DOCSLIB.ORG

Ph. D. Programs for the Past Departments Or Schools Would Drop Colleagues Twenty Years and Are Not Typical

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ph. D. Programs for the Past Departments Or Schools Would Drop Colleagues Twenty Years and Are Not Typical ---------------------------------------------------------~hlIMPORTANT ADVISORY FOR GRADUATE STUDENTS IN MATHEMATICS In the past several years the jolJ prospects for ,1 recent recipient of a Ph D. degree in the mathematical sciences have changed from excellent to poor. All indications are that this rapid deterioration will accelerate over the next few years, and that the decade 1974-19H5 will be exceedingly bleak. The report of the chairman of the AMS Committee on Employment and Educational Policy should be read by everyone who is plfrnning to enter a career in mathematics in order to become acquainted with the facts concerning present and future employment. The first part of the report appeared in the November l!J7:l issue of these c\~/m•JJ, pages 34ti-352, and the second part appears in this issue, pages :lfi7-:l72. The action taken in response to these facts must, of course, be determined on an individual basis. Many will feel that for them mathematics is a calling and a way of life that must be followed at whatever cost and sacrifice. They will be undeterred by the prospect of difficulties and, hopefully, will be successful in finding suitable career opportunities. The Americ<m Mathematical Society, whose primary purpose is the fostering of mathematical research, will, of course, make every effort to ameliorate con­ ditions during the difficult period ahead. But it must be realized that the problem has its roots in complex social conditions which the Society has only the most limited powers to control or affect. Hichani S. Palais, Member AMS Committee on Employment and Educational Policv OF THE AMERICAN MATHEMATICAL SOCIETY Everett Pitcher and Gordon L. Walker, Editors WendeD H. Fleming, Associate Editor CONTENTS JOBS AND PH.D.'S IN TilE MATIIEMATICAL SCIENCES, II. • • •• • • • • • • • • • 367 IS TillS DOCTORAL PROGRAM NECESSARY? • • • • •• •• • • • • • • • • • • • • • • • • • • • 371 ASSISTANTSHIPS AND FELLOWSHIPS IN MATHEMATICS IN 1974-1975 I. FOR GRADUATE STUDY AT UNIVERSITIES.................. 373 IT. STIPENDS FOR STUDY AND TRAVEL........................ 437 Graduate Support ...••••••••••••. ~~ ••••• 437 Postdoctoral Support ••••••••••••••••••• 439 Travel and Study Abroad • • • • • • • • • • • • • • • • 444 Study in U.s. for Foreign Nationals • • • • • • 447 Sources of Fellowship Information. • • • • • • • 447 TAX STATUS OF GRANTS • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 449 INDEX OF ABSTRACTS, Volume 20 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • A-693 INDEX, Volume 20 ............................ a • • • • • • • • • • • • • • • • • • • • • A-725 FOREWORD The fifteenth Special Issue of these c}/ofiai) contains lists of assistantships and fellowships available in mathe­ matics and related sciences for the academic year 1974-1975 for graduate study at universities and stipends available for travel and study, Indexes for·abstracts and articles published in these c}/otiJxi) during 1973, a short article on the tax status of grants and two special articles are also included. The list of assistantships and fellowships at universities includes 449 departments of mathematics, applied math­ ematics, statistics, computer science, and related mathematical disciplines; these represent 329 colleges and univer­ sities. Institutions in the United States are listed alphabetically by state, followed by Canadian institutions. Abbreviations used in the section on graduate study Under the DEGREES AWARDED column the following terms have been used: Bachelor' s by inst. Number of bachelor• s degrees awarded J:?y the institution Bachelor• s by dept. Number of degrees awarded by the department Master• s by dept, Number of master• s degrees awarded by the department Ph. D. (1970-1973 incl.) Doctoral degrees awarded during last three years (1970-1971, 1971-1972, and 1972-1973) A&NT Algebra and Number Theory G&T Geometry and Topology L Logic A& FA Analysis and Functional Analysis P&S Probability and Statistics cs Computer Science and Numerical Analysis AM Applied Mathematics The .cJVofiai) of 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. Price per annual volume is $10. Price per copy $3. Special price for copies sold at registration desks of meetings of the Society, $1 per copy. Subscriptions, orders for back numbers (back issues of the last two years only are available), and inquiries should be addressed to the American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940. Second class postage paid at Providence, Rhode Island, and additional mailing offices. Copyright© 1973 by the Americsn Mathematical Society Prioted in the United States of America Jobs and Ph.D.'s in the Mathematical Sciences, II. By R. D. Anderson Part I of this article appeared in these cJioticeiJ, November 1973, pp. 348-352. 