Related papers: On Relations of Hyperelliptic Weierstrass al Funct…
The non-Abelian two-dimensional Toda lattice and matrix sine-Gordon equations with self-consistent sources are established and solved. Two families of quasideterminant solutions are presented for the non-Abelian two-dimensional Toda lattice…
We prove consequences of functional equations of p-adic L-functions for elliptic curves at supersingular primes p. The results include a relationship between the leading and sub-leading terms (for which we use ideas of Wuthrich and…
The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of…
We present an algorithm for the computation of period matrices and the Abel-Jacobi map of complex superelliptic curves given by an equation $y^m=f(x)$. It relies on rigorous numerical integration of differentials between Weierstrass points,…
A problem of current interest, also motivated by applications to Coding theory, is to find explicit equations for \textit{maximal} curves, that are projective, geometrically irreducible, non-singular curves defined over a finite field…
We study the elliptic sinh-Gordon and sine-Gordon equations on the real plane and we introduce new families of solutions. We use a Backlund transformation that connects the elliptic versions of sinh-Gordon and sine-Gordon equations. As an…
We derive estimates relating the values of a solution at any two points to the distance between the points, for quasilinear isotropic elliptic equations on compact Riemannian manifolds, depending only on dimension and a lower bound for the…
We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:\Omega\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the…
Infinite products expansions of the Weierstrass elliptic function \ $\wp(z) = \wp(z,1,\tau)$\ and $n$-order transformations allow us to provide some modular relations.
We study genus $g$ hyperelliptic curves with reduced automorphism group $A_5$ and give equations $y^2=f(x)$ for such curves in both cases where $f(x)$ is a decomposable polynomial in $x^2$ or $x^5$. For any fixed genus the locus of such…
The elliptic sine-Gordon equation in the plane has a family of explicit multiple-end solutions (soliton-like solutions). We show that all the finite Morse index solutions belong to this family. We also prove they are non-degenerate in the…
Adapted pairs and Weierstrass sections are central to the invariant theory associated to the action of an algebraic Lie algebra a on a finite dimensional vector space X. In this a need not be a semisimple Lie algebra. Here their general…
We study several fractal properties of the Weierstrass-type function \[ W(x)=\sum_{n=0} ^\infty \lambda (x) \lambda(\tau x) \cdots \lambda (\tau ^{n-1}x)\, g(\tau ^n x), \] where $\tau :[0,1)\to[0,1)$ is a cookie cutter map with possibly…
We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of…
A minimal Lorentz surface in $\mathbb R^4_2$ is said to be of general type if its corresponding null curves are non-degenerate. These surfaces admit canonical isothermal and canonical isotropic coordinates. It is known that the Gauss…
We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado \cite{D}, we present a description of these…
A. Weinstein has conjectured a nice looking formula for a deformed product of functions on a hermitian symmetric space of non-compact type. We derive such a formula for symmetric symplectic spaces using ideas from geometric quantization and…
An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature…
We consider a transient Brownian motion reflected obliquely in a two-dimensional wedge. A precise asymptotic expansion of Green's functions is found in all directions. To this end, we first determine a kernel functional equation connecting…
Our work proposes a unified approach to three different topics in a general Riemannian setting: splitting theorems, symmetry results and overdetermined elliptic problems. By the existence of a stable solution to the semilinear equation…