Reiterated homogenization of monotone parabolic problems

Reiterated homogenization is studied for divergence structure parabolic problems of the form . It is shown that under standard assumptions on the function a(y ₁,y ₂,t,ξ) the sequence of solutions converges weakly in to the solution u of the homogenized problem .      

Epsilon-inflation with contractive interval functions

For contractive interval functions [g] we show that results from the iterative process after finitely many iterations if one uses the epsilon-inflated vector as input for [g] instead of the original output vector [x] ^k . Applying Brouwer's fixed point theorem, zeros of various mathematical problems can be verified in this way.     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

Fine Properties of the Integrated Density of States and a Quantitative Separation Property of the Dirichlet Eigenvalues

We consider one-dimensional difference Schr?dinger equations with real analytic function V(x). Suppose V(x) is a small perturbation of a trigonometric polynomial V ₀(x) of degree k ₀, and assume positive Lyapunov exponents and Diophantine ω. We prove that the integrated density of states is H?lder continuous for any k > 0. Moreover, we show that is absolutely continuous for a.e. ω. Our approach is via finite volume bounds. I.e., we study the eigenvalues of the problem on a finite interval [1, N] with Dirichlet boundary conditions. Then the averaged number of these Dirichlet eigenvalues which fall into an interval , does not exceed , k > 0. Moreover, for , this averaged number does not exceed exp , for any . For the integrated density of states of the problem this implies that for any . To investigate the distribution of the Dirichlet eigenvalues of on a finite interval [1, N] we study the distribution of the zeros of the characteristic determinants with complexified phase x, and frozen ω, E. We prove equidistribution of these zeros in some annulus and show also that no more than 2k ₀ of them fall into any disk of radius exp. In addition, we obtain the lower bound (with δ > 0 arbitrary) for the separation of the eigenvalues of the Dirichlet eigenvalues over the interval [0, N]. This necessarily requires the removal of a small set of energies. Received: February 2006, Accepted: December 2007     

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

---

Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

Direct Theorem about Approximation on a Family of Two Segments

Introduce the notation: , is the union of two segments [-1,1] and [-1 + ,1+ ], is a noninteger number, is the Hölder class with exponent on The following result announced by the authors in [J. Math. Sci. 117 (2003), No. 3] is proved. There exist numbers a ₁ ( ) , b ₁ ( ) 0 depending only on such that for any there exists a polynomial , such that . Bibliography: 11 titles.     

Subdifferentials with respect to dualities

LetX andW be two sets and: ¯R^X ¯R^W a duality (i.e., a mapping such that for all and all index setsI). We introduce and study the subdifferential of a function at a pointx _o X, with respect to. We also consider the particular cases when is a (Fenchel-Moreau) conjugation, or a -duality, or a -duality, in the sense of [8].     

Discrete spectrum and principal functions of non-selfadjoint differential operator

In this article, we consider the operator L defined by the differential expression in L ₂(–, ), where q is a complex valued function. Discussing the spectrum, we prove that L has a finite number of eigenvalues and spectral singularities, if the condition holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.     

Lexicographic quasiconcave multiobjective programming

Summary Letf _i :A R ben real-valued objective functions on a convex setA -K ^m ,K:=R orC, n, mN. Letg: A R ⁿ be defined by , where for eachxA, (i ₁ (x), ..., i _n (x)) is a permutation of (1, ...,n) such that . In this paper we treat the problem of findingx ^*A such that , wherel-max denotes the lexicographic maximum. If the f_i's are strongly quasiconcave we can reduce the problem stepwise until finally it is in the form of a scalar programming problem. Further, we consider conditions for the existence and uniqueness of a solution and discuss the relationship of the problem to the vector maximum (i.e. Pareto) and maxmin (i.e. Chebychev) problems.

---

Zusammenfassung f _i :AR seienn reellwertige Zielfunktionen über einer konvexen MengeA-K ^m ,K:=R oderC, n, mN. g:AR ⁿ sei definiert durch , wobei für jedesxA (i ₁ (x), ... i _n (x)) eine Permutation von (1, ...,n) derart ist, daß Wir betrachten das Problem, einx ^*A so zu finden, daß , wobeil-max das lexikographische Maximum bedeute. Falls dief _i stark quasikonkav sind, läßt sich das Problem stufenweise reduzieren, bis es schließlich die Gestalt eines skalaren Optimierungsproblems annimmt. Wir geben Existenz- und Eindeutigkeitsbedingungen an und besprechen Zusammenhänge mit dem Vektormaximumproblem (d.h. Pareto-Optimierung) und dem Maxmin-Problem (d.h. Tschebyscheff-Optimierung).

