The Complete Chromatic Number of Maximal Outerplane Graphs

Let G be a maximal outerplane graph and X0(G) the complete chromatic number of G. This paper determines exactly X0(G) for △(G)≠5 and proves 6≤X0.(G)≤7 for △(G) = 5, where △(G) is the maximum degree of vertices of G.     

紧致流形上Lichnerowicz方程的梯度估计(英文)

Let(M,g) be a smooth compact Riemannian manifold of dimension n.Denote△f=△-▽f.▽ the weighted Laplacian operator,where f is a smooth real valued function on M.When N is finite and the N-Bakry-Emery Ricci tensor is bounded from below by a constant,we establish local gradient estimates for positive solutions of the following simple Lichnerowicz equation△fu+cu~(-α)=0 on a compact Riemannian manifold,where α is a positive constant and c is a smooth function.     

Generalized Differential Identities of （Semi-）Prime Rings

Let R be a semiprime ring with characteristic p≥0 and RF be its left Martindale quotient ring. If ф（Xi^△j） is a reduced generalized differential identity for an essential ideal of R, then ф（Zije（△j ）） is a generalized polynomial identity for RF, where e（△j） are idempotents in the extended centroid of R determined by △j. Let R be a prime ring and Q be its symmetric Martindale quotient ring. If ф（Xi△j） is a reduced generalized differential identity for a noncommutative Lie ideal of R, then ф（Zij） is a generalized polynomial identity for [R, R]. Moreover, if ф（Xi△j） is a reduced generalized differential identity, with coefficients in Q, for a large right ideal of R, then ф（Zij） is a generalized polynomial identity for Q.     

Best approximation by downward sets with applications

We develop a theory of downward sets for a class of normed ordered spaces. We study best approximation in a normed ordered space X by elements of downward sets, and give necessary and sufficient conditions for any element of best approximation by a closed downward subset of X. We also characterize strictly downward subsets of X, and prove that a downward subset of X is strictly downward if and only if each its boundary point is Chebyshev. The results obtained are used for examination of some Chebyshev pairs （W,x）, where ∈ X and W is a closed downward subset of X     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.     

Menger空间的收敛性和隔离性

Suppose that (X,F,△) is a Menger spece and that t-norm △ satisfies (?)△(x, a) = a for all a∈[0,l]. The main result of the paper is as follows:     

Edge choosability of planar graphs without short cycles

In this paper we prove that if G is a planar graph with △= 5 and without 4-cycles or 6-cycles, then G is edge-6-choosable. This consequence together with known results show that, for each fixed k ∈{3,4,5,6}, a k-cycle-free planar graph G is edge-(△ 1)-choosable, where △ denotes the maximum degree of G.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.     

ADMISSIBLE ESTIMATES IN THE IMPORTANT CLASS OF ESTIMATES OF THE COVARIANCE MATRIX

Let X1,…XN(where N>m)be independent Nm(μ,∑）random vectors,and put X^-=1/N ∑i=1^N Xi and T‘T=A=∑i=1^N(Xi-X^-)(Xi-X^-)‘,where T is upper-triangular with positive diagonal elements.The author considers the problem of estimating ∑,and restricts his attention to the class of estimates D={T‘△^*T+Nb^*X^-X^‘·△^* is any diagonal matrix and b^* is any nonnegative constant}because it has the following attractive features:(a)Its elements are all quadratic forms of the sufficient and complete statistics(X^-,T).(b)It contains all estimates of the form αA+NbX^-X^-‘(α≥0 and b≥0),which construct a complete subclass of the class of nonnegative quadratic estimates D^8={X‘BX:B≥0}(where X=(X1,…,XN)‘)for any strict convex loss function.(c)It contains all invariant estimates under the transformation group of upper-triangular matrices.The author obtains the characteristics for an estimate of the form.T‘△T+NbX^-X^-‘(△=diag{δ1,…，δm}≥0 and b≥0)of ∑ to be admissible in D when the loss function is chosen as tr(∑^-1∑-I)^2,and shows,by an example,that αA+NX^-X^-‘(α≥0 and b≥0)is admissible in D^* can not imply its admissibility in D.     

