Generalized P(N)-graded Lie superalgebras

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but suffcient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.     

On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify:(a) all such varieties for G = SL(2) × C~×and(b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.     

A stable range for quadratic forms over commutative rings

A commutative ring A has quadratic stable range 1 (qsr(A) = 1) if each primitive binary quadratic form over A represents a unit. It is shown that qsr(A) = 1 implies that every primitive quadratic form over A represents a unit, A has stable range 1 and finitely generated constant rank projectives over A are free. A classification of quadratic forms is provided over Bezout domains with characteristic other than 2, quadratic stable range 1, and a strong approximation property for a certain subset of their maximum spectrum. These domains include rings of holomorphic functions on connected noncompact Riemann surfaces. Examples of localizations of rings of algebraic integers are provided to show that the classical concept of stable range does not behave well in either direction under finite integral extensions and that qsr(A) = 1 does not descend from such extensions.     

The complexity of irredundant sets parameterized by size

An irredundant set of vertices V′V in a graph G=(V,E) has the property that for every vertex uV′, N[V′−{u}] is a proper subset of N[V′]. We investigate the parameterized complexity of determining whether a graph has an irredundant set of size k, where k is the parameter. The interest of this problem is that while most “k-element vertex set” problems are NP-complete, several are known to be fixed-parameter tractable, and others are hard for various levels of the parameterized complexity hierarchy. Complexity classification of vertex set problems in this framework has proved to be both more interesting and more difficult. We prove that the k-element irredundant set problem is complete for W[1], and thus has the same parameterized complexity as the problem of determining whether a graph has a k-clique. We also show that the “parametric dual” problem of determining whether a graph has an irredundant set of size n−k is fixed-parameter tractable.     

Two linear transformations each tridiagonal with respect to an eigenbasis of the other

Let denote a field, and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations A:V→V and A^*:V→V satisfying both conditions below:

1. [(i)] There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

2. [(ii)] There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

We call such a pair a Leonard pair on V. Refining this notion a bit, we introduce the concept of a Leonard system. We give a complete classification of Leonard systems. Integral to our proof is the following result. We show that for any Leonard pair A,A^* on V, there exists a sequence of scalars β,γ,γ^*,,^* taken from such that both

where [r,s] means rs−sr. The sequence is uniquely determined by the Leonard pair if the dimension of V is at least 4. We conclude by showing how Leonard systems correspond to q-Racah and related polynomials from the Askey scheme.     

On strong consistency of the fuzzy generalized nearest neighbor rule

The k nearest neighbor rule (k-NNR) is a well-known nonparametric decision rule in pattern classification. Fuzzy set theory has been widely used to represent the uncertainty of class membership. Several researchers extended conventional k-NNR to fuzzy k-NNR, such as Bezdek et al. [Fuzzy Sets and Systems 18 (1986) 237–256], Keller et al. [IEEE Trans. Syst. Man, and Cybern. 15(4) (1985) 580–585], Béreau and Dubuisson [Fuzzy Sets and Systems 44 (1991) 17–32]. In this paper, we describe a fuzzy generalized k-NN algorithm. This algorithm is a unified approach to a variety of fuzzy k-NNR's. Then we create the strong consistency of posterior risk of the fuzzy generalized NNR.     

Comparison of two norms of matrices

Any complex n × n matrix A satisfies the inequality

A ₁ ≤ n 1/2 A _d

where .₁ is the trace norm and ._d is the norm defined by

Full-size image

,

where B is the set of orthonormal bases in the space of n × 1 matrices. The present work is devoted to the study of matrices A satisfying the identity:

A₁ = n1/2 A _d

This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of matrices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previous study (J. Dazord, Linear Algebra Appl. 254 (1997) 67), this study is a continuation of the work of M. Marcus and M. Sandy (M. Marcus and M. Sandy, Linear and Multilinear Algebra 29 (1991) 283). Also this study is related to the work of R. Gabriel on classification of matrices with respect to unitary similarity (see R. Gabriel, J. Riene Angew, Math. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591).     

Sometimes slow growing is fast growing

The slow growing hierarchy is commonly defined as follows: G₀(x) = 0, G_x−1(x) := G_x(x) + 1 and G_λ(x) := G_λ[x](x) where λ<₀ is a limit and ·[·]:₀∩ Lim × ω → ₀ is a given assignment of fundamental sequences for the limits below ₀. The first obvious question which is encountered when one looks at this definition is: How does this hierarchy depend on the choice of the underlying system of fundamental sequences? Of course, it is well known and easy to prove that for the standard assignment of fundamental sequence the hierarchy (G_x)_x<0 is slow growing, i.e. each G_x is majorized by a Kalmar elementary recursive function.

It is shown in this paper that the slow growing hierarchy (G_x)_x<0 — when it is defined with respect to the norm-based assignment of fundamental sequences which is defined in the article by Cichon (1992, pp. 173–193) — is actually fast growing, i.e. each PA-provably recursive function is eventually dominated by G_x for some <₀. The exact classification of this hierarchy, i.e. the problem whether it is slow or fast growing, has been unsolved since 1992. The somewhat unexpected result of this paper shows that the slow growing hierarchy is extremely sensitive with respect to the choice of the underlying system of fundamental sequences.

The paper is essentially self-contained. Only little knowledge about ordinals less than ₀ — like the existence of Cantor normal forms, etc. and the beginnings of subrecursive hierarchy theory as presented, for example, in the 1984 textbook of Rose — is assumed.     

