Random walks on finitely structured transfinite networks

A general theory for random walks on transfinite networks whose ranks are arbitrary natural numbers is established herein. In such networks, nodes of higher ranks connect together transfinite networks of lower ranks. The probabilities for transitions through such nodes are obtained as extensions of the Nash-Williams rule for random walks on ordinary infinite networks. The analysis is based on the theory of transfinite electrical networks, but it requires that the transfinite network have a structure that generalizes local-finiteness for ordinary infinite networks. The shorting together of nodes of different ranks are allowed; this complicates transitions through such nodes but provides a considerably more general theory. It is shown that, with respect to any finite set of nodes of any ranks, a transfinite random walk can be represented by an irreducible reversible Makov chain, whose state space is that set of nodes.This work was supported by the National Science Foundation under the grants DMS-9200738 and MIP-9200748.     

Security improvement on an anonymous key agreement protocol based on chaotic maps

In 2009, Tseng et al. proposed a password sharing and chaotic map based key agreement protocol (Tseng et al.’s protocol). They claimed that the protocol provided mutual authentication between a server and a user, and allowed the user to anonymously interact with the server to establish a shared session key. However, in 2011, Niu et al. have proved that Tseng et al.’s protocol cannot guarantee user anonymity and protocol security when there is an internal adversary who is a legitimate user. Also it cannot provide perfect forward secrecy. Then Niu et al. introduced a trust third party (TTP) into their protocol designing (Niu et al.’s protocol). But according to our research, Niu et al.’s protocol is found to have several unsatisfactory drawbacks. Based on reconsidering Tseng et al.’s protocol without introducing TTP, we give some improvements to meet the original security and performance requirements. Meanwhile our proposed protocol overcomes the security flaws of Tseng et al.’s protocol.     

An optimal algorithm for geometrical congruence

An algorithm is given which, in time O(n log n), determines all the Euclidean congruences (if any) between two n-point sets in 3-dimensional space. The algorithm is shown to be optimal to within a constant factor.     

An improvement on H design

An H(m, g, 4, 3) is a triple , where X is a set of mg points, is a partition of X into m disjoint sets of size g, and is a set of 4‐element transverses of , such that each 3‐element transverse of is contained in exactly one of them. Such a design was introduced by Hanani 2 , who used it to study Steiner quadruple systems. Mills showed that for , an H(m, g, 4, 3) exists if and only if mg is even and is divisible by 3, and that for , an H(5, g, 4, 3) exists if g is divisible by 4 or 6 10 . In this article, we shall show that an H(5, g, 4, 3) exists if g is even, and . © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 25–35, 2009     

Measure theory based on lattices and transfinite recursion

In this methodological study we develop the foundations of measure theory using lattices as prime structures instead of rings. Topological as well as abstract regularity is incorporated into this approach from the outset. The use of inner and outer measures is replaced by transfinite constructions. Basic extension steps are transfinitely iterated to yield generalizations of Carathéodory’s theorem which are optimal with respect to inner and outer approximations. Received: 5 May 2008     

An interconnection of local maps inducing onto global maps

“Local maps and the global maps induced by them” treated in this paper is a subject not only on cellular or tessellation automata but also on shift dynamical systems. We study an interconnection of local maps inducing onto global maps from a graph-theoretical viewpoint.     

A remark on polynomials and the transfinite diameter

The main result is the following theorem: LetE be the point set for which |Π _v=1 ⁿ (Z-Z _V)|<1. If the zerosz _ν (ν=1, …,n) belong to a bounded, closed and connected set whose transfinite diameter is 1−c (0<c<1), thenE contains a disk of positive radius ρ, dependent only onc.     

An adaptive improvement on PageRank algorithm

In this article, we introduce the Google’s method for quality ranking of web page in a formal mathematical format, use the power iteration to improve the PageRank, and also discuss the effect of different q to the PageRank, as well as how a PageRank will be changed if more links are added to one page or removed from some pages.     

An improvement on the concentration-compactness principle

1. IntroductionThe concenirat ion- compactness lemma proposed by P. L. Lions in [ 1, 21 is a well-knownmethod in treating minimization problems given on Unbounded domains laal RN. Thelemma says that if {p.(x)} is a sequence in L'(R") satisfyingthen there exists a subsequence satisfying one of the three following possibilities: compactness, vanishing and dichotomy. However, there still edests some problem about thisclajssfication: Are they separated from each other, i.e., to one subsequence…     

An anisotropic geometrical approach for non-relativistic extended dynamics

In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and some particular solutions are given. For instance, we obtain as geodesic trajectories some circular helices (depending on an angular velocity $\omega $ ), certain circles situated in some planes (ones are parallel with $xOy $ , and other ones are orthogonal on $xOy$ ), or some straight lines which are parallel with the axis $Oz$ . All these geometrical geodesics are very specific because they are completely determined by the non-constant index of refraction $n(x)$ .     

An anonymous key agreement protocol based on chaotic maps

Recently, Tseng et al. proposed a novel key agreement protocol based on chaotic maps. They claimed that the protocol achieved session key agreement between a server and a user, and allowed the user to anonymously interact with the server. This paper, however, will demonstrate that Tseng et al.’s protocol can not guarantee user anonymity and protocol security against an insider adversary who is a legal user, and it can not provide perfect forward secrecy. Furthermore, the current paper presents a new key agreement protocol based on Chebyshev chaotic map in order to conquer these problems. In contrast with Tseng et al.’s protocol, the proposed protocol is more secure and preserves user anonymity.     

Lines on Del Pezzo surfaces and transfinite heterotic string space-time

It is pointed out that the hierarchy of fractal dimensions characterizing transfinite heterotic string space-times bears a striking resemblance to the sequence of the number of lines lying on Del Pezzo surfaces.     

An existence theorem for planar maps

The main result of this paper is a theorem concerning possible cubic maps on the plane or sphere. The dual approach of spherical triangulations will be used. Every such triangulation must contain a vertex of degree less than 8 and other than 5, or else contain one of a list of 5 configurations. Due to the occurrence of these five configurations in four color reduction arguments, this implies that a minimal five color map must have at least one face with six or seven neighbors. The theorem is given in Heesch [1, p. 129 ff.], but the proof here is much shorter, due to a modification of Heesch's principle of “discharging”. The principle itself is obtained by exploiting the Euler formula, and should be compared with the theory of Euler contributions as developed by Ore in [3], and by Ore and Stemple [4].     

An intersection theorem for multivalued maps

LetE be a Banach space and letV, W ? E be two closed subspaces such thatE=V ⊕ W. LetF, G: E-· E be two multivalued maps. The problem we are concerned with is to give conditions onF andG ensuring the nonemptiness ofF(V) ∩ G(W).     

An existence theorem for harmonic maps

The aim of this article is to prove the existence of a harmonic map in a given homotopy class from a non-compact manifold into another one, while this class fulfills a geometrical simplicity condition.