Władysław Ślebodziński 1931, the definition of the Lie derivative

2012 ◽  
pp. 3-14
Author(s):  
Andrzej Krasiński ◽  
George F. R. Ellis ◽  
Malcolm A. H. MacCallum
Keyword(s):  
Lie Derivative ◽  
Definition Of

scholarly journals A GEOMETRICAL INTERPRETATION OF RENORMALIZATION GROUP FLOW

1994 ◽  
Vol 09 (08) ◽  
pp. 1261-1286 ◽  
Author(s):  
BRIAN P. DOLAN
Keyword(s):  
Renormalization Group ◽  
Euclidean Space ◽  
Point Of View ◽  
Dimensional Euclidean Space ◽  
Lie Derivative ◽  
Coupling Constant ◽  
Four Dimensions ◽  
Anomalous Dimensions ◽  
Definition Of ◽  
Group Flow

The renormalization group (RG) equation in D-dimensional Euclidean space, RD, is analyzed from a geometrical point of view. A general form of the RG equation is derived which is applicable to composite operators as well as tensor operators (on RD) which may depend on the Euclidean metric. It is argued that physical N-point amplitudes should be interpreted as rank N covariant tensors on the space of couplings, [Formula: see text], and that the RG equation can be viewed as an equation for Lie transport on [Formula: see text] with respect to the vector field generated by the β functions of the theory. In one sense it is nothing more than the definition of a Lie derivative. The source of the anomalous dimensions can be interpreted as being due to the change of the basis vectors on [Formula: see text] under Lie transport. The RG equation acts as a bridge between Euclidean space and coupling constant space in that the effect on amplitudes of a diffeomorphism of RD (that of dilations) is completely equivalent to a diffeomorphism of [Formula: see text] generated by the β functions of the theory. A form of the RG equation for operators is also given. These ideas are developed in detail for the example of massive λφ4 theory in four dimensions.

Ignorable coordinates and steady motion in classical mechanics

1965 ◽  
Vol 61 (1) ◽  
pp. 211-222 ◽  
Author(s):  
C. W. Kilmister ◽  
F. A. E. Pirani
Keyword(s):  
Dynamical System ◽  
Classical Mechanics ◽  
Steady Motion ◽  
Lie Derivative ◽  
Covariant Description ◽  
Classical Dynamical System ◽  
Definition Of

AbstractIt is shown, for a classical dynamical system with a Lagrangian, that the existence of an ignorable coordinate is equivalent to the vanishing of a certain Lie derivative. On this covariant description is based a new definition of steady motion. A definition given earlier by Synge is criticized.

The Lie Derivative and Interior Multiplication

2020 ◽  
pp. 81-86
Author(s):  
Loring W. Tu
Keyword(s):  
Vector Field ◽  
Differential Form ◽  
Differential Forms ◽  
Product Formula ◽  
Lie Derivative ◽  
Cartan Model ◽  
Definition Of

This chapter reviews two operations on differential forms, the Lie derivative and interior multiplication. These are necessary to the definition of invariant forms, horizontal forms, and basic forms in the construction of the Cartan model. The chapter then looks at the Lie derivative of a vector field and of a differential form. The Lie derivative of a differential form is defined in a similar way to the Lie derivative of a vector field, but the chapter uses the pullback instead of the pushforward to compare nearby values. One can rearrange the product formula so that it becomes the global formula for the Lie derivative. Meanwhile, the interior multiplication is also called the contraction.

scholarly journals TELEPARALLEL KILLING VECTORS OF THE EINSTEIN UNIVERSE

2008 ◽  
Vol 23 (13) ◽  
pp. 963-969 ◽  
Author(s):  
M. SHARIF ◽  
M. JAMIL AMIR
Keyword(s):  
General Relativity ◽  
Lie Derivative ◽  
Einstein Universe ◽  
Killing Vectors ◽  
Rank Tensor ◽  
Teleparallel Theory ◽  
Theory Of Gravity ◽  
Definition Of

In this paper we establish the definition of the Lie derivative of a second rank tensor in the context of teleparallel theory of gravity and also extend it for a general tensor of rank p + q. This definition is then used to find Killing vectors of the Einstein universe. It turns out that Killing vectors of the Einstein universe in the teleparallel theory are the same as in general relativity.

scholarly journals The Definition of Lie Derivative

1960 ◽  
Vol 12 (1) ◽  
pp. 27-29 ◽  
Author(s):  
T. J. Willmore
Keyword(s):  
Tensor Field ◽  
Applied Mathematics ◽  
Differential Invariant ◽  
Lie Derivative ◽  
Lie Derivation ◽  
Wide Range ◽  
Definition Of ◽  
Recent Monograph ◽  
The Given ◽  
Derivatives Of

The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.

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.

Sign in / Sign up

Export Citation Format

Share Document