On three zero-sum Ramsey-type problems

For a graph G whose number of edges is divisible by k, let R(G,Z_k) denote the minimum integer r such that for every function f: E(K_r) ? Z_k there is a copy G¹ of G in K_r so that Σe∈E(G¹) f(e) = 0 (in Z_k). We prove that for every integer k₁ R(K_n, Z_k) ≤ n + O(k³ log k) provided n is sufficiently large as a function of k and k divides (). If, in addition, k is an odd prime-power then R(K_n, Z_k) ≤ n + 2k - 2 and this is tight if k is a prime that divides n. A related result is obtained for hypergraphs. It is further shown that for every graph G on n vertices with an even number of edges R(G,Z₂) ≤ n + 2. This estimate is sharp. © 1993 John Wiley & Sons, Inc.     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

A note on properties of locally finitely convergent sequences with respect to median filter

Let F_k be a mapping from R^Z to R^Z, satisfying that for x∈R^Z and n∈Z, F_k(x)(n) is the (k+1)th largest value (median value) of the 2k+1 numbers x(n−k),…,x(n),…,x(n+k). In [3] [W.Z. Ye, L. Wang, L.G. Xu, Properties of locally convergent sequences with respect to median filter, Discrete Mathematics 309 (2009) 2775–2781], we conjectured that for k∈{2,3}, if there exists n₀∈Z such that x is locally finitely convergent with respect to F_k on {n₀,…,n₀+k−1}, then x is finitely convergent with respect to F_k. In this paper, we obtain some sufficient conditions for a sequence finitely converging with respect to median filters. Based on these results, we prove that the conjecture is true.     

Bases normales,unités et conjecture faible de leopoldt

Letk be a number field,p an odd prime,R _k the ring ofp-integers ofk. We use Iwasawa theory to study theZ _p -moduleG(R _k ,Z _p ) (resp.NB (R _k ,Z _p )) ofclasses ofZ _p -extensions (resp.Z _p -extensions having a normal basis overR _k ) ofR _k . The rank ofG(G _k ,Z _p ) (resp.NB(R _k ,Z _p )) is related to Leopoldt's conjecture (resp. weak Leopoldt's conjecture) fork andp.      

A Generalization of Engel Conditions with Derivations in Rings

Let R be a prime ring of characteristic different from 2 with Z the center of R and d a nonzero derivation of R. Let k, m, n be fixed positive integers. If ([d(x ^k ), x ^k ] _n ) ^m  ∈ Z for all x ∈ R, then R satisfies S ₄, the standard identity in 4 variables.     

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

On Marcel Riesz's Ultrahyperbolic Kernel

Let t = (t₁, …, t_n) be a point of ?ⁿ. We shall write . We put by definition R_α(u) = u^(α?n)/2/K_n(α); here α is a complex parameter, n the dimension of the space, and K_n(α) is a constant. First we evaluate □R_α(u) = R_α(u), where □ the ultrahyperbolic operator. Then we obtain the following results: R_?2k(u) = □^kδ; R₀(u) = δ; and □^kR_2k(u) = δ, k = 0, 1, …. The first result is the n-dimensional ultrahyperbolic correlative of the well-known one-dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particular case µ = 1 was solved previously.     

The characterization of zero-sum (mod 2) bipartite Ramsey numbers

Let G be a bipartite graph, with k|e(G). The zero-sum bipartite Ramsey number B(G, Z_k) is the smallest integer t such that in every Z_k-coloring of the edges of K_t,t, there is a zero-sum mod k copy of G in K_t,t. In this article we give the first proof that determines B(G, Z₂) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G, Z₂) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in K_n,n, and let F be a field. A function f : E(K_n,n) → F is called G-stable if every copy of G in K_n,n has the same weight (the weight of a copy is the sum of the values of f on its edges). The set of all G-stable functions, denoted by U(G, K_n,n, F) is a linear space, which is called the K_n,n uniformity space of G over F. We determine U(G, K_n,n, F) and its dimension, for all G, n and F. Utilizing this result in the case F = Z₂, we can compute B(G, Z₂), for all bipartite graphs G. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 151–166, 1998     

Power Values of Derivations with Annihilator Conditions on Lie Ideals in Prime Rings

Let R be a prime ring of char R ≠ 2, d a nonzero derivation of R, U a noncentral Lie ideal of R, and a ∈ R. If au ^{n ₁} d(u) ^{n ₂} u ^{n ₃} d(u) ^{n ₄} u ^{n ₅} … d(u) ^{n _k?1} u ^{n _k}  = 0 for all u ∈ U, where n ₁, n ₂,…,n _k are fixed non-negative integers not all zero, then a = 0 and if a(u ^s d(u)u ^t ) ⁿ  ∈ Z(R) for all u ∈ U, where s ≥ 0, t ≥ 0, n ≥ 1 are some fixed integers, then either a = 0 or R satisfies S ₄, the standard identity in four variables.     

Affine cross-polytopes inscribed in a convex body

Let X be an affine cross-polytope, i.e., the convex hull of n segments A ₁ B ₁,…, A _n B _n in \mathbbRⁿ {\mathbb{R}^n} that have a common midpoint O and do not lie in a hyperplane. The affine flag F(X) of X is the chain O ∈ L ₁ ⊂⋯ ⊂ L _n = \mathbbRⁿ {\mathbb{R}^n} , where L _k is the k-dimensional affine hull of the segments A ₁ B ₁,…, A _k B _k , k ≤ n. It is proved that each convex body K ⊂ \mathbbRⁿ {\mathbb{R}^n} is circumscribed about an affine cross-polytope X such that the flag F(X) satisfies the following condition for each k ∈{2,…, n}:the (k−1)-planes of support at A _k and B _k to the body L _k ∩ K in the k-plane L _k are parallel to L _k −1.Each such X has volume at least V(K)/2 ^n(n−1)/2. Bibliography: 5 titles.     

