scholarly journals A New Version of the Hermite–Hadamard Inequality for Riemann–Liouville Fractional Integrals

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 610 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Iver Brevik
Keyword(s):  
Bessel Functions ◽  
Applied Mathematics ◽  
Critical Role ◽  
Strong Relationship ◽  
Fractional Integrals ◽  
Mathematical Methods ◽  
Digamma Function ◽  
Hadamard Inequality ◽  
The Past ◽  
Riemann Integrals

Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite–Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite–Hadamard type for Riemann–Liouville fractional integrals. In addition, we establish a few inequalities of Hermite–Hadamard type for the Riemann integrals and Riemann–Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

scholarly journals On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 595 ◽  
Author(s):  
Pshtiwan Othman Mohammed ◽  
Mehmet Zeki Sarikaya ◽  
Dumitru Baleanu
Keyword(s):  
Bessel Functions ◽  
Applied Mathematics ◽  
Critical Role ◽  
Strong Relationship ◽  
Fractional Integrals ◽  
Modified Bessel Functions ◽  
Mathematical Methods ◽  
Digamma Function ◽  
Riemann Integrals ◽  
Published Result

Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

scholarly journals Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Kottakkaran Sooppy Nisar ◽  
Khadiga Ahmed Ismail ◽  
...  
Keyword(s):  
Applied Mathematics ◽  
Critical Role ◽  
Order Relation ◽  
Strong Relationship ◽  
New Class ◽  
Starting Point ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Definition Of ◽  
Interval Valued

AbstractIn both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued function (LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality (HH-inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality (HH–Fejér inequality)) for LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

scholarly journals Fuzzy Mixed Variational-Like and Integral Inequalities for Strongly Preinvex Fuzzy Mappings

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1816
Author(s):  
Muhammad Bilal Khan ◽  
Hari Mohan Srivastava ◽  
Pshtiwan Othman Mohammed ◽  
Juan L. G. Guirao
Keyword(s):  
Optimality Conditions ◽  
Critical Role ◽  
Fuzzy Optimization ◽  
Strong Relationship ◽  
The Other ◽  
Hadamard Inequality ◽  
New Class ◽  
Invex Set ◽  
Tight Connection ◽  
The Subject

It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings.

scholarly journals Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1485
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Shengda Zeng ◽  
Artion Kashuri
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Applied Mathematics ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Mathematical Methods ◽  
Fractional Integral Operators ◽  
New Class ◽  
Definition Of

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

Proceedings of the 14th Engineering Mathematics and Applications Conference

2020 ◽  
Vol 61 ◽  
Author(s):  
Christopher C Tisdell ◽  
Zlatko Jovanoski ◽  
William Guo ◽  
Judith Bunder
Keyword(s):  
Saudi Arabia ◽  
New Zealand ◽  
Mathematics Education ◽  
Interest Group ◽  
Applied Mathematics ◽  
Special Section ◽  
Mathematical Education ◽  
Mathematical Methods ◽  
Engineering Mathematics ◽  
Technology Group

  EMAC 2019 UNSW Canberra, Australia 26th Nov–29th Nov 2019 This Special Section of the ANZIAM Journal (Electronic Supplement) contains the refereed papers from the 14th Engineering Mathematics and Applications Conference (EMAC2019), which was held at the UNSW Canberra, Australia from 26th November to 29th November 2019. EMAC is held under the auspices of the Engineering Mathematics Group (EMG), which is a special interest group of the Australian and New Zealand Industrial and Applied Mathematics division of the Australian Mathematics Society. This conference provides a forum for researchers interested in the development and use of mathematical methods in engineering and applied mathematics, and aims to foster interactions between mathematicians and engineers, from both academia and industry. A further theme of the conference is the mathematical education of applied mathematicians and engineers. The event attracted participants from around the globe, including: New Zealand, Saudi Arabia, United Kingdom, Japan and Australia. The invited speakers at the 2019 meeting crossed the spectrum of specialities in engineering, mathematics, education and industry. They were: Alexander Kalloniatis (Defence Science and Technology Group), Robert K. Niven (UNSW Canberra), Katherine Seaton (La Trobe University) and Antoinette Tordesillas (University of Melbourne). All of the articles included in the EMAC 2019 Proceedings have been critically peer reviewed to the usual standards of the ANZIAM Journal. EMAC 2019 Organising Committee The conference organising committee were Fiona Richmond, Zlatko Jovanoski (Director), Leesa Sidhu, Duncan Sutherland, Fangbao Tian, Isaac Towers, Timothy Trudgian and Simon Watt. The invited speakers were chosen by a committee of experts including Alys Clark, Jennifer Flegg, Bronwyn Hajek (EMG Chair), Zlatko Jovanoski, Dann Mallet, Robert Niven, Brandon Pincombe, Melanie Roberts (Chair) and Harvinder Sidhu.

