Regularity of Weak Solutions of Nonlinear Equations with Discontinuous Coefficients

In this paper, we prove that the weak solutions u∈Wloc^1, p （Ω） （1 〈p〈∞） of the following equation with vanishing mean oscillation coefficients A（x）： -div[（A（x）△↓u·△↓u）p-2/2 A（x）△↓u＋│F（x）│^p-2 F（x）]=B（x, u, △↓u）, belong to Wloc^1, q （Ω）（A↓q∈（p, ∞）, provided F ∈ Lloc^q（Ω） and B（x, u, h） satisfies proper growth conditions where Ω ∪→R^N（N≥2） is a bounded open set, A（x）=（A^ij（x）） N×N is a symmetric matrix function.     

Fundamental Solution of the Anisotropic Porous Medium Equation

We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1（u^mi）xixi in R^n × （O,∞）, where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n（2 ＋ ∑i^n=1 mi）.     

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|.     

8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let F = Q（√-p1p2） be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol （p1/p2） = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold： （1） The quartic residue symbols （p1/p2）4 = （p2/p1）4 = 1; （2） Either both p1 and p2 are represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=ε（2x^2-p1y^2）,x,y∈Z,ε∈{±1},where h＋（2p1） is the narrow class number of Q（√2p1）,Moreover, we also generalize these results.     

On the Order of Magnitude of the Divisor Function

Let D be an increasing sequence of positive integers, and consider the divisor functions： d（n, D） =∑d｜n,d∈D,d≤√n1, d2（n,D）=∑[d,δ]｜n,d,δ∈D,[d,δ]≤√n1, where [d,δ]=1.c.m.（d,δ）. A probabilistic argument is introduced to evaluate the series ∑n=1^∞and（n,D） and ∑n=1^∞and2（n,D）.     

A Representation of the Lorentz Spin Group and Its Application

For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj （m）, （j = 0, 1,..., m - 1） which satisfy γj （m）γk （m） ＋γk （m）γj （m） = 2ηjk （m）I[m/2], where （ηjk （m）） 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation （m） of the Lorentz Spin group is known. For m≥ 6, we prove that （i） when m = 2n, （n ≥ 3）, （m） is the group generated by the set of matrices {T｜T=1/√ξ（（I＋k） 0 ＋ 0 I-K） （ U 0 0 U）, （ii） when m = 2n ＋ 1 （n≥ 3）, （m） is generated by the set of matrices {T｜T=1/√ξ（I -k^- k I）U,U∈ （m-1）,ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj（m-2）＋a^（m-2） In],K^-=i[m-3∑j=0 a^j γj（m-2）-a^（m-2） In]}     

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.     

The Dual Spaces of Sets of Difference Sequences of Order <Emphasis Type="Italic">m</Emphasis> and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.     

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed(resp．bounded closed)boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X．Let K(X)denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance．Moreover,let KG(X)denote the closure of the set {A∈K(x)：A∩G=0}．We prove that the set of all A∈KG(X)(resp．A∈K(X)),such that the minimization (resp．maximization)problem min(A,G)(resp．max(A,G))is well posed,contains a dense Gδ-subset of KG(X)(resp．K(X))．thus extending the recent results due to Blasi,Myjak and Papini and Li．     

Multilinear Singular Integrals with Rough Kernel

For a class of multilinear singular integral operators T _A ,

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.     

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.     

Effective Reducibility of a Class of Linear Differential Equations with Quasiperiodic Coefficients

In this paper we consider the effective reducibility of the following linear differentialequation: x = (A ∈Q(t,∈))x, |∈| ≤ ∈0, where A is a constant matrix, Q(t,e) is quasiperiodic in t, and e is a small perturbation parameter. We prove that if the eigenvalues of A and the basic frequencies of Q satisfy some non-resonant conditions, the linear differential equation can be reduced to y = (A^*(∈) R^*(t, ∈))y, |∈| ≤ ∈o, where R^* is exponentially small in ∈.     

The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions

Abstract Small-time asymptotics of the trace of the heat semigroup where {μ _ν } are the eigenvalues of the negative Laplacian in the (x ¹, x ²)-plane, is studied for a general bunded domain Ω with a smooth boundary ∂Ω, where a finite number of Dirichlet, Neumann and Robin boundary conditions, on the piecewise smooth parts Γ _i (i = 1, ..., n) of ∂Ω such that , are considered. Some geometrical properties associated with Ω are determined.     

On the Boundary Behaviour,Including Second Order Effects,of Solutions to Singular Elliptic Problems

For γ≥1 we consider the solution u=u（x） of the Dirichlet boundary value problem Δu ＋ u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate u（x）=p（δ（x））[1＋A（x）（log 1/δ（x）^-6], where p（r） ≈ r r√2 log（1/r） near r = 0,δ（x） denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A（x） is a bounded function. For 1 〈 γ 〈 3 we find u（x）=（γ＋1/√2（γ-1）δ（x））^2/γ＋[1＋A（x）（δ（x））2γ-1/γ＋1] For γ3= we prove that u（x）=（2δ（x））^1/2[1＋A（x）δ（x）log 1/δ（x）]     

Universal Similarity Factorization Equalities over Generalized Clifford Algebras

For any element a in a generalized 2^n-dimensional Clifford algebra Lln （F） over an arbitrary field F of characteristic not equal to two, it is shown that there exits a universal invertible matrix Pn over Lln（F） such that Pn^-1DnPn= φ（α）∈F^2n×2n, where φ（a） is a matrix representation of α over and Dα is a diagonal matrix consisting of a or its conjugate.     

Equivalence Bimodule between Spherical Noncommutative Tori

Let n _i ,m _j > 1. In [5], the sperical noncommutative torus was defined by twisting in by a totally skew multiplier p on for T ^pd a pd-homogeneous C*-algebra over . It is shown that is strongly Morita equivalent to . This work is supported by Grant No. 1999-2-102-001-3 from the interdisciplinary research program year of the KOSEF     

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│.     

Asymptotic Approximations of Jacobi Functions and Their Zeros with Error Bounds

The study of zeros of orthogonal functions is an important topic. In this paper, by improving the middle variable x（t）, we＇ve got a new form of asymptotic approximation, completed with error bounds, it is constructed for the Jacobi functions φu^（α,β）（t）（α 〉 -1） as μ→∞. Besides, an accurate approximation with error bounds is also constructed correspondingly for the zeros tμ,s of φu^（α,β）（t）（α≥ 0） as μ→∞, uniformly with respect to s = 1, 2,....