Nondiscrete mathematical induction and iterative existence proofs

The author proves a simple general theorem about complete metric spaces which forms an abstract basis of existence theorem in functional analysis and numerical analysis. He shows that this theorem, the so called induction theorem, contains the classical fixed point theorem for contractive mappings as well as the closed graph theorem.He then explains the principles of application of the induction theorem, the method of nondiscrete mathematical induction which consists in reducing the given problem to a system of functional inequalities, to be satisfied by a certain function, called the rate of convergence. The fact that the rate of convergence is defined as a function and not a number makes it possible to obtain sharp estimates valid for the whole iterative process, not only asymptotically. The method of nondiscrete mathematical induction is then illustrated by means of the example of eigenvalues of almost decomposable matrices.     

Averaging over N-dimensional balls and Cauchy problem for equations of mathematical physics

In this paper, we deal with averaging over balls in E_N and derive an equality describing a relation between scale and space dependence of the average function. We also show connections with equations of mathematical physics.     

A mathematical model for induction hardening including mechanical effects

We investigate a mathematical model for induction hardening of steel. It accounts for electromagnetic effects that lead to the heating of the workpiece as well as thermomechanical effects that cause the hardening of the workpiece. The new contribution of this paper is that we put a special emphasis on the thermomechanical effects caused by the phase transitions. We take care of effects like transformation strain and transformation plasticity induced by the phase transitions and allow for physical parameters depending on the respective phase volume fractions.The coupling between the electromagnetic and the thermomechanical part of the model is given through the temperature-dependent electric conductivity on the one hand and through the Joule heating term on the other hand, which appears in the energy balance and leads to the rise in temperature. Owing to the quadratic Joule heat term and a quadratic mechanical dissipation term in the energy balance, we obtain a parabolic equation with L¹ data. We prove existence of a weak solution to the complete system using a truncation argument.     

Conditions for proving by mathematical induction to be explanatory

In this paper we consider proving to be the activity in search for a proof, whereby proof is the final product of this activity that meets certain criteria. Although there has been considerable research attention on the functions of proof (e.g., explanation), there has been less explicit attention in the literature on those same functions arising in the proving process. Our aim is to identify conditions for proving by mathematical induction to be explanatory for the prover. To identify such conditions, we analyze videos of undergraduate mathematics students working on specially designed problems. Specifically, we examine the role played by: the problem formulation, students’ experience with the utility of examples in proving, and students’ ability to recognize and apply mathematical induction as an appropriate method in their explorations. We conclude that particular combinations of these aspects make it more likely that proving by induction will be explanatory for the prover.     

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.     

Riordan阵与Cauchy多项式

给出了Cauchy多项式c_n~α(z)的定义,并导出它的生成函数.再利用Riordan阵方法得到包含Cauchy多项式的一些恒等式,获得它与广义调和多项式H_n~((r))(z),广义Stirling多项式P_(n,r)(z)的关系式.     

Conditional Cauchy equations and applications

Summary. The purpose of this paper is to develop a new perspective concerning conditional Cauchy equations - namely, that of exploiting a natural algebraic structure on the set of solutions of the equation. We then show how this approach leads to applications in both group and loop theory.     

Hille-Yosida operators and Cauchy problems

A new proof of a temporal regularity result for an abstract Cauchy problem with a Hille-Yosida operator is given having consequences on the existence of other types of solutions. New types of weak solutions are introduced.     

Angular Gaussian and Cauchy estimation

This paper proposes a unified treatment of maximum likelihood estimates of angular Gaussian and multivariate Cauchy distributions in both the real and the complex case. The complex case is relevant in shape analysis. We describe in full generality the set of maxima of the corresponding log-likelihood functions with respect to an arbitrary probability measure. Our tools are the convexity of log-likelihood functions and their behaviour at infinity.     

Generalized Cauchy Spaces

In this paper a unified theory of Cauchy spaces is presented including the classical cases of filter and sequence Cauchy spaces. To by-pass a lattice-theoretical barrier the notion of Urysohn modification of a functor is introduced. Employing this notion for many types of generalized Cauchy spaces a completion method is given.     

The Cauchy numbers

We study many properties of Cauchy numbers in terms of generating functions and Riordan arrays and find several new identities relating these numbers with Stirling, Bernoulli and harmonic numbers. We also reconsider the Laplace summation formula showing some applications involving the Cauchy numbers.