquium 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4ol.4 LRP o o o o o 0 0 o 0 o o o o o o o o o o o o 4o2ol0 Comm-Comm 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 401.5 Legal Aid 0 0 0 0 0 0 0 0 0 0 0 o o 0 0 0 40208 Commission on Professionals in Science and Technology 0 0 5o0o2 Liaison Committee 0 0 0 0 0 0 0 1.1 Committee on Committees 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o3ol Liaison Committee 0 0 0 0 0 0 0 0 0 4o2o9 Committee on the American Mathematics Competition 5o0o3 Liaison Committee with AAAS 0 0 4o7o2 Committee to Monitor Problems in Communication 0 4oi.S Life Sciences 0 0 0 0 0 0 0 0 0 0 0 0 0 408021 Committees 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 Long Range Planning 0 0 0 0 0 0 0 0 4o2ol0 Committees of the Board of Trustees 0 0 0 0 0 0 0 0 0 0 402 Mathematical Reviews 0 0 0 0 0 0 0 4ol.8 Communication, Committee to Monitor Problems in 4oi.S Mathematical Surveys and Monographs 4ol.9 Computer Operations and Facilities, Visiting Committee on 402.4 Mathematics in the Life Sciences 0 0 4o8o2l Conference Board of the Mathematical Sciences 0 50004 Mathematics of Computation 401.10 Contemporary Mathematics 0 0 0 0 0 0 0 0 0 0 0 4olo6 Members-at-Large 0 0 0 0 0 0 20003 Cooperation with the Chinese 0 0 0 0 0 0 0 0 0 0 4o5o9 Membership 4o2oll Corporate Relations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40205 Memoirs 0 0 0 0 0 0 0 0 0 0 0 4ol.l4 Council 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 Monographs 0 0 0 0 0 0 0 0 0 401.9 Data Subcommittee 0 0 0 0 0 0 0 0 0 0 0 0 408011 National Awards and Public Representation 0 4o6ol Developing Countries 0 0 0 0 0 0 0 0 0 0 0 40507 National Meetings 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o4ol Discounted Subscriptions 0 0 0 0 0 0 0 0 0 40202 National Translations Center of the John Crerar Library, EC o o o o o 0 0 0 0 o o o 0 0 0 0 0 0 0 0 0 0 201 Advisory Board of the . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5o0ol Eastern Sectional Meetings (Select Hour Speakers for) 0 4o4o3 Nominating Committee 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o3o3 Editorial and Communications Committees 0 0 0 0 0 0 401 Notices 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4ol.ll Election Scheduling 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40302 Officers 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I Employment Concerns Subcommittee 0 0 0 0 0 0 0 0 0 4o8o12 Officers of the AMS 0 0 0 0 0 0 0 0 0 0 2o0ol Employment Opportunities 0 0 0 0 0 0 0 0 0 0 0 0 0 4o8ol4 PSAM o o o o o o o o o o o o o o o o o 4ol.l3 Employment and Educational Policy 0 0 0 0 0 0 0 408010 Poster Committee 0 4o3o6 Endowment 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40206 Prizes and Awards 0 0 0 4o6 Ethics 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o5.4 Proceedings 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4ol.12 Executive Committee of the Council 0 0 0 0 0 0 0 0 0 201 Proceedings of Symposia in Applied Mathematics 401.13 Far Western Sectional Meetings (Select Hour Speakers for) 0 4.4.4 Professional Ethics 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o5.4 Fellowship Policy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o5ol0 Professionals in Science and Technology, Commission on 0 0 5o0o2 Fulkerson Prize Committee 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 50005 Program Committee for National Meetings 4.401 Gibbs Lecturers for 1989 and 1990, Committee to Select 0 0 4.407 Program and Meetings 0 0 0 0 0 0 0 0 0 0 0 4.4 Graduate Study from the People's Republic of China 0 0 0 0 4o8o22 Proposed Structure of JPBM 0 0 0 0 0 0 0 0 4o3o9 History of Mathematics 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4ol.l7 Public Information Committee 0 0 0 0 0 4o3o7 Human Rights of Mathematicians 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40503 Public Understanding of Mathematics 0 4o8ol8 Index o o o o 0 0 0 o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 Publication Program 0 0 0 0 0 0 0 0 0 0 0 4o2o12 Institutes and Symposia o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 407 Recruitment of Young Mathematicians 0 4o5o5 Institutional Membership o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 402015 Representatives 0 0 0 0 0 0 0 0 0 5 Internal Organization of the American Mathematical Society 403 Representatives of Committees 2o0o2 Investment 0 0 0 0 0 0 0 0 40207 Reprinted Books 0 401.19 JCEO o o o o o o o o o o o o o o o o o o o o o0 0 0 0 0 0 0 0 0 0 0 4o8ol4 Salaries 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o2ol3 JPBM: Science Policy 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4o5o6 AMS-MAA-SIAM Joint Policy Board for Mathematics 4o8ol6 Screen Applicants from the People's Republic of China 4o8o22 Joint Policy Board of Mathematics, Office of Sectional Meetings: Governmental and Public Affairs 0 0 0 0 0 0 0 0 0 0 0 0 408017 Central 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4.402

1076 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Officers and Committee Members

Eastern ... 4.4.3 Tellers, 1987 Election ...... 4.3.10 Far Western 4.4.4 The Publication Program ...... 4.2.12 Southeastern 4.4.5 Theoretical and Applied Mechanics . 5.0.7 Service to Mathematicians in Developing Countries 4.5.7 Transactions ...... 4.1.14 Short Course Subcommittee ...... 4.8.13 Transactions and Memoirs . 4.1.14 Southeastern Sectional Meetings (Select Hour Speakers for) . 4.4.5 Translations: Special Symposia 4.7.1 Chinese ...... 4.1.15 Staff Salaries ...... 4.2.13 Japanese ...... 4.1.16 Staff and Services . . . . 4.2.14 Russian and other Slavic Languages . 4.8.3 Status of the Profession 4.5 Travel Grants for the AMS Centennial . . . 4.3.8 Steele Prizes ...... 4.6.2 U.S. National Committee on Theoretical and Applied Subcommittee on Russian Mathematical History 4.1.18 Mechanics ...... 5.0.7 Subscriptions ...... 4.2.2 Underrepresented Minorities ...... 4.8.1 Summer Institutes and Special Symposia . 4.7.1 University Lecture Series ...... 4.1.20 Summer Research Conferences ...... 4.8.4 Women in the Mathematical Sciences 4.8.2

reviews in GLOBAL ANALYSIS Introduction by 1980-86 Anthony J. Tromba

The term "global analysis" refers to the These five volumes contain the more 1980 Mathematics Subject Classification: 58 general area of analysis on manifolds, in than 18,000 reviews that appeared ISBN 0-8218-0104-X, LC 88-10565 which the methods of modern algebra, in Mathematical Reviews from 1980 5 volumes, 4060 pages (softcover), August 1988 analysis, geometry, and topology are through 1986 and have a primary or List price $295, Institutional member $236, blended. Although the beginnings of a secondary classification in Global Individual member $177, Reviewer $148 these ideas can be traced to the 17th Analysis (classification number 58). To order, please specify REVGL0/86PNA century, major contributions in this Relevant cross-references are provided direction were made by Lie, Riemann, with each review. The fifth volume and Poincare toward the end of the of this set contains ~uthor and key Combination offer. Reviews in Partial last century, followed by the work indexes, making it very easy to locate Differential Equations, 1980-86, and of G. D. Birkhoff, E. Cartan, and items written by a specific author or Reviews in Global Analysis, 1980-86 Morse in the early part of this century. to get information about collections or (REVPDE/86 and REVGL0/86). However, it is only in recent years that conference proceedings dealing with List $472, Institutional member $378, the subject has attained its present global analysis. Individual member $283, Reviewer $236. central position in mathematics. The To order, Please specify subject has many rich appiications to Contents: REVPGL0/86PNA fields outside mathematics--such as . Volume 1: Global analysis, analysis on mechanics, quantum physics, and general manifolds; General theory of differentiable relativity-as well as within mathematics manifolds; Infinite-dimensional manifolds; itself. Calculus on manifolds; nonlinear operators; Shipping/Handling: 1st book $2. each Spaces and manifolds of mappings; add"l $1. $25 max. By air, 1st book $5, Volume 2: Variational problems in infinite­ each add'l $3, $100 max. Today, this vital and active field is Prepayment required. Order from AMS. undergoing a virtual explosion of new dimensional spaces; Ordinary differential P.O. Box 1571, Annex Station, Providence. and important results. Reviews in equations on manifolds; dynamical systems; Rl 02901-9930, or call 800-556-7774 to use Global Analysis makes information about Volume 3: Ordinary differential equations VISA or MasterCard. the most recent contributions to this on manifolds; dynamical systems; Volume 4: rapidly growing field accessible both to Partial differential equations on manifolds; specialists working in global analysis, differential operators; Pseudogroups and and to those in other areas of pure and general structures on manifolds; Volume 5: applied mathematics. Series contents; Author index; Key index.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1077 Miscellaneous

Personal Items has been appointed Professor of Raymond Sidney, a freshman Mathematics at the University of at the California Institute of Tech­ Sheeram S. Abhyankar, Marshall Vermont. He will become Chair­ nology, recently received a Morgan Distinguished Professor of Math­ man of that Department in Jan­ Ward prize. ematics at Purdue University, has uary 1989. Weiping Yin, Professor of Math­ been appointed to the additional Jonathan L. King, of the Uni­ ematics at University of Science positions of Professor of Indus­ versity of Maryland, will be spend­ and Technology of China, will be trial Engineering and Professor of ing the 1988-1989 academic year visiting the Mathematical Sciences Computer Sciences at that institu­ at the Mathematics Department Research Institute, Berkeley, Cali­ tion. at the University of California, fornia, from January 16, 1989, to Laura Anderson, a senior math­ Berkeley, on a National Science April 15, 1989. His field of re­ ematics major at the California Foundation Postdoctoral Research search interest is several complex Institute of Technology, recently Fellowship, starting August 20, received the E. T. Bell Undergrad­ variables. Deaths 1988. uate Mathematics Research Prize. Donald Knuth received an hon­ Orville G. Harrold Jr., of Chico, Eric Babson, a senior math­ orary Doctor of Science degree California, died on May 16, 1988, ematics major at the California from Oxford University on June at the age of 78. He was a member Institute of Technology, recently 22, 1988. of the Society for 55 years. received the E. T. Bell Undergrad­ William I. Layton, of Newberry Maurice Horowitz, of Mobile, uate Mathematics Research Prize. College, was named Professor of Alabama, died on June 6, 1988, at Charles P. Boyer, formerly of the Year at that Awards Day Con­ the age of 67. He was a member Clarkson University, has been ap­ vocation on April 14, 1988. of the Society for 30 years. pointed Professor of Mathematics Lori Nelson, of Rockford Col­ Houston Lewis, of Richardson, and Statistics at the University of lege, is the recipient of that insti­ Texas, died in May 1988, at the New Mexico. tution's 1988 Martha Peirce Mc­ age of 4 7. He was a member of Verena H. Dyson has retired Gavock Prize in Mathematics and the Society for 3 years. fr~m of Ca~gary the University the 1987-1988 Bessie Irving Miller Shu-Tien Li, Chairman and wtth Emeritus status and is spend­ Award. President of the Li Institute of ing the combinatorial group theory Ganapati P. Patil, of Pennsyl­ Science and Technology, died on year at the Mathematical Sciences vania State University, is the re­ March 28, 1988, at the age of 88. Research Institute, Berkeley, Cali­ cipient of a Special Distinguished He was a member of the Society fornia. Statistical Ecologist Award from for 28 years. Martin C. Golumbic, of the the International Association for R. C. Lyndon, of the University IBM Israel Scientific Center and Ecology. He has also received the of Michigan, died on June 8, 1988, Associate Professor at Bar-Ilan most significant paper award from at the age of 70. He was a member University, has been appointed the American Fisheries Society for of the Society for 45 years. founding Editor-in-Chief of the his paper on "Risk Analysis in the Evelyn Nelson, of McMaster Annals of Mathematics and Ar­ Georges Bank Haddock Fishery - University, died on August 1, 1987, tificial Intelligence. He has also A Pragmatic Example of Dealing at the age of 43. She was a member been elected to the board of the with Uncertainty." of the Society for 21 years. Israel Association of Artificial In­ Bruno Remillard, co-winner of Edward S. Quade, of Laguna telligence. the Pierre Robillard Award, has Hills, California, died on June 4, Kenneth I. Gross, formerly of been appointed Professor of Math­ 1988, at the age of 79. He was the University of Wyoming and ematics at the Universite du Quebec, a member of the Society for 53 the National Science Foundation, Montreal, Quebec. years.

1078 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Miscellaneous

Bruce L. Reinhart, of the Uni­ Hiroshima University, Japan, died He was a member of the Society versity of Maryland, died on July on June 7, 1988, at the age of 66. for 10 years. 19, 1988, at the age of 57. He was He was a member of the Society a member of the Society for 35 for 30 years. Erratum years. Ralph M. Warten, Professor In the MayfJune 1988 issue of Verne Schwab, of Johns Hop­ Emeritus of California Polytech­ Notices, page 747, Nils A. Baas kins University, died on April 18, nic State University, died on July (Norway) was incorrectly listed as 1988, at the age of 65. He was 6, 1988, at the age of 62. He was a visiting mathematician at the a member of the Society for 25 a member of the Society for 30 Massachusetts Institute of Tech­ years. years. nology for 1988-1989. Shigeaki Togo, Professor at the Frederick J. Ziegler, of Albu­ Hiroshima Institute of Technol­ querque, New Mexico, died on ogy and Professor Emeritus of the March 15, 1988, at the age of 34.

NINETEEN PAPERS ~~~:~,-. REVIEWS IN NUMERICAL ON ALGEBRAIC SEMIGROUPS (. ~mt~~,·.~ ANALYSIS 1980-86 ~~DEo ,'$ These five volumes bring together a wealth of bibliographic information in the A. Ya. Aizenshtat, A. E. Evseev, N. E. Podran, area of numerical analysis. Containing I. S. Ponizovskii, B. M. Shain (Boris M. Schein), over 17,600 reviews of articles, books, E. G. Shutov, and Yu. M. Vazhenin and conference proceedings, these {American Mathematical Society Translations, Series 2, Volume 139) volumes represent all the numerical analysis entries that appeared in This volume contains papers selected by leading specialists Mathematical Reviews between 1980 in algebraic semigroups in the U.S., the United Kingdom, and 1986. Classified according to the and Australia. Many of the papers strongly influenced the 1980 Mathematics Subject Classification development of algebraic semigroups, but most were virtually scheme, the reviews are listed in each unavailable outside the U.S.S.R. Written by some of the most subsection according to their MR prominent Soviet researchers in the field, the papers have a classification number. Accompanying particular emphasis on semigroups of transformations. Boris each entry is a list of all subsequent Schein of the University of Arkansas is the translator. reviews that cite the entry as a reference. In addition, author and keyword indexes 1980 Mathematics Subject Classifications: 20; 94, 04, 05, 06, 08 and others appear at the end of volume 5. ISBN 0-8218-3115-1, LC 88-10352 1980 Mathematics Subject Classifications: ISSN 0065-9290 65, 76, 41, 35 224 pages (hardcover), June 1988 ISBN 0-8218-0102-3, LC 87-25478 Individual member $41, List price $69, 3750 pages (softcover), January 1988 Institutional member $55 List price $250, Institutional member $200, To order, please specify TRANS2/139NA Individual member $150, Reviewer $125 To order, please specify REVNAN/86NA Shipping/Handling: lot book S2, each add'l Sl, S25 max. By air, lot book $5, each add'l $3, $100 max. Prepayment required. Order from AMS, P.O. Box 1571, Annex Station, SHIPPING AND HANDLING CHARGES WILL BE ADDED. Providence, RI 02901-9930, or call 800-556-7774 to uee VISA USE ORDER FORM IN 1HE BACK OF TillS ISSUE or MasterCard. OR CALL 800-556-7774 TO USE VISA OR MASTER CARD.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1079 Visiting Mathematicians

(Supplementary List)

This list of visiting mathematicians includes only foreign mathematicians visiting in the United States and Canada.

