一元二次方程的根的判别式的应用

一元二次方程的根的判别式是初中代数的重要内容之一 ,它在中学数学中有着广泛的应用 ,成为近几年全国各地中考的热点问题 .为了帮助读者更好地掌握好这部分知识内容 ,现对它在初中数学中的应用进行归纳 ,以餮读者 .应用一 :判断一元二次方程 (或二元二次方程组 )的根的情况 ;或已知根的情况 ,求方程 (或组 )中的待定系数的取值范围 .一元二次方程ａｘ2 +ｂｘ +ｃ =0 (ａ≠ 0 )的根的判别式为△ =ｂ2 - 4ａｃ,它与这个方程的根有着十分密切的关系 :( 1)△ >0 方程有两个不等的实数根 ;( 2 )△ =0 方程有两个相等的实数根 .( 3)△ <0 方程…     

图的邻域复形的同调群的不变性

本文研究了图的邻域复形同调群的不变性质。设G是一个简单连通图,x是G的一个顶点,以G/x表示G中剔去点v及其关联边而得到的图,给出了G和G/x的邻域复形的同阶同调群同构的充要条件。     

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.     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

纯正的群的正则带的构造

本文研究纯正的群的正则带.在给出这类半群的若干特征后,建立了纯正的群的正则带的构造定理.作为应用,同时给出了纯正的群的右拟正规带的构造定理.     

中心对称的多边形的外接圆周上点的性质

笔者发现中心对称的多边形的外接圆周上的点具有性质：中心对称的多边形每组对边上关于它的外接圆心的对称点将各边分为成比例线段,则此圆周上任意点到各个对称点的距离的平方和为定值,即有如下命题。     

Toeplitz算子的谱的某些子集的连续性

本文研究了Hilbert空间上有界线性算子的谱的某些子集的连续性,利用算子谱的精密结构的分析方法,给出了Hilbert空间H上有界线性算子T的谱σ(T)的某些子集如Φn(T),Φ(T),Φ+(T),Φ-(T),σ0p(T)等连续的充要条件.特别在Hardy空间H2(Γ)上,研究了Toeplitz算子Tφ的谱σ(Tφ)的某些子集的连续性.     

随机变量的差的分布函数的积分表达式

本文利用Ｌｅｂｅｓｇｕｅ－Ｓｔｉｅｌｔｊｅｓ积分，把连续型随机变量差的密度函数的积分表达式推广为一般随机变量的分布函数的积分表达式     

乘方的指数与其幂的位数的关系

为了探究乘方的指数与其幂的位数的关系,定义了几个有关的新概念,并且证明了两个关于乘方以及进制进位的定理,由此建立起关于乘方以及进制进位的理论体系,其中包括进位理论中判定乘方的指数与其幂的位数是否存在周期规律的判别法,以及进位规律的求解法和四条相关的性质.     

随机向量的函数的独立性的一个问题

给出了随机变量Ｘ１,Ｘ２,Ｘ３,Ｘ４每三个相互独立,但Ｘ１&;#177;Ｘ２与Ｘ３&;#177;Ｘ４不相互独立的例子,以及Ｘ１,Ｘ２,Ｘ３每两个相互独立,但Ｘ１&;#177;Ｘ２与Ｘ３不相互独立的例子。     

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.     

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.     

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...     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

Determination of asymptotic solutions of problems of nonlinear theories of thermoviscoelasticity

It is shown that the asymptotic solution of a problem of the nonlinear theory of thermoviscoelasticity, if it exists, can be found directly from the solution of the asymptotic boundary-value problem without completely solving the starting problem.M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 3, pp. 395–400, May–June, 1976.     

Emptiness of set of points of lower semicontinuity of Lyapunov exponents

We consider parametric families of differential systems with coefficients that are bounded and continuous on the half-line and uniformly in time continuously depend on a real parameter. For each Lyapunov exponent, we construct a family such that the Lyapunov exponent of its systems treated as a function of the parameter is not a lower semicontinuous function for any value of the parameter.     

图表示范畴的两个子范畴

引进图表示范畴的两个子范畴,研究它们的同调性质。     

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.