Fan‐type theorem for path‐connectivity

For a connected noncomplete graph G, let μ(G):=min{max {d_G(u), d_G(v)}:d_G(u, v)=2}. A well‐known theorem of Fan says that every 2‐connected noncomplete graph has a cycle of length at least min{|V(G)|, 2μ(G)}. In this paper, we prove the following Fan‐type theorem: if G is a 3‐connected noncomplete graph, then each pair of distinct vertices of G is joined by a path of length at least min{|V(G)|?1, 2μ(G)?2}. As consequences, we have: (i) if G is a 3‐connected noncomplete graph with , then G is Hamilton‐connected; (ii) if G is a (s+2)‐connected noncomplete graph, where s≥1 is an integer, then through each path of length s of G there passes a cycle of length≥min{|V(G)|, 2μ(G)?s}. Several results known before are generalized and a conjecture of Enomoto, Hirohata, and Ota is proved. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 265–282, 2002 DOI 10.1002/jgt.10028     

A fixed point theorem for o‐minimal structures

We prove a definable analogue to Brouwer's Fixed Point Theorem for o‐minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o‐minimal functions, with an application of Sperner's Lemma. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Bose‐Burton type theorem for quadrics

Let Q be a non‐degenerate quadric defined by a quadratic form in the finite projective space PG(d,q). Let r be the dimension of the generators of Q. For all k with 2 ≤ k < r we determine the smallest cardinality of a set B of points with the property that every subspace of dimension k that is contained in Q meets B. It turns out that the smallest examples consist of the non‐singular points of quadrics S ∩ Q for suitable subspaces S of codimension k of PG(d,q). For k = 1, the same result was known before. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 317–338, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10051     

An explicit Schilder‐type theorem for super‐Brownian motions

Like ordinary Brownian motion, super‐Brownian motion, a central object in the theory of superprocesses, is a universal object arising in a variety of settings. Schilder‐type theorems and Cramér‐type theorems are two of the major topics for large‐deviation theory. A Schilder‐type (which is also a Cramér‐type) sample large deviation for super‐Brownian motions with a good rate function represented by a variation formula was established in 1993 and 1994; since then there have been very valuable contributions for giving an affirmative answer to the question of whether this sample large deviation holds with an explicit good rate function. In this paper, thanks to previous results on this issue and the Brownian snake, we establish such a large deviation for nonzero finite initial measures. © 2010 Wiley Periodicals, Inc.     

Liouville‐type theorem for the steady compressible Hall‐MHD system

In this paper, we prove a Liouville‐type theorem for the steady compressible Hall‐magnetohydrodynamics system in Π, where Π is whole space or half space . We show that a smooth solution (ρ, u , B ,P) satisfying 1/C₀<ρ<C₀, , and B ∈L^9/2(Π) for some constant C₀>0 is indeed trivial. This generalizes and improves 2 results of Chae.     

A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

LetDR ^N be a region with smooth boundaryD. Letp·q>1,p, q1. We consider the system:u _t=u+v ^p,v _t=u+u ^q inD×[0, ) withu=v=0 inD×[0, ) andu ₀,v ₀ nonnegative. Let=max(p, q). We show that ifD isR ^N, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (+1)/(pq–1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (+1)/(pq–1)<pc(D) both nontrivial global and nonglobal solutions exist. In caseD is a cone orD=R ^N, (a) holds with equality. An explicit formula forpc(D) is given.This research was supported in part by NSF Grant DMS-8822788 and in part by the Air Force Office of Scientific Research.     

A Gallai–Edmonds‐type structure theorem for path‐matchings

As a generalization of matchings, Cunningham and Geelen introduced the notion of path‐matchings. We give a structure theorem for path‐matchings which generalizes the fundamental Gallai–Edmonds structure theorem for matchings. Our proof is purely combinatorial. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 93–102, 2004     

On the strong cell decomposition property for weakly o‐minimal structures

We consider a class of weakly o‐minimal structures admitting an o‐minimal style cell decomposition, for which one can construct certain canonical o‐minimal extension. The paper contains several fundamental facts concerning the structures in question. Among other things, it is proved that the strong cell decomposition property is preserved under elementary equivalences. We also investigate fiberwise properties (of definable sets and definable functions), definable equivalence relations, and conditions implying elimination of imaginaries.     

Filling certain cuts in discrete weakly o‐minimal structures

Discrete weakly o‐minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o‐minimal structures with infinite definable discrete (convex) subsets and we study the Boolean algebra of definable sets of its countable models. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fixed‐point theorem for a generalized almost Hardy‐Rogers‐type F contraction on metric‐like spaces

In this paper, we introduce some fixed‐point theorems for a generalized almost Hardy‐Rogers‐type F contraction in a metric‐like space and give an example to illustrate these main results. Moreover, we show the applications of electric circuit equations, second‐order differential equations, and fractional differential equations. Our results improve, generalize, and extend the corresponding results in literature.     

Ambarzumyan‐type theorem with polynomially dependent eigenparameter

In this paper, we are concerned with the inverse Sturm–Liouville problem with polynomially dependent eigenparameter in discontinuity and boundary conditions. By using a self‐adjoint operator‐theoretic interpretation for this sort of problem, Ambarzumyan theorem is provided for the mentioned Sturm–Liouville operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Central limit theorem for weakly dependent variables

Mathematical Notes -     

A Hille-Yosida theorem for weakly continuous semigroups

We introduce a new class of weakly continuous semigroups and give a characterization of their infinitesimal generators, generalizing the classical Hille-Yosida Theorem for strongly continuous semigroups. The results are illustrated by the example of transition semigroups corresponding to the solutions of certain stochastic differential equations.     

A note on prime models in weakly o‐minimal structures

Let be a weakly o‐minimal structure with the strong cell decomposition property. In this note, we show that the canonical o‐minimal extension of is the unique prime model of the full first order theory of over any set . We also show that if two weakly o‐minimal structures with the strong cell decomposition property are isomorphic then, their canonical o‐minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o‐minimal theory with prime models.