Visiting Foreign Mathematicians Name and Home Countn: Host Institution Field of SQecial Interest Period of Visit Alvarez Lopez, Jesus A. University of Illinois at Differential Geometry 8/88- 5/89 (Spain) Urbana-Champaign Arai, Toshiyasu (Japan) University of Illinois at Logic 8/88- 5/89 Urbana-Champaign Arnoud, Pierre (France) University of Florida Dynamical Systems 8/88- 12/88 Bather, John (United Harvard University Mathematical Statistics 6/88- 10/88 Kingdom) Bhattacharya, Tilak (India) Northwestern University Partial Differential Equations, 9/88- 8/90 Applied Mathematics Billington, Elizabeth Auburn University Combinatorics 7/88- 12/88 (Australia) Cioranescu, Ioana (West Auburn University Analysis 9/88- 6/89 Germany) Erdos, Paul (Hungary) University of Florida Number Theory, Analysis and 2/89- 3/89 Combinatorics Garbey, Marc (France) Argonne National Laboratory Bifurcation Phenomena and 9/88- 8/89 Combustion Guan, Yude (People's University of Illinois at Applied Mathematics 8/88- 5/89 Republic of China) Urbana-Champaign Herzog, Joachim (West University of Illinois at Number Theory 8/88- 5/89 Germany) Urbana-Champaign Herzog, Marcel (Israel) University of Hawaii Group Theory 8/88- 1/89 Jarden, Moshe (Israel) University of Florida Algebra and Logic 8/88- 9/88 Krajicek, Jan University of Illinois at Logic 8/88- 5/89 (Czechoslovakia) Urbana-Champaign Manfredi, Juan Jose (Spain) Northwestern University Partial Differential Equations, 9/88- 8/89 Complex Analysis Manocha, H. L. (India) Auburn University Analysis 9/88- 6/89 Monajemi, Assad (Iran) Harvard University Applied Statistics 6/88- 10/88 Petridis, Nickolas (Greece) University of Illinois at Geometry 8/88- 12/89 Urbana-Champaign Renault, Jean (France) University of Colorado, Boulder C* -algebras, Foliations 9/88- 12/88 Saigo, Megumi (Japan) University of Victoria Special Functions, Fractional 3/88- 3/89 Calculus Schulz, Rainer (West Auburn University Algebra 9/87- 6/89 Germany) Serre, Jean-Pierre (France) Harvard University 9/88- 12/88 Sersouri, Abderrazzak University of Illinois at Analysis 8/88- 5/89 (Morocco) Urbana-Champaign Song, Hyung-Soo (Korea) University of Minnesota Topological Dynamics 8/88- 8/89 Soule, Christoph (France) Harvard University Algebraic K-theory 2/89- 5/89

1080 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Visiting Mathematicians

Szabados, Jossef (Hungary) University of Florida Approximation Theory 8/88- 12/88 Urbas, John I. (Australia) Northwestern University Partial Differential Equations 1/89- 6/89 Villari, Gabriele (Italy) University of Illinois at Ordinary Differential Equations 8/88- 12/89 Urbana-Champaign Wiegmann, Klaus (West University of Rochester Several Complex Variables 9/88- 12/88 Germany) Yang, Seung Kab (South University of Florida Analysis 8/88- 5/89 Korea) Yin, Weiping (People's University of Notre Dame; Several Complex Variables 10/88 - 1/89 Republic of China) Mathematical Sciences 1/89- 4/89 Research Institute Zin, Yuan-xun (People's University of Florida Differential Equations and 8/88- 12/88 Republic of China) Computational Physics

simple equational formalism, Closely logic; semantics; first-order logics with A FORMALIZATION OF related to the formalism of the theory of finitely many variables; definability and relation algebras. There are no variables, axiomatizability questions in set theory, SET THEORY WITHOUT quantifiers, or sentential connectives. Peano arithmetic, and real number VARIABLES Predicates are constructed from two theory; representation and decision atomic binary predicates (which denote problems in the theory of relation Alfred Tarski and Steven Givant the relations of identity and set-theoretic algebras; and decision problems in membership) by repeated applications equational logic. Completed in 1983, this work culminates of four operators that are analogues of nearly half a century of the late Alfred the well-known operations of relative Tarski 's foundational studies in logic, 1980 Mathematics Subject Classification: 03 product, conversion, Boolean addition, ISBN 0-8218-1041-3, LC 86-22168 mathematics, and the philosophy of and complementation. All mathematical ISSN 0065-9258 science. Written in collaboration with statements are expressed as equations 280 pages (hardcover), January 1988 Steven Givant, the book appeals to between predicates. There are ten logical List price $60, Institutional member $48, a very broad audience, and requires Individual member $36 axiom schemata and just one rule of To order, please specify COLL/41P only a familiarity with first-order inference: the one of replacing equals by logic. It is of great interest to logicians · equals, familiar from high school algebra. and mathematicians interested in the foundations of mathematics, but also Though such a simple formalism to philosophers interested in logic, may appear limited in its powers of semantics, algebraic logic, or the expression and proof, this book proves methodology of the deductive sciences, quite the opposite. The authors show and to computer· scientists interested that it provides a framework for the in developing very simple computer formalization of practically all known Shipping/Handling: 1st book $2, each languages rich enough for mathematical systems of set theory, and hence for the add'l $1, $25 max. By air, 1st book $5, and scientific applications. development of all classical mathematics. each add'l $3, $100 max. Prepayment required. Order from AMS. The authors show that set theory and The book contains numerous applications P.O. Box 1571, Annex Station, Providence. number theory can be developed within of the main results to diverse areas of Rl 02901-9930. or call 800-556-7774 to use the framework of a new, different, and foundational research: propositional VISA or MasterCard.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1081 Backlog of Mathematics Research Journals

Backlog. Information on the backlog of papers for revision and its publication may be much shorter than research journals, primarily those published in North is the case otherwise, so these figures are low to that America, is reported to the Providence Office by those extent. editorial boards which elect to participate. The fig­ The observations are made from the latest issue ures are an estimate of the number of printed pages published, before the deadline for this issue of Notices, which have been accepted, but are in excess of the from journals that have actually been received by number required to maintain copy editing and printing a subscriber in the Providence, Rhode Island, area; schedules. in some cases this may be two months later than Observed Waiting Time. The quartiles give a measure publication abroad. If the waiting time as defined above of normal dispersion. They do not include extremes is not given in the journal, if no new issue has been which may be misleading. Waiting times are measured received since the last survey, or if the latest issue is in months from receipt of manuscript in final form to for some reason obviously not typical, no times are publication of the issue. When a paper is revised, the given in this report and such cases are marked NA (not waiting time between an editor's receipt of the final available or not applicable).

Editor's Estimated Observed Waiting Approximate Time for Paper Time in Latest Number Number Backlog of Submitted Currently Published Issue Issues Pages Printed Pages to be Published (in Months) Journal per Year per Year 12/15/87 6/30/88 (in Months) 01 M 03

Acta Inform. 6 720 0 0 5 5 6 7 Aequationes Math. 6 640 0 0 8 10 13 25 Alg. Groups Geom. 4 600 0 0 8 6 8 14 Algorithmica 4 512 128 NR 3 6 6 7 Amer. J. Math. 6 1476 800 900 7 14 24 29 Ann. of Math. 6 1260 500 1000 9 18 19 21 Ann. Probab. 4 1600 0 250 12 13 16 18 Ann. Sci. Ecole Norm. Sup. 4 650 0 0 15 7 8 9 Ann. Statist. 4 1750 300 140 7 11 11 13 Appl. Math. Optim. 6 576 192 NR 4 12 13* 19 Arch. Hist. Exact Scis. 8 800 0 0 8-9 10 11 12 Arch. Math. Logic 2 200 0 NA NA Arch. Rational Mech. Anal. 16 1600 0 0 11-12 11 11 11 Bull. Austral. Math. Soc. 6 1000 200 0 9 11 12 13 Canad. J. Math. 6 1530 400 NR 14 18 20 34 Canad. Math. Bull. 4 NR NR 256 NR 14 19 24 Circuits Systems Signal Proc. 4 512 0 0 0 NA Comm. Algebra 12 2700 2186 2250 14 8 18 18 Comm. Math. Phys. 24 4224 0 0 5 5 7 8 Comm. Partial Diff. Equations 12 NR NR 300 NR 6 8 11 Computing 8 768 NR 7 5 7 9 Constr. Approx. 4 448 112 NR 3 12 13 13 Discrete Comput. Geom. 6 600 370 450 16 19 21 26 Duke Math. J. 6 1500 400 500 9 9 14 15

1082 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Research Journals Backlog

Editor's Estimated Observed Waiting Approximate Time for Paper Time in Latest Number Number Backlog of Submitted Currently Published Issue Issues Pages Printed Pages to be Published (in Months) Journal per Year per Year 12/15/87 6/30/88 (in Months) Ql M Q3

Houston J. Math. 4 600 600 700 24 28 34** 36 Illinois J. Math. 4 704 835 889 21 20 22 24 IMA J. Appl. Math. 6 500 100 88 9 10 12 14 IMA J. Math. Appl. Med. Bioi. 4 350 80 80 9 NA IMA J. Math. Control Inform. 4 350-400 70 NR 3-4 NA IMA J. Numer. Anal. 4 600 0 100 5 7 9 11 Indiana Univ. Math. J. 4 NR NR 100 NR 19 20 21 lnst. Hautes Etudes Sci. Publ. Math. 2 NR NR NR NR 14 21 27 lnternat. J. Math. Math. Sci. 4 832 232 0 6-8 9 14 17 Invent. Math. 12 2688 0 0 7 6 7 8 Israel J. Math. 12 1500 100 NR 6 6 8 10 J. Algorithms 4 NR NR NR NR 17 22 30 J. Amer. Math. Soc. 4 1000 0 NA 9 NA J. Appl. Math. Simulation 4 370 100 0 4 NA J. Assoc. Comput. Mach. 4 1000 100 215 18 10 11 12 J. Austral. Math. Soc. Ser. A 6 844 800 800 24 17 19 21 J. Austral. Math. Soc. Ser. B 4 512 0 43 15 9 11 12 J. Classification 2 300 0 0 0 NA J. Comput. System Sci. 6 1000 300 0 18 10 14 21 J. Differential Geom. 4 1100 1000 900 10 15 16 19 J. Integral Equations Appl. 4 600 0 NA 8 NA J. Math. Bioi. 6 600 100 NR 8 6 8 19 J. Math. Phys. 12 3200 0 0 5 16 16* 16 J. Nigerian Math. Soc. NR NR 0 NR NA J. Operator Theory 4 800 200 NR 12 10 12 15 J. Symbolic Logic 4 1216 500 450 18 16 16 18 Linear Algebra Appl. 14 NR NR 300 NR 11 12 15 Manuscripta Math. 12 1536 0 0 5 4 6 6 Math. Ann. 14 2464 0 0 8-10 10 13 14 Math. Comp. 4 1500 0 0 12 10 11 15 Math. Oper. Res. 4 NR NR NR NR 17 18 19 Math. Programming Ser. A 6 720 190 250 15 16 20 24 Math. Social Sci. 6 600 180 450 11 13 14 14 Math. Systems Theory 4 320 0 0 0 5 7 8 Math. z. 12 1824 0 0 11-12 9 10 12 Mem. Amer. Math. Soc. 6 2000 0 0 1-2 14 16 26 Michigan Math. J. 3 480 30 0 12-16 11 13 13 Monatsh. Math. 8 704 0 NR 10 8 12 13 Numer. Funct. Anal. Optim. 6 700 0 0 5 5 6 6 Numer. Math. 12 1488 0 0 5-7 7 7 8 Oper. Res. 6 1008 575 NR 9 12 13 14 Pacific J. Math. 10 2000 NR NR 11 10 11 15 Probab. Theor. Relat. Fields 12 1920 0 0 12 7 7 8 Proc. Amer. Math. Soc. 12 3850 100 600 13 16 17 18 Proc. London Math. Soc. 6 NR NR 150 NR 10 16 18 Quart. Appl. Math. 4 800 600 600 17 13 15 19 Quart. J. Math. Oxford Ser. (2) 4 512 128 192 12-18 15 17 20 Results Math. 8 800 100 NR 9 9 10 14 Rocky Mountain J. Math.** 4 1000 1845 2000 24 24 25** 29 Semigroup Forum 3 384 0 0 0 5 6 7

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1083 Research Journals Backlog

Editor's Estimated Observed Waiting Approximate Time for Paper Time in Latest Number Number Backlog of Submitted Currently Published Issue Issues Pages Printed Pages to be Published (in Months) Journal per Year per Year 12/15/87 6/30/88 (in Months) Ct M Q3

SIAM J. Appl. Math. 6 . 1480 654 613 20 10 11* 12 SIAM J. Comput. 6 1290 40 190 25 12 13* 14 SIAM J. Control Optim. 6 1500 232 204 26 10 12* 13 SIAM J. Discrete Math. 4 600 0 0 18 6 7* 12 SIAM J. Math. Anal. 6 1500 142 177 27 12 13* 13 SIAM J. Matrix Anal. Appl. 4 600 0 0 16 9 9* 10 SIAM J. Numer. Anal. 6 1450 525 363 28 13 13* 13 SIAM J. Sci. Statist. Comput. 6 1200 86 0 24 8 10* 11 SIAM Rev. 4 680 84 123 30 11 12* 20 Topology Appl. 9 990 825 NR 10 15 16 17 Trans. Amer. Math. Soc. 12 5000 200 100 12 15 16 16

NR means no response received. NA means not available or not applicable. * From date accepted. ** This journal is ending a moratorium on the acceptance of new papers on September 30, 1988. Waiting times for papers submitted after that date should be much shorter. *** Dates of revision not indicated in this journal.

FACTORIZATIONS OF b" ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 UP TO HIGH POWERS, SECOND EDITION John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr. (Contemporary Mathematics, Volume 22, Second Edition) This book is a revised and updated edition of a work that occurred since 1925. An update to the introduction is included originally appeared in 1983. It gives a historical account of the in this edition and discusses the major advances that have various methods and machines that have been used to factor, been made in the five years since the first edition appeared. and prove prime, the numbers bn ± 1. It is a revised version The introduction also discusses the multiplicative structure of an extension of a rare 1925 work by Cunningham and of bn ± 1 and explains the relation between the two kinds of brings together results going back to the seventeenth century. algebraic factorizations of these numbers. The factorizations and the very large primes of special form 1980 M h · s b' Cl 'fi f 11 ~EMil~ are useful in group theory, number theory, discrete Fourier ISBN o-:2~;~~~~_:, (~~3-f~;;ga Ton: s~< transforms, random number generators, and cryptography. ISSN 0271-4132 J/(~~~\ \i The present edition contains more than 2000 large primes ~:~~:~=~ ~~0.::~-~~~~~s~ ~~~: $31 , (~~ lllWlJl:: ~~ which have never been published before. Institutional member $25 £ ;, • The book contains complete factorizations of bn ± 1 for the To order, please specify CONM/22NA • ~Diiififf\~ given values of b and for all n ~ 100, and for many n > 100. Shipping/Handling: 1st book $2, each add'l $1, $25 Included is an extensive and valuable introduction which max. By air, 1st book $5, each add'l $3, $100 max. Prepayment required. Orde.r from AMS, P.O. Box 1571, describes the developments in computing technology and Annex Station, Providence, RI 02901-9930, or call in methods of factoring and primality testing which have 800-556-7774 to use VISA or MasterCard.

