Cone isomorphisms and almost surjective operators

Let E$E$ be a Banach lattice and F$F$ a Banach space. A bounded linear operator T:E→F$T:E\to F$ is an isomorphism on the positive cone of E$E$ if and only if T^∗$T^{\ast }$ is almost surjective. A dual version of this theorem holds also. A bounded linear operator T:F→E$T:F\to E$ is almost surjective if and only if T^∗$T^{\ast }$ is an isomorphism on the positive cone of F^∗$F^{\ast }$.     

Iterative approximation of solutions of nonlinear equations of Hammerstein type

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are Lipschitz ?$?$-strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. An iteration process recently introduced by Chidume and Zegeye is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Approximation of solutions of Hammerstein equations with bounded strongly accretive nonlinear operators

Suppose X$X$ is a real q$q$-uniformly smooth Banach space and F,K:X→X$F,K:X\to X$ are bounded strongly accretive maps with D(K)=F(X)=X$D\left(K\right)=F\left(X\right)=X$. Let u^∗$u^{\ast }$ denote the unique solution of the Hammerstein equation u+KFu=0$u+KFu=0$. A new explicit coupled iteration process is shown to converge strongly to u^∗$u^{\ast }$. No invertibility assumption is imposed on K$K$ and the operators K$K$ and F$F$ need not be defined on compact subsets of X$X$. Furthermore, our new technique of proof is of independent interest. Finally, some interesting open questions are included.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Asymptotic average shadowing property on compact metric spaces

We prove that if for a continuous map f$f$ on a compact metric space X$X$, the chain recurrent set, R(f)$R\left(f\right)$ has more than one chain component, then f$f$ does not satisfy the asymptotic average shadowing property. We also show that if a continuous map f$f$ on a compact metric space X$X$ has the asymptotic average shadowing property and if A$A$ is an attractor for f$f$, then A$A$ is the single attractor for f$f$ and we have A=R(f)$A=R\left(f\right)$. We also study diffeomorphisms with asymptotic average shadowing property and prove that if M$M$ is a compact manifold which is not finite with dimM=2$dimM=2$, then the C¹$C^1$ interior of the set of all C¹$C^1$ diffeomorphisms with the asymptotic average shadowing property is characterized by the set of Ω$\Omega$-stable diffeomorphisms.     

Stable extendibility of vector bundles over real projective spaces

Let F$\mathbb{F}$ be either the real number field R$\mathbb{R}$ or the complex number field C$\mathbb{C}$ and RPⁿ${\mathbb{RP}}^n$ the real projective space of dimension n. Theorems A and C in Hemmi and Kobayashi (2008) [2] give necessary and sufficient conditions for a given F$\mathbb{F}$-vector bundle over RPⁿ${\mathbb{RP}}^n$ to be stably extendible to RP^m${\mathbb{RP}}^m$ for every m?n$m?n$. In this paper, we simplify the theorems and apply them to the tangent bundle of RPⁿ${\mathbb{RP}}^n$, its complexification, the normal bundle associated to an immersion of RPⁿ${\mathbb{RP}}^n$ in R^n+r${\mathbb{R}}^{n+r}$(r>0)$(r>0)$, and its complexification. Our result for the normal bundle is a generalization of Theorem A in Kobayashi et al. (2000) [8] and that for its complexification is a generalization of Theorem 1 in Kobayashi and Yoshida (2003) [5].     

[1, 2]-sets in graphs

A subset S⊆V$S\subseteq V$ in a graph G=(V,E)$G=(V,E)$ is a [j,k]$[j,k]$-set if, for every vertex v∈V?S$v\in V?S$, j≤|N(v)∩S|≤k$j\le |N\left(v\right)\cap S|\le k$ for non-negative integers j$j$ and k$k$, that is, every vertex v∈V?S$v\in V?S$ is adjacent to at least j$j$ but not more than k$k$ vertices in S$S$. In this paper, we focus on small j$j$ and k$k$, and relate the concept of [j,k]$[j,k]$-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k$k$-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G$G$, the restrained domination number is equal to the domination number of G$G$.     

