关于合成图边色数问题的一点注记

本文限于讨论有限简图,故 Vizing 关于简图边色数的定理成立.即,对任意简图G,G∈C~1或 G∈C~2.[2]中指出,尽管已知当 G、H∈C~1时,合成图 G[H]∈C~1,但是,当 G∈C~1、H∈C~2时,是否也有 G[H]∈C~1的问题尚未解决.现在,本文证明下列定理1 若 G∈C~1,则 G[H]∈C~1.     

关于半单李代数的极大环面子代数的中心化子

本文目的是给出以下命题(见[1])一个简单证明,本文沿用[1]中符号.命题.设 L 是 char=0的代数闭域 F 上的半单李代数,H 是其一极大环面子代数,则 H=C_L(H) (这里 C_L(H) 表示 H 的中心化子).证.分几步进行,记 C=C_L(H).(1)C 包含它的元素的半单部分和幂零部分.对任意 x∈C,有 ad_L xH=0,由[1]命题4.2,(ad_Lx)_sH=0,(ad_L x)H=0.由[1]系理6.4,(ad_Lx)_s=ad_L x_s,(ad_Lx)_n=ad_Lx_n.因此 x_s,x_n∈C.(2)C 的所有半单元均在 H 中.     

BRILL－NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Lei X be an arbitrary smooth irreducible complex projective curve, E (?) X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H0(X, [D1] |D2) is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1 + D2, the author constructs a 2g X d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?)Im{H0(X, E) (?) H0(X,[D2]) (?) H0(X,[D2] |D2)}. From this and H0(X,E) = H0(X, [D1]) (?) Im{H0(X, E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.     

THE HARDY TYPE INEQUALITY FOR THE MAXIMAL OPERATOR OF THE ONE-DIMENSIONAL DYADIC DERIVATIVE

In this paper we prove that the maximal operator I of dyadic derivative is not bounded from the Hardy space H p [0, 1] to the Hardy space H p [0, 1], when 0 < p ≤ 1.     

Surjectivity of Generaized Expanding Maps and Some Characterizations of Monotone Operators

In [1] L. Nirenberg stated the following problem: Suppose T is a continuous map H→H (H is a Hilbert space), which is expanding,i.e. and T(0)=0.Suppose T map a neighborhood of the origin onto a neighborhood of the origin. Does T maps H onto H? In [2] Kung-Ching Chang and Li Shujie answered positively the problem when T is differentiable.     

On 2n—th—Order Eigenvalue Problem

In [1], Lu discussed the existence of positive solution for the BVPwhere f: [0, 1] × [0, ∞) → R is continuous and satisfies the following conditions(H1) There exists an M > 0 such that(H2) limx→∞ f(t, x)/x =∞uniformly for t [α,β] (0, 1) for some α < β and λ > 0 is a parameter.In this paper, the result of [1] is generalized to the following 2n-th-order BVPThe result of this paper isTheorem 1 Assume (H1) and (H2) hold. Then the BVP (3)-(4) has a positive solutionifλ > 0 is sm…     

与任意图正交的[0,k_j]_1~m-因子分解

设k1,k2 ,… ,km 是正整数 .证明了 :若G是一个 [0 ,k1 k2 … km -m 1] 图 ,H是G中一个给定的有m条边的子图 ,则G有一个 [0 ,kj]m1 因子分解与H正交 .     

