A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This note deals with one of the most classical quotients, the space of ordered poin...

Neurophysiological evidence is described, showing that some neurons in the macaque temporal cortical visual areas have responses that are invariant with respect to the position, size and view of faces and objects, and that these neurons show rapid processing and rapid learning. A theory is then described of how such invariant representations may be produced in a hierarchically organized set of ...

The Invariant Set Hypothesis proposes that states of physical reality belong to, and are governed by, a non-computable fractal subset I of state space, invariant under the action of some subordinate deterministic causal dynamics I D . The Invariant Set Hypothesis is motivated by key results in nonlinear dynamicalsystems theory, and black-hole thermodynamics. The elements of a reformulation of q...

Given a continuous dynamical system f on a compact metric space X and a continuous potential Φ : X → R, the generalized rotation set is the subset of R consisting of all integrals of Φ with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measuretheoretic entropies over all invariant measures whose integrals pr...

To the memory of Charles C. Conley We generalize Conley's fundamental theorem of dynamical systems in Conley index theory. We also conclude the existence of a regular index filtration for every Morse decomposition. 1. Introduction. Conley is mostly known for his fundamental theorem of dynami-cal systems and his homotopy index theory [1]. In the former, he proved that every continuous flow on a ...

Nearly (or almost) invariant sets for random dynamical systems are subsets of the state space that are left only after long time and, maybe, are visited again after even longer times. The present paper takes up the approach developed in Colonius, Gayer, and Kliemann [8] for Markov diffusions. We develop an analogous theory for random diffeomorphisms and also use results from Zmarrou and Homburg...

The aim of this paper is to propose a mathematical model to determine invariant sets, set covering, orbits and, in particular, attractors in the set of tourism variables. Analysis was carried out based on a pre-designed algorithm and applying our interpretation of chaos theory developed in the context of General Systems Theory. This article sets out the causal relationships associated with tour...

In this paper we summarize known results on claw-free graphs. The paper is subdivided into the following chapters and sections: 1. Introduction 2. Paths, cycles, hamiltonicity a) Preliminaries b) Degree and neighborhood conditions c) Local connectivity conditions d) Further forbidden subgraphs e) Invariants f) Squares g) Regular graphs h) Other hamiltonicity related results and generalizations ...

9. Given an explicit example of a compact, convex set K in `2(Z), containing more than one point, whose extreme points are dense in K. Is such an example possible in Rn? 10. Let X = R2/Z2. Given A ∈ GL2(Z), let fA : X → X be the linear automorphism of the torus induced by A. Find all the ergodic, fA-invariant measures for A = ( 1 1 0 1 ). Show that the set of ergodic probablility measures is no...

A classical theorem of Steinitz [I& p. 1251 states that the characteristic of an algebraically closed field, together with it.s absolute degree of transcendency, uniquely det,ermine the field (up to isomorphism). It is easily seen that the word real-closed cannot be substituted for the words algebraically closed in this theorem. It is therefore natural to inquire what invariants other than the ...