3. JOB PROSPECTS FOR THE NEXT (2) The number of students enrolling in four­ TWENTY YEARS year colleges and universities for the first time has dropped slightly for the past two years, The academic employment prospects for the showing a decrease in the percentage of the age next twenty years are not good. There is now group going on to four-year institutions, revers­ every reason to believe that the total national ing a trend toward greater percentages that had mathematics faculty will increase very little in been observed since 1945. With expected college the rest of the 1970s and will actually decrease enrollments indicating that the percentage will somewhat in the 1980s. The prospects are dom­ probably drop again this fall, an era of relative inated by four factors: (1) the annual size of the stability in the percentage of eighteen-year olds group of eighteen-year olds; (2) the percentage of enrolling in four-year colleges and universities eighteen-year olds enrolling each year for the has presumably been reached, implying that the first time in four-year colleges and universities; annual number of freshmen is likely to be closely (3) the percentage of enrollments in mathematics tied to the size of this age group. courses by freshmen and sophomores; (4) the The percentage, however, will probably go amount of money available for higher education. up slightly in the 1980s, dampening the effect of It should be pointed out that twenty-year projec­ the decreases in the stze of the age group on col­ tions are necessarily subject to considerable un­ lege enrollments. Society probably has relatively certainty. In particular, public priorities could stable numerical needs for college graduates change in favor of substantially more support for over a period of years, and in the latter 1980s, traditional higher education, although the author when the number of twenty-one year olds is de­ knows of no indication of that now. creasing, society can be expected to need a (1) The following table gives the annual number greater percentage of this age group for jobs re­ of live births in the United States from 1950 to quiring college training. Thus, the economic de­ date. These data were obtained from Statistical mand probably has an inverse effect on the re­ Abstract of the United States and the Monthly lationship of the size of the age group and the Vital Statistical Report. percentage of the age group going to college. Perhaps with a large number of eighteen-year Live Births Live Births olds, we are seeing that effect now. (in thousands) Year Year (in thousands) (3) CBMS Survey data and AMS data do not give 1950 3632 1961 4268 explicit percentages of the enrollment of fresh­ men and sophomores in mathematics courses, 3823 1951 1962 4167 but the data do show that approximately 50% of 1952 3913 1963 4098 all undergraduate enrollments in mathematics are in courses below the calculus level, 25% 3965 1953 1964 4027 more are in analytic geometry-calculus courses, 1954 4078 1965 3760 and, in addition, more than half the statistics and computer science course enrollments are in 1955 4104 1966 3606 pre-calculus or introductory courses. The grad­ 1956 4163 1967 3521 uate course enrollments are only about 4% or 5% of the total. Since employment of mathemati­ 1957 4255 1968 3502 cians in academia is tied primarily to course 1958 4204 1969 3571* enrollments, and since 75% of these enrollments are by freshmen and sophomores, we can make 1959 4245 1970 3718* apparently reliable projections concerning the 1960 4258 1971 3559* number of faculty members needed by con­ sidering items (1) and (2) above. In this re­ 1972 3256* spect, academic employment prospects for Ph. D. mathematicians are worse than those for The size of the group of eighteen-year olds is some other potential academicians. The three essentially determined through 1990: It increases potential "growth" areas in higher education: at about 2% a year until 1975, is almost stable two-year colleges, adult or continuing education, until 1979, and then decreases almost 25% until and career or vocational education involve rela­ 1990 .. The figures for the first four months of tively few employment opportunities for Ph. D. 1973 show a further drop of forty-four thousand mathematicians. There is now much evidence from 1972 levels but indicate a probable stabi­ that two-year colleges do not and will not need lizing trend. or hire enough Ph. D. 's to affect the overall job *Preliminary or provisional from 1969 on. 367 market, adult education programs are likely to from tuition.
