李color三系的Frattini子系的性质

给出了李color三系的Frattini子系的定义,得到了李color三系的Frattini子系的一些性质·特别的,证明了李color三系T有分解T=T_1⊕T_2⊕…⊕T_m,则φ(T)有分解φ(T)=φ(T_1)⊕φ(T_2)⊕…⊕φ(T_m).     

组合数学与数论间的一些交叉问题初探

我们证明了有限域上的一类方程组解的个数与图的顶点着色数有密切关系，而这又对许多着色问题的产生了许多应用。另外，我们也用图论的一些技巧解决了数论中一些问题。     

Asymptotic Stability of Large-amplitude Oscillations in Systems with Hysteresis

We study depending on a parameter periodic systems with the main linear part and a hysteresis nonlinearity; the linearized at infinity system has a one-dimensional subspace of periodic solutions for the critical parameter value. We prove theorems on a number, localization, and asymptotic stability of large-amplitude periodic solutions for the nonlinear system. Received March 1998     

Ramsey-Type Results for Geometric Graphs, II

We show that for any two-coloring of the segments determined by n points in the plane, one of the color classes contains noncrossing cycles of lengths . This result is tight up to a multiplicative constant. Under the same assumptions, we also prove that there is a noncrossing path of length Ω(n ^2/3 ) , all of whose edges are of the same color. In the special case when the n points are in convex position, we find longer monochromatic noncrossing paths, of length . This bound cannot be improved. We also discuss some related problems and generalizations. In particular, we give sharp estimates for the largest number of disjoint monochromatic triangles that can always be selected from our segments. Received March 25, 1997, and in revised form March 5, 1998.     

Color cyclic homology and Steinberg Lie color algebras

The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ?-cyclic homology group HC₁(R, ?) of the Γ-graded associative algebra R (which could be seemed as the generalization of cyclic homology group and the ?/2?-graded version of cyclic homology that was introduced by Kassel) to calculate the universal central extension of Steinberg Lie color algebras.     

Nil subsets of graded algebras

We prove that if is a Noetherian -graded algebra, then the Jacobson radical of is nilpotent under mild hypotheses on . We also consider affine PI-algebras graded by torsion groups. Finally we prove a Nullstellensatz-type theorem for enveloping algebras of Lie color algebras.

     

Almost-periodic solutions to systems of differential equations with fast and slow time in the degenerate case

We consider a system of ordinary first-order differential equations. The right-hand sides of the system are proportional to a small parameter and depend almost periodically on fast time and periodically on slow time. With this system, we associate the system averaged over fast time. We assume that the averaged system has a structurally unstable periodic solution. We prove a theorem on the existence and stability of almost periodic solutions of the original system. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 451–456, March, 1998.     

Colorings generated by monotone properties

Given a graph G of order n containing no C₄, color an edge e of the complement of G red if G+e contains a C₄, and blue otherwise. Among other results, we answer a question of Erdős, de la Vina, and Fajtlowicz by showing that neither the red nor the blue graph obtained need contain a large complete subgraph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12 : 1–25, 1998     

Minimum Color Sum of Bipartite Graphs

The problem ofminimum color sumof a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated withinn^{1 − ε}, for any ε > 0, unlessNP = ZPP(Bar-Noyet al., Information and Computation140(1998), 183–202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unlessP = NP. The proof is byL-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.     

A sufficient condition for bipartite graphs to be type one

The total chromatic number χ_T(G) of graph G is the least number of colors assigned to V(G) ∪ E(G) such that no adjacent or incident elements receive the same color. In this article, we give a sufficient condition for a bipartite graph G to have χ_T(G) = Δ(G) + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 133–137, 1998     

Algebraic invariant curves for the Liénard equation

Odani has shown that if then after deleting some trivial cases the polynomial system does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of and . We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system.

     

Perturbation analysis of global error bounds for systems of linear inequalities

This paper studies the existence of a uniform global error bound when a system of linear inequalities is under local arbitrary perturbations. Specifically, given a possibly infinite system of linear inequalities satisfying the Slater’s condition and a certain compactness condition, it is shown that for sufficiently small arbitrary perturbations the perturbed system is solvable and there exists a uniform global error bound if and only if the original system is bounded or its homogeneous system has a strict solution. Received: April 12, 1998 / Accepted: February 11, 2000?Published online July 20, 2000     

A Construction of H 4 without Miracles

We present a new construction of the root system H ₄ . Received January 28, 1998, and in revised form March 23, 1998.     

Hopf Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

This paper is concerned with small quadratic perturbations to one parameter family of generic reversible quadratic vector fields with a simple center. The first objective is to show that this system exhibits two small amplitude limit cycles emerging from a Hopf bifurcation. The second one we prove that the system has no limit cycle around the weak focus of order two. The results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev (1998).     

Uniqueness for a mutually catalytic branching model

The uniqueness in law for a system of SPDEs which describes a mutually catalytic branching model is established. The proof uses a duality technique and requires a linear transformation of the original processes. Received: 16 January 1998 / Revised version: 18 March 1998     

Enveloping algebras of lie color algebras: primeness versus graded-primeness

Let be a finite abelian group and let be a, possibly restricted, -graded Lie color algebra. Then the enveloping algebra is also -graded, and we consider the question of whether being graded-prime implies that it is prime. The first section of this paper is devoted to the special case of Lie superalgebras over a field of characteristic . Specifically, we show that if and if has a unique minimal graded-prime ideal, then this ideal is necessarily prime. As will be apparent, the latter result follows quickly from the existence of an anti-automorphism of whose square is the automorphism of the enveloping algebra associated with its -grading. The second section, which is independent of the first, studies more general Lie color algebras and shows that if is graded-prime and if most homogeneous components of are infinite dimensional over , then is prime. Here we use -methods to study the grading on the extended centroid of . In particular, if is generated by the infinite support of , then we prove that is homogeneous.

     

Local Dissipativity and Attractors for the Evolutionary Superconductivity Problem

In this paper, under the Lorentz gauge, the global existence, local dissipativity and attractors of the evolutionary superconductivity problem which is a parabolic-hyperbolic system are obtained. Received March 4, 1998, Revised October 9, 1998, Accepted March 25, 1999     

On ordering and multiplication of natural numbers

Even if the ordering of all natural number is (known to be) not definable from multiplication of natural numbers and ordering of primes, there is a simple axiom system in the language such that the multiplicative structure of positive integers has a unique expansion by a linear order coinciding with the standard order for primes and satisfying the axioms – namely the standard one. Received: 3 December 1998     

Hypergraphs and the Clar problem in hexagonal systems

We show that for a hypergraph H that is separable and has the Helly property, the perfect matchings of H are the strongly stable sets of the line graph of H. Also, we show that the hypergraph generated by a hexagonal system is separable and has the Helly property. Finally, we note that the Clar problem of a hexagonal system is a minimum cardinality perfect matching problem of the hypergraph generated by the hexagonal system. Hence, the Clar problem of a hexagonal system is a minimum cardinality strongly stable set problem in the line graph of the hypergraph generated by the hexagonal system.     

On b-perfect Chordal Graphs

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph. This research was supported by Algerian-French program CMEP/Tassili 05 MDU 639.