Notices of the American Mathematical

Total Page:16

File Type:pdf, Size:1020Kb

Notices of the American Mathematical Calendar This Calendar lists all of the meetings which have been approved by the Council up to the date this issue of the cNoliaiJ was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Abstracts should be submitted on special forms which are available in most departments of mathematics; forms can also be obtained by writing to the headquarters of the Society. Abstracts to be presented at the meeting in person must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline for the meeting. Meeting Deadline for Abstracts* Number Date Place and News Items 725 June 20-21, 1975 Pullman, Washington Apr. 29, 1975 726 August 18-22, 1975 Kalamazoo, Michigan June 17, 1975 (79th Summer Meeting) 727 October 25, 1975 Cambridge, Massachusetts Sept. 2, 1975 728 November 1, 1975 Chicago, illinois Sept. 2, 1975 729 November 7-8, 1975 Blacks burg, Virginia Sept. 23, 1975 730 November 15, 1975 Los Angeles, California Sept. 23, 1975 731 January 22-26, 1976 San Antonio, Texas Nov. 5, 1975 (82nd Annual Meeting) March 4-5, 1976 Tallahassee, Florida March 15-20, 1976 Urbana, illinois April 23-24, 1976 Reno, Nevada June 18-19, 1976 Portland, Oregon November 19-20, 1976 Columbia, South Carolina November 26-27, 1976 Albuquerque, New Mexico January 27-31, 1977 St. Louis, Missouri (83rd Annual Meeting) *Deadline for abstracts not presented at a meeting (by title) June 1975 issue: April 22 August 1975 issue: June 10 Please affix the peel-off label on these cJYoliai) to correspondence with the Society concerning fiscal matters, changes of address, promotions, or when placing orders for books and journals. ThecJYoliaiJof the American Mathematical Society is published by the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940, in January, February, April, June, August, October, November, and December. Subscription per annual volume is $10. Member subscription of $5 is included in annual dues. Price per copy $3. Special price for copies sold at registration desks of meetings of the Society, $1 per copy. Orders for subscriptions or back numbers (back issues of the last two years only are available) should be sent to the Society at P. 0. Box 1571, Annex Station, Providence, Rhode Island 02901. All orders must be accompanied by payment. Other correspondence should be addressed to P. 0. Box 6248, Providence, Rhode Island 02940. Second class postage paid at Providence, Rhode Island, and additional mailing offices. Copyright© 1975 by the American Mathematical Society Printed in the United States of America OF THE AMERICAN MATHEMATICAL SOCIETY Everett Pitcher and Gordon L. Walker, Editors Hans Samelson, Associate Editor CONTENTS MEETINGS Calendar of Meetings . • . • . • . • . Inside Front Cover Program for the April Meeting in St. Louis, Missouri . • . • • . 130 Abstracts for the Meeting: A-394-A-426 Program for the April Meeting in Monterey, California . • . • . • . • . • • . 139 Abstracts for the Meeting: A-426-A-431 PRELIMINARY ANNOUNCEMENTS OF MEETINGS . • • . • • . • • . • • . • • • • • • . • • . • 141 ORGANIZERS AND TOPICS OF SPECIAL SESSIONS . • . • . • . • . 151 INVITED SPEAKERS AT AMS MEETINGS . • . • . • . • . • . • . • . 151 NONACADEMIC EMPLOYMENT OF PH.D's IN THE MATHEMATICAL SCIENCES . • • . • • • • . 152 COMBINED MEMBERSHIP LIST 1975-1976 . • . • . • . • . • . • . 156 NEW AMS PUBLICATIONS . • . • . • . • . • . • . • . • • . • . 157 LETTERS TO THE EDITOR . • . • . • . • • . • . • . • . • . • 158 NEWS ITEMS AND ANNOUNCEMENTS ....•...••...•...•..•.••.....•.••.. 155, 159 SPECIAL MEETINGS INFORMATION CENTER. • • . • . • . • . • . • 160 SUMMER GRADUATE COURSES (Supplementary List) . • . • . • . • . • . • . • . • • • 161 QUERIES . • . • . • . • . • • . • . • • . • . • . • • . • • • . 162 PERSONAL ITEMS . • . • • . • . • • . • . • . • . • • . • . • • . • • • . 