Load more

Recommended publications
  • Ekeland Variational Principles with Set-Valued Objective Functions And
    Ekeland variational principles with set-valued objective functions and set-valued perturbations1 QIU JingHui School of Mathematical Sciences, Soochow University, Suzhou 215006, China Email: [email protected] Abstract In the setting of real vector spaces, we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective func- tion is a set-valued map taking values in a real vector space quasi-ordered by a convex cone K and the perturbation consists of a K-convex subset H of the or- dering cone K multiplied by the distance function. Here, the assumption on lower boundedness of the objective function is taken to be the weakest kind. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particu- lar, we give several EVPs for approximately efficient solutions in set-valued opti- mization, where a usual assumption for K-boundedness (by scalarization) of the objective function’s range is removed. Moreover, still under the weakest lower boundedness condition, we present a set-valued EVP, where the objective function is a set-valued map taking values in a quasi-ordered topological vector space and the perturbation consists of a σ-convex subset of the ordering cone multiplied by the distance function. arXiv:1708.05126v1 [math.FA] 17 Aug 2017 Keywords nonconvex separation functional, vector closure, Ekeland variational principle, coradiant set, (C, ǫ)-efficient solution, σ-convex set MSC(2010) 46A03, 49J53, 58E30, 65K10, 90C48 1 Introduction Since the variational principle of Ekeland [12, 13] for approximate solutions of nonconvex minimization problems appeared in 1972, there have been various 1This work was supported by the National Natural Science Foundation of China (Grant Nos.
    [Show full text]
  • Techniques of Variational Analysis
    J. M. Borwein and Q. J. Zhu Techniques of Variational Analysis An Introduction October 8, 2004 Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo To Tova, Naomi, Rachel and Judith. To Charles and Lilly. And in fond and respectful memory of Simon Fitzpatrick (1953-2004). Preface Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse ¯elds. The discovery of modern variational principles and nonsmooth analysis further expand the range of applications of these techniques. The motivation to write this book came from a desire to share our pleasure in applying such variational techniques and promoting these powerful tools. Potential readers of this book will be researchers and graduate students who might bene¯t from using variational methods. The only broad prerequisite we anticipate is a working knowledge of un- dergraduate analysis and of the basic principles of functional analysis (e.g., those encountered in a typical introductory functional analysis course). We hope to attract researchers from diverse areas { who may fruitfully use varia- tional techniques { by providing them with a relatively systematical account of the principles of variational analysis. We also hope to give further insight to graduate students whose research already concentrates on variational analysis. Keeping these two di®erent reader groups in mind we arrange the material into relatively independent blocks. We discuss various forms of variational princi- ples early in Chapter 2. We then discuss applications of variational techniques in di®erent areas in Chapters 3{7.
    [Show full text]
  • On the Convexity of Optimal Control Problems Involving Non-Linear Pdes Or Vis and Applications to Nash Games
    Weierstraß-Institut für Angewandte Analysis und Stochastik Leibniz-Institut im Forschungsverbund Berlin e. V. Preprint ISSN 2198-5855 On the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games Michael Hintermüller1,2, Steven-Marian Stengl1,2 submitted: September 11, 2020 1 Weierstrass Institute 2 Humboldt-Universität zu Berlin Mohrenstr. 39 Unter den Linden 6 10117 Berlin 10099 Berlin Germany Germany E-Mail: [email protected] E-Mail: [email protected] [email protected] [email protected] No. 2759 Berlin 2020 2010 Mathematics Subject Classification. 06B99, 49K21, 47H04, 49J40. Key words and phrases. PDE-constrained optimization, K-Convexity, set-valued analysis, subdifferential, semilinear elliptic PDEs, variational inequality. This research was supported by the DFG under the grant HI 1466/10-1 associated to the project “Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion” within the priority pro- gramme SPP1962 “Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization”. Edited by Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) Leibniz-Institut im Forschungsverbund Berlin e. V. Mohrenstraße 39 10117 Berlin Germany Fax: +49 30 20372-303 E-Mail: [email protected] World Wide Web: http://www.wias-berlin.de/ On the convexity of optimal control problems involving non-linear PDEs or VIs and applications to Nash games Michael Hintermüller, Steven-Marian Stengl Abstract Generalized Nash equilibrium problems in function spaces involving PDEs are considered. One of the central issues arising in this context is the question of existence, which requires the topological characterization of the set of minimizers for each player of the associated Nash game.
