Context-free grammars for triangular arrays

We consider context-free grammars of the form G = {f → fb1+b2+1ga1+a2, g → fb1 ga1+1},where ai and bi are integers sub ject to certain positivity conditions. Such a grammar G gives rise to triangular arrays {T(n, k)}0≤k≤n satisfying a three-term recurrence relation. Many combinatorial sequences can be generated in this way. Let Tn (x) =∑nk=0T(n, k)xk. Based on the differential operator with respect to G, we define a sequence of linear operators Pn such that Tn+1(x) = Pn(Tn(x)). Applying the characterization of real stability preserving linear operators on the multivariate polynomials due to Borcea and Br?ndén, we obtain a necessary and sufficient condition for the operator Pn to be real stability preserving for any n. As a consequence, we are led to a sufficient condition for the real-rootedness of the polynomials defined by certain triangular arrays, obtained by Wang and Yeh.Moreover, as special cases we obtain grammars that lead to identities involving the Whitney numbers and the Bessel numbers.     

Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I

We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas ensemble with an external potential V = q2mx2 m. More precisely, let Pn(d H) = Cne-nTrV(H)dH be the distribution of n × n Hermitian random matrices, ρV(x)dx the equilibrium measure, where Cnis a normalization constant, V(x) = q2mx2m with q2m=Γ(m)Γ(12)/Γ(2m+1/2), and m ≥ 1. Let x1 ≤···≤ xnbe the eigenvalues of H. Let k := k(n) be such that k(n)/n∈ [a, 1- a] for n large enough, where a ∈(0,12).Define G(s) :=∫s-1ρV(x)dx,- 1 ≤ s ≤ 1,and set t := G-1(k/n). We prove that, as n →∞,xk- t log n1/2 2π21/2nρV(t)→ N(0, 1)in distribution. Multi-dimensional central limit theorem is also proved. Our results can be viewed as natural extensions of the bulk central limit theorems for GUE ensemble established by J. Gustavsson in 2005.     

Uniform Asymptotics for Orthogonal Polynomials with Exponential Weights—the Riemann–Hilbert Approach

Let   Q ( x ) = q _{2 m} x ^{2 m} + q _{2 m −1} x ^{2 m −1}+⋯  be a polynomial of degree 2 m with   q _{2 m} > 0  , and let  {π _n ( x )} _{n ≥1}  be the sequence of monic polynomials orthogonal with respect to the weight   w ( x ) = e ^{− Q ( x )}  on     . Furthermore, let  α _n   and  β _n   denote the Mhaskar–Rakhmanov–Saff (MRS) numbers associated with Q ( x ). By using the Riemann–Hilbert approach, an asymptotic expansion is constructed for  π _n ( c_nz + d_n )  , which holds uniformly for all z bounded away from  (−∞, −1)  , where     and     .     

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).     

有关广义中心三项式系数的3个超同余式

设b,c为整数,定义广义中心三项式系数Tn(b,c)=[xn](x2+bx+c)n=「n/2」∑k=0(n2k)(2kk)bn-2kck(n∈N={0,1,…}),这里[xn]P(x)表示多项式P(x)中xn项的系数.特别地,中心Delannoy多项式Dn(x)=Tn(2x+1,x2+x)(n ∈ N),中心三项式系数...     

一类量子环面李代数的单性(英文)

令C_q:=C_q[t₁^±1,t₂^±1,t₃^±1]为量子环面结合代数,L_q=C_q/C为量子环面李代数。本文定义了一个与q=（qij）_i,j=1³相关的指数方程体系,称之为C_q的特征方程组。通过这个特征方程组,证明了L_q是非单的当且仅当特征方程组在Z³中有非零解。对于|q|=0,证明了在C_q中存在一个极大交换子代数I,并且I严格包含中心Z（C_q）。同时文中也指出,对于|q|≠0,在C_q中不存在这样的子代数。     

Wm∨Pn的交叉数

在Klesc M给出的联图W_3 V P_n的交叉数的基础上,继续对联图Wm V Pn（m=4,5）的交叉数cr进行了研究,得到了cr（W3 V Pn）=Z（5,n）＋n＋「n/2＋1」以及cr（W5 V Pn）=Z（6,n）＋n＋3[n/2」＋1,n≥2.     

弱分段Koszul模(英文)

任意一个弱分段Koszul模M都被证明存在一个自然的分次子模链0=U₀（?）U₁（?）U₂（?）…（?） U_t=M使得每个商U_i/U_i-1都是分段Koszul模.本文的主要目的是建立M和U_i/U_i-1的极小分次投射解之间的关系.对n≥0,证明了P_n=⊕_i=1^t P_nⁱ,其中P_*ⁱ→U_i/U_i-1→0和P_*→M→0是相应的极小分次投射解,作为其直接推论,有pd（M）=max{pd(U_i/U_i-1)}成立.     

