Recall that a set B with binary operations + and ∗ is a left brace if (B,+) is an abelian group and the following version of distributive combined with associativity holds: (a + b + a ∗ b) ∗ c = a ∗ c + b ∗ c + a ∗ (b ∗ c), a ∗ (b + c) = a ∗ b + a ∗ c for all a, b, c ∈ B. Moreover (B, ◦) is a group where we define a ◦ b = a + b + a ∗ b. Skew braces have connections to several different topics. In particular, skew braces provide the right algebraic framework to study set-theoretic solutions to the Yang–Baxter equation. The goal of this paper is to construct and study braces (resp. skew -braces) with their ideals like strongly prime ideals as a tool for solutions to the Yang-Baxter equation via changing their conditions. We propose several problems in the theory of semi-braces and using new concepts. We hope that these problems will help to strengthen the interest in the theory of skew braces and set-theoretic solutions to the Yang-Baxter equation. Braces were introduced as a generalization of Jacobson radical rings. It turns out those braces allow us to use ring-theoretic and group-theoretic methods for studying involutive solutions to the Yang–Baxter equation. If braces are replaced by skew braces, then one can use similar methods for studying not necessarily involutive solutions. Here we will collect problems related to skew braces (resp. semi-braces) and set-theoretic solutions to the Yang-Baxter equation.

The three-dimensional problem of the modelling of elastic wave propagation in a multi-layered acoustic metamaterial, which is a periodic elastic composite with periodic arrays of interface cracks or planar voids of an arbitrary shape is considered. The boundary integral equation method is extended for this purpose. The unknown crack opening displacement vectors for each array are related using the Floquet theorem and solved using the Galerkin method at the reference delaminations in the arrays. The method provides an efficient tool for fast parametric analysis of the influence of the characteristics of the periodic arrays on the transmission and diffraction of elastic waves. Two modifications of the method are proposed and compared for rectangular cracks. To reduce computational costs, a preliminary analytical evaluation of the arising integral representations in terms of the Fourier transform of Green's matrices and crack opening displacements is presented.

A function is non-differentiable when there is a cusp or a corner point in its graph. To solve this problem we propose a nonlinear optimization model whose objective function is the Euclidean distance function. To identify the maximum points of a function that has corner or cusp points, according to the proposed model, a series of segments are generated which are measured through the Euclidean distance, which are all perpendicular to the abscissa axis. Therefore, by maximizing the Euclidean distance it is possible to identify the segment whose points represent the maximum of the function and its projection on the abscissa axis. The proposed model therefore wants to be an alternative to maximum point search methods in the presence of functions that have points of non-derivability. The proposed method and the consequent model are going to be applied in financial issues connected to the analysis of some stock exchange trends and connected markets’ behaviors. This occasion is useful to present the mathematical approach and structure to a qualified audience in getting important feed-backs in terms of comments, remarks, potential ideas of suitable applications.

We define the class of extended b_v (s)-metric space that properly contains the class of b_v (s)-metric space, and then after we ensure the existence of a fixed point for the self map satisfying the Banach contractive condition in the context of this newly defined space. Moreover, we compare the proved result with the existing fixed point theorems in literature.

In this paper, an eco-epidemiological model of prey refuge and prey harvesting in infected prey populations is discussed. The predator consumes susceptible and infected prey at different rates in the form of a ratio-dependent type of interaction. The existence, positive invariance, and boundedness of the system are addressed. We have also established the stability of equilibrium points. Finally, to support the primary analytical findings, some numerical simulations were also given.