The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.     

A general result on precise asymptotics for linear processes of positively associated sequences

Let {εt; t ∈ Z^＋} be a strictly stationary sequence of associated random variables with mean zeros, let 0〈Eε1^2〈∞ and σ^2=Eε1^2＋1∑j=2^∞ Eε1εj with 0〈σ^2〈∞.{aj;j∈Z^＋} is a sequence of real numbers satisfying ∑j=0^∞｜aj｜〈∞.Define a linear process Xt=∑j=0^∞ ajεt-j,t≥1,and Sn=∑t=1^n Xt,n≥1.Assume that E｜ε1｜^2＋δ′〈 for some δ′〉0 and μ（n）=O（n^-ρ） for some ρ〉0.This paper achieves a general law of precise asymptotics for {Sn}.     

The 3D Inverse Problem of the Wave Equation for a General Multi-connected Vibrating Membrane with a Finite Number of Piecewise Smooth Boundary Conditions

The trace of the wave kernel μ（t） =∑ω=1^∞ exp（-itEω^1/2）, where {Eω}ω^∞=1 are the eigenvalues of the negative Laplacian -△↓2 = -∑k^3=1 （δ/δxk）^2 in the （x^1, x^2, x^3）-space, is studied for a variety of bounded domains, where -∞ 〈 t 〈 ∞ and i= √-1. The dependence of μ （t） on the connectivity of bounded domains and the Dirichlet, Neumann and Robin boundary conditions are analyzed. Particular attention is given for a multi-connected vibrating membrane Ω in Ra surrounded by simply connected bounded domains Ω j with smooth bounding surfaces S j （j = 1,……, n）, where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components Si^＊ （i = 1 ＋ kj-1,……, kj） of the bounding surfaces S j are considered, such that S j = Ui-1＋kj-1^kj Si^＊, where k0=0. The basic problem is to extract information on the geometry Ω by using the wave equation approach from a complete knowledge of its eigenvalues. Some geometrical quantities of Ω （e.g. the volume, the surface area, the mean curvuture and the Gaussian curvature） are determined from the asymptotic expansion ofexpansion of μ（t） for small │t│.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Asymptotic normality and efficiency of a weighted correlogram

For a process X(t)=Σ _j=1 ^M g _j (t)ξ _j (), where g_j(t) are nonrandom given functions, is a stationary vector-valued Gaussian process, Eξ_k(t) = 0, and Eξ_k(0) Eξ_l(τ) = r _kl(τ), we construct an estimate for the functions r _kl(τ) on the basis of observations X(t), t ∈ [0, T]. We establish conditions for the asymptotic normality of as T → ∞. We consider the problem of the optimal choice of parameters of the estimate depending on observations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 937–947, July, 1998.     

Almost orthogonal submatrices of an orthogonal matrix

Lett≥1 and letn, M be natural numbers,n<M. Leta=(a _i,j ) be ann xM matrix whose rows are orthonormal. Suppose that the ℓ₂-norms of the columns ofA are uniformly bounded. Namely, for allj Using majorizing measure estimates we prove that for every ε>0 there exists, a setI ⊃ {1,…,M} of cardinality at most such that the matrix , whereA _I =(a _i,j ) _j∈I , acts as a (1+ε)-isomorphism from ℓ ₂ ⁿ into . Research supported in part by a grant of the US-Israel BSF. Part of this research was performed when the author held a postdoctoral position at MSRI. Research at MSRI was supported in part by NSF grant DMS-9022140.     

On Banach spaces with unconditional bases

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR _n:X→X such thatR _nR_m=R_min(_n,m) ifn≠m and lim _n→∞ R _n x=x for allx∈X. We prove that, ifR _n−R_n −1 factors uniformly through somel _p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL _Λ=closed span , where , has an unconditional basis. Examples include the Hardy space .     

On one extremal problem for numerical series

Let Γ be the set of all permutations of the natural series and let α = {α _j} _j∈ℕ, ν = {ν_j} _j∈ℕ, and η = {η_j} _j∈ℕ be nonnegative number sequences for which

INJECTIVE MAPS ON PRIMITIVE SEQUENCES OVER Z/（p^e）

Let Z/(pe) be the integer residue ring modulo pe with p an odd prime and integer e ≥ 3. For a sequence (a) over Z/(pe), there is a unique p-adic decomposition (a) = (a)0 (a)1·p … (a)e-1 ·pe-1, where each (a)i can be regarded as a sequence over Z/(p), 0 ≤ i ≤ e - 1. Let f(x) be a primitive polynomial over Z/(pe) and G' (f(x), pe) the set of all primitive sequences generated by f(x) over Z/(pe). For μ(x) ∈ Z/(p)[x] with deg(μ(x)) ≥ 2 and gcd(1 deg(μ(x)),p- 1) = 1,set ψe-1 (x0, x1,…, xe-1) = xe-1·[ μ(xe-2) ηe-3 (x0, x1,…, xe-3)] ηe-2 (x0, x1,…, xe-2),which is a function of e variables over Z/(p). Then the compressing map ψe-1: G'(f(x),pe) → (Z/(p))∞,(a) (→)ψe-1((a)0, (a)1,… ,(a)e-1) is injective. That is, for (a), (b) ∈ G' (f(x), pe), (a) = (b) if and only if ψe - 1 ((a)0, (a)1,… , (a)e - 1) =ψe - 1 ((b)0,(b)1,… ,(b)e-1). As for the case of e = 2, similar result is also given. Furthermore, if functions ψe-1 and ψe-1 over Z/(p) are both of the above form and satisfy ψe-1((a)0,(a)1,… ,(a)e-1) = ψe-1((b)0,(b)1,… ,(b)e-1) for (a),(b) ∈ G'(f(x),pe), the relations between (a) and (b), ψe-1 and ψe-1 are discussed.     

