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A mean-value theorem for multiple trigonometric (exponential) sums on the sequence of Bell polynomials is proved. It generalizes I. M. Vinogradov’s and G. I. Arkhipov’s theorems. As is well known, a mean-value theorem of this type is at the core of Vinogradov’s method. The Bell polynomials are very closely related to the Faa di Bruno theorem on higher order derivatives of a composite function. As an application of the mean-value theorem proved in the paper, estimates for the sums $$\sum_{n_1\leq P}\dots\sum_{n_r\leq P}e^{2\pi i(\alpha_1Y_1(n_1)+\dots+\alpha_rY_r(n_1,\dots,n_r))}$$ are obtained, where $$\alpha_s$$ are real numbers and $$Y_s(n_1,\dots,n_s)$$ are the degree $$s$$ Bell polynomials, $$1\leq s\leq r$$ .

We study the sensitivity, essentially the differentiability, of the so-called “intermediate point” c in the classical mean value theorem $ \frac{f(a)-f(b)}{b-a}={f}^{\prime}(c)$we provide the expression of its gradient ∇c(d,d), thus giving the asymptotic behavior of c(a, b) when both a and b tend to the same point d. Under appropriate mild conditions on f, this result is “universal” in the sense that it does not depend on the point d or the function f. The key tool to get at this result turns out to be the Legendre-Fenchel transformation for convex functions.

An online identification method that can simultaneously estimate the unknown system parameter and the unknown time-delay is proposed. Firstly, with the help of Lagrange mean value theorem, the system with time-delay can be transformed into two terms that can be identified by modified least-square algorithm and one term that represents an approximate error. Then, a modified least-square algorithm is introduced to estimate all the unknown parameters in case of external disturbances. Additionally, an restrain term are added in the covariance matrix to enhance the robustness to deal with the approximate error which is related to the estimated error of system parameter and time-delay. Also, the boundedness of the estimation error is guaranteed via Lyapunov stability theory. Finally, the effectivity of the proposed method is verified by simulations results.

In this paper, we use the mean value theorem of Dirichlet L -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.

The mean value theorem of calculus states that, given a differentiable function f on an interval [a,b] , there exists at least one mean value abscissa c such that the slope of the tangent line at (...

We continue our researches concerning the generalization and improvement of R.T.Turganaliev’s result that states an asymptotic formula for the mean value of the Riemann zeta function in the critical strip with power factor saving in the remainder term.We find an asymptotic for the mean value of Dirichlet L-function in the critical strip. This assertion improves R.T.Turganaliev’s theorem for zeta-function in the whole interval $$(1/2 < Re 𝑠 ≤ 1)$$. Our result is based on the special use of the estimation of exponential sums by second derivative test.

In this article, the error term of the mean value theorem for binary Egyptian fractions is studied. An error term of prime number theorem type is obtained unconditionally. Under Riemann hypothesis, a power saving can be obtained. The mean value in short interval is also considered.

This paper provides a mean value theorem for arithmetic functions $f$ defined by $$f(n)=\prod_{d|n}g(d),$$ where $g$ is an arithmetic function taking values in $(0, 1]$ and satisfying some generic conditions. As an application of our main result, we prove that the density $\mu_q(n)$ (resp. $\rho_q(n)$) of normal (resp. primitive) elements in the finite field extension $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ are arithmetic functions of (non zero) mean values., Comment: To appear in IJNT

This note deals with some variants of the integral mean value theorem. Mainly a variant of Sahoo's theorem and a variant of Wayment's theorem were proved. Our approach is rather elementary and does...

We show how the full covering argument can be used to prove some type of Cauchy mean value theorem.

In this note we prove some variants of Lagrange’s mean value theorem. The main tools to prove these results are some elementary auxiliary functions. En esta nota demostramos algunas variaciones del Teorema de valor medio de Lagrange. Las herramientas principales para provar estos resultados son algunas funciones auxiliares elementares.

