Binary Operations in Metric Spaces Satisfying Side Inequalities

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

Dynamical behavior of recurrent neural networks with different external inputs

The main aim of this paper is to study the dynamics of a recurrent neural networks with different input currents in terms of asymptotic point. Under certain circumstances, we studied the existence, the uniqueness of bounded solutions and their homoclinic and heteroclinic motions of the considered system with rectangular currents input. Moreover, we studied the unpredictable behavior of the continuous high-order recurrent neural networks and the discrete high-order recurrent neural networks. Our method was primarily based on Banach’s fixed-point theorem, topology of uniform convergence on compact sets and Gronwall inequality. For the demonstration of theoretical results, we give examples and their numerical simulations.

A non-continuous soft mapping that preserves some structural soft sets

As everyday problems contain a lot of data and ambiguity, it has become necessary to develop new mathematical approaches to address them and soft set theory is the best tool to deal with such problems. Hence, in this article, we introduce a non-continuous mapping in soft settings called soft U -continuous. We mainly focus on studying soft U -continuity and its connection to soft continuity. We further show that soft U -continuity preserves soft compact sets and soft connected sets. The later sets have various applications in computing science and decision making theory. In the end, we show that if each soft U -continuous mapping f from a soft space X into a soft T 0-space Y is soft continuous, then Y is soft T 1.

Metric Spaces

This chapter introduces and illustrates the concept of a metric (distance measure), and the definition of a metric space. Open, closed, and compact sets are discussed in a general context, and the concepts of separability and completeness introduced. It goes on to look at mappings on metric spaces, examines the important case of function spaces, and treats the Arzelà–Ascoli theorem.

Mixed-Integer Convex Representability

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first nonrepresentability results for various nonconvex sets, such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties, such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets.