Classification of surfaces in three-sphere in Lie sphere geometry

Three Sphere ◽

Lie Sphere Geometry ◽

Sphere Geometry ◽

Necessary And Sufficient ◽

Lie Sphere

We studied plane curves in Lie sphere geometry in [YY]. Especially we constructed Lie frames of curves in S2 and classified them by the Lie equivalence. In this paper we are concerned with surfaces in S3. We construct Lie frames and classify them. We moreover obtain the necessary and sufficient condition that two surfaces are Lie equivalent.

The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

Special Matrices ◽

10.1515/spma-2019-0019 ◽

2019 ◽

Vol 7 (1) ◽

pp. 257-262

Author(s):

Kenji Toyonaga

Keyword(s):

Symmetric Matrix ◽

Symmetric Matrices ◽

Sufficient Condition ◽

Geometric Multiplicity ◽

Necessary And Sufficient ◽

Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

Some results on the classification of expansive automorphisms of compact abelian groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008701 ◽

1996 ◽

Vol 16 (1) ◽

pp. 45-50 ◽

Cited By ~ 10

Author(s):

Fabio Fagnani

Keyword(s):

Finite Group ◽

Topological Entropy ◽

Abelian Groups ◽

Topological Classification ◽

Sufficient Condition ◽

Compact Abelian Groups ◽

A Finite Group ◽

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

Cuspidal and noncuspidal robot manipulators

Robotica ◽

10.1017/s0263574707003761 ◽

2007 ◽

Vol 25 (6) ◽

pp. 677-689 ◽

Cited By ~ 22

Author(s):

Philippe Wenger

Keyword(s):

Robot Manipulators ◽

Sufficient Condition ◽

Geometric Conditions ◽

Necessary And Sufficient ◽

Full Classification

SUMMARYThis article synthezises the most important results on the kinematics of cuspidal manipulators i.e. nonredundant manipulators that can change posture without meeting a singularity. The characteristic surfaces, the uniqueness domains and the regions of feasible paths in the workspace are defined. Then, several sufficient geometric conditions for a manipulator to be noncuspidal are enumerated and a general necessary and sufficient condition for a manipulator to be cuspidal is provided. An explicit DH-parameter-based condition for an orthogonal manipulator to be cuspidal is derived. The full classification of 3R orthogonal manipulators is provided and all types of cuspidal and noncuspidal orthogonal manipulators are enumerated. Finally, some facts about cuspidal and noncuspidal 6R manipulators are reported.

On the Differential Invariants of the Laguerre Group

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100002851 ◽

1924 ◽

Vol 22 (2) ◽

pp. 113-123 ◽

Cited By ~ 4

Author(s):

Tadahiko Kubota

Keyword(s):

Analogous Problem ◽

Differential Invariants ◽

Plane Curves ◽

Sufficient Condition ◽

Functional Relations ◽

Laguerre Transformation ◽

1. In a paper “Beiträge zur Inversionsgeometric,” which will appear in the forthcoming volume of the Science Reports of the Tôhoku Imperial University, I have treated the problem of determining the necessary and sufficient condition in order that two plane curves which are given by natural equations should be transformable into each other by inversion. In connection with that paper I propose here to treat the analogous problem for Laguerre transformations:Having given two plane curves, by natural equations, in which the functional relations are all supposed to be analytic, it is required to determine the necessary and sufficient condition in order that the two plane curves should be transformable into each other by a Laguerre transformation.

Plane curves in Lie sphere geometry

Kodai Mathematical Journal ◽

10.2996/kmj/1138043650 ◽

1996 ◽

Vol 19 (3) ◽

pp. 322-340

Author(s):

Takayoshi Yamazaki ◽

Atsuko Yamada Yoshikawa

Keyword(s):

Plane Curves ◽

Lie Sphere Geometry ◽

Sphere Geometry ◽

Lie Sphere

A class of simple tracially AFC*-algebras

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202012917 ◽

2002 ◽

Vol 31 (5) ◽

pp. 271-282

Author(s):

N. E. Livingston

Keyword(s):

Inductive Limit ◽

Broad Class ◽

Sufficient Condition ◽

The concept of a tracially AF (TAF)C*-algebra was introduced recently to aid in the classification of nuclearC*-algebrasHere, we construct and study a broad class of inductive-limitC*-algebras. We give a numerical condition which, when satisfied, ensures that the corresponding algebra in our construction has the TAF property. We further give a necessary and sufficient condition under which certain of theseC*-algebras are TAF.

The classification of free algebras of orthogonal modular forms

Compositio Mathematica ◽

10.1112/s0010437x21007429 ◽

2021 ◽

Vol 157 (9) ◽

pp. 2026-2045

Author(s):

Haowu Wang

Keyword(s):

Modular Forms ◽

Symmetric Domain ◽

Necessary Condition ◽

Sufficient Condition ◽

Classification Result ◽

Type Iv ◽

Hyperbolic Planes ◽

We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being free. From the necessary condition, we derive a classification result. Let $M$ be an even lattice of signature $(2,n)$ splitting two hyperbolic planes. Suppose $\Gamma$ is a subgroup of the integral orthogonal group of $M$ containing the discriminant kernel. It is proved that there are exactly 26 groups $\Gamma$ such that the space of modular forms for $\Gamma$ is a free algebra. Using the sufficient condition, we recover some well-known results.

DISCRETE AND DENSE SUBGROUPS ACTING ON COMPLEX HYPERBOLIC SPACE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972708000622 ◽

2008 ◽

Vol 78 (2) ◽

pp. 211-224 ◽

Cited By ~ 5

Author(s):

WENSHENG CAO

Keyword(s):

Hyperbolic Space ◽

Complex Hyperbolic Space ◽

Sufficient Condition ◽

Dense Subgroup ◽

Discreteness Criteria ◽

Dense Subgroups ◽

AbstractIn this paper, we study the discreteness criteria for nonelementary subgroups of U(1,n;ℂ) acting on complex hyperbolic space. Several discreteness criteria are obtained. As applications, we obtain a classification of nonelementary subgroups of U(1,n;ℂ) and show that any dense subgroup of SU(1,n;ℂ) contains a dense subgroup generated by at most n elements when n≥2. We also obtain a necessary and sufficient condition for the normalizer of a discrete and nonelementary subgroup in SU(1,n;ℂ) to be discrete.