scholarly journals Explicit solutions of certain orientable quadratic equations in free groups

2019 ◽  
Vol 29 (08) ◽  
pp. 1451-1466
Author(s):  
D. Gonçalves ◽  
T. Nasybullov
Keyword(s):  
Free Group ◽  
Homotopy Class ◽  
Explicit Solution ◽  
Orientable Surface ◽  
Explicit Solutions ◽  
Natural Homomorphism ◽  
Quadratic Equations ◽  
Wecken Property ◽  
Number Formula ◽  
Genus Formula

For [Formula: see text] denote by [Formula: see text] the free group on [Formula: see text] generators and let [Formula: see text]. For [Formula: see text] and elements [Formula: see text], we study orientable quadratic equations of the form [Formula: see text] with unknowns [Formula: see text] and provide explicit solutions for them for the minimal possible number [Formula: see text]. In the particular case when [Formula: see text], [Formula: see text] for [Formula: see text] and [Formula: see text] the minimal number which satisfies [Formula: see text], we provide two types of solutions depending on the image of the subgroup [Formula: see text] generated by the solution under the natural homomorphism [Formula: see text]: the first solution, which is called a primitive solution, satisfies [Formula: see text], the second solution satisfies [Formula: see text]. We also provide an explicit solution of the equation [Formula: see text] for [Formula: see text] in [Formula: see text], and prove that if [Formula: see text], then every solution of this equation is primitive. As a geometrical consequence, for every solution, we obtain a map [Formula: see text] from the orientable surface [Formula: see text] of genus [Formula: see text] to the torus [Formula: see text] which has the minimal number of roots among all maps from the homotopy class of [Formula: see text]. Depending on the number [Formula: see text], such maps have fundamentally different geometric properties: in some cases, they satisfy the Wecken property and in other cases not.

On Heisenberg's elementary theory of turbulence

1949 ◽  
Vol 200 (1060) ◽  
pp. 20-33 ◽  
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Explicit Solution ◽  
Order Differential Equation ◽  
Elementary Theory ◽  
Second Order ◽  
Parametric Family ◽  
Explicit Solutions ◽  
Second Order Differential Equation ◽  
Equilibrium Spectrum

In this paper the spectrum of turbulence is considered on the basis of an elementary theory recently developed by Heisenberg. Explicit solutions for the spectrum have been obtained both when the conditions are stationary and an equilibrium spectrum obtains and when the conditions are non-stationary and the turbulence is decaying. In the former case the problem admits of an explicit solution. In the latter case the problem reduces to determining a one-parametric family of solutions of a certain second-order differential equation. The decay spectra for various values of the Reynolds number (which remains constant during the decay) are illustrated.

Geodesic graphs for a homotopy class of virtual knots

2017 ◽  
Vol 26 (13) ◽  
pp. 1750090
Author(s):  
Sumiko Horiuchi ◽  
Yoshiyuki Ohyama
Keyword(s):  
Homotopy Class ◽  
Finite Sequence ◽  
Dimensional Lattice ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Minimum Number ◽  
Vertex Set ◽  
Number Formula ◽  
Lattice Graph ◽  
Knot Diagrams

We consider a local move, denoted by [Formula: see text], on knot diagrams or virtual knot diagrams.If two (virtual) knots [Formula: see text] and [Formula: see text] are transformed into each other by a finite sequence of [Formula: see text] moves, the [Formula: see text] distance between [Formula: see text] and [Formula: see text] is the minimum number of times of [Formula: see text] moves needed to transform [Formula: see text] into [Formula: see text]. By [Formula: see text], we denote the set of all (virtual) knots which can be transformed into a (virtual) knot [Formula: see text] by [Formula: see text] moves. A geodesic graph for [Formula: see text] is the graph which satisfies the following: The vertex set consists of (virtual) knots in [Formula: see text] and for any two vertices [Formula: see text] and [Formula: see text], the distance on the graph from [Formula: see text] to [Formula: see text] coincides with the [Formula: see text] distance between [Formula: see text] and [Formula: see text]. When we consider virtual knots and a crossing change as a local move [Formula: see text], we show that the [Formula: see text]-dimensional lattice graph for any given natural number [Formula: see text] and any tree are geodesic graphs for [Formula: see text].

scholarly journals A combinatorial algorithm for visualizing representatives with minimal self-intersection

2015 ◽  
Vol 24 (11) ◽  
pp. 1550058 ◽  
Author(s):  
Chris Arettines
Keyword(s):  
Homotopy Class ◽  
Orientable Surface ◽  
Closed Curve ◽  
Combinatorial Algorithm ◽  
Minimal Self ◽  
Free Homotopy Class

Given an orientable surface with boundary and a free homotopy class of a closed curve on this surface, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self-intersection.

