Combinatorial invariant for Morse–Smale diffeomorphisms on surfaces with orientable heteroclinic

2021 ◽  
Vol 31 (2) ◽  
pp. 023119
Author(s):  
D. Malyshev ◽  
A. Morozov ◽  
O. Pochinka
Keyword(s):  
Combinatorial Invariant

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.

scholarly journals Systolic geometry and simplicial complexity for groups

2019 ◽  
Vol 2019 (757) ◽  
pp. 247-277
Author(s):  
Ivan Babenko ◽  
Florent Balacheff ◽  
Guillaume Bulteau
Keyword(s):  
Isomorphism Classes ◽  
Combinatorial Invariant ◽  
Satisfactory Answer

AbstractTwenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called simplicial complexity that allows to obtain a quite satisfactory answer to his question. Using this new complexity, we also derive new results on systolic area for groups that specify its topological behaviour.

scholarly journals Nearly free curves and arrangements: a vector bundle point of view

2019 ◽  
pp. 1-24 ◽  
Author(s):  
Simone Marchesi ◽  
Jean Vallès
Keyword(s):  
Single Point ◽  
Point Of View ◽  
Line Bundles ◽  
Zero Section ◽  
Combinatorial Invariant ◽  
Counter Example ◽  
Free Arrangement ◽  
Finite Set ◽  
Interesting Family ◽  
Arrangement Of Lines

Abstract Over the past forty years many papers have studied logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. Terao conjectured thirty years ago that when a curve is a finite set of distinct lines (i.e. a line arrangement) its freeness depends solely on its combinatorics, but this has only been proved for sets of up to 12 lines. In looking for a counter-example to Terao’s conjecture, the nearly free curves introduced by Dimca and Sticlaru arise naturally. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non-zero section that vanishes on one single point, P say, called the jumping point, and that this characterises the bundle. We then give a precise description of the behaviour of P. Based on detailed examples we then show that the position of P relative to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics.

scholarly journals Posets from Admissible Coxeter Sequences

10.37236/684 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Matthew Macauley ◽  
Henning S. Mortveit
Keyword(s):  
Coxeter Groups ◽  
Conjugacy Problem ◽  
Equivalence Classes ◽  
Quiver Representations ◽  
Edge Deletion ◽  
Acyclic Orientations ◽  
Edge Contraction ◽  
Combinatorial Invariant ◽  
Source To Sink ◽  
Admissible Sequences

We study the equivalence relation on the set of acyclic orientations of an undirected graph $\Gamma$ generated by source-to-sink conversions. These conversions arise in the contexts of admissible sequences in Coxeter theory, quiver representations, and asynchronous graph dynamical systems. To each equivalence class we associate a poset, characterize combinatorial properties of these posets, and in turn, the admissible sequences. This allows us to construct an explicit bijection from the equivalence classes over $\Gamma$ to those over $\Gamma'$ and $\Gamma"$, the graphs obtained from $\Gamma$ by edge deletion and edge contraction of a fixed cycle-edge, respectively. This bijection yields quick and elegant proofs of two non-trivial results: $(i)$ A complete combinatorial invariant of the equivalence classes, and $(ii)$ a solution to the conjugacy problem of Coxeter elements for simply-laced Coxeter groups. The latter was recently proven by H. Eriksson and K. Eriksson using a much different approach.

scholarly journals Network theoretic analysis of JAK/STAT pathway and extrapolation to drugs and viruses including COVID-19

2020 ◽  
Author(s):  
Arindam Banerjee ◽  
Rudra Prosad Goswami ◽  
Moumita Chatterjee
Keyword(s):  
Influenza A ◽  
Viral Infections ◽  
Yellow Fever Virus ◽  
Immunosuppressive Agents ◽  
Vertex Cover ◽  
Nipah Virus ◽  
Biological Behaviour ◽  
Combinatorial Invariant ◽  
Syncytial Virus ◽  
Stat Pathway

Abstract Whenever some phenomenon can be represented as a graph or a network it seems pertinent to explore how much the mathematical properties of that network impact the phenomenon. In this study we explore the same philosophy in the context of immunology. Our objective was to assess the correlation of “size” (number of edges and minimum vertex cover) of the JAK/STAT network with treatment effect in rheumatoid arthritis (RA), phenotype of viral infection and effect of immunosuppressive agents on a system infected with the coronavirus. We extracted the JAK/STAT pathway from Kyoto Encyclopedia of Genes and Genomes (KEGG, hsa04630). The effects of the following drugs, and their combinations, commonly used in RA were tested: methotrexate, prednisolone, rituximab, tocilizumab, tofacitinib and baricitinib. Following viral systems were also tested for their ability to evade the JAK/STAT pathway: Measles, Influenza A, West Nile virus, Japanese B virus, Yellow Fever virus, respiratory syncytial virus, Kaposi’s sarcoma virus, Hepatitis B and C virus, cytomegalovirus, Hendra and Nipah virus and Coronavirus. Good correlation of edges and minimum vertex cover with clinical efficacy were observed (for edge, rho= -0.815, R2= 0.676, p=0.007, for vertex cover rho= -0.793, R2= 0.635, p=0.011). In the viral systems both edges and vertex cover were associated with acuteness of viral infections. In the JAK/STAT system already infected with coronavirus, maximum reduction in size was achieved with baricitinib. To conclude, algebraic and combinatorial invariant of a network may explain its biological behaviour. At least theoretically, baricitinib may be an attractive target for treatment of coronavirus infection.

scholarly journals Combinatorial invariant theory of projective reflection groups

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Fabrizio Caselli
Keyword(s):  
Representation Theory ◽  
Invariant Theory ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Combinatorial Invariant ◽  
International Audience ◽  
Natural Description

International audience We introduce the class of projective reflection groups which includes all complex reflection groups. We show that several aspects involving the combinatorics and the representation theory of complex reflection groups find a natural description in this wider setting. On introduit la classe des groupes de réflexions projectifs, ce qui généralise la notion de groupe engendré par des réflexions. On montre que plusieurs aspects concernant la combinatoire et la théorie des représentations des groupes de réflexions complexes trouvent une description naturelle dans ce cadre plus général.

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