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## Enumeration of r-regular Maps on the Torus. Part II: Unsensed Maps

Discrete Mathematics. 2019. Vol. 342. No. 2. P. 600-614.

Omelchenko A., Краско Е. С.

The second part of the paper is devoted to enumeration of r-regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to these homeomorphisms leads to maps on the Klein bottle, the annulus and the M ̈obius band. Using 3- and 4-regular maps as an example we describe the technique of enumerating quotient maps on surfaces with a boundary. Obtained recurrence relations are used to enumerate unsensed r-regular maps on the torus for various r.

Publication based on the results of:

Omelchenko A., Краско Е. С., Discrete Mathematics 2019 Vol. 342 No. 2 P. 584-599

The work that consists of two parts is devoted to the problem of enumerating unrooted r-regular maps on the torus up to all its symmetries. We begin with enumerating near-r- regular rooted maps on the torus, the projective plane and the Klein bottle, as well as some special kinds of maps on the sphere: near-r-regular ...

Added: September 21, 2018

Omelchenko A., Краско Е. С., European Journal of Combinatorics 2017 Vol. 62 P. 167-177

We obtain explicit formulas for enumerating rooted and unrooted 4-regular one-face maps on genus g surfaces. For rooted maps the result is combinatorially derived from Chapuy’s vertex cutting bijection and has a simple sum-free form similar to analogous formulas for general and cubic one-face maps. To enumerate unrooted maps we apply the approach of Liskovets, Mednykh and Nedela ...

Added: August 29, 2018

Omelchenko A., Краско Е. С., Electronic Journal of Combinatorics 2015 Vol. 22 No. 1 P. 1-17

We present new functional equations connecting the counting series of plane and planar (in the sense of Harary and Palmer) dissections. Simple rigorous expressions for counting symmetric rr-dissections of polygons and planar SS-dissections are obtained. ...

Added: August 29, 2018

Gusein-Zade S., Manuscripta Mathematica 2018 Vol. 155 No. 3-4 P. 335-353

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define ...

Added: October 27, 2020

Gusein-Zade S., Функциональный анализ и его приложения 2018 Т. 52 № 4 С. 72-85

The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring ...

Added: October 27, 2020

Kazaryan M., Uribe-Vargas R., Moscow Mathematical Journal 2020 Vol. 20 No. 3 P. 511-530

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. ...

Added: August 24, 2020

Pochinka O., Nozdrinova E., Proceedings of the International Geometry Center 2018 Vol. 11 No. 2 P. 1-15

In this paper it is proved that every orientable surface admits an orientation-preserving Morse-Smale diffeomorphism with one saddle orbit. It is shown that these diffeomorphisms have exactly three node orbits. In addition, all possible types of periodic data for such diffeomorphisms are established. ...

Added: October 14, 2017

N. I. Zhukova, Transformation Groups 2017

We prove an analog of the Lichnerowicz conjecture for compact and noncompact
Riemannian orbifolds. In particular, we prove that any compact Riemannian
orbifold of dimension n >2 with an essential connected Lie group of conformal
transformations is conformally equivalent to the canonical Riemannian orbifold which is the
quotient space of the standard n-dimensional sphere by a finite isometry group ...

Added: April 4, 2017

N. I. Zhukova, Journal of Geometry and Physics 2018 Vol. 132 P. 146-154

We present a new method of investigation of G-structures on orbifolds.
This method is founded on the consideration of a G-structure on an
n-dimensional orbifold as the corresponding transversal
structure of an associated foliation. Using this method we prove the
existence and the uniqueness of a finite dimensional Lie group structures
on the full automorphism group of an elliptic G-structure ...

Added: April 4, 2017

N. I. Zhukova, E. A. Rogozhina .., Siberian Mathematical Journal 2012 Vol. 53 No. 6 P. 1037-1050

Among closed Lorentzian surfaces, only flat tori admit non-compact full isometry groups. Moreover, for every n > 2 the standard n-dimensional flat torus equipped with canonical metric has a non-compact full isometry Lie group. We show that this fails for n= 2 and classify the flat Lorentzian metrics on the torus with a non-compact full ...

Added: October 19, 2014

Gusein-Zade S., Алгебра и анализ 2021 Т. 33 № 3 С. 73-84

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier, there was defined ...

Added: May 2, 2021

Боголепова Е. В., Zhukova N., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2019 № 1 С. 14-28

Using the bundle of pseudo-orthogonal frames some canonical covering map for two-dimensional Lorentzian orbifolds is constructed and applied. The existence of such map shows that any two-dimensional Lorentzian orbifold is very good.
It is proved that there are only two (up to isomorphisms in the category of orbifolds) two-dimensional smooth noncompact orbifolds admitting complete flat Lorentzian ...

Added: April 10, 2019

Bagaev A. V., Zhukova N., Journal of Geometry and Physics 2019 Vol. 142 P. 80-91

S.S. Chern conjectured that the Euler characteristic of every closed affine
manifold has to vanish. We present an analog of this conjecture stating that
the Euler-Satake characteristic of any compact affine orbifold is equal to zero.
We prove that Chern's conjecture is equivalent to its analog for
the Euler-Satake characteristic of compact affine orbifolds, and
orbifolds may be ineffective. This ...

Added: April 26, 2019

Omelchenko A., Краско Е. С., Electronic Journal of Combinatorics 2017 Vol. 24 No. 3 P. 1-23

We enumerate chord diagrams without loops and without both loops and parallel chords. For labelled diagrams we obtain generating functions, for unlabelled ones we derive recurrence relations. ...

Added: August 29, 2018

Gusein-Zade S., Revista Matemática Complutense 2018 Vol. 31 No. 3 P. 595-609

A power structure over a ring is a method to give sense to expressions of the form $(1+a_1t+a_2t^2...)^m$, where $a_i$, $i=1,2, ...$, and $m$ are elements of the ring. The (natural) power structure over the Grothendieck ring of complex quasi-projective varieties appeared to be useful for a number of applications. We discuss new examples of ...

Added: October 27, 2020

Enatskaya N., Промышленные АСУ и контроллеры 2016 № 7 С. 32-36

All investigations of the scheme defined in its name are fulfiled by the direct enumeration of its outcomes, namely: the number of its outcomes of the scheme and its probability distribution are defined, the numbering problem for outcomes of the scheme is solved, this gives the possibility of a quick modeling of its possible values ...

Added: September 13, 2016

Aranson S. K., Belitsky G. R., Zhuzhoma E. V., American Mathematical Society, 1996

The book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achevements in this area obtained in recent times. The reader of this book need to be familiar only with basic courses in differential ...

Added: October 2, 2014

Краско Е. С., Omelchenko A., Acta Mathematica Universitatis Comenianae 2019 Vol. LXXXVIII No. 3 P. 885-890

The work is devoted to the problem of enumerating maps on an orientable or non-orientable surface of a given genus g up to all symmetries (so called unsensed maps). We obtain general formulas which reduce the problem of counting such maps to the problem of enumerating rooted quotient maps on orbifolds. In addition, we solve ...

Added: December 9, 2019

Zhukova N., Moscow Mathematical Journal 2018

We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...

Added: April 2, 2018

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013

Min Namkung, Younghun K., Scientific Reports 2018 Vol. 8 No. 1 P. 16915-1-16915-18

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
states when N receivers are separately located. In this report, we propose optical designs that can
perform sequential state discrimination of two coherent states. For this purpose, we consider not
only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior
probabilities. Since ...

Added: November 16, 2020