Choosability of planar graphs

A graph G = G(V, E) with lists L(v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L(v). We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with L(v) = k v V(G).

Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.     

The countability index of the endomorphism semigroup of a vector space

The countability index C(S) of a semigroup S is the least positive integer n, if such an integer exists, with the property that every countable subset of S is contained in a subsemigroup with n generators. If no such integer exists. C(S) is defined to be infinite. Let V be a vector space over a field F and denote by End V the endomorphism semigroup of V. In the two main results, it is determined precisely when C(End V)=2 and when C(End V)=x SpecificallyC(End V)=2 if and only if V is infinite dimensional or dim V=1 and F is finite and C(End V)=x if and only if F is infinite and dim V is an integer N≥1.     

Primitive graphs with given exponents and minimum number of edges

A graph G = (V, E) on n vertices is primitive if there is a positive integer k such that for each pair of vertices u, v of G, there is a walk of length k from u to v. The minimum value of such an integer, k, is the exponent, exp(G), of G. In this paper, we find the minimum number, h(n, k), of edges of a simple graph G on n vertices with exponent k, and we characterize all graphs which have h(n, k) edges when k is 3 or even.     

Linear circulant matricesover a finite field

Rings of polynomials R_N = Z_p[x]/x^N − 1 which are isomorphic to Z_P^N are studied, where p is prime and N is an integer. If I is an ideal in R_N, the code K whose vectors constitute the isomorphic image of I is a linear cyclic code. If I is a principle ideal and K contains only the trivial cycle 0 and one nontrivial cycle of maximal least period N, then the code words of K/ 0 obtained by removing the zero vector can be arranged in an order which constitutes a linear circulant matrix, C. The distribution of the elements of C is such that it forms the cyclic core of a generalized Hadamard matrix over the additive group of Z_Pp. A necessary condition that C = K/ 0 be linear circulant is that for each row vector v of C, the periodic infinite sequence a(v) produced by cycling the elements of v be period invariant under an arbitrary permutation of the elements of the first period. The necessary and sufficient condition that C be linear circulant is that the dual ideal generated by the parity check polynomial h(χ) of K be maximal (a nontrivial, prime ideal of R_N), with N = p^k − 1 and k = deg (h(χ)).     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).     

An edge version of the matrix-tree theorem and the wiener index

Let T be a tree on n vertices. The Laplacian matrix is L(T)=D(T)-A(T), where D(T) is the diagonal matrix of vertex degrees and A(T) is the adjacency matrix. A special case of the Matrix-Tree Theorem is that the adjugate of L(T) is the n-by-n matrix of l's. The (n-l)-square "edge version" of L(T)is K(T). The main result is a graph-theoretic interpretation of the entries of the adjugate of K(T). As an application, it is shown that the Wiener Index from chemistry is the trace of this adjugate.     

On the Characterization of Maximal Planar Graphs with a Given Signed Cycle Domination Number

Let G =(V, E) be a simple graph. A function f : E → {+1,-1} is called a signed cycle domination function(SCDF) of G if ∑_(e∈E(C))f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ'_(sc)(G) = min{∑_(e∈E)f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ'_(sc)(G) = n.     

On the planarity of jump graphs

For a graph G of size m1 and edge-induced subgraphs F and H of size k (1km), the subgraph H is said to be obtained from F by an edge jump if there exist four distinct vertices u,v,w, and x in G such that uvE(F), wxE(G)−E(F), and H=F−uv+wx. The minimum number of edge jumps required to transform F into H is the k-jump distance from F to H. For a graph G of size m1 and an integer k with 1km, the k-jump graph J_k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J_k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J₂(G) is planar are determined.     

Topological entropy of a graph map

Let G be a graph and f : G → G be a continuous map. Denote by h(f), P(f), AP(f), R(f)and ω(x, f) the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the ω-limit set of x under f, respectively. In this paper,we show that the following statements are equivalent:(1) h(f) 0.(2) There exists an x ∈ G such that ω(x, f) ∩ P(f) = ? and ω(x, f) is an infinite set.(3) There exists an x ∈ G such that ω(x, f)contains two minimal sets.(4) There exist x, y ∈ G such that ω(x, f)-ω(y, f) is an uncountable set and ω(y, f) ∩ω(x, f) = ?.(5) There exist an x ∈ G and a closed subset A ? ω(x, f) with f(A) ? A such that ω(x, f)-A is an uncountable set.(6) R(f)-AP(f) = ?.(7) f |P(f)is not pointwise equicontinuous.     

Magic Labeling of Disjoint Union Graphs

Let G be a graph with vertex set V (G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …,|E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to (1+|E(G)|)/2·d(v), where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V (G),|{e:e ∈ E_v, f(e) ≤ Δ/2}|=|{e:e ∈ E_v, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.     

On size, circumference and circuit removal in 3-connected matroids

This paper proves several extremal results for 3-connected matroids. In particular, it is shown that, for such a matroid M, (i) if the rank r(M) of M is at least six, then the circumference c(M) of M is at least six and, provided |E(M)|4r(M)−5, there is a circuit whose deletion from M leaves a 3-connected matroid; (ii) if r(M)4 and M has a basis B such that Me is not 3-connected for all e in E(M)−B, then |E(M)|3r(M)−4; and (iii) if M is minimally 3-connected but not hamiltonian, then |E(M)|3r(M)−c(M).