Related papers: 5-th order differential equations related to Calab…
The Calabi-Yau differential equations of degree 2 and 3 are listed. The idea is to bring some order into the "big table" into the "big table" (math. AG/0507430). The author has benefitted from an unpublished result by Yifan Yang, which…
We explain an experimental method to find CY-type differential equations of order $3$ related to modular functions of genus zero. We introduce a similar class of Calabi-Yau differential equations of order $5$, show several examples and make…
The main part of this paper is a big table containing what we believe to be a complete list of all fourth order equations of Calabi--Yau type known so far. In the text preceding the tables we explain what a differential equation of…
We show that the solutions to the equations defining the so-called Calabi-Yau condition for fourth order operators of degree two defines a variety that consists of ten irreducible components. These can be described completely in parametric…
Various methods to find Calabi-Yau differential equations are discussed.
This paper contains a preliminary study of the monodromy of certain fourth order differential equations, that were called of Calabi-Yau type in math.NT/0402386. Some of these equations can be interpreted as the Picard-Fuchs equations of a…
Motivated by the relationship of classical modular functions and Picard--Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.
Given a differential operator of geometric origin there exists a list of operations that preserve this property, e.g., tensor products, pull-backs, push-forwards and the middle convolution. We apply certain sequences of these operations to…
We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions, can be generalized to arbitrary order linear…
By imposing special compatible similarity constraints on a class of integrable partial $q$-difference equations of KdV-type we derive a hierarchy of second-degree ordinary $q$-difference equations. The lowest (non-trivial) member of this…
A class of fourth--order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four--dimensional difference system. The…
We study modular differential equations (MDEs) of high orders for weak Jacobi forms and find necessary conditions for weak Jacobi forms to satisfy MDEs of order 3 with respect to the heat operator. We investigate all possible MDEs for the…
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations and third order quintically semi-linear ordinary differential equations, we extend to the fourth order by differentiating…
We investigate the structures of Calabi-Yau differential equations and the relations to the arithmetic of the pencils of Calabi-Yau varieties behind the equations. This provides explanations of some observations and computations in a recent…
Recently, the non-linear Changhee differential equations were introduced in [5] and these differential equations turned out to be very useful for studying special polynomials and mathematical physics. Some interesting identities and…
We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi--Yau varieties. We show that the elliptic genus of any $CY_3$ satisfies a differential…
This paper examines the classification of hyperbolic equations. We study a class of equations of the form $$\frac{\partial^2 u}{\partial x\partial y}=F\left(\frac{\partial u}{\partial x},\frac{\partial u}{\partial y},u\right),$$ where…
We study the Poincare polynomials of all known Calabi-Yau three-folds as constrained polynomials of Littlewood type, thus generalising the well-known investigation into the distribution of the Euler characteristic and Hodge numbers. We find…
We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…