与Gauss测度有关的椭圆方程比较结果中等号成立的条件

In this paper, we deal with a Dirichlet problem for linear elliptic equations related to Gauss measure. For this problem, we study the converse of some inequalities proved by other authors, in the sense that we study the case of equalities and show that equalities are achieved only in the ＂symmetrized＂ situations. In addition, under other assumptions, we give a different form of comparison results and discuss the corresponding case of equalities.     

Uniqueness results for optimization problems with prescribed rearrangement

In this paper we study the case of equalities in some comparison results for L ¹-norm or L -norm of the solutions of Dirichlet elliptic problem or Hamilton-Jacobi equations. We show that equalities are achieved only in spherically symmetric situations.Work partially supported by MURST (40%).     

Approximation of Poisson integrals by de la Valleé-Poussin sums in uniform and integral metrics

On the classes of Poisson integrals of functions belonging to the unit balls of the spaces L _s , 1 ≤ s ≤ ∞, we establish asymptotic equalities for upper bounds of approximations by de la Vallée-Poussin sums in the uniform metric. Asymptotic equalities are also obtained for the case of approximation by de la Vallée-Poussin sums in the metrics of the spaces L _s , 1 ≤ s ≤ ∞, on the classes of Poisson integrals of functions belonging to the unit ball of the space L ₁.     

Approximation of the classes <Emphasis Type="Italic">C</Emphasis><Stack><Subscript>β</Subscript><Superscript>ψ</Superscript></Stack><Emphasis Type="Italic">H</Emphasis><Subscript>ω</Subscript> by generalized Zygmund sums

We obtain asymptotic equalities for the least upper bounds of approximations by Zygmund sums in the uniform metric on the classes of continuous 2π-periodic functions whose (ψ, β)-derivatives belong to the set H _ω in the case where the sequences ψ that generate the classes tend to zero not faster than a power function.     

On the order of relative approximation of classes of differentiable periodic functions by splines

In the case where n → ∞, we obtain order equalities for the best L _q -approximations of the classes W _p ^r , 1 ≤ q ≤ p ≤ 2, of differentiable periodical functions by splines from these classes.     

Evolution Differential Inclusion with Projection for Solving Constrained Nonsmooth Convex Optimization in Hilbert Space

This paper introduces a projection subgradient system modeled by an evolution differential inclusion to solve a class of hierarchical optimization problems in Hilbert space. Basing on the Moreau–Yosida approximation, we prove the global existence and uniqueness of the solution of the proposed evolution differential inclusion with projection and the unique solution of the proposed system is just its “slow solution” when the constrained set is defined by the affine equalities. When the outer layer objective function ψ is strongly convex, any solution of the proposed system is strongly convergent to the unique minimizer of the constrained optimization problem, while, the strongly convergence is also given when the inner layer objective function ϕ is strongly convex. Furthermore, we present some other optimization problem models, which can be solved by the proposed system. All the results obtained are new not only in the infinite dimensional Hilbert space framework but also in the finite dimensional space.     

On some criteria of equality and inequality of random variables

In this paper we propose convenient criteria for equalities or inequalities at levels of random variables. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 5–9, Perm, 1993.     

Exhausters, optimality conditions and related problems

The notions of exhausters were introduced in (Demyanov, Exhauster of a positively homogeneous function, Optimization 45, 13–29 (1999)). These dual tools (upper and lower exhausters) can be employed to describe optimality conditions and to find directions of steepest ascent and descent for a very wide range of nonsmooth functions. What is also important, exhausters enjoy a very good calculus (in the form of equalities). In the present paper we review the constrained and unconstrained optimality conditions in terms of exhausters, introduce necessary and sufficient conditions for the Lipschitzivity and Quasidifferentiability, and also present some new results on relationships between exhausters and other nonsmooth tools (such as the Clarke, Michel-Penot and Fréchet subdifferentials).     

Comments on the general duality survey by J. Tind and L.A. Wolsey

In this note we comment on Tind and Wolsey [11]. It seems that with a number of duality schemes in their paper neither a dual objective function, nor converse duality can properly be defined. Moreover, the paper is restricted to perturbing right-hand sides of (in)equalities only, hence to what is sometimes called ‘Lagrangean duality’. We show how one can remedy these points. In doing so, everything comes close to working with modified Lagrangeans.     

