In this paper, we investigate semirings whose elements are either units or zero-divisors (nilpotents) with many examples. While comparing these semirings with their counterparts in ring theory, we observe that their behavior is different in many cases.

In this paper, we introduce Indigenous semirings and show that they are examples of information algebras. We also attribute a graph to them and discuss their diameters, girths, and clique numbers. On the other hand, we prove that the Zariski topology of any Indigenous semiring is the Sierpiński space. Next, we investigate their algebraic properties (including ideal theory). In the last section, we characterize units and idempotent elements of formal power series over Indigenous semirings.

Travelling wave solutions have been played a vital role in demonstrating the wave character of nonlinear problems arising in the field of ocean engineering and sciences. To describe the propagation of the nonlinear wave phenomenon in the ocean (for example, wind waves, tsunami waves), a variety of evolution equations have been suggested and investigated in the existing literature. This paper studies the dynamic of travelling periodic and solitary wave behavior of a double–dispersive non-linear evolution equation, named the Sharma-Tasso-Olver (STO) equation. Nonlinear evolution equations with double dispersion enable us to describe nonlinear wave propagation in the ocean, hyperplastic rods and other mediums in the field of science and engineering. We analyze the wave solutions of this model using a combination of numerical simulations and Ansatz techniques. Our analysis shows that the travelling wave solutions involve a range of parameters that displays important and very interesting properties of the wave phenomena. The relevance of the parameters in the travelling wave solutions is also discussed. By simulating numerically, we demonstrate how parameters in the solutions influence the phase speed as well as the travelling and solitary waves. Furthermore, we discuss instantaneous streamline patterns among the obtained solutions to explore the local direction of the components of the obtained solitary wave solutions at each point in the coordinate (x,t).

The objective in this article is to extend the applicability of Newton’s method for solving Banach space valued nonlinear equations. In particular, a new semi-local convergence criterion for Newton’s method (NM) based on Kantorovich theorem in Banach space is developed by application of the Heisenberg Uncertainty Principle (HUP). The convergence region given by this theorem is small in general limiting the applicability of NM. But, using HUP and the Fourier transform of the operator involved, we show that it is possible to extend the applicability of NM without additional hypotheses. This is done by enlarging the convergence region of NM and using the concept of epsilon-concentrated operator. Numerical experiments further validate our theoretical results by solving equations in case not covered before by the Newton–Kantorovich theorem.

To help solving intractable nonlinear evolution equations (NLEEs) of waves in the field of fluid dynamics we develop an algorithm to find new high order solutions of the class of Abel, Bernoulli, Chini and Riccati equations of the form (Formula Presented), with constant coefficients a, b, c. The role of this class of equations in NLEEs is explained in the introduction below. The basic algorithm to compute the coefficients of the power series solutions of the class, emerged long ago and is further developed in this paper. Practical application for hitherto unknown solutions is exemplified.

Falls in the geriatric population are one of the most important causes of disabilities in this age group. Its consequences impose a great deal of economic burden on health and insurance systems. This study was conducted by a multidisciplinary team with the aim of evaluating the effect of visuo-spatial-motor training for the prevention of falls in older adults. The subjects consisted of 31 volunteers aged 60 to 92 years who were studied in three groups: (1) A group under standard physical training, (2) a group under visuo-spatial-motor interventions, and (3) a control group (without any intervention). The results of the study showed that visual-spatial motor exercises significantly reduced the risk of falls of the subjects.

The often reported reduction of Reaction Time (RT) by Vision Training) is successfully replicated by 81 athletes across sports. This enabled us to achieve a mean reduction of RTs for athletes eye-hand coordination of more than 10%, with high statistical significance. We explain how such an observed effect of Sensorimotor systems’ plasticity causing reduced RT can last in practice for multiple days and even weeks in subjects, via a proof of principle. Its mathematical neural model can be forced outside a previous stable (but long) RT into a state leading to reduced eye-hand coordination RT, which is, again, in a stable neural state.

Employing the I-concurrence (Ic) measure, entanglement dynamics of superposition of isospin fermionic coherent states (SFCS) in Heisenberg spin chains of Ising, XX, XXX and XXZ models in the presence of Dzyaloshinskii-Moriya (DM) interaction and magnetic field is studied. For the above-mentioned models, the entanglement dynamics of SFCSs is independent of magnetic field effect and the DM interaction effect introduces the quantum fluctuations in the entanglement dynamics of the system. It is shown that depending on the choice of the models in the absence of DM interaction, entanglement dynamics alter by applying and increasing the magnetic field to the first (second) part of the system. We showed that by increasing the spin of the fermionic coherent states (j) and, consequently, increasing their dimension d = 2j + 1, the entanglement dynamics of the SFCS states sharply increases and fluctuates at a higher level. Our results indicate no entanglement sudden death phenomenon under the examined conditions.