Life-span of classical solutions to hyperbolic geometry flow equation in several space dimensions

In this article, we investigate the lower bound of life-span of classical solutions of the hyperbolic geometry flow equations in several space dimensions with “small” initial data. We first present some estimates on solutions of linear wave equations in several space variables. Then, we derive a lower bound of the life-span of the classical solutions to the equations with “small” initial data.     

Common complements of two subspaces of a Hilbert space

In this paper we find a necessary and sufficient condition for two closed subspaces, and , of a Hilbert space to have a common complement, i.e. a subspace having trivial intersection with and and such that .Unlike the finite-dimensional case the condition is significantly more subtle than simple equalities of dimensions and codimensions, and non-trivial examples of subspaces without a common complement are possible.     

The classical limit of the relativistic Vlasov–Maxwell system in two space dimensions

The motion of a collisionless plasma is modelled by the Vlasov–Maxwell system. In this paper, solutions of the relativistic Vlasov–Maxwell system are considered in two space dimensions. The speed of light, c, appears as a parameter in the system. With representations of the electric and magnetic fields, conditions are established under which solutions of the relativistic Vlasov–Maxwell system converge pointwise to solutions of the non‐relativistic Vlasov–Poisson system as c tends to infinity, at the asymptotic rate of 1/c. Copyright © 2004 John Wiley & Sons, Ltd.     

On hyperinvariant subspaces of contraction operators on a Banach space whose spectrum contains the unit circle

In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace. This research is supported by the Natural Science Foundation of P. R. China (No. 10771039)     

The cyclic structure of maximal tori of the finite classical groups

Maximal tori of all finite simple classical groups, as well as of special and general projective linear and unitary groups, are treated. For every such torus, its expression as a direct sum of cyclic groups is obtained in an explicit form. Supported by RFBR grant Nos. 05-01-00797 and 06-01-39001, and by SB RAS Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 129–156, March–April, 2007.     

Lower bounds of the life span of classical solutions to a system of semilinear wave equations in two space dimensions

In this paper we consider a system of semilinear wave equations in two space dimensions and with propagation speeds possibly different from one. Under smallness assumptions on the data, we show lower bounds for the life span of classical solutions.     

Global classical solutions to a kind of quasilinear hyperbolic systems in several space variables

For a kind of quasilinear hyperbolic systems in several space variables whose coefficient matrices commute each other, by means of normalized coordinates, formulas of wave decomposition and pointwise decay estimates, the global existence of classical solution to the Cauchy problem for small and decaying initial data is obtained, under hypotheses of weak linear degeneracy and weakly strict hyperbolicity.     

The affine deformation space of a rank two Schottky group: a picture gallery

The Margulis invariant of an affine hyperbolic element measures the signed Lorentzian displacement along the unique closed geodesic in its class. Given a group of affine transformations of Minkowski spacetime whose linear part is Schottky, the Margulis invariant is a useful tool in determining properness of its action. This paper, based on a presentation given at CGG IV Oostende 2005, describes the affine deformation space of a rank two Schottky group. Several illustrations are included, contrasting the difference between the case of a three-holed sphere and that of a one-holed torus. Paper based on a presentation given at CGG IV Oostende 2005. Attendance of the conference and the subsequent writing of this paper was made possible by an NSERC Discovery Grant.     

On a free group of transformations defined by an automaton

We prove that three automorphisms of the rooted binary tree defined by a certain 3-state automaton generate a free non-Abelian group of rank 3. Both authors are supported by the NSF grants DMS-0308985 and DMS-0456185. Yaroslav Vorobets is supported by a Clay Research Scholarship.     

The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative

Hopscotch, a fast finite difference technique, is used to solve parabolic and elliptic equations in two space dimensions with a mixed derivative. The method is compared numerically with existing alternating direction implicit (A.D.I.) and locally one dimensional (L.O.D.) methods for simple problems.Douglas and Gunn's A.D.I. method is both simplified and improved by reformulating it as a hopscotch method.     

The asymptotic behaviour of solutions for a class of quasilinear wave equations with cubic nonlinearity in two space dimensions

For a class of quasilinear wave equations with small initial data, first we give the lower bound of lifespan of classical solutions, then we discuss the long time asymptotic behaviour of solutions away from the blowup time. This project is supported by the Tianyuan Foundation of China and Laburay of Mathematics for Nonlinear Problems, Fudan University.     

The fundamental group of a compact metric space

We give a forcing free proof of a conjecture of Mycielski that the fundamental group of a connected locally connected compact metric space is either finitely generated or has the power of the continuum.

     

The nearest neighbor random walk on subspaces of a vector space and rate of convergence

LetX be the collection ofk-dimensional subspaces of ann-dimensional vector spaceV _n overGF(q). A metric may be defined onX by letting

Some analytical and numerical investigation of a family of fractional-order Helmholtz equations in two space dimensions

In this article, we aim at solving a family of two-dimensional fractional-order Helmholtz equations by using the Laplace-Adomian Decomposition Method (LADM). The fractional-order derivatives, which we use in this investigation, follows the Liouville-Caputo definition. Our results based upon the LADM are obtained in series form that helps us in analyzing the analytical solutions of the fractional-order Helmholtz equations considered here. For illustration and verification of the analytical procedure using the LADM, several numerical examples and graphical representations are presented for the analytical solution of the fractional-order Helmholtz equations. The mathematical analytic procedure, which we have used here, has shown that the LADM is a fairly accurate and computable method for the solution of problems involving fractional-order Helmholtz equations in two dimensions. In an analogous manner, one can apply the LADM for finding the analytical solution of other classes of fractional-order partial differential equations.     

Global well‐posedness of a damped Schrödinger equation in two space dimensions

We investigate the initial value problem for a semilinear damped Schrödinger equation with exponential growth nonlinearity in two space dimensions. We obtain global well‐posedness in the energy space. Copyright © 2013 John Wiley & Sons, Ltd.