Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

On τ-centralizers of semiprime rings

Let R be a semiprime 2-torsion free ring, and let τ be an endomorphism of R. Under some conditions we prove that a left Jordan τ-centralizer of R is a left τ-centralizer of R. Under the same conditions we also prove that a Jordan τ-centralizer of R is a τ-centralizer of R. We thus generalize Zalar’s results to the case of τ-centralizers of R.     

On stabilizability of evolution systems of partial differential equations on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system

$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $

where D _x =(-i?/?x ₁,...-i?/?x _n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ? ⁿ , w is an unknown function, u(·,t)=P(D _x )w(·,t?h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|²)^γ for some Γ, γ∈? depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e ^-hλ.

     

Some Limit Theorems on the Increments of l^p-valued Multi-Parameter Gaussian Processes

In this paper, we establish some limit theorems on the increments of an l^p-valued multiparameter Gaussian process under weaker conditions than those of Csoergoe-Shao theorems published in Ann. Probab. (1993).     

Necessary and sufficient conditions for a curve to be the gradient range of a <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth function

We find some necessary and sufficient conditions for a plane curve to be the gradient range of a C ¹-smooth function of two variables. As one of the consequences we give the necessary and sufficient conditions on a continuous function ? under which the differential equation \(\frac{{\partial v}}{{\partial t}} = \varphi \left( {\frac{{\partial v}}{{\partial x}}} \right)\) has nontrivial C ¹-smooth solutions.     

Iterative processes based on block <Emphasis Type="Italic">H</Emphasis>-splittings

Block H-splittings of block square matrices (which, in general, have complex entries) are examined. It is shown that block H-matrices are the only ones that admit this type of splittings. Iterative processes corresponding to these splittings are proved to be convergent. The concept of a simple splitting of a block matrix is introduced, and the convergence of iterative processes related to simple splittings of block H-matrices is investigated. Multisplitting and nonstationary iterative processes based on block H-splittings are considered. Sufficient conditions for their convergence are derived, and some estimates for the asymptotic convergence rate are given.     

The Kreps-Yan theorem for Banach ideal spaces

Consider a closed convex cone C in a Banach ideal space X on some measure space with σ-finite measure. We prove that the fulfilment of the conditions C∩X ₊ = {0} and C??X ₊ guarantees the existence of a strictly positive continuous functional on X whose restriction to C is nonpositive.     

Pointwise and integral estimates for the B-Riesz potential in terms of B-maximal and B-fractional maximal functions

We study the maximal and fractional maximal functions and Riesz potentials that are generated by the generalized shift operator associated with the Laplace-Bessel operator. We obtain some pointwise and integral estimates that give a relation between the B-maximal and B-fractional maximal functions and B-Riesz potentials and extend the available results to the objects of a more general nature. Basing on these results, we prove interpolation theorems for the B-fractional maximal functions and B-Riesz potentials.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

Two-stage complex Rosenbrock schemes for stiff systems

New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.     

Bessel (Riesz) potentials on banach function spaces and their applications I theory

In this paper, we shall introduce the concept of the Bessel (Riesz) potential Köthe function spacesX ^s (X ^s ) and give some dual estimates for a class of operators determined by a semi-group in the spacesL ^q (?T, T; X ^s ) (L ^q (?T, T; X ^s )). Moreover, some time-spaceL ^p ?L ^p′ estimates for the semi-group exp(it(-Δ) ^m/2) and the operatorA:=∫ ₀ ^t exp(i(t-τ)(-Δ) ^m/2)·dτ in the Lebesgue-Besov spacesL ^q (?T,T;B _p,2 ^s are given. On the basis of these results, in a subsequent paper we shall present some further applications to a class of nonlinear wave equations.     

Uniform dimension results of multi-parameter stable processes

The problem of uniform dimensions for multi-parameter processes, which may not possess the uniform stochastic Hölder condition, is investigated. The problem of uniform dimension for multi-parameter stable processes is solved. That is, ifZ is a stable (N,d, α)-process and αN ?d, then $\forall E \subseteq \mathbb{R}_ + ^N , \dim Z\left( E \right) = \alpha \cdot \dim E$ holds with probability 1, whereZ(E) = {x : ?t ∈E,Z _t =x} is the image set ofZ onE. The uniform upper bounds for multi-parameter processes with independent increments under general conditions are also given. Most conclusions about uniform dimension can be considered as special cases of our results.     

