scholarly journals Tangent vectors to sets in the theory of geodesics

1987 ◽  
Vol 106 ◽  
pp. 29-47 ◽  
Author(s):  
Dumitru Motreanu
Keyword(s):  
Closed Subset ◽  
Tangent Vector ◽  
General Concept ◽  
Recent Book ◽  
Closed Sets ◽  
Banach Manifolds ◽  
Flow Invariance ◽  
Tangent Vectors

In the setting of Banach manifolds the notion of tangent vector to an arbitrary closed subset has been introduced in [11] by the author and N. H. Pavel, and it has been used in flow-invariance and optimization ([11], [12], [13]). For detailed informations on tangent vectors to closed sets (including historical comments) we refer to the recent book of N. H. Pavel [17].The aim of this paper is to apply this general concept of tangency in the study of geodesies. We are concerned with geodesies which have either the endpoints in given closed subsets or the same angle for a fixed closed subset. This approach allows one to extend important results due to K. Grove [4] and T. Kurogi ([6], [7]).

Complex Approximation and Simultaneous Interpolation on Closed Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 701-706 ◽  
Author(s):  
P. M. Gauthier ◽  
W. Hengartner
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Closed Subset ◽  
Finite Complex ◽  
Complex Approximation ◽  
Closed Sets ◽  
Limit Points ◽  
Simultaneous Interpolation ◽  
Complex Valued

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

Loci Conversion and Corner Smoothing with PH Curves in CNC System

2009 ◽  
Vol 419-420 ◽  
pp. 161-164
Author(s):  
Qi Kui Wang ◽  
You Dong Chen ◽  
Wei Li ◽  
Tian Miao Wang ◽  
Hong Xing Wei
Keyword(s):  
Tangent Vector ◽  
Machining Process ◽  
Free Form ◽  
Surface Interpolation ◽  
Pythagorean Hodograph ◽  
Linear Elements ◽  
Smoothing Process ◽  
Ph Curve ◽  
Interpolation Functions ◽  
Tangent Vectors

Free-form surface interpolation functions give more advantages in machining than the traditional line and circle functions. A method is developed to convert lines and circles into Pythagorean-hodograph (PH) curves. In order to get smooth machining process the PH curve is used to replace the joints of the circular/linear elements by the connection situation. The slope of line is used to get the tangent vector in the line conversion. When converting a circle to a PH curve, points of the divided circle are introduced to compute the vectors. The methods of computing tangent vectors are proposed according to the slope of the line and the quadrant of the circle. The transformation errors from lines and circles to PH curves are computed. In the corner smoothing process the tangent vectors are computed by the connection between lines and circles. Replacement errors at the joints are computed for the use of PH curve. The results demonstrate the feasibility of the conversion from line and circle to PH curve. The PH curves at the joints of the circular and linear elements show continuous trajectory.

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

Lip α Approximation on Closed Sets with Lip α Extension

1995 ◽  
Vol 38 (1) ◽  
pp. 23-33
Author(s):  
A. Bonilla ◽  
J. C. Fariña
Keyword(s):  
Complex Plane ◽  
Closed Subset ◽  
Closed Sets

AbstractLet F be a relatively closed subset of a domain G in the complex plane. Let f be a function that is the limit, in the Lip α norm on F, of functions which are holomorphic or meromorphic on G (0 < α < 1). We prove that, under the same conditions that give Lip α-approximation (0 < α < 1 ) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that f — m can be extended to a function belonging to lip

scholarly journals Base Point Freeness, Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2344
Author(s):  
Edoardo Ballico
Keyword(s):  
Projective Plane ◽  
Linear Systems ◽  
Tangent Vector ◽  
Base Point ◽  
Uniqueness Problem ◽  
Segre Variety ◽  
Double Points ◽  
Segre Varieties ◽  
Tangent Vectors

We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.

