Well-posedness in the energy space for non-linear system of wave equations with critical growth

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure{utt-△u=-F1（｜u｜^2,｜v｜^2）u,utt-△u=-F2（｜u｜^2,｜v｜^2）u where there exists a function F（λ,μ） such that δF（λ,μ）/δλ=F1（λ,μ）.δF（λ,μ）/δμ=F2（λ,μ） By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in ＂Shatah and Struwe＇s paper, Ann. of Math. 138, 503-518 （1993）＂.     

Remarks on the Extremal Functions for the Moser-Trudinger Inequality

We will show in this paper that if A is very close to 1, then I（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dV can be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by Trudinger and Moser and related elliptic equations＂ （Comm. Pure. Appl. Math., 55, 135-152, 2002）.     

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

Many solutions for elliptic equations with critical exponents

Let Ω be an open bounded domain in ℝ^N(N ≥ 3) and . We are concerned with two kinds of critical elliptic problems. The first one is

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.     

ON A CLASS OF BESICOVITCHFUNCTIONS TO HAVE EXACT BOX DIMENSION： A NECESSARY AND SUFFICIENT CONDITION

This paper summarized recent achievements obtained by the authors about the box dimensions of the Besicovitch functions given byB(t) := ∞∑k=1 λs-2k sin(λkt),where 1 ＜ s ＜ 2, λk ＞ 0 tends to infinity as k →∞ and λk satisfies λk 1/λk ≥λ＞ 1. The results show thatlimk→∞ log λk 1/log λk = 1is a necessary and sufficient condition for Graph(B(t)) to have same upper and lower box dimensions.For the fractional Riemann-Liouville differential operator Du and the fractional integral operator D-v,the results show that if λ is sufficiently large, then a necessary and sufficient condition for box dimension of Graph(D-v(B)),0 ＜ v ＜ s - 1, to be s - v and box dimension of Graph(Du(B)),0 ＜ u ＜ 2 - s, to be s uis also lim k→∞logλk 1/log λk = 1.     

On the synthesis of oriented contact circuits with certain restrictions on adjacent contacts

Realization of Boolean functions in the class of oriented contact circuits (OCCs) with certain restrictions on the weight, number, and types of adjacent contacts is studied. Oriented contact circuits are considered in which, from an arbitrary vertex, at most λ arcs issue and at most ν different Boolean variables are used in the marks of the issuing arcs. The weight of a vertex of an OCC is defined as being equal to λ if one arc enters a vertex and equal to λ(1 + ω), where ω > 0, otherwise. Then, as usual, the weight of an OCC is defined as the sum of the weights of its vertices; the weight of a Boolean function, as the minimum weight of OCCs realizing it; and Shannon function W _{λ, ν, ω}(n), as the maximum weight of the Boolean function of n variables. For this Shannon function, the so-called high-accuracy bound

An end point estimate for maximal commutators

In this paper we prove that the maximal commutator of singular integral operator [b, T]^* satisfies the inequality:

Some generalizations of Knopp's identity*

For integers a, b and n > 0, define

FINITE TRAVELING WAVES FOR A REACTION-DIFFUSION SYSTEM WITH N（≥3） COMPONENTS

ＦＩＮＩＴＥＴＲＡＶＥＬＩＮＧＷＡＶＥＳＦＯＲＡＲＥＡＣＴＩＯＮ┐ＤＩＦＦＵＳＩＯＮＳＹＳＴＥＭＷＩＴＨＮ（３）ＣＯＭＰＯＮＥＮＴＳＷＡＮＧＳＨＵ,ＷＥＮＳＨＩＨＬＩＡＮＧＡＮＤＹＥＱＩＸＩＡＯＡｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｅｘ...     

Star number and star arboricity of a complete multigraph

Let G be a multigraph. The star number s(G) of G is the minimum number of stars needed to decompose the edges of G. The star arboricity sa(G) of G is the minimum number of star forests needed to decompose the edges of G. As usual λK _n denote the λ-fold complete graph on n vertices (i.e., the multigraph on n vertices such that there are λ edges between every pair of vertices). In this paper, we prove that for n ⩾ 2

A trace formula for the scalar product of Hecke series and its applications

A trace formula expressing the mean values of the form (k=2,3,...)

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]}     

Asymptotics for the Korteweg-de Vries-Burgers Equation

We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut＋uux-uxx＋uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s （R）∩L^1 （R）, where s 〉 -1/2, then there exists a unique solution u （t, x） ∈C^∞ （（0,∞）;H^∞ （R）） to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u（t）=t^-1/2fM（（·）t^-1/2）＋0（t^-1/2） as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 （x） ∈ L^1 （R）, then the asymptotics are true u（t）=t^-1/2fM（（·）t^-1/2）＋O（t^-1/2-γ） where γ ∈ （0, 1/2）.     

Cauchy-Rassias Stability of Cauchy-Jensen Additive Mappings in Banach Spaces

Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive. Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras.     

Time-periodic solutions of dissipative ε-approximations for modified Navier-Stokes equations

In this paper, the existence “in the large” of time-periodic classical solutions (with period T) is proved for the following two dissipative ε-approximations for the Navier-Stokes equations modified in the sense of O. A. Ladyzhenskaya:

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.     

On the stability of the generalized sine functional equations

The aim of this paper is to study the stability problem of the generalized sine functional equations as follows：
g（x）f（y）=f（x＋y/2）^2-f（x-y/2）^2 f（x）g（y）=f（x＋y/2）^2-f（x-y/2）^2,g（x）g（y）=f（x＋y/2）^-f（x-y/2）^2
Namely, we have generalized the Hyers Ulam stability of the （pexiderized） sine functional equation.     

On Existence and Multiplicity of Solutions for Elliptic Equations Involving the p-Laplacian

In the present paper, the following Dirichlet problem and Neumann problem involving the p-Laplacian

Eigenvalue estimate on a compact Riemann manifold

LetM be a compact Riemann manifold with the Ricci curvature ≽ - R(R = const. > 0) . Denote by d the diameter ofM. Then the first eigenvalue λ₁ ofM satisfies . Moreover if , then