Frobenius groups and retract rationality

Let k$k$ be any field, G$G$ be a finite group acting on the rational function field k(_xg:g∈G)$k(x_g:g\in G)$ by h⋅_xg=_xhg$h\cdot x_g=x_{hg}$ for any h,g∈G$h,g\in G$. Define k(G)=k(_xg:g∈G)^G$k\left(G\right)=k{(x_g:g\in G)}^G$. Noether’s problem asks whether k(G)$k\left(G\right)$ is rational (= purely transcendental) over k$k$. A weaker notion, retract rationality introduced by Saltman, is also very useful for the study of Noether’s problem. We prove that, if G$G$ is a Frobenius group with abelian Frobenius kernel, then k(G)$k\left(G\right)$ is retract k$k$-rational for any field k$k$ satisfying some mild conditions. As an application, we show that, for any algebraic number field k$k$, for any Frobenius group G$G$ with Frobenius complement isomorphic to S_L2(_F5)$S{L}_2\left({\mathbb{F}}_5\right)$, there is a Galois extension field K$K$ over k$k$ whose Galois group is isomorphic to G$G$, i.e. the inverse Galois problem is valid for the pair (G,k)$(G,k)$. The same result is true for any non-solvable Frobenius group if k(_ζ8)$k\left({\zeta }_8\right)$ is a cyclic extension of k$k$.     

On words that are concise in residually finite groups

A group-word w$w$ is called concise if whenever the set of w$w$-values in a group G$G$ is finite it always follows that the verbal subgroup w(G)$w\left(G\right)$ is finite. More generally, a word w$w$ is said to be concise in a class of groups X$X$ if whenever the set of w$w$-values is finite for a group G∈X$G\in X$, it always follows that w(G)$w\left(G\right)$ is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w$w$ is a multilinear commutator and q$q$ is a prime-power, then the word w^q$w^q$ is indeed concise in the class of residually finite groups. Further, we show that in the case where w=_γk$w={\gamma }_k$ the word w^q$w^q$ is boundedly concise in the class of residually finite groups. It remains unknown whether the word w^q$w^q$ is actually concise in the class of all groups.     

Critical edges for the assignment problem: Complexity and exact resolution

This paper investigates two problems related to the determination of critical edges for the minimum cost assignment problem. Given a complete bipartite balanced graph with n$n$ vertices on each part and with costs on its edges, k$k$Most Vital Edges Assignment consists of determining a set of k$k$ edges whose removal results in the largest increase in the cost of a minimum cost assignment. A dual problem, Min Edge Blocker Assignment, consists of removing a subset of edges of minimum cardinality such that the cost of a minimum cost assignment in the remaining graph is larger than or equal to a specified threshold. We show that k$k$Most Vital Edges Assignment is NP$NP$-hard to approximate within a factor c<2$c<2$ and Min Edge Blocker Assignment is NP$NP$-hard to approximate within a factor 1.36$1.36$. We also provide an exact algorithm for k$k$Most Vital Edges Assignment that runs in O(n^k+2)$O\left(n^{k+2}\right)$. This algorithm can also be used to solve exactly Min Edge Blocker Assignment.     

Two remarks on frequent hypercyclicity

We show that if T:X→X$T:X\to X$ is a continuous linear operator on an F$F$-space X≠{0}$X\ne \left\{0\right\}$, then the set of frequently hypercyclic vectors of T$T$ is of first category in X$X$, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→X$T:X\to X$ is a bounded linear operator on a Banach space X≠{0}$X\ne \left\{0\right\}$ and if T$T$ is frequently hypercyclic (or, more generally, syndetically transitive), then the T^∗$T^{\ast }$-orbit of every non-zero element of X^∗$X^{\ast }$ is bounded away from 0, and in particular T^∗$T^{\ast }$ is not hypercyclic.     

