Related papers: Geodesic continued fractions and LLL

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of $p$--adic numbers $\mathbb Q_p$. Here, we study the use of multidimensional continued fractions (MCFs) in this…

The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…

We propose a local Legendre frame (LLF) method for function approximation from equispaced data on a finite interval. Motivated by the difficulty of stable high-order polynomial approximation at equispaced points, especially in the presence…

It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…

In these lectures we give an introduction to the reduction theory of binary forms starting with quadratic forms with real coefficients, Hermitian forms, and then define the Julia quadratic for any degree $n$ binary form. A survey of a…

The Lov\'{a}sz Local Lemma (LLL) is a keystone principle in probability theory, guaranteeing the existence of configurations which avoid a collection $\mathcal B$ of "bad" events which are mostly independent and have low probability. In its…

In this paper, we continue our previous work on the reduction of algebraic lattices over imaginary quadratic fields for the special case when the lattice is spanned over a two dimensional basis. In particular, we show that the…

We present a unified theoretical framework for parametric low-rank approximation, a research area devoted to the development of efficient algorithms that act as adaptive alternatives of traditional methods such as Singular Value…

Cylindrical algebraic decomposition is a classical construction in real algebraic geometry. Although there are many algorithms to compute a cylindrical algebraic decomposition, their practical performance is still very limited. In this…

We give a short introduction to Pade approximation (rational approximation to a function with close contact at one point) and to Hermite-Pade approximation (simultaneous rational approximation to several functions with close contact at one…

Many of today's problems require techniques that involve the solution of arbitrarily large systems $A\mathbf{x}=\mathbf{b}$. A popular numerical approach is the so-called Greedy Rank-One Update Algorithm, based on a particular tensor…

Starting from the Maxwell-Juettner equilibrium distribution, we develop a relativistic lattice Boltzmann (LB) algorithm capable of handling ultrarelativistic systems with flat, but expanding, spacetimes. The algorithm is validated through…

We introduce Parametric Linear Dynamic Logic (PLDL), which extends Linear Dynamic Logic (LDL) by temporal operators equipped with parameters that bound their scope. LDL was proposed as an extension of Linear Temporal Logic (LTL) that is…

This paper extends recursive least squares (RLS) to include time-varying regularization. This extension provides flexibility for updating the least squares regularization term in real time. Existing results with constant regularization…

We consider large linear and nonlinear fixed point problems, and solution with proximal algorithms. We show that there is a close connection between two seemingly different types of methods from distinct fields: 1) Proximal iterations for…

The Sequential Linear Quadratic (SLQ) algorithm is a continuous-time variant of the well-known Differential Dynamic Programming (DDP) technique with a Gauss-Newton Hessian approximation. This family of methods has gained popularity in the…

Low-rank plus diagonal (LRPD) decompositions provide a powerful structural model for large covariance matrices, simultaneously capturing global shared factors and localized corrections that arise in covariance estimation, factor analysis,…

In [Steinerberger, Q. Appl. Math., 79:3, 419-429, 2021] and [Shao, SIAM J. Matrix Anal. Appl. 44(1), 212-239, 2023], two new types of Kaczmarz algorithms, which share some similarities, for consistent linear systems were proposed. These two…

Randomized iterative algorithms for solving a factorized linear system, $\mathbf A\mathbf B\mathbf x=\mathbf b$ with $\mathbf A\in{\mathbb{R}}^{m\times \ell}$, $\mathbf B\in{\mathbb{R}}^{\ell\times n}$, and $\mathbf b\in{\mathbb{R}}^m$,…

We propose a differential analog of the notion of integral closure of algebraic function fields. We present an algorithm for computing the integral closure of the algebra defined by a linear differential operator. Our algorithm is a direct…