1084 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New Members of the AMS

ORDINARY MEMBERS Robert S Cantrell Robert E Jamison II University of Miami Clemson University Al-Khayyal Faiz A Coral Gables, Florida Clemson, South Carolina Georgia Institute of Technology Atlanta, Georgia Toni Carroll Richard Johnson Siena Heights College Los Alamitos, California Allegra Maria Adrian, Michigan Marcel Dekker Incorporated Rahim G Karimpour New York, New York Marte M Carter Southern Illinois University Warrensburg, Missouri Edwardsville, Illinois Mariarosa Allodi Utica College Denver R Childress In Cha Kim Utica, New York New Market, Tennessee East Lansing, Michigan James V Balch Deborah Cohoon Aleksandr Koldobskii Murfreesboro, Tennessee Glen Mills, Pennsylvania Leningrad, U S S R Helen G Bass Deborah Ann Crocker Gerald A Kraus North Haven, Connecticut Banner Elk, North Carolina Gannon University Erie, Pennsylvania Carol Hoffer Bassett Dragan S Cvetkovic Waterville, Maine Belgrade, Yugoslavia Andrea W Lawrence Spelman College Dabagyan Jane F Batten Mihran Atlanta, Georgia DeKalb College Baghdad, Iraq Clarkston, Georgia James D Lewis James Dokas Jr University of Saskatchewan Maryland Kimberly Lane Berger Marriottsville, Saskatoon, Saskatchewan Canada Washington Everett, W Earl Sr Archie Amha T Lisan Virginia Birne T Binegar Suffolk, Silver Spring, Maryland Sciences Research Mathematical Walter Elias Jr Nicholas G Loudin Institute Ettrick, Virginia Berkeley, California Philippi, West Virginia Tamas Erdelyi Geoffrey Martin Curtis John Blanton University of South Carolina Georgia Danielson, Connecticut Stone Mountain, Columbia, South Carolina Elaine D Bohanon David K Masunaga Eugene Fately lolani School Bemidji State University Hopewell Junction, New York Bemidji, Minnesota Honolulu, Hawaii Terry L Gargus Fouad A Mohamed Francis Bonahon Gadsden, Alabama University of Southern California California State University, Fresno Los Angeles, California James M Guffey Fresno, California Rolla, Missouri Ann K Boyle Laura Jean Muller National Science Foundation Xin Kang Guo Bard College Washington, District of Columbia Guangxi University Annandale-on-Hudson, New York Nanning, People's Republic of China Debbie C Calloway A 0 Oganesyan Thomas More College Jeri Rene Hawkins Erevan State University Crestview Hills, Kentucky Pensacola, Florida Erevan, U S S R

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1085 New Members of the AMS

Andrew T Ogielski Luchezar Stojanov Binggen Zhang A T & T Bell Labs Institute of Mathematics Qingdao University of Oceanology Murray Hill, New Jersey Sofia, Bulgaria Qingdao, People's Republic of China Eric Dorsey Partridge Sheila A Sullivan Mohammad S Zouyousefain New Orleans, Louisiana Chicago, Illinois Arlington, Texas John Piepenbrink Jun Tarui Nancy E Zumoff West Georgia College University of Rochester Kenn~saw College Carrolton, Georgia Rochester, New York Marietta, Georgia Arthur B·Powell Jimin Tian Rutgers University Washington State University Newark, New Jersey Pullman, Washington RECIPROCITY MEMBERS Austin F Reller Frank L Tischer Asociaci6n Matemdtica Espanola New York, New York Citrus Heights, California Jose Luis Garcia Douglas J Rosengard Dao-Rong Ton Korean Mathematical Society Acton, Massachusetts Hohai University Jeong Yeo! Choi JangSu, People's Republic of China Dong-Soo Lee David L Ross Deland, Florida Josaphat A Uvah London Mathematical Society Lafayette, Louisiana Martin P Holland Dawn Ross Columbia, Missouri Sherman B Vanaman Mathematical Society ofJapan Carson-Newman College Hironobu Maeda Maciej Sablik Jefferson City, Tennessee Shinichi Suzuki Katowice, Poland Hiromichi Yamada Thomas L Vanden Eynden Mark E Schaefer Thomas More College Societe Mathematique de France Georgia State University Crestview Hill, Kentucky Georges Blanc Atlanta, Georgia Jean-Yves Thibon Jorge X Velasco-Hemandez Phillip H Schmidt Claremont Graduate School Sociedad Matemdtica Mexicana University of Akron Claremont, California Horacio Tapia-Recillas Akron, Ohio Hans W Volkmer Suomen Matemaattinen Yhdistys Andreas Seeger University of Essen Tapani Matala-aho Mathematical Sciences Research Essen, Federal Republic Institute of Germany Unione Matematica ltaliana Berkeley, California Dimitri Vulis Claudio Lazzari Flushing, New York Jang Bahadur Shukla Wiskundig Genootschap Indian Institute of Technology Harriet R Walton Tony C Dorlas Kanpur, India Morehouse College Rene Schoof Atlanta, Georgia James R Smart San Jose State University William C Wickes NOMINEE MEMBERS San Jose, California Hewlett Packard Corvallis, Oregon Arizona State University Karl J Smith Duk-Hyung Lee American Mathematics Association Robert J Wisner Santa Rosa, California New Mexico State University Hofstra University Las Cruces, New Mexico Kimberly S Selby Kathryn Anne Smith Wesleyan College Wing-Chun Godwin Wong Union College Macon, Georgia University of Illinois, at Katharine E Nyhus Arthur David Snider Urbana-Champaign University ofHartford Urbana, University of South Florida Illinois Stephen M Pelletier Tampa, Florida Deborah E Woods University ofMinnesota, Duluth · Joseph E Steams Jr University of Cincinnati Patricia M Kuchta Anderson, South Carolina Cincinnati, Ohio Andre Stern Raymond F Woods Ithaca, New York Bradenton, Florida

1086 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

SUGGESTED USES for classified advertising are books or lecture notes for sale, books being STANFORD UNIVERSITY sought, positions available, exchange or rental of houses, and typing services. THE 1988 RATE IS $.85 per word with a minimum of $42.50. The same ad in 6 consecutive Stanford University solicits application issues is $4.00 per word. The same ad in 10 consecutive issues is $7.00 per word. Type for one or more positions in theoreti­ will be set solid unless centering and spacing are requested. A centered line of any length cal computer science. Appointments in or the equivalent in white space is $10.00 (one-time charge). A word is defined as a group of characters with space at each end. Prepayment is required of individuals but not of the Computer Science Department or, institutions. For an additional $10.00 charge, announcements can be placed anonymously. if appropriate, jointly or fully in a re­ Correspondence will be forwarded. lated department such as Mathematics, DEADLINES are listed on the inside front cover. Operations Research, or Statistics, are u. S. LAWS PROHIBIT discrimination in employment on the basis of color, age, sex, race, possible. Persons at both junior and religion or national origin. ""Positions Available"" advertisements from institutions outside the senior levels will be considered, and u. s. cannot be published unless they are accompanied by a statement that the institution an outstanding record of achievement, does not discriminate on these grounds whether or not it is subject to U.S. laws. Details and commensurate with the proposed level, specific wording may be found following the Classified Advertisements in the January and is expected. July/August issues of the Notices. Candidates are expected to have SITUATIONS WANTED ADVERTISEMENTS from involuntarily unemployed mathematicians are accepted under certain conditions for free publication. Call toll-free 800-556-7774 and a commitment to excellence in teach­ speak to Paula Montella for further information. ing and a record of competence in SEND AD AND CHECK TO: Advertising Department, AMS, P. 0. Box 6248, Providence, at least one field of computation the­ Rhode Island 02940. Individuals are requested to pay in advance, institutions are not required ory, including, but not limited to, such to do so. subjects as: Combinatorial optimization, Complexity theory, Computational ge­ ometry, Database and knowledge sys­ tems, Design and analysis of algorithms, POSITIONS AVAILABLE The University of British Columbia Discrete mathematics, Distributed algo­ Department of Mathematics CARNEGIE MELLON UNIVERSITY rithms, Logic and its applications, Mathe­ Zeev Nehari Assistant The Department is seeking an outstand­ matical programming, Parallel algorithms Professorship in Mathematics ing candidate for a tenure track As­ and architectures, Principles of program­ sistant Professorship to begin 1 July ming languages, or Security of protocols. These positions have been instituted 1989, in one of the following areas: Candidates should send a vita and in the Department of Mathematics of numerical analysis, especially applied to names of at least four references to: Carnegie Mellon University to honor the partial differential equations, combina­ Prof. Jeffrey D. Ullman, Department of memory of Professor Zeev Nehari, a toricsjcombinatorial optimization, partial Computer Science, Stanford Univ., Stan­ member of the Department from 1954 differential equations. For an exception­ ford CA 94305. Stanford is an equal to his death in 1978. Applicants are ally well-qualified candidate, this position opportunity/affirmative action employer. expected to show exceptional research may be upgraded to a junior Asso­ promise as well as clear evidence of ciate Professorship. Applicants should achievement. The position available is have a proven research record of high for two academic years, beginning in THE OHIO STATE UNIVERSITY quality and have demonstrated interest September, 1989 and extendable for one DEPARTMENT OF MATHEMATICS and ability in teaching. Preference will additional year when mutually agreeable. be given to candidates who have one The Department of Mathematics of The It carries a reduced academic year teach­ or more years of postdoctoral experi­ Ohio State University hopes to fill several ing load of six hours per week during ence. This position is subject to final positions, both visiting and permanent, one semester and three hours per week budgetary approval. The salary will be effective Autumn Quarter, 1988. Candi­ during the other. The applicant should commensurate with experience and re­ dates in all areas of applied and pure have research interests which intersect search record. Applicants should send a mathematics are invited to apply. Sig­ those of current faculty of the ·Depart-· C.V. including list of publications, state­ nificant research accomplishments or ment. Applicants should send a vita, list ment of research and teaching interests exceptional research promise, and evi­ of publications, and a statement describ­ and arrange for three letters of recom­ dence of good teaching ability, will be ing current and planned research, and mendation to be sent directly to: Dr. expected of successful applicants. arrange to have at least three letters of David Boyd, Head, Department of Math­ Please send credentials and have recommendation sent to the committee. ematics, University of British Columbia, letters of recommendation sent to Pro­ All communications should be addressed Vancouver, B.C., Canada V6T 1Y4. Ap­ fessor Joseph Farrar, Department of to: Zeev Nehari Assistant Professorship plications must be received before 1 Mathematics, The Ohio State University, Committee, Department of Mathematics, January 1989. In accordance with Cana­ 231 W. 18th Avenue, Columbus, Ohio Carnegie Mellon University, Pittsburgh, dian immigration requirements, priority 43210. Review of resumes will begin PA 15213. Carnegie Mellon University is will be given to Canadian citizens and immediately. an Affirmative Action/Equal Opportunity permanent residents of Canada. The Ohio State University is an Equal Employer. Opportunity 1Affirmative Action Employer.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1087 Classified Advertisements

POSITIONS AVAILABLE Tennessee Technological University THE UNIVERSITY OF Department of Mathematics ALABAMA AT BIRMINGHAM NEW FACULTY Cookeville, Tennessee 38505 DEPARTMENT OF MATHEMATICS in COGNITIVE AND NEURAL SYSTEMS Applications are invited for a tenure­ The Department of Mathematics has fac­ at track position in Statistics at the rank of ulty positions at all ranks. The depart­ BOSTON UNIVERSITY Assistant Professor, available 1 January ment is especially interested in establish­ 1989. Ph.D. in Statistics, or equiva­ ing a group in Numerical POE/Scientific Boston University seeks a tenure track lent, experience in both Applied and computation over the next five years. assistant professor starting in Fall, 1989, Mathematical Statistics, evidence of ex­ Access to the Alabama Super Com­ for its M.A. and Ph.D. Program in Cogni­ cellent teaching ability at all levels, and puter (using a Sun Station and a T-1 tive and Neural Systems. This program strong interest in research are required. line to a Cray X-MP/24) will be avail­ offers an integrated curriculum offering Duties include teaching undergraduate able in the near future. Other areas the full range of psychological, neuro­ and graduate courses, directing grad­ which will enhance our proposed Ph.D. biological, and computational concepts, uate students, consulting and research in Applied Mathematics will be seri­ models, and methods in the broad field activities, and helping develop Statistics ously considered. Applicants for senior variously called neural networks, con­ courses for science and engineering stu­ positions must demonstrate excellence nectionism, parallel distributed process­ dents. Position is open until filled. Send in research, while applicants for junior ing, and biological information process­ transcript and curriculum vitae, and have positions must exhibit the promise of ing, in which Boston University is a three letters of recommendation sent, as excellence. Send as soon as possible a leader. Each faculty member will have soon as possible, to: Chairman, Search curriculum vitae, list of publications, a a joint appointment in the Ph.D. pro­ Committee, Department of Mathematics, few selected reprints, and the names of gram and in one or more of the Box 5054, TIU, Cookeville, TN 38505. three references to Search Committee, departments of mathematics, biology, Department of Mathematics, University computer science, and psychology. Can­ EOE/AA. of Alabama at Birmingham, Birmingham, didates should have extensive analytic AL 35294. UAB is an Affirmative Action/ or computational research experience Equal Opportunity Employer. modelling a broad range of real-time DEPARTMENT OF MATHEMATICS, UNI­ nonlinear neural networks, especially in VERSITY OF OTIAWA. Applications are one or more of the areas: adaptive invited for THREE tenure track positions pattern recognition, speech and lan­ starting July 1, 1988 (one still subject to guage, cognitive information processing, budgetary approval). Candidates must self-organization, and conditioning and have demonstrated research ability. 1. OF MATHEMATICS attention. Send a complete curriculum DEPARTMENT Assistant or associate professor; all ar­ THE UNIVERSITY OF vitae and three letters of recommenda­ eas are invited. Ability to teach in English WESTERN ONTARIO tion to Search Committee, Cognitive and and French is required for tenure but not Neural Systems Program, Room 240, for appointment. 2. Assistant professor; Applications or nominations are invited 111 Cummington Street, Boston Univer­ preference for all areas of analysis or for the position of Chairman, Department sity, Boston, MA 02215 preferably by (P)DEs. Ability to teach in English and of Mathematics, Faculty of Science. The November 15, 1988, but no later than French required for tenure but not for effective date of the appointment is July January 1, 1989. Boston University is appointment. According to Canadian im­ 1' 1989. an Equal Opportunity1 Affirmative Action migration requirements, priority will be Applications or nominations should employer. given to Canadian citizens and perma­ be addressed to: nent residents. 3. Professeur adjoint ou Dr. W. S. Fyfe, agrege de preference en analyse, anal­ Dean, yse numerique, equations differentielles, Faculty of Science, MATHEMATICS FACULTY geometrie algebrique ou differentielle, ou Natural Sciences Centre, statistique. L'enseignement attache a ce The University of Western Ontario, Instructor1 Asst. Prof. to teach a variety poste sera en fran<;ais. Conformement London, Ontario, Canada. of college level and remedial courses. aux exigences prescrites en matiere N6A 5B7 Master's degree required. Teaching ex­ d'immigration au Canada, Ia priorite Applications or nominations should be perience preferred. Hiring range $20,000- sera accordee aux citoyens canadians et submitted by January 1, 1989. In ac­ 25,000. To be filled Nov. 18, 1988. residents permanents. For any of these cordance with Canadian immigration re­ Send resume to John Leddy, Acting Di­ posts, send a resume and have three quirements, priority will be given to rector of Personnel/ Affirmative Action, letters of reference sent before Decem­ Canadian citizens and permanent resi­ State University of New York College of ber 16 to: W. D. Burgess, Dept. Math., dents of Canada. Technology, Delhi, N.Y. 13753 EEO/AA University of Ottawa, Ottawa, CANADA An Equal Opportunity Employer. Employer K1N 6N5

1088 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

POSITIONS AVAILABLE The Department of Mathematics is ac­ The Department of Mathematics is ac­ tively seeking applications in the area tively seeking applications in the area CANISIUS COLLEGE of computational mathematics and nu­ of discrete mathematics and combina­ MATHEMATICS DEPARTMENT OF merical analysis. We anticipate making torics. We anticipate making several at the assis­ A tenure track position (Assistant Pro­ several tenure-track appointments at the tenure-track appointments above beginning fessor) in mathematics is available in late assistant professor level or above be­ tant professor level or 1989. A Ph.D. is required. August, 1989. Applicants must have a ginning in the fall of 1989. A Ph.D. is in the fall of until March Ph.D. in mathematics and a strong com­ required. Applications will be reviewed Applications will be accepted candi­ mitment to quality teaching. The teach­ as they are received and will be accepted 15, 1989, or until a successful letter of ap­ ing load is twelve hours per semester. until the positions are filled. A formal let­ date is found. A formal a resume, Salary and fringe benefits are competi­ ter of application expressing interest, a plication expressing interest, and telephone tive commensurate with credentials and resume, and the names, addresses, and and names, addresses, should experience. telephone numbers of three references numbers of three references Math­ Applicants should send resume, tran­ should be sent to Chairman, Numerical be sent to Chairman, Discrete Department scripts and three letters of recommen­ Analysis Search Committee, Department ematics Search Committee, Tech, Blacks­ dation to Dr. Richard H. Escobales, of Mathematics, Virginia Tech, Blacks­ of Mathematics, Virginia Virginia Tech is Chairman, Department of Mathematics, burg, VA 24061-0123. Virginia Tech is burg, VA 24061-0123. Action Canisius College, Buffalo, New York, an Equal Opportunity I Affirmative Action an Equal Opportunity1 Affirmative 14208. AAIEOE. Employer. Employer.

