Related papers: Local interdefinability of Weierstrass elliptic fu…

We construct a class of companion elliptic functions associated with the even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $\wp(z|\tau)$ as the blueprint, we will derive their…

A class of exact solutions of the Skyrme model are obtained. They are described by the Weierstrass $\wp$-function or the Jacobi elliptic function. They are not solitonic but of wave character. They supply us with examples of the…

Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the expansion…

This work was initially motivated by Miranda's work on elliptic Weierstrass threefolds. Miranda [Mi] describes a smooth equidimensional (flat) model for any elliptic Weierstrass threefold; such models occur naturally in the study of moduli…

We extend the well-known Gelfand-Phillips property for Banach spaces to locally convex spaces, defining a locally convex space $E$ to be Gelfand-Phillips if every limited set in $E$ is precompact in the topology on $E$ defined by barrels.…

In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the…

The $\alpha$-Weierstrass function is defined as $W_g^{\alpha,b}(x) = \sum_{k=0}^{\infty} b^{-\alpha k} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $\alpha$-Weierstrass function, we prove that the upper…

The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss…

Generalized Weierstrass representations for generic surfaces conformally immersed into four-dimensional Euclidean and pseudo-Euclidean spaces of different signatures are presented. Integrable deformations of surfaces in these spaces…

We derive a number of local identities of arbitrary rank involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us…

The aim of this article is to explore in all remaining aspects the spectral theory of locally normal operators. In a previous article we proved the spectral theorem in terms of locally spectral measures. Here we prove the spectral theorem…

We first show that the projection image of a discrete definable set is again discrete for an arbitrary definably complete locally o-minimal structure. This fact together with the results in a previous paper implies tame dimension theory and…

The Stone-Weierstrass approximation theorem is extended to certain unbounded sets in $C^n$. In particular, on a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by…

We develop a wavelet like representation of functions in $L^p(\mathbb{R})$ based on their Fourier--Hermite coefficients; i.e., we describe an expansion of such functions where the local behavior of the terms characterize completely the…

In this paper we continue our study of equivariant minimal Lagrangian surfaces in $\mathbb{C} P^2$, characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally…

This paper addresses a multi-scale finite element method for second order linear elliptic equations with arbitrarily rough coefficient. We propose a local oversampling method to construct basis functions that have optimal local…

We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald…

In this paper two identities involving a function defined by the complete elliptic integrals of the first and second kinds are proved. Some functional inequalities and elementary estimates for this function are also derived from the…

Let $R$ be an o-minimal expansion of the real field. We show that the Hausdorff dimension of an $R$-definable metric space is an $R$-definable function of the parameters defining the metric space. We also show that the Hausdorff dimension…