Divergence points of self-similar measures satisfying the OSC

Recently, Barreira and Schmeling (2000) [1] and Chen and Xiong (1999) [2] have shown, that for self-similar measures satisfying the SSC the set of divergence points typically has the same Hausdorff dimension as the support K. It is natural to ask whether we obtain a similar result for self-similar measures satisfying the OSC. However, with only the OSC satisfied, we cannot do most of the work on a symbolic space and then transfer the results to the subsets of Rd, which makes things more difficult. In this paper, by the box-counting principle we show that the set of divergence points has still the same Hausdorff dimension as the support K for self-similar measures satisfying the OSC.     

On certain spaces of vector measures of bounded variation

If (Σ,X) is a measurable space and X a Banach space we investigate the X-inheritance of copies of ?_∞ in certain subspaces Δ(Σ,X) of bvca(Σ,X), the Banach space of all X-valued countable additive measures of bounded variation equipped with the variation norm. Among the consequences of our main theorem we get a theorem of J. Mendoza on the X-inheritance of copies of ?_∞ in the Bochner space L₁(μ,X) and other of the author on the X-inheritance of copies of ?_∞ in bvca(Σ,X).     

The Cauchy problem for the elliptic-hyperbolic Davey-Stewartson system in Sobolev space

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems

(0.1)     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Eventual differentiability of functional differential equations in Banach spaces

This paper concerns the regularity of a functional differential equation in the form: , t>0, where A is the generator of an analytic semigroup on a Banach space X, and B₁,B₂ are α(γ−A)-bounded linear operator for 0<α<1. By spectral analysis, it is shown that the associated solution semigroup of this equation is eventually differentiable.     

Majorization of regular measures and weights with finite and positive critical exponent

For the sets , 1?p<∞, of positive finite Borel measures μ on the real axis with the set of algebraic polynomials P dense in Lp(R,dμ), we establish a majorization principle of their “boundaries,” i.e. for every there exists such that dμ/dν?1. A corresponding principle holds for the sets , p>0, of non-negative upper semi-continuous on R functions (weights) w such that P is dense in the space : For every there exists such that w?ω.     

The Dirichlet problem of higher order quasilinear elliptic equation

The paper is concerned with the Dirichlet problem of higher order quasilinear elliptic equation:

     

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

Local regularity theorems for the stationary thermistor problem with oscillating degeneracy

In this paper we present some regularity results for solutions to the system −Δu=σ(u)²|∇φ|, div(σ(u)∇φ)=0 in the case where σ(u) is allowed to oscillate between 0 and a positive number as u→∞. In particular, we show that u is locally bounded if σ(u) is bounded below by a suitable exponential function.     

A group-based search for solutions of the n-queens problem

The n-queens problem is a well-known problem in mathematics, yet a full search for n-queens solutions has been tackled until now using only simple algorithms (with the exception of the Rivin-Zabih algorithm). In this article, we discuss optimizations that mainly rely on group actions on the set of n-queens solutions. Most of our arguments deal with the case of toroidal queens; at the end, the application to the regular n-queens problem is discussed, and also the Rivin-Zabih algorithm.     

Estimates of the dilatation function of Beurling-Ahlfors extension

In this paper we estimate the dilatation function of the Beurling-Ahlfors extension in the most general case. By introducing ?h,m-function, we obtain an inequality which is sharp up to a constant.     

On the Alexandrov problem of distance preserving mapping

In this paper the author has studied the Alexandrov problem of area preserving mappings in linear 2-normed spaces and has provided some remarks for the generalization of earlier results of H.Y. Chu, C.G. Park and W.G. Park.In addition the author has introduced the concept of linear (2,p)-normed spaces and for such spaces he has solved the Alexandrov problem.     

On the Cauchy-Rassias stability of a generalized additive functional equation

Let X and Y be Banach spaces and f:X→Y an odd mapping. We solve the following generalized additive functional equation

     

Linear operators with compact supports, probability measures and Milyutin maps

The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for 0-dimensional spaces in terms of regular extension operators having compact supports. Milyutin maps are also considered and it is established that some topological properties, like paracompactness, metrizability and κ-metrizability, are preserved under Milyutin maps.     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

An approximate method via Taylor series for stochastic functional differential equations

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the Lp-norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.     

The generalized nonlinear initial-boundary Riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws

This work is a continuation of our previous work, in the present paper we study the generalized nonlinear initial-boundary Riemann problem with small BV data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space . We prove the global existence and uniqueness of piecewise C¹ solution containing only contact discontinuities to a class of the generalized nonlinear initial-boundary Riemann problem, which can be regarded as a small BV perturbation of the corresponding nonlinear initial-boundary Riemann problem, for general n×n linearly degenerate quasilinear hyperbolic system of conservation laws; moreover, this solution has a global structure similar to the one of the self-similar solution to the corresponding nonlinear initial-boundary Riemann problem. Some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given.     

Fuzzy measures and discreteness

The main result is to show that the space of nonmonotonic fuzzy measures on a measurable space (X,X) with total variation norm is separable if and only if the σ-algebra X is a finite set. Our result is related to fuzzy analysis, functional spaces and discrete mathematics.