,______,/ v MATHEMATICS CONSORTIUM THE ROCKY MOUNTAIN announces publication of THE JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS

The Journal of Integral Equations and Applications is a continuation of the "Journal of Integral Equations", and will be devoted to the theory, applications and numerical analysis of integral equations of all types. The primary aim of this international journal is to publish high-quality research papers in the area of integral equations and their applications, and will be particularly devoted to: The deterministic and probabilistic theory of linear and nonlinear integral equations of various types, integrodifferential equations and related operator equations. Numerical analysis and approximation methods for integral equations. Applications of integral equation~ in the sciences, engineering, and technology.

lhe journal will also publish occasional survey and expository articles presenting, in some depth, recent advances with respect to particular topics. It will serve as a forum for an exchange of i.deas that will stimulate significant contributions in new fields and promote the most salient aspects of the theory of integral equations. The scope and methodologies will embrace classical and complex analysis methods, functional analysis techniques and topological/geometric methods for development of the theory of integral equations. The editors of the Journal of Integral Equations and Applications will be P. M. Anselone, Oregon State University; and M. Z. Nashed, University of Delaware. The institutional subscription price will be $150.00 (rates for individual subscriptions available upon request). Inquiries and orders should be directed to the Rocky Mountain Mathematics Consortium, Dept. of Mathematics, Arizona State University, Tempe, AZ 85287-1904 (phone 602-965·3788). ~ / /

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1089 Classified Advertisements

POSITIONS AVAILABLE STANFORD UNIVERSITY THE UNIVERSITY OF FLORIDA Department of Mathematics NEW SENIOR FACULTY Stanford University solicits applications in for a tenured senior faculty person to The Department of Mathematics is in the COGNITIVE AND NEURAL SYSTEMS lead a proposed new Stanford Institute third year of a five-year program to fill at of Computation Theory. The Institute over 20 new tenure-track faculty posi­ BOSTON UNIVERSITY will initially be a part of the Computer tions with mathematicians of exceptional Boston University seeks a full professor Science Department in the School of caliber. The Department invites appli­ or associate professor starting in Fall, Engineering and will have the participa­ cations for six tenure-track positions 1989, to act as Co-Director for its M.A. tion of the Departments of Mathematics, for the fall semester, 1989. Applications and Ph.D. Program in Cognitive and Neu­ Operations Research, and Statistics. An from junior candidates are especially ral Systems. The Co-Director will play appointment in the Computer Science welcome. a major role in curriculum development, Department or, if appropriate, jointly or Outstanding candidates in all areas administration, and training of mathe­ fully in a related department such as of applied and pure mathematics are in­ matically advanced graduate students. Mathematics, Operations Research, or vited to apply for these positions. Senior The program curriculum offers the full Statistics, is possible. candidates should have distinguished range of psychological, neurobiological, Candidates are expected to have a research records, and junior candidates and computational concepts, models, commitment to excellence in teaching are expected to have made significant and methods in the broad field vari­ and an outstanding record of research research contributions. Every candidate ously called neural networks, connec­ in at least one field of computation the­ is expected to possess a strong com­ tionism, parallel distributed processing, ory, including, but not limited to, such mitment to teaching. and biological information processing, subjects as: Combinatorial optimization, One position will be reserved for in which Boston University is a leader. Complexity theory, Computational ge­ a senior candidate in Partial Differen­ The faculty member will have a joint ometry, Database and knowledge sys­ tial Equations. Strong preference will appointment in the Ph.D. program and tems, Design and analysis of algorithms, be given to Arithmetic Geometers and in one or more of the departments of Discrete mathematics, Distributed al­ Number Theorists in filling a second mathematics, biology, computer science, gorithms, Logic and its applications, position. Among other areas of interest and psychology. Candidates should have Mathematical programming, Parallel al­ to the Department are Dynamical Sys­ extensive analytic and computational re­ gorithms and architectures, Principles tems, Algebraic Geometry and Harmonic search experience and an international of Programming languages, or Security Analysis. reputation for modelling a broad range of of protocols. In addition, demonstrated Candidates should forward a resume real-time nonlinear neural networks, en­ leadership abilities, administrative skills, (including a list of publications) and abling them to teach graduate courses and interpersonal skills are required for should arrange for at least three letters including the areas of adaptive pat­ this position. of recommendation to be sent to: tern recognition and self-organization. Candidates should send a vita and David A. Drake, Chair Send a complete curriculum vitae and names of at least four references to: Department of Mathematics at least three letters of recommenda­ Prof. Jeffrey D. Ullman, Department of University of Florida tion to Search Committee, Cognitive and Computer Science, Stanford Univ ., Stan­ 201 Walker Hall Neural Systems Program, Room 240, ford CA 94305. Stanford is an equal Gainesville, Florida 32611 111 Cummington Street, Boston Univer­ opportunity/affirmative action employer All applications for the academic year sity, Boston, MA 02215, preferably by and energetically solicits applications 1989-1990 should be complete by De­ November 15, 1988, but no later than from women and targeted minorities. cember 31, 1988. The University of January 1, 1989. Boston University is Florida is an equal opportunity employer. an Equal Opportunity 1Affirmative Action employer.

1090 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

POSITIONS AVAILABLE Department of Mathematics The University of British Columbia University of Alberta Department of Mathematics VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY-DEPARTMENT OF Applications are invited for tenure-track The Department is seeking an outstand­ MATHEMATICS. We anticipate making positions, subject to budgetary approval, ing candidate for a tenure track Assistant one or more tenure-track appointments in Approximation Theory (File AP-1), Professorship to begin 1 July 1989, in at the assistant professor level or above Numerical Optimization or Partial Differ­ one of the following areas: mathemat­ beginning in the fall of 1989. A Ph.D. is ential Equations (File NP-1), in Number ical physics, probability, operator alge­ required. Very strong research potential Theory (File NT-1), or closely related ar­ bras, algebra/number theory. Applicants required for junior-level appointments eas and Algebraic or Differential Topol­ should have a proven research record and demonstrated outstanding record ogy (File AT-1) at the Assistant Pro­ of high quality and have demonstrated for senior-level appointments. Applica­ fessor level, beginning July 1, 1989. Re­ interest and ability in teaching. Prefer­ tions will be accepted until March 15, quirements are a Ph.D. and proven ability ence will be given to candidates who 1989, or until the positions are filled. or demonstrated potential for research have one or more years of postdoctoral Applicants should send vita and three and teaching. Current salary range is experience. This position is subject to letters of reference to: Chairman, Search from $33,144 (Canadian) per annum de­ final budgetary approval. The salary will Committee, Department of Mathematics, pending upon qualifications. Send vitae be commensurate with experience and Virginia Tech, Blacksburg, VA 24061- and arrange for three letters of refer­ research record. Applicants should send 0123. Virginia Tech is an Equal Opportu­ ence to be sent to: Professor L. H. a C.V. including list of publications, state­ nity1 Affirmative Action Employer. Erbe, Chairman, Department of Mathe­ ment of research and teaching interests matics, University of Alberta, Edmonton, and arrange for three letters of recom­ Canada, T6G 2G1. In accordance with mendation to be sent directly to: Dr. Canadian Immigration requirements, pri­ David Boyd, Head, Department of Math­ ority will be given to Canadian citizens ematics, University of British Columbia, and permanent residents of Canada. Vancouver, B.C., Canada V6T 1Y4. Ap­ Closing date for applications is Octo­ plications must be received before 1 ber 31, 1988. Please quote file numbers January 1989. In accordance with Cana­ QUEEN'S UNIVERSITY AT KINGSTON when responding to this advertisement. dian immigration requirements, priority DEPARTMENT OF The University of Alberta is committed to will be given to Canadian citizens and MATHEMATICS AND STATISTICS the principle of equity in employment. permanent residents of Canada. Applications are invited for a tenure track position in discrete mathematics beginning July, 1990. The successful applicant must have and maintain a solid research record in such areas as graph theory, enumeration, and combinatorial CARNEGIE MELLON UNIVERSITY designs. Department of Mathematics CARNEGIE MELLON UNIVERSITY Those interested are requested to Department of Mathematics arrange that a curriculum vitae and The Department expects to make one letters of recommendation from three or two tenure-track appointments, to The Department of Mathematics at or more referees be received at the begin in the Fall of 1989. Although Carnegie Mellon University anticipates address below by October 31, 1988. At these appointments are expected to be an opening at the level of full Professor. least one letter should comment on the at the Assistant Professor level, we Applications are invited from mathemati­ candidate's teaching ability. also solicit exceptionally well-qualified cians and applied mathematicians with Professor L. L. Campbell, Head applicants for more advanced positions. exceptionally strong accomplishments in Department of Mathematics and We particularly seek candidates in the research and considerable experience in Statistics area of numerical analysis, but also will teaching. Candidates' research interests Queen's University consider any areas of research which should overlap significantly with those Kingston, Ontario strongly intersect those of the current of Department members. Applications K7L 3N6 faculty of the Department. Candidates should be sent to: Senior Appointments Appointments In accordance with Canadian Immigra­ should send a resume to: Committee, Department of Mathematics, tion requirements, this advertisement is Committee, Department of Mathematics, Carnegie Mellon University, Pittsburgh, Pittsburgh, directed to Canadian citizens and per­ Carnegie Mellon University, PA 15213. Carnegie Mellon University is manent residents. Candidates of either PA 15213. Carnegie Mellon University is an Affirmative Action/Equal Opportunity sex are equally encouraged to apply. an Affirmative Action/Equal Opportunity Employer. Employer.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1091 Classified Advertisements

POSITIONS AVAILABLE The Ohio State University OREGON STATE UNIVERSITY Department of Mathematics Faculty Position-Mathematics. Mesa Research Instructorships in Assistant Professor position in Algebra, State College announces the availability Mathematics (Number Theory), Numerical Analysis, or of three tenure track positions in math­ Geometric-Topology will become avail­ ematics beginning late August, 1989. Applications are invited for the position able September, 1989. Salary depends Specialization in statistics is wanted for of research instructor in mathematics for on qualifications. Closing date January one of them. Required are a Ph.D. the academic year 1989-90. Candidates 20, 1989. Write to: and strong commitment to teaching as should hold a Ph.D. (or equivalent) in Professor Bent Petersen well as interest in professional achieve­ mathematics and show strong research Staff Selection Committee ment beyond teaching. Rank commen­ promise. Department of Mathematics surate with qualifications. Send resume Please send credentials and have Oregon State University with three references to William Put­ letters of recommendation sent to Pro­ Corvallis, Oregon 97331 nam, Dean, School of Natural Sciences fessor Joseph Ferrar, Department of Oregon State University is an Affirma­ and Mathematics, Mesa State College, Mathematics, The Ohio State Univer­ tive Action/Equal Opportunity Employer Grand Junction, Colorado, 81502. To as­ sity, 231 W. 18th Avenue, Columbus, and complies with Section 504 of the sure consideration, applications should Ohio 4321 0. The Ohio State University is Rehabilitation Act of 1973. OSU has a be received by October 17, 1988. Mesa an Equal Opportunity1 Affirmative Action policy of being responsive to the needs State College is an AA/EO employer. Employer. of dual-career couples.

CORNELL UNIVERSITY DIRECTOR, CENTER FOR THEORY AND SIMULATION IN SCIENCE AND ENGINEERING Cornell University seeks qualified individuals for the above position, which will be filled not later than July 1, 1989. The Theory Center The Director • Sponsors interdisciplinary research and graduate instruction in com­ Must be an outstanding scientist of international stature and have broad putational science and engineering. research and administrative experience in computational science. • Provides infrastructure for interdisciplinary collaborations on com­ Will be appointed professor in an appropriate Cornell department and putational issues. expected to maintain his or her research program in computational science. • Contains the CNSF (Cornell National Supercomputer Facility), one of the five NSF sponsored supercomputer centers which is IBM­ The Theory Center provides a national focus on the interface of science based, with current focus on coarse-grained parallelism and user­ and engineering research with computational science. The Director friendly interface. Its present confi!iJuration is especially suitable to is expected to have the personal qualities required to act as a national those with memory intensive applications. spokesperson for computational science. • Contains the ACF (Advanced Computational Facility) for experimen­ tation on prototype highly parallel machines. For consideration please send your resume and a letter highlighting your qualifications to: Joseph Ballantyne, Vice President for Research and Advanced Studies, CORNELL UNIVERSITY, 312 Day Hall, Ithaca, NY 14853-2801. Respond on or before September 30, 1988 CORNELL •••••••• ...... ,."'11"'11"'11. THEORY ...... ~ , ...... ~ .. , .. .. CENTER ...... ·"""""" ·""""

1092 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

POSITIONS AVAILABLE FACULTY APPOINTMENT YORK UNIVERSITY Department of Mathematical Sciences Department of Mathematics UNIV. OF N.C. AT CHAPEL HILL, The Johns Hopkins University Toronto, Canada DEPT. OF MATHEMATICS, CHAPEL HILL, N.C. 27514 Applications are invited for a senior Applications are invited for a tenure­ appointment in the area track position in STATISTICS, rank open, Applications are invited for a senior •Numerical Analysis and Optimization to commence July 1, 1989. Applicants level tenured appointment in the gen­ for Fall 1989. Junior applicants will also should have proven ability or demon­ eral area of applied and computational receive consideration. Selection will be strated potential for research in Sta­ mathematics, effective Fall, 1989. Rank based on demonstration of excellence in tistics and the ability to teach Applied and salary depend on qualifications. research, teaching, and innovative appli­ Statistics at the undergraduate and grad­ A Ph.D. and demonstrated excellence cation. Applicants are asked to furnish uate levels. One or more limited-term or in research and teaching are required. vita together with a letter describing pro­ tenure-track positions, rank and FIELD Applications will be accepted until the fessional interests and aspirations, and OPEN, are also anticipated, subject to position is filled; however, applications to arrange for three letters of recom­ university approval. Resumes and three received by January 15, 1989, are as­ mendation to be sent, by October 15, letters of recommendation should be sured of full consideration. Send 4 letters 1988, to: sent by January 1 , 1989 to: Joan Wick of recommendation, vitae, and abstracts Professor Jong-Shi Pang Pelletier, Chair, Department of Mathe­ of current research to Search Com­ Department of Mathematical matics, 4700 Keele Street, North York, mittee, %Pat Levin, Mathematics De­ Sciences Ontario, M3J 1P3 Canada. York Univer­ partment, Box 3250 Phillips Hall, UNC The Johns Hopkins University sity is implementing a policy of employ­ at Chapel Hill, Chapel Hill, NC 27599. Baltimore, Maryland 21218 ment equity. Qualified women and men EO/AA Employer. Women and minorities The Johns Hopkins University is an Equal are invited to apply. In accordance with are encouraged to identify themselves Opportunity f Affirmative Action Employer. Canadian Immigration requirements, pri­ voluntarily. Employment is offered without discrim­ ority will be given to Canadian citizens ination on the basis of race, color, and permanent residents of Canada. religion, sex or national origin.

