scholarly journals Families of vector fields with many numerical invariants

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nataliya Goncharuk ◽  
Yury Kudryashov
Keyword(s):  
Infinite Number ◽  
Vector Fields ◽  
Topological Classification ◽  
Number Sequence ◽  
Numerical Invariant ◽  
Generic Class ◽  
Numerical Invariants ◽  
Provided Examples

<p style='text-indent:20px;'>We study bifurcations in finite-parameter families of vector fields on <inline-formula><tex-math id="M1">\begin{document}$S^2$\end{document}</tex-math></inline-formula>. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable <inline-formula><tex-math id="M2">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of <inline-formula><tex-math id="M3">\begin{document}$(2D+1)$\end{document}</tex-math></inline-formula>-parameter families such that the topological classification of these families has at least <inline-formula><tex-math id="M4">\begin{document}$D$\end{document}</tex-math></inline-formula> numerical invariants and used those examples to construct families with functional invariants of topological classification.</p><p style='text-indent:20px;'>In this paper, we construct locally generic <inline-formula><tex-math id="M5">\begin{document}$4$\end{document}</tex-math></inline-formula>-parameter families with any prescribed number of numerical invariants and use them to construct <inline-formula><tex-math id="M6">\begin{document}$5$\end{document}</tex-math></inline-formula>-parameter families with functional invariants. We also describe a locally generic class of <inline-formula><tex-math id="M7">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families with a tail of an infinite number sequence as an invariant of topological classification.</p>

scholarly journals On the singularity structure of Kahan discretizations of a class of quadratic vector fields

2021 ◽  
Author(s):  
René Zander
Keyword(s):  
Elliptic Curves ◽  
Vector Fields ◽  
Singularity Structure ◽  
Quadratic Vector Fields ◽  
Parameter Values

AbstractWe discuss the singularity structure of Kahan discretizations of a class of quadratic vector fields and provide a classification of the parameter values such that the corresponding Kahan map is integrable, in particular, admits an invariant pencil of elliptic curves.

scholarly journals Classification of Degenerate Verma Modules for E(5, 10)

2021 ◽  
Author(s):  
Nicoletta Cantarini ◽  
Fabrizio Caselli ◽  
Victor Kac
Keyword(s):  
Infinite Number ◽  
Verma Module ◽  
Lie Superalgebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Finite Dimensional ◽  
De Rham ◽  
Via Classification

AbstractGiven a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

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.

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