A convergence theorem of Rosen’s gradient projection method

Since Rosen’s gradient projection method was published in 1960, a rigorous convergence proof of his method has remained an open question. A convergence theorem is given in this paper. Part of this author’s work was done while he studied at the Department of Mathematics, University of California at Santa Barbara, and was supported by the National Science Foundation under Grant No. MCS83-14977. Part of this author’s work was done while he visited the Computer Science Department, University of Minnesota, Minneapolis, and was supported by the National Science Foundation under Grant No. MCS81-01214.     

Own-company stockholding and work effort preferences of an unconstrained executive

We develop a framework for analyzing an executive’s own-company stockholding and work effort preferences. The executive, characterized by risk aversion and work effectiveness parameters, invests his personal wealth without constraint in the financial market, including the stock of his own company whose value he can directly influence with work effort. The executive’s utility-maximizing personal investment and work effort strategy is derived in closed form, and a utility indifference rationale is applied to determine his required compensation. Being unconstrained by performance contracting, the executive’s work effort strategy establishes a base case for theoretical or empirical assessment of the benefits or otherwise of constraining executives with performance contracting.     

Jacobi and Kummer’s ideal numbers

In this article we give a modern interpretation of Kummer’s ideal numbers and show how they developed from Jacobi’s work on cyclotomy, in particular the methods for studying “Jacobi sums” which he presented in his lectures on number theory and cyclotomy in the winter semester 1836/37.     

Nathan Jacobson (1910-1999)

Nathan Jacobson, who died on 5 December 1999, was an outstandingalgebraist, whose work on almost all aspects of algebra wasof fundamental importance, and whose writings will exercisea lasting influence. He had been an honorary member of the Societysince 1972. Nathan Jacobson (later known as ‘Jake’ to his friends)was born in Warsaw (in what he describes as the ‘Jewishghetto’) on 5 October 1910 (through an error some documentshave the date 8 September); he was the second son of CharlesJacobson (as he would be known later) and his wife Pauline,née Rosenberg. His family emigrated to the USA duringthe First World War, first to Nashville, Tennessee, where hisfather owned a small grocery store, but they then settled inBirmingham, Alabama, where Nathan received most of his schooling.Later the family moved to Columbus, Mississippi, but the youngNathan entered the University of Alabama in 1926 and graduatedin 1930. His initial aim was to follow an uncle and obtain adegree in law, but at the same time he took all the (not verynumerous) mathematics courses, in which he did so well thathe was offered a teaching assistantship in mathematics in hisjunior (3rd) year. This marked a turning point; he now decidedto major in mathematics and pursue this study beyond College.During his final year at Alabama he applied for admission andfinancial aid to three top graduate schools in the country:Princeton, Harvard and Chicago. He was awarded a research assistantshipat Princeton; after the first year he was appointed a part-timeinstructor for two years, and during his fourth year he wasappointed a Procter Fellow. The stipend was enough to enablehim to make a grand tour of Europe by car in 1935, in the companyof two Princeton fellow-students at the time: H. F. Bohnenblustand Robert J. Walker.     

Groupe de Brauer non ramifié d’espaces homogènes à stabilisateurs finis

On s’intéresse au groupe de Brauer d’une compactification lisse d’un espace homogène d’un groupe semi-simple simplement connexe à stabilisateurs finis, sur un corps de nombres. On montre une formule décrivant ce groupe de Brauer et on calcule explicitement celui-ci pour certaines familles d’exemples. On établit des liens entre l’obstruction de Brauer–Manin à l’approximation faible sur cet espace homogène et la cohomologie galoisienne du stabilisateur.     

Purcell’s method, Egerváry and related results

After having been appeared, Egerváry was perhaps the first who responded to Purcell’s paper in 1957. Later in a posthumous paper he returned to the method in 1960, showing that it could be derived from his rank reduction procedure. We review here Purcell’s method in connection with Egerváry’s activity and also, we give a short survey on subsequent developments.     

The life and work of André Cholesky

Cholesky’s method for solving a system of linear equations with a symmetric positive definite matrix is well known. In this paper, I will give an account of the life of Cholesky, analyze an unknown and unpublished paper of him where he explains his method, and review his other scientific works. I dedicated this work to John (Jack) Todd with esteem and respect at the occasion of his 95th anniversary.     

Teiji Takagi, Founder of the Japanese School of Modern Mathematics

This article is a brief historical report on Teiji Takagi which was prepared at the commencement of ‘Takagi Lectures’ of The Mathematical Society of Japan. The first of its two purposes is to give some informations on the circumstances of education and research of mathematics in Japan surrounding Takagi who could finally established himself as the founder of the Japanese school of modern mathematics. The other is a brief overview on Takagi’s works of mathematics some of which are still attractive to and influential on especially ambitious students of mathematics. The author hopes that careful readers may find some hints for the questions how and why Takagi was able to establish his class field theory. At the end of this article the readers will find an English translation of the preface of his book Algebraic theory of numbers (in Japanese) which is the only thing that he left for us to see his total view over class field theory after the establishment of Artin’s reciprocity law.     

