Projective Geometry in the One-Dimensional Affine Group

1964 ◽  
Vol 16 ◽  
pp. 683-700 ◽  
Author(s):  
Hans Schwerdtfeger
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Simple Group ◽  
Final Section ◽  
Theoretical Interpretation ◽  
Special Group ◽  
One Dimensional ◽  
The One ◽  
Straight Lines ◽  
Geometrical Investigation

The idea of considering the set of the elements of a group as a space, provided with a topology, measure, or metric, connected somehow with the group operation, has been used often in the work of E. Cartan and others. In the present paper we shall study a very special group whose space can be embedded naturally into a projective plane and where the straight lines have a very simple group-theoretical interpretation. It remains to be seen whether this geometrical embedding in a projective space can be extended to other classes of groups and whether the method could become an instrument of geometrical investigation, like co-ordinates or reflections. In the final section it is shown how a geometrical theorem may lead to relations within the group.

A Pappus Type Theorem in the Affine Group

1968 ◽  
Vol 11 (4) ◽  
pp. 547-554
Author(s):  
R. Paré
Keyword(s):  
Projective Plane ◽  
Final Section ◽  
Type Theorem ◽  
Affine Group ◽  
Real Numbers ◽  
One Dimensional ◽  
Master's Thesis ◽  
Geometrical Concepts ◽  
The One ◽  
The Relationship

In [3] H. Schwerdtfeger embedded the one-dimensional affine group over the real numbers in the projective plane. The relationship between group-theoretical properties and geometrical concepts was studied.In this paper the methods of [3] are used to prove Pappus' theorem. In the final section we give a similar theorem for (4n+2)-gons.This paper is a generalization of part of my master's thesis, written under the direction of Professor H. Schwerdtfeger.

scholarly journals A general renormalization procedure on the one-dimensional lattice and decay of correlations

2019 ◽  
Vol 19 (01) ◽  
pp. 1950003
Author(s):  
Artur O. Lopes
Keyword(s):  
Fixed Point ◽  
Final Section ◽  
Polynomial Decay ◽  
Renormalization Procedure ◽  
Real Numbers ◽  
One Dimensional ◽  
Point Potential ◽  
Renormalization Operator ◽  
The One ◽  
Natural Way

We present a general form of renormalization operator [Formula: see text] acting on potentials [Formula: see text]. We exhibit the analytical expression of the fixed point potential [Formula: see text] for such operator [Formula: see text]. This potential can be expressed in a natural way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by Baraviera, Leplaideur and Lopes where the fixed point potential [Formula: see text] was of Hofbauer type. For the potentials of Hofbauer type (a well-known case of phase transition) the decay is like [Formula: see text], [Formula: see text]. Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential [Formula: see text] of our general renormalization procedure. In some cases we get polynomial decay like [Formula: see text], [Formula: see text], and in others a decay faster than [Formula: see text], when [Formula: see text]. The potentials [Formula: see text] we consider here are elements of the so-called family of Walters’ potentials on [Formula: see text] which generalizes a family of potentials considered initially by Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice [Formula: see text] of real numbers varying with [Formula: see text] there exists a potential [Formula: see text] on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

scholarly journals DERIVED SCHWARZ MAP OF THE HYPERGEOMETRIC DIFFERENTIAL EQUATION AND A PARALLEL FAMILY OF FLAT FRONTS

2008 ◽  
Vol 19 (07) ◽  
pp. 847-863 ◽  
Author(s):  
TAKESHI SASAKI ◽  
KOTARO YAMADA ◽  
MASAAKI YOSHIDA
Keyword(s):  
Differential Equation ◽  
Projective Space ◽  
Hyperbolic Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Flat Surfaces ◽  
Hypergeometric Differential Equation ◽  
The One ◽  
Dimensional Projective Space

In one of the previous papers, we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce derived Schwarz maps of the hypergeometric differential equation and, second, we construct a parallel family of flat fronts connecting the classical Schwarz map and the derived Schwarz map.

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

Adiabatic large polarons in anisotropic molecular crystals

2011 ◽  
Vol 35 (1) ◽  
pp. 15-27
Author(s):  
Zoran Ivić ◽  
Željko Pržulj
Keyword(s):  
Energy Transfer ◽  
Effective Mass ◽  
Molecular Crystals ◽  
Single Chain ◽  
Proper Understanding ◽  
One Dimensional ◽  
Interchain Coupling ◽  
Large Polaron ◽  
The One

Adiabatic large polarons in anisotropic molecular crystals We study the large polaron whose motion is confined to a single chain in a system composed of the collection of parallel molecular chains embedded in threedimensional lattice. It is found that the interchain coupling has a significant impact on the large polaron characteristics. In particular, its radius is quite larger while its effective mass is considerably lighter than that estimated within the one-dimensional models. We believe that our findings should be taken into account for the proper understanding of the possible role of large polarons in the charge and energy transfer in quasi-one-dimensional substances.

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