For the classical GM(1,1) model, the prediction accuracy is not high, and the optimization of the initial and background values is one-sided. In this paper, the Lagrange mean value theorem is used to construct the background value as a variable related to k. At the same time, the initial value is set as a variable, and the corresponding optimal parameter and the time response formula are determined according to the minimum value of mean relative error (MRE). Combined with the domestic natural gas annual consumption data, the classical model and the improved GM(1,1) model are applied to the calculation and error comparison respectively. It proves that the improved model is better than any other models.

We consider the mean value theorem for the wave equation in the Euclidean space to be a starting point to produce other mean value theorems for the wave equations on a sphere and in a space. We use transformations connecting the wave equation in the Euclidean space and the wave equation in non-Euclidean spaces. Also, we prove two-point mean value formulas for elliptic equations in the Lobachevskii space and on a sphere.

Summary In this paper, we present two new Lyapunov-based observers for the decentralized multiphase flow measurement that are based on the interconnections between the two subsystems to precisely estimate the states of the multiphase flow at the gas refinery. Because the system is composed of two interconnected subsystems, the states of the condensate and gas subsystems were separately estimated using the differential mean value theorem (DMVT), the sliding mode observer (SMOU), and the HYSYS® simulator (Hyprotech, Ltd., Calgary, Alberta, Canada) by considering the relationship between two subsystems, designing an observer, and converting the conditions to linear matrix inequality (LMI). Using the HYSYS simulator with the real process data, we found that both the observers are capable of estimating the states with some differences in performance, and the drift flux model (DFM) is sufficient for states estimation of the multiphase flow entering the gas refinery.

In this paper we study the utility of the functional Pompeiu–Chebyshev in some inequalities. Some results obtained by Alomari will be generalized regarding inequalities with Pompeiu–Chebyshev type functionals, in which linear and positive functionals intervene. We investigate some new inequalities of Grüss type using Pompeiu’s mean value theorem. Improvement of known inequalities is also given.

In this note, some variants of Cauchy's mean value theorem are proved. The main tools to prove these results are some elementary auxiliary functions.

We describe a method for constructing general integrals for some nonstrictly hyperbolic quasilinear equations and prove a nonlinear analog of Asgeirsson’s mean value theorem. Using a general integral, we study the nonlinear version of the Goursat characteristic problem.

In this paper we consider two continuous functions f, g : [a, b] → ℝ and we study for these ones, under which circumstances the intermediate point function is four order di erentiable at the point x = a and we calculate its derivative.

In this paper, a robust $$ H_{\infty } $$ control for a field-oriented control (FOC) of an induction motor (IM) based on mean value theorem (MVT) was presented and compared with a conventional control. Using the MVT approach and sector nonlinearity transformation after representing the nonlinear model of the IM in the Takagi–Sugeno form, the nonlinear error dynamics of the state feedback system was carried out in a linear parameter variable representation. By using Lyapunov theorem, stability conditions were expressed in terms of linear matrix inequalities (LMIs), whereby the offline gains of the controller can be calculated through the solving of these LMIs. To study the effectiveness of the proposed robust controller, a FOC experiment applied to a ¼ HP IM is conducted and compared with conventional controller. Practical results were investigated and gave a good speed tracking with a minimum effect, even in the presence of unpredictable load torque variations on the tracking error and assuring field orientation.

A new observer design for nonlinear systems is considered. The main idea consist of the determination of the estimation error and mean value theorem parameters (β) to introduce them into proposed observer structure. This process is designed on basis of mean value theorem (MVT) and Particle Swarm Optimization algorithm (PSO). This observer does not use Linear Matrix Inequality technique (LMI) for stability study. The stability study relies on the use of a classical quadratic Lyapunov function. The observer's gains are determined systematically. For the validation of theoretical development proposed in this paper. We consider two practical realization that deals the secure communication problem and a statistical performance analysis is realized.

We provide a mean-value theorem for a class of positive linear functionals. As an application, we improve the classical First Mean-value Theorem for Integrals and obtain other related results.

We generalize the Pompeiu mean-value theorem by replacing the graph of a continuous function with a compact set.

