Mean Value Theorem Research Articles

On Weighted Fractional Calculus With Respect to Functions

ABSTRACTThis paper aims to further explore the existing theory of weighted fractional operators with respect to functions. This theory extends some fundamental results of classical Riemann–Liouville and Caputo fractional derivatives to their weighted counterparts involving fractional differentiation and integration with respect to functions. By investigating the fundamental principles of these operators, we establish mean value theorems, Taylor's theorems, and integration by parts formulae. The Leibniz rule is extended for weighted Riemann–Liouville derivatives with respect to functions. Also, we present necessary conditions for the existence and uniqueness of solutions for a class of initial value problems, involving weighted Caputo fractional derivatives with respect to functions, in a Sobolev space.

A Mean Value Theorem for General Dirichlet Series

ABSTRACT In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x + O(x^\beta)$ for some ρ > 0 and β < 1. We prove that $\frac1T\int_0^T |\,f(\sigma+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2\sigma}$ for $\sigma\gt\frac{1+\beta}{2}$ and obtain an upper bound for this moment for $\beta\lt\sigma\le \frac{1+\beta}{2}$. We provide a number of examples indicating the sharpness of our results.

Mean Value Theorem and the Inverse Problem

The mean value theorem of differentiation is the core theorem of differential calculus. It’s an important tool for studying functions, and a bridge between functions and derivatives. This paper briefly reviews the mean value theorem for differentiation. The Lagrange mean value theorem is the core, Rolle theorem is a special case, and the Cauchy theorem is an extension. It’s the foundation of calculus theory and crucial for function research. The paper introduces the specific contents, proof methods, and applicable examples of these theorems. This paper also discusses their inverse problems with proofs. The mean value theorem has a profound impact on academic development, laying the foundation, promoting analysis, and expanding the research field. The theorem can be applied not only in functional analysis but also in biomathematics. It also nurtures scientific thinking, provides problem-solving ideas, and stimulates innovation. Finally, it’s hoped that its application fields will expand, not just in math but also in other disciplines.

The Mean Value Theorem and It’s Application

The Mean Value Theorem (MVT) is the central theorem of differential Calculus. It’s the major tool to research on function, also bridging the gap between function and derivative. There are lot of researchers focus on it from ancient times until now. This article introduces details about two mean theorems include the Rolle’s Theorem and Lagrange Theorem. This paper uses the sample question and refutation to explain the conditions of the theorems, thus addressing the questions that many students will ask when learning definitions of these theorems. The article later examines the application of the MVT. The proof of the unique existence of roots and the inequality is included. The Mean Value Theorem reveals the link between the macroscopic, overall properties for a function on an interval and the microscopic, localized properties of the function at a point. The significance of the application for the mean value theorem is profound, not only promoting the development of mathematical analysis theory, but also playing an important role in multiple fields

The First Mean Value Theorem of integral in Perplex Plane

Perplex numbers are an extension of the real numbers, consisting of pairs of real numbers that form a commutative ring. This article considers the partial order structure of perplex numbers and discusses the concept of perplex interval. Based on the definition of integration of perplex functions of perplex variable, this article we investigate the first mean value theorem of integral in perplex plane which can significantly enrich the mathematical toolbox by providing essential methods for estimating and approximating perplex number functions. It also simplifies the solution of complex integral problems involving perplex numbers and fosters theoretical extensions by exploring the behavior of perplex number functions under various conditions. This, in turn, promotes the development of perplex calculus, enhancing the overall theoretical and practical framework of perplex number analysis. In physics, particularly in relativity, studying the first mean value theorem for perplex numbers can provide more precise methods for describing and estimating physical quantities, enhancing the application of hyperbolic numbers in these fields.

Impulse Controllability for Singular Hybrid Coupled Systems

This study examines the concept of impulse controllability within singular hybrid coupled systems through the utilisation of decentralised proportional plus derivative (P-D) output feedback. By employing the Differential Mean Value Theorem, the nonlinear model can be converted into a linear parameter-varying large-scale system. Our analysis leads to the establishment of algebraic conditions that are both necessary and sufficient for the existence of a decentralised P-D output feedback controller that can guarantee impulse controllability in these complex systems. Moreover, we address the issue of admissibility within these systems by employing matrix trace inequalities. We present a novel sufficient condition for impulse controllability, which offers a new perspective on addressing this challenging problem. To validate our findings, we present numerical examples that demonstrate the effectiveness of the proposed methodologies in practice.