On chain morphisms of commutative rings

Résumé  On dit qu’un homomorphismef :A →B d’anneaux commutatifs est un morphisme de cha?ne (resp., den-cha?ne pour un entiern ≥ 1) si toute cha?ne d’idéaux premiers (resp., d’au plusn idéaux premiers) deA se relève en une cha?ne d’idéaux premiers deB. Sif est un morphisme den-cha?ne, alorsf n’est pas forcément un morphisme de (n + 1)-cha?ne, même siA etB sont des anneaux intègres, doncf n’est pas un morphisme de cha?ne. Sif est un morphisme den-cha?ne pour toutn, alorsf est un morphisme de cha?ne. Un morphisme de cha?ne n’est pas forcément un morphisme de cha?ne universel. Pour tout entiern ≥ 2,f est universellement un morphisme den-cha?ne si et seulement sif est universellement un morphisme de cha?ne. Un morphisme qui est universellement de cha?ne et universellement incomparable n’est pas nécessairement entier, même siA etB sont des anneaux intègres de dimension 1 (au sens de Krull).      

Problems for Broome’s Cognitivist Account of Instrumental Reasoning

In this paper, I examine an account of instrumental reasoning recently put forth by John Broome. His key suggestion is that anyone who engages in reasoning about his intentions also believes that he will do what he intends to do and that combined with a belief about necessary means this creates rational pressure towards believing that one will take the necessary means. I argue that Broome’s model has three significant problems; his key premise is false—the sincere expression of an intention does not entail the belief that one will successfully execute that intention; his account yields a model of instrumental reasoning that is uncomfortably reflective; he seems unable to explain the rational pressure towards taking necessary means that arises directly from having an end and an instrumental belief. All three problems, I argue, are a consequence of Broome’s inadequate position on what it is to intend to do something.     

Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.     

Veber on knowledge and factuality

The article deals with the development of the philosophy of France Veber (1890–1975), the pupil of Meinong and a main Slovene philosopher. One of the most important threads of Veber’s philosophy is the consideration of knowledge and factuality, which may be seen as a driving force of its development. Veber’s philosophical development is usually divided into three phases: the object theory phase, the phase when he created his philosophy of a person as a creature at the crossing of the natural and the spiritual world, who as an active, not merely passive subject possesses her own causal powers, and the third phase, when he supplemented his earlier philosophy with the theory of a special side of our experience which he called hitting-upon-reality. It is a direct experience of reality, a special kind of intentionality, which is however fundamentally different from presentational intentionality, which alone is taken into account by object theory or phenomenology. The questions of knowledge and factuality are closely connected in Veber’s philosophy since, pace Veber, knowledge is a kind of, we may say, justified experience the object of which is a factual entity. Hence, if we want to understand what knowledge is, we must face the challenge of comprehending factuality. There are five stages to be noted in the development of his epistemology. The first two belong to his object theory phase, the third to his person phase, the fourth is characterised by his distinguishing and exploring truth and validity with regard to the thought about God, and the basis of the fifth phase lies in his theory of hitting-upon-reality. In Introduction to Philosophy and The System of Philosophy, that is in the year 1921, Veber believed that factuality (“truth,”) was a property of the object, which we do present, but we do not present the factuality of this factuality (that is why he distinguishes between the merely objective truths and truths that are in addition transcendental truths). In 1923, in The Problems of Contemporary Philosophy and in the work Science and Religion, he already rejected such a view. There is something that makes things factual, but that is a complete unknown X. Therefore we cannot even say what kind of an entity this factuality is. Some people would probably demand the following formulation: if X is an ultimate mystery, we should not claim even that it is an entity. In The Problems of Presentation Production (1928) Veber claimed that factuality is not a property since this would lead to a regressum ad infinitum. Philosophy (1930) related internally correct experience to personal will. In The Book about God (1934) he developed the thesis that factuality depends on the act of God. In The Question of Reality (1939) he importantly modified, developed and enriched the thesis that we do not present reality with his theory of immediate experience of (hitting upon) factuality.     

Deformed virasoro algebras andw-algebras from elliptic algebras

Deformedw _n -algebras in the classical case (i.e., Poisson structures) and in the quantum case (exchange algebras) are constructed from the elliptic algebras . In memory of M. Saveliev When I think about Misha, and this has been happening rather frequently these recent months, it is his enthusiasm and his communicative energy that first come to mind. Something very important to Misha was friendship. We scientists have the priviledge to possess friends, real friends, among our colleagues, and Misha was a friend. Something else was fundamental for Misha, his family, i.e., his parents, his wife, and his daughter. As Professor Manin said, summarizing Misha’s behavior very simply and elegantly, Misha was “a good son, a good husband, a good father.” Misha’s parents died about three years ago, and I remember how he was sad when I met him in Georgia in September 1996 and he informed me about his father’s and mother’s deaths a few months before. After that, he concentrated all his forces to protect his wife and daughter, the two Svetlanas, as he used to say! He was so proud of his gifted daughter, a good musician, a good painter, and a brilliant student. I wish to present them my most respectful thoughts. Misha was a great expert in integrable systems. He was deeply interested in developing models, which he did so nicely in the framework of Toda theories. I learned much from him about Toda theories and also continuous Lie algebras. Misha was highly convinced of the importance of symmetries and was especially interested in the use of infinite symmetries, the W-algebras among them. The landscape of continuous symmetries, and particularly of infinite-dimensional symmetries, is becoming richer and richer these years. I dare to think that Misha would have been interested in the modest seminar I now present. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 311–322, May, 2000.     

