German mathematicians in exile in Turkey: Richard von Mises,William Prager,Hilda Geiringer,and their impact on Turkish mathematics

There is a sizable and growing literature on scholars who fled from the Nazi regime, a literature which often focuses on the periods before leaving Germany and after settling permanently in the USA, but relatively less work on the interim period in which many of them found temporary homes in countries such as Turkey. In this article we would like to discuss the scholarly work, activities and the impact of mathematicians Richard von Mises, William Prager and Hilda Geiringer during their stay in Turkey. We argue that the establishment and the development of applied mathematics and mechanics in Turkey owe much to them.     

Straddling centuries: The struggles of a mathematician and his university to enter the ranks of research mathematics, 1870–1950

This paper weaves two interlocking histories together. One strand of the fabric traces the development of the American mathematician Joseph B. Reynolds from a peripheral player to an active contributor to mathematics, astronomy, and engineering and to the founding of a sectional association of mathematicians. The other piece describes the evolution of his institution, Lehigh University, from its founding in 1865 to a full-fledged research department that began producing doctorates in 1939. Both Reynolds and Lehigh straddled the line between the pre- and post-Chicago eras in American mathematics.     

The Society for the Protection of Science and Learning as a patron of refugee mathematicians

The Society for the Protection of Science and Learning (SPSL, which started out as the Academic Assistance Council (AAC) 1933–36) distinguished itself from many other aid organizations set up in response to Nazi policies towards Jews and political dissidents in its focus on academic excellence as a criterion for support. Its archives are deposited in the Bodleian Library. Today, the organization is known as the Council for At-Risk Academics (CARA, http://www.cara1933.org/).

In the archives of the SPSL, there are files on two and a half thousand scientists, victims of persecution in their home countries, who appealed to the Society for assistance in finding refuge and work in Britain. Almost a hundred are mathematicians, of which about one in four found temporary or permanent employment in British academia. Many others found their way to the US, in most cases aided by the Emergency Committee, an American organization with priorities similar to those of the SPSL. The paths of some of the most successful SPSL grantees are described, and contrasted with those of some less fortunate applicants.     

E.W. Chittenden and the early history of general topology

E.W. Chittenden's work and its influence on the early history of general topology are examined. Particular attention is given to his work in metrization theory and its role in the background of the Aleksandrov-Uryson Metrization Theorem. A recounting of Professor Chittenden's career, a list of his students and his publications and a chronology in the early history of General Topology are also included.     

Mathematical recreations of Dénes König and his work on graph theory

Dénes König (1884–1944) is a Hungarian mathematician well known for his treatise on graph theory (König, 1936). When he was a student, he published two books on mathematical recreations ( and ). Does his work on mathematical recreations have any relation to his work on graph theory? If yes, how are they connected? To answer these questions, we will examine his books of 1902, 1905 and 1936, and compare them with each other. We will see that the books of 1905 and 1936 include many common topics, and that the treatment of these topics is different between 1905 and 1936.     

Calculations of faith: mathematics, philosophy, and sanctity in 18th-century Italy (new work on Maria Gaetana Agnesi)

The recent publication of three books on Maria Gaetana Agnesi (1718-1799) offers an opportunity to reflect on how we have understood and misunderstood her legacy to the history of mathematics, as the author of an important vernacular textbook, Instituzioni analitiche ad uso della gioventú italiana (Milan, 1748), and one of the best-known women natural philosophers and mathematicians of her generation. This article discusses the work of Antonella Cupillari, Franco Minonzio, and Massimo Mazzotti in relation to earlier studies of Agnesi and reflects on the current state of this subject in light of the author’s own research on Agnesi.     

Non-Classical Stems from Classical: N. A. Vasiliev’s Approach to Logic and his Reassessment of the Square of Opposition

In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it?      

Sets versus trial sequences,Hausdorff versus von Mises: “Pure” mathematics prevails in the foundations of probability around 1920

The paper discusses the tension which occurred between the notions of set (with measure) and (trial-) sequence (or—to a certain degree—between nondenumerable and denumerable sets) when used in the foundations of probability theory around 1920. The main mathematical point was the logical need for measures in order to describe general nondiscrete distributions, which had been tentatively introduced before (1919) based on von Mises’s notion of the “Kollektiv.” In the background there was a tension between the standpoints of pure mathematics and “real world probability” (in the words of J.L. Doob) at the time. The discussion and publication in English translation (in Appendix) of two critical letters of November 1919 by the “pure” mathematician Felix Hausdorff to the engineer and applied mathematician Richard von Mises compose about one third of the paper. The article also investigates von Mises’s ill-conceived effort to adopt measures and his misinterpretation of an influential book of Constantin Carathéodory. A short and sketchy look at the subsequent development of the standpoints of the pure and the applied mathematician—here represented by Hausdorff and von Mises—in the probability theory of the 1920s and 1930s concludes the paper.     

