Absolute extensors and absolute neighborhood extensors in asymptotic categories

This paper deals with extending maps in asymptotic categories, i.e., in categories consisting of metric spaces and asymptotically Lipschitz coarsely proper maps. We demonstrate certain examples of absolute extensors and absolute neighborhood extensors. We give some conditions under which a version of Borsuk's homotopy extension theorem holds in these categories, and in answer to a problem posed by Dranishnikov in [Russian Math. Surveys 55 (2000) 1085] we show the failure of a general homotopy extension theorem. Finally, we show that a pair of an Hadamard space and its convex subspace has the homotopy extension property.     

A tauberian theorem for random walk

LetX ₁,X ₂,... be independent random variables, all with the same distribution symmetric about 0; $$S_n = \sum\limits_{i = 1}^n {X_i } $$ It is shown that if for some fixed intervalI, constant 1<a≦2 and slowly varying functionM one has $$\sum\limits_{k = 1}^n {P\{ S_k \in I\} \sim \frac{{n^{1 - 1/\alpha } }}{{M(n)}}} (n \to \infty )$$ then theX _i belong to the domain of attraction of a symmetric stable law.     

Necessary and sufficient conditions for inclusion relations for absolute summability

We obtain a set of necessary and sufficient conditions for to imply for 1 <k ≤ s < ∞. Using this result we establish several inclusion theorems as well as conditions for the equivalence of and .     

Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.     

Necessary conditions for absolute double matrix summability factors

We obtain necessary conditions for a doubly triangular matrix A to have the property that a double series ΣΣ λ _mn b _mn is summable |A| _k whenever the series ΣΣb _mn is bounded |A| _k .     

A tauberian theorem for distributional point values

We give a tauberian theorem for boundary values of analytic functions. We prove that if is the distributional limit of the analytic function F defined in a region of the form (a, b) × (0, R), if F (x ₀ + iy)→ γ as y → 0⁺, and if f is distributionally bounded at x = x ₀, then f (x ₀) = γ distributionally. As a consequence of our tauberian theorem, we obtain a new proof of a tauberian theorem of Hardy and Littlewood. Received: 10 December 2007     

The hausdorff metric and measurable selections

We construct measurable selections for closed set-valued maps into arbitrary complete metric spaces. We do not need to make any separability assumptions. We view the set-valued maps as point-valued maps into the hyperspace and our measurability assumptions arethe usual kinds of measurability of point-valued maps in this setting. We also discuss relationship of these measurability conditions to the ones usually considered in the theory of measurable selections.     

Two CSCS-based iteration methods for solving absolute value equations

Recently, two families of HSS-based iteration methods are constructed for solving the system of absolute value equations (AVEs), which is a class of non-differentiable NP-hard problems. In this study, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for AVEs involving the Toeplitz matrix. Then, we analyze the convergence of the Picard-CSCS iteration method for solving AVEs. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iteration solver for AVEs. The advantage of these methods is that they do not require the storage of coefficient matrices at all, and the sub-system of linear equations can be solved efficiently via the fast Fourier transforms (FFTs). Therefore, computational cost and storage can be saved in practical implementations. Numerical examples including numerical solutions of nonlinear fractional diffusion equations are reported to show the effectiveness of the proposed methods in comparison with some existing methods.     

Absolute cyclicity, Lyapunov quantities and center conditions

In this paper we consider analytic vector fields X₀ having a non-degenerate center point e. We estimate the maximum number of small amplitude limit cycles, i.e., limit cycles that arise after small perturbations of X₀ from e. When the perturbation (Xλ) is fixed, this number is referred to as the cyclicity of Xλ at e for λ near 0. In this paper, we study the so-called absolute cyclicity; i.e., an upper bound for the cyclicity of any perturbation Xλ for which the set defined by the center conditions is a fixed linear variety. It is known that the zero-set of the Lyapunov quantities correspond to the center conditions (Caubergh and Dumortier (2004) [6]). If the ideal generated by the Lyapunov quantities is regular, then the absolute cyclicity is the dimension of this so-called Lyapunov ideal minus 1. Here we study the absolute cyclicity in case that the Lyapunov ideal is not regular.     

Continuity of the hausdorff dimension for piecewise monotonic maps

In this paper a piecewise monotonic mapT:X→?, whereX is a finite union of intervals, is considered. DefineR(T)= \(\mathop \cap \limits_{n = 0}^\infty \overline {T^{ - n} X} \) . The influence of small perturbations ofT on the Hausdorff dimension HD(R(T)) ofR(T) is investigated. It is shown, that HD(R(T)) is lower semi-continuous, and an upper bound of the jumps up is given. Furthermore a similar result is shown for the topological pressure.     

Convergence conditions for secant-type methods

We provide new sufficient convergence conditions for the convergence of the secant-type methods to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses recurrent functions, and Lipschitz-type and center-Lipschitz-type instead of just Lipschitz-type conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than earlier ones and under our convergence hypotheses we can cover cases where earlier conditions are violated. Numerical examples are also provided.     

Sufficient conditions for the unsolvability and solvability of the absolute value equation

We give linear-inequalities-based sufficient conditions for the unsolvability and solvability of the NP-hard absolute value equation: \(Ax-|x|=b\), where A is an \(n\times n\) square matrix. The satisfaction of the linear inequalities is easily verified using a linear program.