The definition of the unknown parameters in the conformal mapping of the upper half-plane onto an arbitrary curvilinear rectangle

1972 ◽  
Vol 23 (2) ◽  
pp. 227-232
Author(s):  
L. M. Nepritvorennaya
Keyword(s):  
Conformal Mapping ◽  
Half Plane ◽  
Unknown Parameters ◽  
Upper Half Plane ◽  
Definition Of

ELLIPTIC AND AUTOMORPHIC DYNAMICAL SYSTEMS ON SURFACES

2006 ◽  
Vol 16 (04) ◽  
pp. 911-923 ◽  
Author(s):  
S. P. BANKS ◽  
SONG YI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Half Plane ◽  
Complex Plane ◽  
Fundamental Domain ◽  
Function Theory ◽  
Theta Series ◽  
The Hyperbolic Metric ◽  
Upper Half Plane ◽  
Definition Of

We derive explicit differential equations for dynamical systems defined on generic surfaces applying elliptic and automorphic function theory to make uniform the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series we will determine general meromorphic systems on a fundamental domain in the upper half plane the solution trajectories of which "roll up" onto an appropriate surface of any given genus.

scholarly journals Cuspidal Modular Symbols are Transportable

2001 ◽  
Vol 4 ◽  
pp. 170-181 ◽  
Author(s):  
William A. Stein ◽  
Helena A. Verrill
Keyword(s):  
Half Plane ◽  
Congruence Subgroup ◽  
Modular Symbols ◽  
Upper Half Plane ◽  
Definition Of

AbstractModular symbols of weight 2 for a congruence subgroup Γ satisfy the identity {α,γ,(α)}={β,γ(β)} for all α,β in the extended upper half plane and γ ∊ Γ. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cuspforms.

scholarly journals On Tangential Principal Cluster Sets of Normal Meromorphic Functions

1970 ◽  
Vol 40 ◽  
pp. 133-137 ◽  
Author(s):  
Theodore A. Vessey
Keyword(s):  
Meromorphic Function ◽  
Conformal Mapping ◽  
Finite Order ◽  
Real Axis ◽  
Half Plane ◽  
Meromorphic Functions ◽  
The Family ◽  
Normal Meromorphic Function ◽  
Upper Half Plane ◽  
Principal Cluster

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

scholarly journals The meromorphic functions of completely regular growth on the upper half-plane

2020 ◽  
Vol 30 (3) ◽  
pp. 396-409
Author(s):  
K.G. Malyutin ◽  
M.V. Kabanko
Keyword(s):  
Meromorphic Function ◽  
Half Plane ◽  
Meromorphic Functions ◽  
Growth Function ◽  
Regular Growth ◽  
Completely Regular ◽  
Upper Half Plane ◽  
Definition Of ◽  
Half Axis ◽  
Increasing Function

A strictly positive continuous unbounded increasing function $\gamma(r)$ on the half-axis $[0,+\infty)$ is called growth function. Let the growth function $\gamma(r)$ satisfies the condition $\gamma(2r)\leq M\gamma(r)$ for some $M>0$ and for all $r>0$. In the paper, the class $JM(\gamma(r))^o$ of meromorphic functions of completely regular growth on the upper half-plane with respect to the growth function $\gamma$ is considered. The criterion for the meromorphic function $f$ to belong to the space $JM(\gamma(r))^o$ is obtained. The definition of the indicator of function from the space $JM(\gamma(r))^o$ is introduced. It is proved that the indicator belongs to the space $\mathbf{L}^p[0,\pi]$ for all $p>1$.

scholarly journals Conformal mapping of the Misner–Sharp mass from gravitational collapse

2016 ◽  
Vol 25 (07) ◽  
pp. 1650081 ◽  
Author(s):  
Fayçal Hammad
Keyword(s):  
Conformal Mapping ◽  
Gravitational Collapse ◽  
Conformal Transformation ◽  
Conformal Mappings ◽  
Conformal Transformations ◽  
Theories Of Gravity ◽  
Definition Of ◽  
Geometric Definition

The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.

scholarly journals On Complex Explicit Formulae Connected with the Möbius Function of an Elliptic Curve

2014 ◽  
Vol 57 (2) ◽  
pp. 381-389
Author(s):  
Adrian Łydka
Keyword(s):  
Functional Equation ◽  
Meromorphic Function ◽  
Elliptic Curve ◽  
Explicit Formula ◽  
Half Plane ◽  
Complex Plane ◽  
Möbius Function ◽  
Mobius Function ◽  
Upper Half Plane ◽  
Explicit Formulae

AbstractWe study analytic properties function m(z, E), which is defined on the upper half-plane as an integral from the shifted L-function of an elliptic curve. We show that m(z, E) analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for m(z, E) in the strip |ℑz| < 2π.

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