A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical

1999 ◽  
Vol 4 (1) ◽  
pp. 35-52 ◽  
Author(s):  
L. Hille ◽  
G. R�hrle
Keyword(s):  
Finite Number ◽  
Unipotent Radical ◽  
Classical Groups ◽  
Parabolic Subgroups

Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

2021 ◽  
Vol 29 (6) ◽  
pp. 835-850
Author(s):  
Vladislav Kruglov ◽  
◽  
Olga Pochinka ◽  
◽  
Keyword(s):  
Finite Number ◽  
Limit Cycle ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Smooth Flow ◽  
Direction Of Movement ◽  
Smale Flows ◽  
Phase Surface ◽  
Flows On Surfaces

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another.

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

scholarly journals Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

Finiteness conditions for CW complexes. Il

1966 ◽  
Vol 295 (1441) ◽  
pp. 129-139 ◽  
Keyword(s):  
Topological Space ◽  
Finite Number ◽  
Simple Form ◽  
Finite Dimension ◽  
Homotopy Equivalent ◽  
Finite Complex ◽  
Finiteness Conditions ◽  
Cw Complexes ◽  
Number Of Cells

A CW complex is a topological space which is built up in an inductive way by a process of attaching cells. Spaces homotopy equivalent to CW complexes play a fundamental role in topology. In the previous paper with the same title we gave criteria (in terms of more-or-less standard invariants of the space) for a CW complex to be homotopy equivalent to one of finite dimension, or to one with a finite number of cells in each dimension, or to a finite complex. This paper contains some simplification of these results. In addition, algebraic machinery is developed which provides a rough classification of CW complexes homotopy equivalent to a given one (the existence clause of the classification is the interesting one). The results would take a particularly simple form if a certain (rather implausible) conjecture could be established.

On certain types of isolated s-ple points on algebraic primals in Sd

1962 ◽  
Vol 58 (3) ◽  
pp. 465-475
Author(s):  
J. Herszberg
Keyword(s):  
Finite Number ◽  
Tangent Cone ◽  
Double Point ◽  
Complete Classification ◽  
Singular Points ◽  
Multiple Points ◽  
Rank Zero ◽  
Double Points

Singular points on irreducible primals were investigated briefly by C. Segre(8), where the author classified multiple points by the nature of the nodal tangent cone. For surfaces the problem of classification was investigated by, amongst others, Du Val(1) and a complete classification of isolated double points of surfaces lying on non-singular threefolds was given by Kirby(5). In (3) we classified certain types of double points on algebraic primals in Sn. An isolated double point which after a finite number of resolutions gave rise to at most a finite number of isolated double points was called a double point of rank zero. We found that the only isolated double points of rank zero are those which are analogous to the binodes, unodes and exceptional unodes (2) of surfaces.

scholarly journals On holomorphic flows on Stein surfaces: Transversality, dicriticalness and stability

2014 ◽  
Vol 25 (10) ◽  
pp. 1450093
Author(s):  
T. Ito ◽  
B. Scárdua ◽  
Y. Yamagishi
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Finite Number ◽  
Sufficient Conditions ◽  
First Integral ◽  
Holomorphic Vector Field ◽  
Necessary And Sufficient ◽  
Holonomy Map

We study the classification of the pairs (N, X) where N is a Stein surface and X is a ℂ-complete holomorphic vector field with isolated singularities on N. We describe the role of transverse sections in the classification of X and give necessary and sufficient conditions on X in order to have N biholomorphic to ℂ2. As a sample of our results, we prove that N is biholomorphic to ℂ2 if H2(N, ℤ) = 0, X has a finite number of singularities and exhibits a singularity with three separatrices or, equivalently, a singularity with first jet of the form [Formula: see text] where λ1/λ2 ∈ ℚ+. We also study flows with many periodic orbits (i.e. orbits diffeomorphic to ℂ*), in a sense we will make clear, proving they admit a meromorphic first integral or they exhibit some special periodic orbit, whose holonomy map is a non-resonant nonlinearizable diffeomorphism map.

The Plancherel formula for parabolic subgroups of the classical groups

1978 ◽  
Vol 34 (1) ◽  
pp. 120-161 ◽  
Author(s):  
Ronald L. Lipsman ◽  
Joseph A. Wolf
Keyword(s):  
Plancherel Formula ◽  
Classical Groups ◽  
Parabolic Subgroups
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