圆内接闭折线垂心的一个性质

众所周知 ,三角形的垂心有如下性质[1] :定理 1　设△ ABC的外接圆半径为 R,垂心为 H ,则 ( AB2 BC2 CA2 ) ( H A2 H B2 H C2 ) =1 2 R2 .将这个定理推广到一般圆内接闭折线中 ,可得定理 2　设闭折线 A1A2 A3 … An A1内接于⊙ ( O,R) ,其垂心为 H ,则　 ∑ni=1Ai A2i 1 ∑ni=1H A2i =n( n 1 ) R2 ,( * )其中 An 1为 A1.证明　以圆心 O为原点建立直角坐标系x Oy(图略 ) ,设顶点 Ai 的坐标为 ( xi,yi) ( i =1 ,2 ,… ,n) ,垂心 H的坐标为 ( x H,y H) ,则有[2 ]x H =∑ni=1xi,　 y H =∑ni=1yi. 1由两点间的距…     

三角形垂心的一个性质推广

文[1]给出了三角形垂心的一个性质: 定理若△ABC的垂心为H,且D、E、F分别为H在BC、CA、AB边所在直线上的射影,H1、H2、H3分别为△AEF、△BDF、△CDE的垂心,则△DEF≌△H1H2H3.     

与任意图正交的[0,kj]m1-因子分解

设kl,k2,…,km是正整数.证明了若G是一个[0,k1+k2+…+km-m+1]-图,H是G中一个给定的有m条边的子图,则G有一个[0,kj]m1-因子分解与H正交.     

Structure and Properties of Nanocrystalline Oxide Powders

The crystalline and porous structure of superfine powders and fibers made of alumina and zirconia doped with yttria is investigated. The particle sizes were determined by the methods of coherent dispersion and scanning transmission electron microscopy. Based on data on the sizes of crystallites of the metal oxides, which were obtained by different methods, it is found that crystalline transformations of the oxides lead to dispersion of the materials. The structural transformations of the superfine oxides are accompanied by changes in crystallite sizes and in the character and size of pores. The laws established allow one to purposefully control the process of producing nanostructured oxide powders and fibers, which can be utilized as active fillers for composite materials based on various matrices. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 831–840, November–December, 2005.     

Effects of permeability and viscosity in linear polymeric gels

We propose and analyze a mathematical model of the mechanics of gels, consisting of the laws of balance of mass and linear momentum of the polymer and liquid components of the gel. We consider a gel to be an immiscible and incompressible mixture of a nonlinearly elastic polymer and a fluid. The problems that we study are motivated by predictions of the life cycle of body‐implantable medical devices. Scaling arguments suggest neglecting inertia terms, and therefore, we consider the quasi‐static approximation to the dynamics. We focus on the linearized system about stress‐free states, uniform expansions, and compressions and derive sufficient conditions for the solvability of the time‐dependent problems. These turn out to be conditions that guarantee local stability of the equilibrium solutions. We also consider non‐stress free equilibria and states with residual stress and derive an energy law for the corresponding time‐dependent system. The conditions that guarantee stability of solutions provide a selection criteria of the material parameters of devices. The boundary conditions that we consider are of two types, displacement‐traction and permeability of the gel surface to the fluid. We address the cases of viscous and inviscid solvent, assume Newtonian dissipation for the polymer component, and establish existence of weak solutions for the different boundary permeability conditions and viscosity assumptions. We present two‐dimensional, finite element numerical simulations to study stress concentration on edges, this being the precursor to debonding of the gel from its substrate. Copyright © 2015 John Wiley & Sons, Ltd.     

Cocyclic Codes of Length 40

There are 5 groups of order 20. This paper reports on the search for binary self-dual codes of length 40, cocyclic over any one of the first four groups, using cocyclic Hadamard matrices and the [I, A] construction. The fifth group is not investigated here. A total of 28 classes of extremal cocyclic self-dual codes were found—27 of these are doubly-even and one singly-even. The majority of these classes arise from the dihedral-cocyclic Hadamard matrices. There is also a class of dihedral-cocyclic Hadamard matrices which gives a large collection of [40, 20] codes with only one codeword of length 4.     

Survey on path and cycle embedding in some networks

To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.      

