Certain Inequalities Pertaining to Some New Generalized Fractional Integral Operators

In this paper, we introduce the generalized left-side and right-side fractional integral operators with a certain modified ML kernel. We investigate the Chebyshev inequality via this general family of fractional integral operators. Moreover, we derive new results of this type of inequalities for finite products of functions. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. From our above-mentioned results, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view. Furthermore, two important results and some interesting consequences for convex functions in the framework of the defined class of generalized fractional integral operators are established. Finally, two basic examples demonstrated the significance of our results.

Marrying Wright's Law to Thermodynamics for an Ideal Relative Final Cost-predicting Model of Carbon-fuel Substitution

The problem of assessing the cost of C-fuel substitution is approached by means of a general interpretation of Wright's law and the introduction of the concept of thermodynamic utility which derives from energy carrier specific energy and storage state conditions. Via the Bienaymé–Chebyshev inequality, the ideal final cost ratio is determined at three different probabilities each with regard to compressed hydrogen, liquid hydrogen and Li-ion technology. The 96 % probability values supposedly balance insight and interval size best: however, cost-parity is not a result in any case. This paper points out evidence for the thesis that the subject of C-fuel substitution is governed by an intrinsic thermodynamic causality beyond economic and human factors, ultimately defining hidden pre-existing ideal baseline thresholds to the achievable in reality.<br>

New General Variants of Chebyshev Type Inequalities via Generalized Fractional Integral Operators

In this study, new and general variants have been obtained on Chebyshev’s inequality, which is quite old in inequality theory but also a useful and effective type of inequality. The main findings obtained by using integrable functions and generalized fractional integral operators have generalized many existing results as well as iterating the Chebyshev inequality in special cases.

New Chebyshev type inequalities via a general family of fractional integral operators with a modified Mittag-Leffler kernel

<abstract><p>The main goal of this article is first to introduce a new generalization of the fractional integral operators with a certain modified Mittag-Leffler kernel and then investigate the Chebyshev inequality via this general family of fractional integral operators. We improve our results and we investigate the Chebyshev inequality for more than two functions. We also derive some inequalities of this type for functions whose derivatives are bounded above and bounded below. In addition, we establish an estimate for the Chebyshev functional by using the new fractional integral operators. Finally, we find similar inequalities for some specialized fractional integrals keeping some of the earlier results in view.</p></abstract>

Data-based polyhedron model for optimization of engineering structures involving uncertainties

This paper studies the data-based polyhedron model and its application in uncertain linear optimization of engineering structures, especially in the absence of information either on probabilistic properties or about membership functions in the fussy sets-based approach, in which situation it is more appropriate to quantify the uncertainties by convex polyhedra. Firstly, we introduce the uncertainty quantification method of the convex polyhedron approach and the model modification method by Chebyshev inequality. Secondly, the characteristics of the optimal solution of convex polyhedron linear programming are investigated. Then the vertex solution of convex polyhedron linear programming is presented and proven. Next, the application of convex polyhedron linear programming in the static load-bearing capacity problem is introduced. Finally, the effectiveness of the vertex solution is verified by an example of the plane truss bearing problem, and the efficiency is verified by a load-bearing problem of stiffened composite plates.