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.     

First cohomology group of rank two Witt algebra to its Larsson modules

Let U = ℂ², Γ = ℤ², and let ℂ[x ₁^±1, x ₂^±1] be the ring of Laurent polynomials. The Witt algebra L is the Lie algebra of derivations over ℂ[x ₁^±1, x ₂^±1], which is spanned by elements of the form D(u, r) = x ^r (u ₁ d ₁ + u ₂ d ₂), u = (u ₁, u ₂) ∈ U, r ∈ Γ, where d ₁ and d ₂ are the degree derivations of ℂ[x ₁^±1, x ₂^±1]. The image of gl ₂-module V under Larsson functor F ^α , denoted by W = F ^α (V), gives a class of L-modules, often called the Larsson-modules of L. In this paper, we study the derivations from the Witt algebra L to its Larsson-modules W, and we determine the first cohomology group H ¹(L,W).     

Bipartite Distance-regular Graphs, Part I

Let Γ=(X,E) denote a bipartite distance-regular graph with diameter D≥4, and fix a vertex x of Γ. The Terwilliger algebra T=T(x) is the subalgebra of Mat _X(C) generated by A, E ^* ₀, E ^* ₁,…,E ^* _D, where A denotes the adjacency matrix for Γ and E ^* _i denotes the projection onto the i ^TH subconstituent of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE ^* _i W≤1 for 0≤i≤Di. The endpoint of W is min{i|E ^* _i W≠0}. We determine the structure of the (unique) irreducible T-module of endpoint 0 in terms of the intersection numbers of Γ. We show that up to isomorphism there is a unique irreducible T-module of endpoint 1 and it is thin. We determine its structure in terms of the intersection numbers of Γ. We determine the structure of each thin irreducible T-module W of endpoint 2 in terms of the intersection numbers of Γ and an additional real parameter ψ=ψ(W), which we refer to as the type of W. We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we show that the intersection numbers of Γ are determined by the diameter D of Γ and the set of ordered pairs

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     

Disjoint homotopic paths and trees in a planar graph

In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ², a subset {I ₁, …,I _p} of the faces ofG, and pathsC ₁, …,C _k inG, with endpoints on the boundary ofI ₁ ∪ … ∪I _p; find: pairwise disjoint simple pathsP ₁, …,P _k inG so that, for eachi=1, …,k, P _i is homotopic toC _i in the space ℝ²\(I ₁ ∪ … ∪I _p). Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ² and pairwise disjoint setsW ₁, …,W _k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT ₁, …,T _k ofG whereT _i (i=1, …, k).     

Andrews–Gordon type identities from combinations of Virasoro characters

For p∈{3,4} and all p′>p, with p′ coprime to p, we obtain fermionic expressions for the combination χ _1,s ^p,p′+q ^Δ χ _p−1,s ^p,p′ of Virasoro (W ₂) characters for various values of s, and particular choices of Δ. Equating these expressions with known product expressions, we obtain q-series identities which are akin to the Andrews–Gordon identities. For p=3, these identities were conjectured by Bytsko. For p=4, we obtain identities whose form is a variation on that of the p=3 cases. These identities appear to be new. The case (p,p′)=(3,14) is particularly interesting because it relates not only to W ₂, but also to W ₃ characters, and offers W ₃ analogues of the original Andrews–Gordon identities. Our fermionic expressions for these characters differ from those of Andrews et al. which involve Gaussian polynomials. BF is partially supported by grant number RFBR 05-01-01007, and OF by the Australian Research Council (ARC).     

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     

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.     

Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X ₁ be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F _* W is bounded by instability of W⊗T^ℓ(Ω¹ _X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F _* W being stable whenever W is stable. Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.     

On a stationary transport equation

Summary Let Ω, Γ,v, a andX be as described at the beginning of the introduction below, letp∈]1, +∞[, and setq=p/(p-1). Ifp>2, we also assume that the mean curvature {itx}{su(itx)} of Γ is everywhere nonnegative. In this paper we solve the existence problem in spacesX, for equation (1.1) below, ifX=W ₀ ^1,q , orX=W ⁻¹,p. As a by-product, the solvability of (1.1) in spacesW ^1,pandL ^pfollows (without any assumption on {itx}{su(itx)}). For more general results on the above problem, see ref. [1].     

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Two inverse problems for the Sturm-Liouville operator Ly = s-y″ + q(x)y on the interval [0, fy] are studied. For θ ⩾ 0, there is a mapping F:W ₂^θ → l _B ^θ, F(σ) = {s _k }₁^∞, related to the first of these problems, where W ₂^∞ = W ₂^∞[0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q, and l _B ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂^θ, where we place the regularized spectral data s = {s _k }₁^∞ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for ∥σ - σ₁∥_θ via the l _B ^θ-norm ∥s − s₁∥_θ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case q ∈ L ₂, which corresponds to θ = 1.     

Quadratically Hyponormal Recursively Generated Weighted Shifts Need Not Be Positively Quadratically Hyponormal

We study a class of weighted shifts W _α defined by a recursively generated sequence α ≡ α₀, … , α _m−2, (α _m−1, α _m , α _m+1)^∧ and characterize the difference between quadratic hyponormality and positive quadratic hyponormality. We show that a shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisfies a finite number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, W _α is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for quadratic hyponormality of recursively generated weighted shift W _α (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift W _α need not be positively quadratically hyponormal.     

