In this paper, we develop a new application of the Laplace transform method (LTM) using the series expansion of the dependent variable for solving fractional logistic growth models in a population as well as fractional prey-predator models. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving fractional-order nonlinear differential equations.

The generation of coefficients of terms of positive and negative powers of \(n\) and \(-n\) of Kifilideen trinomial theorem as the terms are progress is stressful and time-consuming which the same problem is identified with coefficients of terms of binomial theorem of positive and negative powers of \(n\) and \(-n\). This slows the process of producing the series of any particular trinomial expansion. This study established Kifilideen coefficient tables for positive and negative powers of \(n\) and \(-n\) of the Kifilideen trinomial theorem and other developments based on matrix and standardized methods. A Kifilideen theorem of matrix transformation of the positive power of \(n\) of trinomial expression in which three variables \(x,y\), and \(z\) are found in parts of the trinomial expression was originated. The development would ease evaluating the trinomial expression’s positive power of \(n\). The Kifilideen coefficient tables are handy and effective in generating the coefficients of terms and series of the Kifilideen expansion of trinomial expression of positive and negative powers of \(n\) and \(-n.\)

The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor’s classical theorem is often needed, but only tacitly assumed. In this note, we provide an elementary proof of this measurable Taylor’s theorem, which guarantees that the interpolating point in the Lagrange form of the remainder can be chosen to depend measurably on the independent variable.

In this paper, we establish sharp inequalities for trigonometric functions. We prove in particular for \(0 < x < \frac{\pi}{2}\) and any \(n \geq 5\) \[0 < P_n(x)\ <\ (\sin x)^2- x^3\cot x < P_{n-1}(x) + \left[\left(\frac{2}{\pi}\right)^{2n} – \sum_{k=3}^{n-1} a_k \left(\frac{2}{\pi}\right)^{2n-2k}\right] x^{2n} \] where \(P_n(x) = \sum_{3=k}^n a_k x^{2k+1}\) is a \(n\)-polynomial, with positive coefficients (\(k \geq 5\)), \(a_{{k}}=\frac{{2}^{2\,k-2}}{\ \left( 2\,k-2 \right) ! } \left( \left| {B}_{ 2\,k-2} \right| +{\frac { \left( -1\right) ^{k+1}}{ \left( 2\,k-1 \right) k}} \right),\) \( B_{2k} \) are Bernoulli numbers. This improves a lot of lower bounds of \( \frac{\sin(x)}{x}\) and generalizes inequalities chains.
Moreover, bounds are obtained for other trigonometric inequalities as Huygens and Cusa inequalities as well as for the function
\[g_n(x) = \left(\frac{\sin(x)}{x}\right)^2 \left( 1 – \frac{2\left(\frac{2 x}{\pi}\right)^{2n+2}}{1-(\frac{2x}{\pi})^2}\right) +\frac{\tan(x)}{x}, \ n\geq 1 \].

In this manuscript, our primary focus revolves around extending the inequalities associated with the Quadratic \(\varphi(\delta_{1},\delta_{2})-\)function. Our approach involves leveraging the general quadratic functional equation encompassing \(2k\)-variables within the context of the fuzzy Banach space. Our main contribution lies in the expansion of these inequalities, representing a significant result within this study.

We classify particle paths for systems in thermal equilibrium satisfying the usual relations and prove that the only solutions are given by straight line parallel paths with speed \(c\).

In this paper, we obtain the bounds for the Laplacian eigenvalues of a weighted graph using traces. Then, we find the bounds for the Kirchhoff and Laplacian Estrada indices of a weighted graph. Finally, we define the Laplacian energy of a weighted graph and get the upper bound for this energy.

A number of results on claw-free, paw-free graphs have been presented in the literature. Although the proofs of such results are elegant, sound and valid, it has gone unnoticed that all the results about claw-free, paw-free graphs in the literature are a consequence of a result by Olariu [1]. The note, apart from covering the aforementioned gap, also provides an alternate proof to a result by Faudree and Gould found in [2] in that, an unnoticed consequence resulted in the characterization of claw-free, paw-free graphs.

A graph \(G\) is said to be super edge-magic if there exists a bijective function \(f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}\) such that \(f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}\) and \(f\left(u\right) + f\left(v\right) + f\left(uv\right)\) is a constant for each \(uv\in E\left( G\right)\). In this paper, we study the super edge-magicness of graphs of order \(n\) with degree sequence \(s:4, 2, 2, \ldots, 2\). We also investigate the super edge-magic properties of certain families of graphs. This leads us to propose some open problems.

Coronoids are nice chemical structures that may be represented mathematically in the planar hexagonal lattice. They have been well-studied both for their chemical properties and also their enumerative aspects. Typical approaches to the latter type of questions often include classification and algorithmic techniques. Here we study one simple class of coronoids called hollow hexagons. Notably, hollow hexagons may be represented with a collection of partitions on the set \(\{2,3,4,6\}\). The hollow hexagons are used to classify another family of primitive coronoids, which we introduce here, called lattice path coronoids. Techniques from lattice path enumeration are used to count these newly-defined structures within equivalence classes indexed by enclosing hollow hexagons.