带有尖形式Fourier系数的指数和估计

设λ_f/(n)是全模群Γ上权为k的全纯Hecke特征形f的第n个Fourier系数,Λ(n)是Mangoldt函数.本文得到了如下估计∑_(Xn≤2X)Λ(n)λ_f(n)e(n~(1/2)α)■f,αX~(5/6)(logX)~(13/2),(α0),改进了Zhao的结果。     

图的k部划分的一个注记

设k≥2是一个整数。本文证明了任意有m条边的图都存在一个顶点的划分V_1,V_2…,V_k,使得e(V_1,V_2…,V_k)≥k-1/k m+k-1/2k((2m+1/4)~1/2-1/2)-(k-2)~2/8k,且max{e(V_i):1≤i≤k}≤m/k~2+(k-1)/2k~2((2m+1/4)~1/2-1/2+3/8-7k-4/8k~2.我们的结果改进了[Fan G.,Hou J.,Zeng Q.,A bound for judicious k-partitions of graphs,Discrete Appl.Math.,2014,179:86—99]的主要结论.     

振荡超奇性Hilbert变换的Sobolev有界性

主要研究R~n上沿曲线Γ(t)=(t~(p_1),t~(p_2),…,t~(p_n))的振荡超奇性Hilbert变换H_(n,α,β)=∫_0~1 f(x-Γ(t))e~(it-β)t~(-1-α),在Sobolev空间上的有界性,其中0p_1P_2…P_n,αβ0.证明了对于0γ(nα)/((n+1))(p_1+α),当|1/p-1/2|(β-(n+1)[α-(β+p_1)γ])/(2β)时,H_(n,α,β)是从L_γ~2(R~n))到L~2(R~n)的有界算子.特别地,当β≥(α-γp_1)/(γ+1/(n+1))等时,H_(n,α,β)是从L_γ~2(R~n)到L~2(R~n)的有界算子·     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

非齐型空间上分数型Marcinkiewicz积分算子的加权估计

令（X，d，μ）为满足所谓上倍双倍条件和几何双倍条件的度量测度空间.设M_β，ρ，q为（X，d，μ）上的分数型Marcinkiewicz积分算子.在本文中，作者证明了若β ∈[0，∞），ρ ∈（0，∞），q ∈（1，∞）且M_β，ρ，q在L²（μ）上有界，则M_β，ρ，q是从加权Lebesgue空间L^p（w）到加权弱Lebesgue空间L^p，∞（w）上有界和从加权Morrey空间L^p，κ，η（ω）到加权弱Morrey空间WL^p，κ，η（ω）上有界.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

余Frame范畴中余积的构造

首先,在并半格中引入了上覆盖关系的概念,并以此为基础引入强并半格以及强并半格中上覆盖和C-滤子的概念,证明了强并半格S中全体C-滤子之族C Fil(S)是余Frame,讨论了简单上集值映射u:S→C Fil(S)的相关并半格同态性质;其次,证明了由一族余Frame{A_λ|λ∈Γ}的直积Π_(λ∈Γ)A_λ中只有有限个坐标非零的元素构成的子集A是强并半格,还证明了A是余Frame族{A_λ|λ∈Γ}在并半格范畴中的余积对象;最后,通过各个坐标集中的上覆盖关系在A中定义了上覆盖C~*,再结合简单上集值映射u:A→C~*Fil(A)和标准入射qλ:Aλ→Π_(λ∈Γ)A_λ(λ∈Γ),证明了强并半格A中由上覆盖C~*诱导的余Frame C~*Fil(A)是余Frame族{A_λ|λ∈Γ}在余Frame范畴中的余积对象.     

von Neumann代数上的局部可乘Lie n-导子

A₁，…，A_n的（n-1）-换位子记为p_n（A₁，…，A_n）.令M是von Neumann代数，n ≥ 2是任意正整数，L：M → M是一个映射.本文证明了，若M不含I₁型中心直和项，且L满足L（p_n（A₁，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足条件A₁A₂=0的A₁，A₂，…，A_n ∈ M成立，则L（A）=φ（A）+f（A）对所有A ∈ M成立，其中φ：M → M和f：M → Z（M）（M的中心）是两个映射，且满足φ在P_iMP_j上是可加导子，f（p_n（A₁，A₂，…，A_n））=0对所有满足A₁A₂=0的A₁，A₂，…，A_n ∈ P_iMP_j成立（1 ≤ i，j ≤ 2），P₁ ∈ M是core-free投影，P₂=I-P₁；若M还是因子且n ≥ 3，则L满足条件L（p_n（A₁，A₂，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立当且仅当L（A）=φ（A）+h（A）I对所有A ∈ M成立，其中φ是M上的可加导子，h是M上的泛函且满足h（p_n（A₁，A₂，…，A_n））=0对所有满足条件A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立.     