In this paper, a matrix method is developed to solve quadratic non-linear differential equations. It is assumed that the approximate solutions of main problem which we handle primarily, is in terms of Bernoulli polynomials. Both the approximate solution and the main problem are written in matrix form to obtain the solution. The absolute errors are applied to numeric examples to demonstrate efficiency and accuracy of this technique. The obtained tables and figures in the numeric examples show that this method is very sufficient and reliable for solution of non-linear equations. Also, a formula is utilized based on residual functions and mean value theorem to seek error bounds.

In this article, benefiting from the nabla h − fractional functions and nabla h − Taylor polynomials, some properties of the nabla h − discrete version of Mittag-Leffler ( h − ML) function are studied. The monotonicity of the nabla h − fractional difference operator with h − ML kernel (Atangana–Baleanu fractional differences) is discussed. As an application, the Mean Value Theorem (MVT) on h Z is proved.

In this paper, we present the monotonicity analysis for the nabla fractional differences with discrete generalized Mittag-Leffler kernels$( {}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta )$(a−1ABR∇δ,γy)(η)of order$00<δ<0.5,$\beta =1$β=1,$00<γ≤1starting at$a-1$a−1. If$({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y ) ( \eta )\geq 0$(a−1ABR∇δ,γy)(η)≥0, then we deduce that$y(\eta )$y(η)is$\delta ^{2}\gamma $δ2γ-increasing. That is,$y(\eta +1)\geq \delta ^{2} \gamma y(\eta )$y(η+1)≥δ2γy(η)for each$\eta \in \mathcal{N}_{a}:=\{a,a+1,\ldots\}$η∈Na:={a,a+1,…}. Conversely, if$y(\eta )$y(η)is increasing with$y(a)\geq 0$y(a)≥0, then we deduce that$({}^{ABR}_{a-1}{\nabla }^{\delta ,\gamma }y )(\eta ) \geq 0$(a−1ABR∇δ,γy)(η)≥0. Furthermore, the monotonicity properties of the Caputo and right fractional differences are concluded to. Finally, we find a fractional difference version of the mean value theorem as an application of our results. One can see that our results cover some existing results in the literature.

The purpose of the present paper is to discuss the integrability of the Fourier transform of $$L^\infty $$ -functions subject to a very weak decay condition. This will include the negative power of the logarithm. For a start, the case when $$d=1$$ is investigated by the use of the second mean value theorem. Based on the discussion of this case, a passage to the weighted Hankel transform is done, which covers the integrability of the d-dimensional Fourier transform of the radial functions.

Here is introduced and studied the left fractional local general M-derivative of various orders. All basic properties of an ordinary derivative are established here. We also define the corresponding left fractional M-integrals. Important theorems are established such as: the inversion theorem, the fundamental theorem of fractional calculus, the mean value theorem, the extended mean value theorem, the Taylor’s formula with integral remainder, the integration by parts. Our left fractional derivative generalizes the alternative fractional derivative and the local M-fractional derivative. See also [3].

Differentiable functions are defined and their main properties obtained. Some applications include the problem of maxima and minima, and the use of Taylor polynomials for asymptotic approximation. Principal results include the mean value theorem and L’Hopital’s rule. Nuggets include Jensen’s inequality; rates of convergence.

In this paper, we prove some inequalities satisfied by the modified degenerate gamma function which was recently introduced. The tools employed include Holder’s inequality, mean value theorem, Hermite–Hadamard’s inequality, and Young’s inequality. By some parameter variations, the established results reduce to the corresponding results for the classical gamma function.

By employing the classical mean value theorem, Hermite-Hadamard inequality and some other analytical techniques, we establish some new inequalities for Nielsen's beta function. Some of these inequalities provide bounds for certain ratios of the gamma function.

The aim of this paper is to describe the solution (f, g) of the equation $$\begin{aligned}{}[f(x)-f(y)]g'(\alpha x+(1-\alpha )y)= [g(x)-g(y)]f'(\alpha x+(1-\alpha )y),\ x,y\in I, \end{aligned}$$ where $$I\subset \mathbb {R}$$ is an open interval, $$f,g:I\rightarrow \mathbb {R}$$ are differentiable, $$\alpha $$ is a fixed number from (0, 1).