162 ABSTRACTS . • • • . • . • . • • . • . • . • . • . • • • . A-377 SITUATIONS WANTED A-432 RESERVATION FORM . • • . • • . • • • . • . • . • A-438 The Seven Hundred Twenty-Third Meeting University of Missouri St. Louis, Missouri April11-12, 1975 The seven hundred twenty-third meeting of Charles P. Lanski, Donald S. Passman, Gary L. the American Mathematical Society will be held Peterson, Richard E. Phillips, Derek J. S. at the University of Missouri, st. Louis, Missouri, Robinson, Martha K. Smith, Keoneth W. Weston, on Friday and Saturday, April 11 and 12, 1975. Julian s. Williams, and Cleon R. Yohe. Profes­ The University of Missouri at st. Louis is lo­ sor Richard P. Jerrard of the University of illi­ cated about ten miles northwest of downtown st. nois at Urbana-Champaign has arranged a spe­ Louis and about four miles east of st. Louis Lam­ cial session on Geometric Topology, to be held bert Airport. All sessions will be held in the Friday morning; the speakers will be Joan S. J. C. Penney Building of the university. Birman, John E. Connett, Robert J. Daverman, By invitation of the Committee to Select Mary-Elizabeth Hamstrom, and Jan Jaworowski. Hour Speakers for Western Sectional Meetings, Professor Rangachary Kannan of the University there will be four one-hour addresses. Profes­ of Missouri at st. Louis and Michigan state Uni­ sor A. 0. L. Atkin of the University of illinois versity has arranged a special session on Non­ at Chicago Circle will speak on Friday, April11, linear Functional Analysis, to be held Saturday; at 11:00 a.m. ; the title of his talk will be "Super­ the speakers will be Lamberto Cesari, Michael singular games". Professor Kuo-Tsai Chen of G. Crandall, Jack K. Hale, Roger D. Nussbaum, the University of illinois at Urbana-Champaign Paul H. Rabinowitz, Duane P. Sather, Luc Tar­ will address the Society on Friday, April11, at tar, and Hans F. Weinberger. Professor Walter 1:45 p.m.; his subject will be "Iterated path in­ Leighton of the University of Missouri at Colum­ tegrals". Professor Kenneth Kunen of the Uni­ bia has arranged a special session on Ordinary versity of Wisconsin at Madison will speak on Differential Equations: Oscillation Theory, Saturday, April 12, at 11:00 a.m.; the title of Boundary Value Problems, to be held all day his talk will be "What good are ultrafilters ? " Friday and Saturday morning; the speakers will Professor Guido L. Weiss of Washington Univer­ be Sui-Sun Cheng, William J. Coles, Arlington sity at st. Louis will address the Society on M. Fink, Louis J. Grimm, Heinrich W. Guggen­ Saturday, April12, at 1:45 p.m.; his topic will heimer, Don B. Hinton, Lloyd K. Jackson, Mar­ be "The use of Hardy spaces and their generali­ vin S. Keener, Kurt Kreith, Alan C. Lazer, zations in harmonic analysis". All four hour William T. Reid, Jerry R. Ridenhour, Curtis C. talks will be held in the auditorium of the J. C. Travis, and W. Roy Utz, Jr. Professor Marian Penney Building. Boykan Pour-El of the University of Minnesota By invitation of the same committee there has arranged a special session on Recursion will be nine sessions of selected twenty-minute Theory, to be held Saturday; the speakers will be papers. Professor David Drasin of Purdue Uni­ Harvey Friedman, Carl G. Jockusch, Jr., versity has arranged a special session on Classi­ Manuel Lerman, Thomas G. McLaughlin, Anil cal Function Theory, to be held Friday afternoon Nerode, Hilary Putnam, Wayne H. Richter, and Saturday morning; the speakers will be Al­ Gerald E. Sacks, and Robert I. Soare. Profes­ bert Baernstein IT, Burgess J. Davis, Frederick sor Grant V. Weiland of the University of Mis­ W. Gehring, Simon Hellerstein, James A. Jen­ souri at St. Louis has arranged a special session kins, John L. Lewis, Richard H. Rochberg, on Harmonic Analysis and Related Topics, to be Donald C. Rung, Glenn E. Schober, Ted J. Suf­ held Friday; the speakers will be David R. Adams, fridge, and Allen W. Weitsman. Professor Richard J. Bagby, Richard A. Hunt, Alexander J. David L. Elliott of Washington University has ar­ Nagel, Victor L. Shapiro, Elias M. stein, ranged a special session on Differential Geomet­ Mitchell H. Taibleson, Alberto Torchinsky, S. ric Problems in Control Theory, to be held all Vagi, Wo-Sang Young, and William P. Ziemer. day Friday and Saturday morning; the speakers Professor David J. Winter of the University of will be John B. Baillieul, William M. Boothby, Michigan has arranged a special session on Roger w. Brockett, Jan M. Gronski, Robert Finite Dimensional Field Extensions, to be held Hermann, Henry G. Hermes, Ronald M. Hir­ Friday; the speakers will be stephen u. Chase, schorn, Velimir Jurdjevic, Arthur J. Krener, Lindsay N. Childs, James K. Deveney, Ray­ Deborah Rebhuhn, Jackson L. Sedwick, Jr., mond T. Hoobler, Herbert F. Kreimer, Jr., M. B.Suryanarayana, and Hector J. Sussman. Andy R. Magid, Moss E. Sweedler, and David J. Professor Franklin Haimo of Washington Univer­ Winter. There will also be five sessions of con­ sity has arranged a special session on Applica­ tributed ten-minute papers. tions of Ring Theory to Groups, to be held both On Thursday, April10, the day before the Friday and Saturday; the speakers will be Paul F. meeting itself, the University of Missouri at st. Conrad, Vance Faber, Burton I. Fein, Charles E. Louis and Washington University will sponsor a Ford, Brian Hartley, Jutta Hausen, Israel N. brief symposium on Harmonic Analysis and Re­ Herstein, Arun V. Jategaonkar, Everett L. Lady, lated Topics, which will supplement Professor 130 Weiland's special session. This symposium will Single $16 be held in the Auditorium of the J. C. Penney Twin/Double $22 Building according to the following schedule: Triple $28 9:30 a.m. , Ronald R. Coifman, "On multilinear Requests for room reservations at the Eight Days singular integrals" (Abstract 723-Bl8); 11:00 Inn should be made directly with the Inn using the a.m. , Charles L. Fefferman, "On several com­ room reservation form which appeared on the plex variables"; 1:30 p.m., Nestor M. Riviere, last page of the January c}(otit:tLJ.
Recommended publications
  • “What Happened to the Post-War Dream?”: Nostalgia, Trauma, and Affect in British Rock of the 1960S and 1970S by Kathryn B. C
    “What Happened to the Post-War Dream?”: Nostalgia, Trauma, and Affect in British Rock of the 1960s and 1970s by Kathryn B. Cox A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Music Musicology: History) in the University of Michigan 2018 Doctoral Committee: Professor Charles Hiroshi Garrett, Chair Professor James M. Borders Professor Walter T. Everett Professor Jane Fair Fulcher Associate Professor Kali A. K. Israel Kathryn B. Cox [email protected] ORCID iD: 0000-0002-6359-1835 © Kathryn B. Cox 2018 DEDICATION For Charles and Bené S. Cox, whose unwavering faith in me has always shone through, even in the hardest times. The world is a better place because you both are in it. And for Laura Ingram Ellis: as much as I wanted this dissertation to spring forth from my head fully formed, like Athena from Zeus’s forehead, it did not happen that way. It happened one sentence at a time, some more excruciatingly wrought than others, and you were there for every single sentence. So these sentences I have written especially for you, Laura, with my deepest and most profound gratitude. ii ACKNOWLEDGMENTS Although it sometimes felt like a solitary process, I wrote this dissertation with the help and support of several different people, all of whom I deeply appreciate. First and foremost on this list is Prof. Charles Hiroshi Garrett, whom I learned so much from and whose patience and wisdom helped shape this project. I am very grateful to committee members Prof. James Borders, Prof. Walter Everett, Prof.
