Equidistribution and Counting of Common Perpendiculars in Quotient Spaces

2019 ◽  
pp. 253-285
Author(s):  
Anne Broise-Alamichel ◽  
Jouni Parkkonen ◽  
Frédéric Paulin
Keyword(s):  
Quotient Spaces

scholarly journals Quotient spaces defined by linear relations

1982 ◽  
Vol 32 (2) ◽  
pp. 227-232
Author(s):  
Árpád Száz ◽  
Géza Száz
Keyword(s):  
Linear Relations ◽  
Quotient Spaces

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

2015 ◽  
Vol 25 (11) ◽  
pp. 1550150 ◽  
Author(s):  
Oxana Cerba Diaconescu ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Parameter Space ◽  
Group Action ◽  
Sufficient Conditions ◽  
Phase Portraits ◽  
Topological Spaces ◽  
Canonical Forms ◽  
Affine Transformations ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Quotient Spaces

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

Quotient Spaces Modulo Reductive Algebraic Groups

1972 ◽  
Vol 95 (3) ◽  
pp. 511 ◽  
Author(s):  
C. S. Seshadri
Keyword(s):  
Algebraic Groups ◽  
Quotient Spaces ◽  
Reductive Algebraic Groups

Locally Compact Hjelmslev Planes and Rings

1981 ◽  
Vol 33 (4) ◽  
pp. 988-1021 ◽  
Author(s):  
J. W. Lorimer
Keyword(s):  
Projective Planes ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Hjelmslev Plane ◽  
Projective Hjelmslev Plane ◽  
Quotient Spaces ◽  
Continuous Maps ◽  
Topological Plane ◽  
Line Point ◽  
Canonical Image

Affine and projective Hjelmslev planes are generalizations of ordinary affine and projective planes where two points (lines) may be joined by (may intersect in) more than one line (point). The elements involved in multiple joinings or intersections are neighbours, and the neighbour relations on points respectively lines are equivalence relations whose quotient spaces define an ordinary affine or projective plane called the canonical image of the Hjelmslev plane. An affine or projective Hjelmslev plane is a topological plane (briefly a TH-plane and specifically a TAH-plane respectively a TPH-plane) if its point and line sets are topological spaces so that the joining of non-neighbouring points, the intersection of non-neighbouring lines and (in the affine case) parallelism are continuous maps, and the neighbour relations are closed.In this paper we continue our investigation of such planes initiated by the author in [38] and [39].

scholarly journals Harmonic analysis on the quotient spaces of Heisenberg groups

1991 ◽  
Vol 123 ◽  
pp. 103-117 ◽  
Author(s):  
Jae-Hyun Yang
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Unitary Representation ◽  
Lie Group ◽  
Theta Series ◽  
Foundations Of Quantum Mechanics ◽  
Nilpotent Lie Group ◽  
Heisenberg Groups ◽  
Quotient Spaces

A certain nilpotent Lie group plays an important role in the study of the foundations of quantum mechanics ([Wey]) and of the theory of theta series (see [C], [I] and [Wei]). This work shows how theta series are applied to decompose the natural unitary representation of a Heisenberg group.

scholarly journals Quantum homogeneous spaces as quantum quotient spaces

1996 ◽  
Vol 37 (5) ◽  
pp. 2388-2399 ◽  
Author(s):  
Tomasz Brzeziński
Keyword(s):  
Homogeneous Spaces ◽  
Quantum Homogeneous Spaces ◽  
Quotient Spaces
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