scholarly journals Small Molecule Inhibitors of Influenza Virus Entry

2021 ◽  
Vol 14 (6) ◽  
pp. 587
Author(s):  
Zhaoyu Chen ◽  
Qinghua Cui ◽  
Michael Caffrey ◽  
Lijun Rong ◽  
Ruikun Du
Keyword(s):  
Influenza Virus ◽  
Membrane Fusion ◽  
Small Molecule ◽  
Drug Target ◽  
Small Molecule Inhibitors ◽  
Virus Entry ◽  
Critical Role ◽  
Structural Basis ◽  
Future Directions ◽  
The Past

Hemagglutinin (HA) plays a critical role during influenza virus receptor binding and subsequent membrane fusion process, thus HA has become a promising drug target. For the past several decades, we and other researchers have discovered a series of HA inhibitors mainly targeting its fusion machinery. In this review, we summarize the advances in HA-targeted development of small molecule inhibitors. Moreover, we discuss the structural basis and mode of action of these inhibitors, and speculate upon future directions toward more potent inhibitors of membrane fusion and potential anti-influenza drugs.

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.

Disorders of Musical Cognition

2012 ◽  
Author(s):  
Lauren Stewart ◽  
Katharina von Kriegstein ◽  
Simone Dalla Bella ◽  
Jason D. Warren ◽  
Timothy D. Griffiths
Keyword(s):  
Case Studies ◽  
Cognitive Neuroscience ◽  
Cognitive Architecture ◽  
Critical Role ◽  
Theoretical Models ◽  
Systematic Evaluation ◽  
Neural Basis ◽  
The Past ◽  
Musical Cognition ◽  
Musical Listening

This article presents an overview of case studies of acquired disorders of musical listening. Like any cognitive faculty, music is multifaceted, and the identification of the neural basis of any complex faculty must proceed, hand in hand, with an elucidation of its cognitive architecture. The past decade has seen an evolution in the theoretical models of musical processing, allowing the development of theoretically motivated instruments for the systematic evaluation of musical disorders. Such developments have allowed reports of musical disorders to evolve from historical anecdotes to systematic, verifiable accounts that can play a critical role in contributing to our understanding of the cognitive neuroscience of music.

scholarly journals Comparison of Juvenile Feed Protocols on Growth and Spawning in Zebrafish

2021 ◽  
Author(s):  
Stephen C Frederickson ◽  
Mark D Steinmiller ◽  
Tiffany Rae Blaylock ◽  
Mike E Wisnieski II ◽  
James D Malley ◽  
...  
Keyword(s):  
Reproductive Performance ◽  
Critical Role ◽  
Fertilization Rate ◽  
Long Term Effects ◽  
The Past ◽  
Adult Feeding ◽  
Freeze Dried ◽  
Feeding Protocol ◽  
Spawning Success

Over the past 2 decades, zebrafish, Danio rerio, have become a mainstream laboratory animal model, yet zebrafish husbandrypractices remain far from standardized. Feeding protocols play a critical role in the health, wellbeing, and productivity ofzebrafish laboratories, yet they vary significantly between facilities. In this study, we compared our current feeding protocol for juvenile zebrafish (30 dpf to 75 dpf), a 3:1mixture of fish flake and freeze-dried krill fed twice per day with live artemia twice per day (FKA), to a diet of Gemma Micro 300 fed once per day with live artemia once per day (GMA). Our results showed that juvenile EK wild-type zebrafish fed GMA were longer and heavier than juveniles fed FKA. As compared with FKA-fed juveniles, fish fed GMA as juveniles showed better reproductive performance as measured by spawning success, fertilization rate, and clutch size. As adults, fish from both feeding protocols were acclimated to our standard adult feeding protocol, and the long-term effects of juvenile diet were assessed. At 2 y of age, the groups showed no difference in mortality or fecundity. Reproductive performance is a crucial aspect of zebrafish research, as much of the research focuses on the developing embryo. Here we show that switching juvenile zebrafish from a mixture of flake and krill to Gemma Micro 300 improves reproductive performance, even with fewer feedings of live artemia, thus simplifying husbandry practices.

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