On solutions of quasilinear wave equations with nonlinear damping terms

In this paper we consider the existence and asymptotic behavior of solutions of the following problem:

On a Problem of P. Gallagher

Let p be an odd prime. For each integer a with x < a x + u and (a,p) = 1, we define by a 1 (mod p) and 1 p - 1. Let r(p,u,x) be the number of integers with x < a x + u and (a,p) = 1 for which a and are of opposite parity, and let E(n,u,x) = r(n,u,x) - 1/2 1, where denotes summation over all a such that (a,p) = 1. The main purpose of this paper is to prove that for any positive integer 1 u we have the asymptotic formula

Class Numbers of Indefinite Binary Quadratic Forms

Let be the class number of properly equivalent primitive binary quadratic forms of discriminant . The case of indefinite forms is considered. Assuming that the extended Riemann hypothesis for some fields of algebraic numbers holds, the following results are proved. 1. Let be an arbitrarily slow monotonically increasing function such that . Then

Infinite time optimal control and periodicity

For smooth nonlinear systems

Effects Connected with the Coincidence of Velocities in the Two-Velocity Dynamical System

The paper deals with the system

On the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Singular Hyperbolic Equations

In the rectangle D = (0,

The type set for some measures on \mathbb{R}^{2n} with n-dimensional support

Let be realhomogeneous functions in ofdegree and let bethe Borel measure on given by

On Plane Maximal Curves

The number N of rational points on an algebraic curve of genus g over a finite field satisfies the Hasse–Weil bound . A curve that attains this bound is called maximal. With and , it is known that maximalcurves have . Maximal curves with have been characterized up to isomorphism. A natural genus to be studied is and for this genus there are two non-isomorphic maximal curves known when . Here, a maximal curve with genus g ₂ and a non-singular plane model is characterized as a Fermat curve of degree .     

Intersections of Convex Bodies

Let be nonempty convex bodies in . Let be vectors in , let , and let . Then is a convex set, and the family of sets is concave. Let . Then for the mean cross-sectional measures W_v (\Phi (\rho )), , the functions are concave on D. (Note that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca% WGxbWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabfA6agjaacIcacqaH% bpGCcaGGPaGaaiykaiabg2da9iaabAfacaqGVbGaaeiBamaaBaaale% aatCvAUfKttLearyqr1ngBPrgaiuGacqWFRbWAaeqaaOGaeuOPdyKa% aiikaiabeg8aYjaacMcaaaa!4EE7!\[W_0 (\Phi (\rho )) = {\text{Vol}}_k\Phi (\rho )\] is the k-volume.) Bibliography: 2 titles.     

When all trajectories in the Liénard plane cross the vertical isocline?

In this paper we study the problem whether all trajectories of the system =y–F(x) and =–g(x) cross the vertical isocline which is very important for the existence of periodic solutions and oscillation theory. The problem has not been solved for the critical case:

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

On the Lax Solution to the Hamilton–Jacobi Equation and Its Generalizations. Part II

This article is a continuation of [J. Math. Sci., 99, No.5, 1541–1547 (2000)] devoted to the validity of the Lax formula (cited in the article of Crandall, Ishii, and Lions [Bull. AMS, 27, No.1, 1–67 (2000)])

Adaptive Procedures for Numerical Methods

A generalized interpolation polynomial with a base function and node coefficients , is a polynomial of the form

Asymptotic Formulas for the Solutions of Integro-Differential Equations

We study the problem of asymptotic integration of the linear integro-differential equation

A fourth order method for a singular two-point boundary value problem

Recently, Chawla et al. described a second order finite difference method for the class of singular two-point boundary value problems:

The Multidimensional Weyl Theorem and Covering Families

The well-known theorem of Weyl about the essential self-adjointness of the Sturm--Liouville operator in with , and is generalized to second-order elliptic operators in . The multidimensional Weyl theorem is derived from a more general theorem; to state and prove the latter, a special covering family is constructed. The results obtained imply the known multidimensional analogs of the Weyl theorem and, unlike these analogs, apply to open proper subsets in .     

On Sparse Parity Check Matrices

We consider the extremal problem to determine the maximal number of columns of a 0-1 matrix with rows and at most ones in each column such that each columns are linearly independent modulo . For fixed integers and , we shall prove the probabilistic lower bound = ; for a power of , we prove the upper bound which matches the lower bound for infinitely many values of . We give some explicit constructions.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

On some properties of the metric subalgebras ofℓ ∞

The objects studied are the subalgebras of which contain c_o. These are isometrically isomorphic to the algebras C( ) where is a compactification of a discrete denumerable set N . It is shown: 1) If is metric then there is a projection of norm 1, P: C( ) C( ) with kernel c_o defined by PF = f o where is a retraction of onto = – N . 2) If is metric, then the group of homeomorphisms of is isomorphic to a complete group of permutations of the natural numbers . 3) The group of homeomorphisms of a compact metric space is the homomorphic image of a complete group of permutations of ("complete" means "no outer automorphisms, trivial center").