Use of the Domain Derivative to Prove Symmetry Results in Partial Differential Equations

In this paper we present a different method for proving symmetry results in partial differential equations. The principal idea is to use the classical tool of shape optimization, namely, the differentiation with respect to the domain. The aim is to derive, from an optimality condition, information on the boundary of the domain.     

A construction of commutative nilsemigroups

All commutative nilsemigroups are constructed from congruences on free commutative semigroups.     

Nonlinear vibrations of loudspeaker-like structures

During high amplitudes of vibration, nonlinearities affect the electroacoustical behavior of electrodynamic transducers and are responsible for audible distortions. We distinguish two types of nonlinearities: electrical and mechanical. In this study, attention is paid to the mechanical and geometrical properties of loudspeaker-like structures. The loudspeaker is viewed as a combination of an annular plate with a circular plate. Nonlinear vibrations of such a structure are investigated, using the dynamic analog of the Von-Kármán equations. Furthermore, the influence of both material properties and geometrical parameters is studied. It is shown that nonlinear effects can be substantially reduced by choosing appropriate material and geometrical parameters.     

Sparse geometric graphs with small dilation

Given a set S of n points in , and an integer k such that 0k<n, we show that a geometric graph with vertex set S, at most n−1+k edges, maximum degree five, and dilation O(n/(k+1)) can be computed in time O(nlogn). For any k, we also construct planar n-point sets for which any geometric graph with n−1+k edges has dilation Ω(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.     

Partial inner product spaces. II. Operators

We study linear operators between nondegenerate partial inner product spaces and their relationships to selfadjoint operators in a “middle” Hilbert space.     

Equal signs additive sequences in Banach spaces

A sequence (e_n) spanning a Banach space E is called ESA or equal signs additive if the norm of a linear combination of the e_i's does not change when adjacent coefficients of equal sign are combined. Call the sequence (e_n) regular if neither E nor its dual contain an isomorphic copy of c₀. It is shown that a regular ESA sequence is a boundedly complete and 1-shrinking basis for its span which is thus quasi-reflexive. It is further possible to replace a regular ESA norm by an equivalent ESA norm rendering E isometrically isomorphic to its second dual. A sequence (e_n) is called IS or invariant under spreading if the norm of a linear combination of the e_i's does not change when the mutual distances of the terms in the sequence (but not their relative positions) change. We give a simple construction of an unconditional norm for an IS sequence, hence, in particular, for an ESA sequence. Also, an inverse construction is obtained: We prove that each unconditional IS basis gives rise to an ESA basis by means of an inversion formula; to nonequivalent IS unconditional bases correspond nonequivalent ESA bases. It follows that nonisomorphic ESA bases are plentiful.     

Computing finite commutative semigroups

Precedence results are used to improve existing algorithms for the enumeration of finite commutative semigroups. As an application 11,545,843 distinct commutative semigroups of order 9 were found.