Related papers: Computations with p-adic numbers
These lectures about lattice field theory were written for, and given at, TASI 2019, ``The many dimensions of quantum field theory.'' The students at this TASI were mostly interested in formal things, and so these are slightly unusual…
These are expanded and revised notes for a minicourse entitled "Affine W-algebras", which took place as part of the thematic month "Quantum Symmetries" at the Centre de Recherches Mathematiques in Montreal, Canada in October 2022. The first…
We discuss the relation between p-adic numbers and kernels in view of a recent large deviation theory for mean-field spin glasses. As an application we show several fundamental properties of numerical bases in kernel language. In…
Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…
In recent years, as fractional calculus becomes more and more broadly used in research across different academic disciplines, there are increasing demands for the numerical tools for the computation of fractional…
Some question about representations of $p$-adic groups are discussed.
$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…
This chapter provides a hands-on tutorial on the important technique known as self-reducibility. Through a series of "Challenge Problems" that are theorems that the reader will---after being given definitions and tools---try to prove, the…
We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…
I'll outline the latest version of my limits of math course. The purpose of this course is to illustrate the proofs of the key information-theoretic incompleteness theorems of algorithmic information theory by means of algorithms written in…
In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be…
Lagrange coded computation (LCC) is essential to solving problems about matrix polynomials in a coded distributed fashion; nevertheless, it can only solve the problems that are representable as matrix polynomials. In this paper, we propose…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
A survey of real differential geometry and loop theory is given in order to introduce the construction of an analytic loop associated to p-adic differential manifold.
This paper works out the structure of singular points of p-adic differential equations (i.e. differential modules over the ring of functions analytic in some annulus with external radius 1). Surprisingly results look like in the formal case…
This document is a pedagogical introduction to statistics for particle physics. Emphasis is placed on the terminology, concepts, and methods being used at the Large Hadron Collider. The document addresses both the statistical tests applied…
Recently, several striking advances have taken place regarding the discrete logarithm problem (DLP) in finite fields of small characteristic, despite progress having remained essentially static for nearly thirty years, with the best known…
Summation of a large class of the functional series, which terms contain factorials, is considered. We first investigated finite partial sums for integer arguments. These sums have the same values in real and all p-adic cases. The…
A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.
On example of tasks of class NP the questions concerning accuracy of work of already existing and possible in the future algorithms for the solution of tasks on discrete structures are considered.