On L(d,1)-labelings of graphs

Given a graph G and a positive integer d, an L(d,1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u)−f(v)|d; if u and v are not adjacent but there is a two-edge path between them, then |f(u)−f(v)|1. The L(d,1)-number of G, λ_d(G), is defined as the minimum m such that there is an L(d,1)-labeling f of G with f(V){0,1,2,…,m}. Motivated by the channel assignment problem introduced by Hale (Proc. IEEE 68 (1980) 1497–1514), the L(2,1)-labeling and the L(1,1)-labeling (as d=2 and 1, respectively) have been studied extensively in the past decade. This article extends the study to all positive integers d. We prove that λ_d(G)Δ²+(d−1)Δ for any graph G with maximum degree Δ. Different lower and upper bounds of λ_d(G) for some families of graphs including trees and chordal graphs are presented. In particular, we show that the lower and the upper bounds for trees are both attainable, and the upper bound for chordal graphs can be improved for several subclasses of chordal graphs.     

No-hole 2-distant colorings for Cayley graphs on finitely generated abelian groups

A no-hole 2-distant coloring of a graph Γ is an assignment c of nonnegative integers to the vertices of Γ such that |c(v)-c(w)|?2 for any two adjacent vertices v and w, and the integers used are consecutive. Whenever such a coloring exists, define nsp(Γ) to be the minimum difference (over all c) between the largest and smallest integers used. In this paper we study the no-hole 2-distant coloring problem for Cayley graphs over finitely generated abelian groups. We give sufficient conditions for the existence of no-hole 2-distant colorings of such graphs, and obtain upper bounds for the minimum span nsp(Γ) by using a group-theoretic approach.     

Radio number for trees

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f:V(G)→{0,1,2,…,k} with the following satisfied for all vertices u and v: |f(u)-f(v)|?diam(G)-d_G(u,v)+1, where d_G(u,v) is the distance between u and v. We prove a lower bound for the radio number of trees, and characterize the trees achieving this bound. Moreover, we prove another lower bound for the radio number of spiders (trees with at most one vertex of degree more than two) and characterize the spiders achieving this bound. Our results generalize the radio number for paths obtained by Liu and Zhu.     

Best Simultaneous Approximation in L(I,E)

Let G be a reflexive subspace of the Banach space E and let L^p(I,E) denote the space of all p-Bochner integrable functions on the interval I=[0,1] with values in E, 1p∞. Given any norm N( , ) on R², N nondecreasing in each coordinate on the set R²₊, we prove that L^p(I,G) is N-simultaneously proximinal in L^p(I,E). Other results are also obtained.     

(Almost) primitivity of Hecke L -functions

Let f be a cusp form of the Hecke space \frak M₀(l,k,e){\frak M}_0(\lambda,k,\epsilon) and let L _f be the normalized L-function associated to f. Recently it has been proved that L _f belongs to an axiomatically defined class of functions [`(S)]<sup>\sharp</sup>\bar{\cal S}^\sharp . We prove that when λ ≤ 2, L <sub>f</sub> is always almost primitive, i.e., that if L <sub>f</sub> is written as product of functions in [`(S)]^\sharp\bar{\cal S}^\sharp , then one factor, at least, has degree zeros and hence is a Dirichlet polynomial. Moreover, we prove that if l ? {?2,?3,2}\lambda\notin\{\sqrt{2},\sqrt{3},2\} then L _f is also primitive, i.e., that if L _f = F ₁ F ₂ then F ₁ (or F ₂) is constant; for l ? {?2,?3,2}\lambda\in\{\sqrt{2},\sqrt{3},2\} the factorization of non-primitive functions is studied and examples of non-primitive functions are given. At last, the subset of functions f for which L _f belongs to the more familiar extended Selberg class S^\sharp{\cal S}^\sharp is characterized and for these functions we obtain analogous conclusions about their (almost) primitivity in S^\sharp{\cal S}^\sharp .     

The existence of inner invariant means on L(G)

The purpose of this paper is to generalize the concepts of amenability for locally compact groups and inner amenability for discrete groups by considering the existence of inner invariant means. Based on this generalization, locally compact groups can be classified as so called [IA] groups or non-[IA] groups. A number of equivalent conditions characterizing [IA] groups are given. Also the possibility of inner invariant extension of the Dirac measure δ_e is discussed.     

Archimedean L -factors on GL ( n ) × GL ( n ) and generalized Barnes integrals

The Rankin-Selberg method associates, to each local factorL(s, π _v × π _v ^′ ) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π _v and _v ^′ . In this paper we show that, if ν is archimedean, and π _v and _v ^′ are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one. We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.     

（L^p,L^q） Estimates for Multilinear Oscillatory Singular Integralswith Smooth Phases

In this paper,the authors establish the weighted (L^p,L^q) estimates for a class of multilinear oscillatory singular integrals with smooth phases.Certain endpoint estimates are also considered.     