UNIVERSITY OF MINNESOTA, DULUTH Department of Mathematics and DEPARTMENT OF STATISTICS Statistics DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, THE UNIVERSITY OF BERKELEY Tenure-track Asst. Prof. & tenure-track WESTERN ONTARIO or tenured Assoc. Prof. starting 9/1/89. Pending final budgetary approval, appli­ Teach 2 courses per quarter at grad & Applications or nominations are invited cations are invited for several special undergrad level; assist in master's pro­ for the position of Chairman, Department Neyman Visiting Assistant Professor po­ gram in applied & computational math; of Mathematics, Faculty of Science. The sitions, beginning Fall, 1989 or Spring, do research. Required: Ph.D. in ap­ effective date of the appointment is July 1990. The appointment is of two-year plied math or related field by 9/1/89. 1, 1989. duration (but can be shorter by mutual Desired: 5 years professional experi­ Applications or nominations should agreement) and is not renewable. Appli­ ence, appropriate research area, ef­ be addressed to: cants should have exhibited exceptional fective teaching & advising experience, Dr. W. S. Fyfe, research potential in any of the following publications, industrial or governmental Dean, areas: Theoretical or applied statistics, experience in math. Salary competitive. Faculty of Science, computational statistics, probability the­ Send resume, 3 letters of recommen­ Natural Sciences Centre, ory, applied probability. Appointees will dation, & transcripts (if degree received The University of Western Ontario, be expected to teach as well as to carry within past 5 years) to Harlan Stech, London, Ontario, Canada. out a vigorous program of research. MG 1 08, 10 University Drive, Duluth, N6A 5B7 Send applications or inquiries (including MN 55812, by 1/15/89. 218-726-8272 Applications or nominations should be resume and names of three references) THE UNIVERSITY OF MINNESOTA IS submitted by January 1, 1989. In ac­ by March 15, 1989 to: R. J. Beran, AN EQUAL OPPORTUNITY EDUCATOR cordance with Canadian immigration re­ Chairman, Department of Statistics, Uni­ AND EMPLOYER AND SPECIFICALLY quirements, this advertisement is di­ versity of California, Berkeley, California INVITES AND ENCOURAGES APPLICA­ rected to Canadian citizens and perma­ 94720. The University of California is TIONS FROM WOMEN AND MINORI­ nent residents of Canada. an Equal Opportunity, Affirmative Action TIES. An Equal Opportunity Employer Employer.

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1093 Classified Advertisements

AVAILABLE POSITIONS POSTE D'ATTACHE DE RECHERCHE Head, School of Civil Engineering EN MATHEMATIQUES NUMERIQUES Purdue University TRINITY UNIVERSITY SAN ANTONIO, TEXAS Le Centre de recherche en sciences et The Schools of Engineering at Purdue ASSISTANT/ ASSOCIATE PROFESSOR en ingenierie des molecules (CERSIM) University invite nominations and appli­ OF et le Departement de mathematiques cations for the position of Head, School MATHEMATICS et de statistique sollicitent des can­ of Civil Engineering. The successful can­ didatures pour un poste d'attache de didate will possess outstanding leader­ Trinity University invites applications and recherche pour une duree possible ship qualities and administrative abilities. nominations for a tenure-track position de 3 ans debutante le 1er janvier The candidate shall be eligible for aP­ in mathematics, appointment beginning 1989. Ce poste d'attache de recherche pointment as full professor with imme­ August, 1989. The appointment will be pourra ~tre integra au Department de diate tenure based on a distinguished made at the rank of Assistant Profes­ mathematiques et de statistique. record of scholarly activity to include sor or Associate Professor, depending La personne retenue teaching, research and service in the on qualifications. Responsibilities include * fera de Ia recherche portant sur Ia engineering profession. teaching nine credit hours per semester, simulation numerique des ecoulements Purdue is a land grant institution. continuing scholarly activity, assisting in polymeres et ce, dans le cadre d'une The Schools of Engineering constitute curriculum development as appropriate de action structurante; one of the largest and highest quality to the needs of the department and * contribuera Ia .direction d'etudi­ engineering instructional and research the university, advising and committee a ant(e)s des 2eme et 3eme cycles. organizations in the United States. The service. La personne interessee doit School of Civil Engineering currently has Minimum qualifications are the Ph.D. * detenir un doctorat en mathematiques 58 faculty members and over 600 gradu­ in Mathematics or Applied Mathematics ou un diplOma juga equivalent avec ate and undergraduate students exclud­ with excellence in and strong commit­ specialisation en resolution numerique ing freshmen. Research activities cover ment to teaching. Preference given to des equations aux derivees partielles a broad range of topics and account candidates with teaching and research par Ia ml!'thode des elements finis; for an annual expenditure of approxi­ interests in one or more of the following * manifester de l'inter~t pour Ia re­ mately $3 million. The curriculum spans areas: applied mathematics, numerical cherche multidisciplinaire et le travail a wide spectrum of Civil Engineering analysis, classical analysis, differential disciplines, with emerging technologies equations. d'equipe. Dans Ia perspective d'une integration being continuously integrated into both Founded in 1869, Trinity University au Departement de mathematiques et de instruction and research. The candidate occupies a modern campus overlook­ statistique, les qualites d'enseignant de selected shall be an innovative individual ing the San Antonio skyline. Purposely personne interessee et ses capacites with a firm grasp and understanding of small and selective, with about 2700 Ia s'exprimer en franyais seront prises the current and future needs of the Civil students, Trinity stresses a high quality, a consideration. Engineering profession. undergraduate liberal arts and science en Conformement aux exigences rela­ The position will be available as early program. San Antonio is a city of ap­ tives !'immigration au Canada, cet avis as 1 July 1989. Applications will be proximately 850,000 people situated in a a concours s'adresse en premier lieu considered until the position is filled. metropolitan area of 1.2 million. de citoyen(ne)s canadien(ne)s et aux Screening of applications will begin 1 Closing date for applications is Jan­ aux permanent(e)s du Canada. August 1988. Nominations and applica­ uary 27, 1989. Send vita, transcripts and resident(e)s Le curriculum vita doit parvenir avant tions should be sent to: three letters of reference to: 1988 l'adresse suiv­ Dr. Henry T. Yang Dr. Donald F. Bailey, Chairman le 30 septembre a Dean, Schools of Engineering Department of Mathematics ante: Purdue University Trinity University Robert COte de mathematiques et West Lafayette, IN 47907 715 Stadium Drive Department de statistique Purdue University is an Equal Opportu­ San Antonio, Texas 78284 Faculte des sciences et de genie nity I Affirmative Action employer. Trinity University is an equal opportunity Universite Laval, Quebec affirmative action employer. G1K 7P4 Le 23 juin 1988

1094 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

POSITIONS AVAILABLE WILLIAMS COLLEGE University of Puerto Rico DEPARTMENT OF MATHEMATICS Department of Mathematics GROVE CITY COLLEGE POSITION WILLIAMSTOWN, MASSACHUSETTS Rio Piedras, PR 00931 IN MATHEMATICS 01267 Applications are invited for at least 4 Grove City College, an independent, We anticipate three positions, probably tenure-track positions at the Assistant Christian college of liberal arts and sci­ at the rank of assistant professor, for Professor level beginning August, 1989. ences, affiliated with the Presbyterian Fall, 1989. For one position, there is a Candidates must have a Ph.D., a com­ Church (U.S.A.), seeks a Ph.D. in math­ preference for statistics or operations mitment to undergraduate teaching and ematics for the fall of 1989. ABO's research. potential for continuing research. Con­ are invited to apply. Rank and salary Strong commitment to both teaching versational knowledge of Spanish is a are open. A strong evangelical Christian and scholarship is essential. The teach­ plus. Positions are open in Computer commitment is expected. This is a teach­ ing load is five 1-semester courses per Science, Probability, Statistics, Opera­ ing position; scholarly activity of genuine year, plus a "Winter Study" in alternate tions Research, Numerical Analysis, Or­ interest to the teacher (not publication Januaries, beginning in the second year. dinary and Partial Differential Equations, for publications' sake) is encouraged. Please send a vita and three let­ Functional Analysis, Algebra and Topol­ Calculus and upper level teaching as­ ters of recommendation on teaching and ogy. Salary around $23,820 for 9 months signments. Good locale, excellent facil­ research to Frank Morgan, Chair. Evalu­ of teaching. Applications considered un­ ities, top-rated students and Christian ation of applications will begin November til June 1, 1989. Send resume and environment. Send vitae to Dr. Jerry H. 15 and continue until positions are filled. three letters of recommendation to J. M. Combee, Vice President for Academic Since Williams is an affirmative ac­ L6pez, Head, Department of Mathemat­ Affairs, Grove City College, Grove City, tion/equal opportunity employer, we en­ ics, University of Puerto Rico, P.O. Box PA 16127. Grove City College is an equal courage applications from women and BF, Rio Piedras, PR 00931. EOE/AA. opportunity employer. minorities.

Let f(z)'s animated screens shoiN you what a textbook cannot.

The Complete Complex Variables Graphing Package

Lltscaux Graphics 3220 Steuben Ave. Bronx, NY 10467 (212) 654-7429 .1 log (z) z .2 z Explain this apparent convergence

If you have a PC compatible, wnte for a FREE demo disc!

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1095 Classified Advertisements

POSITIONS AVAILABLE THE UNIVERSITY OF ADELAIDE It is University policy to encourage SOUTH AUSTRALIA women to apply for consideration for DEPARTMENT HEAD invites applications from both women appointment to tenurable academic ap­ MATHEMATICS AND STATISTICS and men for the following position: pointments. Holders of full-time tenured LOUISIANA TECH UNIVERSITY PROFESSOR OF MATHEMATICAL or tenurable academic appointments PHYSICS have the opportunity to take leave with­ Louisiana Tech University invites appli­ (Tenurable) cations for Head, Department of Mathe­ out pay on a half time basis for a specific matics and Statistics. The Departments (Ref: 1516) in the DEPARTMENT OF period of up to ten years where this is of Chemistry, Mathematics and Statis­ PHYSICS AND MATHEMATICAL PHYS­ necessary for the care of children. tics, and Physics make up the School of ICS. The appointment follows the retire­ INFORMATION about the general Science within the College of Arts and ment of Professors H. S. Green, F. A. A. conditions of appointment may be ob­ Sciences. The Department Head has and C. A. Hurst, F. A. A. and is available tained from the Senior Assistant Regis­ responsibility for all facets of depart­ from June, 1989. The Department was trar (Personnel) at the University. mental activity, which include curricu­ formed from the Departments of Physics SALARY per annum: AUS $59,183 lum development, budgeting, recruiting, and Mathematical Physics by the amal­ per annum (subject to second tier wage scheduling, and faculty evaluation. As gamation of the two Departments on increase) the Department's chief administrative of­ 1 January 1988. The new Department APPLICATIONS, IN DUPLICATE, quot­ ficer, he/she will report to the Dean of is responsible for teaching of physics ing reference number 1516 and giving full the College through the Director of the and theoretical physics in the Faculty personal particulars (including whether School. The Department is comprised of of Science and mathematical physics in candidates hold Australian permanent approximately twenty-five full-time fac­ the Faculty of Mathematical Sciences. residency status), details of academic ulty members. Salary is commensurate The Department teaches undergraduate qualifications and names and addresses with qualifications. The position will be physics subjects to physics majors and of three referees should reach the Se­ available onjatter December 1, 1988. Honours students and has a substan­ nior Assistant Registrar (Personnel) at Applicants must hold a Ph.D. in Math­ tial responsibility for teaching physics to the University of Adelaide, GPO Box ematics or Statistics, have an established the professional faculties and to other 498, Adelaide, South Australia, 5001, record of research and scholarly activity, Departments in the Faculties of Science Telex UNIVAD AA 89141/FAX NO. (618) and be able to provide strong academic and Mathematical Sciences. 224 0464 not later than 30 September leadership. The current research interests of the 1988. Please submit application, resume, Department are atmospheric physics, The University reserves the right to and three letters of reference by Novem­ cosmic rays and high energy astro­ make enquiries of any person regarding ber 1, 1988 to: physics, mathematical physics, physical any candidate's suitability for appoint­ Dean archaeometry, theoretical nuclear and ment, not to make an appointment or to College of Arts and Sciences particle physics and ultra-violet physics. appoint by invitation. Louisiana Tech University There are two other Professors in the THE UNIVERSITY OF ADELAIDE IS Ruston, LA 71272 Department, Professors J. R. Prescott AN EQUAL OPPORTUNITY EMPLOYER. An Affirmative Action/Equal Opportunity and A. W. Thomas. Employer The new professor will be expected of Notre Dame to contribute to the academic leadership University Department of Mathematics of the whole Department but will have Notre Dame, ln. 46556 a particular responsibility for teaching Florida State University. Department of subjects in the discipline of mathemati­ Applications are invited for a tenure­ Mathematics, Tallahassee, FL 32306. cal physics in second, third and Honours track or tenured associate professorship Applications are invited for a tenure­ year, leading to degrees in the Fac­ in the area of global analysis and differ­ track assistant professorship with ap­ ulty of Mathematical Sciences. The new ential geometry. Applicants should have pointment beginning in August, 1989. professor will be expected to promote substantial teaching ability and an out­ The candidate should be a numerical and extend a strong programme of re­ standing record of publications in areas analyst with experience in the develop­ search and postgraduate studies within related to geodesics and the theory of ment of the vortex method for the Euler the new Department, and in cooperation submanifolds. Send by September 30, equations of fluid mechanics, and should with other Departments, particularly in 1988, a curriculum vitae, list of pub­ have potential for excellence in research the Faculty of Mathematical Sciences. lications, a few selected reprints or and teaching. Please send resume, and A commitment to excellence in teaching preprints, and the names of three refer­ arrange for three letters of recommenda­ and research is expected. ences to Andrew Sommese, Chairman, tion to be sent, to Ralph D. McWilliams, Further information concerning the at the above address. The University Chairman. Application deadline is Octo­ duties of the position may be obtained of Notre Dame is an equal opportunity ber 15, 1988. Florida State University is from Dr. L. R. Dodd, telephone (618) affirmative action employer. an EEO fAA employer. 228 5113.