Reinhardt域上一类推广的Roper-Suffridge算子

研究一类推广的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)∑_(k=2)~nakz_k~pk,f′(z)1)(~1/p2)z_2,…,f′(z_1)~(1/pn)z_n)′,证明该算子在复欧氏空间中的Reinhardt域Ω_(n,p2,%…,pn)={z=(z_1,…,z_n)∈C~n:|z_|~2+∑_(k=2)~n|zk|~(pk)1,Pk∈N~+,k=2,…,n}上分别保持α次的殆β型螺形性,α次的β型螺形性及强β型螺形性.     

Infinitely Many Solutions for Schr?dinger–Choquard–Kirchhoff Equations Involving the Fractional p-Laplacian

In this article,we show the existence of infinitely many solutions for the fractional pLaplacian equations of Schr?dinger-Kirchhoff type equation ■ ,where(-△)_p~s is the fractional p-Laplacian operator,[u]_(s,p) is the Gagliardo p-seminorm,0 s 1 q p N/s,α∈(0,N),M and V are continuous and positive functions,and k(x) is a non-negative function in an appropriate Lebesgue space.Combining the concentration-compactness principle in fractional Sobolev space and Kajikiya's new version of the symmetric mountain pass lemma,we obtain the existence of infinitely many solutions which tend to zero for suitable positive parameters λ and β.     

Justification of a Perturbation Approximation of the Klein–Gordon Equation

Consider the nonlinear wave equation
u_tt − γ ² u_xx + f(u) = 0
with the initial conditions
u ( x ,0) = εφ ( x ), u _t( x ,0) = εψ ( x ),
where f ( u ) is either of the form f ( u )= c ² u −σ u ^{2 s +1}, s =1, 2,…, or an odd smooth function with f '(0)>0 and | f '( u )|≤ C ₀².The initial data φ( x )∈ C ² and ψ( x )∈ C ¹ are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u ( x ,  t ; ɛ), and prove its boundedness in x ∈ R and t >0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u ( x ,  t ; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ³ uniformly for x ∈ R and 0≤ t ≤ T /ɛ², as long as ɛ is sufficiently small, T being an arbitrary positive number.     

A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop–Phelps–Bollobás theorem: Suppose that X is a Banach space,T : X → C(K) is an Asplund operator with ║T║= 1, and that x_0 ∈ S_X, 0 ε satisfy ║T(x_0)║ 1-ε~2/2.Then there exist x_ε∈ S_X and an Asplund operator S : X → C(K) of norm one so that ║S(x_ε)║ = 1, x_0-x_ε ε and ║T-S║ ε.Making use of this theorem, we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T : ?_1 → Y of norm one: Suppose that y_0~*∈ S_(Y~*), ε≥ 0 satisfy T~*(y_0~*) 1-ε~2/2. Then there exist y_ε~*∈ S_(Y~*), x_ε∈(±e_n), y_ε∈ S_Y, and a strong Radon–Nikodym operator S : ?_1 → Y of norm one so that (ⅰ)║S(x_ε)║= 1;(ⅱ) S(x_ε) = y_ε;(ⅲ)║T-S║ ε;(ⅳ)║S~*(y_ε~*)║=y_ε~*, y_ε= 1;(ⅴ)║y_0~*-y_ε~*║ ε and (ⅵ)║T~*-S~*║ ε,where(e_n) denotes the standard unit vector basis of ?_1.     

