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About 2,963 papers for “Modular equation”

On a modular equation II

Tadeusz Jędryka, Julian Musielak · Repository of Kazimierz Wielki University · 1977

Cited by 0

On a modular Fermat equation

Jonathan Pila · Commentarii Mathematici Helvetici · 2017

We study some diophantine problems suggested by the analogy between multiplicative groups and powers of the modular curve in problems of “unlikely intersections”. We prove a special case of the Zilber–Pink conjecture for curves.

Cited by 0Open access

EXPLICIT FORMULAS FOR THE MODULAR EQUATION

Paul Baginski, Elena Fuchs · 2015

Abstract. We determine a linear time algorithm for calculating the modular equation ΦN (X, J) for N = p1p2, where p1 and p2 are distinct primes. We provide N = 10 as an example. 1.

Cited by 0

Modular Schrödinger equation and dynamical duality

Garbaczewski, Piotr · arXiv (Cornell University) · 2008

We discuss quite surprising properties of the one-parameter family of modular (Auberson and Sabatier (1994)) nonlinear Schrödinger equations. We develop a unified theoretical framework for this family. Special attention is paid to the emergent \it dual \rm time evolution scenarios which, albeit running in the \it real time \rm parameter of the pertinent nonlinear equation, in each considered case, may be mapped among each other by means of an "imaginary time" transformation (more seriously, an analytic continuation in time procedure).

Cited by 0Open access

Modular Schrödinger equation and dynamical duality

Piotr Garbaczewski · Physical Review E · 2008

We discuss quite surprising properties of the one-parameter family of modular nonlinear Schrödinger equations [G. Auberson and P. G. Sabatier, J. Math. Phys. 35, 4028 (1994)]. We develop a unified theoretical framework for this family. Special attention is paid to the emergent dual time evolution scenarios which, albeit running in the real time parameter of the pertinent nonlinear equation, in each considered case may be mapped among each other by means of a suitable analytic continuation-in-time procedure. This dynamical duality is characteristic for nondissipative quantum motions and their dissipative (diffusion-type processes) partners, and naturally extends to classical motions in confining and scattering potentials.

Cited by 10Open access

On Explicit Formulas for the Modular Equation

Shamita Dutta Gupta, Xiaotie She · Rocky Mountain Journal of Mathematics · 2001

An algorithm is given to determine explicitly the modular equation n (X, J) = 0 of degree n, n = p 2 . 9 (X, J) is used as an example.

Cited by 1Open access

Eigenvalue equation for genus two modular graphs

Anirban Basu · Journal of High Energy Physics · 2019

A bstract We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by analyzing the variations of these graphs under the variation of the Beltrami differentials. This equation involves seven distinct graphs, three of which appear in the integrand of the D 8 $$ \mathrm{\mathcal{R}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℛ</mml:mi> </mml:math> 4 term in the low momentum expansion of the four graviton amplitude at genus two in type II string theory.

Cited by 16Open access

Stability of quadratic functional equation in modular spaces

Baza, Abderrahman, Rossafi, Mohamed, Gnanaprakasam, Arul Joseph · ArXiv.org · 2025

In this paper, we study the Hyers-Ulam stability of the following equation \begin{multline*} \phi(x+y-z)+\phi(x+z-y)+\phi(y+z-x)=\phi (x-y)+\phi(x-z)+\phi(z-y) +\phi(x)+\phi(y) +\phi(z) \end{multline*} in modular space, with or without $\Delta_2$-condition, and in $\beta$-homogeneous Banach space.

Cited by 0Open access

On the Galois group of the modular equation

Catherine Jane Barry · Durham e-Theses (Durham University) · 1992

This thesis looks at a method of generating infinitely many extensions of the rationals with Galois group PGL(_2)Z(_n)). Firstly, the Galois group of the modular equation over Q(j) is shown to be PGL(_2)(Z(_n) by considering the n-th division points on an elliptic curve. Then, using Hilbert's Irreducibility Theorem and work discussed by Lang, we show that there are infinitely many rational values of such that this Galois group does not reduce in size. Finally, an equation whose roots generate the same extension as the modular equation but which has much smaller coefficients is found, based on work by Cohn.

Cited by 0Open access

Stability of quadratic functional equation in modular spaces

Baza, Abderrahman, Rossafi, Mohamed, Gnanaprakasam, Arul Joseph · arXiv (Cornell University) · 2025

In this paper, we study the Hyers-Ulam stability of the following equation \begin{multline*} ϕ(x+y-z)+ϕ(x+z-y)+ϕ(y+z-x)=ϕ(x-y)+ϕ(x-z)+ϕ(z-y) +ϕ(x)+ϕ(y) +ϕ(z) \end{multline*} in modular space, with or without $Δ_2$-condition, and in $β$-homogeneous Banach space.

Cited by 0Open access