High levelm-harmonic cardinal B-splines

We generalize the notion ofm-harmonic cardinal B-spline defined in [Rabut, [6c]] to obtain B-splines on an infinite regular grid, which are halfway between elementary B-splines and the Lagrangean cardinal spline function. We give the main properties of these functions: Fourier transform, decay when x , integration,P _k -reproduction (fork<-2m–1) of the associated B-spline approximation, etc. We show that, in some sense, high levelm-harmonic B-splines may be considered as a finer regular approximation of the Dirac distribution than the elementarym-harmonic B-splines are.     

Tuning the mesh of a mixed method for the stream function Vorticity formulation of the Navier-Stokes equations

Summary In this article we study a new mixed method for the Stokes and Navier-Stokes equations. The method uses two meshes, one very fine for and a coarser one for . Error estimates show that boundary layers do not require to refine the mesh for the stream function as much as for the vorticity when the Reynolds number is large. We prove estimates and study implementation problems.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Implicit elliptic boundary-value problems with discontinuous nonlinearities

Let be a bounded domain in ⁿ (n3) having a smooth boundary, let be an essentially bounded real-valued function defined on × ^h, and let be a continuous real-valued function defined on a given subset Y of Y ^h. In this paper, the existence of strong solutions u W ^2,p (, ^h) W _o ^1,p (n/2<p<+) to the implicit elliptic equation (–u)=(x,u), with u=(u₁, u₂, ..., u_h) and u=(u ₁, u ₂, ..., u _h), is established. The abstract framework where the problem is placed is that of set-valued analysis.     

A remark on fixed points of functors in topological categories

For a topological category over Set we prove that if a functor T: has a fixed cardinal (i.e. for each object K with card (UK)= we have card (UTK)), then T has a least fixed point, and if T has a successive pair of fixed cardinals and ⁺, then T has a greatest fixed point. This extends results of Adámek and Koubek.Partial financial support of the Grant Agency of the Czech Republic under Grant No. 201/93/0950 is gratefully acknowledged.     

Über die Anzahl der Lösungen der diskreten Theodorsen-Gleichung

Summary Discretization of the Theodorsen integral equation (T) yields the discrete Theodorsen-equation (T _d ), a system of 2N nonlinear equations. A so-called -condition may be fulfilled. It is known that (T) has exactly one continuous solution. This solution gives the boundary correspondence of the normalized conformal map of the unit disc onto a given domainG. It is also known that (T _d ) has one and only one solution if <1 and at least one solution if 1. We show here that for every 1 and N\ {1} there is a domainG satisfying an -condition such that (T _d ) has an infinite number of solutions. Moreover, givenK>0 and any domainG that fulfills an -condition, we will construct a domainG ₁ in the neighbourhood ofG that fulfills a max (1, +K)-condition such that (T _d ) forG ₁ has an infinite number of solutions. The underlying idea of the construction of those domains allows also to give important new facts about iterative methods for the solution of (T _d ), even in the case <1.

     

Moduli of smoothness and best approximation in the spacesL p , 0<p<1

L ^p , 0<<1, .     

Martingale Hardy spaces,BMO and VMO spaces with nonlinearly ordered stochastic basis

VMO. , ⁺, «» VMO . « » .     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

Computing the Hilbert transform of a Jacobi weight function

We discuss the evaluation of the Hilbert transformf _–1 ¹ (t-)^–1 w(, )(t)dt,–1<<1, of the Jacobi weight functionw(, )(t)=(1–t))(1+t) by analytic and numerical means and also comment on the recursive computation of the quantitiesf _–1 ¹ )(t–)^–1 _n (t;w ^{(, ))} w ^{(, )}(t)dt,n=0, 1, 2, ..., where _n (·;w ^{(, )}) is the Jacobi polynomial of degreen.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561. The work of the second author was supported by the National Science Foundation under grant DMS-8419086.     

Parallel continuation-based global optimization for molecular conformation and protein folding

This paper presents our recent work on developing parallel algorithms and software for solving the global minimization problem for molecular conformation, especially protein folding. Global minimization problems are difficult to solve when the objective functions have many local minimizers, such as the energy functions for protein folding. In our approach, to avoid directly minimizing a difficult function, a special integral transformation is introduced to transform the function into a class of gradually deformed, but smoother or easier functions. An optimization procedure is then applied to the new functions successively, to trace their solutions back to the original function. The method can be applied to a large class of nonlinear partially separable functions including energy functions for molecular conformation and protein folding. Mathematical theory for the method, as a special continuation approach to global optimization, is established. Algorithms with different solution tracing strategies are developed. Different levels of parallelism are exploited for the implementation of the algorithms on massively parallel architectures.     

On the discrete mean square of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions at 1

There have been many results obtained so far for the mean square of the (absolute) value of the Dirichlet L-function L(s,) in the critical strip 0<<1, especially on the critical line , but relatively few results were known for discrete mean value of |L(1,)|² till W. Zhang had published papers improving the error term step by step, which have recently been superseded by M. Katsurada and K.Matsumoto in which they succeeded in deriving an asymptotic formula for ₀|L(1,)|². The object of our paper is to point out a structural property contained in the formation of the mean square, to find out the niryana–the true body of the above sum.Dedicated to Professor Jean Louis Nicolás on his sixtieth birthdayin final form: 7 October 2003     

О предельном распределении для сумм независимых слууайных велиуин на бикомпактных коммутативных топологиуеских группах

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Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs y=6y ²–x , which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1.     

Orthonormal systems satisfying an inequality of S. M. Nikol'skii

N- (p, q) (1 pN-, L _p - L _q -. , , , L L _q - , , .     

An error analysis for absolute and relative approximation

We consider the vectorial algorithm for finding best polynomial approximationsp P _n to a given functionf C[a, b], with respect to the norm · _s , defined byp – f _s =w ₁ (p – f)+w ₂ (p – f) A bound for the modulus of continuity of the best vectorial approximation operator is given, and using the floating point calculus of J. H. Wilkinson, a bound for the rounding error in the algorithm is derived. For givenf, these estimates provide an indication of the conditioning of the problem, an estimate of the obtainable accuracy, and a practical method for terminating the iteration.This paper was supported in part by the Canadian NCR A-8108, FCAC 74-09 and G.E.T.M.A.Part of this research was done during the first-named author's visit to theB! Chair of Applied Mathematics, University of Athens, Spring term, 1975.     

On aU-set for multiple Walsh series

m- (m2) , - , .     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

A quantitative refinement of Rado's theorem

The fundamental result of the paper is the following. Theorem: Let be a k-quasiconformal Jordan curve and let be another Jordan curve (not necessarily quasiconformal). Assume that f maps conformallyext ontoext , f()=, f()>0. We assume that there exists a homeomorphism between and such that Then there exist numbers =(k)>0 and A=A(k), such that f(())– A, .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 103–112, 1987.