Generalizations of the image conjecture and the Mathieu conjecture

We first propose a generalization of the image conjecture Zhao (submitted for publication) [31] for the commuting differential operators related with classical orthogonal polynomials. We then show that the non-trivial case of this generalized image conjecture is equivalent to a variation of the Mathieu conjecture Mathieu (1997) [21] from integrals of G-finite functions over reductive Lie groups G to integrals of polynomials over open subsets of Rⁿ with any positive measures. Via this equivalence, the generalized image conjecture can also be viewed as a natural variation of the Duistermaat and van der Kallen theorem Duistermaat and van der Kallen (1998) [14] on Laurent polynomials with no constant terms. To put all the conjectures above in a common setting, we introduce what we call the Mathieu subspaces of associative algebras. We also discuss some examples of Mathieu subspaces from other sources and derive some general results on this newly introduced notion.     

Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture

Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K _r,r (for odd r) and K _r+1. If true, this would be a strengthening of the Hajnal-Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r ≥ 6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains K _r,r and V(G) partitions into subsets V ₀, …, V _t such that G[V ₀] = K _r,r and for each 1 ≤ i ≤ t, G[V _i ] = K _r . We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen-Lih-Wu Conjecture by induction.     

Near-proper vertex 2-colorings of sparse graphs

A graph G is (2, 1)-colorable if its vertices can be partitioned into subsets V ₁ and V ₂ such that each component in G[V ₁] contains at most two vertices while G[V ₂] is edgeless. We prove that every graph with maximum average degree mad(G) < 7/3 is (2, 1)-colorable. It follows that every planar graph with girth at least 14 is (2, 1)-colorable. We also construct a planar graph G _n with mad (G _n ) = (18n − 2)/(7n − 1) that is not (2, 1)-colorable.     

The Bass-Quillen problem on a class of local rings with weak global dimension two

Let (R,m) be a local GCD domain. R is called a U2 ring if there is an element u ∈ m-m2 such that R/(u) is a valuation domain and Ru is a B′ezout domain. In this case u is called a normal element of R. In this paper we prove that if R is a U2 ring, then R and R[x] are coherent; moreover, if R has a normal element u and dim(R/(u)) = 1, then every finitely generated projective module over R[X] is free.     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G = （V, E） be a graph without isolated vertices. A set S lohtain in V is a domination set of G if every vertex in V - S is adjacent to a vertex in S, that is N[S] = V. The domination number of G, denoted by γ（G）, is the minimum cardinality of a domination set of G. A set S lohtain in V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S] has a perfect matching. The paired-domination number, denoted by γpr（G）, is defined to be the minimum cardinality of a paired-domination set S in G. A subset S lohtain in V is a power domination set of G if all vertices of V can be observed recursively by the following rules： （i） all vertices in N[S] are observed initially, and （ii） if an observed vertex u has all neighbors observed except one neighbor v, then v is observed （by u）. The power domination number, denoted by γp（G）, is the minimum cardinality of a power domination set of G. In this paper, the constructive characterizations for trees with γp=γ and γpr = γp are provided respectively.     

Isomorphisms preserving invariants

Let V and W be finite dimensional real vector spaces and let G ì GL(V){G \subset {\rm GL}(V)} and H ì GL(W){H \subset {\rm GL}(W)} be finite subgroups. Assume for simplicity that the actions contain no reflections. Let Y and Z denote the real algebraic varieties corresponding to \mathbbR[V]^G{\mathbb{R}[V]^G} and \mathbbR[W]^H{\mathbb{R}[W]^H}, respectively. If V and W are quasi-isomorphic, i.e., if there is a linear isomorphism L : V → W such that L sends G-orbits to H-orbits and L ⁻¹ sends H-orbits to G-orbits, then L induces an isomorphism of Y and Z. Conversely, suppose that f : Y → Z is a germ of a diffeomorphism sending the origin of Y to the origin of Z. Then we show that V and W are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.     

The augmentation ideal in group near-rings

The definition of the group near-ring R[G] of the near-ring R over the group G as a near-ring of mappings from R ^(G) to itself is due to Le Riche et al. (Arch Math 52:132–139, 1989). In this paper we consider the augmentation ideal Δ of R[G]. If the exponent of G is not 2, then the structure of ΔR ^(G) is determined in terms of commutators and distributors. This is then used to show that Δ is nilpotent if and only if R is weakly distributive, has characteristic p ⁿ for some prime p and G is a finite p-group for the same prime p.      

