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.

Symmetry sets

1985 ◽  
Vol 101 (1-2) ◽  
pp. 163-186 ◽  
Author(s):  
J. W. Bruce ◽  
P. J. Giblin ◽  
C. G. Gibson
Keyword(s):  
Smooth Manifold ◽  
Normal Forms ◽  
Shape Recognition ◽  
Uniqueness Result ◽  
Potential Functions ◽  
Plane Curves ◽  
Symmetry Set ◽  
Space Curves ◽  
The Family ◽  
Set Of Points

SynopsisFor a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).

scholarly journals Centre Symmetry Sets of Families of Plane Curves

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Peter Giblin ◽  
Graham Reeve
Keyword(s):  
Normal Forms ◽  
Direct Calculation ◽  
Parameter Family ◽  
Plane Curves ◽  
The Family ◽  
Study Centre

AbstractWe study centre symmetry sets and equidistants for a I-parameter family of plane curves where, for a special member of the family, there exist two inflexions with parallel tangents. Some results can be obtained by reducing a generating family to normal forms, but others require direct calculation from the generating family.

scholarly journals The diminished base locus is not always closed

2014 ◽  
Vol 150 (10) ◽  
pp. 1729-1741 ◽  
Author(s):  
John Lesieutre
Keyword(s):  
Blow Up ◽  
Weak Sense ◽  
Prime Divisors ◽  
General Fiber ◽  
Base Locus ◽  
The Family

AbstractWe exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

scholarly journals General-affine invariants of plane curves and space curves

2019 ◽  
Vol 70 (1) ◽  
pp. 67-104
Author(s):  
Shimpei Kobayashi ◽  
Takeshi Sasaki
Keyword(s):  
Plane Curves ◽  
Space Curves ◽  
Affine Invariants ◽  
General Affine

2. Applications of the Theorem that Two Closed Plane Curves intersect an even number of times

1878 ◽  
Vol 9 ◽  
pp. 237-246 ◽  
Author(s):  
Tait
Keyword(s):  
Special Class ◽  
Double Point ◽  
Closed Curve ◽  
British Association ◽  
Plane Curves ◽  
Double Points ◽  
The Wire ◽  
Closed Plane

The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation.

Double points in families of map germs from ℝ2 to ℝ3

2020 ◽  
pp. 1-18
Author(s):  
J. A. Moya-Pérez ◽  
J. J. Nuño-Ballesteros
Keyword(s):  
Double Point ◽  
Parameter Family ◽  
Milnor Number ◽  
Double Points ◽  
The Family ◽  
Number Formula ◽  
Real Analytic

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

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.

scholarly journals Computing from projections of random points

2019 ◽  
Vol 20 (01) ◽  
pp. 1950014
Author(s):  
Noam Greenberg ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Cost Function ◽  
Random Sequence ◽  
Turing Degrees ◽  
Orthogonal Projections ◽  
Open Set ◽  
Proper Subclass ◽  
The Family ◽  
Function Characterization ◽  
The Cost ◽  
The Ideal

We study the sets that are computable from both halves of some (Martin–Löf) random sequence, which we call [Formula: see text]-bases. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e. elements. It is a proper subideal of the [Formula: see text]-trivial sets. We characterize [Formula: see text]-bases as the sets computable from both halves of Chaitin’s [Formula: see text], and as the sets that obey the cost function [Formula: see text]. Generalizing these results yields a dense hierarchy of subideals in the [Formula: see text]-trivial degrees: For [Formula: see text], let [Formula: see text] be the collection of sets that are below any [Formula: see text] out of [Formula: see text] columns of some random sequence. As before, this is an ideal generated by its c.e. elements and the random sequence in the definition can always be taken to be [Formula: see text]. Furthermore, the corresponding cost function characterization reveals that [Formula: see text] is independent of the particular representation of the rational [Formula: see text], and that [Formula: see text] is properly contained in [Formula: see text] for rational numbers [Formula: see text]. These results are proved using a generalization of the Loomis–Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyze arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the [Formula: see text]-trivial sets; we can calculate from the family which [Formula: see text] it characterizes. We finish by studying the union of [Formula: see text] for [Formula: see text]; we prove that this ideal consists of the sets that are robustly computable from some random sequence. This class was previously studied by Hirschfeldt [D. R. Hirschfeldt, C. G. Jockusch, R. Kuyper and P. E. Schupp, Coarse reducibility and algorithmic randomness, J. Symbolic Logic 81(3) (2016) 1028–1046], who showed that it is a proper subclass of the [Formula: see text]-trivial sets. We prove that all such sets are robustly computable from [Formula: see text], and that they form a proper subideal of the sets computable from every (weakly) LR-hard random sequence. We also show that the ideal cannot be characterized by a cost function, giving the first such example of a [Formula: see text] subideal of the [Formula: see text]-trivial sets.

scholarly journals Parameterizations of 1-Bridge Torus Knots

2003 ◽  
Vol 12 (04) ◽  
pp. 463-491 ◽  
Author(s):  
Doo Ho Choi ◽  
Ki Hyoung Ko
Keyword(s):  
Normal Form ◽  
Fundamental Group ◽  
Explicit Formula ◽  
Normal Forms ◽  
Double Cover ◽  
Torus Knots ◽  
Number Of Components ◽  
Torus Knot ◽  
Branched Cover ◽  
The Family

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

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