Measure preserving actions of affine semigroups and patterns

Ergodic and combinatorial results obtained in Bergelson and Moreira [Ergodic theorem involving additive and multiplicative groups of a field and$\{x+y,xy\}$patterns.Ergod. Th. & Dynam. Sys.to appear, published online 6 October 2015, doi:10.1017/etds.2015.68], involved measure preserving actions of the affine group of a countable field$K$. In this paper, we develop a new approach, based on ultrafilter limits, which allows one to refine and extend the results obtained in Bergelson and Moreira,op. cit., to a more general situation involving measure preserving actions of thenon-amenableaffine semigroups of a large class of integral domains. (The results and methods in Bergelson and Moreira,op. cit., heavily depend on the amenability of the affine group of a field.) Among other things, we obtain, as a corollary of an ultrafilter ergodic theorem, the following result. Let$K$be a number field and let${\mathcal{O}}_{K}$be the ring of integers of$K$. For any finite partition$K=C_{1}\cup \cdots \cup C_{r}$, there exists$i\in \{1,\ldots ,r\}$such that, for many$x\in K$and many$y\in {\mathcal{O}}_{K}$,$\{x+y,xy\}\subset C_{i}$.

New copulas based on general partitions-of-unity and their applications to risk management

We construct new multivariate copulas on the basis of a generalized infinite partition-of-unity approach. This approach allows, in contrast to finite partition-of-unity copulas, for tail-dependence as well as for asymmetry. A possibility of fitting such copulas to real data from quantitative risk management is also pointed out.

EVOLUTIONARY GAME THEORETICAL MODEL OF THE EVOLUTION OF THE CONCEPT OF HUE, A HUE STRUCTURE, AND COLOR CATEGORIZATION IN NOVICE AND STABLE LEARNERS

Evolutionary game theory is used to form a finite partition of a continuous hue circle in which perceptually similar hues are each represented by an icon chip and the circle by a finite but game dynamically determined number of icon chips. On the basis of such icon chip structures, a color categorization for both an individual learner and a population of learners is then evolved. These results remove limitations of some particular previous color categorization simulation work which assumed a fixed number of color stimuli and a maximal number of predefined color categories. These simulations are extended to demonstrate that learners need neither to share the same icon chip structures, nor do these structures have to be fully developed for a population of learners to produce a stable color categorization system. Additionally, when a naïve learner is introduced into a population with a stable color categorization, the game dynamics result in the learner's adopting the existing categorization. All results are shown to hold while the underlying icon chip structures evolve continuously in response to novel stimuli. The usefulness of the approach as well as some of the potential implications of the results for human learning of color categories are discussed.

Diophantine approximation by orbits of expanding Markov maps

In 1995, Hill and Velani introduced the ‘shrinking targets’ theory. Given a dynamical system ([0,1],T), they investigated the Hausdorff dimension of sets of points whose orbits are close to some fixed point. In this paper, we study the sets of points well approximated by orbits {Tnx}n≥0, where Tis an expanding Markov map with a finite partition supported by [0,1]. The dimensions of these sets are described using the multifractal properties of invariant Gibbs measures.

Modeling and Simulation of a Externally Pressurized Spherical Gas Bearing

Because of many advantages that other kinds of bearings can not match, for example high precision and high speed, gas lubricated bearing has been used widely in areas such as ultra-precision machine tool and inertia apparatus. The paper introduced an orifice restrictor externally pressurized spherical gas bearing, which is easily shaped and has bigger stiffness in both axial and radial directions. The paper also introduced a bearing force calculation model based on finite partition of the bearing, in which the influence of the uniform pressure slot also considered.