1096 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

AVAILABLE FOR SALE POSITIONS 90024-1555. Attn: Staff Search. UCLA is action an equal opportunity 1affirmative PASCAL-SC: Compilers for personal UNIVERSITY OF CALIFORNIA, employer. LOS ANGELES computers, documentation, software. Department of Mathematics FBSoftware, 5101 Odana Road, Madi­ son, Wisconsin 53711. (608) 273-3702. TEMPORARY POSITIONS ( 1) One or two E. R. Hedrick Assistant Professorships. Applicants must show Advertisement very strong promise in research and The Jordan University of Science and teaching. Salary $36,000. Three year Technology announces vacancies for Journals from 1961, American Mathemat­ appointment. Teaching load: four quarter staff members in the Faculty of Science ical Monthly, Bulletin AMS. courses per year, which may include one in the following areas: For detailed information write advanced course in the candidate's field. 1. Mathematics Professor Bert Ross Preference will be given to applications 2. Computer Science Mathematics Dept. completed by January 1, 1989. Candidates must have an earned Ph.D. University of New Haven (2) Two or three Research Assistant degree, preferably with some academic West Haven, Conn. 06516 Professorships in Computational and experience. The candidate must have a Applied Mathematics. Applicants must university degree (Bachelor of Science) show very strong promise in research in the same field. and teaching. Salary $36,000. Three year Applications with full curriculum vitae WYSIWYG for Macintosh. appointment. Teaching load: four quarter should be sent to: include one courses per year, which may Personnel Department With SCIPAGE, type, reposition, re­ the candidate's field. advanced course in Jordan University of Science and shape: complex fractions, roots, inte­ given to applications Preference will be Technology grals, matrices, frames and grids, ... 1, 1989. completed by January lrbid P.O. Box (3030) Anywhere on a page! Ideal for lecture Assistant Profes­ (3) One or two Jordan notes, exams, ... $40 postpaid. Write for in Comput­ sorships in the Program samples. METAMATH, Box 26, Chris­ show very ing (PIC). Applicants must tiana, PA 17509. strong promise in teaching and resear~h, preferably in the general area of Log1c, Language and Computation. Teaching UNIVERSITY OF CALIFORNIA, load: four quarter programming courses LOS ANGELES and an advanced quarter course of Department of Mathematics the candidate's choice per year. Two REGULAR POSITIONS IN PURE AND PUBLICATIONS year appointment, renewable once or APPLIED MATHEMATICS Salary range: $36,000-$42,000. MATH SCI PRESS, 53 Jordan Rd., twice. Six to eight regular positions in pure and will be given to applications Brookline, MA 02146, (617)738-0307. Preference applied mathematics. Specific fields of 1 , 1989. SPECIAL SALE: INTERDISCIPLINARY completed by January interest include algebra/number theory Lectureships i_n the MATHEMATICS, Vols. 1-9, 11-16, 18, (4) One or two (including algebraic geometry), analy­ Computing (PIC). Applicants 19. LIE GROUPS: 1-3, 6-8, 10, 11. SYS­ Program in sis, applied and computational mat_he­ promise in the TEMS AND CONTROL: 1, 2. $10 plus must show very strong matics, differential equations, dynam1cal Teaching load: postage for each volume. teaching of programming. systems, game theory, logic, mat_he­ courses per five quarter programming matical computer science, mathematical year appointment, renewable year. One physics, probability and statistics. V~ry depends on up to four times. Salary strong promise in research and teach1ng experience, begins at $30,300. Just appeared: required. Positions initially budg~t~d at (5) Subject to administrative appro~al, the assistant professor level. SuffiCiently Frolik (ed): General Topology and its a few adjunct assistant professorships. z. outstanding candidates at higher levels Relations ..., Proc. of the Sixth Prague Two year appointments. Strong re­ will also be considered. Teaching load: Topol. Symp. 1986, 736 pp., DM 108.­ search and teaching background re­ Averaging 1.5 courses per Quarter, or or $66.- quired. Salary $31,500-$35,400 per year. 4.5 Quarter courses per year. To apply, K. G. Murty: Linear complementarity, Teaching load: five quarter courses per write to Alfred W. Hales, Chair, De­ linear and nonlinear programming, 680 year. partment of Mathematics, University of pp., OM 148.- or $89.- (6) Several positions for visitors and California, Los Angeles, CA 90024-1555. 0rder directly from: Heldermann Ver­ lecturers. Attn: Staff Search. UCLA is an equal lag, Nassauische Str. 26, D-1 000 Berlin­ To apply, write to Alfred W. Hales, opportunity/affirmative action employer. West 31. Chair, Department of Mathematics, Uni­ versity of California, Los Angeles, CA

SEPTEMBER 1988, VOLUME 35, NUMBER 7 1097 Classified Advertisements

ANNOUNCEMENTS

CALL FOR PAPERS Colby College PRIZE ($1 ,000) for PHILOSOPHIA Mathematics Department MATHEMATICA (circulating in some 50 Waterville. Maine 04901 nations since 1964) on such topics as: 1. Ethics and Mathematics (in particu­ lar, Morals of Knowledge in general); 2. Carter Professor of Sociology or Social Theory of Mathemat­ Mathematics and Chair ics; 3. The Topological vs the Algebraic; Colby invites nominations and applications for the 4. Others, related to the study on the Carter Professor of Mathematics and Chair of the Nature of Mathematics (definitely not Mathematics Department, effective September 1, 1989. via Logic alone). For details, write to: Ph.D. in mathematical sciences required. Desirable [Prof.] J. Fang, ODU (Phil), Norfolk, VA qualifications include a distinguished record as teacher and scholar; demonstrated departmental and collegial 23529-0083. leadership, including the ability to nurture faculty development and research programs; commitment to liberal arts and undergraduate math education. POSITIONS WANTED Colby is a highly selective college of 1700 students and 165 faculty. Its Mathematics Department, within which Seeking Full Professorship with tenure at are taught computer science and statistics courses, has Ph.D. granting institutions where normal 9 FTEs. Colby is an affirmative action/equal opportunity employer, and encourages applications from women teaching load is 0 to 6 cr. hrs. per term. and minorities. Seeking seniority since Jan. 1st '76. Send applications Age: 42, Speciality: Algebraic (letter of application and resume) and Topology nominations to: H.T. Hayslett, Jr., Chair, Mathematics (Lefschetz Type Fixed Point Theorems), Department, by December 1, 1988. M. Experience: '67 to '83. C. Vora, 1178 Milford Str., Backporch, Johnstown, PA 15905. Available immediately.

THEORIE DES VARiETES MINIMAI-ES ET APPLICATIONS (MINIMAL SUBMANIFOLDS) SEMINAIRE PALAISEAU (Asterisque, Number 154-155)

The study of minimal submanifolds is by now established Jr. at Ecole Polytechnique in Palaiseau, presents recent as one of the deep and esthetically appealing parts of contributions to the theory of minimal submanifolds in their mathematics. It combines in an exemplary fashion geometric diversity. It starts with an elementary approach to the subject, and analytical techniques both of a classical and of a more hence is appropriate as a source book for a graduate seminar. modern nature. In recent years it became a powerful tool to investigate the internal geometry of manifolds, a subject SOCIETE MATHEMATIQUE DE FRANCE, ASTERISQUE of interest today to both mathematicians and theoretical The AMS distributes Asterisque only in the U.S., Canada, and Mexico. physicists. Orders from other countries should be sent to the SMF, B.P. 126-05, 75226 Paris Cedex 05, France, or to OFFILIB, 48 rue Gay-Lussac, This volume, devoted to notes of a seminar held from October 75240 Paris Cedex 05, France. Individual members of either AMS or 1983 to June 1984 under the direction of H. B. Lawson SMF are entitled to the member price. (ISSN 0303-1179)

1980 Mathematics Subject Classifications: 57, 53, 58 Shipping/Handling: 1st book 12. each add'l 11. S25 354 pages (softcover), 1988 max. By air, 1st book 15. each add'l 13, 1100 max. Individual member $27, List price $39 Prepayment required. Order from AMS. P.O. Box To order, please specify AST/154/155NA 1571, Annex Station, Providence, Rl 02901-9930. or call 800-556-7774 to use VISA or MasterCard.

1098 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Systems governed by non-linear differential equations are of fundamental The Geometry importance in all branches of science, but our understanding of them is still extremely limited. In this book a particular system, describing the interaction and Dynamics of magnetic monopoles, is investigated in detail. The use of new geometrical methods produces a reasonably clear picture of the dynamics for slowly of Magnetic moving monopoles. This picture clarifies the important notion of solitons, which has attracted Monopoles much attention in recent years. The soliton idea bridges the gap between the concepts of "fields" and "particles," and is here explored in a fully three­ Michael Atiyah dimensional context. and Nigel Hitchin M. B. Porter Lectures, Rice University, Department of Mathematics Cloth: $25.00 ISBN 0-691-08480-7

This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of Lie Groups, cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational tech­ Lie Algebras, niques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook and in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory. Cohomology These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first con­ Anthony Knapp sequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor. Mathematical Notes, 34 William Browder, Robert Langlands, John Milnor, and Elias M. Stein, Editors Paper: $29.50 ISBN 0-691-08498-X

The classical pseudodifferential calculus is well adapted to detailed study of elliptic operators such as the Laplacian associated to the De Rham complex. Calculus on This book develops a full asymptotic calculus adapted to certain second order operators which are hypoelliptic but not elliptic. The motivating example is Heisenberg the operator Db associated to the ab-complex on a CR-manifold. Like the laplacian, Db is a natural operator of intrinsic interest, a prototype of a general Manifolds class, and a test case. Principal terms of parametrices and other operators Richard Beals associated to Db are calculated on both the symbol side and the kernel side. It is hoped that this viewpoint on pseudodifferential operators will be fruitful in and Peter Greiner attacking other nonelliptic problems including more degenerate cases of Db. Annals of Mathematics Studies, 119 William Browder, Robert Langlands, John Milnor, Elias M. Stein, Editors Paper: $15.95 ISBN 0-691-08501-3 Cloth: $40.00 ISBN 0-691-08500-5 Not available in Japan.

AT YOUR BOOKSTORE OR Princeton University Press 41 WILLIAM sr. • PRINCElON, NJ 08540 • (609) 452-4900 I ORDERS 800-PRS-ISBN (777-4726) WHY SETTLE FOR THIS? •••

••• WHEN WHAT YOU REALLY WANT IS THIS.

TRY THIS:

Quality work gets a quality response-when pc'Jl:X FORMATTING/TYPESETTING SYSTEM: the work looks high-quality. What better reason to try PC"IEX•-the best-selling FINE TYPESET QUALITY from dot matrix or laser printers, full implementation of Prof. D. Knuth's or phototypesetters. revolutionary lEX formatting/typesetting A COMPLETE PRODUCT. Includes • our specially writ· program. ten PCTEX Manual that lets you use TeX immediately • custom PClEX offers professional typesetting 'macro package' formats for letters, manuals, technical doc­ & to PC users. In uments, etc. • the LaTeX document preparation system (with capabilities advantages user's manual) macro package for article, book, report prepara­ a word, it gives you 'control'. Control-of tion • AMS· TeX. developed by the American Mathematical So­ format, type, symbols, quality-for complex ciety for professional mathematical typesetting. mathematical & engineering material, OUTPUT DEVICE DRIVERS for • Epson FX, LQ statistical tables or straight matter. You • Toshiba • Cordata LP-300 Series • HP LaserJet Series get camera/publisher-ready manuscripts INC • Apple LaserWriter • APS-5 phototypesetter • Linotronic • to be proud of, quick & simple. 1b order or for information, call: Compugraphic • Screen preview, with EGA or Hercules card. So whether you're writing the next 415/388-8853 REQUIRES: IBM PC/XT, AT or compatible, DOS 2.0 or higher starshot manual, a thesis on relativity or the or write: Personal 'lEX, Inc. & 512K RAM; hard disk for printer drivers & fonts. great American novel, when a professional 12 Madrona Avenue presentation is your priority, don't settle for Mill Valley, CA 94941 USA less than PClEX. Distributed iD 14 coUBtries; (Also available: Printer drivers; interfaces to Bitstream Fontware; From Personal lEX, Inc., distributoriDquirieswelc:ome. Metafont, to design-your-own-type, PC Paintbrush, PC Palette, PC'll;X ;1 a registered TM of Penonal ~ Inc. FancyFont & Fontrix. Complete packages, including laser printer, starting at $249; VISA/MC welcome. 'lEX ia an American Matbemalical Societ¥ ™ printer driver, PCTeX. and screen previewer from $2850. Site Satisfaction guaranteed. Manufacturen' product names are lbeir TM& licenses and volume discounts available.) This ad Is typeset & composed using: PC TeX. Bllltream fonts & 1-pdrUt logclype & black backgrolllds done ~· W WADSWORTH & BROOKS/COLE ADVANCED BOOKS & SOFTWARE

Announces Lecture Notes on Particle Systems and Percolation Richard Durrett

This book provides an informal introduction to interacting particle systems, percolation, and first passage percolation. The style is that of an informal seminar talk with more than 90 pictures that illustrate the main results. The book will be useful for those interested in probability theory, cellular automata, and dynamical systems. Contents: 1. The Simplest Growth Models. 2. The Voter Model. 3. The Biased Voter Model. 4. The Contact Process. 5. One Dimensional Discrete Time Models. 6. Percolation in Two Dimensions. 7. Mandelbrot's Percolation Process. 8. First Passage Percolation. 9. Epidemic Models. 10. The Voter Model, II. 11. The Contact Process, II. • 1988. 335 pp. Cloth. ISBN 0-534-09462-7. $39.95. EXP: The Scientific Word Processor, Release 1.11 Simon L. Smith and Walter L. Smith

EXP is a complete and very fast. word processing program for the IBM PC, PC/XT, PC/AT, and compatibles, and the AT&T 6300. EXP includes the ability to create hundreds of macros, automatic positioning, sizing, and centering of technical expressions; automatic reformatting; find and search­ replace; soft-hyphenation; proportional spacing on/off; even/odd headers-footers; right-hand justification on/off; footnotes; left-hand side equation justification; automatic page numbering; automatic numbering for equations, exercises, etc.; windows to view and edit up to four files at once; column printing; and "what you see is what you get" printing for all technical symbols, fonts, and formatting. Release 1.11 contains a macro editor and a driver for the HP LaserJet PLUS/Series II printers as well as drivers for 8-, 9-, and 24-pin dot-matrix printers from Epson, NEC, Toshiba, and Tandy. Specify whether 51" or 3~" disks desired. Not copy-protected. • 1987. ISBN 0-534-09096-6. $150. University, department, and nonacademic site licenses available. This advertisement (except for the logos) was written with EXP Release 1.11 and printed on an HP Series II. Also Available Enumerative Combinatorics, Volume I Mathematical Statistics and Richard P. Stanley Data Analysis Graduate text. John A. Rice •1986. 306 pp. Cloth. Undergraduate text. ISBN 0-534-06546-5. •1988. 594 pp. Cloth. ISBN 0-534-08247-5.

Algebras, Lattices, Varieties, Volume I Introduction to Mathematical Logic, Ralph N. McKenzie, George F. McNulty, Third Edition and Walter F. Tayior Elliott Mendelson Graduate text. Advanced introduction to logic. •1987. 361 pp. Cloth. •1987. 339 pp. Cloth. ISBN 0-534-07651-3. ISBN 0-534-06624-0.

Graphs &; Digraphs, Second Edition Graphical Aids for Stochastic Processes Gary Chartrand and Linda Lesniak Bob Fisch and David Griffeath Graduaie text Six disks for the IBM PC. •1986. 359 pp. Cloth. •1988. 6 disks and 7 pp. doc. ISBN 0-534-06324-1. ISBN 0-534-09042-7. $99.

