Cauchy and the infinitely small

We consider Cauchy's use of the infinitely small in his textbooks. He never examined fully his concept of variables with limit zero, and he sometimes argued as if he were using actual infinitesimals. Occasionally he adopted an epsilon-delta approach. The author argues that historical evaluations of mathematical analysis may and should be made in the light of both standard and non-standard analysis. From this point of view, Cauchy's move toward founding analysis entirely on the standard real number system does not seem to have been inevitable. Some historical observations by the founder of non-standard analysis, Abraham Robinson, are extended, and in one case contested. It is shown that some of Cauchy's alleged errors are explained if he is admitted to have been thinking of actual infinitesimals and infinitely large integers. Cauchy's definitions of differential in his textbooks are examined, and the author shows that the earlier of his two definitions of total differential works well, but the later does not.     

Neo-Kantian foundations of geometry in the German Romantic period

While mathematics received relatively little attention in the idealistic systems of most of the German Romantics, it served as the foundation in the thought of the Neo-Kantian philosopher/mathematician Jakob Friedrich Fries (1773–1843). It fell to Fries to work out in detail the implications of Kant's declaration that all mathematical knowledge was synthetic a priori. In the process Fries called for a new science of the philosophy of mathematics, which he worked out in greatest detail in his Mathematische Naturphilosophie of 1822. In this work he analyzed the foundations of geometry with an eye to clearing up the historical controversy over Euclid's theory of parallels. Contrary to what might be expected, Fries' Kantian perspective provoked rather than inhibited a reexamination of Euclid's axioms. Fries' attempt to make explicit through axioms what was being implicitly assumed by Euclid while at the same time wishing to eliminate unnecessary axioms belies the claim that there was no concern to improve Euclid prior to the discovery of non-Euclidean geometry. Fries' work therefore serves as an important historical example of the difficulties facing those who wanted to provide geometry with a logically secure foundation in the era prior to the published work of Gauss, Bolyai, and others.     

Peirce the logician

This paper traces the influence of the Boolean school, and more specifically of Peirce and his students, on the development of modern logic. In the 1890s it was Schröder's Algebra derLogik that represented the state of the art. This work mentions Frege, but the quantifier notation it adopts (a variant of the modern notation) is credited to Peirce and his students O. H. Mitchell and Christine Ladd-Franklin. This notation was widely adopted; both Zermelo and Löwenheim wrote famous papers in Peirce-Schröder notation. Even Whitehead (in 1908, in his Universal Algebra) fails to mention Frege, but cites the “suggestive papers” by Mitchell and Ladd-Franklin. (Russell credits Frege, with many things, but nowhere credits him with the quantifer; if the quantifiers in Principia were devised by Whitehead, they probably come from Peirce). The aim of this paper is not to detract from our appreciation of Frege's great work, but to emphasize that its influence came largely after 1900 (after Russell pointed out its significance). Although Frege discovered the quantifier in 1879 and Peirce's student Mitchell independently discovered it only in 1883, it was Mitchell's discovery (as modified and disseminated by Peirce) that made the quantifier part of logic. And neither Löwenheim's theorem nor Zermelo set-theory depended on Frege's work at all, but only on the work of the Boole-Peirce school.     

American attempts to solve Newton's pasturage problem

Frederick Emerson's North American Arithmetic contained a “pasturage problem” which baffled his compatriots. Actually, as the Americans discovered forty-two years later, this problem was taken from Isaac Newton's Arithmetica Universalis. The history of this problem illuminates the tradition of standard artificial exercises, the isolation of American mathematics, a chain of mathematical and historical plagiarisms, and changing patterns of arithmetical reasoning.     

Wigner-type theorem on transition probability preserving maps in semifinite factors

The Wigner's theorem, which is one of the cornerstones of the mathematical formulation of quantum mechanics, asserts that every symmetry of quantum system is unitary or anti-unitary. This classical result was first given by Wigner in 1931. Thereafter it has been proved and generalized in various ways by many authors. Recently, G.P. Gehér extended Wigner's and Molnár's theorems and characterized the transformations on the Grassmann space of all rank-n projections which preserve the transition probability. The aim of this paper is to provide a new approach to describe the general form of the transition probability preserving (not necessarily bijective) maps between Grassmann spaces. As a byproduct, we are able to generalize the results of Molnár and G.P. Gehér.     

