In this paper, we extend previous results on average commute-times for undirected graphs to fully-connected directed graphs, corresponding to irreducible Markov chains. We introduce an unsymmetrized generalized Laplacian matrix and show how its pseudo-inverse directly yields the one-way first-transit times and round-trip commute times with formulas almost matching those for the undirected graph...

Let B be the closed unit ball in R and S its boundary. We define a family of pseudo metrics on B. As an application, we prove that for any countable-to-one function f : S → [0, a], the set NMnf = {x ∈ S 2 | there exists y ∈ S2 such that f(x)−f(y) > ndE(x, y)} is uncountable for all n ∈ N, where dE is the Euclidean metric on R . 2000 Mathematics Subject Classification ; 57N05, 57M40 1 The family...

We review the geodesic motion of pseudo-classical spinning particles in curved spaces. Investigating the generalized Killing equations for spinning spaces, we express the constants of motion in terms of KillingYano tensors. The general results are applied to the case of the fourdimensional Euclidean Taub-NUT spinning space. A simple exact solution, corresponding to trajectories lying on a cone,...

Generalized Weierstrass representations for generic surfaces confor-mally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces generated by the Davey-Stewartson hierarchy of integrable equations are proposed. Willmore functional of a surface is invariant under such deformations.

For any two points P = (p (1) ,p (2) ,...,p (n)) and Q = (q (1) ,q (2) ,...,q (n)) of R n , we define the crisp vector → PQ = (q (1) −p (1) ,q (2) −p (2) ,...,q (n) −p (n)) = Q(−)P. Then we obtain an n-dimensional vector space E n = { → PQ | for all P,Q ∈ R n }. Further, we extend the crisp vector into the fuzzy vector on fuzzy sets of R n. Let D, E be any two fuzzy sets on R n and define the f...

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Itô algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular,...

We demonstrate that random walks embed connected undirected graphs into Euclidean space, in which distances and angles acquire the clear statistical interpretation. Being defined on the spatial network of Venetian canals, random walks afford a portrait of the city space syntax and measure its intelligibility. In the present Letter, we investigate the spatial network of 96 canals in Venice (that...

A maximal inequality is an which involves the (absolute) supremum sups⩽t|Xs| or running maximum sups⩽tXs of a stochastic process (Xt)t⩾0. We discuss inequalities for several classes processes with values in Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller (compound) pseudo Poisson stable-like and solutions to SDEs driven by –, strong Markov Gaussian processes. Using Bu...

The integration procedure for multidimensional cosmological models with multicomponent perfect fluid in spaces of constant curvature is developed. Reduction of pseudo-Euclidean Toda-like systems to the Euclidean ones is done. Some known solutions are singled out from those obtained. The existence of the wormholes is proved. PACS numbers: 04.20.J, 04.60.+n, 03.65.Ge