Approximation theorem of quaternion-valued almost periodic functions of two variables

In this paper, the quaternion Fourier-Stieltjes transform of two variables is studied. By orthogonal-split method, some fundamental properties of the quaternion-valued almost periodic functions of two variables are obtained. Through introducing the almost periodic approximation via quaternion Fourier-Stieltjes integrals, we establish the quaternion orthogonal-split mean-value theorem. Finally, the approximation theorem of quaternion-valued almost periodic functions of two variables is established.

Lyapunov-derived fuzzy temperature control for thermoelectric cooling system using dynamic updating law

In this paper, an intelligent temperature control strategy is proposed for thermoelectric cooling systems. The Lyapunov-based control method allows a dynamic updating law to be employed in the fuzzy logic system to improve the system performance in the presence of unmodeled dynamics. A new smooth function is also integrated into the control loop to limit input voltage, effectively addressing potential energy waste issue. Furthermore, the controller design challenges posed by even-power input are resolved using the adaptive fuzzy control method and Lagrange’s mean value theorem. Performance analysis indicates that the control function and state signals of the refrigeration system are bounded under our proposed method. Compared with the traditional intelligent control strategy, we only need to design a dynamic update law, which effectively reduces the complexity of the algorithm and is favorable for implementation. Finally, simulation and experimental results demonstrate the effectiveness of the method proposed in this paper. Compared to the traditional PID method, our control method reduces the maximum error by 29.4% and the setting time by 35.7% in the experiment.

Adaptive output feedback command filter control based on extreme learning machine for nonaffine time delay systems with input saturation

This paper is concerned with the tracking control problem of nonlinear time delay systems. For time delay systems, we consider nonaffine pure-feedback structure. Additionally, the case where the time delay systems are subject to input saturation and unknown states is also taken into account. To begin with, a mean value theorem and an implicit function theorem are exploited to transform the systems into an affine form. Afterwards, a state observer is developed to estimate the unknown state variables and an extreme learning machine (ELM) is used to approximate the unknown functions. The output of the command filter replaces the virtual control signal'derivative, thereby circumventing the problem of ‘explosion of complexity’. Meanwhile, compensation signals are introduced to eliminate the filtering errors in a dynamic surface control (DSC). A Lyapunov–Krasovskii functional is designed to mitigate the unknown constant state time delay. During the controller design process, combining a prescribed performance control (PPC) with a command filter control, the tracking error can converge to a predetermined bound. Additionally, the effect of input saturation is eliminated with the aid of an auxiliary system. Finally, the effectiveness of such controller is further proved by taking an electromechanical system as an application object.

Late Night Thoughts on Rolle’s Theorem

Summary Rolle’s theorem plays a key role in every modern calculus text as a prelude to the mean-value theorem. We discuss whether it ought to, how it might be better expressed, how best to present and prove the multitude of ideas related to this theorem, and whether to allow Michel Rolle to appear in our textbooks in any case.

Proving the AM–GM Inequality Via the Mean Value Theorem

Summary We will give two proofs of the AM–GM inequality using a lemma that is proved by Lagrange’s mean value theorem.

The Dirac-Dolbeault Operator Approach to the Hodge Conjecture

The Dirac-Dolbeault operator for a compact Kähler manifold is a special case of Dirac operator. The Green function for the Dirac Laplacian over a Riemannian manifold with boundary allows the expression of the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash–Moser generalized inverse function theorem, we prove the existence of complex submanifolds of a complex projective manifold satisfying globally a certain partial differential equation under a certain injectivity assumption. Thereby, internal symmetries of Dolbeault and rational Hodge cohomologies play a crucial role. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for complex projective manifolds.

On Some Cauchy Type Mean-Value Theorems with Applications

Some Cauchy type mean-value theorems for Chebychev’s inequality, Steffensen’s inequality, midpoint rule and Simpson’s rule are presented. Furthermore, we give some applications for the obtained results using the exponential and logarithmic functions, their Taylor polynomials and for some trigonometric functions. Further, we obtain some exponential, logarithmic and trigonometric equations and give two inequalities for midpoint and Simpson’s rules.