Ensuring the boundedness of the core of games with restricted cooperation

The core of a cooperative game on a set of players N is one of the most popular concepts of solution. When cooperation is restricted (feasible coalitions form a subcollection \(\mathcal{F}\) of 2 ^N ), the core may become unbounded, which makes its usage questionable in practice. Our proposal is to make the core bounded by turning some of the inequalities defining the core into equalities (additional efficiency constraints). We address the following mathematical problem: can we find a minimal set of inequalities in the core such that, if turned into equalities, the core becomes bounded? The new core obtained is called the restricted core. We completely solve the question when \(\mathcal{F}\) is a distributive lattice, introducing also the notion of restricted Weber set. We show that the case of regular set systems amounts more or less to the case of distributive lattices. We also study the case of weakly union-closed systems and give some results for the general case.     

Approximation of functions from the classes $ C_{\beta, \infty }^\psi $ by biharmonic Poisson integrals

We deduce asymptotic equalities for the upper bounds of deviations of biharmonic Poisson integrals on the classes of (ψ, β)-differentiable periodic functions in the uniform metric.     

Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

Approximation of (ψ, β)-differentiable functions by Poisson integrals in the uniform metric

We obtain asymptotic equalities for upper bounds of approximations of functions from the class C _β,∞^ψ by Poisson integrals in the metric of the space C.     

Universal similarity factorization equalities for commutative quaternions and their matrices

In this work, we have established universal similarity factorization equalities over the commutative quaternions and their matrices. Based on these equalities, real matrix representations of commutative quaternions and their matrices have been derived, and their algebraic properties and fundamental equations have been determined. Moreover, illustrative examples are provided to support our results.     

On Chern characters of algebraic hypersurface with arbitrary singularities

In this paper, we discuss the Chern characters of hypersurfaces with arbitrary singularities. When the hypersurface is smooth, the Chern characters are just the usual Chern numbers. For any given dimension, we prove that the Chern characters satisfy a kind of inequalities. And we discover that the Chern characters satisfy some kind of equalities when the dimension is greater than 3. Therefore we obtain some more inequalities which are satisfied by the Chern characters of hypersurfaces with arbitrary singularities. The definition of Chern characters see [5]—From editor.     

Inequalities in completely convex stochastic programming

A two-stage stochastic programming problem in which the random variable enters in a convex manner is called completely convex. For such problems we give a sequence of inequalities and equalities showing the equivalence of optimality over plans and optimality of a two-stage procedure related to dynamic programming and giving upper bounds on the expected value of perfect information. Our assumptions are the weakest possible to guarantee the results in the completely convex case and supersede previous related results which have received erroneous proofs or have been established under highly restrictive conditions. In the course of our argument we exhibit a new measurable selection theorem and a rather general form of Jensen's inequality. We also present a multistage generalization of our central theorem.     

Some remarks on the study of good contractions

LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK _x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.     

On the best polynomial approximation of entire transcendental functions of generalized order

We prove a Hadamard-type theorem that associates the generalized order of growth of an entire transcendental function ƒ with the coefficients of its expansion in a Faber series. This theorem is an extension of one result of Balashov to the case of a finite simply connected domain G with boundary γ belonging to the Al'per class Λ*. Using this theorem, we obtain limit equalities that associate with a sequence of the best polynomial approximations of ƒ in certain Banach spaces of functions analytic in G. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1011–1026, August, 2008.     

<Emphasis Type="Italic">I</Emphasis>-projection onto isotonic cones and its applications to maximum likelihood estimation for log-linear models

A frequently occurring problem is to find a probability vector,p∈D, which minimizes theI-divergence between it and a given probability vector π. This is referred to as theI-projection of π ontoD. Darroch and Ratcliff (1972,Ann. Math. Statist.,43, 1470–1480) gave an algorithm whenD is defined by some linear equalities and in this paper, for simplicity of exposition, we propose an iterative procedure whenD is defined by some linear inequalities. We also discuss the relationship betweenI-projection and the maximum likelihood estimation for multinomial distribution. All of the results can be applied to isotonic cone.     

Commute times of random walks on trees

In this paper we provide exact formula for the commute times of random walks on spherically symmetric random trees. Using this formula we sharpen some of the results presented in Al-Awadhi et al. to the form of equalities rather than inequalities.