The domain of the function $ f(x)=\cos \sqrt{x} $ is the set of all nonnegative real numbers. In this article, we will show how to use power series to extend this function to an analytic function defined on the entire real line. We will then show how this analytic extension of $ f(x) $ makes it easier and quicker for calculus students to compute derivatives of $ f(x) $ at the origin. We will moreover describe the process of extending the domain of any function of the form $ g(\sqrt{x}) $ for a given even analytic function $ g(x) $.

In [6] we have shown that the Frobenius-Perron operators associated with a one parameter family of piecewise linear chaotic maps have a common invariant (fixed) density map. In this paper we show the uniqueness of the common invariant density map and analyze the corresponding fixed point algorithm.

In this study, the notion of $AB$-homomorphism of an $AB$-algebra is introduced and some of its properties are obtained. Moreover, the first and third isomorphism theorems for $AB$-algebras are investigated.

In this article we will introduce different types of topological foldings on chaotic projective space. The limit of topological foldings on the fundamental group of the real projective plane will be obtained. The chain of folding on the chaotic projective spaces will induce a chain of fundamental groups. The relations between these chains will be achieved.

The height of a Pythagorean triple $(a,b,c)$ is defined as the difference $c - b$. In this paper, building on the Height-Excess Enumeration Theorem, we show how to generate primitive Pythagorean triples of a given height using powers of a single matrix. Then, we briefly discuss other matrices that both map any Pythagorean triple to another and also preserve a triple's height. We also note how Pythagorean triples with a given leg difference can be generated using matrices.

In graph theory, many topological indices based on vertex and edge distances are utilized in theoretical chemistry. Even though there are many such indices, we restrict our attention in this paper to the vertex Padmakar-Ivan index $PI_v(G)$ and the edge Padmakar-Ivan index $PI_e(G)$ of a simple connected graph $G$ without directed edges. We shall emphasize the computation of the vertex and edge Padmakar-Ivan indices of unitary Cayley graphs, $X_n$.

The notion of an int-soft (weak) implicative hyper BCK-ideal is introduced, and related properties are investigated. Relations between an int-soft weak implicative hyper BCK-ideal, an int-soft implicative hyper BCK-ideal, an int-soft hyper BCK-ideal, and an int-soft strong hyper BCK-ideal are considered, and characterizations of an int-soft weak implicative hyper BCK-ideal are discussed. Conditions for an int-soft hyper BCK-ideal to be an int-soft weak implicative hyper BCK-ideal are provided. Using an int-soft weak implicative hyper BCK-ideal, a new int-soft weak implicative hyper BCK-ideal is established. Finally, we show the hyper homomorphic preimage of an int-soft implicative hyper BCK-ideal is also an int-soft implicative hyper BCK-ideal.

In this paper, we investigate and describe all the minimal completions and covers of the symmetric, Hankel symmetric, and centrosymmetric doubly substochastic matrices, respectively.

In this paper, aspects of generalized continuity and generalized closedness are explored. The standard material on the notions of $*g$-open, $\mathbf{g}$-open sets and some definitions and results that are needed are presented first. Then the class of $\mathcal{I}_\mathbf{g}^*$-closed sets is introduced and its fundamental properties are studied. Also, $\mathcal{I}_\mathbf{g}^*$-regular, $^*$-additive, $^*$-multiplicative, $\mathcal{I}_\mathbf{g}^*$-additive, and $\mathcal{I}_\mathbf{g}^*$-multiplicative spaces are introduced and their properties are investigated.

We study lower and upper bounds of the isoperimetric deficit of equichordal curves in $\mathbb{R}^2$. An equichordal curve is defined by its polar radius $r(t)=p(t)+c$ with $t\in \mathbb{R}$, constant $c$, and $p(t)+p(t+\pi)=0$, such that chords through the origin are always of length $2c$. In the convex case, the isoperimetric deficit of equichordal curves has a non-attained constant upper bound. Otherwise, non-convex equichordal curves have an unbounded deficit. However, we prove that for any sufficiently regular equichordal curves $r(t)$, even non-convex, we can find associated curves of same enclosed area whose isoperimetric deficits bound from above and below the isoperimetric deficit of the equichordal curve $$0<L_{y_0}^2-4\pi A_{y_0} < L_r^2-4\pi A \leq L_{y_{max}}^2-4\pi A_{y_{max}} ,$$ where $L_{y_0}$ and $L_{y_{max}}$ refer to lengths of curves with radii $y_0(t)=\sqrt{p(t)^2+c^2}$ and, respectively, $y_{max}(t)=\sqrt{p(t)^2+c^2+\frac{p_m^2+c^2}{p_m}p(t)}$ with $p_m=\max\ |p|$.

A module is called coneat injective if it is injective with respect to all coneat exact sequences. The class of such modules is enveloping and falls properly between injectives and pure injectives. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. Using properties of such classes of modules, we characterize certain types of rings like von Neumann regular and right SF-rings. For instance, $R$ is a right SF-ring if and only if every coneat injective left $R$-module is injective.