Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact.This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC.Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.     

Banach spaces that admit support sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.

     

Decidability and Specker sequences in intuitionistic mathematics

A bounded monotone sequence of reals without a limit is called a Specker sequence. In Russian constructive analysis, Church's Thesis permits the existence of a Specker sequence. In intuitionistic mathematics, Brouwer's Continuity Principle implies it is false that every bounded monotone sequence of real numbers has a limit. We claim that the existence of Specker sequences crucially depends on the properties of intuitionistic decidable sets. We propose a schema (which we call ED ) about intuitionistic decidability that asserts “there exists an intuitionistic enumerable set that is not intuitionistic decidable” and show that the existence of a Specker sequence is equivalent to ED . We show that ED is consistent with some certain well known axioms of intuitionistic analysis as Weak Continuity Principle, bar induction, and Kripke Schema. Thus, the assumption of the existence of a Specker sequence is conceivable in intuitionistic analysis. We will also introduce the notion of double Specker sequence and study the existence of them (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

线性增长条件下一类非线性梁方程解的存在性

考察了一类非线性四阶两点边值问题的解的存在性.通过构造适当的Banach空间并且利用积分方程建立了一个存在定理.主要工具是Leray-Schauder非线性抉择.本文表明,如果非线性项在其定义域的某个有界子集上的增长速度是线性的,该问题可能存在解.     

On the theorem of T. Yamada and S. Watanabe

The well-known theorem of T. Yamada and S. Watanabe asserts that (weak) existence and pathwise uniqueness of the solution of a stochastic equation implies the existence of a strong solution. This is the most powerful tool for proving that a stochastic equation possesses a strong solution. However, pathwise uniqueness is far from being a necessary condition for this. Even if the solution is not unique in law it is also of interest to ask for strong solutions. In the present note, we will discuss in more detail the connection between pathwise uniqueness and the existence of a strong solution. We will state a condition which is not only sufficient but also necessary for the existence of a strong solution.     

Variance minimization for continuous-time Markov decision processes: two approaches

This paper studies the limit average variance criterion for continuous-time Markov decision processes in Polish spaces. Based on two approaches, this paper proves not only the existence of solutions to the variance minimization optimality equation and the existence of a variance minimal policy that is canonical, but also the existence of solutions to the two variance minimization optimality inequalities and the existence of a variance minimal policy which may not be canonical. An example is given to illustrate all of our conditions.     

一类重特征方程柯西问题的存在唯一性

讨论了一类高维重特征方程栖西问题存在唯一性中的离散现象，证明发池方程的参数取某些值时存在唯一性有离散现象发生。并建立了存在唯一性与黎曼问题的联系。     

Topologically‐based characterizations of the existence of solutions of optimization‐related problems

Necessary and sufficient conditions for the existence of solutions of optimization‐related problems, defined on general sets, are established by using topologically‐based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM‐structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two‐way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia‐Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)     

Traveling waves in delayed lattice dynamical systems with competition interactions

This paper deals with the existence of traveling wave solutions of a class of delayed system of lattice differential equations, which formulates the invasion process when two competitive species are invaders. Employing the comparison principle of competitive systems, a new cross-iteration scheme is given to establish the existence of traveling wave solutions. More precisely, by the cross-iteration, the existence of traveling wave solutions is reduced to the existence of an admissible pair of upper and lower solutions. To illustrate our main results, we prove the existence of traveling wave solutions in two delayed two-species competition systems with spatial discretization. Our results imply that the delay appeared in the interspecific competition terms do not affect the existence of traveling wave solutions.     

On the role of supercompact and extendible cardinals in logic

It is proved that the existence of supercompact cardinal is equivalent to a certain Skolem-Löwenheim Theorem for second order logic, whereas the existence of extendible cardinal is equivalent to a certain compactness theorem for that logic. It is also proved that a certain axiom schema related to model theory implies the existence of many extendible cardinals.     