A new upper bound on the acyclic chromatic indices of planar graphs

An acyclic edge coloring of a graph G$G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G)$a^{\prime }\left(G\right)$ of G$G$ is the smallest integer k$k$ such that G$G$ has an acyclic edge coloring using k$k$ colors. It was conjectured that a^′(G)≤Δ+2$a^{\prime }\left(G\right)\le \Delta +2$ for any simple graph G$G$ with maximum degree Δ$\Delta$. In this paper, we prove that if G$G$ is a planar graph, then a^′(G)≤Δ+7$a^{\prime }\left(G\right)\le \Delta +7$. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Müller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463–478], which says that every planar graph G$G$ satisfies a^′(G)≤Δ+12$a^{\prime }\left(G\right)\le \Delta +12$.     

Growth order for the size of smallest hamiltonian chain saturated uniform hypergraphs

We say that a hypergraph H$H$ is hamiltonian chain saturated if H$H$ does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H$H$. In this paper, for each k≥3$k\ge 3$, we establish the right order of magnitude n^k−1$n^{k-1}$ for the size of the smallest k$k$-uniform hamiltonian chain saturated hypergraph. This solves an open problem of G.Y. Katona.     

Feedback vertex set on graphs of low clique-width

The Feedback Vertex Set problem asks whether a graph contains q$q$ vertices meeting all its cycles. This is not a local property, in the sense that we cannot check if q$q$ vertices meet all cycles by looking only at their neighbors. Dynamic programming algorithms for problems based on non-local properties are usually more complicated. In this paper, given a graph G$G$ of clique-width cw$cw$ and a cw$cw$-expression of G$G$, we solve the Minimum Feedback Vertex Set problem in time O(n²2^O(cwlogcw))$O\left(n^2{2}^{O(cwlogcw)}\right)$. Our algorithm applies dynamic programming on a so-called k$k$-module decomposition of a graph, as defined by Rao (2008) [29], which is easily derivable from ak$k$-expression of the graph. The related notion of module-width of a graph is tightly linked to both clique-width and NLC-width, and in this paper we give an alternative equivalent characterization of module-width.     

On the sum of two closed subspaces

If U,V$U,V$ are closed subspaces of a Fréchet space, then E$E$ is the direct sum of U$U$ and V$V$ if and only if E^′$E^{\prime }$ is the algebraic direct sum of the annihilators U^°$U^{°}$ and V^°$V^{°}$. We provide a simple proof of this (possibly well-known) result.     

A note on distance-regular graphs with a small number of vertices compared to the valency

In this note we study distance-regular graphs with a small number of vertices compared to the valency. We show that for a given α>2$\alpha >2$, there are finitely many distance-regular graphs Γ$\Gamma$ with valency k$k$, diameter D≥3$D\ge 3$ and v$v$ vertices satisfying v≤αk$v\le \alpha k$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite). We also show, as a consequence of this result, that there are finitely many distance-regular graphs with valency k≥3$k\ge 3$, diameter D≥3$D\ge 3$ and _c2≥εk$c_2\ge \epsilon k$ for a given 0<ε<1$0<\epsilon <1$ unless (D=3$D=3$ and Γ$\Gamma$ is imprimitive) or (D=4$D=4$ and Γ$\Gamma$ is antipodal and bipartite).     

Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Embedding cycles of given length in oriented graphs

Kelly, Kühn and Osthus conjectured that for any ?≥4$?\ge 4$ and the smallest number k≥3$k\ge 3$ that does not divide ?$?$, any large enough oriented graph G$G$ with δ⁺(G),δ⁻(G)≥⌊|V(G)|/k⌋+1${\delta }^{+}\left(G\right),{\delta }^{-}\left(G\right)\ge \lfloor \left|V\left(G\right)\right|/k\rfloor +1$ contains a directed cycle of length ?$?$. We prove this conjecture asymptotically for the case when ?$?$ is large enough compared to k$k$ and k≥7$k\ge 7$. The case when k≤6$k\le 6$ was already settled asymptotically by Kelly, Kühn and Osthus.