     

Minihypers and Linear Codes Meeting the Griesmer Bound: Improvements to Results of Hamada, Helleseth and Maekawa

This article improves results of Hamada, Helleseth and Maekawa on minihypers in projective spaces and linear codes meeting the Griesmer bound.In [10,12],it was shown that any -minihyper, with , where , is the disjoint union of points, lines,..., -dimensional subspaces. For q large, we improve on this result by increasing the upper bound on non-square, to non-square, square, , and (4) for square, p prime, p<3, to . In the case q non-square, the conclusion is the same as written above; the minihyper is the disjoint union of subspaces. When q is square however, the minihyper is either the disjoint union of subspaces, or the disjoint union of subspaces and one subgeometry . For the coding-theoretical problem, our results classify the corresponding codes meeting the Griesmer bound.     

Comparison results in a class of reaction-diffusion problems

In this paper pointwise bounds for the solutionsu (x, t) of some reaction-diffusion problems are derived. They are of the form wherez (t) is the solution of the associated kinetic equation and (x, t) is the solution of a pure diffusion problem.

---

Zusammenfassung In der vorliegenden Arbeit werden punktweise Schranken für die Lösungenu (x, t) gewisser Reaktions-Diffusions-Probleme hergeleitet. Sie haben die Form wobeiz (t) die Lösung der zugehörigen kinetischen Gleichung und (x, t) die Lösung eines reinen Diffusionsproblems ist.

     

On a singular perturbation problem related to optimal lifting in BV-space

We prove a Γ-convergence result for the family of functionals defined on H ¹(Ω) by for a given and a parameter . We show that in either of the two cases, p = 2 or , any limit of the minimizers is an optimal lifting.     

Construction of asymptotic solutions of linear singularly perturbed systems of second order with degeneracy

The paper discusses the asymptotic behavior of the general solution of a linear singularly perturbed system

Asymptotic properties of local densities of measures generated by semimartingales

For families of probability measures (P , )) generated by semimartingales, we consider the local density)(y, )= _t (y, )) _t0 of a, measureP _y with respect to the measureP whose logarithm is the difference of a local martingale and a positive predictable increasing locally bounded process. Conditions are obtained under which the relations and hold, wherey _t depends in some way ont, while _t ast . Applications of these relations are exhibited and an example is given when the hypotheses of the theorems proved can be verified.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 48–55, 1986.     

On the ε-optimal control of quasilinear integral equations

We consider an optimal control problem for the stochastic quasilinear integro-functional equation , with cost functional . Two controls are introduced: a program control and a feedback control, generated by the optimal control of the problem under consideration when=0. It is proved that both controls are-optimal for the original problem.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 96–100, 1986.     

On a Family of Hyperplane Arrangements Related to the Affine Weyl Groups

Let be an irreducible crystallographic rootsystem in a Euclidean space V, with ⁺ theset of positive roots. For , , let be the hyperplane . We define a set of hyperplanes . This hyperplane arrangement is significant inthe study of the affine Weyl groups. In this paper it is shown that thePoincaré polynomial of is , where n is the rank of and h is the Coxeter number of the finiteCoxeter group corresponding to .     

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

On the riemann zeta-function and the divisor problem II

Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of . If E ^*(t)=E(t)-2πΔ^*(t/2π) with , then we obtain

On functions having Cauchy differences in some prescribed sets

Summary AssumeE is a real topological vector space,F is a real Banach space,K is a discrete subgroup ofF andC is a symmetric, convex and compact subset ofF such thatK (6C) = {0}. If a functionh:E F is continuous at at least one point andh(x + y) – h(x) – h(y) K + C for allx, y E, then there exists a continuous linear functiona:E F such thath(x) – a(x) K + C for everyx E.     

Strong topologies on vector-valued function spaces

Let be a real Banach space and let E be an ideal of L ⁰ over a -finite measure space (, , ). Let (X) be the space of all strongly -measurable functions f: X such that the scalar function , defined by , belongs to E. The paper deals with strong topologies on E(X). In particular, the strong topology the order continuous dual of E(X)) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.