Related papers: An Algorithm to Find Rational Points on Elliptic C…
We study the iterative algorithm proposed by S. Armstrong, A. Hannukainen, T. Kuusi, J.-C. Mourrat to solve elliptic equations in divergence form with stochastic stationary coefficients. Such equations display rapidly oscillating…
We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…
In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional…
We present a detailed analysis of how to implement the computation of modular symbols for a given elliptic curve by using numerical approximations. This method turns out to be more efficient than current implementation as the conductor of…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic…
Quantum algorithms for Hamiltonian simulation and linear differential equations more generally have provided promising exponential speed-ups over classical computers on a set of problems with high real-world interest. However, extending…
We describe how the Mordell-Weil group of rational points on a certain family of elliptic curves give rise to solutions to a conjecture of Erd\"{o}s on $3$-powerful numbers, and state a related conjecture which can be viewed as an elliptic…
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
In a previous paper, a point of order 8 on an elliptic curve was calculated. Exploiting the well-known correspondence of the points on an elliptic curve with the points of a respective period parallelogram, we proceed to calculating all…
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.
We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.
Ideal class pairings map the rational points of rank $r\geq 1$ elliptic curves $E/\Q$ to the ideal class groups $\CL(-D)$ of certain imaginary quadratic fields. These pairings imply that $$h(-D) \geq \frac{1}{2}(c(E)-\varepsilon)(\log…
We show how the Eulcidean algorithm for polynomials can be used to find the intersection points, with multiplicities, of two plane algebraic curves.
In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of…
In a former paper it has been shown that the elliptic Gau{\ss} sums, whose use has been proposed in the context of counting points on elliptic curves and primality tests, can be computed by using modular functions. In this work we give…
In this work, we investigate a neural network based solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. It utilizes a coupled system derived from the first-order…
In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We…
We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem…