Complete topological classification of the disposition of ovals of a sixth order curve in the projective plane

1978 ◽  
pp. 97-122
Author(s):  
D. A. Gudkov
Keyword(s):  
Projective Plane ◽  
Sixth Order ◽  
Topological Classification ◽  
Order Curve

scholarly journals On the disposition of cubic and pair of conics in a real projective plane

2020 ◽  
Vol 22 (1) ◽  
pp. 24-37
Author(s):  
Victoriya A. Gorskaya ◽  
Grigory M. Polotovskiy
Keyword(s):  
Projective Plane ◽  
General Position ◽  
Algebraic Curves ◽  
Hilbert Problem ◽  
Topological Classification ◽  
Real Projective Plane ◽  
16Th Hilbert Problem ◽  
Algebraic Manifolds ◽  
Manifolds With Singularities

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

scholarly journals Wavefront dislocations reveal the topology of quasi-1D photonic insulators

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Clément Dutreix ◽  
Matthieu Bellec ◽  
Pierre Delplace ◽  
Fabrice Mortessagne
Keyword(s):  
Local Density ◽  
Topological Defects ◽  
Wave Functions ◽  
Topological Classification ◽  
Local Density Of States ◽  
Topological Phase Transition ◽  
Interference Fringes ◽  
Classical Wave ◽  
Phase Singularities

AbstractPhase singularities appear ubiquitously in wavefields, regardless of the wave equation. Such topological defects can lead to wavefront dislocations, as observed in a humongous number of classical wave experiments. Phase singularities of wave functions are also at the heart of the topological classification of the gapped phases of matter. Despite identical singular features, topological insulators and topological defects in waves remain two distinct fields. Realising 1D microwave insulators, we experimentally observe a wavefront dislocation – a 2D phase singularity – in the local density of states when the systems undergo a topological phase transition. We show theoretically that the change in the number of interference fringes at the transition reveals the topological index that characterises the band topology in the insulator.

Topological classification of nodal-line semimetals in square-net materials

2021 ◽  
Vol 103 (16) ◽  
Author(s):  
Inho Lee ◽  
S. I. Hyun ◽  
J. H. Shim
Keyword(s):  
Nodal Line ◽  
Topological Classification

Invariants of weak equivalence in primitive matrices

2000 ◽  
Vol 20 (2) ◽  
pp. 611-626 ◽  
Author(s):  
RICHARD SWANSON ◽  
HANS VOLKMER
Keyword(s):  
Inverse Limit ◽  
Direct Application ◽  
Topological Classification ◽  
Weak Equivalence ◽  
Transition Matrices ◽  
Inverse Limit Spaces ◽  
Positive Dimension ◽  
One Dimensional ◽  
Dimension Group

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

2021 ◽  
Vol 29 (6) ◽  
pp. 835-850
Author(s):  
Vladislav Kruglov ◽  
◽  
Olga Pochinka ◽  
◽  
Keyword(s):  
Finite Number ◽  
Limit Cycle ◽  
Topological Classification ◽  
Topological Conjugacy ◽  
Smooth Flow ◽  
Direction Of Movement ◽  
Smale Flows ◽  
Phase Surface ◽  
Flows On Surfaces

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another.

scholarly journals Classification of rough transformations of a circle from a modern point of view

2018 ◽  
Vol 20 (4) ◽  
pp. 408-418
Author(s):  
A. E. Kolobyanina ◽  
E. V. Nozdrinova ◽  
O. V. Pochinka
Keyword(s):  
Invariant Manifolds ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Point Of View ◽  
Topological Classification ◽  
Topological Invariants ◽  
Modern Theory ◽  
Ambient Manifold ◽  
Periodic Data

In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

Sign in / Sign up

Export Citation Format

Share Document