On the strength of marriage theorems and uniformity

Kierstead showed that every computable marriage problem has a computable matching under the assumption of computable expanding Hall condition and computable local finiteness for boys and girls. The strength of the marriage theorem reaches or if computable expanding Hall condition or computable local finiteness for girls is weakened. In contrast, the provability of the marriage theorem is maintained in even if local finiteness for boys is completely removed. Using these conditions, we classify the strength of variants of marriage theorems in the context of reverse mathematics. Furthermore, we introduce another condition that also makes the marriage theorem provable in , and investigate the sequential and Weihrauch strength of marriage theorems under that condition.     

Stability theorems for infinitely constrained mathematical programs

The primary concern of this paper is to investigate stability conditions for the mathematical program: findx E ⁿ that maximizesf(x):g ^j(x)0 for somej J, wheref is a real scalarvalued function and eachg is a real vector-valued function of possibly infinite dimension. It should be noted that we allow, possibly infinitely many, disjunctive forms. In an earlier work, Evans and Gould established stability theorems wheng is a continuous finite-dimensional real-vector function andJ=1. It is pointed out that the results of this paper reduce to the Evans-Gould results under their assumptions. Furthermore, since we use a slightly more general definition of lower and upper semicontinuous point-to-set mappings, we can dispense with the continuity ofg (except in a few instances where it is implied by convexity assumptions).     

Unification and extensions of some stability theorems in mathematical programming

In this paper, we provide the stability theorem for the program: inf{f(x, t)|xH(t)}, using the uniformlyN-type functions (also called -chainable functions^[10]). This theorem generalizes the results of Dantzig^[1], Hogan^[2], Greenberg^[3], Ying Mei-qian^[4] et al.Project supported by the Science Foundation of the Chinese Academy of Science.     

Implicit function theorems for mathematical programming and for systems of inequalities

Implicit function formulas for differentiating the solutions of mathematical programming problems satisfying the conditions of the Kuhn—Tucker theorem are motivated and rigorously demonstrated. The special case of a convex objective function with linear constraints is also treated with emphasis on computational details. An example, an application to chemical equililibrium problems, is given.Implicit function formulas for differentiating the unique solution of a system of simultaneous inequalities are also derived.     

Pauli problem and related mathematical problems

We formulate several mathematical modifications of the Pauli problem of recovering the particle state from the results of measuring its coordinates and momenta, including the most abstract statement of the problem for two projection-valued measures. We describe how the Pauli problem reduces to the problem of finding the eigenvectors of certain operators. The performed analysis introduces the problems of modifications of Pauli pairs, and the considered topics and formulated problems require further study. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 3–7, October, 2008.     

Applications of Tauberian theorems in some problems in mathematical physics

We give some multidimensional Tauberian theorems for generalized functions and show examples of their application in mathematical physics. In particular, we consider the problems of stabilizing the solutions of the Cauchy problem for the heat kernel equation, multicomponent gas diffusion, and the asymptotic Cauchy problem for a free Schrödinger equation in the norms of different Banach spaces among others.     

Physical aspect of mathematical description of the Doppler effect

We present the Doppler effect as a solution to a wave equation concerning waves generated by a moving in space source of the wave that periodically changes it's signal in time. The solution has been reached by basing on the Cagniard-de Hoop's method. The reached conclusion of the analysis proves that the Doppler effect presents the change in the frequency of the received signal as well as the change of it's phase. In conclusion these changes are inhomogeitic and anisotropic. This research has been successfully applied in telecommunication and in locating sources of radiation of electromagnetic waves. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some angularity and inertia theorems related to normal matrices

Motivated by the definition of the inertia, introduced by Ostrowski and Schneider, a notion of angularity of a matrix is defined. The angularity characterizes the distribution of arguments of eigenvalues of a matrix. It is proved that if B and C are nonsingular matrices, then B¹AB and C¹AC have the same angularity provided they are diagonal. Some well-known inertia theorems (e.g. Sylvester's law) have been deduced as corollaries of this result. The case when C is permitted to be singular is discussed next. Finally, we prove that (a) any linear transformation T, on the set of n by n complex matrices, mapping Hermitian matrices into themselves and preserving the inertia of each Hermitian matrix is of the form T(A)=C¹AC or T(A)=C^1LA′C where C is some nonsingular matrix, and (b) any linear transformation T mapping normal matrices into normal matrices and preserving the angularity of each normal matrix is also of one of the above forms, but with C=kU where k≠0 and U is unitary.     

Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements. Research supported by an NSF Graduate Research Fellowship and a Princeton Centennial Fellowship. Research supported in part by NSF CAREER award DMS-0812005 and by USA-Israeli BSF grant.     

Existences theorems of systems of vector quasi-equillibrium problems and mathematical programs with equilibrium constraint

In this paper, we introduce systems of vector quasi-equilibrium problems and prove the existence of their solutions. As applications of our results, we derive the existence theorems for solution of system of vector quasi-saddle point problem, the existences theorems of a solution of system of generalized quasi-minimax inequalities, the mathematical program with equilibrium constraint, semi-infinite and bilevel problems.     

Strong and absolute almost-convergence of vector functions and related Tauberian theorems

We discuss relations among notions of (C,1)-convergence, almost-convergence, absolute almost-convergence, and -convergence of a continuous vector-valued function f(t) as t tends to infinity. Equivalent conditions and Tauberian criterions are obtained, and some examples are exhibited.     

Aspects of mathematical modelling related to optimization

Many practical optimization problems involved mathematical models of complex real-world phenomena. This paper discusses some aspects of modelling that influence the performance of optimization methods. Information and advice are given concerning the construction of smooth models, the transformation of an optimization problem from one category to another, scaling, formulation of constraints, and techniques for special types of models.     

Metric theorems related to the Kronecker-Weyl theorem modm

Let (Ω, ?,P) be the infinite product of identical copies of the unit interval probability space. For a Lebesgue measurable subsetI of the unit interval, let \(A(N,I,\omega ) = \# \left\{ {n \leqslant N|\omega _n \varepsilon I} \right\}\) , where ω=(ω₁,ω₂,...). For integersm>1, and 0≤r<m, define $$\varepsilon (k,r,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,A(k,I,\omega ) \equiv r(\bmod m)} \\ {0\,otherwise} \\ \end{array} } \right.$$ and $$\eta (k,m,I,\omega ) = \left\{ {\begin{array}{*{20}c} {1\,if\,(A(k,I,\omega ),m) \equiv 1} \\ {0\,otherwise.} \\ \end{array} } \right.$$ A theorem ofK. L. Chung yields an iterated logarithm law and a central limit theorem for sums of the variables ε(k) and η(k).     

The blow-up rate for positive solutions of indefinite parabolic problems and related Liouville type theorems

In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.     

Complementarity forms of theorems of Lyapunov and Stein, and related results

The well-known Lyapunov's theorem in matrix theory / continuous dynamical systems asserts that a (complex) square matrix A is positive stable (i.e., all eigenvalues lie in the open right-half plane) if and only if there exists a positive definite matrix X such that AX+XA^* is positive definite. In this paper, we prove a complementarity form of this theorem: A is positive stable if and only if for any Hermitian matrix Q, there exists a positive semidefinite matrix X such that AX+XA^*+Q is positive semidefinite and X[AX+XA^*+Q]=0. By considering cone complementarity problems corresponding to linear transformations of the form I−S, we show that a (complex) matrix A has all eigenvalues in the open unit disk of the complex plane if and only if for every Hermitian matrix Q, there exists a positive semidefinite matrix X such that X−AXA^*+Q is positive semidefinite and X[X−AXA^*+Q]=0. By specializing Q (to −I), we deduce the well known Stein's theorem in discrete linear dynamical systems: A has all eigenvalues in the open unit disk if and only if there exists a positive definite matrix X such that X−AXA^* is positive definite.