An alternative method for q-counting directed column-convex polyominoes

The area+perimeter generating function of directed column-convex polyominoes will be written as a quotient of two expressions, each of which involves powers of q of all kinds: positive, zero and negative. The method used in the proof applies to some other classes of column-convex polyominoes as well. At least occasionally, that method can do the case q=1 too.     

Regular automorphisms of order p2

Let G be a group, and let α be a regular automorphism of order p² of G, where p is a prime. If G is polycyclic-by-finite and the map ϕ : G →G defined by g^ϕ= [g,α] is surjective, then G is soluble. If G is polycyclic, then C_G(α^p) and G/[G,α^p] are both nilpotent-by-finite.     

On p-CAP-subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. H is said to be a p-CAP-subgroup of G if H either covers or avoids each pd-chief factor of G. We give some characterizations for a group G to be p-solvable under the assumption that some subgroups of G are p-CAP-subgroups of G.     

OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns

The prime graph of a finite group G, which is denoted by GK(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let p₁2<?<p_k be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p₁), degG(p₂), ? , degG(p_k)), where degG(p) signifies the degree of the vertex p in GK(G). A finite group H is said to be OD-characterizable if G? H for every finite group G such that |G| = |H| and D(G) = D(H). The purpose of this article is threefold. First, it finds sharp upper and lower bounds on ?(G), the sum of degrees of all vertices in GK(G), for any finite group G (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups L₄(q), q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable (Theorem 4.2).     

THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU（2,2）/S（U（2） × U（2））

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.     

An Object Grammar for Column-Convex Polyominoes

In this paper we propose an object grammar decomposition for the classes of columnconvex, and directed column-convex polyominoes. As a consequence, we obtain the enumeration of such classes according to the semi-perimeter, thus giving a natural explanation of the fact that the generating functions of both the classes are algebraic.AMS Subject Classification: 05A15.     

On the Tiling System Recognizability of Various Classes of Convex Polyominoes

We consider some problems concerning two relevant classes of two-dimensional languages, i.e., the tiling recognizable languages, and the local languages, recently introduced by Giammarresi and Restivo and already extensively studied. We show that various classes of convex and column-convex polyominoes can be naturally represented as two-dimensional words of tiling recognizable languages. Moreover, we investigate the nature of the generating function of a tiling recognizable language, providing evidence that such a generating function need not be D-finite.     

Nonsolvable groups whose irreducible character degrees have special 2-parts

Let G be a nonsolvable group and Irr(G) the set of irreducible complex characters of G. We consider the nonsolvable groups whose character degrees have special 2-parts and prove that if χ(1)₂ = 1 or |G|₂ for every χ ∈ Irr(G), then there exists a minimal normal subgroup N of G such that N ≅ PSL(2, 2ⁿ) and G/N is an odd order group.     

Trace forms of normal extensions over local fields

Let k be a local field of char(k)≠2 and K/k a finite field extension of degree n. Then K can be viewed as a quadratic space of k under the quadratic form T(X) =tr_K/k(x²). The invariants of this form are given in the case when K/k is a Galois extension, except for Galois extensions K/k with k dyadicn divisible by 4 and the 2-Sylowgroups of the Galois group are non-cyclic. Conversely all quadratic forms of a local field k of char(k)≠ 2 which appear as trace forms of Galois extensions of k are determined.     

Hinged dissection of polyominoes and polyforms

A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be rotated into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P. This construction uses kn pieces, where k is the number of vertices of P. When P is a regular polygon, we show how to reduce the number of pieces to k/2(n−1). In particular, we consider polyominoes (made up of unit squares), polyiamonds (made up of equilateral triangles), and polyhexes (made up of regular hexagons). We also give a hinged dissection of all polyabolos (made up of right isosceles triangles), which do not fall under the general result mentioned above. Finally, we show that if P can be hinged into Q, then any edge-to-edge gluing of n congruent copies of P can be hinged into any edge-to-edge gluing of n congruent copies of Q.     

覆盖数不超过3的图上渡河问题

1000多年前,英国著名学者Alcuin曾提出一个古老的渡河问题,即狼、羊和卷心菜的渡河问题。2006年,Prisner把该问题推广到任意的冲突图上,考虑了一类情况更一般的渡河运输问题。所谓冲突图是指一个图G=(V,E),这里V代表某些物品的集合,V中的两个点有边连结当且仅当这两个点是冲突的,即在无人监管的情况下不允许留在一起的点。图G=(V,E)的一个可行运输方案是指在保证不发生任何冲突的前提下,把V的点所代表的物品全部摆渡到河对岸的一个运输方案。图G的Alcuin数定义为它存在可行运输方案时所需船的最小容量。本文讨论了覆盖数不超过3的连通图的Alcuin数,给出了该类图Alcuin数的完全刻画。     

