The Stieltjes Function--Definition and Properties

1988 ◽  
Vol 51 (183) ◽  
pp. 281
Author(s):  
Jan Bohman ◽  
Carl-Erik Froberg
Keyword(s):  
Stieltjes Function ◽  
Function Definition

scholarly journals The Stieltjes function―definition and properties

1988 ◽  
Vol 51 (183) ◽  
pp. 281-281
Author(s):  
Jan Bohman ◽  
Carl-Erik Fr{öberg
Keyword(s):  
Stieltjes Function ◽  
Function Definition

scholarly journals Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 477
Author(s):  
Katarzyna Górska ◽  
Andrzej Horzela
Keyword(s):  
Stochastic Processes ◽  
Spectral Function ◽  
Evolution Equations ◽  
Memory Function ◽  
Relaxation Phenomena ◽  
Stieltjes Function ◽  
Infinitely Divisible ◽  
Spectral Functions ◽  
Debye Relaxation ◽  
Relaxation Functions

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function M(t), which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function M(t) with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory k(t). Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process.

A function definition operator

1981 ◽  
Vol 12 (1) ◽  
pp. 142-145 ◽  
Author(s):  
Kenneth E. Iverson ◽  
Peter K. Wooster
Keyword(s):  
Function Definition

scholarly journals Modelowanie układów sterowania z użyciem pochodnych ułamkowych

2020 ◽  
pp. 33-50
Author(s):  
Grzegorz Dec ◽  
Keyword(s):  
Experimental Data ◽  
System Theory ◽  
Control System ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
System Control ◽  
Current State ◽  
Function Definition ◽  
Definition Of ◽  
State Of Art

In the paper is presented review of some approaches corelated with subject of using fractional derivatives in control system theory. Popular algorithms used in the industry are presented, along with relating designing methodology. Using of fractional derivatives calculations is relatively new concept, but constantly getting increasing interest. Deliberation in recent years indicate that many scientific problems like thermodynamic or biology problems can be well considered and modeled by fractional order derivatives. On the market there is available tools that support a processes of identification and regulators designing, based on experimental data. One of such tools are toolbox CRONE for MATLAB, which contains three modules: mathematical, identifying, system control designing. That toolbox allows implementation of CRONE regulators with different level of complexity. Other tool is FOMCON, which also is a toolbox for MATLAB and it is based on already existed toolbox FOTF. FOMCON allows to identifying of control system and PIλDµ regulator designing. This article is aiming to present current state of art, discussion about existing tools and concepts correlated with fractional order derivatives and their usage in control system theory, like: gamma function, definition of fractional derivative, Laplace transform and basics of control system theory.

London Underground Deep Tube Upgrade Programme: System Function Definition Model Case Study

2019 ◽  
Vol 29 (1) ◽  
pp. 988-1002
Author(s):  
Nathan Everett ◽  
Michael Coultharde-Steer ◽  
Luke Fischer
Keyword(s):  
System Function ◽  
Model Case ◽  
London Underground ◽  
Function Definition
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