Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Testing for Unit Roots in a Nearly Nonstationary Spatial Autoregressive Process

The limiting distribution of the normalized periodogram ordinate is used to test for unit roots in the first-order autoregressive model _st= _s-1,t+_s,t-1- _s-1,t-1+_st. Moreover, for the sequence _n = e ^c/n , _n = e ^d/n of local Pitman-type alternatives, the limiting distribution of the normalized periodogram ordinate is shown to be a linear combination of two independent chi-square random variables whose coefficients depend on c and d. This result is used to tabulate the asymptotic power of a test for various values of c and d. A comparison is made between the periodogram test and a spatial domain test.     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

Comparison of four algorithms accelerating the convergence of a subset of logarithmic fixed point sequences

The behaviour of four algorithms accelerating the convergence of a subset of LOG is compared (LOG is the set of logarithmic sequences). This subset, denoted LOGF ₁ , is that of fixed point sequences whose associated error sequence,e _n =S _n –S, verifiese _n+1 =e _n + ₂ e _n ² + ₃ e _n ³ +... , where _{3 2} ² , ₂ < 0. The algorithms are modifications of the -algorithm and of Aitken's ² adapted to LOGF ₁ , the iterated ₂-algorithm, or Lubkin's transform, and the -algorithm of Brezinski. All of them accelerate the convergence of sequences in LOGF ₁ , but precise results are given on their relative convergence speed. This comparison is illustrated by numerical examples.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

A preconditioner for constrained and weighted least squares problems with Toeplitz structure

We study methods for solving the constrained and weighted least squares problem min _x by the preconditioned conjugate gradient (PCG) method. HereW = diag (₁, , _m ) with _{1 m} 0, andA ^T = [T ₁ ^T , ,T _k ^T ] with Toeplitz blocksT _l R ^{n × n} ,l = 1, ,k. It is well-known that this problem can be solved by solving anaugmented linear 2 × 2 block linear systemM +Ax =b, A ^T = 0, whereM =W ^–1. We will use the PCG method with circulant-like preconditioner for solving the system. We show that the spectrum of the preconditioned matrix is clustered around one. When the PCG method is applied to solve the system, we can expect a fast convergence rate.Research supported by HKRGC grants no. CUHK 178/93E and CUHK 316/94E.     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

Интегральные средни е и сходимость рядов Ф урье в точке

Interrelations between such local properties of a functionfL(0, 2) as convergence of its Fourier series, differentiability of its indefinite integral, approximative continuity and Lebesgue-point property are considered in the paper. Besides certain restrictions of structural character are imposed either on the integral modulus of continuity or on the behaviour of the deviations of the function from the integral means taken over all possible segments. In particular, it is shown that if _p , (f,)=O( ^1/p ) (1p<), then the above mentioned properties are pairwise equivalent. The results obtained are exact in a certain sense.     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

Linear Networks and Convex Polytopes

It is shown that the set [G,] of immersed linear networks in that are parallel to a given immersed linear network and have the same boundary as is a convex polyhedral subset of the configuration space of movable vertices of the graph G. The dimension of [G,] is calculated, and the number of its maximal faces is estimated. As an application, the spaces of all locally minimal and weighted minimal networks with fixed boundary and topology in are described. Bibliography: 21 titles.     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.