On finite order invariants of triple point free plane curves

1999 ◽  
pp. 275-300 ◽  
Author(s):  
V. A. Vassiliev
Keyword(s):  
Finite Order ◽  
Triple Point ◽  
Plane Curves ◽  
Free Plane

scholarly journals Freeness versus maximal global Tjurina number for plane curves

2016 ◽  
Vol 163 (1) ◽  
pp. 161-172 ◽  
Author(s):  
ALEXANDRU DIMCA
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Tjurina Number ◽  
Free Plane

AbstractWe give a characterisation of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterisation of free curves as curves with a maximal Tjurina number, given by A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterisation of free curves and a simple characterisation of nearly free curves in terms of their syzygies conclude this paper.

scholarly journals On the proportion of transverse-free plane curves

2021 ◽  
Vol 72 ◽  
pp. 101833
Author(s):  
Shamil Asgarli ◽  
Brian Freidin
Keyword(s):  
Plane Curves ◽  
Free Plane

FINITE ORDER TOPOLOGICAL INVARIANTS OF PLANE CURVES

1999 ◽  
Vol 08 (01) ◽  
pp. 33-47
Author(s):  
TETSUYA OZAWA
Keyword(s):  
Finite Order ◽  
Plane Curves ◽  
Topological Invariants ◽  
Mutually Independent ◽  
Closed Plane

We introduce three families of topological invariants of stable closed plane curves, which contain infinitely many mutually independent invariants among them. We study the order of these invariants in the sense of Vassiliev. As a consequence, we conclude that there exist infinitely many independent topological invariants for stable closed plane curves with order equal to 1.

A GENERALIZATION OF ARNOLD'S STRANGENESS INVARIANT

1999 ◽  
Vol 08 (05) ◽  
pp. 551-567
Author(s):  
HIDEYO ARAKAWA ◽  
TETSUYA OZAWA
Keyword(s):  
Triple Point ◽  
Infinite Sequence ◽  
Natural Extension ◽  
Plane Curves ◽  
Topological Invariants ◽  
Cusp Point ◽  
Tangent Point ◽  
Mutually Independent ◽  
Point Triple ◽  
Closed Plane

The purpose of this paper is to introduce an infinite sequence {Stk}k of mutually independent topological invariants of smooth closed plane curves, which is proved to be a natural extension of the rotation number and the strangeness invariant defined by Arnold in [Ar]. We prove a formula to express Stk by using the invariants [Formula: see text] which are defined in [Oz]. The jumps of Stk at perestroikas of three types (namely cusp point, triple point, and self tangent point perestroika) are investigated, and as a consequence we find that Stk have the order in the sense of Vassiliev equal to 1.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

On Commutation Properties of the Composition Relation of Convergent and Divergent Permutations (Part I)

2014 ◽  
Vol 58 (1) ◽  
pp. 13-22
Author(s):  
Roman Wituła ◽  
Edyta Hetmaniok ◽  
Damian Słota
Keyword(s):  
Finite Order ◽  
Convergent Series ◽  
The Other ◽  
Other Hand ◽  
The Family ◽  
Composition Relation ◽  
The Continuum

Abstract In the paper we present the selected properties of composition relation of the convergent and divergent permutations connected with commutation. We note that a permutation on ℕ is called the convergent permutation if for each convergent series ∑an of real terms, the p-rearranged series ∑ap(n) is also convergent. All the other permutations on ℕ are called the divergent permutations. We have proven, among others, that, for many permutations p on ℕ, the family of divergent permutations q on ℕ commuting with p possesses cardinality of the continuum. For example, the permutations p on ℕ having finite order possess this property. On the other hand, an example of a convergent permutation which commutes only with some convergent permutations is also presented.

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