The genus of a module

Let R be a Dedekind domain satisfying the Jordan-Zassenhaus theorem (e.g., the ring of integers in a number field) and Λ a module finite R-algebra. We extend classical results of Jacobinski, Roiter, and Drozd on orders and lattices. In particular, it is shown that the genus of a finitely generated Λ-module M is finite. Moreover, given M, there exist a positive integer t and a finite extension S of R such that a Λ-module N is the genus of M if and only if M^(t) ? N^(t) if and only if M ? S ? N ? S.     

The Maslov form in non-invariant slant submanifolds of S-space-forms

We present a characterization theorem for the Maslov form in certain non-invariant slant submanifolds of S-space-forms to be closed and, from it, we deduce a topological obstruction for these types of non-invariant slant immersions. Moreover, we also give conditions for an anti-invariant submanifolds of an S-manifold, tangent to the structure vector fields, to have closed and conformal Maslov form.     

A limiting absorption principle for Schrödinger operators with spherically symmetric exploding potentials

LetH=?Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V _S(r)+V _L(r), whereV _S(r) is short-range and the exploding partV _L(r) satisfies the following assumptions: (a) Λ=lim sup _r→∞ V _L(r)<∞ (but Λ=?∞ is possible). Denote Λ⁺= max(Λ,0). (b)V _L(r)∈C ^2k (r ₀, ∞) and, with someδ>0 such that 2kδ>1: (d/dr) ^j V _L(r) · (Λ⁺?V _L(r))^?1=O(r ^jδ) asr → ∞,j=1, ..., 2k. (c) ∫ _r0 ^∞ dr|V _L(r|^1/2 dr|V _L(r)|^1/2=∞. (d) (d/dr)V _L(r)≦0. Under these assumptions a limiting absorption principle forR(z)=(H?z)^?1 is established. More specifically, ifK ?C ⁺={zImz≧0} is compact andK ∩ (?∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ?L ²(R ⁿ) is a weighted-L ² space. To ensure uniqueness of solutions of (H?z)u=f, z ∈K, a suitable radiation condition is introduced.     

Necessary condition for the existence of the S matrix outside the scope of perturbation theory

Using the Maslov—Shvedov complex-germ method due to Maslov—Shvedov, we obtain a necessary condition for the existence of the quantum-field S matrix outside the scope of perturbation theory in the leading order of semiclassical approximation. This condition consists in that the tangent symplectic transformation to the evolution operator of the nonlinear classical field equation is realized by a unitary transformation of Fock space. It follows from the results of the book of Maslov and Shvedov that this condition always holds.     

Independent domination in chordal graphs

A graph chordal if it does not contain any cycle of length greater than three as an induced subgraph. A set of S of vertices of a graph G = (V,E) is independent if not two vertices in S are adjacent, and is dominating if every vertex in V?S is adjacent to some vertex in S. We present a linear algorithm to locate a minimum weight independent dominating set in a chordal graph with 0–1 vertex weights.     

Analyse de Fourier,multiplicateurs de Schur sur Sp et ensembles Λ(p)cb non commutatifs

This work deals with various questions concerning Fourier multipliers on L^p, Schur multipliers on the Schatten class S^p as well as their completely bounded versions when L^p and S^p are viewed as operator spaces. We use for this aim subsets ofenjoying the Λ(p)_cb-property which is much stronger than the usual Λ(p)-property. We start by studying the notion of Λ(p)_cb-sets in the general case of an arbitrary discrete group before turning to .     

On degree sum conditions for long cycles and cycles through specified vertices

Let G be a graph. For S⊂V(G), let Δ_k(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ₂(S)≥d for every independent set S of order κ(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If Δ₂(S)≥|V(G)| for every independent set S of order κ(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles.     

Relative length of longest paths and longest cycles in triangle-free graphs

In this paper, we study triangle-free graphs. Let G=(V,E) be an arbitrary triangle-free graph with minimum degree at least two and σ₄(G)?|V(G)|+2. We first show that either for any path P in G there exists a cycle C such that |V_P?V_C|?1, or G is isomorphic to exactly one exception. Using this result, we show that for any set S of at most δ vertices in G there is a cycle C such that S⊆V_C.     

Minimality of planes in normed spaces

We prove that a region in a two-dimensional affine subspace of a normed space V has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary. Furthermore, the 2-dimensional Hausdorff area density admits a convex extension to Λ² V. The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon.     

On the existence of an invariant non-degenerate bilinear form under a linear map

Let V be a vector space over a field F. Assume that the characteristic of F is large, i.e. char(F)>dimV. Let T:V→V be an invertible linear map. We answer the following question in this paper. When doesVadmit a T-invariant non-degenerate symmetric (resp. skew-symmetric) bilinear form? We also answer the infinitesimal version of this question.Following Feit and Zuckerman 2, an element g in a group G is called real if it is conjugate in G to its own inverse. So it is important to characterize real elements in GL(V,F). As a consequence of the answers to the above question, we offer a characterization of the real elements in GL(V,F).Suppose V is equipped with a non-degenerate symmetric (resp. skew-symmetric) bilinear form B. Let S be an element in the isometry group I(V,B). A non-degenerate S-invariant subspace W of (V,B) is called orthogonally indecomposable with respect to S if it is not an orthogonal sum of proper S-invariant subspaces. We classify the orthogonally indecomposable subspaces. This problem is non-trivial for the unipotent elements in I(V,B). The level of a unipotent T is the least integer k such that (T-I)^k=0. We also classify the levels of unipotents in I(V,B).     

