Uniformity of dynamic inequalities constituted on time Scales

2020 ◽  
Vol 3 (4) ◽  
pp. 19-27
Author(s):  
Muhammad Jibril Shahab Sahir ◽  
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Time Scale ◽  
Fractional Integrals ◽  
Young’S Inequality ◽  
Liouville Type ◽  
Young's Inequality ◽  
Hybrid Theory

In this article, we present extensions of some well-known inequalities such as Young's inequality and Qi's inequality on fractional calculus of time scales. To find generalizations of such types of dynamic inequalities, we apply the time scale Riemann-Liouville type fractional integrals. We investigate dynamic inequalities on delta calculus and their symmetric nabla results. The theory of time scales is utilized to combine versions in one comprehensive form. The calculus of time scales unifies and extends some continuous forms and their discrete and quantum inequalities. By applying the calculus of time scales, results can be generated in more general form. This hybrid theory is also extensively practiced on dynamic inequalities.

scholarly journals Объединение классических и динамических неравенств, возникающих при анализе временных масштабов

2020 ◽  
pp. 26-36
Author(s):  
M.J.S. Sahir
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Liouville Type ◽  
Lyapunov’S Inequality ◽  
Discrete Analogues ◽  
Radon’S Inequality

In this paper, we present an extension of dynamic Renyi’s inequality on time scales by using the time scale Riemann–Liouville type fractional integral. Furthermore, we find generalizations of the well–known Lyapunov’s inequality and Radon’s inequality on time scales by using the time scale Riemann–Liouville type fractional integrals. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. В этой статье мы представляем расширение динамического неравенства Реньи на шкалы времени с помощью дробного интеграла типа Римана-Лиувилля. Кроме того, мы находим обобщения хорошо известного неравенства Ляпунова и неравенства Радона на шкалах времени с помощью дробных интегралов типа Римана-Лиувилля на шкале. Наши исследования объединяют и расширяют некоторые непрерывные неравенства и соответствующие им дискретные аналоги.

scholarly journals Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Usama Hanif ◽  
Ammara Nosheen ◽  
Rabia Bibi ◽  
Khuram Ali Khan ◽  
Hamid Reza Moradi
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Fractional Integrals ◽  
Discrete Fractional Calculus ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Superquadratic Functions

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

scholarly journals Consensus of classical and fractional inequalities having congruity on time scale calculus

2019 ◽  
Vol 27 (1) ◽  
pp. 57-69
Author(s):  
Muhammad Jibril Shahab Sahir
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Hölder’S Inequality ◽  
Fractional Integrals ◽  
Power Mean ◽  
Extended Form ◽  
Hölder's Inequality ◽  
Time Scale Calculus ◽  
Power Mean Inequality ◽  
Radon’S Inequality

Abstract In this paper, we find accordance of some classical inequalities and fractional dynamic inequalities. We find inequalities such as Radon’s inequality, Bergström’s inequality, Rogers-Hölder’s inequality, Cauchy-Schwarz’s inequality, the weighted power mean inequality and Schlömilch’s inequality in generalized and extended form by using the Riemann-Liouville fractional integrals on time scales.

scholarly journals Young’s inequality and related results on time scales

2005 ◽  
Vol 18 (9) ◽  
pp. 983-988 ◽  
Author(s):  
Fu-Hsiang Wong ◽  
Cheh-Chih Yeh ◽  
Shiueh-Ling Yu ◽  
Chen-Huang Hong
Keyword(s):  
Time Scales ◽  
Young’S Inequality ◽  
Young's Inequality

scholarly journals SOME HARDY-TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA DELTA FRACTIONAL INTEGRALS

Fractals ◽  
2021 ◽  
pp. 2240004
Author(s):  
FUZHANG WANG ◽  
USAMA HANIF ◽  
AMMARA NOSHEEN ◽  
KHURAM ALI KHAN ◽  
HIJAZ AHMAD ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Convex Functions ◽  
Type Inequality ◽  
Fractional Integrals ◽  
Discrete Fractional Calculus ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Hilbert Type

In this paper, some Jensen- and Hardy-type inequalities for convex functions are extended by using Riemann–Liouville delta fractional integrals. Further, some Pólya–Knopp-type inequalities and Hardy–Hilbert-type inequality for convex functions are also proved. Moreover, some related inequalities are proved by using special kernels. Particular cases of resulting inequalities provide the results on fractional calculus, time scales calculus, quantum fractional calculus and discrete fractional calculus.

scholarly journals Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
S. Manikandan ◽  
V. Muthulakshmi ◽  
S. Harikrishnan ◽  
Porpattama Hammachukiattikul
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Fractional Derivatives ◽  
Dynamic Equations ◽  
Riccati Transformation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Interval Oscillation

