Edge vibration of a pre-stressed semi-infinite strip with traction-free edge and mixed face boundary conditions

Edge-vibration, and associated resonance phenomena, is investigated in respect of a semi-infinite strip composed of pre-stressed incompressible elastic material. The strip is assumed to have a traction free outer edge, with the upper and lower edges subject to some simple mixed boundary conditions. The frequency of the modes of free edge-vibration are shown to depend on the surface wave speed. Moreover, when the normal pre-stress approaches one of two critical values, associated with the vanishing of the surface wave speed, the edge spectrum density of the boundary value problem increases significantly. This problem then provides an example for which the famous Weyls hypothesis, stating the edge spectrum is secondary in comparison with the whole bodys spectrum, is not true. However, the corresponding theorems statement is valid only with imposition of the Shapiro-Lopatinsky condition, which is not satisfied in this case. Variation of the pre-stress is also shown to greatly influence the resonance frequency arising in the forced vibration problem, to the extent that the phenomenon of resonance may be totally removed.     

Brownian motion with killing and reflection and the ‘‘hot–spots’’ problem

We investigate the hot–spots property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the hot–spots property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other.Supported in part by NSF Grant # 9700585-DMSSupported in part by NSF Grant # 0203961-DMS     

The Hodge Filtration and Cyclic Homology

We relate the Hodge filtration of the cohomology of a complex algebraic variety X to the Hodge decomposition of its cyclic homology. If X is smooth and projective, is the quotient of the Betti cohomology by the piece of the Hodge filtration.     

Some applications of the maximum principle to a variety of fully nonlinear elliptic PDE’s

The authors derive best possible maximum principles for some combinations of solutions of a class of fully nonlinear elliptic PDEs and their gradients. These maximum principles are then applied to establish various inequalities of interest in the theory of Weingarten surfaces.     

Line Hypergraphs: A Survey

The survey is devoted to line graphs and a new multivalued function called the line hypergraph. This function generalizes two classical concepts at once, namely the line graph and the dual hypergraph. In a certain sense, line graphs and dual hypergraphs are extreme values of the function . There are many publications about line graphs, but our considerations are restricted to papers concerning Krausz global characterization of line graphs or Whitneys theorem on edge isomorphisms. The survey covers almost all known results on the function because they are concentrated around Krausz and Whitneys theorems. These results provide evidence that the notion of the line hypergraph is quite natural. It enables one to unify the classical theorems on line graphs and to obtain their more general versions in a simpler way.     

An Edge Point Gibbs Phenomenon for Fourier Series in Two Dimensions

Let f be a function periodic in its two variables x, y with period 2 and having at the origin an edge point jump discontinuity (its graph may be roughly described by a cliff with a smooth but possibly inclined rim). The Fourier series of f exhibits a Gibbs phenomenon as in the one-dimensional case but with an affine horizontal deformation depending on the compass direction of the rim at the origin.     

Local Lipschitz property of the perturbation function in infinite-dimensional nonconvex extremal problems with operator constraints

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, Sbornik Trudov VNIISI, No. 14, pp. 52–58, 1990.     

Necessary conditions of parametric extremum for nash equilibrium in mixed strategies in differential games

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, No. 19, pp. 33–35, Vsesoyuznyi Nauchno-Issledovatel'skii Institut Sistemnykh Issledovanii, Moscow (1988).     

‘Outside’ as a primitive notion in constructive projective geometry

In intuitionistic (or constructive) geometry there are positive counterparts, apart and outside, of the relations = and incident. In this paper it is shown that the relation outside suffices to define incident, apart and equality. The equivalence of the new system with Heyting's system is shown and as a simple corollary one obtains duality for intuitionistic projective geometry.     

On the uniqueness of the solution of the inverse exact interpolation problem

Kazan Institute for Chemical Technology and Kazan University, Research Institute of Mathematics and Mechanics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 2, pp. 67–68, April–June, 1992.     

Invariant Subspaces for Positive Operators Acting on a Banach Space with Markushevich Basis

We introduce weak quasinilpotence for operators. Then, by substituting Markushevich basis and weak quasinilpotence at a nonzero vector for Schauder basis and quasinilpotence at a nonzero vector, respectively, we answer a question on the invariant subspaces of positive operators in [3].     

Conformal spaces

A conformal space is a non-singular metric vector space to which has been adjoined a null-cone of points at infinity. We define a conformal space in terms of a higher dimensional coordinate space, and then state and prove a fundamental theorem of conformal geometry.     

Pairs of Convex Bodies with Centrally Symmetric Intersections of Translates

For a pair of convex bodies $K$ and $K$ in $E^d$, the $d$-dimensional intersections $K \cap (x + K),$ $x \in E^d,$ are centrally symmetric if and only if $K$ and $K$ are represented as direct sums $K = R \oplus P$ and $K = R \oplus P$ such that: (i) $R$ is a compact convex set of some dimension $m$, $0 \le m \le d,$ and $R = z - R$ for a suitable vector $z \in E^d$, (ii) $P$ and $P$ are isothetic parallelotopes, both of dimension $d-m$.     

Investigation of some parametric convex programming problems

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, No. 19, pp. 20–29, Vsesoyuznyi Nauchno-Issledovatel'skii Institut Sistemnykh Issledovanii, Moscow (1988).     

Branching of extremals of some variational problems with phase constraints

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, No. 19, pp. 54–62, Vsesoyuznyi Nauchno-Issledovatel'skii Institut Sistemnykh Issledovanii, Moscow (1988).     

Imperfect and Nonideal Clutters: A Common Approach

We prove three theorems. First, Lovászs theorem about minimal imperfect clutters, including also Padbergs corollaries. Second, Lehmans result on minimal nonideal clutters. Third, a common generalization of these two. The endeavor of working out a common denominator for Lovászs and Lehmans theorems leads, besides the common generalization, to a better understanding and simple polyhedral proofs of both.* Visiting of the French Ministry of Research and Technology, laboratoire LEIBNIZ, Grenoble, November 1995—April 1996.     

L p bounds for a maximal dyadic sum operator

The authors prove L ^p bounds in the range 1<p< for a maximal dyadic sum operator on R ⁿ . This maximal operator provides a discrete multidimensional model of Carlesons operator. Its boundedness is obtained by a simple twist of the proof of Carlesons theorem given by Lacey and Thiele [7] adapted in higher dimensions [9]. In dimension one, the L ^p boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunts extension [4] of Carlesons theorem on almost everywhere convergence of Fourier integrals. Mathematics Subject Classification (2000):Primary 42A20, Secondary 42A24Grafakos is supported by the NSF. Tao is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.     

On the Singular Spectrum for Adiabatic Quasi-Periodic Schrödinger Operators on the Real Line

In this paper, we study spectral properties of a family of quasi-periodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular.Résumé. Cet article est consacré à létude du spectre dune famille dopérateurs de Schrödinger quasi-périodiques sur laxe réel lorsque les courbes iso-énergétiques adiabatiques sont non bornées dans la direction des moments. Dans des intervalles dénergies où cette propriété est vérifiée, nous obtenons une formule asymptotique pour lexposant de Lyapounoff, et nous démontrons que le spectre est purement singulier.Communicated by Bernard Helffersubmitted 17/06/03, accepted 05/03/04