Mean Value Theorem

Bombieri and Pila gave sharp estimates for the number of integer points (m, n) on a given arc of a curve y = F (x) , enlarged by some size parameter M , for algebraic curves and for transcendental analytic curves. The transcendental case involves the maximum number of intersections of the given arc by algebraic curves of bounded degree. We obtain an analogous result for functions F (x) of some class C k that satisfy certain differential inequalities that control the intersection number. We allow enlargement by different size parameters M and N in the x-and y-directions, and we also estimate integer points close to the curve, with n − N F " m M "˛ δ, for δ sufficiently small in terms of M and N. As an appendix we obtain a determinant mean value theorem which is a quantitative version of a linear independence theorem of Pólya.

This survey is an account of the current status of subdi erential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might pro tably use subdi erentials as tools.

In this paper we study improper integrals on time scales. We also give some mean value theorems for integrals on time scales, which are used in the proof of an analogue of the classical Dirichlet{Abel test for improper integrals. AMS (MOS) Subject Classication. 39A10.

We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if $f$ is locally distributionally integrable over the real line and $\psi\in\mathcal{D}(\mathbb{R}%) $ is a test

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Take any function f(z) analytic in the disk I z I < r, r > 0, with f" (0) r 0. exist r o, 0 < r 0 < r, and a point ~ in the disk I zl < I z01 = I zl I for which for any points Zo and Zl UDC 517.5 Cinquini [1] proved that there f(z!)-f(Zo) = f'(~)(Zl-z0) (1)

We obtain a new generalization of the Flett theorem and several new mean value theorems. We give condensed representations of the Flett and generalized Flett theorems in terms of divided differences.  2004 Elsevier Inc. All rights reserved.

The usefulness of fluorescence techniques for the study of macromolecular structure and dynamics depends on the accuracy and sensitivity of the methods used for data analysis. Many methods for data analysis have been proposed and used, but little attention has been paid to the maximum likelihood method, generally known as the most powerful statistical method for parameter estimation. In this paper we study the properties and behavior of maximum likelihood estimates by using simulated fluorescence intensity decay data. We show that the maximum likelihood method provides generally more accurate estimates of lifetimes and fractions than does the standard least-squares approach especially when the lifetime ratios between individual components are small. Three novelties to the field of fluorescence decay analysis are also introduced and studied in this paper: a) discretization of the convolution integral based on the generalized integral mean value theorem: b) the likelihood ratio test as a tool to determine the number of exponential decay components in a given decay profile; and c) separability and detectability indices which provide measures on how accurately, a particular decay component can be detected. Based on the experience gained from this and from our previous study of the Pad6-Laplace method, we make some recommendations on how the complex problem of deconvolution and parameter estimation of multiexponential functions might be approached in an experimental setting.

A procedure has been developed to determine the reliable form of the glass-crystal transformation function, and to deduce the kinetic parameters by using differential scanning calorimetry data, obtained from experiments performed under non-isothermal conditions. It is an integral method, which is based on a transformation rate independent of the thermal history and expressed as the product of two separable functions of absolute temperature and the fraction transformed. Considering the same temperatures for the different heating rates, one obtains a constant value for temperature integral, and, therefore, a plot of a function of the volume fraction transformed versus the reciprocal of the heating rate leads to a straight line with an intercept of zero, if the reaction mechanism is correctly chosen. Besides, by using the first mean value theorem to approach the temperature integral, one obtains a relationship between a function of the temperature and other function of the volume fraction transformed. The logarithmic form of the quoted relationship leads to a straight line, whose slope and intercept allow to obtain the activation energy and the frequency factor. The method developed has been applied to the crystallization kinetics of the Sb 0.12 As 0.36 Se 0.52 glassy alloy and it has been found that the kinetic model of normal grain growth is the most suitable to describe the crystallization of the quoted alloy. The mean values obtained for the activation energy and the frequency factor have been 27.36 kcal mol À1 and 1.5 Â 10 9 s À1 , respectively.

The Mean Value Theorem is a great theory and guide for any body who deals with the rate of change. This paper aims to add something to the theory. Even a small addition is better than none, only time will tell the significance of the addition.

A B-spline backstepping controller is proposed for a class of multiple-input multiple-output (MIMO) nonlinear systems. The control scheme incorporates the backstepping design technique with a B-spline neural network which is utilized to estimate the system dynamics. The B-spline neural network has the advantage of locally controlling its output behavior compared with other neural networks; therefore, it is very suitable to

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon-Nikodým property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-upper semicontinuity, and a mean-value theorem. A connection with known notions is provided and chain rules are proved using key results developed. This included the vectorization and restriction theorem, and the extension theorem. Therefore, the generalized Jacobian introduced in this paper is proved to enjoy all the properties required of a derivative like-set.

