scholarly journals Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Thabet Abdeljawad ◽  
Saima Rashid ◽  
Zakia Hammouch ◽  
İmdat İşcan ◽  
Yu-Ming Chu
Keyword(s):  
Convex Functions ◽  
Fractional Derivatives ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Auxiliary Result ◽  
Fractal Sets ◽  
Different Types ◽  
Novel Applications ◽  
Class Of Functions ◽  
Harmonically Convex Functions

Abstract The present article addresses the concept of p-convex functions on fractal sets. We are able to prove a novel auxiliary result. In the application aspect, the fidelity of the local fractional is used to establish the generalization of Simpson-type inequalities for the class of functions whose local fractional derivatives in absolute values at certain powers are p-convex. The method we present is an alternative in showing the classical variants associated with generalized p-convex functions. Some parts of our results cover the classical convex functions and classical harmonically convex functions. Some novel applications in random variables, cumulative distribution functions and generalized bivariate means are obtained to ensure the correctness of the present results. The present approach is efficient, reliable, and it can be used as an alternative to establishing new solutions for different types of fractals in computer graphics.

scholarly journals On new generalized unified bounds via generalized exponentially harmonically s-convex functions on fractal sets

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yu-Ming Chu ◽  
Saima Rashid ◽  
Thabet Abdeljawad ◽  
Aasma Khalid ◽  
Humaira Kalsoom
Keyword(s):  
Convex Functions ◽  
Fractal Theory ◽  
Type Inequality ◽  
Auxiliary Result ◽  
Fractal Sets ◽  
Special Cases ◽  
Mathematical Sciences ◽  
Novel Strategy ◽  
Real Linear ◽  
Novel Applications

AbstractThe visual beauty reflects the practicability and superiority of design dependent on the fractal theory. Based on the applicability in practice, it shows that it is the completely feasible, self-comparability and multifaceted nature of fractal sets that made it an appealing field of research. There is a strong correlation between fractal sets and convexity due to its intriguing nature in the mathematical sciences. This paper investigates the notions of generalized exponentially harmonically ($GEH$ G E H ) convex and $GEH$ G E H s-convex functions on a real linear fractal sets $\mathbb{R}^{\alpha }$ R α ($0<\alpha \leq 1$ 0 < α ≤ 1 ). Based on these novel ideas, we derive the generalized Hermite–Hadamard inequality, generalized Fejér–Hermite–Hadamard type inequality and Pachpatte type inequalities for $GEH$ G E H s-convex functions. Taking into account the local fractal identity; we establish a certain generalized Hermite–Hadamard type inequalities for local differentiable $GEH$ G E H s-convex functions. Meanwhile, another auxiliary result is employed to obtain the generalized Ostrowski type inequalities for the proposed techniques. Several special cases of the proposed concept are presented in the light of generalized exponentially harmonically convex, generalized harmonically convex and generalized harmonically s-convex. Meanwhile, an illustrative example and some novel applications in generalized special means are obtained to ensure the correctness of the present results. This novel strategy captures several existing results in the corresponding literature. Finally, we suppose that the consequences of this paper can stimulate those who are interested in fractal analysis.

scholarly journals Accurate relationships between fractals and fractional integrals: New approaches and evaluations

2017 ◽  
Vol 20 (5) ◽  
Author(s):  
Raoul R. Nigmatullin ◽  
Wei Zhang ◽  
Iskander Gubaidullin
Keyword(s):  
Physical Meaning ◽  
Smooth Function ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Fractal Dimensions ◽  
Fractional Integrals ◽  
Fractal Sets ◽  
Fractional Derivatives And Integrals ◽  
Different Types ◽  
Physical Phenomena

AbstractIn this paper the accurate relationships between the averaging procedure of a smooth function over 1D- fractal sets and the fractional integral of the RL-type are established. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used.

scholarly journals Some new Hermite-Hadamard type inequalities for generalized harmonically convex functions involving local fractional integrals

2021 ◽  
Vol 6 (10) ◽  
pp. 10679-10695
Author(s):  
Wenbing Sun ◽  
◽  
Rui Xu
Keyword(s):  
Convex Functions ◽  
Integral Identity ◽  
Fractional Integral ◽  
Fractional Integrals ◽  
Fractal Sets ◽  
Special Means ◽  
Local Fractional Integral ◽  
Harmonically Convex Functions

<abstract><p>In this paper, we establish a new integral identity involving local fractional integral on Yang's fractal sets. Using this integral identity, some new generalized Hermite-Hadamard type inequalities whose function is monotonically increasing and generalized harmonically convex are obtained. Finally, we construct some generalized special means to explain the applications of these inequalities.</p></abstract>

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

scholarly journals Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

2021 ◽  
Vol 5 (3) ◽  
pp. 80
Author(s):  
Hari Mohan Srivastava ◽  
Artion Kashuri ◽  
Pshtiwan Othman Mohammed ◽  
Dumitru Baleanu ◽  
Y. S. Hamed
Keyword(s):  
Integral Inequalities ◽  
Fractional Integrals ◽  
Auxiliary Result ◽  
Wright Function ◽  
Nonconvex Functions ◽  
Special Cases ◽  
Type Integral ◽  
Generic Class ◽  
Two Parameters ◽  
Class Of Functions

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well.

scholarly journals Do galactic bars depend on environment?: an information theoretic analysis of Galaxy Zoo 2

2020 ◽  
Vol 501 (1) ◽  
pp. 994-1001
Author(s):  
Suman Sarkar ◽  
Biswajit Pandey ◽  
Snehasish Bhattacharjee
Keyword(s):  
Spatial Distribution ◽  
Mutual Information ◽  
Local Density ◽  
Statistical Significance ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Host Galaxy ◽  
Data Sets ◽  
Data Set ◽  
Information Theoretic

ABSTRACT We use an information theoretic framework to analyse data from the Galaxy Zoo 2 project and study if there are any statistically significant correlations between the presence of bars in spiral galaxies and their environment. We measure the mutual information between the barredness of galaxies and their environments in a volume limited sample (Mr ≤ −21) and compare it with the same in data sets where (i) the bar/unbar classifications are randomized and (ii) the spatial distribution of galaxies are shuffled on different length scales. We assess the statistical significance of the differences in the mutual information using a t-test and find that both randomization of morphological classifications and shuffling of spatial distribution do not alter the mutual information in a statistically significant way. The non-zero mutual information between the barredness and environment arises due to the finite and discrete nature of the data set that can be entirely explained by mock Poisson distributions. We also separately compare the cumulative distribution functions of the barred and unbarred galaxies as a function of their local density. Using a Kolmogorov–Smirnov test, we find that the null hypothesis cannot be rejected even at $75{{\ \rm per\ cent}}$ confidence level. Our analysis indicates that environments do not play a significant role in the formation of a bar, which is largely determined by the internal processes of the host galaxy.

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals Composite Aerosol Optical Depth Mapping over Northeast Asia from GEO-LEO Satellite Observations

2021 ◽  
Vol 13 (6) ◽  
pp. 1096
Author(s):  
Soi Ahn ◽  
Sung-Rae Chung ◽  
Hyun-Jong Oh ◽  
Chu-Yong Chung
Keyword(s):  
Aerosol Optical Depth ◽  
Optical Depth ◽  
Northeast Asia ◽  
Low Earth Orbit ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Radiometric Calibration ◽  
Climate Data ◽  
Error Statistics ◽  
Spatiotemporal Resolution

This study aimed to generate a near real time composite of aerosol optical depth (AOD) to improve predictive model ability and provide current conditions of aerosol spatial distribution and transportation across Northeast Asia. AOD, a proxy for aerosol loading, is estimated remotely by various spaceborne imaging sensors capturing visible and infrared spectra. Nevertheless, differences in satellite-based retrieval algorithms, spatiotemporal resolution, sampling, radiometric calibration, and cloud-screening procedures create significant variability among AOD products. Satellite products, however, can be complementary in terms of their accuracy and spatiotemporal comprehensiveness. Thus, composite AOD products were derived for Northeast Asia based on data from four sensors: Advanced Himawari Imager (AHI), Geostationary Ocean Color Imager (GOCI), Moderate Infrared Spectroradiometer (MODIS), and Visible Infrared Imaging Radiometer Suite (VIIRS). Cumulative distribution functions were employed to estimate error statistics using measurements from the Aerosol Robotic Network (AERONET). In order to apply the AERONET point-specific error, coefficients of each satellite were calculated using inverse distance weighting. Finally, the root mean square error (RMSE) for each satellite AOD product was calculated based on the inverse composite weighting (ICW). Hourly AOD composites were generated (00:00–09:00 UTC, 2017) using the regression equation derived from the comparison of the composite AOD error statistics to AERONET measurements, and the results showed that the correlation coefficient and RMSE values of composite were close to those of the low earth orbit satellite products (MODIS and VIIRS). The methodology and the resulting dataset derived here are relevant for the demonstrated successful merging of multi-sensor retrievals to produce long-term satellite-based climate data records.

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