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Journal of Analysis & Number Theory
An International Journal



Some symmetric identities for the generalized Hermite-Euler and Hermite-Genocchi polynomials

M. Ghayasuddin,
Abstract :
In this paper, we introduce a new class of generalized Hermite-Euler and Hermite-Genocchi polynomials and derive some symmetric identities by applying the generating functions. Also, we obtain some potentially useful relations for the Bernoulli polynomials, Euler polynomials, power sum, alternating sum and Genocchi numbers. These results extend some known summations and identities of generalized Hermite-Euler and Hermite-Genocchi polynomials studied by Dattoli et al. [3], Pathan and Khan [9], Khan [5].


Some transcendence results from a harmless irrationality theorem

Fabio Menezes de Souza Lima,
Abstract :
The arithmetic nature of values of some functions of a single variable, particularly, $\sin{z}$, $\cos{z}$, $\sinh{z}$, $\cosh{z}$, $e^z$, and $\ln{z}$, is a relevant topic in number theory. For instance, all those functions return transcendental values for non-zero algebraic values of $z$ ($z \ne 1$ in the case of $\ln{z}$). On the other hand, not even an irrationality proof is known for some numbers like $\,\ln{\pi}$, $\,\pi + e\,$ and $\pi \, e$, though it is well-known that at least one of the last two numbers is irrational. In this note, I first generalize the last result, showing that at least one of the sum and product of any two transcendental numbers is transcendental. I then use this to show that, given any non-null complex number $\,z \ne 1/e$, at least two of the numbers $\,\ln{z}$, $\,z +e\,$ and $\,z \, e\,$ are transcendental. I also show that $\,\cosh{z}$, $\sinh{z}\,$ and $\,\tanh{z}\,$ return transcendental values for all $\,z = r \, \ln{t}$, $\,r \in \mathbb{Q}$, $r \ne 0$, $\,t\,$ being any transcendental number. The analogue for common trigonometric functions is also proved.


Existence of Solutions to a Three-point Entrainment of Frequency Problem

Mesliza Mohamed,
Abstract :
In this paper, we deal with the existence of solutions to the frequency problem of a perturbed system, x−A(t)x = εf(x,sinwt, coswt, ε) with three-point boundary condition. The topological technique is used to obtain existence theorem. Two examples are given to illustrate our results.


Ascertainment of The Certain Fundamental Unit in a Specific Type of Real Quadratic Fields

Zubair Nisar,
Abstract :
The aim of the paper is to classify the real quadratic number fields Q(√ d) having specific form of continued fraction expansions of algebraic integer wd and is to determine the general explicit parametric representation of the fundamental unit d for such real quadratic number fields where d ≡ 2,3(mod4) is a square free positive integer. Also, Yokoiís dinvariants nd and md will be calculated in the relation to continued fraction expansion of wd for such real quadratic fields.


Intuitionistic fuzzy (ψ,η)-contractive mapping and Fixed points

Abstract :
In this article, using the definition of fuzzy ψ-contractive map- ping, we introduce intuitionistic fuzzy (ψ,η)-contractive mapping and extend the fixed point results to intuitionistic fuzzy metric spaces.


Solutions for certain classes of Non-linear Diophantine Eqautions

Abstract :
ABSTRACT: The main aim of this paper is to introduce a method, to solve certain class of non-Linear Diophantine equations and investigate various properties using the well-known Eulerís theorem and the theory of congruence. Some of the interesting special cases of our main results have been discussed. KEY WORDS: Diophantine Equations, Non-linear Diophantine equations, Euler phi-function, Eulerís theorem.


Uniform Exponential Stability for Time Varying Linear Dynamic Systems over Time Scales

Syed Omar Shah,
Abstract :
This paper proves the uniform exponential stability of the time varying linear dynamic system $x^\Delta(r)=G(r)x(r),\ r \in \overline{T}$ in terms of bounded-ness of solution of the following Cauchy problem: \begin{equation*} \left\{ \begin{split} W^{\Delta}(r)&=G(r)W(r) + \omega(r), \ 0\leq r \in \overline{T}, \ W(0) &= v_0, \end{split} \right. \end{equation*} where $\overline{T}$ denotes time scale, $G(r)$ is a matrix valued function, $\omega(r)$ is a bounded function on $\overline{T}$ and $v_0 \in \mathbb{C}^m$. In this note we prove the results that have the above result as an immediate corollaries.


On cyclic (α,β)-admissible generalized contraction mappings in generalized metric spaces

Eskandar Ameer,
Abstract :
The aim of this paper is to present new fixed point results in the framework of Branciari metric spaces. Some examples are presented to support the results proved herein. These results unify, generalize and complement the results of Jleli and Samet [J. Inequal. Appl. 2014, Article ID38(2014)]. We also provide an example of our result, where the comparable result in the existing literature is not applicable. As an application of our results, we obtain a fixed point results involving a cyclic mapping, not necessarily continuous.


Two Dimensional Laplace Transforms for Solving Systems of Fractional Partial Differential Equations

M.R. Masomi,
Abstract :
Abstract In this work, the authors implemented two dimensional Laplace transform to solve certain in homogenous sub ballistic fractional PDE and homogeneous systems of time fractional heat equations which is a generalization to the problem of thermal effects on fluid flow and also the problem of the effect of a uniform overburden on the passage of a thermal wave and the temperatures in the underlying rock. Constructive examples are also provided.

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