Wadsworth&; Brooks/Cole Advanced Books & Software 511 Forest Lodge Rd., Pacific Grove, CA 93950-5098, ( 408)373-0728. For orders only from within the U.S. call our toll-free number (800)354-9706. American Mathematical Society (r= JOURNALS OF DISTINCTION

D MATHEMATICAL REVIEWS Since 1940, Mathematical Reviews has been the recognized reviewing and ab­ stracting journal covering published mathematical research literature. Over 40,000 reviews or abstracts, by 12,000 international reviewers, are published each year. The reviews in each issue are arranged according to the 1980 Math­ ematics Subject Classification (1985 Revision). Subscriptions include annual author and subject indexes. Published monthly. 1989 subscription rate: Either paper or microfiche subscriptions: List 13510; Inst member 12808; Indiv member 1421*; MR Reviewer 1281* Postage surcharge: India 1153; other foreign 160 Combination paper and microfiche subscriptions: List 14668; Inst member 13735 $200 late charge applies Optional delivery: First class $70; airmail $190 *Add $10 for subscriptions in the US; $60 for subscriptions outside the US. D CURRENT MATHEMATICAL PUBLICATIONS This current awareness journal, which is published every three weeks, is a sub­ ject index of recent and forthcoming mathematical publications which have been classified by the editors of Mathematical Reviews. (The classification scheme used is the 1980 Mathematics Subject Classification (1985 Revision), published in the most recent annual index of Mathematical Reviews.) Each issue contains an author and key index; author and key indexes covering a half year are in­ cluded in issues 9 and 17. Each issue contains a list of the serials represented in that issue and a separate listing of serial additions and changes, as well as a section containing the tables of contents of certain journals. Volume 21 is the 1989 volume. 1989 subscription rate: List 1275; Inst member 1220; Indiv 1165; MR Reviewer 1110 Postage surcharge: India 138; other foreign 125 Optional delivery: First class $30; airmail $60 10% late charge applies.

Subscriptions are sold only on a calendar year basis; split year and multiple year subscriptions are not accepted. Subscription orders received from nonmember institutions and organizations after January 1 of the subscription year will be sold at list price plus a 10% late charge (rounded); the maximum late charge for any subscription is $200. Microfiche Subscriptions: Microfiche editions of subscriptions are mailed (first class or airmail) before the issue is sent by the printer. All fiche are negative, 105mm by 148mm, have a 24 to 1 reduction ratio, and contain 98 frames per fiche.

PREPAYMENT REQUIRED. Send orders to: American Mathematical Society, P. 0. Box 1571, Annex Station, Providence, Rhode Island 02901-9930 USA, or call 800-5~7774 in the continental US to use VISA or MasterCard. Can you identify these prominent members of the mathematical community ? Learn their identities in I HAVE A PHOTOGRAPHIC MEMORY by Paul R. Halmos

Paul R. Halmos, eminent mathematician and brilliant expositor, is also a snapshot addict. For the past 45 years, Halmos has snapped mathe­ maticians, their spouses, their brothers and sisters and other relatives, their offices, their dogs, and their carillon towers. He knows about cohomology groups of algebras From 6000 or so photographs, Halmos has chosen about 600 for this book. The pictures are candid shots showing mathe­ maticians just being them­ selves, and the accompanying captions, in addition to identi­ fying the subjects, include anecdotes and bits of history that reveal Halmos' inimitable wit, charm, and insight. This delightful collection of mathe­ matical memorabilia is certain to become a favorite browsing book as well as a valuable An algebraist and number theorist historical record. with more than one name ·

~i\\EMA~~ 1988, 336 pages, Hardcover, ISBN 0-8218-0115-5, LC 87-33450 ~ TPHToi MH ~ List $58, All individuals $35 . JJ ~ ~ ~ To order I HAVE A PHOTOGRAPHIC MEMORY, please spec1fy PHOMEM/NA. ;;? @ S ;; To use VISA or MasterCard, call (401) 272-9500 or (800) 556-7774 in the continen- ~ ~ 0 ~ tal United States. Or write: American Mathematical Society, Annex Station, P.O. "' • Box 1571, Providence, Rhode Island 02901 -9930 USA. Prepayment is required. • ~O(j '\~ Add postage and handling: Surface: 1st book $2, each additional $1, max. $25; lVDED Air: 1st book $5, each additional $3, max. $100. Look what's new in Mathematical Surveys and Monographs The Mathematical Surveys and Monographs series is one you're already familiar with. It's been around since 1943, when The Problem of Moments byJ. A. Shohat andJ. D. Tamarkin was pub­ lished. The series has grown over the years and includes important mathematical monographs that cover a broad spectrum of areas such as function theory, analysis, approximation theory, algebra, and topology. Each of the books is designed to give a survey of the subject and a brief introduction to its recent developments and unsolved problems. The editorial committee••• Victor W. Guillemin M. Susan Montgomery IIWinKra R 0. Wells, Jr. (Chairman)

------•• invites JIOU to participate in the development of this classic series bJI

• consulting them as reference books • using books in this series in your courses • recommending them to your library • submitting manuscripts for publication to begin a standing order to the Director of Publication at the AMS

New and Recent Publications in Mathematical Surveys and Monographs Each new title is exceptional. All are the high quality mathematical publications you've grown to expect in the Mathematical Sutveys and Monographs series. Yet the variety of topics and areas of mathematical interest make them perfect reference books or possible textbooks for graduate level courses.

0 AmenablUty by Alan L. T. Paterson, 1988, 400 pp. 0 Introduction to varloua upects of degree theory in (ISBN 0-8218-1529-6). List $90, Inst. mem. $72, Banach spaces by E. H. Rothe, 1986, 254 pp. (LC 86- Indiv. mem. $54, Code SURV/29NA 8038: ISBN 0-8218-1522-9). List $62, Inst. mem. $50, Indiv. mem. $37, Code SURV/23NA 0 Direct and inverse scattering on the Une by Richard Beals, Percy Delft, and Carlos Tomei, 1988, 0 Noncommutatlve harmonic analysts by Michael E. Taylor, 200 pp. (LC 88-14487; ISBN 0-8218-1530-X). List 1986, 344 pp. (LC 86-10924: ISBN 0-8218-1523-7). List $53, lnst. mem. $42, Indiv. mem. $32, Code SURV /28NA $70, Inst. mem. $56, Indiv. mem. $42, Code SURV/22NA

0 Baste hyperaeometrlc series and appUcatlons by 0 The Bleberbach col\fecture: Proceedings of the Nathan J. Fine, 1988, 144 pp. (LC 88-6235; ISBN 0-8218- symposium on the occulon of the proof, edited by 1524-5). List $39, Inst. mem. $31, Indiv. mein. $23, Albert Baernstein, David Drasin, Peter Duren, and Code SURV/27NA Albert Marden, 1986: reprinted 1987, 260 pp. (LC 86-10843; ISBN 0-8218-1521-0). List $46, lnst. mem. $37, Indiv 0 Operator theory and arithmetic in B by Hart Bercovici, mem. $28, Code SURV/21NA 1988, 280 pp. (LC 88-10344; ISBN 0-8218-1528-8). List $67, lnst. mem. $54, Indiv. mem. $40, Code SURV /26NA 0 Partially ordered abeUan groups with interpolation by Kenneth R Goodearl, 1986, 358 pp. (LC 86-7876; 0 Aaymptotlc behavior of dlsslpatlve syatema by ISBN 0-8218-1520-2). List $70, Inst. mem. $56, Indiv. Jack K. Hale, 1988, 200 pp. (LC 87-33495; ISBN 0-8218- mem. $42, Code SURV/20NA 1527-X). List $54, Inst. mem. $43, Indiv. mem. $32, Code SURV/25NA

0 Noetherian rings and their appUcatlons, edited by For information on numbers 1-19, please see the AMS Lance W. Small, 1987, 230 pp. (LC 87-14997: ISBN Catalogue of Publications. 0-8218-1525-3). List $38, Inst. mem. $30, Indiv. mem. $23, Code SURV/24NA

Standing orders are accepted for this and any other series published by the Society. Proforma invoices are sent to standing order customers prior to the publication of each new volume. Shipment is made upon receipt of payment and publication. To begin a standing order, please contact the Membership and Sales Department of the AMS.

PREPAYMENT REQUIRED. Order from the American Mathematical Society, P. 0. Box 1571, Annex Station, Providence, RI 02901-1571, or callB00-556-7774 from within the continental U.S. (401-272-9500, elsewhere) to use VISA or MasterCard. Prices subject to change. Please add for SHIPPING AND HANDLING: first book $2, each additional $1, $25 maximum. By air, first book $5, each additional $3, $100 maximum. MathDoc™ Your Best Access to Inaccessible Items in the Mathematical Literature

What is MathDoc? How to Use MathDoc MathDoc is the document delivery service offered This labor-intensive service is a very low-cost by the AMS, designed for users of CMP, MR, and method of locating documents. The AMS makes MathSci who wish to obtain copies of original MathDoc available essentially at cost as a service to articles. the mathematical community.

MathDoc is particularly useful for obtaining "inacces­ The current price for each item is sible" items that cannot easily be obtained through $12 for the first ten pages copied interlibrary loan. For example, you might want one $2 for each succeeding ten pages or less of these: copyright royalties* (if applicable) Acta Sci. Natur. Univ. Sunyatseni (1986), No. 1, plus postage. pages90-93 Many Chinese journals are available only in a An invoice is sent with the requested item. There is a very small number of U.S. libraries .. 5% discount for VISA and MasterCard prepayments. Documents are mailed first class within North Amer­ Methods and Algorithms in Numerical Analysis (1984), ica and airmail elsewhere. Moskov. Gos. Univ., pages 154-162 Russian collections are often printed in limited *MathDoc complies with copyright laws and adds copyright quantities and MathDoc may be the only U.S. royalties, when applicable. MathDoc is affiliated with the source. Copyright Clearance Center (CCC), Salem, MA.

SurikaisekikenJcyOsho KokyDroku (1986), No. 581, ™MathDoc is a trademark of the American Mathematical Society pages 85-100 This Japanese journal is available only through a limited number of sources in the U.S. Ordering Information

600,000+ Item Access For further information or to place an order through MathDoc, write to MathDoc has access to over 600,000 items at present MathDoc Document Delivery in mathematics and related subjects, including 70,000 Mathematical Reviews articles from Soviet journals. 416 Fourth Street P. 0. Box 8604 95% Success Rate in Two Ann Arbor, MI 48107-8604 USA Working Days OR telephone (313) 996-5268 and ask for MathDoc. And MathDoc has over a 95% success rate in quickly locating and supplying the articles in journals, collec­ tions, and conference proceedings listed in CMP, MR, OR send electronic mail (Internet) to and MathSci...usually within two worldng days. MA [email protected] All electronic mail will be acknowledged. Although MathDoc cannot ordinarily supply an entire book, it often can give you information about the OR order through the DIALOG DIALORDER or book's availability through other sources. the BRS systems. Application for Membership 1988

AMERICAN MATHEMATICAL SOCIETY Date ...... 19 ......

Please read the reverse side of this form to determine what member­ Fields of Interest ship category you are eligible for. Then fill out this application and If you wish to be on the mailing lists to receive information return it as soon as possible. Your name will be added to our mailing about publications in fields of mathematics in which you lists upon our receipt of your completed application, and payment for have an interest, please consult the list of major headings member dues. of the 1980 Mathematics Subject Classification below. Subscriptions to the Notices and the Bulletin (New Series) are Select no more than five category numbers and fill in the included as part of your membership. numbers where indicated on the left. These categories will be added to your computer record so that you will be informed of new pu~lications or special sales in the fields Family Name First Middle you have indicated.

Please indicate below the way your name should appear in the Combined Mem­ bership List. 00 General 01 History and biography 03 Mathematical logic and foundations Family Name First Middle or lnHial 04 Set theory 05 Combinatorics Place of Birth ...... 06 Order, lattices, ordered algebraic structures City Slate Zip/Country 08 General mathematical systems 11 Number theory Date of Birth ...... 12 Field theory and polynomials Day Month Year 13 Commutative rings and algebras 14 Algebraic geometry If formerly a member of AMS, please indicate dates ...... 15 Linear and muttilinear algebra; matrix theory 16 Associative rings and algebras Check here if you are now a member of either MAA or SIAM 0 17 Nonassociative rings and algebras 18 Category theory, homological algebra Degrees, with institutions and dates ...... 19 K·theory 20 Group theory and generalizations 22 Topological groups, Lie groups 26 Real functions 28 Measure and integration 30 Functions of a complex variable Present position ...... 31 Potential theory 32 Several complex variables and analytic spaces Firm or institution ...... 33 Special functions 34 Ordinary differential equations 35 Partial differential equations State Zip/Country City 39 Finite differences and functional equations 40 Sequences. series, summability five from the list at right) Primary Fields of Interest (choose 41 Approximations and expansions 42 Fourier analysis 43 Abstract harmonic analysis 44 Integral transforms, operational calculus (choose five from the list at right) Secondary Fields of Interest 45 Integral equations 46 Functional analysis 4 7 Operator theory 49 Calculus ol variations and optimal control; Address for all mail optimization 51 Geometry 52 Convex sets and related geometric topics 53 Differential geometry 54 Generaltopology 55 Algebraic topology 57 Manifolds and cell complexes 58 Globel analysis, analysis on manifolds 60 Probability theory and stochastic processes ...... ~ ...... 62 Statistics 65 Numerical analysis 68 Computer science 70 Mechanics of particles and systems Prepayment Methods and Mailing Addresses 73 Mechanics of solids All payments must be In U.S. Funds. 76 Fluid mechanics 78 Optics, electromagnetic theory 80 Classical thermodynamics, heat transfer checks, money orders, UNESCO coupons to American Mathematical Send 81 Quantum mechanics 1571, Annex Station, Providence, AI 02901-9930 Society, P.O. Box 82 Statistical physics, structure of matter 83 Relativity and mail to American To use VISA or MasterCard, fill in information requested 85 Astronomy and astrophysics Mathematical Society, P.O. Box 6248, Providence, AI 02940. 86 Geophysics 90 Economics, operations research, programming, For Foreign Bank Transfers: The name and address of the AMS bank is games · Rhode Island Hospital Trust National Bank, Account #000-753-111, One 92 Biology and behavioral sciences Hospital Trust Plaza, Providence, AI 02903, U.S.A. 93 Systems theory; control 94 Information and communication, circuits VISA 0 MasterCard 0 ...... Account number M8NO Membership Categories Reciprocating Societies Please read the following to determine what membership category you are eligi­ 0 Allahabad Mathematical Society ble tor, and then indicate below the category tor which you are applying. 0 Asociaci6n Matematica Espanola 0 Australian Mathematical Society 0 Berliner Mathematische Gessellschaft e.V. For ordinary members whose annual professional income is below $30,000, Calcutta Mathematical Society the dues are $66; for those whose annual professional income is $30,000 or 0 more, the dues are $88. 0 Dansk Matematisk Forening For a joint family membership, one pays ordinary dues, based on his or her 0 Deutsche Mathematiker-Vereinigung e.V. income, and the other pays ordinary dues based on his or her income, less $20. 0 Edinburgh Mathematical Society (Only the member paying full dues will receive the Notices and the Bulletin as a 0 Gesellschatt tor Angewandte privilege of membership, but both members will be accorded all other privileges Mathematik und Mechanik of membership.) 0 Glasgow Mathematical Association Minimum dues for contributing members are $132. 0 Indian Mathematical Society For either students or unemployed individuals, dues are $22, and annual 0 Iranian Mathematical Society verification is required. 0 Irish Mathematical Society The annual dues for reciprocity members who reside outside the U.S. and lslenzka Staerdfraedafelagid Canada are $44. To be eligible for this classification, members must belong to 0 one of those foreign societies with which the AMS has established a reciprocity 0 Israel Mathematical Union agreement, and annual verification is required. Reciprocity members who 0 Korean Mathematical Society reside in the U.S. or Canada must pay ordinary member dues ($66 or $88). 0 London Mathematical Society The annual dues for external members, those who reside in developing 0 Malaysian Mathematical Society countries which do not have any mathematical society, are $47. 0 Mathematical Society of Japan Members can purchase a multi-year membership by prepaying their cur­ 0 Mathematical Society of the Philippines rent dues rate for either two, three, four or five years. This option is not available Mathematical Society of the Republic of China to either unemployed or student members. 0 0 New Zealand Mathematical Society 0 Nigerian Mathematical Society 1988 Dues Schedule 0 Norsk Matematisk Forening For any category of membership where no dues prices are given, the higher one 0 Osterreichische Mathematische Gesellschafl is to be paid by persons whose annual professional income is $30,000 or more. 0 Polskie Towarzystwo Matematyczne 0 Punjab Mathematical Socie!y 0 Real Sociedad Matematica Espanola Ordinary member ...... U$66CJ$88 0 Sociedad Colombiana de Matematica Joint family member (full rate). ()$66 () $88 0 Sociedad de Matematica de Chile Sociedad Matematica de Ia Joint family member (reduced rate). 0$46CJ$68 0 Republica Dominicana Contributing member (minimum $132). () 0 Sociedad Matematica Mexicana 0 Sociedade Brasileira Matematica Student member (please verify)' ...... \)$22 0 Sociedade Brasileira de Matematica Unemployed member (please verify)•. . U$22 Aplicada e Computacional 0 Sociedade Paranaense de Matematica Reciprocity member (please verify)3 . ( ) $44 () $66 () $88 0 Sociedade Portuguesa de Matematica External member. ()$47 0 Societal Catalana de CiEmcies Fisiques, Quimiques i Matematiques Multi-year membership. $. . .for. . years 0 Societe de Mathematiques Appliquees et lndustrielles 0 Societe Mathematique de Belgique 0 Societe Mathematique de France 1 Student Verification (sign below) 0 Societe Mathematique Suisse I am a full-time student at 0 Southeast Asian Mathematical Society 0 Suomen Matemaattinen Yhdistys currently working toward a degree. 0 Svenska Matematikersamfundet 0 Union Matematica Argentina 2 Unemployed Verification (sign below) I am currently unemployed and 0 Unione Matematica ltaliana actively seeking employment. My unemployment status is not a result of volun­ 0 Vijnana Parishad of India tary resignation or of retirement from my last position. 0 Wiskundig Genootschap

