A uniformly distributed hitting position for a two-dimensional anisotropic diffusion process: The limit normed curve

Let W₁ and W₂ be two independent Wiener processes on a half-line, and let W^(a)=(W₁, aW₂) (a≥1). We consider open neighborhoods of the initial point with uniform hitting distribution. This property uniquely determines the form of a neighborhood. The main result is as follows: such a neighborhood has a limit form as a→∞. Properties of the limit form are studied. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 228, 1996, pp. 333–348.     

A generalized solution in W2 1(0,1) of a problem concerning the deflection of a beam

A new method is proposed for formulating a boundary-value problem for a fourth-order ordinary differential equation with a solution in W₂ ¹(0, 1). This generalized formulation is based on a system of second-order equations with coefficients in W₂ ^–1 (0, 1). The existence and uniqueness of the indicated solution in this class is proven.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 90–96, 1989.     

The initial boundary-value problem with a free surface condition for the penalized equations of aqueous solutions of polymers

In this paper, we study some nonlocal problems for the Kelvin-Voight equations (1) and the penalized Kelvin-Voight equations (2): the first and second initial boundary-value problems and the first and second time periodic boundary problems. We prove that these problems have global smooth solutions of the classW _∞ ¹ (ℝ⁺;W ₂ ^2+k (Ω)),k=1,2,...;Ω⊂ℝ³. Bibliography: 25 titles. Dedicated to N. N. Uraltseva on her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 185–207. Translated by N. A. Karazeeva.     

Type II Matrices of Size Five

. A type II matrix is an n×n matrix W with non-zero entries W _i,j which satisfies , i, j=1, …, n. Two type II matrices W, W′ are said to be equivalent if W′=P ₁Δ₁ WΔ₂ P ₂ holds for some permutation matrices P ₁, P ₂ and for some non-singular diagonal matrices Δ₁, Δ₂. In the present paper, it is shown that there are up to equivalence exactly three type II matrices in M ₅(C). Received: August 15, 1996 Revised: May 16, 1997     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

Discrete spectrum of hamiltonians of some quantum system models

We study the discrete spectrum of the Hamiltonian H ₀ [Z ₁ ] of relative motion of an n-particle quantum system Z ₁ consisting of mutually identical particles of two types. The interaction of the first-type particles is described by a short-range potential W ₁ , the interaction of the second-type particles is described by a long-range potential W ₂ , and the interaction of particles of different types is described by a negative long-range potential W ₃ . Under some assumptions about the potentials W ₂ and W ₃ , we demonstrate that the discrete spectrum of the operator H ₀ [Z ₁ ] is infinite both with and without taking the permutation symmetry into account.     

On an approach to solving problems of optimal control of elliptic systems in domains of an arbitrary shape

We use the method of dummy domains to solve the problem of optimal control for elliptic systems in domains that have an arbitrary shape. An estimate of the convergence rate in the norms W₂ ¹ and L₂ is obtained. Bibliography: 4 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 72–78.     

Symmetric Designs Attached to Four-Weight Spin Models

H. Guo and T. Huang studied the four-weight spin models (X, W ₁, W ₂, W ₃, W ₄;D) with the property that the entries of the matrix W ₂ (or equivalently W ₄) consist of exactly two distinct values. They found that such spin models are always related to symmetric designs whose derived design with respect to any block is a quasi symmetric design. In this paper we show that such a symmetric design admits a four-weight spin model with exactly two values on W ₂ if and only if it has some kind of duality between the set of points and the set of blocks. We also give some examples of parameters of symmetric designs which possibly admit four-weight spin models with exactly two values on W ₂.     

One-sided approximation of functions

The best one-sided approximation of periodic functions by trigonometric polynomials of classW _p ⁰ (K) in the metric ofL is obtained. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 136–140, January, 2000.     

An estimate of the error for eigen functions in the eigenfunction problem for the operator of the linear elasticity theory

We consider an eigenfunction problem for a system of Lamé equations in a three-dimensional parallelepiped in the case of a mixed boundary-value condition on the boundary. By using Steklov averaging operators, the approximation error is given in divergent form. The accuracy of the difference scheme is studied for generalized solutions from the spaceW ₂ ³(Ω). An O(|h|^1.5)-estimate is obtained for eigenfunctions in the grid metric ofW ₂ ¹(ω). Bibliography: 5 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 100–109.     

An isomorphism extension theorem for Landau-Ginzburg B-models

Landau-Ginzburg mirror symmetry studies isomorphisms between A- and B-models, which are graded Frobenius algebras that are constructed using a weighted homogeneous polynomial W and a related symmetry group G. Given two polynomials W₁, W₂ with the same weights and same group G, the corresponding A-models built with (W₁,G) and (W₂,G) are isomorphic. Though the same result cannot hold in full generality for B-models, which correspond to orbifolded Milnor rings, we provide a partial analogue. In particular, we exhibit conditions where isomorphisms between unorbifolded B-models (or Milnor rings) can extend to isomorphisms between their corresponding orbifolded B-models (or orbifolded Milnor rings).     

