scholarly journals Dynamic Hilbert-Type Inequalities with Fenchel-Legendre Transform

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 582 ◽  
Author(s):  
Ahmed A. El-Deeb ◽  
Samer D. Makharesh ◽  
Dumitru Baleanu
Keyword(s):  
Time Scale ◽  
Legendre Transform ◽  
Special Cases ◽  
Hilbert Type ◽  
General Time

Our work is based on the multiple inequalities illustrated in 2020 by Hamiaz and Abuelela. With the help of a Fenchel-Legendre transform, which is used in various problems involving symmetry, we generalize a number of those inequalities to a general time scale. Besides that, in order to get new results as special cases, we will extend our results to continuous and discrete calculus.

scholarly journals Some Dynamic Hilbert-Type Inequalities on Time Scales

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1410 ◽  
Author(s):  
Ghada AlNemer ◽  
Mohammed Zakarya ◽  
Hoda A. Abd El-Hamid ◽  
Praveen Agarwal ◽  
Haytham M. Rezk
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Discrete Inequalities ◽  
Hilbert Type ◽  
General Time

Throughout this article, we will demonstrate some new generalizations of dynamic Hilbert type inequalities, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. From these inequalities, as particular cases, we formulate some integral and discrete inequalities that have been demonstrated in the literature and also extend some of the dynamic inequalities that have been achieved in time scales.

scholarly journals Some new dynamic Steffensen-type inequalities on a general time scale measure space

2021 ◽  
Vol 7 (3) ◽  
pp. 4326-4337
Author(s):  
Ahmed A. El-Deeb ◽  
◽  
Inho Hwang ◽  
Choonkil Park ◽  
Omar Bazighifan ◽  
...  
Keyword(s):  
Time Scale ◽  
Measure Space ◽  
Finite Measure ◽  
Special Cases ◽  
Scale Measure ◽  
New Inequalities ◽  
General Time

<abstract><p>Our work is based on the multiple inequalities illustrated by Josip Pečarić in 2013, 1982 and Srivastava in 2017. With the help of a positive $ \sigma $-finite measure, we generalize a number of those inequalities to a general time scale measure space. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete and continuous calculus.</p></abstract>

Abel dynamic equations of first and second kind

2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Martin Bohner ◽  
Sabrina H. Streipert
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Dynamic Equations ◽  
Specific Class ◽  
Special Cases ◽  
Abel Differential Equations ◽  
General Time

AbstractThis paper gives the definition and analysis of Abel dynamic equations on a general time scale. As such, the results contain as special cases results for classical Abel differential equations and results for new Abel difference equations. By using appropriate transformations, expressions of Abel dynamic equations of second kind are derived on the general time scale. This also leads to a specific class of Abel dynamic equations of first kind. Finally, the canonical Abel dynamic equation is defined and examined.

scholarly journals More on Hölder’s Inequality and It’s Reverse via the Diamond-Alpha Integral

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1716
Author(s):  
M. Zakarya ◽  
H. A. Abd El-Hamid ◽  
Ghada AlNemer ◽  
H. M. Rezk
Keyword(s):  
Linear Combination ◽  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Hölder's Inequality ◽  
Special Cases ◽  
General Time

In this paper, we investigate some new generalizations and refinements for Hölder’s inequality and it’s reverse on time scales through the diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals, which are used in various problems involving symmetry. We develop a number of those symmetric inequalities to a general time scale. Our results as special cases extend some integral dynamic inequalities and Qi’s inequalities achieved on time scales and also include some integral disparities as particular cases when T=R.

scholarly journals New Weighted Opial-Type Inequalities on Time Scales for Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 842
Author(s):  
Ahmed A. El-Deeb ◽  
Dumitru Baleanu
Keyword(s):  
Time Scales ◽  
Convex Functions ◽  
Time Scale ◽  
Hölder Inequality ◽  
Special Cases ◽  
Discrete Inequalities ◽  
General Time

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results.

scholarly journals New dynamic Hilbert-type inequalities in two independent variables involving Fenchel–Legendre transform

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
A. A. El-Deeb ◽  
Saima Rashid ◽  
Zareen A. Khan ◽  
S. D. Makharesh
Keyword(s):  
Time Scales ◽  
Legendre Transform ◽  
Independent Variables ◽  
Hilbert Type ◽  
Two Independent Variables ◽  
New Inequalities

AbstractIn this paper, we establish some dynamic Hilbert-type inequalities in two independent variables on time scales by using the Fenchel–Legendre transform. We also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as particular cases. Our results give more general forms of several previously established inequalities.