On the rate of convergence of a regular martingale related to a branching random walk

Let M _n , n = 1, 2, ..., be a supercritical branching random walk in which the number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale W _n related to M _n is regular and W is a limit random variable. Let a(x) be a nonnegative function regularly varying at infinity with index greater than −1. We present sufficient conditions for the almost-sure convergence of the series . We also establish criteria for the finiteness of EW ln⁺ Wa(ln⁺ W) and E ln⁺|Z _∞|a(ln⁺|Z _∞|), where and (M _n , Q _n ) are independent identically distributed random vectors not necessarily related to M _n . __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 3, pp. 326–342, March, 2006.     

Limit correlation functions at zero for fixed trace random matrix ensembles

The limit as N → ∞ of the eigenvalue correlation function is studied in a neighborhood of zero for N × N Hermitian matrices chosen at random from the Hilbert-Schmidt sphere of an appropriate radius. Dyson’s famous sin π(t₁ − t₂)/π(t₁ − t₂)-kernel asymptotic is extended to this class of matrix ensembles. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 68–80.     

On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2 ^X of a compactumX is a Borel monotone mapD: 2 ^X →2 ^X . The derivative determines a Cantor-Bendixson type rank δ:2^X → ω₁ ∪ {∞} . We show that ifA⊂2 ^X is analytic andZ⊂A intersects stationary many layers δ₋₁({ξ}), then for almost all σ,F∩δ₋₁({ξ}) cannot be separated fromZ ∩∪ _a<ξ ^δ−1({a}) (and also fromZ ∩∪ _a>ξ ^δ−1({a}) by anyF _σ-set. Another main result involves a natural partial order on 2 ^X related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper. During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would like to express his gratitude to the Department of Mathematics and Statistics for the hospitality.     

Root differences for A n

Suppose W is a Weyl group with Φ = Φ(W) a root system of W. The set D of root differences is given by D = {α − β : α, β, ∈ Φ}. We define t(Φ) to be the maximum exponent of the torsion subgroup of for any In this article we show that if W is of type A_n, then t(Φ) = 2ⁿ. Received: 25 November 2004     

Analytic sets extending the graphs of holomorphic mappings

Let D, D′ ⊂ ℂⁿ be bounded domains with smooth real analytic boundaries and ƒ: D → D′ be a proper holomorphic map. Our main result implies that if the graph of ƒ extends as an analytic set to a neighborhood of a poìnt (a, a′) ∈ ∂D × 3D′ with a′ ∈ cl_ƒ(a), then ƒ extends holomorphically to a neighborhood of a.     

Submanifolds with Parallel Second Fundamental Form Studied via the Gauss Map

For an arbitrary n-dimensional Riemannian manifold N and an integer m ∈ {1,…,n−1} a covariant derivative on the Grassmann bundle ^ := G_m(T N) is introduced which has the property that an m-dimensional submanifold M ⊂ N has parallel second fundamental form if and only if its Gauss map M → ^ is affine. (For N Rⁿ this result was already obtained by J. Vilms in 1972.) By means of this relation a generalization of Cartan's theorem on the total geodesy of a geodesic umbrella can be derived: Suppose, initial data (p,W,b) prescribing a tangent space W ∈ G_m(T_pN) and a second fundamental form b at p ∈ N are given; for these data we construct an m-dimensional ‘umbrella’ M = M(p,W,b) ⊂ N the rays of which are helical arcs of N; moreover, we present tensorial conditions (not involving ) which guarantee that the umbrella M has parallel second fundamental form. These conditions are as well necessary, and locally every submanifold with parallel second fundamental form can be obtained in this way. Mathematics Subject Classifications (2000): 53B25, 53B20, 53B21.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

Return times, recurrence densities and entropy for actions of some discrete amenable groups

It is a theorem of Wyner and Ziv and Ornstein and Weiss that if one observes the initialk symbolsX ₀,…,X _k−1 of a typical realization of a finite valued ergodic process with entropyh, the waiting time until this sequence appears again in the same realization grows asymptotically like 2 ^hk [7, 12]. A similar result for random fields was obtained in [8]: in this case, one observes cubes in ℤ ^d instead of initial segments. In the present paper, we describe generalizations of this. We examine what happens when the set of possible return times is restricted. Fix an increasing sequence of sets of possible times {W _n } and defineR _k to be the firstn such thatX ₀,…,X _k−1 recurs at some time inW _n . It turns out that |W _{R k} | cannot drop below 2 ^hk asymptotically. We obtain conditions on the sequence {W _n } which ensure that |W _{R k} | is asymptotically equal to 2 ^hk . We consider also recurrence densities of initial blocks and derive a uniform Shannon-McMillan-Breiman theorem. Informally, ifU _k,n is the density of recurrences of the blockX ₀,…,X _k−1 inX _−n ,…,X _n , thenU _k,n grows at a rate of 2 ^hk , uniformly inn. We examine the conditions under which this is true when the recurrence times are again restricted to some sequence of sets {W _n }. The above questions are examined in the general context of finite-valued processes parametrized by discrete amenable groups. We show that many classes of groups have time-sequences {W _n } along which return times and recurrence densities behave as expected. An interesting feature here is that this can happen also when the time sequence lies in a small subgroup of the parameter group.