On strong solutions of Poisson's equation in Beppo Levi spaces

LetG sun (n 2) be an unbounded open set having a compact complement and a smooth boundary G of classC ². InG we consider the equations — u=f,u¦G= and prove the existence of a solutionu L ^2,q(G) providedf L ^q(G) and W ^{2 —1/q-q}(G) (1 <q < ). HereL ^2,q(G) is the space of all functionsu L _Ioc ^q (G) having all second order distributional derivatives inL ^q(G). Concerning the uniqueness of this solution we show that the corresponding nullspace has dimensionn + 1 (n 2).

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Zusammenfassung SciG ⁿ (n 2) eine unbeschränkte offene Menge mit kompaktem Komplement und mit glattem Rand G der KlasseC ². InG betrachten wir das Randwertproblem — u=f,u¦g= und beweisen die Existenz einer Lösungu L ^2,q(G) für beliebigef L ^q(G) und Randwerte W ^2-1/q,q(G) (1 <q < ). Dabei istL ^2,q(G) der Raum aller Funktionenu L _Ioc ^q (G), die Distributionsableitungen zweiter Ordnung inL ^q (G) besitzen. Bezüglich der Eindeutigkeit solcher Lösungen zeigen wir, daß der entsprechende Nullraum die Dimensionn + 1 (n 2) besitzt.

     

A counterexample to the theorem of Beppo Levi in three dimensions

Supported by the Society of Fellows and NSF Grant DMS-86-06160     

Generalizations of Sylow's Theorem

We study the properties of maximal -subgroups of a finite group.     

Integral Representations,Differentiability Properties and Limits at Infinity for Beppo Levi Functions

The first aim in the present paper is to give an integral representation for Beppo Levi functions on R ⁿ. Our integral representation is an extension of Sobolev's integral representation given for infinitely differentiable functions with compact support. As applications, continuity and differentiability properties of Beppo Levi functions are studied.Our second aim in this paper is to study the existence of limits at infinity for Beppo Levi functions. We also consider the existence of fine-type limits at infinity with respect to Bessel capacities, which yields the radial limit result at infinity.     

New Generalizations of the Scorza-Dragoni Theorem

We consider Carathéodory functions f : T × X Y, where T is a topological space with regular -finite measure, the spaces X and Y are metrizable and separable, and X is locally compact. We show that every function of this sort possesses the Scorza-Dragoni property. A similar result is also established in the case where the space T is locally compact and X = .     

On Generalizations of the Frank-Wolfe Theorem to Convex and Quasi-Convex Programmes

In this paper we are concerned with the problem of boundedness and the existence of optimal solutions to the constrained optimization problem. We present necessary and sufficient conditions for boundedness of either a faithfully convex or a quasi-convex polynomial function over the feasible set defined by a system of faithfully convex inequality constraints and/or quasi-convex polynomial inequalities, where the faithfully convex functions satisfy some mild assumption. The conditions are provided in the form of an algorithm, terminating after a finite number of iterations, the implementation of which requires the identification of implicit equality constraints in a homogeneous linear system. We prove that the optimal solution set of the considered problem is nonempty, this way extending the attainability result well known as the so-called Frank-Wolfe theorem. Finally we show that our extension of the Frank-Wolfe theorem immediately implies continuity of the solution set defined by the considered system of (quasi)convex inequalities.     

Generalizations of a Theorem of Datko and Pazy

This note gives necessary and sufficient conditions for exponential stability of semigroups of linear operators in Banach spaces. Generalizations of a well-known result due to Datko, Pazy and Neerven are obtained for the case of semigroups of operators that are not strongly continuous.     

Krasnoselskii定理几种新的推广

利用随机不动点指数,研究了随机半闭1-集压缩算子方程的随机解,同时,从不同角度推广了著名的Krasnoselskii定理,得出若干新的结果.     

Several Generalizations of the Wedderburn-Artin Theorem with Applications

We say that an R-module M is virtually semisimple if each submodule of M is isomorphic to a direct summand of M. A nonzero indecomposable virtually semisimple module is then called a virtually simple module. We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules . Our study provides several natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-module is virtually semisimple; (ii) Every finitely generated left (right) R-module is a direct sum of virtually simple R-modules; (iii) \(R\cong {\prod }_{i = 1}^{k} M_{n_{i}}(D_{i})\) where k,n ₁,…,n _k ∈ ? and each D _i is a principal ideal V-domain; and (iv) Every nonzero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form R m ₁ ⊕… ⊕ R m _k where each R m _i is either a simple R-module or a virtually simple direct summand of R.     

Generalizations of Marriage Theorem for Degree Factors

Let G be a bipartite graph with bipartition (A, B). We give new criteria for a bipartite graph to have an f -factor, a (g, f)-factor and other factors together with some applications of these criteria. These criteria can be considered as direct generalizations of Hall’s marriage theorem. Among some results, we prove that for a function \(h: A\cup B \rightarrow \{0,1,2, \ldots \}\), G has a factor F such that \(\deg _F(x)=h(x)\) for \(x\in A\) and \(\deg _H(y) \le h(y)\) for \(y\in B\) if and only if \(h(X) \le \sum _{x\in N_G(X)}\min \{h(x), e_G(x,X)\}\) for all \(X\subseteq A\).     

Generalizations of Krasnoselskii’s Theorem and Petryshyn’s Theorem

In this work, the fixed point theorems of Krasnoselskii and Petryshyn are generalized to inner product spaces. An example is given to show the applications of the theorem.     

On the descent of Levi factors

Let G be a linear algebraic group over a field k of characteristic p > 0, and suppose that the unipotent radical R of G is defined and split over k. By a Levi factor of G, one means a closed subgroup M which is a complement to R in G. In this paper, we give two results related to the descent of Levi factors. First, suppose ? is a finite Galois extension of k for which the extension degree [? : k] is relatively prime to p. If G _/? has a Levi decomposition, we show that G has a Levi decomposition. Second, suppose that there is a G-equivariant isomorphism of algebraic groups ${R \simeq Lie(R)}$ – i.e. R is a vector group with a linear action of the reductive quotient G/R. If ${G_{{/k}_{sep}}}$ has a Levi decomposition for a separable closure k _sep of k, then G has a Levi decomposition. Finally, we give an example of a disconnected, abelian, linear algebraic group G for which ${G_{{/k}_{sep}}}$ has a Levi decomposition, but G itself has no Levi decomposition.     

Generalizations of the Instability Theorem of Chetaev to Nonlinear Control Systems

In this paper, we generalize the well-known instability theoremof N.G. Chetaev concering instability of a differential equation,to derive sufficient conditions for instability at a specifiedequilibrium point of nonlinear systems that are affine in control.     

On the Levi and Lebesgue properties

The paper is devoted to a question if the Levi property is preserved by direct sums and quotients. The three-space problem for the Levi and Lebesgue properties in topological Riesz spaces is also investigated.     

On Generalizations of Commutativity

This note is concerned with generalizations of commutativity. We introduce identity-symmetric and right near-commutative, and study basic structures of rings with such ring properties. It is shown that if R is an identity-symmetric ring, then the set of all nilpotent elements forms a commutative subring of R. Moreover, identity-symmetric regular rings are proved to be commutative. The near-commutativity is shown to be not left-right symmetric, and we study some conditions under which the near-commutativity is left-right symmetric. We also examine the near-commutativity of skew-trivial extensions, which has a role in this note.     

On a Ramanujan Identity and Its Generalizations

Mathematical Notes - In the present paper, we propose a new method of derivation of number-theoretic identities which is applied to the proof of the multidimensional analogue of one of the...