Related papers: Transseries for beginners
The differential field of transseries extends the field of real Laurent series, and occurs in various context: asymptotic expansions, analytic vector fields, o-minimal structures, to name a few. We give an overview of the algebraic and…
We show that \'Ecalle's transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called…
Germs of real-valued functions, surreal numbers, and transseries are three ways to enrich the real continuum by infinitesimal and infinite quantities. Each of these comes with naturally interacting notions of ordering and derivative. The…
The present article surveys surreal numbers with an informal approach, from their very first definition to their structure of universal real closed analytic and exponential field. Then we proceed to give an overview of the recent…
We study subfields of surreal numbers, called hyperseries fields, that are suited to be equipped with derivations and composition laws. We show how to define embeddings on hyperseries fields that commute with transfinite sums and all…
More remarks and questions on transseries. In particular we deal with the system of ratio sets and grids used in the grid-based formulation of transseries. This involves a "witness" concept that keeps track of the ratios required for each…
Additional remarks and questions for transseries. In particular: properties of composition for transseries; the recursive nature of the construction of R[[[ x ]]]; modes of convergence for transseries. There are, at this stage, questions…
We define the field $\mathbb{L}$ of logarithmic hyperseries, construct on $\mathbb{L}$ natural operations of differentiation, integration, and composition, establish the basic properties of these operations, and characterize these…
Formal Laurent-Puiseux series are important in many branches of mathematics. This paper presents a {\it Mathematica} implementation of algorithms developed by the author for converting between certain classes of functions and their…
We study the existence of formal Taylor expansions for functions defined on fields of generalised series. We prove a general result for the existence and convergence of those expansions for fields equipped with a derivation and an…
Transformer architecture has widespread applications, particularly in Natural Language Processing and computer vision. Recently Transformers have been employed in various aspects of time-series analysis. This tutorial provides an overview…
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this…
The transformer is a neural network component that can be used to learn useful representations of sequences or sets of data-points. The transformer has driven recent advances in natural language processing, computer vision, and…
Surreal numbers form the ultimate extension of the field of real numbers with infinitely large and small quantities and in particular with all ordinal numbers. Hyperseries can be regarded as the ultimate formal device for representing…
We develop here the algebra of the differential field of transseries and of related valued differential fields. This book contains in particular our recently obtained decisive positive results on the model theory of these structures.
Transformers have achieved superior performances in many tasks in natural language processing and computer vision, which also triggered great interest in the time series community. Among multiple advantages of Transformers, the ability to…
Transcendental functions, such as exponentials and logarithms, appear in a broad array of computational domains: from simulations in curvilinear coordinates, to interpolation, to machine learning. Unfortunately they are typically expensive…
We have discovered three non-power infinite series representations for Bessel functions of the first kind of integer orders and real arguments. These series contain only elementary functions and are remarkably simple. Each series was…
We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper.
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…