The meaning of Ramanujan now and for the future

December 22, 2010 marks the 123th anniversary of Ramanujan’s birth. In this paper we pay homage to this towering figure whose mathematical discoveries so affected mathematics throughout the twentieth century and into the twenty-first.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Ky Fan (1914�C2010), he spent every waking moment thinking about mathematics

This survey article on Dr. Ky Fan summarizes his versatile achievements and fundamental contributions in the fields of topological groups, nonlinear and convex analysis, operator theory, linear algebra and matrix theory, mathematical programming, and approximation theory, etc., and as well reveals Fan’s exemplary mathematical formation opening up the beauty of pure mathematics, with natural conditions, concise statements and elegant proofs. This article contains a brief biography of Dr. Fan and epitomizes his life. He was not only a great mathematician, but also a very serious teacher known to be extremely strict to his students. He loved his motherland and made generous donations for promoting mathematical development in China. He devoted his life to mathematics, continued his research and published papers till 85 years old.     

Contributions of Lo Yang to Value Distribution Theory

Professor Lo Yang is a world famous mathematician of our country. He made a lot of outstanding achievements in the value distribution theory of function theory, which are highly rated and widely quoted by domestic and foreign scholars. He also did a lot of work to develop Chinese mathematics. It can be said that Professor Yang is one of the mathematicians who made main influences on the mathematical development in modern China. This paper briefly introduces Professor Yang’s life, mainly discusses his academic achievement and influence, and briefly describes his contributions to the Chinese mathematics community.     

Robert Richard Anstice (1813–1853): a Hertfordshire bicentenary

R R Anstice was a little known English clergyman who made significant contributions to Combinatorics, anticipating results in modern design theory. The year 2013 is the 200th anniversary of his birth. Anstice took the first steps in treating Combinatorial Design Theory as part of Algebra, in retrospect anticipating many later developments. In particular he touched on an important structure now known as a ‘Room square’, named after the twentieth century Anglo-Australian mathematician T G Room. Here we review Anstice's short life and give an account of his mathematics.     

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.     

Solving generalized equations via homotopies

Global Newton methods for computing solutions of nonlinear systems of equations have recently received a great deal of attention. By using the theory of generalized equations, a homotopy method is proposed to solve problems arising in complementarity and mathematical programming, as well as in variational inequalities. We introduce the concepts of generalized homotopies and regular values, characterize the solution sets of such generalized homotopies and prove, under boundary conditions similar to Smale’s [10], the existence of a homotopy path which contains an odd number of solutions to the problem. We related our homotopy path to the Newton method for generalized equations developed by Josephy [3]. An interpretation of our results for the nonlinear programming problem will be given.     

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.     

A Non-Standard Analysis of a Cultural Icon: The Case of Paul Halmos

We examine Paul Halmos’ comments on category theory, Dedekind cuts, devil worship, logic, and Robinson’s infinitesimals. Halmos’ scepticism about category theory derives from his philosophical position of naive set-theoretic realism. In the words of an MAA biography, Halmos thought that mathematics is “certainty” and “architecture” yet 20th century logic teaches us is that mathematics is full of uncertainty or more precisely incompleteness. If the term architecture meant to imply that mathematics is one great solid castle, then modern logic tends to teach us the opposite lesson, namely that the castle is floating in midair. Halmos’ realism tends to color his judgment of purely scientific aspects of logic and the way it is practiced and applied. He often expressed distaste for nonstandard models, and made a sustained effort to eliminate first-order logic, the logicians’ concept of interpretation, and the syntactic vs semantic distinction. He felt that these were vague, and sought to replace them all by his polyadic algebra. Halmos claimed that Robinson’s framework is “unnecessary” but Henson and Keisler argue that Robinson’s framework allows one to dig deeper into set-theoretic resources than is common in Archimedean mathematics. This can potentially prove theorems not accessible by standard methods, undermining Halmos’ criticisms.     

Tracking a reference solution of a control system of phase field equations

We consider the problem of tracking a reference solution of a dynamical system described by a pair of distributed differential equations, the phase field equations. To solve this problem, we propose an algorithm based on Yu.S. Osipov’s theory of dynamic inversion and on N.N. Krasovskii’s extremal shift method developed in the theory of positional differential games.     

Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany

Remainder problems have a long tradition and were widely disseminated in books on calculation, algebra, and recreational mathematics from the 13th century until the 18th century. Many singular solution methods for particular cases were known, but Bachet de Méziriac was the first to see how these methods connected with the Euclidean algorithm and with Diophantine analysis (1624). His general solution method contributed to the theory of equations in France, but went largely unnoticed elsewhere. Later Euler independently rediscovered similar methods, while von Clausberg generalized and systematized methods that used the greatest common divisor procedure. These were followed by Euler's and Lagrange's continued fraction solution methods and Hindenburg's combinatorial solution. Shortly afterwards, Gauss, in the Disquisitiones Arithmeticae, proposed a new formalism based on his method of congruences and created the modular arithmetic framework in which these problems are posed today.     

Discrete monodromy, pentagrams, and the method of condensation

This paper studies the pentagram map, a projectively natural iteration on the space of polygons. Inspired by a method from the theory of ordinary differential equations, the paper constructs roughly n algebraically independent invariants for the map, when it is defined on the space of n-gons. These invariants strongly suggest that the pentagram map is a discrete completely integrable system. The paper also relates the pentagram map to Dodgson’s method of condensation for computing determinants, also known as the octahedral recurrence. I dedicate this paper to Professor V. I. Arnold on the occasion of his 70th birthday     

Special functions in academician A.A. Dorodnicyn’s scientific legacy: Special functions arising in the solution of boundary value problems for second-order singular ordinary differential equations by asymptotic methods

On the occasion of the centenary of Academician A.A. Dorodnicyn’s birth, his main results concerning asymptotic methods for solving second-order singular ordinary differential equations and special functions related to these equations are presented (with certain corrections and additions).     

Petrarch's canzoniere: Rational addiction and amorous cycles

The paper is concerned with a celebrated collection of love poems, the 14th century Italian poet Francis Petrarch's Canzoniere. A striking feature of these poems is the emotional ups and downs experienced by Petrarch and his platonic mistress Laura. Recently, attempts have been made to model these emotional swings by catastrophe theory or nonlinear differential equations. This paper takes a different approach. Starting with a pair of differential equations that model the dynamics of the emotions of the two individuals, we formulate an optimal control problem. A key hypothesis of this problem is that Petrarch was a rational addict of his desire for Laura. With specific functional forms and parameter values we identify a stable limit cycle that gives a representation of the oscillating emotions of Laura and Petrarch.     

The contributions of W.A. harris Jr. to difference equations

William Ashton Harris Jr. was born December 18, 1930 in New Orleans, Louisiana, USA. He studied mathematics with minors in physics and appl ied mechanics (elasticity) at the University of Minnesota receiving his Ph.D. in mathematics in 1955 with the thesis "A boundary value problem for a system of ordinary linear differential equations involving powers of a parameter". His thesis advisor was Professor H.L. Turrittin.He remained at the University of Minnesota and became professor in 1968. In 1970 he moved to the University of Southern California where he stayed until his untimely death on January 8, 1998. He has held visiting appointments at several other universities.     

Differential equations satisfied by Eisenstein series of level 2

Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.     

追忆我的大学老师华罗庚先生

华罗庚先生离开我们已经25年了.作为他的一个老学生,对60多年前他所给予的种种指导和启发,我仍记忆犹新.谨以此文追忆华罗庚先生在大学任教期间的部分轶闻趣事,以勾勒其非凡的人生阅历、展现其特有的学术风范、彰显其崇高的个人品质.     

Faraday’s Form and Maxwell’s Equations in the Heisenberg Group

In this note we present a geometric formulation of Maxwell’s equations in Carnot groups (connected simply connected nilpotent Lie groups with stratified Lie algebra) in the setting of the intrinsic complex of differential forms defined by M. Rumin. Restricting ourselves to the first Heisenberg group \mathbbH¹{\mathbb{H}^{1}}, we show that these equations are invariant under the action of suitably defined Lorentz transformations, and we prove the equivalence of these equations with differential equations “in coordinates”. Moreover, we analyze the notion of “vector potential”, and we show that it satisfies a new class of 4th order evolution differential equations.     

W. Charles Holland, 75<Superscript>th</Superscript> birthday

This issue of Mathematica Slovaca is in honour of W. Charles Holland’s 75th birthday. We present here a brief account of some of his research (to date) and a couple of brief personal sketches of the man.