Riesz multiwavelet bases generated by vector refinement equation

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L ₂(ℝ ^s ). Suppose ψ = (ψ¹,..., ψ ^r ) ^T and are two compactly supported vectors of functions in the Sobolev space (H ^μ(ℝ ^s )) ^r for some μ > 0. We provide a characterization for the sequences {ψ _jk ^l : l = 1,...,r, j ε ℤ, k ε ℤ ^s } and to form two Riesz sequences for L ₂(ℝ ^s ), where ψ _jk ^l = m ^j/2ψ ^l (M ^j ·−k) and , M is an s × s integer matrix such that lim _n→∞ M ⁻ⁿ = 0 and m = |detM|. Furthermore, let ϕ = (ϕ¹,...,ϕ ^r ) ^T and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψ^ν = (ψ^ν1,...,ψ^νr ) ^T and , ν = 1,..., m − 1 such that two sequences {ψ _jk ^νl : ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ ^s } and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ ^s } form two Riesz multiwavelet bases for ^L ₂(ℝ ^s ). The bracket product [f, g] of two vectors of functions f, g in (L ₂(ℝ ^s )) ^r is an indispensable tool for our characterization. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123)     

On the product of the distances of a point from the vertices of a polytope

Letx ₁,...,x _m be points in the solid unit sphere ofE _n and letx belong to the convex hull ofx ₁,...,x _m. Then . This implies that all such products are bounded by (2/m) ^m (m −1) ^m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z ₀)| where andp′(z ₀)=0.     

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

Problems of almost everywhere convergence related to harmonic analysis and number theory

The object of this paper is to discuss certain methods for studying almost everywhere convergence problems. We consider the generalization of the Riesz-Raikov theorem where the dilation numberϑ>1 is not necessarily an integer. It is known (see [B2]) that the averages (1/N)Σ ₁ ^N f(ϑ ⁿ x) converge a.e. to ∝ ₀ ¹ fdx wheneverϑ is algebraic andf a 1-periodic function onR satisfying ∝ ₀ ¹ |f(x)|² dx<∞. Here the particular case of rational dilation is treated. The reader is referred to [B2] for the general (algebraic) case. The following definitive relation between a.e. convergence and algebraic numbers is proved. Let {μ _j} be the sequence of measures converging weak* to the natural measureμ on the Cantor set of dissection ratioϑ. Thenf*μ _j→f*μ a.e. for allL ^∞ (T) functions iffϑ is algebraic. This fact depends on [B3] and a variant of Rota’s theorem [Ro] on a.e. convergence of certain compositions of operators. Further applications of this result in ergodic theory are presented in the last section of the paper. In section 4, a.e. convergence of Riemann sums of periodicL ²-functions is investigated. It is shown that almost surelyR _n f has a logarithmic density, where . This result complements the work of R. Salem on the subject.     

Biorthogonal bases and multiple interpolation in weighted Hardy spaces

Let {z_k=x_k+iy_k} be a sequence on upper half plane and {s_i} be the number of appearence of z_k in {z₁,z₂,...,z_k}. Suppose sup s_i<+∞. Let ω(x) be a weight belonging to A^∞ and . We Consider the weighted Hardy space and operator T_p mapping f(z)∈H _+w ^p into a sequence defined by , 0<p≤+∞, j=1,2,.... Then T_p(H _+w ^p )=l^p if and only if {z_k} is uniformly separated. Besides the effective solution for interpolation is obtained. Supported by National Science Foundation of China and Shanghai Youth Science Foundation     

On extremal problems of graphs and generalized graphs

Anr-graph is a graph whose basic elements are its vertices and r-tuples. It is proved that to everyl andr there is anε(l, r) so that forn>n ₀ everyr-graph ofn vertices andn ^{r−ε(l, r)} r-tuples containsr. l verticesx ^(j), 1≦j≦r, 1≦i≦l, so that all ther-tuples occur in ther-graph.     

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Let M ⁿ be an n-dimensional compact C ^∞-differentiable manifold, n ≥ 2, and let S be a C ¹-differential system on M ⁿ . The system induces a one-parameter C ¹ transformation group φ _t (−∞ < t < ∞) over M ⁿ and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M ⁿ . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M ⁿ of the differential system S. With respect to a given C ^∞ Riemannian metric of M ⁿ , we consider the bundle of all (n−2) spheres Q _x ⁿ⁻², x∈M, where Q _x ⁿ⁻² for each x consists of all unit tangent vectors of M ⁿ orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ _t ^# (−∞<t<∞) of . For an l-frame α = (u ₁, u ₂,⋯, u _l ) of M ⁿ at a point x in M, 1 ≥ l ≥ n−1, each u _i being in , we shall denote the volume of the parallelotope in the tangent space of M ⁿ at x with edges u ₁, u ₂,⋯, u _l by υ(α), and let . This is a continuous real function of t. Let

Quasi-reflection algebras and cyclotomic associators

We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism , where B _n ¹ is a braid group of type B. The formality isomorphism depends algebraically on a series Ψ_KZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded (N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue (N, k) of (N, k); it is a graded Lie algebra with an action of , and we give a lower bound for the character of its space of generators.      

Concentration and influences

Consider the discrete cube Ω={0,1} ^N , provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT _i (x) obtained by changing the value of theith coordinate. Then We prove that we always have Since it is easy to see that , this recovers the well known fact that ∫_Ω d(x, A)dP(x) is at most of order whenP(A)≥1/2. The new information is that ∫_Ω d(x, A)dP(x) can be of order only ifA reassembles the Hamming ball {x; ∑_1≤N x _i ≥N/2}.