Convergence theorems for semigroup-type families of non-self mappings

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm, where G is an unbounded subset of ℝ⁺. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.      

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001     

Finding almost squares IV

In this paper, we continue the study of almost squares; these are integers n representable as n = a · b for some . We show that almost all (in the measure–theoretic sense) short intervals [x, x + (log x)¹²] contain at least one almost square, and we consider related questions. Moreover, a result of Erdős shows that the exponent 12 cannot be smaller than 0.086 .... Received: 26 June 2008; Revised: 22 December 2008     

Two-phase Stefan problem with vanishing specific heat

The unique solvability of the two-phase Stefan problem with a small parameter ε ∈ [0; ε ₀] at the time derivative in the heat equations is proved. The solution is obtained on a certain time interval [0; t ₀] independent of ε. The solution of the Stefan problem is compared with the solution to the Hele–Shaw problem corresponding to the case ε = 0. The solutions of both problems are not assumed to coincide at the initial moment of time. Bibliography: 18 titles. Dedicated to Vsevolod Alekseevich Solonnikov on the occasion of his jubilee Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 337–363.     

Universal meromorphic approximation on Vitushkin sets

The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ _n } ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n _k } of ℕ such that converges to f(z) uniformly on K. A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ _n }={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3]. Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp. 66–75.     

Talbot quadratures and rational approximations

Many computational problems can be solved with the aid of contour integrals containing e ^z in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3^-N ), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903^-N ), can be achieved by using a different approach: best supremum-norm rational approximants to e ^z for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Carathéodory–Fejér approximation. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65D30, 41A20     

Radial Multiresolution, Cuntz Algebras Representations and an Application to Fractals

We study Bernoulli type convolution measures on attractor sets arising from iterated function systems on R. In particular we examine orthogonality for Hankel frequencies in the Hilbert space of square integrable functions on the attractor coming from a radial multiresolution analysis on R³. A class of fractals emerges from a finite system of contractive affine mappings on the zeros of Bessel functions. We have then fractal measures on one hand and the geometry of radial wavelets on the other hand. More generally, multiresolutions serve as an operator theoretic framework for the study of such selfsimilar structures as wavelets, fractals, and recursive basis algorithms. The purpose of the present paper is to show that this can be done for a certain Bessel–Hankel transform. Submitted: February 20, 2008., Accepted: March 6, 2008.     

Doubly Invariant Submodules in L �� -Spaces

A classical theorem of Wiener (Ann Math 33:1–100, 1932) on the form of a doubly invariant subspace of the shift operator in L ² over (-π, π] is generalized in three directions: The interval (-π, π] is replaced by a locally compact abelian group, L ² is replaced by L^a, a ? (0, ￥){L^{\alpha}, \alpha \in (0, \infty)}, and the measure as well as the functions of L ^α may be operator-valued.     

Generation of cosine families via Lord Kelvin’s method of images

We show that generation theorems for cosine families related to one-dimensional Laplacians in C[0, ∞] may be obtained by Lord Kelvin’s method of images, linking them with existence of invariant subspaces of the basic cosine family. This allows us to deal with boundary conditions more general than those considered before (Bátkal and Engel in J Differ Equ 207:1–20, 2004; Chill et al. in Functional analysis and evolution equations. The Günter Lumer volume, Birkhauser, Basel, pp 113–130, 2007; Xiao and Liang in J Funct Anal 254:1467–1486, 2008) and to give explicit formulae for transition kernels of related Brownian motions on [0, ∞). As another application we exhibit an example of a family of equibounded cosine operator functions in C[0, ∞] that converge merely on C ₀(0, ∞] while the corresponding semigroups converge on the whole of C[0, ∞].     

Measurable Sets With Excluded Distances

For a set of distances D = {d ₁,..., d _k } a set A is called D-avoiding if no pair of points of A is at distance d _i for some i. We show that the density of A is exponentially small in k provided the ratios d ₁/d ₂, d ₂/d ₃, …, d _k-1/d _k are all small enough. This resolves a question of Székely, and generalizes a theorem of Furstenberg–Katznelson–Weiss, Falconer–Marstrand, and Bourgain. Several more results on D-avoiding sets are presented. Received: January 2007, Revision: February 2008, Accepted: February 2008     

Hysteresis nonstationary nonlinearities

We consider an operator (variable hysteron) used to describe a nonstationary hysteresis nonlinearity (whose characteristics vary under the action of external forces) according to the Krasnosel’skii-Pokrovskii scheme. Sufficient conditions under which the operator is defined for the inputs from the class of functions H ¹[t ₀, T] satisfying the Lipschitz condition in the segment [t ₀, T] are established. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 295–309, March, 2008.     