An explorative study on "Differentiation, Rolle's theorem, and the mean value theorem: In Vedic and modern methods

An explorative study on "Differentiation, Rolle's theorem, and the mean value theorem: In Vedic and modern methods

GeoGebra—A great platform for experiential learning, explorations and creativity in mathematics

In this paper we are presenting some examples of how GeoGebra is used in: a) explaining concepts of the first derivative, monotony, extremums; b) studying the properties of the function (strictly increasing/decreasing); c) demonstrating the mean value theorem. The results and the conclusions are based on the experiment carried out in the teaching and learning process in the chapter of derivatives in a third-year class of a secondary school in Albania. Also, there are some encouraging facts got by the use of GeoGebra: the double representation and the dynamic features of GeoGebra allow the students to quickly grasp the mathematical concepts and properties and be actively involved in further explorations. The use of GeoGebra tools is similar to the use of the tools of virtual games, and this is a great advantage to stimulate the students to learn mathematics and master their mathematical performance the same way they play games. On the other side, using GeoGebra, it is easier for the teachers to explain mathematical concepts, the properties of algebraic objects, to discuss about different situations and aspects of the subject under study and to methodically reason the results got. GeoGebra provides a very commodious environment for the students to effectively interoperate with each other. Our results showed that GeoGebra is effective in teaching and learning mathematics. GeoGebra software contributed in enhancing students’ understanding of mathematical concepts and improved students’ interest to learn mathematics. Also, we admit that not all the stages of implementing GeoGebra software in the classroom are flowing smoothly. Based on our experience and the other researchers, it is observed that the effectiveness is dependent on the way GeoGebra is integrated in the teaching and learning process, implying that the research must continue with the commitment of many researchers of mathematics, physics and other sciences.

A new fractional derivative extending classical concepts: Theory and applications

In this paper, a novel general definition for the fractional derivative and fractional integral based on an undefined kernel function is introduced. For 0<α≤1, this definition aligns with classical interpretations and is applicable for calculating the derivative in an open negative interval I⊆[a,+∞),a∈R. Additionally, when α=1, the definition coincides with the classical derivative. Fundamental properties of the fractional integral and derivative, including the product rule, quotient rule, chain rule, Rolle’s theorem, and the mean value theorem, are derived. These properties are illustrated through various applications to demonstrate their applicability. Furthermore, some applications of solving fractional nonlinear systems of integro-differential equations using framelets are presented.

Good functions, measures, and the Kleinbock–Tomanov conjecture

Abstract In this paper, we prove a conjecture of Kleinbock and Tomanov (2007) on Diophantine properties of a large class of fractal measures on Q p n \mathbb{Q}_{p}^{n} . More generally, we establish the 𝑝-adic analogues of the influential results of Kleinbock, Lindenstrauss and Weiss (2004) on Diophantine properties of friendly measures. We further prove the 𝑝-adic analogue of one of the main results of Kleinbock (2008) concerning Diophantine inheritance of affine subspaces, which answers a question of Kleinbock. One of the key ingredients in the proofs of Kleinbock, Lindenstrauss and Weiss is a result on ( C , α ) (C,\alpha) -good functions whose proof crucially uses the Mean Value Theorem. Our main technical innovation is an alternative approach to establishing that certain functions are ( C , α ) (C,\alpha) -good in the 𝑝-adic setting. We believe this result will be of independent interest.

A quadratic Vinogradov mean value theorem in finite fields

A quadratic Vinogradov mean value theorem in finite fields

Lagrange's Mean Value Theorem and Taylor's Theorem and Their Applications

Lagrange's Mean Value Theorem and Taylor's Theorem and Their Applications

Узагальнені обернені нерівності Єнсена-Штеффенсена та пов’язані нерівності

We compare two linear functionals that are negative on convex functions. Further, using Green's functions we give some new conditions for reversed Jensen-Steffensen and related inequalities to hold. Using Green's function we also give refinement of Levinson type generalization of reversed Jensen-Steffensen and related inequalities. The acquired results are then used for constructing mean-value theorems.