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Mean-value type equalities with interchanged function and derivative

100%

Matkowski J.

tom Nr 47

19--27

According to a new mean value-theorem, if a func¬tion f satisfies the classical conditions ensuring the existence and uniqueness of Lagrange’s mean, then there also exists a unique mean M such that ...[wzór]. The main result gives necessary and sufficient conditions for the equality ...[wzór] The relevant equality for the Lagrange mean-value theorem is also considered.

A note on a Cauchy-type mean value theorem

80%

Ivan M.

2002

tom Vol. 35, nr 3

493-494

We prove that a Cauchy-type mean value theorem [E. Wachnicki, Une variante du theoreme de Cauchy de la valeur moyenne, Demonstratio Math., 33 (4) (2000), 737-740] is a particular case of Flett's Mean Value Theorem [T. M. Flett, A mean value theorem, Math. Gazette 42 (1958), 38-39].

Local cone approximations in optimization

60%

Castellani M. , Pappalardo M.

Control and Cybernetics

2007

tom Vol. 36, no 3

583-600

We show how to use intensively local cone approximations to obtain results in some fields of optimization theory such as optimality conditions, constraint qualifications, mean value theorems and error bound.

Generalizations of Lagrange and Cauchy Mean-Value theorems

51%

Matkowski J.

2010

tom Vol. 43, nr 4

765-774

Some generalizations of the Lagrange Mean-Value Theorem and Cauchy Mean-Value Theorem are proved and the extensions of the corresponding classes of means are presented.

Numerical treatment of the generalized time-fractional Huxley-Burgers’ equation and its stability examination

51%

Hadhoud A. R. , Alaal F. E. A. , Abdelaziz A. A. , Radwan T.

tom Vol. 54, nr 1

436--451

In this paper, we show how to approximate the solution to the generalized time-fractional Huxley-Burgers’ equation by a numerical method based on the cubic B-spline collocation method and the mean value theorem for integrals. We use the mean value theorem for integrals to replace the time-fractional derivative with a suitable approximation. The approximate solution is constructed by the cubic B-spline. The stability of the proposed method is discussed by applying the von Neumann technique. The proposed method is shown to be conditionally stable. Several numerical examples are introduced to show the efficiency and accuracy of the method.

Towards historical roots of necessary conditions of optimality: Regula of Peano

41%

Dolecki S. , Greco G. H.

Control and Cybernetics

2007

tom Vol. 36, no 3

491-518

At the end of 19th century Peano discerned vector spaces, differentiability, convex sets, limits of families of sets, tangent cones, and many other concepts, in a modern perfect form. He applied these notions to solve numerous problems. The theorem on necessary conditions of optimality (Regula) is one of these. The formal language of logic that he developed, enabled him to perceive mathematics with great precision and depth. Actually he built mathematics axiomatically based exclusively on logical and set-theoretic primitive terms and properties, which was a revolutionary turning point in the development of mathematics.

Mean-value theorems and some symmetric means

41%

Matkowski J.

Demonstratio Mathematica

2013

tom Vol. 46, nr 3

461--474

Some variants of the Lagrange and Cauchy mean-value theorems lead to the conclusion that means, in general, are not symmetric. They are symmetric iff they coincide (respectively) with the Lagrange and Cauchy means. Under some regularity assumptions, we determine the form of all the relevant symmetric means.