On certain types of isolated s-ple points on algebraic primals in Sd

1962 ◽  
Vol 58 (3) ◽  
pp. 465-475
Author(s):  
J. Herszberg
Keyword(s):  
Finite Number ◽  
Tangent Cone ◽  
Double Point ◽  
Complete Classification ◽  
Singular Points ◽  
Multiple Points ◽  
Rank Zero ◽  
Double Points

Singular points on irreducible primals were investigated briefly by C. Segre(8), where the author classified multiple points by the nature of the nodal tangent cone. For surfaces the problem of classification was investigated by, amongst others, Du Val(1) and a complete classification of isolated double points of surfaces lying on non-singular threefolds was given by Kirby(5). In (3) we classified certain types of double points on algebraic primals in Sn. An isolated double point which after a finite number of resolutions gave rise to at most a finite number of isolated double points was called a double point of rank zero. We found that the only isolated double points of rank zero are those which are analogous to the binodes, unodes and exceptional unodes (2) of surfaces.

On Prime Immersions of S1 into R2

1976 ◽  
Vol 28 (3) ◽  
pp. 589-593
Author(s):  
John R. Martin
Keyword(s):  
Finite Number ◽  
Open Subset ◽  
Double Point ◽  
Dense Open Subset ◽  
Double Points ◽  
Combinatorial Invariant ◽  
Linearly Independent ◽  
Oriented Plane ◽  
Intersection Sequence ◽  
Element Set

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.

Note in regard to surfaces in space of four dimensions, in particular rational surfaces

1932 ◽  
Vol 28 (1) ◽  
pp. 62-82 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Algebraic Surface ◽  
Double Point ◽  
Arbitrary Point ◽  
Common Point ◽  
Multiple Points ◽  
Five Dimensions ◽  
Rational Surfaces ◽  
Double Points ◽  
Four Dimensions ◽  
Surface Passing

To a non-singular algebraic surface in space of five dimensions there can generally (the Veronese surface, of order 4, and cones are exceptional) be drawn, from an arbitrary point, a finite number of chords. If such a surface be projected from a point into space of four dimensions, there will, therefore, in general, be a certain number of points upon the resulting surface, at which two sheets of this surface, with distinct tangent planes, have an isolated common point. Such points have been called improper double points. We consider an algebraic surface ψ, in space of four dimensions [4], with no other multiple points than such double points, which we shall call accidental double points. The chords of the surface ψ, drawn from an arbitrary point O of the space [4], form a surface, or conical sheet, of which a general generator meets the surface in two points. The locus of these points is a curve which we shall call the chord curve. This curve has an actual double point at each of the accidental double points of ψ There will also, generally, be a certain number of points of the surface which are points of contact of tangent planes of the surface passing through O (and therefore also points of contact of tangent lines through O, these tangent lines being generally tangent lines of the chord curve).

scholarly journals Products of locally cyclic groups

2021 ◽  
Author(s):  
Bernhard Amberg ◽  
Yaroslav Sysak
Keyword(s):  
Finite Number ◽  
Finite Groups ◽  
Complete Classification ◽  
The Other ◽  
Supersoluble Group ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Prüfer Rank ◽  
Torsion Free

AbstractWe consider groups of the form $${G} = {AB}$$ G = AB with two locally cyclic subgroups A and B. The structure of these groups is determined in the cases when A and B are both periodic or when one of them is periodic and the other is not. Together with a previous study of the case where A and B are torsion-free, this gives a complete classification of all groups that are the product of two locally cyclic subgroups. As an application, it is shown that the Prüfer rank of a periodic product of two locally cyclic subgroups does not exceed 3, and this bound is sharp. It is also proved that a product of a finite number of pairwise permutable periodic locally cyclic subgroups is a locally supersoluble group. This generalizes a well-known theorem of B. Huppert for finite groups.

SINGULAR POINTS OF QUADRATIC SYSTEMS: A COMPLETE CLASSIFICATION IN THE COEFFICIENT SPACE ℝ12

2008 ◽  
Vol 18 (02) ◽  
pp. 313-362 ◽  
Author(s):  
JOAN C. ARTES ◽  
JAUME LLIBRE ◽  
NICOLAE VULPE
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Complete Classification ◽  
Quadratic System ◽  
Singular Points ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Algebraic Functions ◽  
Algebraic Invariants

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert,1900], are still open for this class. Even when not dealing with limit cycles, still some problems have remained unsolved like a complete classification of different phase portraits without limit cycles. For some time it was thought (see [Coppel, 1966]) there could exist a set of algebraic functions whose signs would completely determine the phase portrait of a quadratic system. Nowadays we already know that this is not so, and that there are some analytical, nonalgebraic functions that also play a role when dealing with limit cycles and separatrix connections. However, it is possible to find out a set of algebraic functions whose signs determine the characteristics of all finite and infinite singular points. Most of the work up to now has dealt with this problem studying it using different normal forms adapted to some subclasses of quadratic systems. A general work useful for any quadratic system regardless of affine changes has only been done for the study of infinite singular points [Schlomiuk et al., 2005]. In this paper, we give a complete global classification of quadratic differential systems according to their topological behavior in the vicinity of the finite singular points. Our classification Main Theorem gives us a complete dictionary describing the local behavior of finite singular points using algebraic invariants and comitants which are a powerful tool for algebraic computations. Linking the result of this paper with the main one of [Schlomiuk et al., 2005] which uses the same algebraic invariants, it is possible to complete the algebraic classification of singular points (finite and infinite) for quadratic differential systems.

On the Singularities of Plane Curves

1986 ◽  
Vol 38 (4) ◽  
pp. 947-968 ◽  
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Finite Number ◽  
Projective Plane ◽  
Singular Points ◽  
Plane Curves ◽  
Multiple Point ◽  
Real Projective Plane ◽  
Multiple Points ◽  
Minimum Value ◽  
Even Order ◽  
Image Position

Let Γ be a differentiable curve in a real projective plane P2 met by every line of P2 at a finite number of points. The singular points of Γ are inflections, cusps (cusps of the first kind) and beaks (cusps of the second kind). Let n1(Γ), n2(Γ) and n3(Γ) be the number of these points in Γ respectively. Then Γ is non-singular ifotherwise, Γ is singular.We wish to determine when T is singular and then find the minimum value of n(Γ). A history and an analysis of this problem were presented in [1] and [2]. It was shown that we may assume that Γ is a curve of even order (even degree if Γ is algebraic), met by every line in P2. Then if Γ does not contain any multiple points or if Γ contains only a certain type of multiple point, Γ is singular. Presently, we complete this investigation

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

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