Envelope Theorems in Economics: Historical Development and Modern Cost-Benefit Applications

This paper reviews some historical development and modern applications of the envelope theorems in economics from a static to a dynamic context. First, we show how the static version of the theorem surfaced in economics, which had eventually lead to the well-known Shephard’s lemma in microeconomics. Second, we present its dynamic version in terms of the classical calculus of variations and optimal control theory via the optimized Hamiltonian function. Third, we show some applications of the theorem for deriving dynamic cost-benefifit rules with special reference to environmental projects involving the green or comprehensive net national product (CNNP). Finally, we illustrate how to extend the cost-benefifit rules to a stochastic economic growth setting.

Fractional order stagnation point flow of the hybrid nanofluid towards a stretching sheet

Fractional calculus characterizes a function at those points, where classical calculus failed. In the current study, we explored the fractional behavior of the stagnation point flow of hybrid nano liquid consisting of TiO2 and Ag nanoparticles across a stretching sheet. Silver Ag and Titanium dioxide TiO2 nanocomposites are one of the most significant and fascinating nanocomposites perform an important role in nanobiotechnology, especially in nanomedicine and for cancer cell therapy since these metal nanoparticles are thought to improve photocatalytic operation. The fluid movement over a stretching layer is subjected to electric and magnetic fields. The problem has been formulated in the form of the system of PDEs, which are reduced to the system of fractional-order ODEs by implementing the fractional similarity framework. The obtained fractional order differential equations are further solved via fractional code FDE-12 based on Caputo derivative. It has been perceived that the drifting velocity generated by the electric field E significantly improves the velocity and heat transition rate of blood. The fractional model is more generalized and applicable than the classical one.

Generalized Conformable Mean Value Theorems with Applications to Multivariable Calculus

The conformable derivative and its properties have been recently introduced. In this research work, we propose and prove some new results on the conformable calculus. By using the definitions and results on conformable derivatives of higher order, we generalize the theorems of the mean value which follow the same argument as in the classical calculus. The value of conformable Taylor remainder is obtained through the generalized conformable theorem of the mean value. Finally, we introduce the conformable version of two interesting results of classical multivariable calculus via the conformable formula of finite increments.

Optimal Control Theory: Introduction to the Special Issue

Optimal control theory is a modern extension of the classical calculus of variations [...]

A novel exact solution for the fractional Ambartsumian equation

Fractional calculus (FC) is useful in studying physical phenomena with memory effect. In this paper, a fractional form of Ambartsumian equation is considered utilizing the Caputo fractional derivative. The Heaviside expansion formula in classical calculus (CC) is extended/developed in view of FC. Then, the extended Heaviside expansion formula is applied to obtain the exact solution in a simplest form. Several theorems and lemmas are proved to facilitate the evaluation of the inverse Laplace transform of specific expressions in fractional forms. The exact solution is established in terms of a one-parameter Mittag-Leffler function which is provided for the first time for the Ambartsumian equation in FC. The present solution reduces to the corresponding one in the relevant literature as the fractional order tends to one. Moreover, the convergence of the obtained solution is theoretically proved. Comparisons with another approach in the literature are performed. The advantage of the present analysis over the existing one in the relevant literature is discussed and analyzed.

Novel higher order iterative schemes based on the $ q- $Calculus for solving nonlinear equations

<abstract><p>The conventional infinitesimal calculus that concentrates on the idea of navigating the $ q- $symmetrical outcomes free from the limits is known as Quantum calculus (or $ q- $calculus). It focuses on the logical rationalization of differentiation and integration operations. Quantum calculus arouses interest in the modern era due to its broad range of applications in diversified disciplines of the mathematical sciences. In this paper, we instigate the analysis of Quantum calculus on the iterative methods for solving one-variable nonlinear equations. We introduce the new iterative methods called $ q- $iterative methods by employing the $ q- $analogue of Taylor's series together with the inclusion of an auxiliary function. We also investigate the convergence order of our newly suggested methods. Multiple numerical examples are utilized to demonstrate the performance of new methods with an acceptable accuracy. In addition, approximate solutions obtained are comparable to the analogous solutions in the classical calculus when the quantum parameter $ q $ tends to one. Furthermore, a potential correlation is established by uniting the $ q- $iterative methods and traditional iterative methods.</p></abstract>

ON THE CONSTRUCTION OF ENERGY-EFFICIENT ALGORTIMS OF THE WALKING ROBOT RAISING AND BRAKING

An energetically effective motion of a walking robot on a hard ground is considered. With the help of the methods of the classical calculus of variations, laws of the optimal acceleration and deceleration energy of the robot are calculated based on the minimum criterion. The resulting motion laws correspond to the extremals of the investigated quality functional.

Deleuze’s Theory of Dialectical Ideas: The Influence of Lautman and Heidegger

In Différence et répétition Deleuze’s overall ontology is structured by his theory of dialectical Ideas or problems. This theory draws features from Plato, Kant, and classical calculus. However, Deleuze bring those features together by fitting them into a theory of Ideas/problems developed by the mathematician and philosopher Albert Lautman. Lautman sought to explain the nature of the problems or dialectical Ideas with which mathematics engages and the solutions or mathematical theories which attempt to comprehend them. Lautman drew heavily upon Martin Heidegger’s early ontology to develop his theory of Ideas/problems. Although Deleuze seldom cited Heidegger, understanding how Lautman serves as a mediator between the two shows that certain elements in Heidegger’s ontology indirectly shaped Deleuze’s. This line of Heidegger’s influence has been largely unrecognized and unexplored in Deleuze scholarship. In this article the author seeks (1) to clarify Deleuze’s theory of dialectical Ideas or problems through an analysis of its debts to Lautman and Heidegger and (2) to demonstrate Heidegger’s crucial influence via Lautman on Deleuze’s ontology. In order to do this he focuses on five core claims that Deleuze’s theory of dialectical Ideas adopts from Lautman’s. The article provides an extensive reconstruction of what those claims mean in Lautman’s theory and discusses Lautman’s use of Heidegger to explain key parts of his position. The five core claims of Deleuze/Lautman that the author outlines are: (1) Ideas/problems are different in kind from solutions and do not disappear with solutions; (2) Ideas/problems are dialectical; (3) Ideas/problems are transcendent in relation to solutions; (4) Ideas/problems are simultaneously immanent in those solutions; (5) the relation between Ideas/problems and solutions is genetic, that is, solutions are generated on the basis of the determining conditions of Ideas/problems.