    [Show full text]
  • On Norm Attaining Operators and Multilinear Maps
    On norm attaining operators and multilinear maps Francisco J. Falcó Benavent Supervised by Domingo García, Manuel Maestre & Pilar Rueda Facultat de Ciències Matemàtiques Universitat de València This dissertation is submitted for the degree of Doctor in Mathematics (University of Valencia) September 2014 I, Francisco J. Falcó Benavent, declare that this thesis titled, “On norm attaining operators and multilinear maps” and the work presented in it are my own. I confirm that: ■ This work was done wholly or mainly while in candidature for a degree of Doctor in Mathematics (University of Valencia) at this University. ■ Where I have consulted the published work of others, this is always clearly attributed. ■ Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. ■ I have acknowledged all main sources of help. Signed: Date: i To Maika for all the cups of coffee during the long nights. To Domingo, Manolo and Pilar for their dedication and wise advice. Acknowledgements “If I have seen further it is by standing on the shoulders of giants.” — Isaac Newton I would like to thank my supervisors for their support throughout these years, keeping me going when times were tough, asking insightful questions, and offering invaluable advice. They allowed me the space and freedom I needed to work and continued supporting and guiding me from day one. Their advice was essential to the completion of this dissertation and has taught me innumerable lessons and insights on the workings of academic research in general. I acknowledge with great gratitude Richard and Eleanor Aron, for so many things that if I had to write all them here, this thesis will be written in two volumes.
    [Show full text]
  • Juan Carlos Ferrando Manuel López-Pellicer Editors Descriptive Topology and Functional Analysis in Honour of Jerzy Kąkol’S 60Th Birthday
    16 Springer Proceedings in Mathematics & Statistics Justificación de ser editores. Referencias al Prometeo en referencias 17 y 20, capítulos 1 y 6 del libro. Juan Carlos Ferrando Manuel López-Pellicer Editors Descriptive Topology and Functional Analysis In Honour of Jerzy Kąkol’s 60th Birthday Datos del libro al dorso Editors Juan Carlos Ferrando Manuel López-Pellicer The University Research Institute Center Matematica Aplicada of Operations Research Universidad Politecnica de Valencia Universidad Miguel Hernandez Valencia Elche Spain Alicante Spain ISSN 2194-1009 ISSN 2194-1017 (electronic) ISBN 978-3-319-05223-6 ISBN 978-3-319-05224-3 (eBook) DOI 10.1007/978-3-319-05224-3 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014942526 Ó Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer.
    [Show full text]
  • Boundaries for Spaces of Holomorphic Functions on C(K)
    Publ. RIMS, Kyoto Univ. 42 (2006), 27–44 Boundaries for Spaces of Holomorphic Functions on C(K) By Mar´ıa D. Acosta∗ Abstract We consider the Banach space Au(X) of holomorphic functions on the open unit ball of a (complex) Banach space X which are uniformly continuous on the closed unit ball, endowed with the supremum norm. A subset B of the unit ball of X is a boundary A ∈A | | for u(X) if for every F u(X), the norm of F is given by F =supx∈B F (x) . We prove that for every compact K, the subset of extreme points in the unit ball of C(K) is a boundary for Au(C(K)). If the covering dimension of K is at most one, then every norm attaining function in Au(C(K)) must attain its norm at an extreme point of the unit ball of C(K). We also show that for any infinite K, there is no Shilov boundary for Au(C(K)), that is, there is no minimal closed boundary, a result known before for K scattered. §1. Introduction A classical result by Silovˇ [Lo, p. 80] states that if A is a separating algebra of continuous functions on a compact Hausdorff space K, then there is a smallest closed subset F ⊂ K with the property that every function of A attains its maximum absolute value at some point of F . Bishop [Bi] proved that for every compact metrizable Hausdorff space K, any separating Banach algebra A ⊂C(K) has a minimal boundary, that is, there is M ⊂ K such that every element in A attains its norm at M and M is minimal with such a property.