Suppose that in a random sample of size n from a population with probability density function (p.d.f.) f(x) the order statistics are X1≤X(2)&lt;…&lt; X(n). It is proved that a necessary and sufficient condition far f(x) to be the p.d.f. of the pareto distribution is that the statistics are independent.

Using the mathematical induction and Cauchy's mean-value theorem, for any positive number r, we prove that , where n and m are natural numbers, k is a nonnegative integer. The lower bound is best possible. This inequality generalizes the Alzer's inequality [J. Math. Anal. Appl.179 (1993), 396–402]. An open problem is proposed.

This paper considers the noncooperative maximization of mutual information in the vector Gaussian interference channel in a fully distributed fashion via game theory. This problem has been widely studied in a number of works during the past decade for frequency-selective channels, and recently for the more general multiple-input multiple-output (MIMO) case, for which the state-of-the art results are valid only for nonsingular square channel matrices. Surprisingly, these results do not hold true when the channel matrices are rectangular and/or rank deficient matrices. The goal of this paper is to provide a complete characterization of the MIMO game for arbitrary channel matrices, in terms of conditions guaranteeing both the uniqueness of the Nash equilibrium and the convergence of asynchronous distributed iterative waterfilling algorithms. Our analysis hinges on new technical intermediate results, such as a new expression for the MIMO waterfilling projection valid (also) for singular matrices, a mean-value theorem for complex matrix-valued functions, and a general contraction theorem for the multiuser MIMO watefilling mapping valid for arbitrary channel matrices. The quite surprising result is that uniqueness/convergence conditions in the case of tall (possibly singular) channel matrices are more restrictive than those required in the case of (full rank) fat channel matrices. We also propose a modified game and algorithm with milder conditions for the uniqueness of the equilibrium and convergence, and virtually the same performance (in terms of Nash equilibria) of the original game.

This survey is an account of the current status of subdi erential research. It is intended to serve as an entry point for researchers and graduate students in a wide variety of pure and applied analysis areas who might pro tably use subdi erentials as tools.

From Bombieri's mean value theorem one can deduce the prime number theorem π(x) = Li(x) + O(x 1/2 ln 15 x), which is equivalent to the Riemann hypothesis, and the least prime P(q) satisfying P(q) = O{[ϕ(q)] 2 [lnϕ(q)] 32 } in arithmetic progressions with common difference q, where ϕ(q) is the Euler function.

We prove that if f is a real valued lower semicontinuous function on a Banach space X and if there exists a C1, real valued Lipschitz continuous function on X with bounded support and which is not identically equal to zero, then f is Lipschitz continuous of constant K provided all lower subgradients of f are bounded by K. As

We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitzsmooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinüous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the "smooth" variational principle of Bonvein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, conemonotonicity, quasiconvexity, and convexity of lower semicontinüous functions which clarify, unify and extend many existing results for specific subdifferentials.

Asymptotic expansion has been used to simplify the transport of high charge and energy ions for broad beam applications in the laboratory and space. The solution of the lowest order asymptotic term is then related to a GreenÕs function for energy loss and straggling coupled to nuclear attenuation providing the lowest order term in a rapidly converging Neumann series for which higher order collisions terms are related to the fragmentation events including energy dispersion and downshift. The first and second Neumann corrections were evaluated numerically as a standard for further analytic approximation. The first Neumann correction is accurately evaluated over the saddle point whose width is determined by the energy dispersion and located at the downshifted ion collision energy. Introduction of the first Neumann correction leads to significant simplification of the second correction term allowing application of the mean value theorem and a second saddle point approximation. The regular dependence of the second correction spectral dependence lends hope to simple approximation to higher corrections. At sufficiently high energy nuclear crosssection variations are small allowing non-perturbative methods to all orders and renormalization of the second corrections allow accurate evaluation of the full Neumann series.

We describe two new ways of efficiently deriving continuous reach set information for hybrid systems. In both cases, we overapproximate the differential equations to constraints that are then solved using a solver for first-order predicate logical formulae over the reals. We prove some properties about the amount of introduced over-approximations. Moreover, we embed the results in our earlier method for verification of hybrid systems using abstraction refinement and provide some timings that document the resulting improvement.