与双调和方程相关的连续模和Heinz-Schwarz型不等式

对于给定的两个正整数n ≥ 2和m ≥ 1，假设函数f满足如下条件：（1）在Bⁿ内满足非齐次双调和方程△（△f）=g（g ∈ C（Bⁿ，R^m））；（2）在S^n-1上满足f=ψ₁（ψ₁ ∈ C（S^n-1，R^m）），以及∂f/∂n=ψ₂（ψ₂ ∈ C（S^n-1，R^m）），其中∂/∂n表示内法线方向导数，Bⁿ表示Rⁿ中的单位球以及S^n-1表示Bⁿ的边界.本文主要研究f的连续模和Heinz-Schwarz型不等式.     

On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs.Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques.This connection provided a new proof of Turán classical result on the Turán density of complete graphs.Since then,Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs.Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range.They showed that if G is a 3-uniform graph with m edges containing a clique of order t-1,then λ(G)=λ([t-1]~((3))) provided (t-13)≤m≤(t-13)+_(t-22).They also conjectured:If G is an r-uniform graph with m edges not containing a clique of order t-1,then λ(G)λ([t-1]~((r))) provided (t-1r)≤ m ≤(t-1r)+(t-2r-1).It has been shown that to verify this conjecture for 3-uniform graphs,it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m=t-13+t-22.Regarding this conjecture,we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t-1]~((3))\E(G)|=p,then λ(G)λ([t-1]~((3))) provided m=(t-13)+(t-22) and t≥17p/2+11.     

球面稳定同伦群中非平凡元素γ_sξ_n

当p≥5, n≥0时,(i_1i_0)_*(h_n)∈Ext_■~(1,p~nq)(H~*K,Z_p)在Adams谱序列中是永久循环,并且收敛到π_(p~nq-1)K中的非零元.本文在此基础上,考虑了涉及第三希腊字母类乘积元素的收敛性,并且扩大了球面稳定同伦群中非平凡元素滤子s+1的取值范围,即当p+1 s+1 2p时,■_sh_n∈Ext_■~(s+1,t)(Z_p,Z_p)在Adams谱序列中是永久循环,并且收敛到π_(t-s-1)S中的非零元γ_sξ_n,其中p≥7, n≥3, t=p~nq+sp~2q+(s-1)pq+(s-2)q+s-3,q=2(p-1).     

双线性Fourier乘子交换子的有界性与紧性

假定T_σ是关于乘子σ的双线性Fourier乘子算子,其中σ满足如下Sobolev正则条件:对某个s∈(n,2n],有sup_(κ∈Z)‖σ_k‖W~s(R~(2m))∞.对于p_1,p_2,p∈(1,∞)且满足1/p=1/p_1+1/p_2和ω=(ω_1,ω_2)∈A_(p/t)(R~(2n)),建立了T_σ及其与函数b=(b_1,b_2)∈(BMO(R~n))~2生成的交换子T_(σ,b)由L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的有界性;同时,在b_1,b_2∈CMO(R~n)(C_c~∞(R~n)在BMO拓扑下的闭包)的条件下,证明交换子T_(σ,b)是L~(p_1,λ)(ω_1)×L~(p_2,λ)(ω_2)到L~(p,λ)(v_w)的紧算子.为了得到主要结果,我们先后建立了几个双(次)线性极大函数在加多权Morrey空间上的有界性以及该空间中准紧集的判定.     

p-adic 超几何函数与 Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟，与许多数论问题都有联系.设F_q是q元有限域，λ∈F_q，n为正整数.本文研究了Dwork超曲面D_λⁿ：x₁ⁿ+x₂ⁿ+…+x_nⁿ=nλx₁x₂…x_n及其推广形式上的F_q-有理点，并在n与q（q-1）互素时给出了由p-adic超几何函数表示的各种F_q-有理点个数的公式，从而修正和改进了Barman与Goodson等人的结论.     

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.     

Bounded Fatou components of transcendental entire functions with order less than 1/2

Let f be a transcendental entire function with order ρ 12and let σ be a sufficiently large constant. We prove that if there exists r0 1 such that, for all r r0 and any small ε 0,M(rσ, f) ≥ M(r, f)σ+ε,then every component of the Fatou set F(f) is bounded.     

Complexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier term

In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization(SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q1 q2 1, the algorithm has O((q1 + 1) nq1+1/2(q1-q2)logn/ε)and O((q1 + 1)3q1-2q2+1/2(q1-q2)n~1/2 logn/ε) complexity results for large- and small-update methods, respectively.