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.     

Georg Hamel and Richard von Mises in Brno

The article focuses on two outstanding mathematicians who worked at Brno German Technical University, Georg Hamel and Richard von Mises. After their biographies are summarized, their Brno stays are described in detail. A major part of the article is devoted to the process of appointing professors at Brno German Technical University and the difficulties Richard von Mises as a candidate, or those who wanted him to become professor, had to face: he participated in four consecutive competitions for a chair, but without success. © 2002 Elsevier Science (USA).Der Beitrag ist Georg Hamel und Richard von Mises gewidmet—zwei Mathematikern von Weltrang, welche an der deutschen technischen Hochschule in Brünn tätig waren. Auf eine kurze Zusammenfassung ihrer Lebensläufe folgt eine ausführliche Beschreibung ihrer Aufenthalte in Brünn unter besonderer Beachtung der erfolglosen Bemühungen um die Ernennung von Richard von Mises zum Professor für Mathematik und Mechanik. © 2002 Elsevier Science (USA).Tento článek je věnován dvěma mimořádným osobnostem světové matematiky, které působily na německé technice v Brně—Georgu Hamelovi a Richardu von Misesovi. V úvodu jsou krátce zmı́něny jejich životnı́ osudy a poté je podrobně zachycen jejich brněnský pobyt. Je připomenuto úsilı́ o Misesovo jmenovánı́ profesorem matematiky a mechaniky v Brně. © 2002 Elsevier Science (USA).MSC 1991 subject classifications: 01A70, 01A73, 01A60.     

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.     

Stable étale realization and étale cobordism

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A¹-homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero.On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.     

Comments on: Recent progress on the combinatorial diameter of polytopes and simplicial complexes

This paper features two main contributions. On the one hand, it gives an impressive survey on the progress on the diameter problem, including the breakthrough of the author with his disproof of the Hirsch conjecture among many other recent results. On the other hand, it features new results (exponential lower bounds) on the diameter of simplicial complexes and re-interprets the recent activities of the polymath 3 project that has been coordinated by Gil Kalai.     

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.     

Level crossing probabilities II: Polygonal recurrence of multidimensional random walks

In part I we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n−1/2). In higher dimensions we call a random walk ‘polygonally recurrent’ if there is a bounded set, hit by infinitely many of the straight lines between two consecutive sites a.s. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z³ in the sense of Mauldin, Monticino and von Weizsäcker [R.D. Mauldin, M. Monticino, H. von Weizsäcker, Directionally reinforced random walks, Adv. Math. 117 (1996) 239-252] is transient. On the other hand, we construct an example of a transient but polygonally recurrent random walk with independent components on Z².     

A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.     

Asymptotic behavior of functionals of empirical distribution functions for the two-sample problem

Summary Mises functional is extended for the two-sample problem. It is shown that the extended Mises functional also has the asymptotic property given by von Mises (1947,Ann. Math. Statist.,18, 309–348) and by Filippova (1962,Theory Prob. Appl.,7, 24–57) in the one-sample case. Asymptotic behavior ofU-statistic in the two-sample case, the statistic of Cramér-von Mises type for testing homogeneity and so forth are investigated as important examples of the theory. The Institute of Statistical Mathematics     

Die Habilitation von John von Neumann an der Friedrich-Wilhelms-Universität in Berlin: Urteile über einen ungarisch-jüdischen Mathematiker in Deutschland im Jahr 1927

The mathematician John von Neumann was born in Hungary but principally received his scientific education and socialization in the German science system. He received his Habilitation from the Friedrich-Wilhelms–Universität in Berlin in 1927, where he lectured as a Privatdozent until his emigration to the USA. This article aims at making a contribution to this early part of Neumann’s scientific biography by analyzing in detail the procedure that led to his Habilitation as well as the beginnings of Neumann’s research on functional analysis. An analysis of the relevant sources shows that in Berlin in the year 1927 Neumann was not yet regarded as the outstanding mathematical genius of the 20th century. Furthermore it will be seen that Neumann had great difficulties in developing the fundamental concepts for his path breaking work in spectral theory and only managed to do so with the support of the Berlin mathematician Erhard Schmidt.     

Three-way analysis of imprecise data

Data are often affected by uncertainty. Uncertainty is usually referred to as randomness. Nonetheless, other sources of uncertainty may occur. In particular, the empirical information may also be affected by imprecision. Also in these cases it can be fruitful to analyze the underlying structure of the data. In this paper we address the problem of summarizing a sample of three-way imprecise data. In order to manage the different sources of uncertainty a twofold strategy is adopted. On the one hand, imprecise data are transformed into fuzzy sets by means of the so-called fuzzification process. The so-obtained fuzzy data are then analyzed by suitable generalizations of the Tucker3 and CANDECOMP/PARAFAC models, which are the two most popular three-way extensions of Principal Component Analysis. On the other hand, the statistical validity of the obtained underlying structure is evaluated by (nonparametric) bootstrapping. A simulation experiment is performed for assessing whether the use of fuzzy data is helpful in order to summarize three-way uncertain data. Finally, to show how our models work in practice, an application to real data is discussed.     

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

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A constructive hypothesis test for the single-index models with two groups

Comparison of two-sample heteroscedastic single-index models, where both the scale and location functions are modeled as single-index models, is studied in this paper. We propose a test for checking the equality of single-index parameters when dimensions of covariates of the two samples are equal. Further, we propose two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals. These statistics evaluate the difference of the empirical residual processes to test the equality of mean functions of two single-index models. Asymptotic distributions of estimators and test statistics are derived. The Kolmogorov–Smirnov and Cramér–von Mises test statistics can detect local alternatives that converge to the null hypothesis at a parametric convergence rate. To calculate the critical values of Kolmogorov–Smirnov and Cramér–von Mises test statistics, a bootstrap procedure is proposed. Simulation studies and an empirical study demonstrate the performance of the proposed procedures.     

Devaney’s chaos on uniform limit maps

Let (X, d) be a compact metric space and f_n : X → X a sequence of continuous maps such that (f_n) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all f_n mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

Second Order Hadamard Differentiability in Statistical Applications

A formulation of the second-order Hadamard differentiability of (extended) statistical functionals and some related theoretical results are established. These results are applied to derive the limiting distributions of a class of generalized Cramér–von Mises type test statistics, which include some proposed new ones for the tests of goodness of fit in the 3-sample problems, the tests in linear regression models, and the tests of bivariate independence, as special cases.     

The distribution of length of N random unit vectors for a von Mises population

Greenwood and Durand (1955) have expressed the distribution function (d.f.) of the length $\text{R}$ of N random unit vectors for a von Mises population as a double integral. This double integral is simplified herein by calculating analytically one of the integrals. Hence a numerical calculation for large parameter values now becomes possible.     

Limit theorems for the canonical von Mises statistics with dependent data

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.     

Tests of independence among continuous random vectors based on Cramér–von Mises functionals of the empirical copula process

A decomposition of the independence empirical copula process into a finite number of asymptotically independent sub-processes was studied by Deheuvels. Starting from this decomposition, Genest and Rémillard recently investigated tests of independence among random variables based on Cramér–von Mises statistics derived from the sub-processes. A generalization of Deheuvels’ decomposition to the case where independence is to be tested among continuous random vectors is presented. The asymptotic behavior of the resulting collection of Cramér–von Mises statistics is derived. It is shown that they are not distribution-free. One way of carrying out the resulting tests of independence then involves using the bootstrap or the permutation methodology. The former is shown to behave consistently, while the latter is employed in practice. Finally, simulations are used to study the finite-sample behavior of the tests.