Topological Chaos for Differential Inclusions with Multivalued Impulses on Tori

A multivalued version of the Ivanov inequality for the lower estimate of topological entropy of admissible maps is applied to differential inclusions with multivalued impulses on tori via the associated Poincaré translation operators along their trajectories. The topological chaos in the sense of a positive topological entropy is established in terms of the asymptotic Nielsen numbers of the impulsive maps being greater than 1. This condition implies at the same time the existence of subharmonic periodic solutions with infinitely many variety of periods. Under a similar condition, the coexistence of subharmonic periodic solutions of all natural orders is also carried out.

Joint creative process in translation

In this article, we explore socially distributed cognition (SDC) as a theoretical model of translation and investigate it empirically as an aspect of the collaborative and creative translation workflow. With the aim of developing a better understanding of SDC and collaborative workflows in translation, we analyzed two different settings where more than one person works on a translation: commercial specialized translation (CST) services, and the production of audio descriptions (AD) as teamwork between blind and sighted describers. The analysis focuses on how the process of co-creation unfolds in the communication that binds together the systems of SDC. While the process of co-creation was strikingly similar in the two different translation contexts, the differences were bound to channels of communication (with or without direct contact between participants), and the draft translation was identified as a central artifact that carries much of the communication when the participants do not work in the same space. With an emphasis on socially distributed cognition, our study provides a framework for both the cognitive and social aspects of translation and develops the understanding of collaborative translation processes. It also contributes to the development of translation practices by helping translation operators and trainers make choices between alternative workflows.

Chaos for Differential Equations with Multivalued Impulses

The deterministic chaos in the sense of a positive topological entropy is investigated for differential equations with multivalued impulses. Two definitions of topological entropy are examined for three classes of multivalued maps: [Formula: see text]-valued maps, [Formula: see text]-maps and admissible maps in the sense of Górniewicz. The principal tool for its lower estimates and, in particular, its positivity are the Ivanov-type inequalities in terms of the asymptotic Nielsen numbers. The obtained results are then applied to impulsive differential equations via the associated Poincaré translation operators along their trajectories. The main theorems for chaotic differential equations with multivalued impulses are formulated separately on compact subsets of Euclidean spaces and on tori. Several illustrative examples are supplied.

Quantum calculus of Fibonacci divisors and infinite hierarchy of Bosonic–Fermionic Golden quantum oscillators

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd [Formula: see text] describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number [Formula: see text]. In the limit [Formula: see text], Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, [Formula: see text]-matrices, geometry of hydrodynamic images and quantum computations are discussed.

Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).