On approximation properties of Balázs-Szabados operators their Kantorovich extension

In this paper we deal with a sequence of positive linear operatorsR _n^[β] approximating functions on the unbounded interval [0, t8) which were firstly used by K. Balázs and J. Szabados. We give pointwise estimates in the framework of polynomial weighted function spaces. Also we establish a Voronovskaja type theorem in the same weighted spaces for K_n^[β] operators, representing the integral generalization in Kantorovich sense of the R_n^[β].     

Reciprocal Polynomials and p-Group Cohomology

Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ?-algebra R _G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring $H^\ast(G, \mathbb{F}_p)Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ℤ-algebra R _G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra R[G] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring H^*(G, \mathbbF_p)H^\ast(G, \mathbb{F}_p) of G has the same spectrum as the ring of invariants of R _G mod p (R_G ?_\mathbbZ \mathbbF_p)^G(R_G \otimes_{\mathbb{Z}} \mathbb{F}_p)^G where the action of G is induced by conjugation.     

Cycles in Partially Square Graphs

. In this work we consider finite undirected simple graphs. If G=(V,E) is a graph we denote by α(G) the stability number of G. For any vertex x let N[x] be the union of x and the neighborhood N(x). For each pair of vertices ab of G we associate the set J(a,b) as follows. J(a,b)={u∈N[a]∩N[b]∣N(u)⊆N[a]∪N[b]}. Given a graph G, its partially squareG ^* is the graph obtained by adding an edge uv for each pair u,v of vertices of G at distance 2 whenever J(u,v) is not empty. In the case G is a claw-free graph, G ^* is equal to G ². If G is k-connected, we cover the vertices of G by at most ⌈α(G ^*)/k⌉ cycles, where α(G ^*) is the stability number of the partially square graph of G. On the other hand we consider in G ^* conditions on the sum of the degrees. Let G be any 2-connected graph and t be any integer (t≥2). If ∑ _{x ∈ S} deg _G (x)≥|G|, for every t-stable set S⊆V(G) of G ^* then the vertex set of G can be covered with t−1 cycles. Different corollaries on covering by paths are given. Received: January 22, 1997 Final version received: February 15, 2000     

Alcuni risultati sugli ideali massimali di anelli di polinomi

Riassunto In questo lavoro viene studiata la cardinalità minima dei sistemi di generatori di ideali massimaliM dell'anello di polinomiR[X ₁, …,X _n] (R dominio d'integrità) tali cheM⊃R=0. In particolare è dimostrato che sen≥2 edR è unS-dominio di dimensione 1, ogni siffatto ideale massimale può essere generato dan elementi.

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Summary This paper is concerned with estimates of the minimal number of generators for maximal idealsM in the polynomial ringR[X ₁, …,X _n] (R an integral domain) such thatM⊃R=0. In particular it is proved that ifn≥2 andR is a 1-dimensionalS-domain, every such maximal ideal can be generated byn elements.

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Lavoro eseguito nell'ambito dell'attività del G.N.S.A.G.A. del C.N.R.     

Conjugacy Classes and Invariant Subrings of R-Automorphisms of R[x]

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail examples involving the ring of integers modulo n. If R is a local ring with maximal ideal m such that R/m = ?₂ and m ² = (0), we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.     

Lifting chains of prime ideals to paravaluation rings

Résumé  SoitR ⊂T une extension des anneaux commutatifs et soit {P _α :α ∈I} une cha?ne croissante des idéaux premiers deR (I étant un ensemble totalement ordonné, peut-être infini). Alors il existe un anneau de paravaluationV deT et une cha?ne {Q _α} des idéaux premiers deV de sorte queR ⊂V etQ _α ∩R =P _α pour toutα ∈I. Tout d’abord, on établit le cas spécial dans lequelT est un corps; dans ce cas, on trouve en effet un tel anneau de valuationV deT. Ensuite, l’assertion ci-dessus pour le cas général découle comme conséquence. Dans le cas général, on peut aussi remplacer le mot “paravaluation” avec le mot “valuation” siR est un anneau de Marot etT est son anneau total de fractions.      

