Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

(g, f)-factorizations orthogonal to a subgraph in graphs

LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)⩽(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively. Project supported by the National Natural Science Foundation of China.     

Inequalities for the interpolation points in Chebyshev approximation by polynomials

Letf andg be approximated in the Chebyshev sense by polynomials of degree n and n–1, respectively. It is shown that if the sum and difference of the normalized (n+1)-st derivatives off andg do not change sign, then the interpolation points ofg separate those off. A corollary is that the zeros of the Chebyshev polynomialT _n separate the interpolation points off iff ⁽ⁿ⁺¹⁾ does not change sign. The sharpness of this result is demonstrated.     

Lower degree bounds for modular invariants and a question of I. Hughes

We prove two statements. The first one is a conjecture of Ian Hughes which states that iff ₁, ..., f_n are primary invariants of a finite linear groupG, then the least common multiple of the degrees of thef _i is a multiple of the exponent ofG.The second statement is about vector invariants: IfG is a permutation group andK a field of positive characteristicp such thatp divides |G|, then the invariant ringK[V ^m]^G ofm copies of the permutation moduleV overK requires a generator of degreem(p–1). This improves a bound given by Richman [6], and implies that there exists no degree bound for the invariants ofG that is independent of the representation.     

On the isometries of the Lorentz function spaces

LetT be an (into linear) isometry on a (real or complex) Lorentz function spaceL _w,p,1≤p<∞. We show that iff andg have disjoint support, thenT f andT g also have disjoint support. Using this result, we give a characterization of the isometries ofL _w,p.     

Decomposition of positive definite functions defined on a neighbourhood of the identity

LetV be a symmetric open neighbourhood of the identity of a topological groupG. We show that every positive definite functionf onV can be written asf=f _c +f _s wheref _c andf _s are positive definite functions onV, f _c is continuous andf _s averages to zero. IfG is locally compact with Haar measurem _G andf ism _G -measurable thenf _s =0m _G -almost everywhere.     

Group-graded algebras with polynomial identity

LetG be a finite group and letR=Σ _g∈G R _g be any associative algebra over a field such that the subspacesR _g satisfyR _g R _h ⊆R _gh . We prove that ifR ₁ satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withR ^H satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.     

A new characterization of frobenius complements

LetH, G be finite groups such thatH acts onG and each non-trivial element ofH fixes at mostf elements ofG. It is shown that, ifG is sufficiently large, thenH has the structure of a Frobenius complement. This result depends on the classification of finite simple groups. We conclude that, ifG is a finite group andA ⊆G is any non-cyclic abelian subgroup, then the order ofG is bounded above in terms of the maximal order of a centralizerC _G(a) for 1≠a ∈A.     

Teichmüller sequences on trajectories invariant under a Kleinian group

We extend the results of [T2] to the situation where there is a compatibility with the action of a Kleinian group. A classical Techmüller sequence is a sequence of quasiconformal mapsf _i with complex dilatations of the form , where ϕ is a quadratic differential and 0<-k _i<1 are numbers such thatk _i→1 asi→∞. We proved in [T2] that if τ is a vertical trajectory associated to ϕ, then there is often, for instance if the sequence is normalized so thatf _i fix 3 points, a subsequence such thatf _i tend either toward a constant or an injective map of τ. If there is compatibility with the action of a non-elementary finitely generated Kleinian groupG, we can given a precise characterization which of these cases occurs. Suppose thatf _i induce isomorphisms ϕ_i ofG onto another Kleinian group and that ϕ_i have algebraic limit ϕ. If the quadratic differential is defined on a component of the ordinary set ofG, if there are no parabolic elements, and if τ is extended maximally so that all branches coming together at a singular point are included, then we can state the main result as follows. The limit is a constantc if the stabilizerG _τ of τ is elementary; and, if it is non-elementary, then the limit is injective. In the first case, ϕ(g) is parabolic with fixpointc wheneverg∈G _τ is of infinite order; and in the latter case, the limitf is an embedding of τ in a natural topology of τ, andf embeds τ into a component of the limit set of ϕG whose stabilizer is ϕG _τ. Various extensions and generalizations are presented. The research for this paper has been supported by the project 51749 of the Academy of Finland.     

Local linear operators and multicontour solutions of homogeneous coupled map lattices

Theorems are proved establishing a relationship between the spectra of the linear operators of the formA+Ωg _iB_ig_i ⁻¹ andA+B _i, whereg _i∈G, andG is a group acting by linear isometric operators. It is assumed that the closed operatorsA andB _i possess the following property: ‖B _iA⁻¹gB_jA⁻¹‖→0 asd(e,g)→∞. Hered is a left-invariant metric onG ande is the unit ofG. Moreover, the operatorA is invariant with respect to the action of the groupG. These theorems are applied to the proof of the existence of multicontour solutions of dynamical systems on lattices. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 37–47, January, 1999.     

(g,f)-Factorizations of graphs orthogonal to [1,2]-subgraph

LetG be a simple graph. Letg(x) andf(x) be integer-valued functions defined onV(G) withf(x)g(x)1 for allxV(G). It is proved that ifG is an (mg+m–1,mf–m+1)-graph andH is a [1,2]-subgraph withm edges, then there exists a (g,f)-factorization ofG orthogonal toH.This work is supported by China Postdoctoral Science Foundation and Shandong Youth Science Foundation.     