    [Show full text]
  • Report on the 1975 Survey of Users of the Services of Radio Stations Wwv and Wwvh
    'JM ^^ t*.;: .,-.;, .'-ti ^^#' • J* .^: '•^i'-^v'-' '- \ • REFERENCE N B S V U) NBS TECHNICAL NOTE 674 ^''fff AU O* * U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards Report On The 1975 Survey of Users of the Services of Radio Stations WWY and WWYH /OO ' .6/5753 . no,(o7i . /975, NATIONAL BUREAU OF STANDARDS The National Bureau of Standards' was established by an act of Congress March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau consists of the Institute for Basic Standards, the Institute for Materials Research, the Institute for Applied Technology, the Institute for Computer Sciences and Technology, and the Office for Information Programs. THE INSTITUTE FOR BASIC STANDARDS provides the central basis within the United States of a complete and consistent system of physical measurement; coordinates that system with measurement systems of other nations; and furnishes essential services leading to accurate and uniform physical measurements throughout the Nation's scientific community, industry, and commerce. The Institute consists of the Office of Measurement Services, the Office of Radiation Measurement and the following Center and divisions: Applied Mathematics — Electricity — Mechanics — Heat — Optical Physics — Center for Radiation Research: Nuclear Sciences; Applied Radiation — Laboratory Astrophysics ° — Cryogenics" — Electromagnetics" — Time and Frequency".
    [Show full text]
  • Record Store Day 2015 Releases Price Record
    Record Store Day 2015 Releases Price Record Store Day 2015 Releases Price Record Store Day 2015 Releases Price Record Store Day 2015 Releases Price 7" SINGLES 7" J Dilla : Fuck The Police - Badge Shaped 7" Edition29.99 7" Soko : Ocean Of Tears 6.99 10" The Replacements : Alex Chilton 12.99 7" Adam And The Ants : Kings Of The Wild Frontier / Ant8.99 Music7" Jagaara : In The Dark 6.49 7" The Spaceape Feat Kode9 & The Bug : Ghost Town8.99 / At War10" With Time Roxy- Gold Music Vinyl :Edition Ladytron 14.99 7" Ryan Adams : Come Pick Me Up 10.99 7" Bert Jansch : Needle Of Death EP 9.99 7" Dusty Springfield : What's It Gonna Be / Spooky 11.99 10" Tracey Thorn : Songs From The Falling 7.99 7" A-ha : Take On Me 13.99 7" Jay-Z / Ghostface Killah : U Don't Know / Whip You11.99 With A7" Strap Stealing Sheep / The Voyeurs : Split 7" 7.99 10" The Waterboys : Puck's Blues 8.99 7" Air : Playground Love 12.99 7" Sharon Jones & The Dap-Kings : Little Boys With Shiny5.49 Toys7" The Subways : Taking All The Blame 7.49 10" Young Knives : Something Awful 12.99 7" A$ap Rocky : LPFJ2 / Multiply 7.99 7" Tom Jones : Chills & Fever / Breathless 11.99 7" Supergrass : Sofa (Of My Lethargy) 11.99 10" Various : An On-U Sound Journey Through Time & 8.99Space 7" B-Movie : They Forget / Trash & Mystery 6.49 7" Joy Division : Love Will Tear Us Apart / Leaders Of14.99 Men 7" Temples / Fever The Ghost : Split 7" 7.99 12" SINGLES 7" Syd Barrett / R.E.M.
