Related papers: Class Number and Regulator Computation in Purely C…
In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete…
In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D'Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics,…
This paper addresses the optimal control problem known as the Linear Quadratic Regulator in the case when the dynamics are unknown. We propose a multi-stage procedure, called Coarse-ID control, that estimates a model from a few experimental…
The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows and methods employing Leopoldt's decomposition of the class number…
This paper introduces new constructions of sum-rank metric codes derived from algebraic function fields, as existing results on such codes remain limited. A major challenge lies in the determination of their parameters. We address this…
It was believed that modular data are enough to distinguish different modular categories (and topological orders in 2+1-dimensions). Then counterexamples to this conjecture were found by Mignard and Schauenburg in 2017. In this work, we…
We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems…
We implement and interpret various supervised learning experiments involving real quadratic fields with class numbers 1, 2 and 3. We quantify the relative difficulties in separating class numbers of matching/different parity from a…
We prove the existence of secondary terms of order $X^{5/6}$ in the asymptotic formulas for the average size of the genus number of cubic fields and for the number of cubic fields with a given genus number, establishing improved error…
In this paper, we investigate the unit groups, the $2$-class groups, the $2$-class field towers and the structures of the second $2$-class groups of some multiquadratic number fields of degree $8$ and $16$.
Genus 2 curves have been an object of much mathematical interest since eighteenth century and continued interest to date. They have become an important tool in many algorithms in cryptographic applications, such as factoring large numbers,…
We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic…
In this paper, we present a method to control a quadrotor with a neural network trained using reinforcement learning techniques. With reinforcement learning, a common network can be trained to directly map state to actuator command making…
We compute the class of the locus in M_g of curves having a pencil with two unspecified triple ramification points. This is the first example of a geometric divisor on M_g which is not the pull-back of a divisor on the moduli space of…
Current research suggests the use of a liner quadratic performance index for optimal control of regulators in various applications. Some examples include correcting the trajectory of rocket and air vehicles, vibration suppression of…
We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common…
We give an asymptotic formula for class numbers of orders in cubic number fields.
The theory of continued fractions of functions $ \sqrt D $ is used to give lower bound for class numbers $h(D)$ of general real quadratic function fields $K=k(\sqrt D)$ over $k={\bf F}_q(T)$. For five series of real quadratic function…
Our work addresses the well-known open problem of distributed computing of bilinear functions of two correlated sources ${\bf A}$ and ${\bf B}$. In a setting with two nodes, with the first node having access to ${\bf A}$ and the second to…
The optimal controller design problem for a linear, first-order spatially-invariant distributed parameter system is considered. Through a case study of the Linear Quadratic Regulator (LQR) problem for the diffusion equation over the torus,…