Closed ideals of homogeneous algebras

LetG denote a compact group andB a homogeneous Banach algebra of pseudomeasures onG (B is left translation invariant with continuous shifts; multiplication is convolution). It is shown that the closed ideal structure ofB is precisely that ofL ¹ or of one of the closed translation invariant subspaces,L _g ¹ , ofL ¹. The closed ideal structure ofL _g ¹ is presented.     

On Lie algebra decompositions related to spherical homogeneous spaces

LetG be a connected, reductive, linear algebraic group over an algebraically closed fieldk of characteristik zero. LetH ₁ andH ₂ be two spherical subgroups ofG. It is shown that for allg in a Zariski open subset ofG one has a Lie algebra decomposition g = h₁ + Adg ? h₂, where a is the Lie algebra of a torus and dim a ≤ min (rankG/H ₁,rankG/H ₂). As an application one obtains an estimate of the transcendence degree of the fieldk(G/H ₁ xG/H ₂) ^G for the diagonal action ofG. Ifk = ? andG _a is a real form ofG defined by an antiholomorphic involution σ :G→G then for a spherical subgroup H ? G and for allg in a Hausdorff open subset ofG one has a decomposition g = g_a + a Adg ? h, where a is the Lie algebra of σ-invariant torus and dim a ≤ rankG/H.     

Irregularity of certain algebraic fiber spaces

LetX, Y be smooth complex projective varieties, andf: X →Y be a fiber space whose general fiber is a curve of genusg. Denote byq _f the relative irregularity off. It is proved thatq _f ≤5g+1 / 6, iff is not generically trivial; moreover, if either a)f is non-constant and the general fiber is either hyperelliptic or bielliptic or b)q(Y)=0, thenq _f ≤g+1 / 2, and the bound is best possible. A classification of fiber surfaces of genus 3 withq _f =2 is also given in this note. Project supported by China Postdoctoral Science Foundation     

Correlation inequalities and a conjecture for permanents

This paper presents conditions on nonnegative real valued functionsf ₁,f ₂,...,f _m andg ₁,g ₂,...g _m implying an inequality of the type

Decomposition of Graphs into (g, f)-Factors

Let G be a multigraph, g and f be integer-valued functions defined on V(G). Then a graph G is called a (g, f)-graph if g(x)≤deg _G(x)≤f(x) for each x∈V(G), and a (g, f)-factor is a spanning (g, f)-subgraph. If the edges of graph G can be decomposed into (g, f)-factors, then we say that G is (g, f)-factorable. In this paper, we obtained some sufficient conditions for a graph to be (g, f)-factorable. One of them is the following: Let m be a positive integer, l be an integer with l=m (mod 4) and 0≤l≤3. If G is an -graph, then G is (g, f)-factorable. Our results imply several previous (g, f)-factorization results. Revised: June 11, 1998     

Greatest common divisors and least common multiples of graphs

A graphH divides a graphG, writtenH|G, ifG isH-decomposable. A graphG without isolated vertices is a greatest common divisor of two graphsG ₁ andG ₂ ifG is a graph of maximum size for whichG|G ₁ andG|G ₂, while a graphH without isolated vertices is a least common multiple ofG ₁ andG ₂ ifH is a graph of minimum size for whichG ₁|H andG ₂|H. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple ofG ₁ andG ₂ to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairsG ₁,G ₂ of graphs are determined, including when one ofG ₁ andG ₂ is a cycle of even length and the other is a star.G. C's research was supported in part by the Office of Naval Research, under Grant N00014-91-I-1060     

Unitary units in modular group algebras

Letp be a prime,K a field of characteristicp, G a locally finitep-group,KG the group algebra, andV the group of the units ofKG with augmentation 1. The anti-automorphismg→g ⁻¹ ofG extends linearly toKG; this extension leavesV setwise invariant, and its restriction toV followed byv→v ⁻¹ gives an automorphism ofV. The elements ofV fixed by this automorphism are calledunitary; they form a subgroup. Our first theorem describes theK andG for which this subgroup is normal inV. For each elementg inG, let denote the sum (inKG) of the distinct powers ofg. The elements 1+(g-1) withh,hεG are thebicyclic units ofKG. Our second theorem describes theK andG for which all bicyclic units are unitary. Research partly supported by the Hungarian National Foundation for Scientific Research grant no. T4265. The second author is indebted to the ‘Universitas’ Foundation and the Lajos Kossuth University of Debrecen, Hungary, for warm hospitality and generous support during the period when this work began. This article was processed by the authors using the Springer-Verlag T_EX mamath macro package 1990.     

On the dynamics of composite entire functions

Letf andg be nonlinear entire functions. The relations between the dynamics off⊗g andg⊗f are discussed. Denote byℐ (·) andF(·) the Julia and Fatou sets. It is proved that ifz∈C, thenz∈ℐ8464 (f⊗g) if and only ifg(z)∈ℐ8464 (g⊗f); ifU is a component ofF(f○g) andV is the component ofF(g○g) that containsg(U), thenU is wandering if and only ifV is wandering; ifU is periodic, then so isV and moreover,V is of the same type according to the classification of periodic components asU. These results are used to show that certain new classes of entire functions do not have wandering domains. The second author was supported by Max-Planck-Gessellschaft ZFDW, and by Tian Yuan Foundation, NSFC.