Bari bases of subspaces

We shall establish certain characteristic properties of Bari^* bases of subspaces. We shall show that a complete sequence of finite-dimensional subspaces {N _j}₁ is a Bari basis if and only if each sequence {_j{₁ (_j€N _j, _j=1) is a Bari basis of its own closed linear hull.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 461–469, April, 1969.     

High density limit theorems for nonlinear chemical reactions with diffusion

Summary The solutionX of a nonlinear reaction-diffusion equation on then-dimensional unit cubeS is approximated by a space-time jump Markov processX _v,N (law of large numbers (LLN)).X _v,N is constructed on a gridS _N onS ofN cells, wherev is proportional to the initial number of particles in each cell. The deviation ofX _v,N fromX is computed by a central limit theorem (CLT). The assumptions on the parametersv, N are for the LLN: , asN , and for the CLT: , asN . The limitY =Y ^X in the CLT, which is a generalized Ornstein-Uhlenbeck process, is represented as the mild solution of a linear stochastic partial differential equation (SPDE) and its best possible state spaces are described. The problem of stationary solutions ofY ^X in dependence ofX is also investigated.On leave from Universität Bremen. This work was supported by the Stiftung Volkswagenwerk and a grant from ONR     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

Optimal quadrature formula of Markov's and Locher's type with weight function

Summary A quadrature formula of Markov's type with a weight functionx (1–x), which has properties of formulas exact for polynomials of a given degree and properties of optimal formulas on some sets of functions, is given. The particular case of formula (where ==p=q=0) is the formula of Locher [1, 2].     

Asymptotic limit law for the close approach of two trajectories in expanding maps of the circle

Summary Given two pointsx, yS ¹ randomly chosen independently by a mixing absolutely continuous invariant measure of a piecewise expanding and smooth mapf of the circle, we consider for each >0 the point process obtained by recording the timesn>0 such that |f ⁿ (x)–f ⁿ (y)|. With the further assumption that the density of is bounded away from zero, we show that when tends to zero the above point process scaled by –1 converges in law to a marked Poisson point process with constant parameter measure. This parameter measure is given explicity by an average on the rate of expansion off.Partially supported by FAPESP grant number 90/3918-5     

Large-Deviation Local Theorem for Additive Functionals of a Markov Gaussian Field. I

We prove a local limit theorem (LLT) on Cramer-type large deviations for sums S _V = _t V ( _t ), where _t , t Z , 1, is a Markov Gaussian random field, V Z , and is a bounded Borel function. We get an estimate from below for the variance of S _V and construct two classes of functions , for which the LLT of large deviations holds.     

Reiter’s Condition <Emphasis Type="Italic">P</Emphasis><Subscript>1</Subscript> and Approximate Identities for Polynomial Hypergroups

Let K be a commutative hypergroup with the Haar measure . In the present paper we investigate whether the maximal ideals in L¹(K,) have bounded approximate identities. We will show that the existence of a bounded approximate identity is equivalent to the existence of certain functionals on the space L(K,). Finally we apply the results to polynomial hypergroups and obtain a rather complete solution for this class.The third author was partially supported by KBN (Poland) under grant 5 P03A 034 20 and by Research Training Network Harmonic Analysis and Related Problems Contract HPRN-CT-2001-00273.     

Nonlinear programming,approximation, and optimization on infinitely differentiable functions

A nonnegative, infinitely differentiable function defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ₀ ¹ (t)dt=1. In this article, the following problem is considered. Determine _k =inf ₀ ¹ |^(k)(t)|dt,k=1, 2, ..., where ^(k) denotes thekth derivative of and the infimum is taken over the set of all mollifier functions , which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that _k =k!2^2k–1,k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function ofk. Some inequalities of independent interest are also derived.This research was supported in part by the National Science Foundation, Grant No. GK-32712.     

From Ginzburg-Landau to Gross-Pitaevskii

In this note we consider the Gross-Pitaevskii equation i _t ++(1–²)=0, where is a complex-valued function defined on ^N×, and study the following 2-parameters family of solitary waves: (x, t)=e ^it v(x ₁–ct, x), where and x denotes the vector of the last N–1 variables in ^N . We prove that every distribution solution , of the considered form, satisfies the following universal (and sharp) L -bound:

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.     

Local partial regularity theorems for suitable weak solutions of a class of degenerate systems

A partial regularity theorem is established for a particular class of weak solutions to the systemu/t– div(K(u)u)=(u)¦¦², div((u))=0 on a bounded domain inR ^N . Under our assumptions, (u) may exhibit exponential decay, and thus the system may be degenerate. Our proof is based upon a blow-up argument.This work was supported in part by NSF Grant DMS9424448.     

Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.     

On the domain dependence of solutions to the two-phase Stefan problem

We prove that solutions to the two-phase Stefan problem defined on a sequence of spatial domains _n ^N converge to a solution of the same problem on a domain where is the limit of _n in the sense of Mosco. The corresponding free boundaries converge in the sense of Lebesgue measure on ^N.     

Bochner-Riesz means of functions in weak-L p

The Bochner-Riesz means of order 0 for suitable test functions on ^N are defined via the Fourier transform by . We show that the means of the critical index , do not mapL ^p,( ^N ) intoL ^p,( ^N ), but they map radial functions ofL ^p,( ^N ) intoL ^p,( ^N ). Moreover, iff is radial and in theL ^p,( ^N ) closure of test functions,S _R f(x) converges, asR+, tof(x) in norm and for almost everyx in ^N . We also observe that the means of the function|x| ^–N/p, which belongs toL ^p,( ^N ) but not to the closure of test functions, converge for nox.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

On lower bounds in radial basis approximation

We investigate the approximation by manifolds _n() generated by linear combinations of n radial basis functions on R^d of the form (|–a|), where is the thin-plate spline type function. We obtain exact asymptotic estimates for the approximation of Sobolev classes W^r(B^d) in the space L(B^d) on the unit ball B^d. AMS subject classification 41A25, 41A63, 65D07, 41A15     

On a nonlinear eigenvalue problem

In the present paper we consider a selfadjoint and nonsmooth operator-valued function on (c, d)R ¹. We suppose that the equation (L()x, x)=0,x0, has exactly one rootp(x) (c, d) and the functionf()=(L()x, x) is increasing at the pointp(x). We discuss questions of the variational theory of the spectrum. Some theorems on the variational properties of the spectrum are proved.     

Montgomery's weighted sieve for dimension two

The most well-known application of Montgomery's weighted sieve is to the so-called Brun-Titchmarsh inequality, which was proved byH. L. Montgomery andR. C. Vaughan in the form (x, k, l)2x((k)log(x/k))^–1 for 1k<x, (k, l)=1, (x, k, l) being the number of primespx andpl modk, (k) being Euler's function. In this paper an upper estimate is given for a certain class of two-dimensional sieve problems, among them bounds for the number of twin primes and the number of Goldbach representations.     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

Spinc Structures and Scalar Curvature Estimates

In this note, we look at estimates for the scalar curvature of a compact, connected Riemannian manifold Mwhich are related to spin ^c Dirac operators.We show that one may not enlarge a Kähler metric with positiveRicci curvature without making smaller somewhere on M.More generally, if f: N M is an area-nonincreasing map of a certain topological type,then the scalar curvature k of Ncannot be everywhere larger than f.If k f, then N is isometric to M × F, where F possesses a parallel untwisted spinor.We also give explicit upper bounds for min for arbitrary Riemannian metrics on certainsubmanifolds of complex projective space.In certain cases, these estimates are sharp:we give examples where equality is obtained.