In this brief, robust adaptive neural network (NN) control is presented for helicopters in vertical flight, with dynamics in single-input-single-output (SISO) nonlinear nonaffine form. Based on the use of the implicit function theorem and the mean value theorem, we propose a constructive approach for adaptive NN control design with guaranteed stability. Considering both full-state and output feedback cases, it is shown that the output tracking error converges to a small neighborhood of the origin, while the remaining closed-loop signals remain bounded. The simulation study demonstrates the effectiveness of the proposed control.

In this paper, an active FTC scheme is proposed. First, it is developed in the context of linear systems and then it is extended to non-linear systems with the differential mean value theorem. The key contribution of the proposed approach is an integrated FTC design procedure of the fault identification and fault-tolerant control schemes. Fault identification is based on the use of an observer. While, the FTC controller is implemented as a state feedback controller. This controller is designed such that it can stabilize the faulty plant using Lyapunov theory and LMIs. Finally, the last part of the paper shows application results regarding the Twin-Rotor MIMO System (TRMS) that confirm the high performance of the proposed approach.

In this paper, we characterize all the functions that attain their Flett mean value at a particular point between the endpoints of the interval under consideration. These functions turn out to be cubic polynomials and thus, we also characterize these.

In this paper, an active FTC scheme is proposed. First, it is developed in the context of linear systems and then it is extended to non-linear systems with the differential mean value theorem. The key contribution of the proposed approach is an integrated FTC design procedure of the fault identification and fault-tolerant control schemes. Fault identification is based on the use of an observer. While, the FTC controller is implemented as a state feedback controller. This controller is designed such that it can stabilize the faulty plant using Lyapunov theory and LMIs. Finally, the last part of the paper shows application results regarding the Twin-Rotor MIMO System (TRMS) that confirm the high performance of the proposed approach.

and coeditor of La Gaceta de la R.S.M.E. His main interests are: approximation theory, differential equations, new proofs, and the history of mathematics. M. Jiménez graduated at Granada University in 1989. He also obtained a degree in management sciences at U.N.E.D. University in 2001. Presently, he works with J.M. Almira in his Ph.D. thesis. He is interested in mathematical modelling and optimization techniques. N. Del Toro graduated at La Laguna University in 1999. Presently, she has a research grant from Junta de Andalucía to complete her Ph.D. thesis on approximation theory. In her paper [3] I. Rosenholtz has proved, as a consequence of the Jordan curve theorem, the following interesting mean value theorem for the plane.

A B-spline backstepping controller is proposed for a class of multiple-input multiple-output (MIMO) nonlinear systems. The control scheme incorporates the backstepping design technique with a B-spline neural network which is utilized to estimate the system dynamics. The B-spline neural network has the advantage of locally controlling its output behavior compared with other neural networks; therefore, it is very suitable to online estimate the system dynamics by tuning its interior parameters, including control points and knot points. Based on the mean-value theorem, the derivative of B-spline basis functions in relation to parameters can be estimated to online adjust these parameters. In addition, the validity of the proposed scheme is verified through an experiment on a servo motor system which is controlled by the output voltage of the Buck DC-DC converter.

This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists in two su cient conditions that can be tested on a digital computer. If the rst condition is satis ed then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution. If the second condition is satis ed then the multidimensional interval is guaranteed to contain no exact solution. This study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations.

This paper proposes a validation method for solutions of linear complementarity problems. The validation procedure consists of two sufficient conditions that can be tested on a digital computer. If the first condition is satisfied then a given multidimensional interval centered at an approximate solution of the problem is guaranteed to contain an exact solution. If the second condition is satisfied then the multidimensional interval is guaranteed to contain no exact solution. This study is based on the mean value theorem for absolutely continuous functions and the reformulation of linear complementarity problems as nonsmooth nonlinear systems of equations. Classification (1991): 65K10, 90C33

In this paper, we propose a method for state estimation of nonlinear systems represented by Takagi-Sugeno (T-S) models with unmeasurable premise variables. The main result is established using the differential mean value theorem which provides a T-S representation of the differential equation generating the state estimation error. This allows to extend some results obtained in the case of measurable premise variables to the unmeasurable one. Using the Lyapunov theory, stability conditions are obtained and expressed in term of linear matrix inequalities. Furthermore, an extension for observer design with disturbance attenuation performance is proposed. Finally, this approach is illustrated on a DC series motor and compared to the existing approaches.

We describe two new ways of efficiently deriving continuous reach set information for hybrid systems. In both cases, we overapproximate the differential equations to constraints that are then solved using a solver for first-order predicate logical formulae over the reals. We prove some properties about the amount of introduced over-approximations. Moreover, we embed the results in our earlier method for verification of hybrid systems using abstraction refinement and provide some timings that document the resulting improvement.