Automatic choice of parameters at approximation of functions by rational splines

Automatic Choice ◽

Derivatives Of ◽

Best Parameters ◽

Approximation Of A Function

We obtain necessary and sufficient conditions put on the parameters of rational splines that provide given order of approximation of smooth functions. We point out the formulas of asymptotically the best parameters of rational splines that, while providing the best order of approximation of a function by rational splines, do not contain information about the values of higher derivatives of a function.

The classical theory of calculus of variations for generalized functions

Advances in Nonlinear Analysis ◽

10.1515/anona-2017-0150 ◽

2017 ◽

Vol 8 (1) ◽

pp. 779-808 ◽

Cited By ~ 1

Author(s):

Alexander Lecke ◽

Lorenzo Luperi Baglini ◽

Paolo Giordano

Keyword(s):

Calculus Of Variations ◽

Classical Theory ◽

Sufficient Conditions ◽

Generalized Functions ◽

Conjugate Points ◽

Lagrange Equations ◽

Smooth Functions ◽

Nonlinear Properties ◽

Schwartz Distributions ◽

Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

Monotonicity and complete monotonicity of two functions constituted via three derivatives of a function involving trigamma function

10.31219/osf.io/whb2q ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Sufficient Conditions ◽

Laplace Transforms ◽

Complete Monotonicity ◽

Convolution Theorem ◽

Monotonic Functions ◽

Completely Monotonic ◽

In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author (1) presents the decreasing monotonicity of a ratio constituted via three derivatives of a function involving trigamma function; (2) discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. These results conform previous guesses posed by the author.

Lagrangian for Circuits with Higher-Order Elements

Entropy ◽

10.3390/e21111059 ◽

2019 ◽

Vol 21 (11) ◽

pp. 1059 ◽

Cited By ~ 2

Author(s):

Zdenek Biolek ◽

Dalibor Biolek ◽

Viera Biolkova

Keyword(s):

Variational Principle ◽

Equations Of Motion ◽

Sufficient Conditions ◽

Higher Order ◽

The State ◽

Even Order ◽

Independent Variable ◽

Derivatives Of ◽

State Functions

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.

Higher-order conditions for strict local Pareto minima for problems with partial order introduced by a polyhedral cone

Journal of Applied Analysis ◽

10.1515/jaa-2018-0005 ◽

2018 ◽

Vol 24 (1) ◽

pp. 45-54

Author(s):

Aleksandra Stasiak

Keyword(s):

Partial Order ◽

Optimization Problems ◽

Sufficient Conditions ◽

Polyhedral Cone ◽

Directional Derivatives ◽

Pareto Minima ◽

Vector Valued Functions ◽

Vector Valued ◽

Abstract Using the definitions of μ-th order lower and upper directional derivatives of vector-valued functions, introduced in Rahmo and Studniarski (J. Math. Anal. Appl. 393 (2012), 212–221), we provide some necessary and sufficient conditions for strict local Pareto minimizers of order μ for optimization problems where the partial order is introduced by a pointed polyhedral cone with non-empty interior.

Necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic

10.31219/osf.io/6ar4p ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Exponential Function ◽

Sufficient Conditions ◽

Laplace Transforms ◽

Convolution Theorem ◽

Completely Monotonic ◽

In the paper, by convolution theorem for the Laplace transforms, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic.

Complete monotonicity of a difference constituted by four derivatives of a function involving trigamma function

10.20944/preprints202011.0343.v1 ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Sufficient Conditions ◽

Laplace Transforms ◽

Complete Monotonicity ◽

Convolution Theorem ◽

Monotonic Functions ◽

Completely Monotonic ◽

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic.

Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm191111014q ◽

2021 ◽

pp. 14-14

Author(s):

Feng Qi

Keyword(s):

Exponential Function ◽

Sufficient Conditions ◽

Laplace Transforms ◽

Complete Monotonicity ◽

Monotonic Functions ◽

Completely Monotonic ◽

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic. These results generalize corresponding known ones.

Disturbance Decoupling Problem: Logic-Dynamic Approach-Based Solution

Symmetry ◽

10.3390/sym11040555 ◽

2019 ◽

Vol 11 (4) ◽

pp. 555 ◽

Cited By ~ 1

Author(s):

Alexey Zhirabok

Keyword(s):

Control Systems ◽

Sufficient Conditions ◽

Algebraic Approach ◽

Dynamic Measurement ◽

Dynamic Approach ◽

Smooth Functions ◽

System Description ◽

Disturbance Decoupling ◽

Measurement Feedback ◽

Necessary And Sufficient

This paper considers the disturbance decoupling problem by the dynamic measurement feedback for discrete-time nonlinear control systems. To solve this problem, the algebraic approach, called the logic-dynamic approach, is used. Such an approach assumes that the system description may contain non-smooth functions. Necessary and sufficient conditions are obtained in terms of matrices similar to controlled and ( h , f ) -invariant functions. Furthermore, procedures are developed to determine the corresponding matrices and the dynamic measurement feedback.

VARIATION PROBLEM CONTAINING SECOND DERIVATIVES OF UNKNOWN FUNCTIONS

Chronos: natural and technical sciences ◽

10.52013/2712-9691-34-1-6 ◽

2021 ◽

Vol 6 (1(34)) ◽

pp. 30-42

Author(s):

Misraddin Allahverdi oglu Sadigov

Keyword(s):

Variational Problem ◽

Sufficient Conditions ◽

Continuous Functions ◽

Necessary Condition ◽

Absolutely Continuous Functions ◽

Absolutely Continuous ◽

Variation Problem ◽

Second Derivatives ◽

The property subdifferential of an integral and terminal functional in a space of the type of absolutely continuous functions is studied. Necessary and sufficient conditions for an extremum for a variational problem containing the second derivatives of unknown functions are obtained. With the help of the subdifferential introduced by the author, a nonconvex generalized variational problem containing the second derivatives of unknown functions is considered, and the necessary condition for an extremum is obtained.

Complete monotonicity of a difference defined by four derivatives of a function containing trigamma function

10.31219/osf.io/56c2s ◽

2020 ◽

Author(s):

Feng Qi

Keyword(s):

Gamma Function ◽

Integral Inequality ◽

Sufficient Conditions ◽

Complete Monotonicity ◽

Logarithmic Convexity ◽

Monotonic Functions ◽

Completely Monotonic ◽

In the paper, by virtue of convolution theorem for the Laplace transforms, logarithmic convexity of the gamma function, Bernstein's theorem for completely monotonic functions, and other techniques, the author finds necessary and sufficient conditions for a difference defined by four derivatives of a function containing trigamma function to be completely monotonic. Moreover, by virtue of Cebysev integral inequality, the author presents logarithmic convexity of the sequence of polygamma functions.