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Comparison principle and analogues of the phragmen—lindelöf theorem for solutions of parabolic inequalities

It is well known that a Monotonicity Condition and a Coerciveness Condition principally lie in the basis of most results of the Theory of PDE's. The necessity of these important assumptions for the validity of a comparison principle and analogues of the Phragmen-Lindelöf theorem for solutions of quasilinear parabolic inequalities is discussed in the paper. In the first part of the work we introduce a new concept of monotonicity for nonlinear differential operators-nonlinear monotonicity concept-and on its basis we obtain new phenomena for solutions, subsolutions and supersolutions of the well-known quasilinear differential equations. In the second part we omit the current coerciveness condition and change it by a weaker one. In spite of this we obtain a series of new qualitative properties of solutions for wide classes of quasilinear parabolic inequalities. Most of these properties are also new for solutions of the well-known equations, which we consider in the paper.     

Finding Optimal Targets for Change Agents: A Computer Simulation of Innovation Diffusion

We introduce a diffusion of innovation model based on a network threshold approach. Realistic network and threshold data were gathered regarding the diffusion of new software tools within part of a large organization. Novel model features are a second threshold for innovation rejection and a memory that allows actors to take trends into account. Computer simulations produce expected outcomes, such as the S-shaped diffusion curve, but also diffusion breakdown and oscillations. We define and compute the quality of change agent targets in terms of the impact targeted actors have on the diffusion process. Our simulations reveal considerable variance in the quality of actors as change agent targets. Certain actors can be singled out as especially important to the diffusion process. Small changes in the distribution of thresholds and changes in some parameters, such as the sensitivity for trends, lead to significant changes in the target quality measure. To illustrate these interdependencies we outline how the impact of an actor targeted by a change agent spreads through the network. We thus can explain why a good change agent target does not necessarily need to be an opinion leader. Simulations comparing the effectiveness of randomly selected targets versus a group of good change agent targets indicate that the selection of good targets can accelerate innovation diffusion.     

Torsional vibrations and dispersion of torsional waves in orthotropic composite rods

The Barr’s refined theory of torsional vibrations of isotropic rods of noncircular cross section is generalized for an orthotropic material. An analysis of natural frequencies of torsional vibration of free-free orthotropic prismatic rods of rectangular cross section is carried out with the help of an exact solution of the frequency equation. For orthotropic CFRP and GFRP rods, the improved theory, which takes into account the normal stresses and inertia forces in the axial direction, in some cases, predicts a noticeable raise in the natural frequencies compared with those following from the Saint-Venant classical theory. A good agreement is obtained between the experimental and calculated values of natural frequencies of torsional vibrations of rectangular quartz and fiber glass rods. The dispersion of torsional waves in an orthotropic quasi-homogeneous rod is considered. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 165–182, March–April, 2008.     

Structural constraints on communication: The diffusion of the homomorphic signal analysis technique through scientific fields

Within an organizational environment there are a variety of communication channels through which information can and does diffuse. One's view about the nature of information affects which type of communication channel is expected to be most effective for diffusion. Further, each type of communication channel defines a structure within the organization. In order to determine the relative impact of various communication channels on the diffusion of information within science the diffusion of a particular method, Homomorphic Signal Analysis, is explored relative to a set of organizational structures for the relevant scientific community. Findings include the following. Methods, as information, have a map rather than a grammatical production nature. Consequently, organizational structures that combine both shared knowledge and potential for interaction are most effective in the diffusion of methods. Generalist journals and the working connections between opinion leaders play dominant roles in the diffusion of methods.     

Paths and cycles of hypergraphs

Hypergraphs are the most general structures in discrete mathematics. Acyclic hypergraphs have been proved very useful in relational databases. New systems of axioms for paths, connectivity and cycles of hypergraphs are constructed. The systems suit the structure properties of relational databases. The concepts of pseudo cycles and essential cycles of hypergraphs are introduced. They are relative to each other. Whether a family of cycles of a hypergraph is dependent or independent is defined. An enumeration formula for the maximum number of independent essential cycles of a hypergraph is given.