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|.     

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│.     

Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.     

INVERSE PROBLEMS FOR A GENERAL MULTI－CONNECTED BOUNDED DRUM WITH APPLICATIONS IN PHYSICS

this paper studies the influence of a finite container on an ideal gas,The trace of the heat kernel Θ(t)=∑(μ=1)^∞ exp(-tλμ），where{λμ}(μ=1)^∞ are the eigenvalues of the negative Laplacian-△n=-∑(p=1)^n (a/axp)^2 in R^n(n=2 or 3) ,is studied for a general mult-connected bounded drum Ω which is surrounded by simply connected bounded domains Ωi with smooth boundaries aΩi(i=1,……,m) where the Dirichlet ,Neumann and Robin boundary Conditions on aΩi(i=1,……,m) are considered.Some geometrical properties of Ω are determined ,The theremodynamic quantities for an ideal gas encolosed in Ω are examined by using the asymptotic expansions of Θ(t) for short-sime t.It is shown that the ideal gas can not feel the shape of its container Ω，althought it can feel some geometrical properties of it.     

BLOWING UP AND MULTIPLICITY OF SOLUTIONS FOR A FOURTH-ORDER EQUATION WITH CRITICAL NONLINEARITY

In this paper,we consider the following nonlinear elliptic problem:△~2u=|u|~(8/(n-4))u+μ|u|~(q-1)u,in Ω,△u = u = 0 on δΩ,where Ω is a bounded and smooth domain in R~n,n ∈ {5,6,7},μ is a parameter and q ∈]4/(n- 4),(12- n)/(n- 4)[.We study the solutions which concentrate around two points of Ω.We prove that the concentration speeds are the same order and the distances of the concentration points from each other and from the boundary are bounded.For Ω =(Ω_α)α a smooth ringshaped open set,we establish the existence of positive solutions which concentrate at two points of Ω.Finally,we show that for μ 0,large enough,the problem has at least many positive solutions as the LjusternikSchnirelman category of Ω.     

POSITIVE SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS WITH MIXED BOUNDARY CONDITIONS

We consider the nonlinear elliptic equation with mixed boundary conditions where Ω is a smooth bounded domain in R~N (N≥3), whose boundary Ω is made of two open sets Γ_0 and Γ_1, p= N+2/N-2. We prove the existence of a positive solution of (P)when b (x) and λ satisfy suitable conditions.     

Existence and Nonexistence of Weak Positive Solution for a Class of p-Laplacian Systems

In this work, we are interested to obtain some result of existence and nonex- istence of positive weak solution for the following p-Laplacian system
{-△piui=λifi（u1,^…,um）,inΩ, i=1,...,m, ui=0,
on δΩ,Vi=1,…,m,
where △piz = div（｜△z｜^pi-2△Z）, Pi ≥ 1,λi,1 ≤ i ≤ m are a positive parameter, and Ω is a bounded domain in IR^N with smooth boundary δΩ. The proof of the main results is based to the method of sub-supersolutions.     

THE EXISTENCE OF INFINITELY MANY SOLUTIONS OF NONLOCAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR ELLIPTIC EQUATIONS

Let Ω be a bounded domain in R~n with smooth boundary, a uniformly elliptio and self-adjoint operator, whose coefficients are real and smooth; M an arbitrary close subspace ia H~(1/2)((?)Ω), and     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

WEIGHTED A PRIORI ESTIMATES FOR SOLUTION OF （-△）^mu = f WITH HOMOGENEOUS DIRICHLET CONDITIONS

Let u be a weak solution of （-△）mu = f with Dirichlet boundary conditions in a smooth bounded domain Ω C Rn. Then, the main goal of this paper is to prove the following a priori estimate：｜｜u｜｜w2m/ω·p（Ω）≤C｜｜f｜｜L^pω（Ω）,where ω is a weight in the Muckenhoupt class Ap.     

Wavelets and Geometric Structure for Function Spaces

Abstract With Littlewood–Paley analysis, Peetre and Triebel classified, systematically, almost all the usual function spaces into two classes of spaces: Besov spaces and Triebel–Lizorkin spaces ; but the structure of dual spaces of is very different from that of Besov spaces or that of Triebel–Lizorkin spaces, and their structure cannot be analysed easily in the Littlewood–Paley analysis. Our main goal is to characterize in tent spaces with wavelets. By the way, some applications are given: (i) Triebel–Lizorkin spaces for p = ∞ defined by Littlewood–Paley analysis cannot serve as the dual spaces of Triebel–Lizorkin spaces for p = 1; (ii) Some inclusion relations among these above spaces and some relations among and L ¹ are studied. Supported by NNSF of China (Grant No. 10001027)     

Parameter Estimates in Random Intercept Mixed Effects Model for Repeated Measures

In this article the following random intercept mixed effects model will be considered： yij = vi =v^τijβ＋ εij,i=1,…,m;j=1,2,…,ni, where {vi} are i.i.d, random effects with mean α 2. 2 and finite variance σ^2 v, {εij} are i.i.d, random errors with finite variance ε^2 ε. Here we will estimate α,σ^2 v,σ^2 ε,β and study their large sample properties, such as strong consistency, strong convergence rates and asymptotic normality.     

More on Bounding Introspection in Modal Nonmonotonic Logics

Abstract By we denote the set of all propositional formulas. Let be the set of all clauses. Define . In Sec. 2 of this paper we prove that for normal modal logics , the notions of -expansions and -expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of -expansions for I and -expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of -expansions for I is the same as that of -expansions for I. The project is supported by NSFC     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

Some Limit Theorems for a Particle System of Single Point Catalytic Branching Random Walks

We study the scaling limit for a catalytic branching particle system whose particles perform random walks on Z and can branch at 0 only. Varying the initial （finite） number of particles, we get for this system different limiting distributions. To be more specific, suppose that initially there are n^β particles and consider the scaled process Zt^n（·） = Znt（√n·）, where Zt is the measure-valued process 1 and to a representing the original particle system. We prove that Ztn converges to 0 when β 〈1/4 and to a nondegenerate discrete distribution when β=1/4.In addition,if 1/4〈β〈1/2 then n-^（2β-1/2）Zt^n converges to a random limit,while if β 〉21then n^-βZtn converges to a deterministic limit.     

Strong Approximation by Cesàro Means with Critical Index in the Hardy Spaces H p (\mathbb{S}^{{d - 1}} ) (0 < p ≤ 1)

Let be a unit sphere of the d–dimensional Euclidean space ℝ ^d and let (0 < p ≤ 1) denote the real Hardy space on For 0 < p ≤ 1 and let E _j (f,H ^p ) (j = 0, 1, ...) be the best approximation of f by spherical polynomials of degree less than or equal to j, in the space Given a distribution f on its Cesàro mean of order δ > –1 is denoted by For 0 < p ≤ 1, it is known that is the critical index for the uniform summability of in the metric H ^p . In this paper, the following result is proved: Theorem Let 0<p<1 and Then for

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

Weak Hopf Algebras Corresponding to Borcherds-Cartan Matrices

Let y be a generalized Kac-Moody algebra with an integral Borcherds-Cartan matrix. In this paper, we define a d-type weak quantum generalized Kac-Moody algebra wUq^d（y）, which is a weak Hopf algebra. We also study the highest weight module over the weak quantum algebra wUdq^d（y） and weak A-forms of wUq^d（y）.