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We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of learning theory. These questions concern: the outstanding…
We propose the extension of the complex numbers to be the new domain where new concepts, like negative and imaginary probabilities, can be defined. The unit of the new space is defined as the solution of the unsolvable equation in the…
The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose difference is $d=2$. Generalizing this idea, we define new similar product mappings,…
In the present paper, we propose a new axiomatic approach to nonstandard analysis and its application to the general theory of spatial structures in terms of category theory. Our framework is based on the idea of internal set theory, while…
In this paper, we construct a higher dimensional generalization of affine buildings and introduce a new structure, which we call Babel buildings. These buildings are non-connected, non-convex metric spaces of non-positive curvature. Despite…
In the presence of model risk, it is well-established to replace classical expected values by worst-case expectations over all models within a fixed radius from a given reference model. This is the "robustness" approach. We show that…
This book presents our case in defense of a constructivist epistemological framework and the use of compatible statistical theory and inference tools. The basic metaphor of decision theory is the maximization of a gambler's expected…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This…
One often sees a sharp distinction in mathematics between descriptions from the outside and from the inside. Think of defining a set in the plane through an algebraic equation, or dynamically as the closure of the orbit of some point under…
We introduce a general theory of quantitative and metric rewriting systems, namely systems with a rewriting relation enriched over quantales modelling abstract quantities. We develop theories of abstract and term-based systems, refining…
In this paper, we prove a structure theorem for the infinite union of $n$-adic doubling measures via techniques which involve far numbers. Our approach extends the results of Wu in 1998, and as a by product, we also prove a classification…
We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point. After introducing…
We summarise known algebraic and model theoretic results on the ring $\mathscr{A}$ of integers modulo infinitely large primes for number theorists, and share topics in transcendental number theory with algebraists and model theorists. In…
The constructive approach to mathematics has the advantage that witnesses can be extracted from statements of existence and theorems can be unwound to give algorithms. Even better, constructive theorems can be interpreted in any topos,…
Leibniz entertained various conceptions of infinitesimals, considering them sometimes as ideal things and other times as fictions. But in both cases, he compares infinitesimals favorably to imaginary roots. We agree with the majority of…
The structure entropy is one of the most important parameters to describe the structure property of the complex networks. Most of the existing struc- ture entropies are based on the degree or the betweenness centrality. In order to describe…
This is an overview of the basic tools of nonsmooth analysis which are grounded on nonstandard models of set theory. By way of illustration we give a criterion for an infinitesimally optimal path of a general discrete dynamic system.
We study tilings of the plane that combine strong properties of different nature: combinatorial and algorithmic. We prove existence of a tile set that accepts only quasiperiodic and non-recursive tilings. Our construction is based on the…
CZF is a system of set theory which, over classical logic, is equivalent to ZF, while over intuitionistic logic, it has a well-known constructive type-theoretic interpretation. This article introduces a simpler, intuitive family of…