Moment vanishing problem and positivity: Some examples

We consider the problem of vanishing of the momentswith Ω a compact domain in Rn and P(x), q(x) complex polynomials in x∈Ω (MVP). The main stress is on relations of this general vanishing problem to the following conjecture which has been studied recently in Mathieu (1997) [22], Duistermaat and van der Kallen (1998) [17], Zhao (2010) [34] and [35] and in other publications in connection with the vanishing problem for differential operators and with the Jacobian conjecture:

Conjecture A. 

For positive μ ifmk(P,1)=0fork=1,2,… , thenmk(P,q)=0fork?1for any q.     

Un Algorithme pour le Calcul des Syzygies sur V[X] dans le cas où V est un Domaine de Valuation

Soit V un domaine de valuation. Nous donnons un algorithme pour calculer une base du V-saturé d'un sous-module de type fini d'un V-module libre (avec une base éventuellement infinie). Nous l'appliquons pour calculer le V-saturé d'un sous-V[X]-module de type fini de V[X] ⁿ (n ∈ ?*). Ceci permet enfin de calculer un système générateur fini pour les syzygies sur V[X] d'une famille finie de vecteurs de V[X] ^k .

We give an algorithm for computing the V-saturation of any finitely generated submodule of V[X] ⁿ (n ∈ ?*), where V is a valuation domain. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of V[X] ^k .     

A generalization of Mathieu subspaces to modules of associative algebras

We first propose a generalization of the notion of Mathieu subspaces of associative algebras $ \mathcal{A} $ \mathcal{A} , which was introduced recently in [Zhao W., Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra, 2010, 214(7), 1200–1216] and [Zhao W., Mathieu subspaces of associative algebras], to $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} . The newly introduced notion in a certain sense also generalizes the notion of submodules. Related with this new notion, we also introduce the sets σ(N) and τ(N) of stable elements and quasi-stable elements, respectively, for all R-subspaces N of $ \mathcal{A} $ \mathcal{A} -modules $ \mathcal{M} $ \mathcal{M} , where R is the base ring of $ \mathcal{A} $ \mathcal{A} . We then prove some general properties of the sets σ(N) and τ(N). Furthermore, examples from certain modules of the quasi-stable algebras [Zhao W., Mathieu subspaces of associative algebras], matrix algebras over fields and polynomial algebras are also studied.     

Ore Extensions over Total Valuation Rings

It is shown that any Ore extension R = V[x;σ,δ] over a total valuation ring V is always v-Bezout which is a generalization of commutative GCD domains. By using this result, a necessary and sufficient condition are given for R to be fully left bounded.     

PARTITION A GRAPH WITH SMALL DIAMETER INTO TWO INDUCED MATCHINGS

§1　IntroductionGraphs considered in this paper are finite and simple.For a graph G,its vertex setandedge set are denoted by V(G) and E(G) ,respectively.If vertices u and v are connected inG,the distance between u and v,denoted by d G(u,v) ,is the length ofa shortest(u,v) -pathin G.The diameter of a connected graph G is the maximum distance between two verticesof G.For X V(G) ,the neighbor set NG(X) of X is defined byNG(X) ={ y∈V(G) \X:there is x∈X such thatxy∈E(G) } .NG({ x} )…     

Global secure sets of grid-like graphs

Let G=(V,E) be a graph and S⊆V. The set S is a secure set if ∀X⊆S,|N[X]∩S|≥|N[X]−S|, and S is a global secure set if S is a secure set and a dominating set. The cardinality of a minimum global secure set of G is the global security number of G, denoted γ_s(G). The sets studied in this paper are different from secure dominating sets studied in Cockayne et al. (2003) [3], Grobler and Mynhardt (2009) [8], or Klostermeyer and Mynhardt (2008) [13], which are also denoted by γ_s.In this paper, we provide results on the global security numbers of paths, cycles and their Cartesian products.     

Lax Operator for the Quantised Orthosymplectic Superalgebra Uq[osp(m|n)]

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U _q [osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U _q [osp(m|n)] module. The case W = V is studied as an example. Presented by A. Verschoren.