The Ladies' Diary or Woman's Almanack, 1704–1841

The existence of the Ladies' Diary or the Woman's Almanack, an 18th century English magazine devoted largely to problems and puzzles in mathematics, indicates that stereotypes about the inability of women to understand and enjoy mathematics were less strongly believed in the 18th century than they are today. The beginning of the Ladies' Diary coincides with the popularization of mathematics and the growth of mathematical literacy. However, as mathematical literacy spread in response to developing technology's requirements for more mathematically sophisticated workers, women, not part of this need, were left behind. This effect is reflected in the decline in the number of women contributors over the life of the publication.     

Nicolai Nicolaevich Luzin and the Moscow school of the theory of functions

Shortly before the revolution of 1917, four papers written by participants in N. N. Luzin's analysis seminar at Moscow University appeared in the Comptes Rendus of the Paris Academy of Sciences. The publication of these papers--written by A. Ya. Khinchin, D. E. Menshov, P. S. Aleksandrov and M. Ya. Suslin--and Luzin's monograph, The Integral and Trigono-metric Series 1915, marked the emergence of Moscow University as a center of research in the theory of functions of a real variable. This paper describes Luzin's early mathematical education at Moscow University and the three year period he spent abroad (mainly in Paris) where he wrote a series of papers whose results form the core of his influential and widely praised monograph. Finally, we will show how Luzin's ideas formed the basis for the early investigations of a series of young Moscow mathematicians.     

Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact K¨ahler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space. The conjecture for the case of maximal volume growth has been recently confirmed by G. Liu in [23]. In the first part, we will give a survey on the progress.In the second part, we will consider Yau's conjecture for manifolds with non-maximal volume growth. We will show that the finiteness of the first Chern number C_1~n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that,under bounded curvature conditions, C_1~n is automatically finite provided that there exists a positive line bundle with finite Chern number. In particular, we obtain a partial answer to Yau's uniformization conjecture on K¨ahler manifolds with minimal volume growth.     

Mathematical formalization and diagrammatic reasoning: the case study of the braid group between 1925 and 1950

The standard historical narrative regarding formalism during the twentieth century indicates the 1920s as a highpoint in the mathematical formalization project. This was marked by Hilbert’s statement that the sign stood at the beginning of pure mathematics [‘Neubegründung der Mathematik. Erste Mitteilung’, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 1 (1922), 157–177]. If one takes the braid group as a case study of research whose official goal was to symbolically formalize braids and weaving patterns, a reconsideration of this strict definition of formalism is nevertheless required. For example, does it reflect what actually occurred in practice in the mathematical research of this period? As this article shows, the research on the braid group between 1926 and 1950, led among others by Artin, Burau, Fröhlich and Bohnenblust, was characterized by a variety of practices and reasoning techniques. These were not only symbolic and deductive, but also diagrammatic and visual. Against the historical narrative of formalism as based on a well-defined chain of graphic signs that has freedom of interpretation, this article presents how these different ways of reasoning—which were not only sign based—functioned together within the research of the braid group; it will be shown how they are simultaneously necessary and complementary for each other.     

Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.     

On relaxation methods for systems of linear inequalities

In their classical papers Agmon and Motzkin and Schoenberg introduced a relaxation method to find a feasible solution for a system of linear inequalities. So far the method was believed to require infinitely many iterations on some problem instances since it could (depending on the dimension of the set of feasible soltions) converge asymptotically to a feasible solution, if one exists. Hence it could not be used to determine infeasibility.Using two lemma's basic to Khachian's polynomially bounded algorithm we can show that the relaxation method is finite in all cases and thus can handle infeasible systems as well. In spite of more refined stopping criteria the worst case behaviour of the relaxation method is not polynomially bounded as examplified by a class of problems constructed here.     

Solitons and other solutions to a new coupled nonlinear Schrodinger type equation

In this paper, the first integral method combined with Liu's theorem is applied to integrate a new coupled nonlinear Schrodinger type equation. Using this combination, more new exact traveling wave solutions are obtained for the considered equation using ideas from the theory of commutative algebra. In addition, more solutions are also obtained via the application of semi-inverse variational principle due to Ji-Huan He. The used approaches with the help of symbolic computations via Mathematica 9, may provide a straightforward effective and powerful mathematical tools for solving nonlinear partial differential equations in mathematical physics.     

