On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

Constructions and Cohomology of Hom–Lie Color Algebras

The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α ^k -generalized derivations and provide examples.     

Operations on maps,and outer automorphisms

By representing maps on surfaces as transitive permutation representations of a certain group Γ, it is shown that there are exactly six invertible operations (such as duality) on maps; they are induced by the outer automorphisms of Γ, and form a group isomorphic to S₃. Various consequences are deduced, such as the result that each finite map has a finite reflexible cover which is invariant under all six operations.     

On the Genus of a Subfield of a Field

In 1977 Babai showed that if Γ₁ is a subgroup of a finite group Γ₂, then the genus of Γ₁ is at most the genus of Γ₂. We establish a similar result for finite fields.     

The euclidean bianchi group

The Bianchi Groups Γ_d are PSL₂(0_d) where 0_d is the ring of integers in the quadratic imaginary number field Q√-d with d a positive square-free rational integer. If d=1,2,3,7,11 0_d has a Euclidean algorithm and the corresponding groups are called the Euclidean Bianchi Groups. The group Γ₁ is the Picard Group has been studied independently. Here the subgroup structure of the remaining Euclidean Bianchi Groups is investigated. We show that they have a unique normal subgroup of index in if(n,6)=1 among other results. We also classify the abelian and nilpotent subgroups and discuss the structure of both congruence and non-congruence subgroups.     

Commuting Graphs of Matrix Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R, and two distinct vertices x and y are adjacent if and only if xy = yx. The commuting graph of a group G, denoted by Γ(G), is similarly defined. In this article we investigate some graph-theoretic properties of Γ(M _n (F)), where F is a field and n ≥ 2. Also we study the commuting graphs of some classical groups such as GL _n (F) and SL _n (F). We show that Γ(M _n (F)) is a connected graph if and only if every field extension of F of degree n contains a proper intermediate field. We prove that apart from finitely many fields, a similar result is true for Γ(GL _n (F)) and Γ(SL _n (F)). Also we show that for two fields F and E and integers n, m ≥ 2, if Γ(M _n (F))?Γ(M _m (E)), then n = m and |F|=|E|.     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

An accessibility theorem for infinite graph minors

We prove the following recent conjecture of Halin. Let Γ₀ be the class of all graphs, and for every ordinal μ > 0 let Γ_μ be the class of all graphs containing infinitely many disjoint connected graphs from Γ_λ, for every λ < μ. Then a graph lies in all these classes Γ_μ if and only if it contains a subdivision of the infinite binary tree. Published by John Wiley & Sons, Inc., 2000 J Graph Theory 35: 273–277, 2000     

A novel approach towards fuzzy Γ-ideals in ordered Γ-semigroups

In many applied disciplines like computer science, coding theory and formal languages, the use of fuzzified algebraic structures especially ordered semigroups play a remarkable role. In this paper, we introduce a new concept of fuzzy Γ-ideal of an ordered Γ-semigroup G called an (∈, ∈ ?q _k )-fuzzy Γ-ideal of G. Fuzzy Γ-ideal of type (∈, ∈ ∨q _k ) are the generalization of ordinary fuzzy Γ-ideals of an ordered Γ-semigroup G. A new characterization of ordered Γ-semigroups in terms of an (∈, ∈ ∨q _k )-fuzzy Γ-ideal is given. We show that a fuzzy subset λ of an ordered Γ-semigroup G is an (∈, ∈ ∨q _k )-fuzzy Γ-ideal of G if and only if U (λ; t) is a Γ-ideal of G for all \(t \in \left( {0,\frac{{1 - k}} {2}} \right]\) . We also investigate some important characterization theorems in terms of this notion. Finally, regular ordered Γ-semigroups are characterized by the properties of their (∈, ∈ ∨q _k )-fuzzy Γ-ideals.     

Infinite Components of An Auslander-Reiten Quiver with Finite DTr-Orbits

Abstract

Let A be a basic connected finite dimensional algebra over an algebraically closed field. We show that if Γ is an infinite connected component of the Auslander-Reiten quiver Γ_A of A in which each Γ_A-orbit contains only finitely many vertices, then the number of indecomposable direct summands of the middle term of any mesh, whose starting vertex belongs to the infinite stable part of Γ, is less than or equal to 3. Moreover, if the nonstable vertices belong to τ_A-orbits of exceptional projectives in Γ, then Γ can be obtained from a stable tube by a finite number of multiple coray-ray insertions of type α?γ and multiple coray-ray insertions of type α?γ.     

Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory

We establish an isomorphism between the vertex and spinor representations of affine Lie algebras for types D_l⁽¹⁾and D_{l + 1}⁽²⁾. We also study decomposition of spinor representations using the infinite family of Casimir operators and prove that they are either irreducible or have two irreducible components. We show that the vertex and spinor constructions of the representations can be reformulated in the language of two-dimensional quantum field theory. In this physical context, the two constructions yield the generalized sine-Gordon and Thirring models, respectively, already in renormalized form. The isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson-fermion correspondence     

On approximation of Lie groups by discrete subgroups

A locally compact group G is said to be approximated by discrete subgroups (in the sense of Tôyama) if there is a sequence of discrete subgroups of G that converges to G in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if G has a rational structure. On the other hand, if Γ is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group G then G is approximated by discrete subgroups Γ _n containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γ _n . The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γ _n ) _n?≥?0 in terms of Γ. Several applications are given.     

On the integral equation method for the plane mixed boundary value problem of the Laplacian

Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ₂ and given Neumann data on the remaining part Γ₂ of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ₂ and the also desired values of the normal derivatie on Γ₁. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ₁ ∩ Γ₂ even for C^∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ₂ providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ₁ and Γ₂ augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).     

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)?{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ _I (R), with vertices {x ∈ R?I | xy ∈ I for some y ∈ R?I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ _I (R). We also determine when Γ _I (R) is either a complete graph or a complete bipartite graph and investigate when Γ _I (R) ? Γ(S) for some commutative ring S.     

Finite Groups with a Disconnected p -Regular Conjugacy Class Graph

Abstract

Let G be a finite p-solvable group for a fixed prime p. Attach to G a graph Γ _p (G) whose vertices are the non-central p-regular conjugacy classes of G and connect two vertices by an edge if their cardinalities have a common prime divisor. In this note we study the structure and arithmetical properties of the p-regular class sizes in p-solvable groups G having Γ _p (G) disconnected.     

Structure of cubic mapping graphs for the ring of Gaussian integers modulo n

Let ? _n [i] be the ring of Gaussian integers modulo n. We construct for ?n[i] a cubic mapping graph Γ(n) whose vertex set is all the elements of ?n[i] and for which there is a directed edge from a ∈ ?n[i] to b ∈ ?n[i] if b = a ³. This article investigates in detail the structure of Γ(n). We give suffcient and necessary conditions for the existence of cycles with length t. The number of t-cycles in Γ₁(n) is obtained and we also examine when a vertex lies on a t-cycle of Γ₂(n), where Γ₁(n) is induced by all the units of ?n[i] while Γ₂(n) is induced by all the zero-divisors of ?n[i]. In addition, formulas on the heights of components and vertices in Γ(n) are presented.     

Groups with the same prime graph as the orthogonal group B n (3)

Let G be a finite group. The prime graph of G is denoted by Γ(G). It is proved in [1] that if G is a finite group such that Γ(G) = Γ(B _p (3)), where p > 3 is an odd prime, then G ? B _p (3) or C _p (3). In this paper we prove the main result that if G is a finite group such that Γ(G) = Γ(B _n (3)), where n ≥ 6, then G has a unique nonabelian composition factor isomorphic to B _n (3) or C _n (3). Also if Γ(G) = Γ(B ₄(3)), then G has a unique nonabelian composition factor isomorphic to B ₄(3), C ₄(3), or ² D ₄(3). It is proved in [2] that if p is an odd prime, then B _p (3) is recognizable by element orders. We give a corollary of our result, generalize the result of [2], and prove that B _2k+1(3) is recognizable by the set of element orders. Also the quasirecognition of B _2k (3) by the set of element orders is obtained.     

Two applications of semi-stable graphs

A Michigan graph G on a vertex set V is called semi-stable if for some υ?V, Γ(G_υ) = Γ(G)_υ. It can be shown that all regular graphs are semi-stable and this fact is used to show (i) that if Γ(G) is doubly transitive then G = K_n or K?_n, and (ii) that Γ(G) can be recovered from Γ(G_υ). The second result is extended to the case of stable graphs.     

On the composition factors of a group with the same prime graph as B n (5)

Let G be a finite group. The prime graph of G is a graph whose vertex set is the set of prime divisors of |G| and two distinct primes p and q are joined by an edge, whenever G contains an element of order pq. The prime graph of G is denoted by Γ(G). It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if G is a finite group such that Γ(G) = Γ(B _n (5)), where n ? 6, then G has a unique nonabelian composition factor isomorphic to B _n (5) or C _n (5).