LOWER BOUNDS OF DIRICHLET EIGENVALUES FOR A CLASS OF FINITELY DEGENERATE GRUSHIN TYPE ELLIPTIC OPERATORS

Let ? be a bounded open domain in Rnwith smooth boundary ??,X =(X_1,X_2,···,X_m) be a system of real smooth vector fields defined on ? and the boundary ?? is non-characteristic for X. If X satisfies the H¨ormander's condition,then the vector field is finitely degenerate and the sum of square operator △X =Σ_(j=1)~mX_j~2 is a finitely degenerate elliptic operator. In this paper,we shall study the sharp estimate of the Dirichlet eigenvalue for a class of general Grushin type degenerate elliptic operators △X on ?.     

Almansi decomposition for Dunkl operators

Let Ωbe a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ωwhich are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x) |x|2f1(x) … |x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.     

Variational Principle for Topological Pressure on Subsets of Free Semigroup Actions

We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions. Let(X, G) be a system, where X is a compact metric space and G is a finite family of continuous maps on X. Given a continuous function f on X, we define Pesin–Pitskel topological pressure P_G(Z, f)for any subset Z ■ X and measure-theoretical pressure P_(μ,G)(X, f) for any μ∈ M(X), where M(X)denotes the set of all Borel probability measures on X. For any non-empty compact subset Z of X, we show that P_G(Z, f) = sup{P_(μ,G)(X, f) : μ∈ M(X), μ(Z) = 1}.     

GENERAL DECAY OF SOLUTIONS FOR A VISCOELASTIC EQUATION WITH NONLINEAR DAMPING AND SOURCE TERMS

The initial boundary value problem for a viscoelastic equation | u t | ρ u tt △u-△u tt + t 0 g(ts)△u(s)ds + | u t | m u t = | u | p u in a bounded domain is considered, where ρ, m, p > 0 and g is a nonnegative and decaying function. The general uniform decay of solution energy is discussed under some conditions on the relaxation function g and the initial data by adopting the method of [14, 15, 19]. This work generalizes and improves earlier results in the literature.     

Scalar curvature type problem on the three dimensional bounded domain

In this paper we prove an existence result for the nonlinear elliptic problem:-△u = Ku~5,u 0 in Ω,u = 0 on?Ω,where Ω is a smooth bounded domain of R~3 and K is a positive function in Ω.Our method relies on studying its corresponding subcritical approximation problem and then using a topological argument.     

NOTES ON POLYNOMIAL LEVELABILITY

We consider the polynomial levelability with respect to approximation algorithms (PLAA).A set A is PLAA if given any approximation algorithm a for A and a polynomial p,there are another approximation algorithm β for A and a polynomial q such that for infinitely many inputs x,a accepts x but has ruuning time greater than p(|x|) and β accepts x within time q(|x|).In this paper,an algorithm a is called an approximation algorithm for A if the symmetric difference A△L(a) is sparse,where L(a) is the set of strings recegnized by a.We prove that all natural NP-complete sets are PLAA unless P=NP and all EXP-complete sets are PLAA.     

Strong law of large number for branching hunt processes

In this paper,we prove that,under certain conditions,a strong law of large number holds for a class of branching particle systems X corresponding to the parameters(Y,β,ψ),where Y is a Hunt process andψis the generating function for the offspring.The main tool of this paper is the spine decomposition and we only need an L log L condition.     

Gradient Estimates for a Nonlinear Parabolic Equation with Diffusion on Complete Noncompact Manifolds

The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0σ1 is a real constant,and b(x,t) is a function which is C~2 in the x-variable and C~1 in the t-variable.     

ADDITIVE FUNCTIONALS AND PERTURBATION OF SEMIGROUP

g1. Introduction and PreliminariesSuPpose that X is a right Markov process with state space E and transition sendgrouP(Pt). Given a multiplicative functional M of X, we define for any nonnegative measurablefunction f on E,Qti(x):= p"(f(X)M), x E E, t 2 0.It follows from the multiplicativity of M that (Qt) is also a semigrouP of transition functionson E and usuaJly called a perturbation semigroap of (Pt). The wellknOWn FeynmanKacsendgrouP is a special case where X is a Brownian moti…     

Lower Bounds of Dirichlet Eigenvalues for General Grushin Type Bi-Subelliptic Operators

Let Q be a bounded open domain in R^n with smooth boundaryаΩ.Let X=(X1,X2…,Xm)be a system of general Grushin type vector fields defined onΩand the boundaryаΩis non-characteristic for X.For△x=∑j=1^mXj^2,we denoteλk as the k-th eigenvalue for the bi-subelliptic operator△X2^2 onΩ.In this paper,by using the sharp sub-elliptic estimates and maximally hypoeliptic estimates,we give the optimal lower bound estimates ofλk for the operatork△X^2.     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.