The mathematics of F. J. Almgren, Jr.

Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field. He solved many of them and, in the process, discovered ideas which turned out to be useful for many other problems. This article is a more-or-less chronological survey of Almgren’s mathematical research. (Excerpts from this article appeared in the December 1997 issue of theNotices of the American Mathematical Society.) Almgren was also an outstanding educator, and he supervised the thesis work of nineteen PhD students; the 1997 volume 6 issue of the journalExperimental Mathematics is dedicated to Almgren and contains reminiscences by two of his PhD students and by various colleagues. A general article about Almgren’s life appeared in the October 1997Notices of the American Mathematical Society [MD]. See [T3]for a brief biography.     

Contribution a Deux Problemes,Concernant Les Fonctions de la Classe A

Construction explicite d’une fonction de la classeA=F(l ¹) lipschitzienne d’ordrea>0, ne satisfaisant pas la synthèse spectrale. Nouveaux exemples d’ensembles qui ne sont pas de résolution au sens de Malliavin (en particulier, l’ensemble triadique de Cantor). Nouveaux examples d’ensemblesE telles que les seules fonctions d’une variable réelle opérant dans l’algèbreA(E) des restrictions aE des fonctions de la clesseA, soient les fonctions analytiques.

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The research reported in this document has been sponsored in part by the Air Force office of Scientific Research, OAR through the European Office, Aerospace Research, United States Air Force”.     

Probability no longer a stepchild

S. R. S. Varadhan was awarded the Abel Prize for the year 2007. I met him on 14 and 15 May—one week before the prize ceremony in Oslo—in his office at the Courant Institute to interview him for the Mathematical Intelligencer. My qualifications to interview him were that he and I are Ph.D.’s from the same institute, my Varadhan number is 2, and his was the first research talk that I attended as a graduate student. My major disqualification was that I know little of probability, and I felt like someone destitute of geometry daring to enter Plato’s Academy. Though we had planned to talk for two or three hours, our conversation was spread over nearly eight hours. What follows is the record of this with very minor editing. To help the reader I have added a few “box items” that explain some of the mathematical ideas alluded to in the conversation.     

Curry’s Formalism as Structuralism

In 1939, Curry proposed a philosophy of mathematics he called formalism. He made this proposal in two works originally written then, although one of them was not published until 1951. These are the two philosophical works for which Curry is known, and they have left a false impression of his views. In this article, I propose to clarify Curry’s views by referring to some of his later writings on the subject. I claim that Curry’s philosophy was not what is now usually called formalism, but is really a form of structuralism.     

Experiments with an interactive paired comparison simplex method for molp problems

In this paper, an interactive paired comparison simplex based method formultiple objective linear programming (MOLP) problems is developed and compared to other interactive MOLP methods. The decision maker (DM)’s utility function is assumed to be unknown, but is an additive function of his known linearized objective functions. A test for ‘utility efficiency’ for MOLP problems is developed to reduce the number of efficient extreme points generated and the number of questions posed to the DM. The notion of ‘strength of preference ’ is developed for the assessment of the DM’s unknown utility function where he can express his preference for a pair of extreme points as ‘strong ’, ‘weak ’, or ‘almost indifferent ’. The problem of ‘inconsistency of the DM’ is formalized and its resolution is discussed. An example of the method and detailed computational results comparing it with other interactive MOLP methods are presented. Several performance measures for comparative evaluations of interactive multiple objective programming methods are also discussed. All rights reserved. This study, or parts thereof, may not be reproduced in any form without written permission of the authors.     

Advertising coordination games of a manufacturer and a retailer while introducing a new product

Two kinds of vertical cooperative advertising program are considered in a distribution channel constituted by a manufacturer and a retailer, where the manufacturer pays part of the retailer’s advertising costs. In the first participation scheme, the manufacturer chooses his/her advertising participation rate in the retailer’s advertising effort and then each player determines the advertising effort that maximizes his/her profit. In the second scheme, the retailer chooses the manufacturer’s participation rate and then the manufacturer determines the advertising efforts of both players with the objective of maximizing the manufacturer’s profit. Each participation scheme corresponds to a special Stackelberg game: the manufacturer is the leader of the first, while the retailer is the leader of the second. The Stackelberg equilibrium advertising efforts and participation rate in both games are provided. Then the equilibrium strategies of the two players in the analyzed scenarios are compared with the Nash equilibrium in the competitive framework. Finally, the conditions which suggest a special kind of agreement to a player are analyzed. This work was supported by the Italian Ministry of University and Research and the University of Padua.