一类新型BMO空间

引进了弱型有界平均震荡函数空间WBMO_q,1q∞,它是类似于弱型勒贝格空间L~(q,∞)所对应的BMO空间.证明了‖·‖*(BMO范数)与‖·‖_(WBMO_q)之间的等价特征刻画.作为应用,对于p∈(1,∞)和1/q=1/p-α/n,交换子[b,I_α]是从Lp到L~(q,∞)的有界算子,当且仅当局部可积函数b属于BMO空间,其中I_α表示分数次积分算子.另外,还引进以及学习了弱型的中心有界平均震荡空间W_q.     

有限域上一类负循环码

设F_q为一个阶为q的有限域,其中q为奇数.本文研究了x~n+1在F_q上的不可约分解及环F_q[x]/x~n+1中所有本原幂等元,这里的n是素因子整除q-1的某些正整数.进一步,得到了F_q上所有长度为n的不可约负循环码的检验多项式及极小汉明距离.     

某类非线性微分方程超越亚纯解的进一步结果

本文研究非线性微分方程f~n+Q_d(z,f)=P_1(z)e~(α_1(z))+p_2(z)e~(α_2(z))超越亚纯解的存在性和形式,其中n≥4是整数,Q_d(z,f)是关于f的次数d≤n-3且系数为有理函数的微分多项式,p_1,p_2是非零的有理函数,α_1,α_2是非常数的多项式.运用Nevanlinna值分布理论,能够得到该方程存在超越亚纯解时p_1,p_2,α_1及α_2所满足的条件.特别地,还考虑了当Q_d(z,f)=a(z)ff'且n=4时方程的超越亚纯解的存在性和形式,其中a(z)是一个非零的有理函数.     

A generalization of Simpson's Rule

Let and let , where P _c ⁿ denoles the Taylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of the Trapezoidal rule and the midpoint rule, respectively, and are each exact for all polynomials of degree ≤n+1. We let L(f)=αTM(f)+(1−α)TA(f), where , to obtain a numerical integration rule L which is exact for all polynomials of degree≤n+3 (see Theorem 1). The case n=0 is just the classical Simpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacing P _(a+b) ^2/n+1 (x) by the Hermite cubic interpolant at a and b, we obtain some known formulae by a different approach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by taking piecewise polynomials of odd degree, each piece being the Taylor polynomial of f at a and b, respectively. Of course all of our formulae can be compounded over subintervals of [a,b].     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

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·     

Multivariate normality via conditional specification

If X is a k-dimensional random vector, we denote by X_(i) the vector X with coordinate i deleted and by X_(i,j) the vector X with coordinates i and j deleted. If for each i the conditional distribution of X_i given X_(i) = x_(i) is univariate normal for each x_(i) ^K−1 and if for each i, j the conditional distribution of X_i given X_(i,j) = x_(i,j) is univariate normal for each x_(i,j) ^k−2 then it is shown that X has a classical k-variate normal distribution.     

Movable Singularities of a Class of Nonlinear Ordinary Differential Equations of Arbitrary Order

In this paper we consider nonlinear ordinary differential equations   y ^{( n )}= F ( y ', y , x )  of arbitrary order   n ≥ 3  , where F is algebraic in   y , y '  and locally analytic in x . We prove that for   n > 3  these equations always admit movable branch points. In the case   n = 3  these equations admit movable branch points unless they are of the known class   y '= a ( x )( y ')²+ ( b ₂( x ) y ²+ b ₁( x ) y + b ₀( x )) y '+ ( c ₄( x ) y ⁴+ c ₃( x ) y ³+ c ₂( x ) y ²+ c ₁( x ) y + c ₀( x ))  , where   a ,  b_j ,  c_j   are locally analytic in x .     

On the Differential Polynomial of a Graph

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial B(G; x) :=∑?(G)k=-nB_k(G) x~(n+k), where B_k(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G;x) and its coefficients.In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.     

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.     

玛欣凯维奇函数与泊松核的一个新微分性质

将Stein[On the functions of Littlewood-Paley,Lusin,and Marcinkiewicz,Trans.Amer.Math.Soc.,1958,88:430-466]中的玛欣凯维奇函数的逆向不等式推广到一般情形.主要结果是对于n-维欧几里得空间k-阶球面调和函数空间的任意一基底,得到玛欣凯维奇函数的一般性的逆向不等式,即存在不依赖于函数f正常数C_p,使得||f||_p≤C_pΣ_(j=1)~N=1||μ_j(f)||_p,其中{μ_j(f)}_(j=1)~N是f的由这些球面调和函数生成的玛欣凯维奇函数.此外,对于任意的n-变元的k-阶调和多项式Q(x)以及泊松核P_t(x),有Q(D)P_t(x)=C_n k(tQ(x))/((|x|)~2+t~2~(n+2k+1)/2).