A Marchenko equation for complex interactions with a regular analytic continuation

Using Marchenko's own method, it is shown that three elements are required for the existence of a Marchenko fundamental equation. These are a convergent sum over the discrete spectrum, a bounded translation operator, and sometimes when there are “spectral singularities,” a domain in the complex plane of the momentum k where the representation of the regular solution as a linear combination of the two Jost solutions is meaningful. Meanwhile, we prove that for a class of complex potentials that will be called regular, a variant of Marchenko's equation exists. Clarification of the relationship between the completeness of the two sets of solutions for the unperturbed and the perturbed equation on one hand and the existence of a fundamental equation on the other hand is also achieved.     

Global well-posedness of the velocity–vorticity-Voigt model of the 3D Navier–Stokes equations

The velocity–vorticity formulation of the 3D Navier–Stokes equations was recently found to give excellent numerical results for flows with strong rotation. In this work, we propose a new regularization of the 3D Navier–Stokes equations, which we call the 3D velocity–vorticity-Voigt (VVV) model, with a Voigt regularization term added to momentum equation in velocity–vorticity form, but with no regularizing term in the vorticity equation. We prove global well-posedness and regularity of this model under periodic boundary conditions. We prove convergence of the model's velocity and vorticity to their counterparts in the 3D Navier–Stokes equations as the Voigt modeling parameter tends to zero. We prove that the curl of the model's velocity converges to the model vorticity (which is solved for directly), as the Voigt modeling parameter tends to zero. Finally, we provide a criterion for finite-time blow-up of the 3D Navier–Stokes equations based on this inviscid regularization.     

Cauchy and the spectral theory of matrices

It is well-known that Cauchy (1829) provided the first general proof that the eigenvalues of a symmetric matrix are real. Furthermore, Cauchy's paper initiated the developments that resulted in the creation of a substantial spectral theory of matrices by the early 1870's. The following essay relates Cauchy's work and its historical significance to the consideration of eigenvalue problems during the 18th century.     

Truth versus validity in mathematical proof

In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology and mathematics education emphasize the difference between common sense and mathematical logic. In this paper, we provide both epistemological and didactic arguments to reconsider this point of view, taking into account the distinction made in logic between truth and validity on one hand, and syntax and semantics on the other. In the first part, we provide epistemological arguments showing that a central concern for logicians working with a semantic approach has been finding an appropriate distance between common sense and their formal systems. In the second part, we turn from these epistemological considerations to a didactic analysis. Supported by empirical results, we argue for the relevance of the distinction and the relationship between truth and validity in mathematical proof for mathematics education.     

Helmut Hasse in 1934

In 1934 Helmut Hasse became Professor of Mathematics at Göttingen. Hasse's attitudes and behavior during the Nazi period were representative of the ambiguous position of much of the mathematical community at that time. Various aspects of Hasse's situation make him appear to be almost the ideal example of the apolitical conservative and ideologically naive German academic in extremis. The purpose of this article is to present him as such an example.     

Fixed point theorems for precomplete numberings

In the context of his theory of numberings, Ershov showed that Kleene's recursion theorem holds for any precomplete numbering. We discuss various generalizations of this result. Among other things, we show that Arslanov's completeness criterion also holds for every precomplete numbering, and we discuss the relation with Visser's ADN theorem, as well as the uniformity or nonuniformity of the various fixed point theorems. Finally, we base numberings on partial combinatory algebras and prove a generalization of Ershov's theorem in this context.     

Inequalities induced by network connections II. Hybrid connections

It has been found that interesting mathematical relationships arise from a vectorial generalization of Kirchhoff's and Ohm's laws, in which the “resistors” become Hermitian positive semidefinite (PSD) linear operators. In analogy to the parallel connection of resistors Anderson and Duffin studied the parallel sum R : S of two PSD operators on a finite dimensional space, defined by $\text{R : S = R(R + S)}{}^2\text{S}$. Duffin and Trapp then studied the hybrid connection. This paper generalizes some of their results to a much broader class of electrical connections.