On discrete α-unimodality

A continuous composition semigroup of probability generating functionsF≔(F _t ,t≥0) and the corresponding multiplication ⊙ _F of van Harnet al. (1982,Z. Wahrsch. Verw. Gebiete,61, 97–118) are used to introduce the concept of [F; α]-unimodality which generalizes the discrete α-unimodality due to Abouammoh (1987,Statist. Neerlandica,41, 239–244) and Alamatsaz (1993,Statist. Neerlandica,47, 245–252). We offer various characterizations and other properties of [F;α]-unimodality. Notably, several convolution results are presented. Moreover, we explore the relationship between [F;α]-unimodality and the concepts of discrete self-decomposability and stability. Finally, lower bounds for variances of [F;α]-monotone and [F;α]-unimodal random variables are derived and some examples are also mentioned. Research supported by Grant SS024 of the Research Center of Kuwait University.     

Reconstruction of Function Fields

For k an algebraic closure of the finite field , ℓ prime distinct from p and X a surface over k, we prove that the field of rational functions k(X) can be recovered from the maximal pro-ℓ-quotient of its absolute Galois group – in fact already from the second central descending series quotient of . Submitted: July 2004, Revision: October 2005, Final revision: February 2008, Accepted: February 2008     

Cauchy problem F(t, x(t), x′(t)) = 0, x(0) = 0: solvability and asymptotics of solutions

For the singular Cauchy problem, the authors find some sufficient conditions for the existence of continuously differentiable solutions x: (0, ρ] → ℝ (ρ > 0 is sufficiently small) of the form

The negative cycles polyhedron and hardness of checking some polyhedral properties

Given a graph G=(V,E) and a weight function on the edges w:E→ℝ, we consider the polyhedron P(G,w) of negative-weight flows on G, and get a complete characterization of the vertices and extreme directions of P(G,w). Based on this characterization, and using a construction developed in Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008), we show that, unless P=NP, there is no output polynomial-time algorithm to generate all the vertices of a 0/1-polyhedron. This strengthens the NP-hardness result of Khachiyan et al. (Discrete Comput. Geom. 39(1–3):174–190, 2008) for non 0/1-polyhedra, and comes in contrast with the polynomiality of vertex enumeration for 0/1-polytopes (Bussiech and Lübbecke in Comput. Geom., Theory Appl. 11(2):103–109, 1998). As further applications, we show that it is NP-hard to check if a given integral polyhedron is 0/1, or if a given polyhedron is half-integral. Finally, we also show that it is NP-hard to approximate the maximum support of a vertex of a polyhedron in ℝ ⁿ within a factor of 12/n.     

Rational points on the unit sphere

It is known that the unit sphere, centered at the origin in ℝ ⁿ , has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝ ⁿ , and every ν > 0; there is a point r = (r ₁; r ₂;…;r _n) such that:

Several modifications of DPR estimator of the tail index

In this paper, we continue the investigation of an estimator proposed in [Yu. Davydov, V. Paulauskas, and A. Račkauskas, More on p-stable convex sets in Banach spaces, J. Theor. Probab., 13:39–64, 2000] and [V. Paulauskas, A new estimator for tail index, Acta Appl. Math., 79:55–67, 2003] and considered in [V. Paulauskas and M. Vaičiulis, Once more on comparison of tail index estimators, preprint, 2010]. We propose a class of modifications of the so-called DPR estimator and demonstrate that these modifications can have better asymptotic properties than the original DPR estimator.     

The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from L P( · )(Γ)

We consider the Riemann–Hilbert problem in the following setting: find a function whose boundary values ϕ⁺(t) satisfy the condition a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and K ^{p( · )}(D;ω) is the set of functions ϕ(z) representable in the form , where ω(z) is a weight function and (K _Γφ )(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-H?lder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points. Submitted: May 13, 2007. Revised: August 8, 2007 and August 28, 2007. Accepted: November 8, 2007.     

Square-free orders for CM elliptic curves modulo <Emphasis Type="Italic">p</Emphasis>

Let E be an elliptic curve defined over , of conductor N, and with complex multiplication. We prove unconditional and conditional asymptotic formulae for the number of ordinary primes , p ≤ x, for which the group of points of the reduction of E modulo p has square-free order. These results are related to the problem of finding an asymptotic formula for the number of primes p for which the group of points of E modulo p is cyclic, first studied by Serre (1977). They are also related to the stronger problem about primitive points on E modulo p, formulated by Lang and Trotter (Bull Am Math Soc 83:289–292, 1977), and the one about the primality of the order of E modulo p, formulated by Koblitz [Pacific J. Math. 131(1):157–165, 1988].