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This research presents a novel analytical approach to explore the fractional analysis of the one-dimensional time-fractional Schrödinger model (TFSM) using Caputo fractional derivatives. By integrating the Mohand transform (MT) with the residual power series method (RPSM), we develop the Mohand residual power series method (MT-RPSM) that provides results in the form of convergent series without assumptions on variables. Initially, we employ MT to reduce the fractional order, and then we transfer the fractional problem into the Mohand space formulation. Second, we use the RPSM concept to derive the iterative series formula for the Mohand space formulation. We analyze these findings using visual layouts to show the physical representation of the TFSM, which matches the precise results very well. The results indicate that the proposed technique is a reliable and practical method for identifying and analyzing various nonlinear models of physical phenomena. [ABSTRACT FROM AUTHOR]

Reflecting on existence as a linguistic practice has always been one of the sublime features of literature. This issue has had a significant appearance in Khayyam, so that Khayyam's work can be considered as a kind of problematic project of questioning existence. In the current research, it was tried to achieve new ontological perceptions of Khayyamian Thing with a materialistic approach and from a post-structuralist perspective. Here, identifying a convergent series of objects on one hand, and proposing the Conversion and Transmission idea as an underlying mechanism in Khayyam on the other hand, prepare strong analytical possibilities for new ontological formulations of Khayyamian Thing. Designing the Conversion and Transmission idea showed that the convergent series of objects are continuously connected with each other, and by producing new ratios, it displays a process of physical inner expansion. Finally, by theoretical analysis of these extended and expanding objects’ relations, four ontological ideas including inexhaustibility of existence, the object having memory, inner-expansion of desire (Oedipal economy of desire), and fetishism, as the result of the symptomatic encounter with Khayyam, are presented.

Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for θ=0.5,0.7,0.9,1.0. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative.

Airy and Bessel functions are one of the most popular and important special functions in various branches of physics, mathematics, and engineering. An observation to their behavior for the real argument suggest that they are related. This relation was studied earlier, but were accompanied by a number of assumptions, approximations, and sometimes even misconceptions. This motivated us to develop a fresh and transparent method to establish these relations. As the continuation of our study of the two papers published in resonance already, here we have used the general asymptotic series and the convergent series of these functions and thereby developed two new methods which throw light on the subtle interrelationships between these functions. Numerical evidences of our claims are provided for better clarity and understanding. [ABSTRACT FROM AUTHOR]

Health organizations are working to reduce the outbreak of infectious diseases with the help of several techniques so that exposure to infectious diseases can be minimized. Mathematics is also an important tool in the study of epidemiology. Mathematical modeling presents mathematical expressions and offers a clear view of how variables and interactions between variables affect the results. The objective of this work is to solve the mathematical model of MERS-CoV with the simplest, easiest and most proficient techniques considering the fractional Caputo derivative. To acquire the approximate solution, we apply the Adomian decomposition technique coupled with the Laplace transformation. Also, a convergence analysis of the method is conducted. For the comparison of the obtained results, we apply another semi-analytic technique called the homotopy perturbation method and compare the results. We also investigate the positivity and boundedness of the selected model. The dynamics and solution of the MERS-CoV compartmental mathematical fractional order model and its transmission between the human populace and the camels are investigated graphically for ϑ = 0:5; 0:7; 0:9; 1:0. It is seen that the recommended schemes are proficient and powerful for the given model considering the fractional Caputo derivative. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize the minimality criterion for a convergent power series f on Z p in terms of its coefficients for the cases p = 2 or 3. For an arbitrary prime p ≥ 5 , the minimality criterion of such a series can be obtained explicitly provided that the prescribed minimal conditions for the reduction of f modulo p are found. Second, we provide the minimality criterion for a rational map of at least degree 2 with good reduction on the projective line P 1 ( Q p) over Q p . This criterion enables us to obtain a complete description of minimal conditions for such a map on P 1 ( Q p) in terms of its coefficients for p = 2 or 3. For an arbitrary prime p ≥ 5 , we present a method of characterizing minimal rational maps ϕ of degree ≥ 2 on P 1 ( Q p) , provided that the prescribed conditions for the reduction of ϕ on P 1 (F p) to be transitive are known. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize the minimality criterion for a convergent power series f on Z p in terms of its coefficients for the cases p = 2 or 3. For an arbitrary prime p ≥ 5 , the minimality criterion of such a series can be obtained explicitly provided that the prescribed minimal conditions for the reduction of f modulo p are found. Second, we provide the minimality criterion for a rational map of at least degree 2 with good reduction on the projective line P 1 ( Q p) over Q p . This criterion enables us to obtain a complete description of minimal conditions for such a map on P 1 ( Q p) in terms of its coefficients for p = 2 or 3. For an arbitrary prime p ≥ 5 , we present a method of characterizing minimal rational maps ϕ of degree ≥ 2 on P 1 ( Q p) , provided that the prescribed conditions for the reduction of ϕ on P 1 (F p) to be transitive are known. [ABSTRACT FROM AUTHOR]

