scholarly journals The Geometric Theory of the Fundamental Germ

2008 ◽  
Vol 190 ◽  
pp. 1-34 ◽  
Author(s):  
T. M. Gendron
Keyword(s):  
Group Actions ◽  
Geometric Theory ◽  
Universal Cover ◽  
Closed Surface ◽  
Topological Invariant ◽  
Hyperbolic Type ◽  
Smooth Structure

The fundamental germ is a generalization of π1, first defined for laminations which arise through group actions [4]. In this paper, the fundamental germ is extended to any lamination having a dense leaf admitting a smooth structure. In addition, an amplification of the fundamental germ called the mother germ is constructed, which is, unlike the fundamental germ, a topological invariant. The fundamental germs of the antenna lamination and the PSL(2,ℤ) lamination are calculated, laminations for which the definition in [4] was not available. The mother germ is used to give a new proof of a Nielsen theorem for the algebraic universal cover of a closed surface of hyperbolic type.

Maximum orders of cyclic group actions on surfaces extending over the 3-dimensional torus

2017 ◽  
Vol 26 (06) ◽  
pp. 1742005
Author(s):  
Chao Wang ◽  
Shicheng Wang ◽  
Yimu Zhang
Keyword(s):  
Cyclic Group ◽  
Group Actions ◽  
Closed Surface ◽  
Dimensional Torus ◽  
Maximum Order ◽  
3 Dimensional

We determine the maximum order of cyclic group actions on the pair [Formula: see text] among all embeddings of closed surface [Formula: see text] into the 3-dimensional torus [Formula: see text] in the orientable category.

scholarly journals A COUNTEREXAMPLE TO GUILLEMIN'S ZOLLFREI CONJECTURE

2013 ◽  
Vol 05 (03) ◽  
pp. 251-260 ◽  
Author(s):  
STEFAN SUHR
Keyword(s):  
Universal Cover ◽  
Closed Surface ◽  
Global Hyperbolicity ◽  
Circle Bundle ◽  
Lorentzian Metrics

We construct Zollfrei Lorentzian metrics on every non-trivial orientable circle bundle over an orientable closed surface. Further we prove a weaker version of Guillemin's conjecture assuming global hyperbolicity of the universal cover.

scholarly journals Distinguishing Maps

10.37236/537 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Thomas W. Tucker
Keyword(s):  
Automorphism Group ◽  
Chromatic Number ◽  
Group Actions ◽  
Closed Surface ◽  
General Group ◽  
Distinguishing Number ◽  
Nonidentity Element ◽  
Vertex Set ◽  
Group A ◽  
Planar Maps

The distinguishing number of a group $A$ acting faithfully on a set $X$, denoted $D(A,X)$, is the least number of colors needed to color the elements of $X$ so that no nonidentity element of $A$ preserves the coloring. Given a map $M$ (an embedding of a graph in a closed surface) with vertex set $V$ and without loops or multiples edges, let $D(M)=D({\rm Aut}(M),V)$, where ${\rm Aut(M)}$ is the automorphism group of $M$; if $M$ is orientable, define $D^+(M)$ similarly, using only orientation-preserving automorphisms. It is immediate that $D(M)\leq 4$ and $D^+(M)\leq 3$. We use Russell and Sundaram's Motion Lemma to show that there are only finitely many maps $M$ with $D(M)>2$. We show that if a group $A$ of automorphisms of a graph $G$ fixes no edges, then $D(A,V)=2$, with five exceptions. That result is used to find the four maps with $D^+(M)=3$. We also consider the distinguishing chromatic number $\chi_D(M)$, where adjacent vertices get different colors. We show $\chi_D(M)\leq \chi(M)+3$ with equality in only finitely many cases, where $\chi(M)$ is the chromatic number of the graph underlying $M$. We also show that $\chi_D(M)\leq 6$ for planar maps, answering a question of Collins and Trenk. Finally, we discuss the implications for general group actions and give numerous problems for further study.

scholarly journals Topology of optimal flows with collective dynamics on closed orientable surfaces

2020 ◽  
Vol 13 (2) ◽  
pp. 50-67
Author(s):  
Alexandr Olegovich Prishlyak ◽  
Mariya Viktorovna Loseva
Keyword(s):  
Morse Function ◽  
Hamiltonian Dynamics ◽  
Closed Surface ◽  
Topological Invariant ◽  
The Other ◽  
Heteroclinic Cycles ◽  
Collective Dynamics ◽  
Gradient Dynamics ◽  
Orientable Surfaces

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

Holographic part integral and exemal closed surface

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Closed Surface

Holographic part integral and exemal closed surface

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