Quadratic Central Differential Identities on a Multilinear Polynomial

Let K be a commutative ring with unity, R a prime K-algebra, Z(R) the center of R, d and δ nonzero derivations of R, and f(x ₁,…, x _n ) a multilinear polynomial over K. If [d(f(r ₁,…, r _n )), δ (f(r ₁,…, r _n ))] ? Z(R), for all r ₁,…, r _n  ? R, then either f(x ₁,…, x _n ) is central valued on R or {d, δ} are linearly dependent over C, the extended centroid of R, except when char(R) = 2 and dim _C RC = 4.     

Grassmannians of 2-Sided Vector Spaces

Let k ? K be an extension of fields, and let A ? M _n (K) be a k-algebra. We study parameter spaces of m-dimensional subspaces of K ⁿ which are invariant under A. The space  _A (m, n), whose R-rational points are A-invariant, free rank m summands of R ⁿ , is well known. We construct a distinct parameter space,  _A (m, n), which is a fiber product of a Grassmannian and the projectivization of a vector space. We then study the intersection  _A (m, n) ∩  _A (m, n), which we denote by  _A (m, n). Under suitable hypotheses on A, we construct affine open subschemes of  _A (m, n) and  _A (m, n) which cover their K-rational points. We conclude by using  _A (m, n),  _A (m, n), and  _A (m, n) to construct parameter spaces of 2-sided subspaces of 2-sided vector spaces.     

Partitioning complete multipartite graphs by monochromatic trees

The tree partition number of an r‐edge‐colored graph G, denoted by t_r(G), is the minimum number k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex‐disjoint monochromatic trees. We determine t₂(K(n₁, n₂,…, n_k)) of the complete k‐partite graph K(n₁, n₂,…, n_k). In particular, we prove that t₂(K(n, m)) = ? (m‐2)/2ⁿ? + 2, where 1 ≤ n ≤ m. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 133–141, 2005     

Central localization and Gelfand-Kirillov dimension

LetR be a factor ring of the enveloping algebra of a finite dimensional Lie algebra over a fieldk. If the centre ofR, Z, consists of non-zero divisors inR, the ringR _z obtained by localizing at the non-zero elements ofZ becomes a finitely generated algebra over the fieldK which arises as the field of fractions ofZ. The Gelfand-Kirillov dimension of anR-moduleM is denotedd(M). In this paper it is shown that ifR _Z ⊗ _R M ≠ 0 thend(M) ≧d(R _Z ⊗ _R M) + tr. deg _k Z, whered (R _z ⊗M) is the Gelfand-Kirillov dimension ofR _z ⊗M) viewed as anR _z -module andR _z is viewed as a finitely generatedK-algebra (not as ak-algebra). The result is primarily of a technical nature.     

Strong Sperner property of the subgroup lattice of an Abelianp-group

Letn andk be arbitrary positive integers,p a prime number and L_(k ⁿ)(p) the subgroup lattice of the Abelianp-group (Z/p ^k ) ⁿ . Then there is a positive integerN(n,k) such that whenp N(n,k),L _(k ^N )(p) has the strong Sperner property.     

On semi-fredholm properties of a boundary value problem inR + n

The paper considers a boundary value problem with the help of the smallest closed extensionL ^∼ :H _k →H _{k 0}×B _{h 1}×...×B _{h N} of a linear operatorL :C ₍₀₎ ^∞ (R ₊ ⁿ ) →L(R ₊ ⁿ )×L(R ⁿ⁻¹)×...×L(R ⁿ⁻¹). Here the spacesH _k (the spaces ℬ _h ) are appropriate subspaces ofD′(R ₊ ⁿ ) (ofD′(R ⁿ⁻¹), resp.),L(R ₊ ⁿ ) andC ₍₀₎ ^∞ (R ₊ ⁿ )) denotes the linear space of smooth functionsR ⁿ →C, which are restrictions onR ₊ ⁿ of a function from the Schwartz classL (fromC ₀ ^∞ , resp.),L(R ⁿ⁻¹) is the Schwartz class of functionsR ⁿ⁻¹ →C andL is constructed by pseudo-differential operators. Criteria for the closedness of the rangeR(L ^∼) and for the uniqueness of solutionsL ^∼ U=F are expressed. In addition, ana priori estimate for the corresponding boundary value problem is established.     

Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R ⁿ , where K is the cone of nonnegative vectors in R ⁿ . A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k (I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^k ρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

Resolvable Coverings of 2-Paths by Cycles

 Let K _n be the complete graph on n vertices. A C(n,k,λ) design is a multiset of k-cycles in K _n in which each 2-path (path of length 2) of K _n occurs exactly λ times. A C(lk,k,1) design is resolvable if its k-cycles can be partitioned into classes so that every vertex appears exactly once in each class. A C(n,n,1) design gives a solution of Dudeney's round table problem. It is known that there exists a C(n,n,1) design when n is even and there exists a C(n,n,2) design when n is odd. In general the problem of constructing a C(n,n,1) design is still open when n is odd. Necessary and sufficient conditions for the existence of C(n,k,λ) designs and resolvable C(lk,k,1) designs are known when k=3,4. In this paper, we construct a resolvable C(n,k,1) design when n=p ^e +1 ( p is a prime number and e≥1) and k is any divisor of n with k≠1,2. Received: October, 2001 Final version received: September 4, 2002 RID="*" ID="*" This research was supported in part by Grant-in-Aid for Scientific Research (C) Japan     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .