An algebraic definition of fractional differentiation

1975 ◽  
pp. 226-231
Author(s):  
Peter D. Johnson
Keyword(s):  
Fractional Differentiation ◽  
Definition Of

scholarly journals Initial value problems should not be associated to fractional model descriptions whatever the derivative definition used

2021 ◽  
Vol 6 (10) ◽  
pp. 11318-11329
Author(s):  
Jocelyn SABATIER ◽  
◽  
Christophe FARGES
Keyword(s):  
Initial Value Problems ◽  
Initial Conditions ◽  
Model Analysis ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Validity Of Results ◽  
Fractional Model ◽  
Definition Of

<abstract> <p>The paper shows that the Caputo definition of fractional differentiation is problematic if it is used in the definition of a time fractional model and if initial conditions are taken into account. The demonstration is done using simple examples (or counterexamples). The analysis is extended to the Riemann-Liouville and Grünwald-Letnikov definitions. These results thus question the validity of results produced in the field of time fractional model analysis in which initial conditions are involved.</p> </abstract>

scholarly journals ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE

2017 ◽  
Vol 23 (2) ◽  
pp. 32-43
Author(s):  
M. V. Kukushkin
Keyword(s):  
Fractional Calculus ◽  
Fractional Integral ◽  
Adjoint Operator ◽  
Sufficient Conditions ◽  
Fractional Differentiation ◽  
Qualitative Properties ◽  
Dimensional Theory ◽  
One Dimensional ◽  
Integral Equality ◽  
Definition Of

In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

ON STABILITY OF COMMENSURATE FRACTIONAL ORDER SYSTEMS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250084 ◽  
Author(s):  
JOCELYN SABATIER ◽  
CHRISTOPHE FARGES
Keyword(s):  
Fractional Order ◽  
Original Work ◽  
Closed Loop ◽  
Initial Conditions ◽  
Stability Theorem ◽  
Fractional Order Systems ◽  
Fractional Differentiation ◽  
Starting Point ◽  
Definition Of

This paper proposes a new proof of the Matignon's stability theorem. This theorem is the starting point of numerous results in the field of fractional order systems. However, in the original work, its proof is limited to a fractional order ν such that 0 < ν < 1. Moreover, it relies on Caputo's definition for fractional differentiation and the study of system trajectories for non-null initial conditions which is now questionable in regard of recent works. The new proof proposed here is based on a closed loop realization and the application of the Nyquist theorem. It does not rely on a peculiar definition of fractional differentiation and is valid for orders ν such that 1 < ν < 2.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

1975 ◽  
Vol 26 ◽  
pp. 21-26
Keyword(s):  
Coordinate System ◽  
Working Group ◽  
General Character ◽  
Coordinate Systems ◽  
Reference Coordinate System ◽  
Group 2 ◽  
Definition Of ◽  
Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

Symbiotic Stars at Optical, Infrared and Radio Wavelengths

1979 ◽  
Vol 46 ◽  
pp. 125-149 ◽  
Author(s):  
David A. Allen
Keyword(s):  
Temperature Regime ◽  
Type Star ◽  
Stellar Systems ◽  
Symbiotic Stars ◽  
Lower Excitation ◽  
Definition Of ◽  
Original Type

No paper of this nature should begin without a definition of symbiotic stars. It was Paul Merrill who, borrowing on his botanical background, coined the termsymbioticto describe apparently single stellar systems which combine the TiO absorption of M giants (temperature regime ≲ 3500 K) with He II emission (temperature regime ≳ 100,000 K). He and Milton Humason had in 1932 first drawn attention to three such stars: AX Per, CI Cyg and RW Hya. At the conclusion of the Mount Wilson Ha emission survey nearly a dozen had been identified, and Z And had become their type star. The numbers slowly grew, as much because the definition widened to include lower-excitation specimens as because new examples of the original type were found. In 1970 Wackerling listed 30; this was the last compendium of symbiotic stars published.

