Global well-posedness of the fractional Klein-Gordon-Schr¨odinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schr¨odinger system with fractional Laplacian in the Schr¨odinger equation in R~(1+1). We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schr¨odinger data in H~(s_1) and wave data in H~(s_2) × H~(s_2-1)for 3/4- α s_1≤0 and-1/2 s_2 3/2, where α is the fractional power of Laplacian which satisfies 3/4 α≤1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s_1 = 0 and-1/2 s_2 1/2 by using the conservation law for the L~2 norm of u.     

有限域上一类方程组的解数公式

设F_q为一个q元有限域,其中q=p~s(s≥1),p是一个奇素数.本文给出下列方程组在F_q上的解数公式:a_(k1)x_1~(d_(11)~((k)))...x_(n_1)~(d_(1n_1)~((k)))+...+a_(k,s_1)x_1~(d_(s_1,1)~((k)))...x_(n_1)~(d_(s_1,n_1)~((k)))+a_(k,s_1)+1x_1~(d_(s_1+1,1)~((k)))...x_(n_2)~(d_(s_1+1,n_2)~((k)))+...a_(k,s_2)x_1~(d_(s_2,1)~((k)))...x_(n_2)~(d_(s_2,1)~((k)))...x_(n_2)~(d_(s_2,n_2)~((k)))=b_k,k=1,...,m,其中0s_1s_2,0n_1n_2,a_(ki)∈F_q~*,b_k∈F_q,d_(ij)~(k)0(k=l,...,m,i=1,...,s_2,j=1,...,n_2).特别当ms_1≤n_1,ms_2≤n_2,d_(ij)~(k)满足一定条件时,得到了明确的解数公式.     

Derivative formulas and Poincaré inequality for Kohn-Laplacian type semigroups

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator L :=1/2 sum from i=1 to m X_i~2 on R~(m+d):= R~m× R~d is investigated, where X_i(x, y) = sum (σki?xk) from k=1 to m+sum (((A_lx)_i?_(yl)) from t=1 to d,(x, y) ∈ R~(m+d), 1 ≤ i ≤ m for σ an invertible m × m-matrix and {A_l}_1 ≤ l ≤d some m × m-matrices such that the Hrmander condition holds.We first establish Bismut-type and Driver-type derivative formulas with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.     

有限域上一类方程组解数的直接公式

本文给出有限域F=F_q(q=p~f,f≥1,p是一个奇素数)上一类方程组∑_(i=s_(r-1)+1~(s_r)∑_(j=1)~(m_i-m_(i-1))a_(m_(i-1)+j)x_1~(d_m(i-1)+j,1)…x_(n_i)~d_(m_(i-1)+j,n_i)=b_r,r=1,…,k当指数满足一定条件时,在F~(n_s_k)上解数的一个直接公式,这里d_(ij)＞0,a_i∈F~*,b_i∈F,0= s_0＜s_1＜…＜s_k,0=m_0＜m_1＜…＜m_(s_k),0=n_0＜n_1＜…＜n_(s_k), m_1≤n_1,…,m_(s_k)≤n_(s_k)．     

CLASSIFICATION OF POSITIVE SOLUTIONS TO A SYSTEM OF HARDY-SOBOLEV TYPE EQUATIONS

In this paper, we are concerned with the following Hardy-Sobolev type system{(-?)~(α/2) u(x) =v~q(x)/|y|~(t_2) (-?)α/2 v(x) =u~p(x)/|y|~(t_1),x =(y, z) ∈(R ~k\{0}) × R~(n-k),(0.1)where 0 α n, 0 t_1, t_2 min{α, k}, and 1 p ≤τ_1 :=(n+α-2t_1)/( n-α), 1 q ≤τ_2 :=(n+α-2 t_2)/( n-α).We first establish the equivalence of classical and weak solutions between PDE system(0.1)and the following integral equations(IE) system{u(x) =∫_( R~n) G_α(x, ξ)v~q(ξ)/|η|t~2 dξ v(x) =∫_(R~n) G_α(x, ξ)(u~p(ξ))/|η|~(t_1) dξ,(0.2)where Gα(x, ξ) =(c n,α)/(|x-ξ|~(n-α))is the Green's function of(-?)~(α/2) in R~n. Then, by the method of moving planes in the integral forms, in the critical case p = τ_1 and q = τ_2, we prove that each pair of nonnegative solutions(u, v) of(0.1) is radially symmetric and monotone decreasing about the origin in R~k and some point z0 in R~(n-k). In the subcritical case (n-t_1)/(p+1)+(n-t_2)/(q+1) n-α,1 p ≤τ_1 and 1 q ≤τ_2, we derive the nonexistence of nontrivial nonnegative solutions for(0.1).     