In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.

scholarly journals On some generalizations of nonlinear dynamic inequalities on time scales and their applications

2019 ◽  
Vol 13 (2) ◽  
pp. 440-462 ◽  
Author(s):  
A. El-Deeb
Keyword(s):  
Time Scales ◽  
Nonlinear Dynamic ◽  
Dynamic Equations ◽  
Discrete Version ◽  
Young’S Inequality ◽  
Qualitative Properties ◽  
Young's Inequality

In this article, by using Young's inequality, we prove some new nonlinear dynamic inequalities of Gronwall-Bellman-Pachpatte type on time scales. These inequalities give us the integral and discrete version, and also extend some known dynamic inequalities in certain papers. We can also say, the inequalities proved in this paper can be used as handy tools in the study of qualitative properties of some dynamic equations on time scales. Some examples are introduced to demonstrate the applications of these inequalities.

scholarly journals New fractional approaches for n-polynomial P-convexity with applications in special function theory

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shu-Bo Chen ◽  
Saima Rashid ◽  
Muhammad Aslam Noor ◽  
Zakia Hammouch ◽  
Yu-Ming Chu
Keyword(s):  
Fractional Calculus ◽  
Convex Functions ◽  
Special Functions ◽  
Real Life ◽  
Fractional Integrals ◽  
Digamma Function ◽  
Novel Approach ◽  
Inequality Theory ◽  
New Strategy ◽  
Significant Mechanism

Abstract Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

scholarly journals GNSS-to-GNSS time offsets: study on the broadcast of a common reference time

GPS Solutions ◽  
2021 ◽  
Vol 25 (2) ◽  
Author(s):  
Ilaria Sesia ◽  
Giovanna Signorile ◽  
Tung Thanh Thai ◽  
Pascale Defraigne ◽  
Patrizia Tavella
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Reference Time ◽  
Single Time ◽  
Common Reference ◽  
Prediction Quality ◽  
Common Time ◽  
Time Offset ◽  
The Impact ◽  
Low Uncertainty

AbstractWe present two different approaches to broadcasting information to retrieve the GNSS-to-GNSS time offsets needed by users of multi-GNSS signals. Both approaches rely on the broadcast of a single time offset of each GNSS time versus one common time scale instead of broadcasting the time offsets between each of the constellation pairs. The first common time scale is the average of the GNSS time scales, and the second time scale is the prediction of UTC already broadcast by the different systems. We show that the average GNSS time scale allows the estimation of the GNSS-to-GNSS time offset at the user level with the very low uncertainty of a few nanoseconds when the receivers at both the provider and user levels are fully calibrated. The use of broadcast UTC prediction as a common time scale has a slightly larger uncertainty, which depends on the broadcast UTC prediction quality, which could be improved in the future. This study focuses on the evaluation of two different common time scales, not considering the impact of receiver calibration, at the user and provider levels, which can nevertheless have an important impact on GNSS-to-GNSS time offset estimation.

Sensor-Based Estimation of Dim Light Melatonin Onset Using Features of Two Time Scales

2021 ◽  
Vol 2 (3) ◽  
pp. 1-15
Author(s):  
Cheng Wan ◽  
Andrew W. Mchill ◽  
Elizabeth B. Klerman ◽  
Akane Sano
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Light Exposure ◽  
Biological Activities ◽  
Second Step ◽  
Sampled Data ◽  
Dim Light Melatonin Onset ◽  
Dim Light ◽  
Using Data ◽  
Mean Square Errors

Circadian rhythms influence multiple essential biological activities, including sleep, performance, and mood. The dim light melatonin onset (DLMO) is the gold standard for measuring human circadian phase (i.e., timing). The collection of DLMO is expensive and time consuming since multiple saliva or blood samples are required overnight in special conditions, and the samples must then be assayed for melatonin. Recently, several computational approaches have been designed for estimating DLMO. These methods collect daily sampled data (e.g., sleep onset/offset times) or frequently sampled data (e.g., light exposure/skin temperature/physical activity collected every minute) to train learning models for estimating DLMO. One limitation of these studies is that they only leverage one time-scale data. We propose a two-step framework for estimating DLMO using data from both time scales. The first step summarizes data from before the current day, whereas the second step combines this summary with frequently sampled data of the current day. We evaluate three moving average models that input sleep timing data as the first step and use recurrent neural network models as the second step. The results using data from 207 undergraduates show that our two-step model with two time-scale features has statistically significantly lower root-mean-square errors than models that use either daily sampled data or frequently sampled data.

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