Immunoferritin localization of intracellular antigens in positively stained frozen sections

1978 ◽  
Vol 36 (2) ◽  
pp. 168-169
Author(s):  
K. T. Tokuyasu
Keyword(s):  
Surface Tension ◽  
Water Surface ◽  
Thin Layers ◽  
The Other ◽  
Positive Staining ◽  
Frozen Sections ◽  
The Past ◽  
Ultrathin Frozen Sections ◽  
Definition Of

During the past investigations of immunoferritin localization of intracellular antigens in ultrathin frozen sections, we found that the degree of negative staining required to delineate u1trastructural details was often too dense for the recognition of ferritin particles. The quality of positive staining of ultrathin frozen sections, on the other hand, has generally been far inferior to that attainable in conventional plastic embedded sections, particularly in the definition of membranes. As we discussed before, a main cause of this difficulty seemed to be the vulnerability of frozen sections to the damaging effects of air-water surface tension at the time of drying of the sections.Indeed, we found that the quality of positive staining is greatly improved when positively stained frozen sections are protected against the effects of surface tension by embedding them in thin layers of mechanically stable materials at the time of drying (unpublished).

Initial Polymerization Sites Indicated During Mitochondrial Oxidation of Diaminobenzidine after Toluene Treatment

1977 ◽  
Vol 35 ◽  
pp. 408-409
Author(s):  
W. A. Shannon ◽  
M. A. Matlib
Keyword(s):  
Membrane Permeability ◽  
Cytochrome Oxidase ◽  
Rat Liver ◽  
Precise Definition ◽  
Cytochemical Localization ◽  
Oxidative Reaction ◽  
Mitochondrial Oxidation ◽  
Definition Of ◽  
Exogenous Substrates

Numerous studies have dealt with the cytochemical localization of cytochrome oxidase via cytochrome c. More recent studies have dealt with indicating initial foci of this reaction by altering incubation pH (1) or postosmication procedure (2,3). The following study is an attempt to locate such foci by altering membrane permeability. It is thought that such alterations within the limits of maintaining morphological integrity of the membranes will ease the entry of exogenous substrates resulting in a much quicker oxidation and subsequently a more precise definition of the oxidative reaction.The diaminobenzidine (DAB) method of Seligman et al. (4) was used. Minced pieces of rat liver were incubated for 1 hr following toluene treatment (5,6). Experimental variations consisted of incubating fixed or unfixed tissues treated with toluene and unfixed tissues treated with toluene and subsequently fixed.

Application of Scanning Electron Microscopy to Medical Research

1969 ◽  
Vol 27 ◽  
pp. 12-13
Author(s):  
J. D. Hutchison
Keyword(s):  
Electron Microscopy ◽  
Electron Microscope ◽  
Medical Research ◽  
Prosthetic Heart Valve ◽  
School Of Medicine ◽  
Transmission Electron ◽  
Definition Of ◽  
Mechanism Of Failure ◽  
The Brain ◽  
Scanning Electron

When the transmission electron microscope was commercially introduced a few years ago, it was heralded as one of the most significant aids to medical research of the century. It continues to occupy that niche; however, the scanning electron microscope is gaining rapidly in relative importance as it fills the gap between conventional optical microscopy and transmission electron microscopy.IBM Boulder is conducting three major programs in cooperation with the Colorado School of Medicine. These are the study of the mechanism of failure of the prosthetic heart valve, the study of the ultrastructure of lung tissue, and the definition of the function of the cilia of the ventricular ependyma of the brain.

Selection and Preparation of Specific Tissue Regions For Tem Using Large Epoxy-Embedded Blocks

1973 ◽  
Vol 31 ◽  
pp. 324-325 ◽  
Author(s):  
P. M. Lowrie ◽  
W. S. Tyler
Keyword(s):  
Electron Microscopy ◽  
Anatomical Structures ◽  
Ultrastructural Studies ◽  
Transmission Electron ◽  
Microscopic Evaluation ◽  
Embedded Blocks ◽  
Specific Tissue ◽  
Definition Of ◽  
Areas Of Interest ◽  
Selection Of

The importance of examining stained 1 to 2μ plastic sections by light microscopy has long been recognized, both for increased definition of many histologic features and for selection of specimen samples to be used in ultrastructural studies. Selection of specimens with specific orien ation relative to anatomical structures becomes of critical importance in ultrastructural investigations of organs such as the lung. The uantity of blocks necessary to locate special areas of interest by random sampling is large, however, and the method is lacking in precision. Several methods have been described for selection of specific areas for electron microscopy using light microscopic evaluation of paraffin, epoxy-infiltrated, or epoxy-embedded large blocks from which thick sections were cut. Selected areas from these thick sections were subsequently removed and re-embedded or attached to blank precasted blocks and resectioned for transmission electron microscopy (TEM).

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