双圈连通图的L(2,1)-labelling

给定图G,G的一个L(2,1)-labelling是指一个映射f:V(G)→{0,1,2,…},满足:当dG(u,v)=1时,f(u)-f(v)≥2;当dG(u,v)=2时,f(u)-f(v)≥1.如果G的一个L(2,1)-labelling的像集合中没有元素超过k,则称之为一个k-L(2,1)-labelling.G的L(2,1)-labelling数记作l(G),是指使得G存在k-L(2,1)-labelling的最小整数k.如果G的一个L(2,1)-labelling中的像元素是连续的,则称之为一个no-holeL(2,1)-labelling.本文证明了对每个双圈连通图G,l(G)=△ 1或△ 2.这个工作推广了[1]中的一个结果.此外,我们还给出了双圈连通图的no-hole L(2,1)-labelling的存在性.     

Separation properties of induced I(L)-topological spaces

The concept of induced I(L)-topological spaces has been introduced by Kubiak (Ph.D. Thesis, UAM, Poznan, 1985) and independently by Wang (Kexue Tongbao 34 (5) (1989) 333). In this paper, the separation properties in the sense of Hutton–Reilly of induced I(L)-topological spaces are investigated. The main result of the paper is a characterization of L-topological spaces by means of the appropriate Hutton–Reilly separation properties of its induced I(L)-topological space.     

MULTIPLIERS AND TENSOR PRODUCTS OF L（p, q） LORENTZ SPACES

Let G be a locally compact abelian group. The main purpose of this article is to find the space of multipliers from the Lorentz space L(p1, q1)(G) to L(p21,q21)(G). For this reason, the authors define the space Ap1,q1p2,q2(G), discuss its properties and prove that the space of multipliers from L(p1,q1)(G) to L(p21,q21)(G) is isometrically isomorphic to the dual of Ap1,q1p2q2(G).     

L p -dual Quermassintegral sums

In this paper, we first introduce a concept of L _p -dual Quermassintegral sum function of convex bodies and establish the polar projection Minkowski inequality and the polar projection Aleksandrov-Fenchel inequality for L _p -dual Quermassintegral sums. Moreover, by using Lutwak’s width-integral of index i, we establish the L _p -Brunn-Minkowski inequality for the polar mixed projection bodies. As applications, we prove some interrelated results. This work was partially supported by the National Natural Science Foundation of China (Grant No. 10271071), Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605065) and Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Weighted Uncertainty Principles in L

We determine sufficient conditions on positive weights W and V such that there exists continuous, strictly increasing functions Φ and Ψ on [0, ∞) such that Φ(0)=0=Ψ(0) and whenever f: R→R is a continuous integrable function. We also give an example that shows the optimality of our conditions.     

Best polynomial and durrmeyer approximation in Lp(S)

On a simplex SR^d, the best polynomial approximation is E_n()_Lp(S)=Inf{P_n−_Lp(S): P_n of total degree n}. The Durrmeyer modification, M_n, of the Bernstein operator is a bounded operator on L_p(S) and has many “nice” properties, most notably commutativity and self-adjointness. In this paper, relations between M_n−z.dfnc;_Lp(S) and E_[√n]()_Lp(S) will be given by weak inequalities will imply, for 0<α<1 and 1≤p≤∞, E_n()_Lp(S)=O(n^-2α)M_n−z.dfnc;_Lp(S)=O(n^-α). We also see how the fact that P(D)εL_p(S) for the appropriate P(D) affects directional smoothness.     

H(λ)-completely Hausdorff axiom on L-topological spaces

This paper defines the new concept of completely Hausdorff axiom of an L-topological space by means of L-continuous mappings from an L-topological space to the refined Hutton's unit L-interval by Wang. Some characterizations of the completely Hausdorff axiom, defined in this paper, are given, and many nice properties of this kind of completely Hausdorff axiom are proved. For example, it is hereditary and product invariant; the refined Hutton's unit L-interval satisfy this kind of completely Hausdorff axiom, and when an L-topological space satisfy this kind of completely Hausdorff axiom, every f-convergent ideal does not have f-limit points with different supports etc. The relation between the completely Hausdorff axiom defined in the paper and other separation axioms is discussed also.     

Estimating L ∞ Norms by L 2k Norms for Functions on Orbits

Abstract. Let G be a compact group acting in a real vector space V . We obtain a number of inequalities relating the L ^∞ norm of a matrix element of the representation of G with its L ^2k norm for a positive integer k . As an application, we obtain approximation algorithms to find the maximum absolute value of a given multivariate polynomial over the unit sphere (in which case G is the orthogonal group) and for the assignment problem of degree d , a hard problem of combinatorial optimization generalizing the quadratic assignment problem (in which case G is the symmetric group).