Powerful alliances in graphs

For a graph G=(V,E), a non-empty set S⊆V is a defensive alliance if for every vertex v in S, v has at most one more neighbor in V−S than it has in S, and S is an offensive alliance if for every v∈V−S that has a neighbor in S, v has more neighbors in S than in V−S. A powerful alliance is both defensive and offensive. We initiate the study of powerful alliances in graphs.     

Join of two graphs admits a nowhere-zero 3-flow

Let G be a graph, and λ the smallest integer for which G has a nowherezero λ-flow, i.e., an integer λ for which G admits a nowhere-zero λ-flow, but it does not admit a (λ ? 1)-flow. We denote the minimum flow number of G by Λ(G). In this paper we show that if G and H are two arbitrary graphs and G has no isolated vertex, then Λ(G ∨ H) ? 3 except two cases: (i) One of the graphs G and H is K ₂ and the other is 1-regular. (ii) H = K ₁ and G is a graph with at least one isolated vertex or a component whose every block is an odd cycle. Among other results, we prove that for every two graphs G and H with at least 4 vertices, Λ(G ∨ H) ? 3.     

On the combinatorial rank of a graded braided bialgebra

Let B be a graded braided bialgebra. Let S(B) denote the algebra obtained dividing out B by the two sided ideal generated by homogeneous primitive elements in B of degree at least two. We prove that S(B) is indeed a graded braided bialgebra quotient of B. It is then natural to compute S(S(B)), S(S(S(B))) and so on. This process yields a direct system whose direct limit comes out to be a graded braided bialgebra which is strongly N-graded as a coalgebra. Following V.K. Kharchenko, if the direct system is stationary exactly after n steps, we say that B has combinatorial rank n and we write κ(B)=n. We investigate conditions guaranteeing that κ(B) is finite. In particular, we focus on the case when B is the braided tensor algebra T(V,c) associated to a braided vector space (V,c), providing meaningful examples such that κ(T(V,c))≤1.     

On inverse scattering for the wave equation with a potential term via the Lax-Phillips theory: A simple proof

The inverse scattering problem for the perturbed wave equation (1) □u + V(x)u = 0 in $\text{Ω=R}{}^{\mathrm{n}}$ (n = odd ? 3) is considered. Here the potentials V(x) are real, smooth, with compact support and non-negative. We apply the Lax and Phillips theory, together with some properties of solutions of a Dirichlet problem associated with the operator ?Δ + V(x) to show, in a very simple way, that the scattering operator S(V) associated with (1) determines uniquely the scatterer, provided that a fixed sign condition on the potentials is satisfied. We also show that the map V → S(V) is once-differentiable.     

Algebraic density property of Danilov–Gizatullin surfaces

A Danilov–Gizatullin surface is an affine surface V which is the complement of an ample section S for the ruling of a Hirzebruch surface. The remarkable theorem of Danilov and Gizatullin states that the isomorphism class of V depends only on the self-intersection number (S.S). In this paper we apply the theorem of Danilov–Gizatullin to prove that the Lie algebra generated by the complete algebraic vector fields on V coincides with the set of all algebraic vector fields of V.     

Bounds on a graph's security number

Let G=(V,E) be a graph. A set S⊆V is a defensive alliance if |N[x]∩S|?|N[x]-S| for every x∈S. Thus, each vertex of a defensive alliance can, with the aid of its neighbors in S, be defended from attack by its neighbors outside of S. An entire set S is secure if any subset X⊆S, not just singletons, can be defended from an attack from outside of S, under an appropriate definition of what such a defense implies. The security number s(G) of G is the cardinality of a smallest secure set. Bounds on s(G) are presented.     

An upper bound for the restrained domination number of a graph with minimum degree at least two in terms of order and minimum degree

Let G=(V,E) be a graph. A set S⊆V is a restrained dominating set if every vertex in V−S is adjacent to a vertex in S and to a vertex in V−S. The restrained domination number of G, denoted γ_r(G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is a connected graph of order n and minimum degree δ and not isomorphic to one of nine exceptional graphs, then .     

Error bounds for asymptotic expansions of Wilks’ lambda distribution

Let Λ=|S_e|/|S_e+S_h|, where S_h and S_e are independently distributed as Wishart distributions W_p(q,Σ) and W_p(n,Σ), respectively. Then Λ has Wilks’ lambda distribution Λ_p,q,n which appears as the distributions of various multivariate likelihood ratio tests. This paper is concerned with theoretical accuracy for asymptotic expansions of the distribution of T=-nlogΛ. We derive error bounds for the approximations. It is necessary to underline that our error bounds are given in explicit and computable forms.     

Dynamical behavior of endomorphisms on certain invariant sets

We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ?². We consider the perturbation (z ² +?z +b?w ², w ²) of (z ², w ²) and we prove that, for b sufficiently small, it is injective on its basic set Λ_? close to Λ:= {0} × S ¹. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ _? , in the case of these maps.     

A sufficient condition for pancyclability of graphs

Let G be a graph of order n and S be a vertex set of q vertices. We call G,S-pancyclable, if for every integer i with 3≤i≤q there exists a cycle C in G such that |V(C)∩S|=i. For any two nonadjacent vertices u,v of S, we say that u,v are of distance two in S, denoted by d_S(u,v)=2, if there is a path P in G connecting u and v such that |V(P)∩S|≤3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u,v of S with d_S(u,v)=2, , then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u,v of S with d_S(u,v)=2, , then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S=V(G).