W.W. Rouse Ball and the mathematics of string figures

This article examines a chapter of the popular book Mathematical Recreations and Essays (5th to 9th editions) written by the Cambridge mathematician Walter William Rouse Ball (1850–1925). This chapter is devoted to “String Figures”, a procedural activity which consists in producing geometrical forms with a loop of string and which is carried out in many traditional societies throughout the world. By analyzing the way in which Ball selected some string figures within ethnographical publications and conceived the structure of this chapter, it appears that he implicitly brought to light the mathematical dimension of this practice.     

Commentary on Robert Riley’s article “A personal account of the discovery of hyperbolic structures on some knot complements”

We give some background and biographical commentary on the posthumous article  [4] that appears in this journal issue by Robert Riley on his part of the early history of hyperbolic structures on some compact 3-manifolds. A complete list of Riley’s publications appears at the end of this article.     

The dating of Ptolemy's Almagest based on the coverings of the stars and on lunar eclipses

This paper is a natural extension and continuation of the authors' studies of the astronomical dating problem of Ptolemy's famous Almagest. In previous papers, the authors suggested and developed a new geometrical-statistical method for dating ancient star catalogues. This method was then applied to Ptolemy's Almagest. The results obtained do not confirm the traditional dating of the Almagest (2nd century AD or 2nd century BC) but shift it to the epoch AD 600–1300. In this paper, we extend our analysis to other parts of the Almagest and study the dating problem for series of lunar eclipses described in the Almagest and for the covering of stars by planets. The results obtained completely agree with our previous results and give the same time interval, AD 600–1300.     

The emergence of open sets,closed sets,and limit points in analysis and topology

General topology has its roots in real and complex analysis, which made important uses of the interrelated concepts of open set, of closed set, and of a limit point of a set. This article examines how those three concepts emerged and evolved during the late 19th and early 20th centuries, thanks especially to Weierstrass, Cantor, and Lebesgue. Particular attention is paid to the different forms of the Bolzano–Weierstrass Theorem found in the latter's unpublished lectures. An abortive early, unpublished introduction of open sets by Dedekind is examined, as well as how Peano and Jordan almost introduced that concept. At the same time we study the interplay of those three concepts (together with those of the closure of a set and of the derived set of a set) in the struggle to determine the ultimate foundations on which general topology was built, during the first half of the 20th century.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Prediction for two processes and the nehari problem

We exploit an analogy between the trigonometric moment problem and prediction theory for a stationary stochastic process. Extending this theory, we show how to use correlations between two processes to predict one from the other. In turn, this gives rise to a simple and unified treatment of the Caratheodory and Nehari moment problems.     

On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions

Recently, Mandelbrot has encountered and numerically investigated a probability densityp _d (t) on the nonnegative reals, where, 0D<1. this=" density=" has=" fourier=" transform=">f _d (-is), wheref _d (z)=–Dz ^d (–D, z) and (·.·) is an incomplete gamma function. Previously, Darling had met this density, but had not studied its form. We expressf _d (z) as a confluent hypergeometric function, then locate and approximate its zeros, thereby improving some results of Buchholz. Via properties of Laplace transforms, we approximatep _d (t) asymptotically ast0+ and +, then note some implications asD0+ and 1–.Communicated by Mourad Ismail.     

Convergence and formal manipulation in the theory of series from 1730 to 1815

In the early calculus mathematicians used convergent series to represent geometrical quantities and solve geometrical problems. However, series were also manipulated formally using procedures that were the infinitary extension of finite procedures. By the 1720s results were being published that could not be reduced to the original conceptions of convergence and geometrical representation. This situation led Euler to develop explicitly a more formal approach which generalized the early theory. Formal analysis, which was predominant during the second half of the 18th century despite criticisms of it by some researchers, contributed to the enlargement of mathematics and even led to a new branch of analysis: the calculus of operations. However, formal methods could not give an adequate treatment of trigonometric series and series that were not the expansions of elementary functions. The need to use trigonometric series and introduce nonelementary functions led Fourier and Gauss to reject the formal concept of series and adopt a different, purely quantitative notion of series.     

Fundamental solutions, transition densities and the integration of Lie symmetries

In this paper we present some new applications of Lie symmetry analysis to problems in stochastic calculus. The major focus is on using Lie symmetries of parabolic PDEs to obtain fundamental solutions and transition densities. The method we use relies upon the fact that Lie symmetries can be integrated with respect to the group parameter. We obtain new results which show that for PDEs with nontrivial Lie symmetry algebras, the Lie symmetries naturally yield Fourier and Laplace transforms of fundamental solutions, and we derive explicit formulas for such transforms in terms of the coefficients of the PDE.     

Limit distribution of the sum and maximum from multivariate Gaussian sequences

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.     

Exact Distribution of the Occurrence Number for K-tuples Over an Alphabet of Non-Equal Probability Letters

A nucleotide sequence can be considered as a realization of the non-equal-probability independently and identically distributed (niid) model. In this paper we derive the exact distribution of the occurrence number for each K-tuple with respect to the niid model by means of the Goulden-Jackson cluster method. An application of the probability function to get exact expectation curves [9] is presented, accompanied by comparison between the exact approach and the approximate solution.Received October 31, 2004