Related papers: Modular correspondences and replicable functions (…

We give examples of infinite order rational transformations that leave linear differential equations covariant. These examples are non-trivial yet simple enough illustrations of exact representations of the renormalization group. We first…

Modular equations occur in number theory, but it is less known that such equations also occur in the study of deformation properties of quasiconformal mappings. The authors study two important plane quasiconformal distortion functions,…

The Springer correspondence makes a link between the characters of a Weyl group and the geometry of the nilpotent cone of the corresponding semisimple Lie algebra. In this article, we consider a modular version of the theory, and show that…

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…

The general idea of this paper is to start from a classical integrable (partial differential) equation which arises as a compatibility condition for a matrix linear differential problem. For definitiveness' sake, a generalised sinh-Gordon…

We give a new construction of noncommutative surfaces via elliptic difference operators, attaching a 1-parameter noncommutative deformation to any projective rational surface with smooth anticanonical curve. The construction agrees with one…

Differential equations are derived for a continuous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

A set of nonlinear differential equations associated with the Eisenstein series of the congruent subgroup $\Gamma_0(2)$ of the modular group $SL_2(\mathbb{Z})$ is constructed. These nonlinear equations are analogues of the well known…

We develop an algebraic approach to studying the spectral properties of the stationary Schr\"odinger equation in one dimension based on its high order conditional symmetries. This approach makes it possible to obtain in explicit form…

In this paper, we will apply the tools from number theory and modular forms to the study of the Seiberg-Witten theory. We will express the holomorphic functions $a, a_D$, which generate the lattice $Z=n_e a+n_m a_D, (n_e, n_m) \in…

We study modular theory in hyperfinite von Neumann algebras, i.e. in those of type II or type III, from the viewpoint of a subregion charge sector decomposition. We address this symmetry resolution by considering infinite tensor products of…

This note considers linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \emph{reduced} linear recurrence with the same solutions as a given linear recurrence, and…

In this paper, we introduce the symmetric multiple Eisenstein series, a variant of the multiple Eisenstein series. As a fundamental result, we show that they satisfy the linear shuffle relation. As a case study, we investigate the vector…

In 1976, Springer defined a correspondence making a link between the irreducible ordinary (characteristic zero) representations of a Weyl group and the geometry of the associated nilpotent variety. In this thesis, we define a modular…

Chen has recently shown how the response matrix of a two-phase composite can be written as linear combinations of products of the component matrices. We elaborate on Chen's expansions by deriving them in a different way, which a. shows them…

We study the class of non-holonomic power series with integer coefficients that reduce, modulo primes, or powers of primes, to algebraic functions. In particular we try to determine whether the susceptibility of the square-lattice Ising…

We suggest a generalization of the Lie algebraic approach for constructing quasi-exactly solvable one-dimensional Schroedinger equations which is due to Shifman and Turbiner in order to include into consideration matrix models. This…

This paper analyzes the nonlinear correspondence between the reflectivity profile (model) and the plane wave impulse response at the boundary (data) for a three-dimensional half space consisting of a sequence of homogeneous horizontal…

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a "determinant" map from this moduli surface to (Z/NZ)*; its fibers are the components of the…

We compare the distribution function and the maximum of solutions of nonlinear elliptic equations defined in general domains with solutions of similar problems defined in a ball using Schwarz symmetrization. As an application, we prove the…