Bose-Mesner Algebras Associated with Four-Weight Spin Models

It is known that for each matrix W _i and it's transpose ^t W _i in any four-weight spin model (X, W ₁, W ₂, W ₃, W ₄; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W _i ) and N( ^t W _i ) = N′(W _i ) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W ₁) = N′(W ₁) = N(W ₃) = N′(W ₃) holds. In this paper we show that all of them coincide, that is, N(W _i ), N′(W _i ), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra. Received: June 17, 1999 Final version received: Januray 17, 2000     

On the spanning trees of weighted graphs

Given a weighted graph, letW ₁,W ₂,W ₃,... denote the increasing sequence of all possible distinct spanning tree weights. Settling a conjecture due to Kano, we prove that every spanning tree of weightW ₁ is at mostk–1 edge swaps away from some spanning tree of weightW _k . Three other conjectures posed by Kano are proven for two special classes of graphs. Finally, we consider the algorithmic complexity of generating a spanning tree of weightW _k .This work was supported in part by a grant from the AT&T foundation and NSF grant DCR-8351757.Primarily supported by a 1967 Science and Engineering Scholarship from the Natural Sciences and Engineering Research Council of Canada.     

Comparison of approximation properties of generalized polynomials and splines

We establish that, for p ∈ [2, ∞), q = 1 or p = ∞, q ∈ [ 1, 2], the classes W _p^rof functions of many variables defined by restrictions on the L _p-norms of mixed derivatives of order r = (r ₁, r ₂, ..., r _m) are better approximated in the L _q-metric by periodic generalized splines than by generalized trigonometric polynomials. In these cases, the best approximations of the Sobolev classes of functions of one variable by trigonometric polynomials and by periodic splines coincide. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1011–1020, August, 1998.     

On the identifiability of multivariate survival distribution functions

Let (T₁, T₂) be a non-negative random vector which is subjected to censoring random intervals [X₁, Y₁] and [X₂, Y₂]. The censoring mechanism is such that the available informations on T₁ and T₂ are expressed by a pair of random vectors W=(W₁, W₂) and δ=(δ₁, δ₂), where W_i=max(min(Y_i, T_i), X_i) and In this paper we will show that under some mild conditions the joint survival function of T₁ and T₂ can be expressed uniquely as functional of observable joint survival functions. Our results extend recent works on the randomly right censored bivariate data case and on the univariate problem with double censoring to the bivariate data with double censoring.     

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (W,A,m){(\Omega ,\mathcal{A},\mu)} and investigating the main properties of both the Banach space

Weak convergence to the matrix stochastic integral ∫0 B dB′

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form ∫₀¹ W dW, where W(r) is standard Brownian motion. In multiple regressions and vector autoregressions with vector ARIMA processes, the theory involves weak convergence to matrix stochastic integrals of the form ∫₀¹ B dB′, where B(r) is vector Brownian motion with a non-scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to ∫₀¹ B dB′ under quite general conditions. The theory is applied to vector autoregressions with integrated processes.     

Fixed points of the smoothing transformation

Summary Let W ₁,..., W _N be N nonnegative random variables and let be the class of all probability measures on [0, ∞). Define a transformation T on by letting Tμ be the distribution of W ₁X₁+ ... + W _N X _N , where the X _i are independent random variables with distribution μ, which are independent of W ₁,..., W _N as well. In earlier work, first Kahane and Peyriere, and then Holley and Liggett, obtained necessary and sufficient conditions for T to have a nontrivial fixed point of finite mean in the special cases that the W _i are independent and identically distributed, or are fixed multiples of one random variable. In this paper we study the transformation in general. Assuming only that for some γ>1, EW _i ^γ <∞ for all i, we determine exactly when T has a nontrivial fixed point (of finite or infinite mean). When it does, we find all fixed points and prove a convergence result. In particular, it turns out that in the previously considered cases, T always has a nontrivial fixed point. Our results were motivated by a number of open problems in infinite particle systems. The basic question is: in those cases in which an infinite particle system has no invariant measures of finite mean, does it have invariant measures of infinite mean? Our results suggest possible answers to this question for the generalized potlatch and smoothing processes studied by Holley and Liggett. The research of both authors was supported in part by NSF Grant MCS 80-02732. The first author is an Alfred P. Sloan fellow     

On the accuracy of a difference scheme for solving the nonstationary coupled thermoelastic problem in stresses

For numerical solution of the coupled one-dimensional problem of dynamic thermoelasticity in stresses (strains) we construct a second-order approximating difference scheme. We study its stability and obtain an a priori estimate. We prove that the solution of the scheme converges to a generalized solution of the original problem in the Sobolev class W ₂ ² (Q_T).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 95–99.