scholarly journals Some quadratic equations in the free group of rank 2

2008 ◽  
Author(s):  
Daciberg Gonçalves ◽  
Elena Kudryavtseva ◽  
Heiner Zieschang
Keyword(s):  
Free Group ◽  
Quadratic Equations ◽  
Rank 2

scholarly journals Explicit Solutions and Bifurcations for a System of Rational Difference Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 69
Author(s):  
Bashir Al-Hdaibat ◽  
Saleem Al-Ashhab ◽  
Ramadan Sabra
Keyword(s):  
Hypergeometric Function ◽  
Difference Equations ◽  
Explicit Solution ◽  
Initial Conditions ◽  
Global Dynamics ◽  
Explicit Solutions ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equations

In this paper, we consider the explicit solution of the following system of nonlinear rational difference equations: x n + 1 = x n - 1 / x n - 1 + r , y n + 1 = x n - 1 y n / x n - 1 y n + r , with initial conditions x - 1 , x 0 and y 0 , which are arbitrary positive real numbers. By doing this, we encounter the hypergeometric function. We also investigate global dynamics of this system. The global dynamics of this system consists of two kind of bifurcations.

Explicit solutions for a class of dual integral equations

1982 ◽  
Vol 91 (3-4) ◽  
pp. 277-285 ◽  
Author(s):  
R. S. Anderssen ◽  
F. R. de Hoog ◽  
L. R. F. Rose
Keyword(s):  
Integral Equations ◽  
Explicit Solution ◽  
Explicit Solutions ◽  
Dual Integral Equations ◽  
Linear Combinations ◽  
Basic Technique ◽  
Image Position

SynopsisAn explicit solution is derived for the dual integral equationswhen A(ξ) takes the formIt is then shown how the basic technique can be adapted to derive explicit solutions for more general forms of A(ξ) such asandand linear combinations of such sums.

scholarly journals Numerical semigroups of small and large type

2021 ◽  
pp. 1-20
Author(s):  
Deepesh Singhal
Keyword(s):  
Numerical Semigroup ◽  
Frobenius Number ◽  
Numerical Semigroups ◽  
Natural Numbers ◽  
Type Formula ◽  
Explicit Characterization ◽  
Large Type ◽  
Number Formula ◽  
Genus Formula

A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number [Formula: see text], genus [Formula: see text] and type [Formula: see text]. It is known that for any numerical semigroup [Formula: see text]. Numerical semigroups with [Formula: see text] are called almost symmetric, we introduce a new property that characterizes them. We give an explicit characterization of numerical semigroups with [Formula: see text]. We show that for a fixed [Formula: see text] the number of numerical semigroups with Frobenius number [Formula: see text] and type [Formula: see text] is eventually constant for large [Formula: see text]. The number of numerical semigroups with genus [Formula: see text] and type [Formula: see text] is also eventually constant for large [Formula: see text].

scholarly journals Edge-preserving maps of the nonseparating curve graphs, curve graphs and rectangle preserving maps of the Hatcher–Thurston graphs

2020 ◽  
Vol 29 (11) ◽  
pp. 2050078
Author(s):  
Elmas Irmak
Keyword(s):  
Orientable Surface ◽  
Edge Preserving ◽  
Curve Graph ◽  
Genus Formula

Let [Formula: see text] be a compact, connected, orientable surface of genus [Formula: see text] with [Formula: see text] boundary components with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the nonseparating curve graph, [Formula: see text] be the curve graph and [Formula: see text] be the Hatcher–Thurston graph of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is closed and [Formula: see text] is a rectangle preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We also prove that these homeomorphisms are unique up to isotopy when [Formula: see text].

scholarly journals LINEAR ESTIMATES FOR SOLUTIONS OF QUADRATIC EQUATIONS IN FREE GROUPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250004 ◽  
Author(s):  
OLGA KHARLAMPOVICH ◽  
ALINA VDOVINA
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Quadratic Equation ◽  
Minimal Solution ◽  
Quadratic Equations

We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by 2s for an orientable equation and by 12s4 for a non-orientable equation, where s is the sum of the lengths of the coefficients.

scholarly journals Embeddings of free groups into asymptotic cones of Hamiltonian diffeomorphisms

2019 ◽  
Vol 11 (02) ◽  
pp. 467-498
Author(s):  
D. Alvarez-Gavela ◽  
V. Kaminker ◽  
A. Kislev ◽  
K. Kliakhandler ◽  
A. Pavlichenko ◽  
...  
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Asymptotic Cone ◽  
Asymptotic Cones ◽  
Hamiltonian Diffeomorphisms ◽  
Symplectic Surface ◽  
Genus Formula ◽  
Hofer Metric ◽  
Aspherical Manifolds ◽  
Symplectically Aspherical

Given a symplectic surface [Formula: see text] of genus [Formula: see text], we show that the free group with two generators embeds into every asymptotic cone of [Formula: see text], where [Formula: see text] is the Hofer metric. The result stabilizes to products with symplectically aspherical manifolds.

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