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A Cauchy-type generalization of Flett's theorem

Markov Lubomir

Demonstratio Mathematica

2021

Vol. 54, nr 1

500--509

We prove a Cauchy-type generalization of Flett’s theorem and note its geometric interpretations. Several other mean value theorems extending further the result, which involve both real and complex functions, are also proved.

Dodatnia pochodna Grünwalda-Letnikova jako pochodna funkcji drogi

Cioć R.

Autobusy : technika, eksploatacja, systemy transportowe

2017

R. 18, nr 12

786--791, CD

W artykule przeanalizowano pochodną Grünwalda-Letnikova ƒ(ƞ)(t) w odniesieniu do klasycznego zagadnienia prędkości, jako pierwszej pochodnej funkcji drogi w czasie ƒ(1)(t). Autor argumentuje, że dodatnia pochodna Grünwalda-Letnikova nie spełnia twierdzenia Lagrange’a, co wiąże się z problemami jednoznacznej fizycznej jej interpretacji.

The paper analyses Grünwald-Letnikov ƒ(ƞ)(t) derivative in space of first order derivative ƒ(1)(t) and also analyses the classical interpretation of derivative of path function as velocity. The author argues that the Grünwald-Letnikov positive derivative does not fulfil the Lagrange Theorem (Mean-Value Theorem for Derivatives) and this problem causes not clear physical interpretation of the Grünwald-Letnikov positive derivative.

Integrals of the one-dimensional continuity equation

Dowkontt S., Dowkontt G.

TTS Technika Transportu Szynowego

2016

R. 23, nr 12

77--81

The authors analyze the method used by Cauchy and Lagrange to obtain the integral of continuity equation. The authors propose their own method of integration using Schwarz’ theorem. As a result, the authors obtain a greater number of possible solutions with a higher level of generality while also being able to identify the basic disadvantages of the Cauchy-Lagrangian method. Further, the authors conducted a detailed interpretation of the results of their solution.

Mean-value theorems and some symmetric means

Matkowski J.

Demonstratio Mathematica

2013

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.