Related papers: Hyperelliptic tangential covers and even elliptic …

Let $\Omega\subset \R^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -\Delta…

Suppose that $\Omega$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T $ which preserves support in $\bar{\Omega$}, is…

A double covering of the proper orthochronous Lorentz group is understood as a complexification of the special unimodular group of second order (a double covering of the 3-dimensional rotation group). In virtue of such an interpretation the…

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…

Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically…

In the present paper, a hierarchy of the mKdV equation is integrated by the methods of algebraic geometry. The mKdV hierarchy in question arises on coadjoint orbits in the loop algebra of $\mathfrak{sl}(2)$, and employs a family of…

The potential of the $BC_1$ quantum elliptic model is a superposition of two Weierstrass functions with doubling of both periods (two coupling constants). The $BC_1$ elliptic model degenerates to $A_1$ elliptic model characterized by the…

Let $\lambda \in \mathbb{Q}\setminus \{0, 1\}$ and $l \geq 2$, and denote by $C_{l,\lambda}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=x(x-1)(x-\lambda).$$ In this paper we define…

Along recent decades, an intensive worldwide research activity is focusing both black holes and cosmos (e.g. the dark-energy phenomenon) on the basis of entropic approaches. The Boltzmann-Gibbs-based Bekenstein-Hawking entropy…

We observe that approximate copies of the function $\Lambda _{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} \Lambda _{n}(x)=\exp \left( -x_{1}-\pi \sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails…

This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a…

For a lattice \Lambda in the complex plane, let K_{\Lambda} be the field of \Lambda-elliptic functions. For two relatively prime integers p (respectively q) greater than 1, consider the endomorphisms \psi (resp. \phi) of K_{\Lambda} given…

We show how certain non-perturbative superpotentials W, which are the two-dimensional analogs of the Seiberg-Witten prepotential in 4d, can be computed via geometric engineering from 4-folds. We analyze an explicit example for which the…

A set of complex numbers $\Lambda=\{\lambda_n,\mu_n\}_{n=1}^{\infty}$ with multiple terms \[ \{\lambda_n,\mu_n\}_{n=1}^{\infty}:= \{\underbrace{\lambda_1,\lambda_1,\dots,\lambda_1}_{\mu_1 - times},…

We provide two kinds of representations for the Taylor coefficients of the Weierstrass $\sigma$-function $\sigma(\cdot;\Gamma)$ associated to an arbitrary lattice $\Gamma$ in the complex plane $\mathbb{C}=\mathbb{R}^2$ - the first one in…

We lay out the foundations of the theory of second-order conformal superintegrable systems. Such systems are essentially Laplace equations on a manifold with an added potential: $(\Delta_n+V({\bf x}))\Psi=0$. Distinct families of…

This paper addresses the following problem. \begin{equation} \left\{ \begin{array}{lr} -{\Delta}u=\lambda I_\alpha*_\Omega u+|u|^{2^*-2}u\mbox{ in }\Omega ,\nonumber u\in H_0^1(\Omega).\nonumber \end{array} \right. \end{equation} Here,…

In a previous paper [Nijhoff,Puttock,2003], a 2-parameter extension of the lattice potential KdV equation was derived, associated with an elliptic curve. This comprises a rather complicated 3-component system on the quad lattice which…

$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1, $ bound of the values, and criterion of reaching the bound are given.…

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger…