Random walks on stochastic hyperbolic half planar triangulations

We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in (Angel and Ray, Ann Probab, in press). We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 213–234, 2016     

Degenerations of ideal hyperbolic triangulations

Let M be a cusped 3-manifold, and let ${\mathcal{T}}$ be an ideal triangulation of M. The deformation variety ${\mathfrak{D}(\mathcal{T})}$ , a subset of which parameterises (incomplete) hyperbolic structures obtained on M using ${\mathcal{T}}$ , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.     

Positively oriented ideal triangulations on hyperbolic three-manifolds

Let M³ be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M³. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.     

Minimum ideal triangulations of hyperbolic 3-manifolds

Let σ(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volumen-cusped hyperbolic 3-manifold, orientable or not. Let σ_or(n) be the corresponding number when we restrict ourselves to orientable manifolds. The correct values of σ(n) and σ_or(n) and the corresponding manifolds are given forn=1,2,3,4 and 5. We then show that 2n−1≤σ(n)≤σ_or(n)≤4n−4 forn≥5 and that σ_or(n)≥2n for alln. Both authors were supported by NSF Grants DMS-8711495, DMS-8802266 and Williams College Research Funds.     

Fan triangulations of a hyperbolic plane of positive curvature

We study the families (?_λ) of normal partitions of a 3-(1)-contour F of a hyperbolic plane \(\hat H\) of positive curvature into simple 4-contours whose hyperbolic diagonal lines are parallel to the base of F. A 3-(1)-contour with a given partition from a family (?_λ) (or some its normal subpartition) is called a fan. We construct fan partitions P _e, P _h, and P _p of \(\hat H\) whose symmetry groups are generated by a shift along an elliptic (respectively, hyperbolic and parabolic) straight line. It is proved that the partitions P _h and P _p are normal. The partitions P _h and P _p whose cells are trihedrals present examples of the first triangulations of \(\hat H\) .     

Planar Eulerian triangulations are equivalent to spherical Latin bitrades

Given a pair of Latin squares, we may remove from both squares those cells that contain the same symbol in corresponding positions. The resulting pair T={P₁,P₂} of partial Latin squares is called a Latin bitrade. The number of filled cells in P₁ is called the size of T. There are at least two natural ways to define the genus of a Latin bitrade; the bitrades of genus 0 are called spherical. We construct a simple bijection between the isomorphism classes of planar Eulerian triangulations on v vertices and the main classes of spherical Latin bitrades of size v−2. Since there exists a fast algorithm (due to Batagelj, Brinkmann and McKay) for generating planar Eulerian triangulations up to isomorphism, our result implies that also spherical Latin bitrades can be generated very efficiently.     

Planar random evolutions governed by hyperbolic equations

Planar random evolutions governed by hyperbolic equations are considered. Two cases are presented in detail: motion with three directions and the model with four perpendicular directions. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part I.     

An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds

It is not known whether every noncompact hyperbolic 3-manifold of finite volume admits a decomposition into ideal tetrahedra. We give a partial solution to this problem: Let be a hyperbolic 3-manifold obtained by identifying the faces of convex ideal polyhedra . If the faces of are glued to , then can be decomposed into ideal tetrahedra by subdividing the 's.

     

On a stochastic hyperbolic integro-differential equation

In this paper we study an initial-boundary-value problem for a hyperbolic integro-differential equation with random memory and a random noise. We establish the existence, uniqueness and exponential stability of solutions. Our method consists of finite-dimensional approximation and energy estimates.     

On the stochastic quasi-linear symmetric hyperbolic system

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly.     

Homogenization of hyperbolic damped stochastic wave equations

For a family of linear hyperbolic damped stochastic wave equations with rapidly oscillating coefficients, we establish the homogenization result by using the sigma-convergence method. This is achieved under an abstract assumption covering special cases like the periodicity, the almost periodicity and some others.     

Optimal control of certain hyperbolic and integral stochastic equations

Using Gateaux differentiation of the quality functional we obtain necessary conditions for optimality of a control for stochastic differential equations of hyperbolic type containing two-parameter white noise and for stochastic integral equations.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 110–116, 1987.     

Investigation of properties of the solutions of hyperbolic stochastic PDEs

In this paper we investigate the compact support property of the solutions of hyperbolic Stochastic PDE (SPDE) providing that initial condition function is deterministic and has compact support property. First, to approach this problem, we consider semi-SPDE. It turns out that in the semi-SPDE case solution u (t, x) preserve compact support property. When we consider SPDE, we use the stochastic differential-difference equations (SDDE) approach. It turns out that in SPDE case solution u (t, x) does not preserve compact support property. So, if we compare the semi-SPDE and SPDE then it becomes obvious that differentiation in space in SPDE plays crucial role and influence the behavior of the solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Maximum likelihood estimate for discontinuous parameter in stochastic hyperbolic systems

The purpose of this paper is to study the identification problem of a spatially varying discontinuous parameter in stochastic hyperbolic equations. In previous works, the consistency property of the maximum likelihood estimate (MLE) was explored and the generating algorithm for MLE proposed under the condition that an unknown parameter is in a sufficiently regular space with respect to spatial variables.In order to show the consistency property of the MLE for a discontinuous coefficient, we use the method of sieves, i.e. the admissible class of unknown parameters is projected into a finite-dimensional space. For hyperbolic systems, we cannot obtain a regularity property of the solution with respect to a parameter. So in this paper, the parabolic regularization technique is used. The convergence of the derived finite-dimensional MLE to the infinite-dimensional MLE is justified under some conditions.     

Clean triangulations

A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS _p (resp.N _p ) be the closed orientable (resp. nonorlentable) surface of genusp. If (S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: (N ₁)=20, (S ₁)=24, lim(S _p )p ^–1=4, lim(N _p )p ^–1=2 forp.     

The method of lines for hyperbolic stochastic functional partial differential equations

We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small L²-perturbations.