Lifting a circular membrane by unitary forces

Let Ω be a convex membrane. We lift certain parts Γ of its boundary by means of unitary forces while the remaining parts are maintained at level 0. Call u[Γ] the position that the such lifted membrane assumes. When the parts Γ are varying on ∂Ω so that their total lenght C is preserved, it has been conjectured that the functional Γ ||u(Γ)||p attain its maximum value for a certain conected arc of lenght C. In this paper we present a proof of this conjecture for the case in which Ω is a circle and p = 1

Engranajes y mejores aproximaciones racionales

Motivado por la construcción de sistemas de ruedas dentadas con determinada relación de transmisión, se discute el problema de aproximar un número real E por una fracción racional P/Q de manera que el error de aproximación sea menor que el que resultaría de tomar cualquier otra fracción con denominador más pequeño

Generalized Cauchy means

Given two means M and N, the operator MM,NMM,N assigning to a given mean μ the mean MM,N(μ)(x,y)=M(μ(x,N(x,y)),μ(N(x,y),y)) was defined in Berrone and Moro (Aequationes Math 60:1–14, 2000) in connection with Cauchy means: the Cauchy mean generated by the pair f, g of continuous and strictly monotonic functions is the unique solution μ to the fixed point equation MA(f),A(g)(μ)=μ, where A(f) and A(g) are the quasiarithmetic means respectively generated by f and g. In this article, the operator MM,NMM,N is studied under less restrictive conditions and a general fixed point theorem is derived from an explicit formula for the iterates MnM,NMM,Nn . The concept of class of generalized Cauchy means associated to a given family of mixing pairs of means is introduced and some distinguished families of pairs are presented. The question of equality in these classes of means remains a challenging open problem.Fil: Berrone, Lucio Renato. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Coalescence of measures and f-rearrangements of a function

This paper addresses the question of characterizing optimum values in the problem sup{n(E) : m(E)£C}, {1} where m and n are measures defined on a s-finite measurable space X. With this purpose, the f-rearrangement of a function g is introduced so as to formalize the idea of rearranging the level sets of the function g according to how these sets are arranged in a given function f. A characterization of optima of problem (1) is then obtained in terms of dn/dm-rearrangements, dn/dm being the Radon-Nikodym derivative of the measure n with respect to m. When X is a topological space and m, n are Borel measures, we say that n is coalescent with respect to m when, for every C>0, there exist connected optima solving problem (1). A general criterion for coalescence is given and some simple examples are discussed

Invariance of the Cauchy mean-value expression with an application to the problem of representation of Cauchy means

The notion of invariance under transformations (changes of coordinates) of the Cauchy mean-value expression is introduced and then used in furnishing a suitable two-variable version of a result by L. Losonczi on equality of many-variable Cauchy means. An assessment of the methods used by Losonczi and Matkowski is made and an alternative way is proposed to solve the problem of representation of two-variable Cauchy means

On the number of finite topological spaces

In this paper we deal with the problem of enumerating the finite topological spaces, studying the enumeration of a restrictive class of them. By employing simple techniques, we obtain a recursive lower bound for the number of topological spaces on a set of n elements. Besides we prove some collateral results, among which we can bring a new proof (Cor. 1.5) of the fact that p(n) – the number of partitions of the integer n – is the number of non-isomorphic Boolean algebras on a set of n elements