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To any solution of a linear system of differential equations, we associate a kernel, correlators satisfying a set of loop equations, and in presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion…
Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The…
Hyperdeterminants are generalizations of determinants from matrices to multi-dimensional hypermatrices. They were discovered in the 19th century by Arthur Cayley but were largely ignored over a period of 100 years before once again being…
On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the…
We develop a boundary integral equation solver for elliptic partial differential equations on complex \threed geometries. Our method is efficient, high-order accurate and robustly handles complex geometries. A key component is our singular…
In this work, we introduce two novel classes of quasilinear elliptic equations, each driven by the double phase operator with variable exponents. The first class features a new double phase equation where exponents depend on the gradient of…
To decide upon the arithmetic nature of some numbers may be a non-trivial problem. Some cases are well know, for example exp(1) and W(1), where W is the Lambert function, are transcendental numbers. The Tsallis q-exponential, e_q (z), and…
By using representation theory of the elliptic quantum group U_{q,p}(sl_N^), we present a systematic method of deriving the weight functions. The resultant sl_N type elliptic weight functions are new and give elliptic and dynamical…
In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in…
We extend Sklyanin's method of separation of variables to quantum integrable models associated to elliptic curves. After reviewing the differential case, the elliptic Gaudin model studied by Enriquez, Feigin and Rubtsov, we consider the…
We study the complex powers $A^{z}$ of an elliptic, strictly positive pseudodifferential operator $A$ using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras,…
We define elliptic sequences over a commutative ring as sequences indexed by the (positive) integers satisfying a 4-parameter, highly symmetric family of homogeneous quartic relations among terms which we call elliptic relations. We…
For an elliptic curve $E$ over a finite field $\F_q$, where $q$ is a prime power, we propose new algorithms for testing the supersingularity of $E$. Our algorithms are based on the Polynomial Identity Testing (PIT) problem for the $p$-th…
We study about solutions of certain kind of non-linear differential difference equations $$f^{n}(z)+wf^{n-1}(z)f^{'}(z)+f^{(k)}(z+c)=p_{1}e^{\alpha_{1}z}+p_{2}e^{\alpha_{2}z}$$ and…
In this paper, by making use of properties of elliptic functions, we describe meromorphic solutions of Fermat-type functional equations $f(z)^{n}+f(L(z))^{m}=1$ over the complex plane $\mathbb{C}$, where $L(z)$ is a nonconstant entire…
We study algebro-geometric (finite-gap) and elliptic solutions of fully discretized KP or 2D Toda equations. In bilinear form they are Hirota's difference equation for $\tau$-functions. Starting from a given algebraic curve, we express the…
In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as $p$-{\it ellipticity}. Specifically, let $\Omega$ be a chord-arc domain in $\mathbb R^n$…
In this paper, a theory of hyperelliptic functions based on multidimensional sigma functions is developed and explicit formulas for hyperelliptic solutions to the Kadomtsev-Petviashvili equations KP-I and KP-II are obtained. The…
The classical concept of $Q$-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be…
For smooth finite fields $F_q$ (i.e., when $q-1$ factors into small primes) the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division,…