The simultaneous conjugacy problem in groups of piecewise linear functions

Guba and Sapir asked, in their joint paper [8], if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit. The techniques we develop allow us also to solve the ordinary conjugacy problem as well, and we can compute roots and centralizers. Moreover, these techniques can be generalized to solve the same questions in larger groups of piecewise-linear homeomorphisms.

A public key encryption based on third order linear sequences

Based on third order linear sequences, an improvement version of the Diffie-Hellman distribution key scheme and the ElGamal public key cryptosystem scheme are proposed, together with an implementation and computational cost. The security relies on the difficulty of factoring an RSA integer and on the difficulty of computing the discrete logarithm.

Conjugacy of piecewise linear Lorenz map that expand on average

For piecewise linear Lorenz map that expand on average, we show that it admits a dichotomy: it is either periodic renormalizable or prime. As a result, such a map is conjugate to a ß-transformation.

Quasiconformal maps, analytic capacity, and non linear potentials

"Vegeu el resum a l'inici del document del fitxer adjunt."

Near-linear dynamics in KdV with periodic boundary conditions

Near linear evolution in Korteweg de Vries (KdV) equation with periodic boundary conditions is established under the assumption of high frequency initial data. This result is obtained by the method of normal form reduction.

The non-linear Dirichlet problem in Hadamard manifolds

We prove existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two iterations of the Perron method. The a-priori estimates used in the continuity method are valid in any ambient manifold.

Geometry and physics of knots

KNOTS are usually categorized in terms of topological properties that are invariant under changes in a knot's spatial configuration(1-4). Here we approach knot identification from a different angle, by considering the properties of particular geometrical forms which we define as 'ideal'. For a knot with a given topology and assembled from a tube of uniform diameter, the ideal form is the geometrical configuration having the highest ratio of volume to surface area. Practically, this is equivalent to determining the shortest piece of tube that can be closed to form the knot. Because the notion of an ideal form is independent of absolute spatial scale, the length-to-diameter ratio of a tube providing an ideal representation is constant, irrespective of the tube's actual dimensions. We report the results of computer simulations which show that these ideal representations of knots have surprisingly simple geometrical properties. In particular, there is a simple linear relationship between the length-to-diameter ratio and the crossing number-the number of intersections in a two-dimensional projection of the knot averaged over all directions. We have also found that the average shape of knotted polymeric chains in thermal equilibrium is closely related to the ideal representation of the corresponding knot type. Our observations provide a link between ideal geometrical objects and the behaviour of seemingly disordered systems, and allow the prediction of properties of knotted polymers such as their electrophoretic mobility(5).

A critical analysis of dimensions and curve fitting practice in economics

When dealing with sustainability we are concerned with the biophysical as well as the monetary aspects of economic and ecological interactions. This multidimensional approach requires that special attention is given to dimensional issues in relation to curve fitting practice in economics. Unfortunately, many empirical and theoretical studies in economics, as well as in ecological economics, apply dimensional numbers in exponential or logarithmic functions. We show that it is an analytical error to put a dimensional unit x into exponential functions ( a x ) and logarithmic functions ( x a log ). Secondly, we investigate the conditions of data sets under which a particular logarithmic specification is superior to the usual regression specification. This analysis shows that logarithmic specification superiority in terms of least square norm is heavily dependent on the available data set. The last section deals with economists’ “curve fitting fetishism”. We propose that a distinction be made between curve fitting over past observations and the development of a theoretical or empirical law capable of maintaining its fitting power for any future observations. Finally we conclude this paper with several epistemological issues in relation to dimensions and curve fitting practice in economics

Directional discrepancy in two dimensions

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small Minkowski dimension. In each of these cases, extensions of a lemma due to Davenport allow us to construct appropriate rotations of the integer lattice which yield small discrepancy.

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.

Reconsideration of dimensions and curve fitting practice in economics ellaborating on Georgescu-Roegen’s economic methodology

This paper is to examine the proper use of dimensions and curve fitting practices elaborating on Georgescu-Roegen’s economic methodology in relation to the three main concerns of his epistemological orientation. Section 2 introduces two critical issues in relation to dimensions and curve fitting practices in economics in view of Georgescu-Roegen’s economic methodology. Section 3 deals with the logarithmic function (ln z) and shows that z must be a dimensionless pure number, otherwise it is nonsensical. Several unfortunate examples of this analytical error are presented including macroeconomic data analysis conducted by a representative figure in this field. Section 4 deals with the standard Cobb-Douglas function. It is shown that the operational meaning cannot be obtained for capital or labor within the Cobb-Douglas function. Section 4 also deals with economists "curve fitting fetishism". Section 5 concludes thispaper with several epistemological issues in relation to dimensions and curve fitting practices in economics.