    [Show full text]
  • Positively Convex Modules and Ordered Normed Linear Spaces
    Journal of Convex Analysis Volume 10 (2003), No. 1, 109–127 Positively Convex Modules and Ordered Normed Linear Spaces Dieter Pumpl¨un Fachbereich Mathematik, FernUniversit¨at,58084 Hagen, Germany [email protected] Dedicated to Harald Holmann on the occasion of his 70th birthday Received October 11, 2001 A positively convex module is a non-empty set closed under positively convex combinations but not necessarily a subset of a linear space. Positively convex modules are a natural generalization of positively convex subsets of linear spaces. Any positively convex module has a canonical semimetric and there is a universal positively aáne mapping into a regularly ordered normed linear space and a universal completion. 2000 Mathematics Subject Classification: 52A05, 52A01, 46B40, 46A55 1. Introduction In the following all linear spaces will be real. If E is a linear space ordered by a proper cone C; E = C C, and normed by a Riesz norm with respect to C, the positive part of the unit ball− O(E), i.e. O(E) C, is a typical example of a positively convex set or module. Actually any positively\ convex set or module i.e. any non-empty set P closed under positively convex operations, is an aáne quotient of such a positive part of a unit ball. Positively convex modules are natural generalizations of positively convex sets, the postulate that they are subsets of some linear space is dropped. Special positively convex sets play an important part in the theory of order-unit Banach spaces as so-called “universal capsÔ (cp. [21], §9) and also as generators of the topology of locally solid ordered topological linear spaces and the theory of Riesz seminorms (cp.
    [Show full text]
  • The Bishop-Phelps-Bollobás Theorem for L(Lp,Lq)
    THE BISHOP-PHELPS-BOLLOBAS´ THEOREM FOR OPERATORS ON L1(µ) YUN SUNG CHOI, SUN KWANG KIM, HAN JU LEE, AND MIGUEL MART´IN Dedicated to the memory of Robert R. Phelps Abstract. In this paper we show that the Bishop-Phelps-Bollob´astheorem holds for L(L1(µ);L1(ν)) for all measures µ and ν and also holds for L(L1(µ);L1(ν)) for every arbitrary measure µ and every localizable measure ν. Finally, we show that the Bishop-Phelps-Bollob´astheorem holds for two classes of bounded linear operators from a real L1(µ) into a real C(K) if µ is a finite measure and K is a compact Hausdorff space. In particular, one of the classes includes all Bochner representable operators and all weakly compact operators. 1. Introduction The celebrated Bishop-Phelps theorem of 1961 [12] states that for a Banach space X, every element in its dual space X∗ can be approximated by ones that attain their norms. Since then, there has been an extensive research to extend this result to bounded linear operators between Banach spaces [14, 27, 33, 34, 37] and non-linear mappings [2,7, 11, 16, 17, 30]. On the other hand, Bollob´as[13], motivated by problems arising in the theory of numerical ranges, sharpened the Bishop-Phelps theorem in 1970, and got what is nowadays called the Bishop-Phelps-Bollob´astheorem. Previously to presenting this result, let us introduce some notations. Given a (real or complex) Banach space X, we write BX for ∗ the unit ball, SX for its unit sphere, and X for the topological dual space of X.
    [Show full text]
  • A Revised Pre-Order Principle and Set-Valued Ekeland Variational
    A revised pre-order principle and set-valued Ekeland variational principle1 Jing-Hui Qiu School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China Abstract. In my former paper “A pre-order principle and set-valued Ekeland variational principle” (see: arXiv: 1311.4951[math.FA]), we established a general pre-order principle. From the pre-order prin- ciple, we deduced most of the known set-valued Ekeland variational principles (denoted by EVPs) and their improvements. But the pre- order principle could not imply Khanh and Quy’s EVP in [On gener- alized Ekeland’s variational principle and equivalent formulations for set-valued mappings, J. Glob. Optim., 49 (2011), 381-396], where the perturbation contains a weak τ-function. In this paper, we give a re- vised version of the pre-order principle. This revised version not only implies the original pre-order principle, but also can be applied to obtain the above Khanh and Quy’s EVP. Thus, the revised pre-order principle implies all the known set-valued EVPs in set containing forms (to my knowledge). arXiv:1405.1522v1 [math.FA] 7 May 2014 Key words: Pre-order Principle, Ekeland variational principle, Set- valued map, Perturbation, Locally convex space, Vector optimization Mathematics Subject Classifications (2000) 49J53 · 90C48 · 65K10 · 46A03 1. Introduction In [20], we established a general pre-order principle, which consists of a pre- order set (X, ) and a monotone extended real-valued function η on (X, ). The pre-order principle states that there exists a strong minimal point dominated by any 1This work was supported by the National Natural Science Foundation of China (10871141).
