Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

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.     

Ultraproducts of real interpolation spaces between L p -spaces

Let , be a family of compatible couples of L^p-spaces. We show that, given a countably incomplete ultrafilter in , the ultraproduct of interpolation spaces defined by the real method is isomorphic to the direct sum of an interpolation space of type , an intermediate K?the space between and being a purely atomic measure space, and a K?the function space K(Ω₃) defined on some purely non atomic measure space (Ω₃, ν₃) in such a way that Ω₂ ∪ Ω₃ ≠∅. The research of first and third authors is partially supported by the MEC and FEDER project MTM2004-02262 and AVCIT group 03/050.     

On an exponential sum

Let p be a prime number, n be a positive integer, and ƒ(x) = ax^k + bx. We put

Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations

This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt ＋ α1 uδ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x,t）∈ E R^2, and δu/δt ＋ α2 δu/δx δ^2u/δx^2 ＋ βδ^3u/δx^3 ＋ γδ^5u/δx^5 = 0, （x, t） ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0（x） ∈ H^s（R）, and s ≥ 5/4 for the first equation and s≥301/108 for the second equation.     

Exact asymptotic behavior of singular values of a class of integral operators

We find an exact asymptotic formula for the singular values of the integral operator of the form , a Jordan measurable set) where and L is slowly varying function with some additional properties. The formula is an explicit expression in terms of L and T.     

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

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.     

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

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.     

Small gaps between primes

Combining Goldston-Yildirim’s method on k-correlations of the truncated von Mangoldt function with Maier’s matrix method, we show that for all r ≧ 1 where p _n denotes the nth prime number and γ is Euler’s constant. This is the best known result for any r ≧ 11.      

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）.     

A note on periodic solutions of a differential equation with two parameters

In [1] (p. 215), the authors Andronov, Leontovich-Andronova, Gordon, and Maier, consider the following equation: $$\left\{ \begin{gathered} \tfrac{{dx}}{{dt}} = y, \hfill \\ \tfrac{{dy}}{{dt}} = x + x^2 - \left( {\varepsilon _1 + \varepsilon _2 x} \right)y, \hfill \\ \end{gathered} \right.$$ whereε ₁ andε ₂ are real constants andε ₁ andε ₂ are not both zero. They proved that there are no non-trivial periodic solutions except possibly for the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ . They left that case as an open problem. In this note we prove that there are indeed no non-trivial periodic solutions in the case $0< \tfrac{{\varepsilon _1 }}{{\varepsilon _2 }}< \tfrac{3}{2}$ either. Our method of proof consists essentially of constructing a Dulac function (see [6] and [9]) and using the conception of Duff's rotated vector field (see [4], [7], [8], [10], and [11]).     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.     

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type

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.     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

Critical behavior of Gaussian model on diamond-type hierarchical lattices

It is proposed that the Gaussian type distribution constantb _qi in the Gaussian model depends on the coordination numberq _i of sitei, and that the relation holds amongb _qi ’s. The Gaussian model is then studied on a family of the diamond-type hierarchical (or DH) lattices, by the decimation real-space renormalization group following spin-rescaling method. It is found that the magnetic property of the Gaussian model belongs to the same universal class, and that the critical pointK* and the critical exponentv are given by and , respectively. Project supported by the National Natural Science Foundation of China (Grant No. 19775008), the National Basic Research Project supported by the National Natural Science Foundation of China (Grant No. 19775008), the National Basic Research Project supported by the National Natural Science Foundation of China (Grant No. 19775008), the National Basic Research