scholarly journals Hilbert-type inequalities for time scale nabla calculus

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
H. M. Rezk ◽  
Ghada AlNemer ◽  
H. A. Abd El-Hamid ◽  
Abdel-Haleem Abdel-Aty ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Time Scale ◽  
Hölder’S Inequality ◽  
Hölder's Inequality ◽  
Arbitrary Time ◽  
Hilbert Type

Abstract This paper deals with the derivation of some new dynamic Hilbert-type inequalities in time scale nabla calculus. In proving the results, the basic idea is to use some algebraic inequalities, Hölder’s inequality, and Jensen’s time scale inequality. This generalization allows us not only to unify all the related results that exist in the literature on an arbitrary time scale, but also to obtain new outcomes that are analytical to the results of the delta time scale calculation.

scholarly journals Diffusion of dust particles emitted from a fixed source

2015 ◽  
Vol 4 (4) ◽  
pp. 454
Author(s):  
Khaled Al-mashrafi
Keyword(s):  
Source Function ◽  
Strong Dependence ◽  
Vertical Direction ◽  
Dust Particles ◽  
Special Values ◽  
Diffusion Parameters ◽  
Special Cases ◽  
Small Times ◽  
The Mathematical Model ◽  
General Time

<p>In this paper, we investigate the mathematical model for the diffusion of dust particles emitted from a fixed source. Mathematically, the time-dependent diffusion equation in the presence of a point source whose strength is dependent on time is solved. The solution in closed form for a source of general time dependence is obtained. A number of special cases, in which the source function of time is explicitly given and special values of the diffusion parameters are taken are examined in detail. The numerical calculations show the strong dependence of the concentration of dust on the speed of the wind, the source, and its position in the vertical direction. It is also found that the diffusion parameters play an important role in the spread of the dust particles in the atmosphere. When diffusion is present only in the vertical direction, it is found that for small times the dust spreads with a front that travels with the speed of the wind.</p>

Mathematical Morphology and Fourier Analysis on Time Scales

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Image Processing ◽  
Fourier Analysis ◽  
Mathematical Morphology ◽  
Time Scales ◽  
Time Scale ◽  
Time Signal ◽  
Scale Theory ◽  
Black And White ◽  
Introductory Material ◽  
Special Cases

Mathematical morphology, used extensively in image processing, tracks the support domains for the operation of convolution and deconvolution. Morphology is also important in the modelling of signals on time scales. Time scale theory provides a generalization tent under which the operations of discrete and continuous time signal and Fourier analysis rest as special cases. The time scale paradigm provides modelling under which a rich class of hybrid signals and systems can be analyzed. We begin with introductory material on mathematical morphology which is foundational to the development of time scale theory. The support of convolution is related to the operation of dilation in mathematical morphology. Mathematical morphology is most commonly associated with image processing. Applications of morphology was initially applied to binary black and white images by Matheron [966]. The field is richly developed [506, 578]. Here, we outline the fundamentals. In N dimensions, let X and H denote a set of vectors or, in the degenerate case of one dimension, a set of real numbers.

THE QUADRATIC TIME-DEPENDENT HAMILTONIAN: EVOLUTION OPERATOR, SQUEEZING REGIONS IN PHASE SPACE AND TRAJECTORIES

1994 ◽  
Vol 08 (11n12) ◽  
pp. 1563-1576 ◽  
Author(s):  
S.S. MIZRAHI ◽  
M.H.Y. MOUSSA ◽  
B. BASEIA
Keyword(s):  
Phase Space ◽  
Unitary Transformation ◽  
Uniform Magnetic Field ◽  
Evolution Operator ◽  
Single Mode ◽  
Time Dependent ◽  
Special Cases ◽  
Complete Set ◽  
Hamiltonian Evolution ◽  
General Time

We consider the most general Time-Dependent (TD) quadratic Hamiltonian written in terms of the bosonic operators a and a+, which may represent either a charged particle subjected to a harmonic motion, immersed in a TD uniform magnetic field, or a single mode photon field going through a squeezing medium. We solve the TD Schrödinger equation by a method that uses, sequentially, a TD unitary transformation and the diagonalization of a TD invariant, and we verify that the exact solution is a complete set of TD states. We also obtain the evolution operator which is essential to express operators in the Heisenberg picture. The variances of the quadratures are calculated and a phase space of parameters introduced, in which we identify squeezing regions. The results for some special cases are presented and as an illustrative example the parametric oscillator is revisited and the trajectories in phase space drawn.

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