scholarly journals One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-18 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo ◽  
Gonzalo Riera
Keyword(s):  
Dihedral Group ◽  
Meromorphic Functions ◽  
Monodromy Group ◽  
Elliptic Functions ◽  
Hurwitz Spaces ◽  
One Dimensional ◽  
Hurwitz Space ◽  
Upper Half Plane ◽  
Triangular Group ◽  
Real Forms

Hurwitz spaces are spaces of pairs(S,f)whereSis a Riemann surface andf:S→ℂ^a meromorphic function. In this work, we study1-dimensional Hurwitz spacesℋDpof meromorphicp-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of(p−1)/2transpositions and the monodromy group is the dihedral groupDp. We prove that the completionℋDp¯of the Hurwitz spaceℋDpis uniformized by a non-nomal indexp+1subgroup of a triangular group with signature(0;[p,p,p]). We also establish the relation of the meromorphic covers with elliptic functions and show thatℋDpis a quotient of the upper half plane by the modular groupΓ(2)∩Γ0(p). Finally, we study the real forms of the Belyi projectionℋDp¯→ℂ^and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.

scholarly journals Connected Components of H_(r,g)^A (G)

2018 ◽  
Vol 22 (2) ◽  
pp. 106
Author(s):  
Haval M. Mohammed Salih
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Monodromy Group ◽  
Riemann Sphere ◽  
Primitive Group ◽  
Connected Components ◽  
Branch Points ◽  
Genus Zero ◽  
Hurwitz Spaces ◽  
Hurwitz Space

The Hurwitz space   is the space of genus g covers of the Riemann sphere  with  branch points and the monodromy group . In this paper, we enumerate the connected components of the Hurwitz spaces  for a finite primitive group of degree 7 and genus zero except . We achieve this with the aid of the computer algebra system GAP and the MAPCLASS package.

scholarly journals Order parameter profiles in a system with Neumann – Neumann boundary conditions

2018 ◽  
Vol 145 ◽  
pp. 01009 ◽  
Author(s):  
Vassil M. Vassilev ◽  
Daniel M. Dantchev ◽  
Peter A. Djondjorov
Keyword(s):  
Boundary Conditions ◽  
Order Parameter ◽  
Neumann Boundary ◽  
Elliptic Functions ◽  
Thermodynamic System ◽  
Neumann Boundary Conditions ◽  
Ginzburg Landau ◽  
One Dimensional ◽  
The One ◽  
Field Plane

In this article we consider a critical thermodynamic system with the shape of a thin film confined between two parallel planes. It is assumed that the state of the system at a given temperature and external ordering field is described by order-parameter profiles, which minimize the one-dimensional counterpart of the standard ϕ4 Ginzburg–Landau Hamiltonian and meet the so-called Neumann – Neumann boundary conditions. We give analytic representation of the extremals of this variational problem in terms ofWeierstrass elliptic functions. Then, depending on the temperature and ordering field we determine the minimizers and obtain the phase diagram in the temperature-field plane.

scholarly journals Logarithmic derivative estimates of meromorphic functions of finite order in the half-plane

2020 ◽  
Vol 54 (2) ◽  
pp. 172-187
Author(s):  
I.E. Chyzhykov ◽  
A.Z. Mokhon'ko
Keyword(s):  
Meromorphic Function ◽  
Finite Order ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Sharp Estimates ◽  
Exceptional Sets ◽  
Upper Half Plane ◽  
Logarithmic Derivatives

We established new sharp estimates outside exceptional sets for of the logarithmic derivatives $\frac{d^ {k} \log f(z)}{dz^k}$ and its generalization $\frac{f^{(k)}(z)}{f^{(j)}(z)}$, where $f$ is a meromorphic function $f$ in the upper half-plane, $k>j\ge0$ are integers. These estimates improve known estimates due to the second author in the class of meromorphic functions of finite order.Examples show that size of exceptional sets are best possible in some sense.

scholarly journals Extreme problems in the space of meromorphic functions of finite order in the half plane. II

2020 ◽  
Vol 54 (2) ◽  
pp. 154-161
Author(s):  
K.G. Malyutin ◽  
A.A. Revenko
Keyword(s):  
Meromorphic Function ◽  
Half Plane ◽  
Fourier Coefficients ◽  
Meromorphic Functions ◽  
Sharp Estimate ◽  
Extremal Problems ◽  
Parseval Equality ◽  
Upper Limits ◽  
Upper Half Plane ◽  
Space Of Meromorphic Functions

The extremal problems in the space of meromorphic functions of order $\rho>0$ in upper half-plane are studed.The method for studying is based on the theory of Fourier coefficients of meromorphic functions. The concept of just meromorphic function of order $\rho>0$ in upper half-plane is introduced. Using Lemma on the P\'olya peaks and the Parseval equality, sharp estimate from below of the upper limits of relations Nevanlinna characteristics of meromorphic functions in the upper half plane are obtained.