    [Show full text]
  • Review in Studia Universitatis Babes–Bolyai, Series Mathematica, March
    Review in Studia Universitatis Babe»s{Bolyai, Series Mathematica, March 2003 C. Z¸alinescu, Convex Analysis in General Vector Spaces, World Scienti¯c, New Jersey{ London{Singapore{Hong Kong 2002, xx+367 pp, ISBN: 981-238-067-1 This well written book is devoted to convex analysis in in¯nite dimensional spaces. What makes it di®erent from other existing books on convex analysis and optimization is the fact that the results are presented in their most generality, known at this time, as well as the inclusion of new and recent material. The author is a well known specialist in the ¯eld and the book incorporates many of his original results. In order to obtain this generality, the author has included in the ¯rst chapter of the book, Ch. 1, Preliminary results on functional analysis, a detailed study of convex series (cs) closed, lower cs-closed (lcs-closed), ideally convex, lower ideally convex (li-convex) sets and multivalued mappings, allowing him to prove very general open mapping and closed graph theorems of Ursescu{ Robinson type. The chapter contains also a ¯ne study of separation of convex sets and a presentation, with complete proofs, of Ekeland's variational principle and of Borwein{ Preiss smooth variational principle. The second chapter of the book, Ch. 2, Convex analysis in locally convex spaces, beside classical results, contains also the study of some more general classes of functions, corre- sponding to the sets studied in the ¯rst chapter: cs-closed, cs-complete, lcs-closed, ideally convex, bcs-complete and li-convex functions. The conjugate function, duality formulae, the subdi®erential and the "-subdi®erential calculus, are also included.
    [Show full text]
  • Convex Functions: Constructions, Characterizations and Counterexamples C 20081
    Convex Functions: Constructions, Characterizations and Counterexamples c 20081 Jonathan M. Borwein and Jon Vanderwerff August 26, 2008 1Do not circulate without permission 2 To our wives. Judith and Judith. ii Preface This book on convex functions emerges out of fifteen years of collaboration be- tween the authors. It is by far from the first on the subject nor will it be the last. It is neither a book on Convex analysis such as Rockafellar’s foundational 1970 book [342] nor a book on Convex programming such as Boyd and Vanden- berghe’s excellent recent text [121]. There are a number of fine books—both recent and less so—on both those subjects or on Convexity and relatedly on Variational analysis. Books such as [344, 236, 351, 237, 115, 90, 295, 304] com- plement or overlap in various ways with our own focus which is to explore the interplay between the structure of a normed space and the properties of convex functions which can exist thereon. In some ways, among the most similar books to ours are those of Phelps [320] and of Giles [214] in that both also straddle the fields of geometric functional analysis and convex analysis—but without the convex function itself being the central character. We have structured this book so as to accommodate a variety of readers. This leads to some intentional repetition. Chapter one makes the case for the ubiquity of convexity, largely by way of example; many but not all of which are followed up in later chapters. Chapter two then provides a foundation for the study of convex functions in Euclidean (finite-dimensional) space, and Chap- ter three reprises important special structures such as polyhedrality, eigenvalue optimization and semidefinite programming.
    [Show full text]
  • Minimal Element Theorems Revisited
    Minimal element theorems revisited Andreas H. Hamel,∗ Constantin Z˘alinescu† June 1, 2018 Abstract Starting with the Brezis-Browder principle, we give stronger versions of many varia- tional principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are discussed and clar- ified, i.e., assumptions to the metric structure of the underlying space and boundedness assumptions. New results involving set-valued maps and the increasingly popular set relations are obtained along the way. Keywords. variational principle, minimal element theorem, set-valued map, set relation, locally convex space MSC2010. Primary 46N10 Secondary 58E30, 49J53 1 Introduction The celebrated variational principle due to Ekeland ensures the existence of minimal elements with respect to (wrt for short) an order relation on a complete metric space (X, d) generated by a lower semicontinuous function f : X → R ∪ {+∞} which is bounded from below; this order is defined by x1 x2 :⇐⇒ f(x1)+ d(x1,x2) ≤ f(x2). (1) The assumptions required from f lead to certain features of whereas the assumptions to X arXiv:1805.12193v1 [math.FA] 30 May 2018 admit a countable induction argument which produces a Cauchy sequence which is decreasing wrt and whose limit then is the desired minimal element. Quite some effort went into attempts to generalize this result. In the first line of research, the assumptions to X, a (complete) metric space, are basically kept, but the order relation (1) is replaced by a more general one. A blueprint result of this type is [3, Theorem 3.2], others include [30, Theorem 1], [9, Theorem 16] (for this, see also [13, Theorem 2.2]) and [19, Lemma 2.2].
    [Show full text]