The structure of transitive ordered permutation groups

We give some necessary and sufficient conditions for transitive l-permutation groups to be 2-transitive. We also discuss primitive components and give necessary and sufficient conditions for transitive l-permutation groups to be normal-valued.     

Factorization of Sums of Polynomials

Let A and B be monic polynomials of the same degree. Given some (or all) information about the individual polynomials A and B, in particular about their factorizations, what can be said about the factorization of their sum C=A+B? We survey results of Fell, Mason and others, and expand upon them. The famous ‘abc conjecture’ of number theory is one obvious motivation for this investigation.     

Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

On the Existence of Elementary Abelian Cycle Systems

We present some necessary and/or sufficient conditions for the existence of an elementary abelian cycle system of the complete graph. We propose a construction for some classes of perfect elementary abelian cycle systems. Finally we consider elementary abelian k-cycle systems of the complete multipartite graph.      

On strictly weakly mixing <Emphasis Type="Italic">C</Emphasis>*-dynamical systems

We consider strictly ergodic and strictly weakly mixing C*-dynamical cystems. We establish that a system is strictly weakly mixing if and only if its tensor product is strictly ergodic and strictly weakly mixing. We also investigate some weighted uniform ergodic theorem with respect to S-Besicovitch sequences for strictly weakly mixing dynamical systems.     

Conditional expectations and operator decompositions

For aC ^*-algebraA with a conditional expectation Φ:A → A onto a subalgebraB we have the linear decompositionA=B⊕H whereH=ker(Φ). Since Φ preserves adjoints, it is also clear that a similar decomposition holds for the selfadjoint parts:A ^s =B ^s ⊕H ^s (we useV ^s ={aεV;a ^*=a} for any subspaceV of A). Apply now the exponential function to each of the three termsA ^s ,B ^s , andH ^s . The results are: the setG ⁺ of positive invertible elements ofA, the setB ⁺ of positive invertible elements ofB, and the setC={e^h;h ^*=h, Φ(h)=0}, respectively. We consider here the question of lifting the decompositionA ^s =B ^s ⊕H ^s to the exponential sets. Concretely, is every element ofG ⁺ the product of elements ofB ⁺ andC, respectively, just as any selfadjoint element ofA is the sum of selfadjoint elements ofB andH? The answer is yes in the following sense: Eacha ε G ⁺ is the positive part of a productbe of elementsb ε B ⁺ and c εC, and bothb andc are uniquely determined and depend analytically ona. This can be rephrased as follows: The map (6, c) →(bc) ⁺ is an analytic diffeomorphism fromB ⁺ x C ontoG ⁺, where for any invertiblex ε A we denote with x⁺ the positive square root ofxx ^*. This result can be expressed equivalently as: The map (b, c) →bcb is a diffeomorphism between the same spaces. Notice that combining the polar decomposition with these results we can write every invertibleg ε A asg=bcu, whereb ε B ⁺,c ε C, andu is unitary. This decomposition is unique and the factorsb, c, u depend analytically ofg. In the case of matrix algebras with Φ=trace/dimension, the factorization corresponds tog=| det(g)|cu withc > 0,det(c)=1, andu unitary. This paper extends some results proved by G. Corach and the authors in [2]. Also, Theorem 2 states that the reductive homogeneous space resulting from a conditional expectation satisfies the regularity hypothesis introduced by L. Mata-Lorenzo and L. Recht in [5], Definition 11.1. The situation considered here is the ”general context” for regularity indicated in the introduction of the last mentioned paper.     

Criteria for Feller transition functions

This paper presents some conditions for the minimal Q-function to be a Feller transition function, for a given q-matrix Q. We derive a sufficient condition that is stated explicitly in terms of the transition rates. Furthermore, some necessary and sufficient conditions are derived of a more implicit nature, namely in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the q-matrix. The criteria lead to some perturbation results. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property directly in terms of the rates. An essential step in the analysis is the idea of associating the Feller property with individual states.