Hausdorff Dimension of Sets of Escaping Points and Escaping Parameters for Elliptic Functions

2015 ◽  
Vol 59 (3) ◽  
pp. 671-690
Author(s):  
Piotr Gałązka ◽  
Janina Kotus
Keyword(s):  
Hausdorff Dimension ◽  
Elliptic Function ◽  
Meromorphic Functions ◽  
Elliptic Functions ◽  
Critical Value ◽  
Parameter Plane ◽  
Dynamical Plane ◽  
The Family ◽  
Maximal Multiplicity ◽  
Additional Assumptions

AbstractLetbe a non-constant elliptic function. We prove that the Hausdorff dimension of the escaping set offequals 2q/(q+1), whereqis the maximal multiplicity of poles off. We also consider theescaping parametersin the familyfβ=βf, i.e. the parametersβfor which the orbit of one critical value offβescapes to infinity. Under additional assumptions onfwe prove that the Hausdorff dimension of the set of escaping parametersεin the familyfβis greater than or equal to the Hausdorff dimension of the escaping set in the dynamical space. This demonstrates an analogy between the dynamical plane and the parameter plane in the class of transcendental meromorphic functions.

LOCALIZED BREATHER-LIKE EXCITATIONS IN THE HELICOIDAL PEYRARD–BISHOP MODEL OF DNA

2009 ◽  
Vol 02 (04) ◽  
pp. 405-417 ◽  
Author(s):  
CONRAD BERTRAND TABI ◽  
ALIDOU MOHAMADOU ◽  
TIMOLEON CREPIN KOFANE
Keyword(s):  
Analytical Solution ◽  
Continuous Wave ◽  
Exact Analytical Solution ◽  
Modulational Instability ◽  
Elliptic Functions ◽  
Instability Criterion ◽  
Dna Dynamics ◽  
Jacobian Elliptic Functions ◽  
One Dimensional ◽  
The One

We consider the one-dimensional helicoidal Peyrard–Bishop (PB) model of DNA dynamics. By means of a method based on the Jacobian elliptic functions, we obtain the exact analytical solution which describes the modulational instability and the propagation of a bright solitary wave on a continuous wave background. It is shown that these solutions depend on the modulational (or Benjamin-Feir) instability criterion. Numerical simulations of their propagation show these excitations to be long-lived and suggest that they are physically relevant for DNA.

THE 2-D SEXTIC HAMILTONIAN OSCILLATOR

2013 ◽  
Vol 23 (06) ◽  
pp. 1330019
Author(s):  
F. J. MOLERO ◽  
J. C. VAN DER MEER ◽  
S. FERRER ◽  
F. J. CÉSPEDES
Keyword(s):  
Phase Space ◽  
Elliptic Functions ◽  
Singular Values ◽  
Nonlinear Oscillators ◽  
Momentum Mapping ◽  
Dimensional Model ◽  
Integrable Hamiltonian Systems ◽  
One Dimensional ◽  
Dimensional Phase Space ◽  
The One

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundamental fibration of the four-dimensional phase space. Special attention is given to the singular values of the energy-momentum mapping connected with rectilinear and circular orbits. They are related to the saddle-center and pitchfork scenarios with the associated homoclinic and heteroclinic trajectories. We also study how the geometry of the phase space evolves during the transition from the one-dimensional to the two-dimensional model. Within an elliptic function approach, the solutions are given using Legendre elliptic integrals of the first and third kind and the corresponding Jacobi elliptic functions.

scholarly journals The Grothendieck--Teichm\"{u}ller group of~PSL(2,q)

2018 ◽  
Vol 21 (2) ◽  
pp. 241-251
Author(s):  
Pierre Guillot
Keyword(s):  
Finite Groups ◽  
Dihedral Group ◽  
Monodromy Group ◽  
Derived Length

AbstractIn this paper, we show that the Grothendieck–Teichmüller group of{\operatorname{PSL}(2,q)}, or more precisely the group{\mathcal{G\kern-0.569055ptT}_{\kern-1.707165pt1}(\operatorname{PSL}(2,q))}as previously defined by the author, is the product of an elementary abelian 2-group and several copies of the dihedral group of order 8. Moreover, whenqis even, we show that it is trivial. We explain how it follows that the moduli field of any “dessin d’enfant” whose monodromy group is{\operatorname{PSL}(2,q)}has derived length{\leq 3}. This paper can serve as an introduction to the general results on the Grothendieck–Teichmüller group of finite groups obtained by the author.

The analytic structure of the reflection coefficient, a sum rule and a complete description of the Weyl m-function of half-line Schrödinger operators with L2-type potentials

2006 ◽  
Vol 136 (3) ◽  
pp. 615-632 ◽  
Author(s):  
Alexei Rybkin
Keyword(s):  
Reflection Coefficient ◽  
Schrödinger Operators ◽  
Half Line ◽  
Analytic Structure ◽  
Schrodinger Operators ◽  
Trace Inequality ◽  
Sum Rule ◽  
One Dimensional ◽  
Upper Half Plane ◽  
Image Position

We prove that the reflection coefficient of one-dimensional Schrödinger operators with potentials supported on a half-line can be represented in the upper half-plane as the quotient of a contractive analytic function and a properly regularized Blaschke product. We apply this fact to obtain a new trace formula and trace inequality for the reflection coefficient that yields a description of the Weyl m-function of Dirichlet half-line Schrödinger operators with slowly decaying potentials q subject to Among others, we also refine the 3/2-Lieb-Thirring inequality.

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