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In this paper, multistability is discussed for delayed recurrent neural networks with ring structure and multi-step piecewise linear activation functions. Sufficient criteria are obtained to check the existence of multiple equilibria. A lemma is proposed to explore the number and the cross-direction of purely imaginary roots for the characteristic equation, which corresponds to the neural network model. Stability of all of equilibria is investigated. The work improves and extends the existing stability results in the literature. Finally, two examples are given to illustrate the effectiveness of the obtained results.

Medical records generated in hospitals often contain private and sensitive information. This privileged information must be prevented from falling into wrong hands. Thus, there is a strong need for developing a secure cryptographic scheme that can be adapted to use in conjunction with transmission and storage of medical information. Previous approaches have proposed the use of the advanced encryption standard (AES) algorithm for this purpose. In this article, we are proposing a new robust, high-speed, and secure cryptographic scheme that has the added advantage of being immune to side-channel attacks. In our article, we have shown that the performance of this scheme is superior in certain aspects to that of the A5/1 system used in global system for mobile (GSM) systems. The parallel architecture employed in this scheme makes it suitable to use in systems where the data-processing operations have to be carried out in real time.

Residue number systems (RNS) based on Chinese remainder theorem (CRT) permits the representation of large integers in terms of combinations of smaller ones. The set of all CRT number system representation of an integer from 0 to M-1 with component wise modular addition and multiplication constitutes a direct sum of smaller commutative rings. An encryption and decryption algorithm based on the properties of direct sum of smaller rings offers distinct advantages over decimal or fixed radix arithmetic. We have utilized the representation of integers using CRT to successfully design additive, multiplicative, and affine stream cipher systems. The use of number system based on CRT allows speeding up the encryption/decryption algorithms, reduces the time complexity, and provides immunity to side-channel, algebraic, and known plain text attacks. In this article, the characteristics of additive, multiplicative, and affine stream cipher systems based on CRT number system representation have been studied and analyzed.

We propose a new theory to explain the formation of spiral arms and of all types of outer rings in barred galaxies. We have extended and applied the technique used in celestial mechanics to compute transfer orbits. Thus, our theory is based on the chaotic orbital motion driven by the invariant manifolds associated to the periodic orbits around the hyperbolic equilibrium points. In particular, spiral arms and outer rings are related to the presence of heteroclinic or homoclinic orbits. Thus, R₁ rings are associated to the presence of heteroclinic orbits, while R₁R₂ rings are associated to the presence of homoclinic orbits. Spiral arms and R₂ rings, however, appear when there exist neither heteroclinic nor homoclinic orbits. We examine the parameter space of three realistic, yet simple, barred galaxy models and discuss the formation of the different morphologies according to the properties of the galaxy model. The different morphologies arise from differences in the dynamical parameters of the galaxy.