    [Show full text]
  • All Around the World the Global Opportunity for British Music
    1 all around around the world all ALL British Music for Global Opportunity The AROUND THE WORLD CONTENTS Foreword by Geoff Taylor 4 Future Trade Agreements: What the British Music Industry Needs The global opportunity for British music 6 Tariffs and Free Movement of Services and Goods 32 Ease of Movement for Musicians and Crews 33 Protection of Intellectual Property 34 How the BPI Supports Exports Enforcement of Copyright Infringement 34 Why Copyright Matters 35 Music Export Growth Scheme 12 BPI Trade Missions 17 British Music Exports: A Worldwide Summary The global music landscape Europe 40 British Music & Global Growth 20 North America 46 Increasing Global Competition 22 Asia 48 British Music Exports 23 South/Central America 52 Record Companies Fuel this Global Success 24 Australasia 54 The Story of Breaking an Artist Globally 28 the future outlook for british music 56 4 5 all around around the world all around the world all The Global Opportunity for British Music for Global Opportunity The BRITISH MUSIC IS GLOBAL, British Music for Global Opportunity The AND SO IS ITS FUTURE FOREWORD BY GEOFF TAYLOR From the British ‘invasion’ of the US in the Sixties to the The global strength of North American music is more recent phenomenal international success of Adele, enhanced by its large population size. With younger Lewis Capaldi and Ed Sheeran, the UK has an almost music fans using streaming platforms as their unrivalled heritage in producing truly global recording THE GLOBAL TOP-SELLING ARTIST principal means of music discovery, the importance stars. We are the world’s leading exporter of music after of algorithmically-programmed playlists on streaming the US – and one of the few net exporters of music in ALBUM HAS COME FROM A BRITISH platforms is growing.
    [Show full text]
  • Notices of the American Mathematical Society
    June 18 and 19)- Page 341 Vl 0 ~ Mathematical Society Calendar of AMS Meetings THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of the Notices was sent to press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the Ameri· can Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and second announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meet­ ing. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society in Providence. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for ab· stracts submitted for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcement and the list of organizers of special sessions. MEETING ABSTRACT NUMBER DATE PLACE DEADLINE
    [Show full text]
  • Generic Global Rigidity in Complex and Pseudo-Euclidean Spaces 3
    GENERIC GLOBAL RIGIDITY IN COMPLEX AND PSEUDO-EUCLIDEAN SPACES STEVEN J. GORTLER AND DYLAN P. THURSTON Abstract. In this paper we study the property of generic global rigidity for frameworks of graphs embedded in d-dimensional complex space and in a d-dimensional pseudo- Euclidean space (Rd with a metric of indefinite signature). We show that a graph is generically globally rigid in Euclidean space iff it is generically globally rigid in a complex or pseudo-Euclidean space. We also establish that global rigidity is always a generic property of a graph in complex space, and give a sufficient condition for it to be a generic property in a pseudo-Euclidean space. Extensions to hyperbolic space are also discussed. 1. Introduction The property of generic global rigidity of a graph in d-dimensional Euclidean space has recently been fully characterized [4, 7]. It is quite natural to study this property in other spaces as well. For example, recent work of Owen and Jackson [8] has studied the number of equivalent realizations of frameworks in C2. In this paper we study the property of generic global rigidity of graphs embedded in Cd as well as graphs embedded in a pseudo Euclidean space (Rd equipped with an indefinite metric signature). We show that a graph Γ is generically globally rigid (GGR) in d-dimensional Euclidean space iff Γ is GGR in d-dimensional complex space. Moreover, for any metric signature, s, We show that a graph Γ is GGR in d-dimensional Euclidean space iff Γ is GGR in d- dimensional real space under the signature s.
    [Show full text]
  • A Generic Property of the Bounded Syzygy Solutions Author(S): Florin N
    A Generic Property of the Bounded Syzygy Solutions Author(s): Florin N. Diacu Source: Proceedings of the American Mathematical Society, Vol. 116, No. 3 (Nov., 1992), pp. 809-812 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2159450 Accessed: 16/11/2009 21:52 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the American Mathematical Society. http://www.jstor.org PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 116, Number 3, November 1992 A GENERIC PROPERTY OF THE BOUNDED SYZYGY SOLUTIONS FLORIN N.
    [Show full text]
  • Newsletter No. 31 March 2004
    Newsletter No. 31 March 2004 Welcome to the second newsletter of 2004. We have lots of goodies for you this month so let's get under way. Maths is commonly said to be useful. The variety of its uses is wide but how many times as teachers have we heard students exclaim, "What use will this be when I leave school?" I guess it's all a matter of perspective. A teacher might say mathematics is useful because it provides him/her with a livelihood. A scientist would probably say it's the language of science and an engineer might use it for calculations necessary to build bridges. What about the rest of us? A number of surveys have shown that the majority of us only need to handle whole numbers in counting, simple addition and subtraction and decimals as they relate to money and domestic measurement. We are adept in avoiding arithmetic - calculators in their various forms can handle that. We prefer to accept so-called ball-park figures rather than make useful estimates in day-to-day dealings and computer software combined with trial-and-error takes care of any design skills we might need. At the same time we know that in our technological world numeracy and computer literacy are vital. Research mathematicians can push boundaries into the esoteric, some of it will be found useful, but we can't leave mathematical expertise to a smaller and smaller proportion of the population, no matter how much our students complain. Approaching mathematics through problem solving - real and abstract - is the philosophy of the nzmaths website.
    [Show full text]
  • Lids-R-904 Robust Stability of Linear Systems
    LIDS-R-904 ROBUST STABILITY OF LINEAR SYSTEMS - SOME COMPUTATIONAL CONSIDERATIONS* by Alan J. Laub** 1. INTRODUCTION In this paper we shall concentrate on some of the computational issues which arise in studying the robust stability of linear systems. Insofar as possible, we shall use notation consistent with Stein's paper [11 and we shall make frequent reference to that work. As we saw in [1] a basic stability question for a linear time-invariant system with transfer matrix G(s) is the following: given that a nominal closed-loop feedbadk system is stable, does the feedback system remain stable when subjected to perturbations and how large can those perturba- tions be? It turned out, through invocation of the Nyquist Criterion, that the size of the allowable perturbations was related to the "nearness to singularity" of the return difference matrix I + G(jW). Closed-loop stability was said to be "robust" if G could tolerate considerable perturbation before I + G became singular. Invited Paper presented at the Second Annual Workshop on the Informa- tion Linkage between Applied Mathematics and Industry held at Naval Postgraduate School, Monterey, California, Feb. 20-24, 1979; this research was partially supported by NASA under grant NGL-22-009-124 and the Department of Energy under grant ET-78-(01-3395). ** Laboratory for Information and Decision Systems, Pm. 35-331, M.I.T., Cambridge, MA 02139. -2- We shall now indulge in a modicum of abstraction and attempt to formalize the notion of robustness. The definition will employ some jargon from algebraic geometry and will be applicable to a variety of situations.
    [Show full text]
  • Generic Flatness for Strongly Noetherian Algebras
    Journal of Algebra 221, 579–610 (1999) Article ID jabr.1999.7997, available online at http://www.idealibrary.com on Generic Flatness for Strongly Noetherian Algebras M. Artin* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 View metadata, citation and similar papersE-mail: at core.ac.uk [email protected] brought to you by CORE provided by Elsevier - Publisher Connector L. W. Small Department of Mathematics, University of California at San Diego, La Jolla, California 92093 E-mail: [email protected] and J. J. Zhang Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195 E-mail: [email protected] Communicated by J. T. Stafford Received November 18, 1999 Key Words: closed set; critically dense; generically flat; graded ring; Nullstellen- satz; strongly noetherian; universally noetherian. © 1999 Academic Press CONTENTS 0. Introduction. 1. Blowing up a critically dense sequence. 2. A characterization of closed sets. 3. Generic flatness. 4. Strongly noetherian and universally noetherian algebras. 5. A partial converse. *All three authors were supported in part by the NSF and the third author was also supported by a Sloan Research Fellowship. 579 0021-8693/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved. 580 artin, small, and zhang 0. INTRODUCTION In general, A will denote a right noetherian associative algebra over a commutative noetherian ring R.LetR0 be a commutative R-algebra. If R0 is finitely generated over R, then a version of the Hilbert basis theorem 0 asserts that A ⊗R R is right noetherian. We call an algebra A strongly right 0 0 noetherian if A ⊗R R is right noetherian whenever R is noetherian.
