scholarly journals Real submanifolds of maximum complex tangent space at a CR singular point, I

2016 ◽  
Vol 206 (2) ◽  
pp. 293-377 ◽  
Author(s):  
Xianghong Gong ◽  
Laurent Stolovitch
Keyword(s):  
Singular Point ◽  
Tangent Space ◽  
Real Submanifolds ◽  
Complex Tangent

scholarly journals An extension theorem for holomorphic mappings - Corrigenda

1983 ◽  
Vol 94 (1) ◽  
pp. 189-189
Keyword(s):  
Tangent Space ◽  
Extension Theorem ◽  
Levi Form ◽  
Holomorphic Mappings ◽  
Additional Hypothesis ◽  
Complex Tangent

J. C. Wood. ‘An extension theorem for holomorphic mappings.’As pointed out by Y.-T. Siu [MR82d:32021] the theorems and corollary require the additional hypothesis that the boundary ∂X, is hyper-(m–1)-convex where m = dim X, i.e. for all p ∈ ∂X, the sum of the eigenvalues of the restriction of the Levi form of ∂X to the complex tangent space Tp(∂X) ∩ JTp(∂X) is non-negative.

scholarly journals Conical singular points and vector fields

2020 ◽  
Vol 65 (4) ◽  
pp. 649-661
Author(s):  
Sergei N. Burian ◽  
Keyword(s):  
Singular Point ◽  
Tangent Space ◽  
Tangent Vector ◽  
Common Point ◽  
Singular Points ◽  
Configuration Spaces ◽  
Dimensional Torus ◽  
Smooth Curves ◽  
Theoretical Approaches ◽  
First Case

In this article, several examples of mechanical systems which configuration spaces are smooth manifolds with a unique singular point are considered. Configuration spaces are the following: two smooth curves with a common point (or tangent) on the two-dimensional torus, four smooth curves on the four-dimensional torus with a common point, twodimensional cone (cusp) in the space R6. The main problem in the article is the calculation of (co)tangent space at a singular point by using different theoretical approaches. Outside of the singular point, the motion could be described in the frames of classical mechanics. But in the neighborhood of the singular points the terms like “tangent vector” and “cotangent vector” must have new conceptual definitions. In this article, the approach of differential spaces is used. Two differential structures for the modeling conical singular point are studied in order to construct (co)tangent space at singular points: locally-constants functions near to the cone vertex and the algebra of the restrictions of smooth functions in the comprehensive Euclidean space on the cone. In the first case, tangent and cotangent spaces at the singular points are zero. In the second case, the value of the functions on the cotangent bundle is constant on the cotangent layer under the singular point.

First order over determined system with singular point

2007 ◽  
Vol 18 (1) ◽  
pp. 65-80
Author(s):  
Adel Nasim Adib ◽  
Nusrat Rajabov
Keyword(s):  
Singular Point ◽  
First Order

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.

Boundary Value Problems of Electroelasticity with Concentrated Singularities

1994 ◽  
Vol 1 (5) ◽  
pp. 459-467
Author(s):  
T. Buchukuri ◽  
D. Yanakidi
Keyword(s):  
Singular Point ◽  
Boundary Value Problems ◽  
Power Function ◽  
Boundary Value ◽  
Linearly Independent

Abstract We investigate the solutions of boundary value problems of linear electroelasticity, having growth as a power function in the neighbourhood of infinity or in the neighbourhood of an isolated singular point. The number of linearly independent solutions of this type is established for homogeneous boundary value problems.

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