Let X be the set of positive real sequences x = (x n) such that the series ∑ n x n is divergent. For each x ∈ X , let I x be the collection of all A ⊆ N such that the subseries ∑ n ∈ A x n is convergent. Moreover, let A be the set of sequences x ∈ X such that lim n x n = 0 and I x ≠ I y for all sequences y = (y n) ∈ X with lim inf n y n + 1 / y n > 0 . We show that A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals I x . This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska. [ABSTRACT FROM AUTHOR]

This paper deals with the sum of reciprocal Fibonacci numbers. Let $ f_0 = 0 $, $ f_1 = 1 $ and $ f_{n+1} = f_n+f_{n-1} $ for any $ n\in\mathbb{N} $. In this paper, we prove new estimates on $ \sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}} $, where $ m\in\mathbb{N} $ and $ 0\leq\ell\leq m-1 $. As a consequence of some inequalities, we prove $ \lim\limits_{n\rightarrow \infty}\left\{\left(\sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}} \right)^{-1} -(f_{mn-\ell}-f_{m(n-1)-\ell})\right\} = 0. $ And we also compute the explicit value of $ \left\lfloor\left(\sum\limits^\infty_{k = n}\frac{1}{f_{mk-\ell}}\right)^{-1}\right\rfloor $. The interesting observation is that the value depends on $ m(n+1)+\ell $.

As a solution to the nonlinear Burger's issue in a single dimension, the homotopy perturbation method (HPM) is what we recommend utilizing according to the findings of this research. We have reached a series solution of the equations in terms of convergent series with easily computable components thanks to the fact that the nonlinear elements of Burger's equations may be addressed by using the homotopy perturbation approach (HPM). HPM is closely associated with the concept of the sum of an infinite series term, which frequently and rapidly converges to the correct response. The intricate equation is simplified into a more understandable form by using the HPM. According to the data that was gathered, the method that was recommended performs better than the state-of-the-art solutions for similar PDEs, and it is also much easier to put into practice. [ABSTRACT FROM AUTHOR]

Abstract Let { f k } k = 1 ∞ $\{f_{k} \} _{k=1}^{\infty}$ be a Fibonacci sequence with f 1 = f 2 = 1 $f_{1}=f_{2}=1$ . In this paper, we find a simple form g n $g_{n}$ such that lim n → ∞ { ( ∑ k = n ∞ a k ) − 1 − g n } = 0 , $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{a_{k}} \Biggr)^{-1}-g_{n} \Biggr\} =0, $$ where a k = 1 f k 2 $a_{k}=\frac{1}{f_{k}^{2}}$ , 1 f k f k + m $\frac{1}{f_{k}f_{k+m}}$ , or 1 f 3 k 2 $\frac{1}{f_{3k}^{2}}$ . For example, we show that lim n → ∞ { ( ∑ k = n ∞ 1 f 3 k 2 ) − 1 − ( f 3 n 2 − f 3 n − 3 2 + 4 9 ( − 1 ) n ) } = 0 . $$\lim_{n\rightarrow\infty} \Biggl\{ \Biggl(\sum^{\infty}_{k=n}{ \frac {1}{f_{3k}^{2}}} \Biggr)^{-1}- \biggl(f_{3n}^{2}-f_{3n-3}^{2}+ \frac {4}{9}(-1)^{n} \biggr) \Biggr\} =0. $$