On Some Classes of Interpolating Splines

In this article, we consider some classes of interpolating spline with difference. derivative and integral interpolating conditions respectively. Let Δ_π:a=x_o相似文献   

PARAMETER ESTIMATION OF SPATIAL AR MODEL

Consider a stable AR model of two parameter spatial series {X_t, t∈N~2}, i. e. {X_(t)t∈N~2} is homogeneous and satisfies the following difference equationX_t-sum from n=s∈相似文献   

Several Results on Systems of Residue Classes

Let (m,n) and a(n) denote the g.c.d, of m, n and the residue class {x∈Z∶x≡α (mod n)} respectively. Any period of the characteristic function ofkU a_i(n_i) is called a covering period of {a_i(n_i)}_(i-1)~k.i-ITheorem Let A = {a_i(n_i)}_(i-1)~k. be a disjoint system (i. e. a_I(n_I,...,a_k(n_k) are pairwise disjoint). Let [n_I,...,n_k] (the I.c.m. of n_1,...,n_k) have the prime faetorization [n_1,...,n_k] = Πp_i~ai and T = Πp_iβi(β_i≥0 be the smallest positive covering period of A. Then     

The L~(p,q)-stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces L~(p,q)(R~(d+1))

The stability is an expected property for functions,which is widely considered in the study of approximation theory and wavelet analysis.In this paper,we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces L~(p,q)(R~(d+1)).We first show that the shiftsφ(·-k)(k∈Z~(d+1))are Lp,q-stable if and only if for anyξ∈R~(d+1),∑_(k∈Z~(d+1))|φ(ξ+2πk)|~20.Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces L~(p,q)(R~(d+1))to be Lp,q-stable which improves some known results.     

Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces

We study the Cauchy problem of a two-species chemotactic model. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish a unique local solution and blow-up criterion of the solution, when the initial data(u0, v0, w0) belongs to homogeneous Besov spaces˙B~(-2+3/p)_(p,1)(R~3) ×˙B~(-2+3/r)_(r,1)(R~3) ×˙B~(3/q)_(q,1)(R~3) for p, q and r satisfying some technical assumptions. Furthermore, we prove that if the initial data is sufficiently small, then the solution is global. Meanwhile, based on the so-called Gevrey estimates, we particularly prove that the solution is analytic in the spatial variable. In addition, we analyze the long time behavior of the solution and obtain some decay estimates for higher derivatives in Besov and Lebesgue spaces.     

Asymtotics of M-estmation in Non-linear Regression

Consider the standard non-linear regression model y_i=g(x_i,θ_0) ε_j,i=1,...,n whereg(x,θ) is a continuous function on a bounded closed region X×θ_0 is the unknown parametervector in R~p,{x_1,x_2,...,x_n} is a deterministic design of experiment and {ε_1,ε_2,...,ε_n} is asequence of independent random variables.This paper establishes the existences of M-estimates andthe asymptotic uniform linearity of M-scores in a family of non-linear regression models when theerrors are independent and identically distributed.This result is then used to obtain the asymptoticdistribution of a class of M-estimators for a large class of non-linear regression models.At the sametime,we point out that Theorem 2 of Wang (1995) (J.of Multivariate Analysis,vol.54,pp.227-238,Corrigenda.vol.55,p.350) is not correct.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

POSITIVE MAPS CONSTRUCTED FROM PERMUTATION PAIRS

A property(C) for permutation pairs is introduced. It is shown that if a pair{π_1, π_2} of permutations of(1,2,…,n) has property(C),then the D-type map Φ_(π_1,π_2) on n× n complex matrices constructed from {π_1,π_2} is positive. A necessary and sufficient condition is obtained for a pair {π_1,π_2} to have property(C),and an easily checked necessary and sufficient condition for the pairs of the form {π~p,π~q} to have property(C) is given, whereπ is the permutation defined by π(i) = i + 1 mod n and 1≤ p q≤ n.     

Duality of weighted multiparameter Triebel-Lizorkin spaces

In this paper, we reintroduce the weighted multi-parameter Triebel-Lizorkin spaces F_p~(α,q) (ω; R~(n_1)× R~(n_2)) based on the Frazier and Jawerth' method in [11]. This space was′firstly introduced in [18]. Then we establish its dual space and get that(F_p~(α,q))*= CMO_p~(-α,q') for 0 p ≤ 1.     

FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES

Necessary and sufficient conditions are studied that a bounded operator T_x =(x_1~*x, x_2~*x,···) on the space ?_∞, where x_n~*∈ ?_∞~*, is lower or upper semi-Fredholm; in particular, topological properties of the set {x_1~*, x_2~*,···} are investigated. Various estimates of the defect d(T) = codim R(T), where R(T) is the range of T, are given. The case of x_n~*= d_nx_(tn)~*,where dn ∈ R and x_(tn)~*≥ 0 are extreme points of the unit ball B_?_∞~*, that is, t_n ∈βN, is considered. In terms of the sequence {t_n}, the conditions of the closedness of the range R(T)are given and the value d(T) is calculated. For example, the condition {n:0 |d_n| δ} = Φ for some δ is sufficient and if for large n points tn are isolated elements of the sequence {t_n},then it is also necessary for the closedness of R(T)(t_(n0) is isolated if there is a neighborhood U of t_(n0) satisfying t_n ■ U for all n ≠ n0). If {n:|d_n| δ} =Φ, then d(T) is equal to the defect δ{_tn} of {t_n}. It is shown that if d(T) = ∞ and R(T) is closed, then there exists a sequence {A_n} of pairwise disjoint subsets of N satisfying χ_(A_n)■R(T).     

Nonlinear time-harmonic Maxwell equations in a bounded domain: Lack of compactness

We survey recent results on ground and bound state solutions E:?→R~3 of the problem {▽(▽×E)+}λE=|E|~(P-2)E in Ω,v×E=0 on Ω on a bounded Lipschitz domain ??R~3,where?×denotes the curl operator in R~3.The equation describes the propagation of the time-harmonic electric field R{E(χ)e~(iwt)}in a nonlinear isotropic material ? withλ=-μεω~2≤0,where μ andεstand for the permeability and the linear part of the permittivity of the material.The nonlinear term|E|~(P-2)E with 2p≤2*=6 comes from the nonlinear polarization and the boundary conditions are those for?surrounded by a perfect conductor.The problem has a variational structure;however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition.We show the underlying difficulties of the problem and enlist some open questions.     

ON SCORE VECTORS AND CONNECTIVITY OF TOURNAMENTS

An n-tournament T is called k-strong (l≤k≤n-2), if every (n+1-k)-subtournament of T is strongly connected. This paper proves that a score vector (s_1, s_2,..., s_n), where s_1≤s_2≤...≤s_n, is the score vector of some k-strong tournament if and only if min{t_1, t_2,..., t_(n-1)}≥k, where t_f=s_1+s_2+...+s_j-j(j-1)/2, j=1, 2,..., n-1.     

混合幂丢番图不等式（英文）

It is proved that if λ_1, λ_2, ···, λ_7are nonzero real numbers, not all of the same sign and not all in rational ratios, then for any given real numbers η and σ, 0 σ 1/16, the inequality |λ_1x~2_1+ λ_2x~2_2+∑7 i=3λ_ix~4_i+ η| ( max1≤i≤7|x_i|)~(-σ)has infinitely many solutions in positive integers x_1, x_2, ···, x_7. Similar result is proved for |λ_1x~2_1+ λ_2x~2_2+ λ_3x~2_3+ λ_4x~4_4+ λ_5x~4_5+ λ_6x~4_6+ η| ( max1≤i≤6|xi|)-σ.These results constitute an improvement upon those of Shi and Li.     

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrödinger equations

The classical critical Trudinger-Moser inequality in R~2 under the constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for any τ 0,it holds that ■ and 4π is sharp.However,if we consider the less restrictive constraint ∫_(R_2)(|▽u|~2+|u|~2)dx≤1,where V(x) is nonnegative and vanishes on an open set in R~2,it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x) makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality ■ when V is nonnegative and vanishes on an open set in R~2.As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u) in R~2,(0.1)where V(x)≥0 and vanishes on an open set of R~2 and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential) has been the standard assumption when one deals with the existence of solutions to the above Schr?dinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.     

On the α-spectra of Uniform Hypergraphs and Its Associated Graphs

For 0 ≤α 1 and a k-uniform hypergraph H, the tensor A_α(H) associated with H is defined as A_α(H) = αD(H) +(1-α)A(H), where D(H) and A(H) are the diagonal tensor of degrees and the adjacency tensor of H, respectively. The α-spectra of H is the set of all eigenvalues of A_α(H) and the α-spectral radius ρ_α(H) is the largest modulus of the elements in the spectrum of A_α(H). In this paper we define the line graph L(H) of a uniform hypergraph H and prove that ρ_α(H) ≤■ρ_α(L(H)) + 1 + α(Δ-1-δ~*/k), where Δ and δ~* are the maximum degree of H and the minimum degree of L(H), respectively. We also generalize some results on α-spectra of G~(k,s), which is obtained from G by blowing up each vertex into an s-set and each edge into a k-set where 1 ≤ s ≤ k/2.