三角域上Bernstein多项式的Lipschitz常数

设T是平面上以T₁,T₂,T₃为顶点的三角形,f(p)为定义在T上的函数,称Bⁿ(f,P):=(?)f(i/n,j/n,k/n)B_i,j,kⁿ(P),为f的n次Bernstein多项式,这儿B_i,j,kⁿ(P):(n!)/(i!j!k!)uⁱv^jω^k是Bernstein基函数,(u,v,w)是P关于T的重心坐标。 B.M.Brown等人对单变量的Bernstein多项式证明了如果f∈Lip_Aλ,0<λ≤1,则对所有的n,都有B^α(f,x)∈Lip_Aλ。本文的目的是对定义在三角域T:{(x,y):x≥0,y≥0,x+y≤1}上的Bernstein多项式证明了类似的结果: 设f(P)∈Lip_Aλ,0<λ≤1,则对所有的n,Bⁿ(f,P)∈Lip_(2^1/2λA)λ,并且,在一定意义上,常数2^1/2λA是最好的。 上述结果对于任意的锐角或直角三角形T,也是成立的。 最后还指出,当T可为钝角三角形时,则不存在同一常数C,使对所有的n和任意三角形T,有Bⁿ(f,P)∈Lip_cλ。     

关于q-树二次整子图和n阶加点q-树色多项式的注记

在这篇文章中我们成功地仅用色多项式表征了最小度不等于q-3的q-树的二次整子图和n阶加点q-树,即当图的最小度δ(G)≠q-3时,n阶图G具有色多项式P(G;λ)=λ(λ-1)…(λ-q+2)(λ-q+1)~3(λ-q)~(n-q-2), n≥q+2,当且仅当G是n阶q-树的二次整子图或n阶加点q-树.     

矩阵指数的偏迹不等式

In this paper it is shown that if C is an n×n complex matrix and C^(k) is the k-th compound of C, 1≤k≤n, N = (n k) and if the eigenvalues of C^(k) are labeled inorder of decreasing magnitude |λ₁(C^(k))|≥λ₂(C^(k))|≥…≥λ_N(C^(k))|define the partial trace tr_i^(k)(C) by (?) Then for two n×n Hermrtian matrices A, B,with equality holds if A, B are commutative or k = n.     

具有色多项式∏∑(ui)/k{k/ui-k}(λ)k的图

本文利用色多项式的性质,讨论了具有色多项式∏_i∑_k(ui)/k{k/u_i-k}(λ)_k的图的结构,给出了具有这种色多项式的全部色等价图.     

多项式的根的一个性质的判定

文献[1]在讨论多项式型的函数迭代方程的局部解析解的存在性时涉及到了多项式的根的一个性质.本文给出了判定该性质是否成立的一个简洁的条件,证明了多项式λ_nzⁿ+…+λ₂z²+λ₁z+λ₀有一个根α满足inf{｜λ_nα^nm+…+λ₂a^2m+λ₁α^m+λ₀｜：m=2,3,…｝>0当且仅当如下两个条件之中至少有一个成立：(i)该多项式有一个根β满足｜β｜>1;(ii)该多项式有一个根β满足｜β｜<1,且λ₀≠0.     

关于图的减控制与符号控制

给定一个图G=(V,E),一个函数f:V→{-1,0,1}被称为G的减控制函数,如果对任意v∈V(G)均有∑_μ∈N[v]f(μ)≥1。G的减控制数定义为γ^-(G)=min{∑_v∈Vf(v)|f是G的减控制函数}。图G的符号控制函数的正如减控制函数,差别是广{-1,0,1}换成{-1,1}。符号控制数γ_s(G)是类似的。本文获得γ^-G)和γ_s(G)的一些下界。同时也证明并推广了 Jean Dunbar等提出的一个猜想,即对任意 n阶 2部图 G,均有γ^-(G)≥ 4(n+1^1/2-1)-n成立。     

Mn(C)中的不可约性

对于M_n(C)(所有n×n矩阵的全体)中的不可约矩阵得到以下结果:对于任意A∈M_n(C),设λ₁, λ₂,…,λ_m为A的所有特征值,这里m≤n而且当i≠j时,λ_i≠λ_j.则A是不可约的当且仅当任意P∈A′(A),P^*=P=P²,有σ(P|ker(A-λ₁))=σ(P|ker(A-λ₂))=…=σ(P|ker(A-λ_m))为单点集.     