From mostly a measure-theoretic consideration, we show that for every nonnegative, finite, and L1 function on a given finite measure space there is some nontrivial sequence of real numbers such that the series, obtained from summing over the term-by-term products of the reals and the summands of any divergent series with positive, vanishing summands such as the harmonic series, is convergent and no greater than the integral of the function. In terms of inequalities, the implications add additional information on mathematical expectation and the behavior of divergent series with positive, vanishing summands, and establish in a broad sense some new, unexpected connections between probability theory and, for instance, number theory. [ABSTRACT FROM AUTHOR]

Our basic aim is to provide a power series representation for b(s), 0 < s < 3, the well-known function satisfying b(s - 1)b(s + 1) = s². We will do this by using integer compositions of n. In the last section, some properties involving the coefficients of sn in the power series expansion of b(s) are given, as well an expression for 4/π. [ABSTRACT FROM AUTHOR]

Let p; q be given positive numbers and a; b non-negative ones. In this paper we study and characterize the class S(a; b; p; q) of all admissible ideals I ⊄ 2N with the following property... for all sequences (an) of positive real numbers. In a series of corollaries we discuss special cases including, also, several previously published theorems on this topic. [ABSTRACT FROM AUTHOR]

Let { f k } k = 1 ∞ be a Fibonacci sequence with f 1 = f 2 = 1 . In this paper, we find a simple form g n such that lim n → ∞ { (∑ k = n ∞ a k) − 1 − g n } = 0 , where a k = 1 f k 2 , 1 f k f k + m , or 1 f 3 k 2 . For example, we show that lim n → ∞ { (∑ k = n ∞ 1 f 3 k 2) − 1 − (f 3 n 2 − f 3 n − 3 2 + 4 9 (− 1) n) } = 0. [ABSTRACT FROM AUTHOR]

An nonhomogeneous integral equation with a singular kernel is considered. A feature of the equation under study is the incompressibility of the integral operator. In the study of the equation, an auxiliary simpler equation is used with the right - hand side equal to 1. The incompressibility of the integral operator for the equation under study is shown. Using the relations for an independent variable, the equation is equivalently reduced to a certain simplified equation. With the help of replacements for independent variables, the equation is reduced to an integral equation with a difference kernel. By applying the Laplace transform, the obtained equation is reduced to an ordinary first - order differential equation (linear). Its solution is found. By using the inverse Laplace transform, a solution of the auxiliary integral equation is obtained in the form of a convergent series in some domain. The solution of the initial equation with an arbitrary right - hand side is written through the solution of the auxiliary equation.

Abstract Many possible definitions have been proposed for fractional derivatives and integrals, starting from the classical Riemann–Liouville formula and its generalisations and modifying it by replacing the power function kernel with other kernel functions. We demonstrate, under some assumptions, how all of these modifications can be considered as special cases of a single, unifying, model of fractional calculus. We provide a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann–Liouville fractional integral operator. We also consider inversion properties of the new operators, prove analogues of the Leibniz and chain rules in this model of fractional calculus, and solve some fractional differential equations using the new operators. [ABSTRACT FROM AUTHOR]

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Let $\sum\limits_{n=1}^\infty a_n$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_n)$ is non-increasing, then $\lim\limits_{n \to\infty} n a_n = 0$. In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $\lim\limits_{n \to\infty} n a_n = 0$; Olivier's theorem is a consequence of our Theorem \ref{import}. (b) We prove properties analogous to Olivier's property when the usual convergence is replaced by the $\mathcal I$-convergence, that is a convergence according to an ideal $\mathcal I$ of subsets of $\mathbb N$. Again, Olivier's theorem is a consequence of our Theorem \ref{Iol}, when one takes as $\mathcal I$ the ideal of all finite subsets of $\mathbb N$.

The main goal in this work is to establish a new and efficient analytical scheme for space fractional telegraph equation (FTE) by means of fractional Sumudu decomposition method (SDM). The fractional SDM gives us an approximate convergent series solution. The stability of the analytical scheme is also studied. The approximate solutions obtained by SDM show that the approach is easy to implement and computationally very much attractive. Further, some numerical examples are presented to illustrate the accuracy and stability for linear and nonlinear cases. [ABSTRACT FROM AUTHOR]