图的κ-覆盖与Grbner基求解

证明图的k-覆盖存在性问题等价于一个多元多项式方程组在{0,1}范围的求解问题,并通过使用Grbner基给出一个图有k-覆盖的有效判别与求解方法,进而求得图的覆盖数和极小覆盖.     

An existence theorem for a generalized self-dual Chern–Simons equation and its application

In this paper, we establish an existence theorem for a generalized self-dual Chern–Simons equation over a doubly periodic domain and use the existence theorem to prove the existence of doubly periodic self-dual vortices in a Maxwell–Chern–Simons model with non-minimal coupling. We find a necessary and sufficient condition for the existence of solutions of the generalized Chern–Simons equation. We prove the existence result by using two methods, a super- and sub-solution method and a constrained minimization method. Our main contribution is that we find a general inequality-type constraint by using the second method and it maybe applied to some related problems with the similar structures.     

A sufficient condition for the existence of a principal eigenvalue for nonlocal diffusion equations with applications

Considerable work has gone into studying the properties of nonlocal diffusion equations. The existence of a principal eigenvalue has been a significant portion of this work. While there are good results for the existence of a principal eigenvalue equations on a bounded domain, few results exist for unbounded domains. On bounded domains, the Krein–Rutman theorem on Banach spaces is a common tool for showing existence. This article shows that generalized Krein–Rutman can be used on unbounded domains and that the theory of positive operators can serve as a powerful tool in the analysis of nonlocal diffusion equations. In particular, a useful sufficient condition for the existence of a principal eigenvalue is given.     

Almost split sequences in tri-exact categories

We study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend a number of existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories by numerous researchers. As applications, we obtain some new results on the existence of almost split triangles in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations.     

Simple formulas defining complicated sets

We consider the question whether large cardinal axioms imply that certain complicated sets cannot be defined by simple formulas. More precisely, we ask whether the existence of larger large cardinals is compatible with the existence of a well-ordering of the real numbers that is definable by a Σ₁-formula that uses a single ordinal as a parameter. This note presents results by Ralf Schindler, Philipp Schlicht and the author showing that the existence of a well-ordering of the reals that is definable by a Σ₁-formula with parameter ω₁ is compatible with the existence of a Woodin cardinal and incompatible with the existence of a Woodin cardinal with a measurable cardinal above it. Moreover, a similar result holds for Σ₁-formulas using certain large cardinals as a parameter. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Equilibria when Agents Have Multiple Priors

We discuss the existence and the qualitative properties of equilibria when agents have multiple priors and there is only one good in each state of the world. We first prove a general existence result in infinite dimension economies. We then fully describe the equilibria in two special cases. We first consider the case of CEU maximizers that have same capacities. We next consider the case of no aggregate uncertainty. We prove that if agents have non-random initial endowments and are uncertainty averse and maximize the minimal expected utility according to a set of possible priors, then the existence of a common prior is equivalent to the existence of a unique equilibrium, the no-trade equilibrium. We lastly give a mild assumption for indeterminacy of equilibria and compute the dimension of indeterminacy.     

A Bernoulli problem with non-constant gradient boundary constraint

In this article, we present a result about the existence and convexity of solutions to a free boundary problem of Bernoulli type, with non-constant gradient boundary constraint depending on the outer unit normal. In particular, we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.     

Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.     

Parabolic regularzation of a first order stochastic partial differential equation

In this paper we use regularization methods for proving the existence and uniqueness of smooth solutions of a first order semilinear stochastic partial differential equation. The regularizations are chosen in such a way so that the known theory of stochastich parabolic Ito equations can be applied. The existence of the generalized solutions and, if the the time parameter is the whole real axis, the existence of mean square bounded generalized solutions, is also considered     

On normal precipitous ideals

An old question of T. Jech and K Prikry asks whether the existence of a precipitous ideal implies the existence of a normal precipitous one. The aim of the paper is to prove some results in the positive direction. In particular, it is shown that under some mild assumptions, the existence of a precipitous ideal over ℵ₁ implies the existence of a normal precipitous ideal over ℵ₁ once a Cohen subset is added to ℵ₂.