Almost sure estimates for the concentration neighborhood of Sinai’s walk

We consider Sinai’s random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is bounded almost surely. We also get that i.o. the maximal distance between two favorite sites is bounded almost surely.     

Suppression of Chaos and Phase Locking in Two Coupled Nonidentical Neurons under Periodic Input

Dynamical behaviour of two coupled neurons with at least one of them being chaotic is presented. Bifurcation diagrams and Lyapunov exponents are calculated to diagnose the dynamical behaviour of the coupled neurons with the increasing coupling strength. It is found that when the coupling strength increases, a chaotic neuron can be controlled by the coupling between neurons. At the same time, phase locking is studied by the maxima of the differences of instantaneous phases and average frequencies between two coupled neurons, and the inherent connection of phase locking and the suppression of chaos is formulated. It is observed that the onset of phase locking is closely related to the suppression of chaos. Finally, a way for suppression of chaos in two coupled nonidentical neurons under periodic input is suggested.     

Completeness and compactness for varieties over a local field

This paper shows that for a local field K, a subfield k ⊂ K and a variety X over k, X is complete if and only if for every finite field extension Kʹ | K, the set X(Kʹ) is compact in its strong topology. The author likes to thank Florian Pop, Jakob Stix, Stefan Wewers, Gunther Cornelissen and his own parents for their support. Received: 13 April 2006     

A study of the effect of ZrO2 on the magnetic properties of FePt/ZrO2 multilayer

Fe_xPt_100−x(30 nm) and [Fe_xPt_100−x(3 nm)/ZrO₂]₁₀ (x = 37, 48, 57, 63, 69) films with different ZrO₂ content were prepared by RF magnetron sputtering technique, then were annealed at 550 °C for 30 min. This work investigates the effect of ZrO₂ doping on the microstructural evolution, magnetic properties, grain size, as well as the ordering kinetics of FePt alloy films. The as-deposited films behaved a disordered state, and the ordered L1₀ structure was obtained by post-annealing. The magnetic properties of the films are changed from soft magnetism to hard magnetism after annealing. The variation of the largest coercivities of [Fe_xPt_100−x/ZrO₂]₁₀ films with the Fe atomic percentage, x and differing amounts of ZrO₂ content reveals that as we increase the ZrO₂ content we must correspondingly increase the amount of Fe. This phenomenon suggests that the Zr or O atoms of ZrO₂ preferentially react with the Fe atoms of FePt alloy to form compounds. In addition, introducing the nonmagnetic ZrO₂ can reduce the intergrain exchange interactions of the FePt/ZrO₂ films, and the interactions are decreased as the ZrO₂ content increases, the dipole interactions are observed in FePt/ZrO₂ films as the ZrO₂ content is more than 15%.     

On dimensional rigidity of bar-and-joint frameworks

Let V={1,2,…,n}. A mapping p:V→R^r, where p¹,…,pⁿ are not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in R^r, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in R^r is said to be dimensionally rigid iff there does not exist a framework G(q) in R^s for s?r+1, such that ∥qⁱ-q^j∥²=∥pⁱ-p^j∥² for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p¹,…,pⁿ of the given r-configuration are in general position, is also investigated.     

Infima and complements in the lattice of quasi-uniformities

We show that the infimum of any family of proximally symmetric quasi-uniformities is proximally symmetric, while the supremum of two proximally symmetric quasi-uniformities need not be proximally symmetric. On the other hand, the supremum of any family of transitive quasi-uniformities is transitive, while there are transitive quasi-uniformities whose infimum with their conjugate quasi-uniformity is not transitive. Moreover we present two examples that show that neither the supremum topology nor the infimum topology of two transitive topologies need be transitive. Finally, we prove that several operations one can perform on and between quasi-uniformities preserve the property of having a complement.     

The minimum span of L(2,1)-labelings of certain generalized Petersen graphs

In the classical channel assignment problem, transmitters that are sufficiently close together are assigned transmission frequencies that differ by prescribed amounts, with the goal of minimizing the span of frequencies required. This problem can be modeled through the use of an L(2,1)-labeling, which is a function f from the vertex set of a graph G to the non-negative integers such that |f(x)-f(y)|? 2 if xand y are adjacent vertices and |f(x)-f(y)|?1 if xand y are at distance two. The goal is to determine the λ-number of G, which is defined as the minimum span over all L(2,1)-labelings of G, or equivalently, the smallest number k such that G has an L(2,1)-labeling using integers from {0,1,…,k}. Recent work has focused on determining the λ-number of generalized Petersen graphs (GPGs) of order n. This paper provides exact values for the λ-numbers of GPGs of orders 5, 7, and 8, closing all remaining open cases for orders at most 8. It is also shown that there are no GPGs of order 4, 5, 8, or 11 with λ-number exactly equal to the known lower bound of 5, however, a construction is provided to obtain examples of GPGs with λ-number 5 for all other orders. This paper also provides an upper bound for the number of distinct isomorphism classes for GPGs of any given order. Finally, the exact values for the λ-number of n-stars, a subclass of the GPGs inspired by the classical Petersen graph, are also determined. These generalized stars have a useful representation on Möebius strips, which is fundamental in verifying our results.     

On the third largest Laplacian eigenvalue of a graph

In this article, a sharp lower bound for the third largest Laplacian eigenvalue of a graph is given in terms of the third largest degree of the graph.     

Prospective and retrospective analyses under logistic regression models

In logistic case-control studies, Prentice and Pyke (Biometrika 66 (1979) 403-411) showed that valid point estimators of the odds-ratio parameters and their standard errors may be obtained by fitting the prospective logistic regression model to case-control data. Wang and Carroll (Biometrika 80 (1993) 237-241; J. Statist. Plann. Inference 43 (1995) 331-340) generalized Prentice and Pyke's (Biometrika 66 (1979) 403-411) results to robust logistic case-control studies. In this paper, we extend the results of Prentice and Pyke (Biometrika 66 (1979) 403-411) and Wang and Carroll (Biometrika 80 (1993) 237-241; J. Statist. Plann. Inference 43 (1995) 331-340) to a class of statistics and a class of unbiased estimating equations. We present some results on simulation and on the analysis of two real datasets.