    [Show full text]
  • Multiplicatively Perfect and Related Numbers
    Journal of Rajasthan Academy of Physical Sciences ISSN : 0972-6306; URL : http://raops.org.in Vol.15, No.4, December, 2016, 345-350 MULTIPLICATIVELY PERFECT AND RELATED NUMBERS Shikha Yadav* and Surendra Yadav** *Department of Mathematics and Astronomy, University of Lucknow, Lucknow- 226007, U.P., India E-mail: [email protected] **Department of Applied Mathematics, Babasaheb Bhimrao Ambedkar University, Lucknow-226025, U.P., India, E-mail: ssyp_ [email protected] Abstract: Sandor [4] has discussed multiplicatively perfect (푀perfect) and multiplicatively +perfect numbers (called as 푀 +perfect). In this paper, we have discussed a new perfect number called as multiplicatively – perfect numbers (also called as 푀 − perfect). Further we study about Abundant, Deficient, Harmonic and Unitary analogue of harmonic numbers. Keywords and Phrases: Multiplicatively perfect numbers, Abundant numbers, Deficient numbers, Harmonic numbers and unitary divisor. 2010 Mathematics Subject Classification: 11A25. 1. Introduction Sandor and Egri [3], Sandor [4] have defined multiplicatively perfect and multiplicatively + perfect numbers as follows, A positive integer 푛 ≥ 1 is called multiplicatively perfect if 푅 푛 = 푛2, (1) where, 푅 푛 is product of divisor function. Let 푑1, 푑2, 푑3, … 푑푟 are the divisors of 푛 then 푅 푛 = 푑1. 푑2. 푑3. … 푑푟 . 푑(푛 ) Also, 푅 푛 = 푛 2 (2) Let 푅+(푛) denotes the product of even divisors of 푛. We say that 푛 is 푀 + perfect number if 2 푅+ 푛 = 푛 . (3) Now, in this paper we define 푀 − perfect numbers as follows, let 푅− 푛 denotes the product of odd divisors of 푛 .We say that 푛 is 푀 − perfect if 346 Shikha Yadav and Surendra Yadav 2 푅− 푛 = 푛 .
    [Show full text]
  • The Humble Sum of Remainders Function
    DRAFT VOL. 78, NO. 4, OCTOBER 2005 1 The Humble Sum of Remainders Function Michael Z. Spivey Samford University Birmingham, Alabama 35229 [email protected] The sum of divisors function is one of the fundamental functions in elementary number theory. In this note, we shine a little light on one of its lesser-known rela- tives, the sum of remainders function. We do this by illustrating how straightforward variations of the sum of remainders can 1) provide an alternative characterization for perfect numbers, and 2) help provide a formula for sums of powers of the first n posi- tive integers. Finally, we give a brief discussion of perhaps why the sum of remainders function, despite its usefulness, is less well known than the sum of divisors function. Some notation is in order. The standard notation for the sum of divisors function is σ(n): X σ(n) = d: djn We denote the sum of remainders function by ½(n), namely, Xn ½(n) = n mod d: d=1 Sums of remainders and perfect numbers A perfect number is a number equal to the sum of its positive divisors excluding itself. Another way to express this is that perfect numbers are those such that σ(n) = 2n. The three smallest perfect numbers are 6; 28, and 496. Euclid proved that every number of the form 2p¡1(2p ¡ 1); where p and 2p ¡ 1 are prime, is perfect. Euler proved the converse, that numbers of this form are the only even perfect numbers [3, p. 59]. One of the famous unsolved problems in number theory is whether or not there are any odd perfect numbers.
    [Show full text]