3 Reciprocity Membership Verification (sign below) I am currently a member of the society indicated on the right and am therefore eligible for reciprocity membership. Signature Order Form For VISA or MasterCard orders, send to: For orders with remittances, send to: American Mathematical Society American Mathematical Society P.O. Box 6248 Annex Station Providence. Rhode Island 02940 P.O. Box 1571 (401) 272-9500 Providence. Rhode Island 02901-1571 Ordered by:------Mail to (if different): ------

Use your peel-off Notices label.

QTY CODE AUTHOR and TITLE PRICE $

Shipping and Handling o Surface o Air

Total due (All orders must be prepaid in U.S. funds) $

Charge by phone in the continental U.S. 800-556-7774 o Check or Money Order OVISA o MasterCard

Card I NumberL--~----L---~---L--~----L---~---L--~L---~---L--~----L---~---L--~----L---~

Card Expiration Date Signature------

Shipping and Handling Prices are subject to change without notice. Books are sent via surface mail (UPS to U.S. addresses and printed matter elsewhere) unless air delivery is requested. The shipping and First Each handling charges for book orders are shown in the table. Journal back numbers, Mathematical Reviews indexes and review volumes Book Additional Maximum are sent via surface mail to any destination unless air delivery is re­ quested. Postage for surface mail is paid by the AMS. Air delivery Surface $2 $1 $ 25 rates. which will be quoted upon request, must be paid by the pur­ chaser. Software: Nonindividual customers need not prepay provided Air $5 $3 $100 a Purchase Order number is given with the order. Software/books are sent via UPS to U.S. addresses, first class mail to Canada, and air delivery elsewhere. Add shipping and handling· for Software/Books: $6 per order in the U.S. and Canada; $25 per order air delivery outside NOT I the U.S. and Canada. Customers in these areas should request price information from and direct their orders to the following distributors: Europe/Middle East/Africa: Barry Emms, Clarke Assoc.­ Europe Ltd .. Unit 2, Pool Rd. Trading Estate, West Molesey, Surrey KT8 OHE. England. Tel. 01-941-6966. Telex 298210 XOCEAN G; exclusive distributor of AMS books. Japan: Maruzen Co. Ltd., P.O. Box 5050, Tokyo Interna­ tional 100-31, Japan. Tel. Tokyo 272-7211, Telex J26516 India: Allied Publishers Pvt. Ltd., 15, J. N. Herdia Marg., Ballard Estate, Bombay 400038, India Staple here

Fold here

NO POSTAGE NECESSARY IF MAILED IN THE UNITED STATES BUSINESS REPLY MAIL FIRST-CLASS MAIL PERMIT NO. 5548 PROVIDENCE, RI

POSTAGE WILL BE PAID BY ADDRESSEE

AMERICAN MATHEMATICAL SOCIETY P. 0. Box 6248 Providence, RI 02940

111 ••••• 1.11.1 ••• 1•• 111 ••• 1.1 •• 1.1 ••• 1•• 1•• 11.11 ••• 1

Fold here

Fasten PAYMENT securely Change of Address

Members of the Society who move or who change When changing their addresses, members are positions are urged to notify the Providence Office urged to cooperate by supplying the information as soon as possible. requested below. The Society's records are of Journal mailing lists must be printed four to value only to the extent that they are current and six weeks before the issue date. Therefore, in accurate. order to avoid disruption of service, members are If your address has changed, or will change requested to provide the required notice well in within the next two or three months, please place advance. the peel-off label from the back cover (which con­ Besides mailing addresses for members, the So­ tains your name, member code, and address) in ciety's records contain information about mem­ the rectangle provided below (or on a copy of this bers' positions and their employers (for publica­ page), supply any other information appropriate tion in the Combined Membership List). In addi­ for the AMS records, and mail to the address tion, the AMS maintains records of members' hon­ given below. ors, awards, and information on Society service. Information of the latter kind appears regularly in Notices.

Place Notices label here

Change effective as of: ------­ New mailing address: ------

New position: ------

If mailing address is not that of your employer, please supply the following information:

New employer: ------­ Location of employer: ------:=--=--=-- City State/Province Country Zip Code

Recent honors and awards: ------

Personal items for publication in Notices: ------

Mail completed form to:

Membership Department, AMS, P.O. Box 6248, Providence, RI 02940 Staple here

Fold here

NO POSTAGE NECESSARY IF MAILED IN THE UNITED STATES BUSINESS REPLY MAIL FIRST-CLASS MAIL PERMIT NO. 5548 PROVIDENCE, RI

POSTAGE WILL BE PAID BY ADDRESSEE

AMERICAN MATHEMATICAL SOCIETY P. 0. Box 6248 Providence, RI 02940

111 ••••• 1.11.1 ••• 1•• 111 ••• 1.1 •• 1.1 ••• 1•• 1•• 11.11 ••• 1

Fold here

Fasten PAYMENT securely Problems in Distributions and Computational Acoustics Partial Differential Equations Proceedings of the 1st IMACS Symposium, Yale By C. Zuily University, New Haven, CT. USA, 6-8 August 1986 North-Holland Mathematics Studies, 143 Edited by D. Lee, R.L. Sternberg and M.H. Schultz Translation of Probtemes des Distributions avec Solutions Volume 1: Wave Propagation Detaittees (Hermann, Paris, 1978) Volume 2: Algorithms and Applications This is an introductory text to distributions theory and modern These two volumes, "Wave Propagation" and "Algorithms and partial differential equations. It is the only book available in Applications". present the results of research into computational English that includes solved problems on these subjects. acoustics, covering the areas of ocean acoustics, seismo­ Contents: acoustics, and aeroacoustics. 1. Preliminaries. Topics examined include mathematical models, numerical 2. Distributions. methods, significant developments in the application of effective 3. Differentiation in the Space of Distributions. methods to solve acoustics problems, solutions to acoustics 4. Convergence in the Space of Distributions. problems by supercomputers, interface acoustics, and other 5. Convolution of Distributions. important developments in related areas. 6. Fourier and Laplace Transforms of Distributions. 1988 xvi + 856 pages (in 2 volumes) 7. Applications. Bibliography. Index. Index of Notations. Volume 1: Wave Propagation ISBN 0-444-70349-7 1988 248 pages. Volume 2: Algorithms and Applications Price: US$ 79.00/Dfl. 150.00 ISBN 0-444-70350-0 ISBN 0-444-70248-2 Price per volume: US$ 97.25/Dfl. 185.00 Set Price: US$ 155.25/Dfl. 295.00 NOW IN PAPERBACK Set ISBN 0-444-70351-9 Handbook of Set-Theoretic Topology Mathematical Elasticity Edited by K. Kunen and J. Vaughan By P.G. Ciarlet " ... an indispensable reference for many years to come ... a Volume 1: Three-Dimensional Elasticity massive undertaking ... I would recommend that anyone Studies in Mathematics and its Applications, 20 interested in general topology take a took at this book .. .many This volume is a thorough introduction to contemporary set-theorists are going to want a copy of their own." research in elasticity, and may be used as a working textbook Stewart Baldwin, The Journal of Symbolic Logic at the graduate level for courses in pure or applied Now available in paperback, this Handbook is an introduction mathematics or in continuum mechanics. It provides a complete to set-theoretic topology for students in the field and for description, with emphasis on the nonlinear aspects, of the two researchers in other areas for whom results in set-theoretic competing mathematical models of three-dimensional elasticity, topology may be relevant. The aim of the editors has been to together with an extensive mathematical analysis of these make it as self-contained as possible without repeating material models. The book is as self-contained as possible. It includes which can easily be found in standard texts. The Handbook over 100 problems and a bibliography of 550 references. contains detailed proofs of core results, and references to the Contents: Description of Three-Dimensional Elasticity. literature for peripheral results where space was insufficient. Geometrical and other Preliminaries. The Equations of Included are many open problems of current interest. Equilibrium and the Principle of Virtual Work. Elastic Materials 1984. 1st repr. as a paperback 1988 and their Constitutive Equations. Hyperelasticity. The Boundary viii+ 127 4 pages Paperback Value Problems of Three-Dimensional Elasticity. Mathematical Price: US$ 73.75/Dfl. 140.00 Methods in Three-Dimensional Elasticity. Existence Theory ISBN 0-444-70431-0 based on the Implicit Function Theorem. Existence Theory Distributed in Japan by Maruzen Co., Ltd. based on the Minimization of the Energy. 1988 472 pages. Price: US$ 115. 75/Dfl. 220.00 Educational Computing in ISBN 0-444-70259-8 Mathematics: ECM/87 In preparation: Proceedings of the International Congress, Rome, Italy, Volume 2: Lower-Dimensional 4-6 June 1987 Theories of Plates and Rods Edited by T.F. Banchoff, I. Capuzzo Dolcetta, M. Dechamps, M. Emmer, H. Kocak and D.L. Salinger This volume examines the use of computers in mathematical education and research. A number of papers report on European and North American North-Holland projects where computer-aided teaching methods have been In the U.S.A. and Canada: Publishing Co. Inc. applied to mathematics education in various fields, including Elsevier Science P.O. Box 1663, Grand Central Station engineering and computer science. Other papers look at New York, NY 10163, U.S.A. situations where current mathematical research takes advantage of the facilities offered by a computer. In all other countries: Elsevier Science Publishers 1988 About 300 pages Book Order Department Price: US$ 79.00/Dfl. 150.00 P.O. Box 211 ISBN 0-444-70456-6 1000 AE Amsterdam, The Netherlands

US $ prices are valid only in the USA and Canada. In all other countries the Dutch Guilder (Oft.) price is definitive. Customers in The Netherlands, please add 6o/o B. T. W. In New York State applicable sates tax should be added. All prices are subject to change without prior notice. NH/407/BK/0976 Start the academic year off right­ with the latest mathematics titles from Springer-Verlag!

An Introduction to Algebraic Topology J.J. Rotman This book is a clear exposition, with exercises, of the basic ideas of algebraic topology: homology (singular, cellular and simplicial), homotopy groups, and cohomology rings. It is suitable for a two semester course at the beginning gradu­ ate level, requiring as prerequisites a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive general­ ity is avoided, and the author explains the geometric or an­ alytic origins of new abstract concepts as they are introduced, making this book of great value to the student. Chapter Table of Contents: Introduction; Some Basic Topological Notions; Simplexes; The Fundamental Group; Singular Homology; Long Exact Sequences; Excision and Applications; Simplicial Complexes; CW Complexes; Natural Transformations; Covering Spaces; Homotopy Groups; Cohomology; Bibliography; Index 1988/approx. 450 pp./92 illus./hardcover/$49.80 ISBN 0-387-96678-1 Graduate Texts in Mathematics, Vol. 119

Groups and Symmetry M.A. Armstrong This text provides a gentle introduction to the highlights of elementary group theory. Throughout the book emphasis is placed on concrete examples, many of them geometrical in nature, so that finite rotation groups and the seventeen Illustrations from Geometrical Methods in the Theory of Ordinary wallpaper groups are treated in detail alongside theoretical Differential Equations. results such as Lagrange's theorem, the Sylow theorems, and the classification theorem for finitely generated abelian groups. A novel feature at this level is a proof of the Niel­ sen-Schreier theorem, using group actions on trees. Con­ Projective Geometry tains over 300 exercises and 60 illustrations to help develop P. Samuel the student's intuition. The purpose of this book is to revive some of the beautiful 1988/app. 192 pp./52 illus.ihardcover/$34.00 results obtained by various geometers of the 19th century ISBN 0-387-96675-7 and to give its readers a taste of concrete algebraic geome­ Undergraduate Texts in Mathematics try. The fundamentals of projective geometry are efficiently dealt with by using a modest amount of linear algebra. Geometrical Methods in the Theory Specific subject headings include: • projective spaces • one­ of Differential Equations dimensional projective geometry • classification of conics 2nd Edition and quadrics • polarity with respect to a quadric. V.I. Arnold From the reviews: Since the first edition of this book, geometrical methods in 'This book of P. Samuel thus fills a gap in the literature. It the theory of ordinary differential equations has become is a little jewel ... he succeeds in 160 pages in giving a co­ very popular, partly due to the computer experiments and herent exposition of all of projective geometry ... one reads theorems that have been proved. Much of this progress is this book like a novel." represented in this revised, expanded edition, including D. Lazard, Gazette des Mathematiciens such topics as the Feigenbaum universality of period dou­ 1988/app. 168 pp./56 illus./softcover/$35.00 bling, the Zoladec solution, the lljashenko proof, the Ecalle ISBN 0-387-96752-4 and Voronin theory, the Varchenko and Hovanski theorems, Undergraduate Texts in Mathematics and the Neistadt theory. In the selection of material for this book, the author expounds on basic ideas and methods ap­ plicable to the study of differential equations. Special efforts were made to keep the basic ideas free from technical de­ ORDER INFORMATION: Write to: Springer-Verlag New tails, thus, the most fundamental questions are considered York, Inc. Attn: I. Cunningham, S955, 175 Fifth Avenue, New in great detail, while the exposition of the more special and York, NY 10010. Checks and Money Orders (plus $2.50 for difficult parts of the theory has been given the character of shipping) are acceptable forms of payment. NY, NJ and CA a survey. Consequently, the reader needs only a general residents, please add state sales tax. To order by credit mathematics knowledge to easily follow this text. It is direct­ card, call TOLL FREE 1-800-SPRINGER (in NJ, ed to mathematicians, as well as to all users of the theory 201-348-4033). of differential equations. 1988/351 pp./162 illus./hardcover/$48.00 ISBN 0-387-96649-8 Grund/ehren der mathematischen Wissenschaften, .£ ~~!"~~,r-~~~~9"'""' Vol. 250 ~ London Paris Tokyo