两个四元数自共轭半正定矩阵乘积的特征估计

设A和B均非0的n阶实四元数自共轭矩阵,λ_i及μ_i分别为共特征值(i=1,…,n),且规定|λ₁|≥|λ₂|≥…≥|λ_n|,|μ₁|≥|μ₂|≥…≥|μ_n|,又λ为AB之任意特征值,则λ为实数,且(1)若A≥0,A(?)GL_n(Q),B≥0,B GL_n(Q),则λ≤λ₁μ₁;(2)若A>0或B>0,则|λ|≤|λ₁μ₁|,特别当A>0且B>0时有λ≤λ₁μ₁;(3)若A>0,B∈GL_n(Q),或B>0,A∈GL_n(Q)则|λ|≥|λ_nμ_n|,特别当A>0且B>0时有λ≥λ_nμ_n。     

关于k阶记录时间与k阶记录值的联合分布

设独立同分布随机变量序列{x_nj n≥1}的分布函数F(x)=p(x₁(k)(n);n≥1},{X^(k)(n);n≥1} 分别为{x_nj n≥1}的K阶记录时间序列和k阶记录值序列.本文我们用直接方法求出了{U^(k)(i),X^(k)(i);1≤i≤n}的联合分布，从而证明了k阶记录时间序列及k阶记录值序列的马氏性，并导出了它们之间的一     

Heisenberg群上一类左不变LPDO的局部基本解及局部可解性

In this paper it is proved that local fundamental solution exists in some space W^m(H_n) (m∈Z), if the left invariant differential operator on the Heisenberg group H_n satisfies certain condition. The main results are:l.Let L be a left invariant differential operator on H_n. If there exist R≥0, r,s∈R and operators {B_λ|λ∈Γ_R} ∈V^s(Γ_R, M^r) such that, for almost all λ∈Γ_R, B_λ is the right inverse of Ⅱ_λ(L), then there exists E∈W^m(H_n) (when m≥0 or m even) or E∈W^m-1(H_n) (when m<0 and odd) such that LE =δ(near the origie) Where m=min([r],-[2s]-n-2); 2. Let L(W,T) be of the form (3.1). If there exist R≥0 and r,s∈R such that when |λ|≥R,(?) and C_λ≥ C|λ|^x(C>0), then the same conclusion as above holds with m=min(-[2r]-n-2,[-2s]-n-2).     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

一类完全正则半群簇子簇格的性质

本文研究了完全正则半群簇的子簇格［Ｖ＋∩ＰＶ，Ｖ＋∩ＰＶ］的某些格运算性质，我们证明了簇Ｖ＋∩ＰＶ可分解为Ｖ与Ｖ＋∩ＰＶ的并；对任意完全正则半群簇Ｗ，有Ｗ∩（Ｖ∨Ｖ＋∩ＰＶ）＝（Ｗ∩Ｖ）∨（Ｗ∩Ｖ＋∩ＰＶ）．特别地，我们得到了等式Ｖ＋∩ＰＶ＝Ｖ成立的若干条件．     

Bases of the identities of products of certain varieties of groups

The following theorem is proved: The product of any variety of two-step solvable groups and a variety having a finite basis of identity relations has a finite basis of identity relations.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 137–144, January, 1969.     

Birational aspects of the geometry of Varieties of Sums of Powers

Varieties of Sums of Powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms. The study of these varieties dates back to Sylvester and Hilbert, but only few of them, for special degrees and number of variables, are concretely identified. In this paper we aim to understand a general birational behavior of VSP. To do this we birationally embed these varieties into Grassmannians and prove the rational connectedness of many VSP in arbitrary degrees and number of variables.     

Estimate of growth of Lebesgue functions of linear methods of summation

Sufficient conditions are obtained in terms of matrix elements for the boundedness of Lebesgue functions of linear methods of summation of expansions in orthogonal systems of polynomial type.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 277–286, September, 1969.