Canonical form of a two-dimensional Riemann manifold

1971 ◽  
Vol 9 (1) ◽  
pp. 38-40
Author(s):  
Yu. M. Rodnyi
Keyword(s):  
Canonical Form ◽  
Two Dimensional ◽  
Riemann Manifold

scholarly journals An Inferentially Many-Valued Two-Dimensional Notion of Entailment

2017 ◽  
Vol 46 (3/4) ◽  
Author(s):  
Carolina Blasio ◽  
João Marcos ◽  
Heinrich Wansing
Keyword(s):  
Canonical Form ◽  
Two Dimensional ◽  
Consequence Relation ◽  
Abstract Characterization ◽  
Suszko’S Thesis

Starting from the notions of q-entailment and p-entailment, a two-dimensional notion of entailment is developed with respect to certain generalized q-matrices referred to as B-matrices. After showing that every purely monotonic singleconclusion consequence relation is characterized by a class of B-matrices with respect to q-entailment as well as with respect to p-entailment, it is observed that, as a result, every such consequence relation has an inferentially four-valued characterization. Next, the canonical form of B-entailment, a two-dimensional multiple-conclusion notion of entailment based on B-matrices, is introduced, providing a uniform framework for studying several different notions of entailment based on designation, antidesignation, and their complements. Moreover, the two-dimensional concept of a B-consequence relation is defined, and an abstract characterization of such relations by classes of B-matrices is obtained. Finally, a contribution to the study of inferential many-valuedness is made by generalizing Suszko’s Thesis and the corresponding reduction to show that any B-consequence relation is, in general, inferentially four-valued.

The bandcount increment scenario. I. Basic structures

2008 ◽  
Vol 464 (2095) ◽  
pp. 1867-1883 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Basic Structure ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Bifurcation Structures ◽  
Multi Band

Bifurcation structures in two-dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In the first part, we report a novel bifurcation scenario formed by crises bifurcations, which includes multi-band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain.

A note regarding ‘On Hamilton's principle for surface waves’

1977 ◽  
Vol 83 (1) ◽  
pp. 159-161 ◽  
Author(s):  
D. Michael Milder
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Canonical Form ◽  
Two Dimensional ◽  
Free Surface Elevation ◽  
Field Equations ◽  
Explicit Reference ◽  
Non Local ◽  
Generalized Coordinate ◽  
Dimensional Surface

The canonical form of the equations for the free-surface elevation and potential of an irrotational fluid is more than a coincidence. The elevation is a ‘generalized coordinate’ field sufficient to define the system Lagrangian without explicit reference to the motion of the fluid interior. The Lagrangian and the associated field equations are complete and self-contained in the two-dimensional surface co-ordinates, but non-local (integro-differential) in form; the canonical equations derived by Miles are just the Hamiltonian counterparts.

scholarly journals Development of Parameters of a Gear Spring Harrow by Planning an Experiment

2019 ◽  
Vol 8 (4) ◽  
pp. 8030-8034
Keyword(s):  
Response Surface ◽  
Response Function ◽  
Canonical Form ◽  
Operating Parameters ◽  
Operating Speed ◽  
Two Dimensional ◽  
Working Process ◽  
Operating Modes ◽  
Optimal Value ◽  
Three Factor

The article presents a method for developing parameters and operating modes of a gear spring harrow using a three-factor experiment of the Bk plan. The regression equation of the machine’s working process, its optimal value (specific traction resistance 432.8 N, angle of inclination of the teeth of the harrow 61.9 degrees, angle of grinding of the tooth 20.98 degrees) and operating parameters (operating speed of the unit 11.7 km / h). The dependences of the response function (specific traction resistance of the tooth-spring harrow) on its parameters and operating modes are represented by mini-max. Hypersurface plots. To determine the nature of the response surface in canonical form near the optimum, two-dimensional sections of its surface are constructed with a family of conjugated isolines, with the help of which the interaction of factors is studied.

The bandcount increment scenario. II. Interior structures

2008 ◽  
Vol 464 (2097) ◽  
pp. 2247-2263 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Parameter Space ◽  
Complex Interaction ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In this second part, we investigate fine substructures nested into the basic structures reported and explained in part I. It is demonstrated that the overall structure of the chaotic domain is caused by a complex interaction of bandcount increment, bandcount adding and bandcount doubling structures, whereby some of them are nested into each other ad infinitum leading to self-similar structures in the parameter space.

The bandcount increment scenario. III. Deformed structures

2008 ◽  
Vol 465 (2101) ◽  
pp. 41-57 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Practical Applications ◽  
Chaotic Region ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

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.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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