多扇图的Pebbling数和Graham猜想

图G的pebbling数f(G)是最小的整数n,使得不论n个pebble如何放置在G的顶点上,总可以通过一系列的pebbling移动把1个pebble移到任意一个顶点上,其中一个pebbling移动是从一个顶点处移走两个pebble而把其中的一个移到与其相邻的一个顶点上。Graham猜想对于任意的连通图G和H有f(G×H)f(G)f(H)。多扇图F_n1,n2,…,nm是指阶为n₁+n₂+…+n_m+1的联图P₁∨(P_n1∪P_n2∪…∪P_nm)。本文首先给出了多扇图的pebbling数,然后证明了多扇图F_n1,n2,…,nm具有2-pebbling性质,最后论述了对于一个多扇图和一个具有2-pebbling性质的图的乘积来说,Graham猜想是成立的。作为一个推论,当G和H都是多扇图时,Graham猜想成立。     

Manin’s conjecture for a class of singular cubic hypersurfaces

Let l > 2 be a fixed positive integer and Q(y) be a positive definite quadratic form in l variables with integral coefficients. The aim of this paper is to count rational points of bounded height on the cubic hypersurface defined by u³ = Q(y)z. We can get a power-saving result for a class of special quadratic forms and improve on some previous work.     

Extensions of n-Hom Lie algebras

n-Hom Lie algebras are twisted by n-Lie algebras by means of twisting maps. n-Hom Lie algebras have close relationships with statistical mechanics and mathematical physics. The paper main concerns structures and representations of n-Hom Lie algebras. The concept of n_ρ-cocycle for an n-Hom Lie algebra (G, [,… , ], α) related to a G-module (V, ρ, β) is proposed, and a sufficient condition for the existence of the dual representation of an n-Hom Lie algebra is provided. From a G-module (V, ρ, β) and an n_ρ-cocycle θ, an n-Hom Lie algebra (T_θ(V ), [, … , ]θ, γ) is constructed on the vector space T_θ(V ) = G⊕V, which is called the T_θ-extension of an n-Hom Lie algebra (G, [, … , ], α) by the G-module (V, ρ, β).     

Decomposition of a rational matrix into partial fractions

In this paper a technique is developed for the expansion of a rational matrix F(λ)=G(λ)L^-1(λ)(orF(λ)=L^-1(λ)G(λ)into block partial fractions when the denominator has multiple roots. The method consists in the construction of interpolating matrix polynomials and their properties. Moreover, the approach is extended when L(λ) has nonlinear divisors.     

Zagreb indices of graphs

The first Zagreb index M₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M₂(G). Also, we present lower and upper bounds on M₂(G) +M₂(G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M₁(G) and second Zagreb coindex M₂(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.     

关于不定方程x3-1=749y2

利用同余式、平方剩余、Pell方程的解的性质、递归序列证明了:不定方程x³-1=749y²仅有整数解(x,y)=(1,0).     

Enumerating several aspects of non-decreasing Dyck paths

A Dyck path is non-decreasing if the $y$-coordinates of its valleys form a non-decreasing sequence. In this paper we give enumerative results and some statistics of several aspects of non-decreasing Dyck paths. We give the number of pyramids at a fixed level that the paths of a given length have, count the number of primitive paths, count how many of the non-primitive paths can be expressed as a product of primitive paths, and count the number of paths of a given height and a given length. We present and prove our results using combinatorial arguments, generating functions (using the symbolic method) and parameterize the results studied here using the Riordan arrays. We use known bijections to connect direct column-convex polyominoes, Elena trees, and non-decreasing Dyck paths.     

Bandwidth of the composition of two graphs

The bandwidth B(G) of a graph G is the minimum of the quantity max{|f(x)−f(y)| : xyE(G)} taken over all proper numberings f of G. The composition of two graphs G and H, written as G[H], is the graph with vertex set V(G)×V(H) and with (u₁,v₁) is adjacent to (u₂,v₂) if either u₁ is adjacent to u₂ in G or u₁=u₂ and v₁ is adjacent to v₂ in H. In this paper, we investigate the bandwidth of the composition of two graphs. Let G be a connected graph. We denote the diameter of G by D(G). For two distinct vertices x,yV(G), we define w_G(x,y) as the maximum number of internally vertex-disjoint (x,y)-paths whose lengths are the distance between x and y. We define w(G) as the minimum of w_G(x,y) over all pairs of vertices x,y of G with the distance between x and y is equal to D(G). Let G be a non-complete connected graph and let H be any graph. Among other results, we prove that if |V(G)|=B(G)D(G)−w(G)+2, then B(G[H])=(B(G)+1)|V(H)|−1. Moreover, we show that this result determines the bandwidth of the composition of some classes of graphs composed with any graph.