On bodies with directly congruent projections and sections

Let K and L be two convex bodies in R⁴, such that their projections onto all 3-dimensional subspaces are directly congruent. We prove that if the set of diameters of the bodies satisfies an additional condition and some projections do not have certain π-symmetries, then K and L coincide up to translation and an orthogonal transformation. We also show that an analogous statement holds for sections of star bodies, and prove the n-dimensional versions of these results.     

Evolving a predator–prey ecosystem of mathematical expressions with grammatical evolution

This article describes the use of grammatical evolution to obtain a predator–prey ecosystem of artificial beings associated with mathematical functions, whose fitness is also defined mathematically. The system supports the simultaneous evolution of several ecological niches and through the use of standard measurements, makes it possible to explore the influence of the number of niches and the values of several parameters on “biological” diversity and similar functions. Sensitivity analysis tests have been made to find the effect of assigning different constant values to the genetic parameters that rule the evolution of the system and the predator–prey interaction or of replacing them by functions of time. One of the parameters (predator efficiency) was found to have a critical range, outside which the ecologies are unstable; two others (genetic shortening rate and predator–prey fitness comparison logistic amplitude) are critical just at one side of the range), the others are not critical. The system seems quite robust, even when one or more parameters are made variable during a single experiment, without leaving their critical ranges. Our results also suggest that some of the features of biological evolution depend more on the genetic substrate and natural selection than on the actual phenotypic expression of that substrate. © 2014 Wiley Periodicals, Inc. Complexity 20: 66–83, 2015     

An extension of a result by Lonke to intersection bodies

In this paper we prove that intersection bodies cannot be direct sums using Fourier analytic techniques. This extends a result by Lonke. We also prove a necessary regularity condition and a convexity condition for a body of revolution to be an intersection body of a star body.     

Strong Type Inequalities and an Almost-Orthogonality Principle for Families of Maximal Operators Along Directions in R2

The paper proves an almost-orthogonality principle for maximaloperators over arbitrary sets of directions in R². Namely, theL^p-bounds for an operator of this type are obtained from thecorresponding L^p-bounds of the maximal functions associatedto a certain partition of the set of directions, and from theparticular structure of this partition. Applications to severaltypes of maximal operators are provided.