Maps generating normals on a manifold

2019 ◽  
pp. 110-125
Author(s):  
K. Polyakova
Keyword(s):  
Tangent Space ◽  
Affine Connection ◽  
Tangent Vector ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
Osculating Space ◽  
Curvature And Torsion ◽  
Order Tangent ◽  
Tangent Vectors

The first- and second-order canonical forms are considered on a smooth m-manifold. The approach connected with coordinate expressions of basic and fiber forms, and first- and second-order tangent vectors on a manifold is implemented. It is shown that the basic first- and secondorder tangent vectors are first- and second-order Pfaffian (generalized) differentiation operators of functions set on a manifold. Normals on a manifold are considered. Differentials of the basic first-order (second-order) tangent vectors are linear maps from the first-order tangent space to the second-order (third-order) tangent space. Differentials of the basic first-order tangent vectors (basic vectors of the first-order normal) set a map which takes a first-order tangent vector into the second-order normal for the first-order tangent space of (into the third-order normal for second-order tangent space). The map given by differentials of the basic first-order tangent vectors takes all tangent vectors to the vectors of the second-order normal. Such splitting the osculating space in the direct sum of the first-order tangent space and second-order normal defines the simplest (canonical) affine connection which horizontal subspace is the indicated normal. This connection is flat and symmetric. The second-order normal is the horizontal subspace for this connection. Derivatives of some basic vectors in the direction of other basic vectors are equal to values of differentials of the first vectors on the second vectors, and the order of differentials is equal to the order of vectors along which differentiation is made. The maps set by differentials of basic vectors of r-order normal for (r – 1)-order tangent space take the basic first-order tangent vectors to the basic vectors of (r + 1)-order normal for r-order tangent space. Horizontal vectors for the simplest second-order connection are constructed. Second-order curvature and torsion tensors vanish in this connection. For horizontal vectors of the simplest second-order connection there is the decomposition by means of the basic vectors of the secondand third-order normals.

Homotopy invariants for tangent vector fields on closed sets

2006 ◽  
Vol 65 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Aleksander Ćwiszewski ◽  
Wojciech Kryszewski
Keyword(s):  
Vector Fields ◽  
Tangent Vector ◽  
Closed Sets ◽  
Homotopy Invariants

A Note on Whitney Maps

1980 ◽  
Vol 23 (3) ◽  
pp. 373-374 ◽  
Author(s):  
L. E. Ward
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Internal Structure ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Fundamental Importance ◽  
Recent Book ◽  
Partially Ordered ◽  
Ordered Space

In his recent book [3] Nadler observes that the property of admitting a Whitney map is of fundamental importance in studying the internal structure of hyperspaces, especially their arc structure. Nadler presents three distinct methods of constructing a Whitney map on the hyperspace 2X of nonempty closed subsets of a continuum.A partially ordered space is a topological space X endowed with a partial order ≤ whose graph is a closed subset of X×X. It is well-known (see, for example, [2], page 167) that if X is a regular Hausdorff space then 2X is a partially ordered space with respect to inclusion.

scholarly journals Local rigidity and group cohomology I: Stowe's theorem for Banach manifolds

1999 ◽  
Vol 59 (2) ◽  
pp. 271-295
Author(s):  
Victor Brunsden
Keyword(s):  
Vector Fields ◽  
Tangent Vector ◽  
Group Cohomology ◽  
Finitely Generated ◽  
Rigidity Result ◽  
Finitely Generated Group ◽  
Local Rigidity ◽  
Banach Manifolds ◽  
The Stability ◽  
Smooth Closed Manifold

Stowe's Theorem on the stability of the fixed points of a C2 action of a finitely generated group Γ is generalised to C1 actions of such groups on Banach manifolds. The result is then used to prove that if φ is a Cr action on a smooth, closed, manifold M satisfying H1(Γ, Dr−1(M)) = 0, then φ is locally rigid. Here, r ≥ 2 and Dk(M) is the space of Ck tangent vector fields on M. This generalises a local rigidity result of Weil for representations of a finitely generated group Γ in a Lie group.

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