On absolute continuity of the spectrum of periodic Schrödinger operators

In this paper we find a new condition on a real periodic potential for which the self-adjoint Schrödinger operator may be defined by a quadratic form and the spectrum of the operator is purely absolutely continuous. This is based on resolvent estimates and spectral projection estimates in weighted \(L^2\) spaces on the torus, and an oscillatory integral theorem.     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

On the positive radial solutions of some semilinear elliptic equations on ℝ n

We study the existence and behavior of positive radial solutions of the equationu + f(u) = 0 in ⁿ . This equation arises in various problems in applied mathematics, e.g. in the study of phase transitions, nuclear cores and more recently in population genetics and solitary waves. The important model casef(u) = – u + u ^p, p>1, describes for instance the pressure distribution in a van der Waals fluid. In this case, we obtain fairly complete knowledge of all positive radial solutions.Supported in part by an NSF grant and a research grant from the Graduate School of the University of Minnesota.     

On the relationship between Semi-Lagrangian and Lagrange–Galerkin schemes

Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange–Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange–Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange–Galerkin schemes represent a “small” (more precisely, an $O(\Delta t$ )) perturbation of exact Lagrange–Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.     

On the representation of biharmonic functions with singularities in ℝ n

Biharmonic functions are defined on Euclidean spaces, Riemannian manifolds, infinite trees, and more generally on abstract harmonic spaces. In this note, we consider biharmonic functions b defined on annular sets Ω \ K and obtain Laurent-type decompositions for b in the Euclidean spaces and in infinite trees. Particular importance is given to the investigation when b extends as a distribution on Ω.     

About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ n|n

In this article, we compute the first space of cohomology of Vect (? ^n|n ), the Lie superalgebra of vector fields on the supermanifold ? ^n|n with coefficients in 𝒻 (? ^n|n ), the space of smooth functions on ? ^n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2 Fuchs , D. B. ( 1986 ). Cohomology of Infinite Dimensional Lie Algebras . New York : Consultants Bureau . [Google Scholar]]. This work allows us to classify the deformations of the action of Vect(? ^n|n ) on 𝒻 (? ^n|n ).     

On the structure of incoming and outgoing subspaces for a wave equations in ℝ n

We investigate the structure of incoming and outgoing subspaces in the Lax-Phillips scheme for the classic wave equation in ℝ ⁿ . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 708–712, May, 1999.     

A uniform estimate for fourier transforms of measures on polynomial curves in ℝ n

We prove a uniform estimate for the decrease of the fourier transforms of smooth measures on polynomial curves in ⁿ . Our estimate improves the estimate recently obtained byW. Stadje.     

On a relationship between solutions of the third Painlevé equation

The direct and inverse Bäcklund transformations for the third Painlevé equation in the case O is used to obtain a nonlinear functional relationship connecting the solutions of this equation for different values of the parameters that occur in it.Belarus State University of Information Technology and Electronics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 364–366, March, 1995.     

Further results on the relationship between μ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains

This note considers continuous-time Markov chains whose state space consists of an irreducible class, C, and an absorbing state which is accessible from C. The purpose is to provide results on μ-invariant and μ-subinvariant measures where absorption occurs with probability less than one. In particular, the well-known premise that the μ-invariant measure, m, for the transition rates be finite is replaced by the more natural premise that m be finite with respect to the absorption probabilities. The relationship between μ-invariant measures and quasi-stationary distributions is discussed.     

Hamiltonization and Integrability of the Chaplygin Sphere in ℝ n

This paper studies a natural n-dimensional generalization of the classical nonholonomic Chaplygin sphere problem. We prove that for a specific choice of the inertia operator, the restriction of the generalized problem onto a zero value of the SO(n−1)-momentum mapping becomes an integrable Hamiltonian system after an appropriate time reparametrization.     

On the Geometry of Similarly Homogeneous ℝ-Trees

Russian Mathematics - We study the geometry of locally complete similarly homogeneous ?-trees. We prove the existence theorem for the introduced earlier class of vertical ?-trees which...     

On subsemigroups of βℕ and absolute coretracts

A semigroup S is called an absolute coretract if for any continuous homomorphism f from a compact Hausdorff right topological semigroup T onto a compact Hausdorff right topological semigroup containing S algebraically there exists a homomorphism g \colon S→ T such that f\circ g=id _S . The semigroup β\ben contains isomorphic copies of any countable absolute coretract. In this article we define a class C of semigroups of idempotents each of which is a decreasing chain of rectangular semigroups. It is proved that every semigroup from C is an absolute coretract and every finite semigroup of idempotents, which is an absolute coretract, belongs to C . July 25, 2000     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Extreme norms on ℝ n

LetN ¹,N be thel ¹ andl norms on ⁿ . We denote by the set of all normsN on ⁿ such thatN NN ¹. The aim of the paper is to present a characterization of the extreme points of .This paper has been written while the author was a research fellow of the Alexander von Humboldt-Stiftung at the Mathematisches Institut der Eberhardt-Karls-Universität, Tübingen.     

On Young hulls of convex curves in ℝ2n

For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls) describe their structure and give a formulas for the volume of the biggest of these domains.To our Teacher Vladimir Igorevich Arnold on the occasion of his 60-th birthday