Distance-Genericity for Real Algebraic Hypersurfaces

1984 ◽  
Vol 36 (2) ◽  
pp. 374-384
Author(s):  
J. W. Bruce ◽  
C. G. Gibson
Keyword(s):  
Catastrophe Theory ◽  
Local Structure ◽  
Height Functions ◽  
Smooth Submanifold ◽  
Smooth Category ◽  
The Family ◽  
Algebraic Hypersurfaces ◽  
Focal Set ◽  
Topological Models

One of the original applications of catastrophe theory envisaged by Thom was that of discussing the local structure of the focal set for a (generic) smooth submanifold M ⊆ Rn + 1. Thom conjectured that for a generic M there would be only finitely many local topological models, a result proved by Looijenga in [4]. The objective of this paper is to extend Looijenga's result from the smooth category to the algebraic category (in a sense explained below), at least in the case when M has codimension 1.Looijenga worked with the compactified family of distance-squared functions on M (defined below), thus including the family of height functions on M whose corresponding catastrophe theory yields the local structure of the focal set at infinity. For the family of height functions the appropriate genericity theorem in the smooth category was extended to the algebraic case in [1], so that the present paper can be viewed as a natural continuation of the first author's work in this direction.

scholarly journals Generic isotopies of space curves

1987 ◽  
Vol 29 (1) ◽  
pp. 41-63 ◽  
Author(s):  
J. W. Bruce ◽  
P. J. Giblin
Keyword(s):  
Point Contact ◽  
Smooth Curve ◽  
Singular Surfaces ◽  
Height Functions ◽  
Space Curves ◽  
Local Structures ◽  
The Family ◽  
Generic Curve ◽  
Geometrical Information ◽  
Focal Set

For a single space curve (that is, a smooth curve embedded in ℝ3) much geometrical information is contained in the dual and the focal set of the curve. These are both (singular) surfaces in ℝ3, the dual being a model of the set of all tangent planes to the curve, and the focal set being the locus of centres of spheres having at least 3-point contact with the curve. The local structures of the dual and the focal set are (for a generic curve) determined by viewing them as (respectively) the discriminant of a family derived from the height functions on the curve, and the bifurcation set of the family of distance-squared functions on the curve. For details of this see for example [6, pp. 123–8].

Quantum Feller semigroup in terms of quantum Bernoulli noises

2020 ◽  
pp. 2150015
Author(s):  
Jinshu Chen
Keyword(s):  
Local Structure ◽  
Identity Operator ◽  
Sufficient Condition ◽  
Markov Semigroups ◽  
Feller Semigroup ◽  
Abelian Subalgebra ◽  
Feller Property ◽  
The Family ◽  
Invariant States ◽  
Creation Operators

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

Geometry of projection-generic space curves

2009 ◽  
Vol 147 (1) ◽  
pp. 115-142
Author(s):  
C. T. C. WALL
Keyword(s):  
Local Structure ◽  
Double Point ◽  
Normal Forms ◽  
Blow Up ◽  
Complex Case ◽  
Plane Curves ◽  
Space Curves ◽  
Open Set ◽  
The Family ◽  
Flat Family

AbstractIn earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

scholarly journals Focal Sets in Certain Riemannian Manifolds

1984 ◽  
Vol 27 (2) ◽  
pp. 209-214
Author(s):  
J. W. Bruce ◽  
D. J. Hurley
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Ambient Space ◽  
Bifurcation Set ◽  
Transversality Theorem ◽  
Generic Submanifolds ◽  
The Family ◽  
Generic Submanifold ◽  
Focal Set

In recent years the geometry of generic submanifolds of Euclidean space has been theobject of much study. Thorn hinted in [7] that the focal set of such a submanifold couldprofitably be studied by using the family of distance squared functions on thesubmanifold from points of the ambient space. For a generic submanifold the focal set isthe catastrophe or bifurcation set of this family. The key to obtaining results on thelocal structure of this focal set is a transversality theorem of Looijenga [5]; for analternative exposition see [8].

scholarly journals Trajectory singularities of general planar motions

1999 ◽  
Vol 129 (1) ◽  
pp. 37-55 ◽  
Author(s):  
P. S. Donelan ◽  
C. G. Gibson ◽  
W. Hawes
Keyword(s):  
Local Structure ◽  
Degrees Of Freedom ◽  
Local Models ◽  
Two Degrees Of Freedom ◽  
The Family ◽  
Small Deformations ◽  
Planar Motions

Local models are given for the singularities that can appear on the trajectories ofgeneral motions of the plane with more than two degrees of freedom. Versal unfoldings of these model singularities give rise to computer-generated pictures describing the family of trajectories arising from small deformations of the tracing point, and determine the local structure of the bifurcation curves.

Generic affine differential geometry of curves in Rn

2006 ◽  
Vol 136 (6) ◽  
pp. 1195-1205 ◽  
Author(s):  
Declan Davis
Keyword(s):  
Differential Geometry ◽  
Distance Functions ◽  
Affine Differential Geometry ◽  
Arc Length ◽  
Height Functions ◽  
The Family

This paper considers curves in Rn. It defines affine arc length and affine curvatures. The family of affine distance functions is generalized, along with the family of affine height functions. A new basis is constructed that makes the conditions for Ak singularity types easier to calculate, and applications are given to geometrical problems.

Triassic sponges (Sphinctozoa) from Hells Canyon, Oregon

1988 ◽  
Vol 62 (03) ◽  
pp. 419-423 ◽  
Author(s):  
Baba Senowbari-Daryan ◽  
George D. Stanley
Keyword(s):  
Endemic Species ◽  
Upper Triassic ◽  
Island Arc ◽  
Tropical Island ◽  
The Family ◽  
Wallowa Terrane ◽  
Unique Condition ◽  
Hells Canyon

Two Upper Triassic sphinctozoan sponges of the family Sebargasiidae were recovered from silicified residues collected in Hells Canyon, Oregon. These sponges areAmblysiphonellacf.A. steinmanni(Haas), known from the Tethys region, andColospongia whalenin. sp., an endemic species. The latter sponge was placed in the superfamily Porata by Seilacher (1962). The presence of well-preserved cribrate plates in this sponge, in addition to pores of the chamber walls, is a unique condition never before reported in any porate sphinctozoans. Aporate counterparts known primarily from the Triassic Alps have similar cribrate plates but lack the pores in the chamber walls. The sponges from Hells Canyon are associated with abundant bivalves and corals of marked Tethyan affinities and come from a displaced terrane known as the Wallowa Terrane. It was a tropical island arc, suspected to have paleogeographic relationships with Wrangellia; however, these sponges have not yet been found in any other Cordilleran terrane.

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