New issues, derivations, and remarks are presented for the subject of continuum damage mechanics. Emphasis is placed on the fundamental and basic aspects of damage mechanics. This work is a continuation of a previous work of the authors and provides for a complete analysis of the issues discussed previously. The topics discussed include: (1) the true nature of the damage–integrity angle, (2) modeling damaged solids as composite systems, (3) hypothesis of strain equivalence as a special case of the hypothesis of elastic energy equivalence, (4) mechanics of damaged beams, (5) a mean value theorem for damage mechanics, (6) partial damage mechanics characterization, (7) micro-damage or nano-damage, (8) crack and void mechanics, and (9) conceptual modeling of interfacial damage. It is hoped that these new and fundamental concepts will pave the way for new, consistent, and holistic avenues in research in damage mechanics and characterization of materials.

Felix Klein and Abraham Fraenkel each formulated a criterion for a theory of infinitesimals to be successful, in terms of the feasibility of implementation of the Mean Value Theorem. We explore the evolution of the idea over the past century, and the role of Abraham Robinson's framework therein., 10 pages, published in Journal of Humanistic Mathematics http://scholarship.claremont.edu/jhm/vol8/iss1/7/

Asymptotics for mean values of complete rational trigonometric sums modulo a power of a prime number are obtained. For polynomials of one variable these asymptotics are not improvable in the degree of averaging of those sums.

This study examines the modification of the integral mean value theorem for discontinuous functions. Modification is studied by proving that a function that is not continuous at a certain and bounded interval can be integrated (finite integral) both in Rieman’s integral and Newton’s integral (integral as antiderivative). The discontinuity of the intended function, namely; f is defined on [a,b] but the value of a function and its limit are not equal at some points or infinite points and countable on (a,b), f is undefined on [a,b] at some points or infinite points and countable on (a,b) but its limit exists there. The results of this study provide a modification of the integral mean value theorem by replacing the value of f in the implication of the theorem with its limit value so that the integral mean value theorem is obtained for the non-continuous function.

The semilocal convergence of a third-order Newton-like method for solving nonlinear equations is considered. Under a weak condition (the so-calledγ-condition) on the derivative of the nonlinear operator, we establish a new semilocal convergence theorem for the Newton-like method and also provide an error estimate. Some numerical examples show the applicability and efficiency of our result, in comparison to other semilocal convergence theorems.

The aim of the paper is to present some mean value theorems obtained as consequences of the intermediate value property. First, we will prove that any nonextremum value of a Darboux function can be represented as an arithmetic, geometric or harmonic mean of some different values of this function. Then, we will present some extensions of the Cauchy or Lagrange Theorem in classical or integral form. Also, we include similar results involving divided differences. The paper was motivated by some problems published in mathematical journals.

The aim of this paper is to explore Karamata’s mean value theorem. In the second section, we reformulate a mean theorem for the convex functions and prove some consequences. Also, we prove an integral version of Karamta theorem. Third section is reserved to the stability results. We establish some conditions for the stability of the intermediate point arising from Karamata, Godner and the integral mean theorem proved in the previous section.

In calculating the total solar energy input into a salt gradient solar pond, the current method is insufficient mainly caused by two problems. Firstly, the existing equations of solar pond energy analysis can be only used for momentary calculations but it is very time-consuming for long time periods. Secondly, the shading effect inside the pond affects significantly the energy storage performance of the pond, especially in small ones. To solve the first problem, the mean values of variable parameters during the concerned time period is proposed in this work and the ‘first mean value theorem for definite integrals’ is used to derive the average values. For the second problem, a rectangular pond with vertical walls is investigated as an example, and the exact sunny areas in different depths of the pond are calculated at different time conditions. The experimental data of a published study is used for the validation. The energy efficiency of the low convective zone of the experimental pond is calculated theoretically, which shows a good agreement with the experimental value. The experimental data and theoretical results for the energy efficiency are 9.68% and 11.38% for January, 17.54% and 18.92% for May, and 28.11% and 30.94% for August, respectively. The modified equations can be used to predict a pond performance before its construction.