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In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and…
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for…
Toy models are highly idealized and deliberately simplified models that retain only the essential features of a system in order to explore specific theoretical questions. Long used in physics and other sciences, they have recently begun to…
We review generalized additive models as a type of ``transparent'' model that has recently seen renewed interest in the deep learning community as neural additive models. We highlight multiple types of nonidentifiability in this model class…
The number of research papers written has been growing at least linearly -- if not exponentially -- in recent years. In proportion, the amount of time a reader allocates per paper has been decreasing. While an accessible paper will be…
In this paper a didactic approach is described which immediately leads to an understanding of those postulates of quantum mechanics used most frequently in quantum computation. Moreover, an interpretation of quantum mechanics is presented…
This book provides an introduction to the mathematical analysis of deep learning. It covers fundamental results in approximation theory, optimization theory, and statistical learning theory, which are the three main pillars of deep neural…
These lecture notes give a pedagogical and (mostly) self-contained review of some basic aspects of the Matrix model of M-theory. The derivations of the model as a regularized supermembrane theory and as the discrete light-cone quantization…
This book provides a compact, derivation-oriented introduction to the mathematical foundations of modern generative artificial intelligence. Rather than surveying every recent architecture or implementation detail, it develops a coherent…
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced…
A knowledge system S describing a part of real world does in general not contain complete information. Reasoning with incomplete information is prone to errors since any belief derived from S may be false in the present state of the world.…
Numerous models for grounded language understanding have been recently proposed, including (i) generic models that can be easily adapted to any given task and (ii) intuitively appealing modular models that require background knowledge to be…
Galois theory is developed using elementary polynomial and group algebra. The method follows closely the original prescription of Galois, and has the benefit of making the theory accessible to a wide audience. The theory is illustrated by a…
Macroeconomics essentially discusses macroeconomic phenomena from the perspectives of various schools of economic thought, each of which takes different views on how macroeconomic agents make decisions and how the corresponding markets…
This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications…
This chapter provides a overview of Bayesian inference, mostly emphasising that it is a universal method for summarising uncertainty and making estimates and predictions using probability statements conditional on observed data and an…
There exists a theory of a single general-purpose learning algorithm which could explain the principles its operation. It assumes the initial rough architecture, a small library of simple innate circuits which are prewired at birth. and…
The main goal of this paper is to give a pedagogical introduction to Quantum Information Theory-to do this in a new way, using network diagrams called Quantum Bayesian Nets. A lesser goal of the paper is to propose a few new ideas, such as…
A multi-decade exploration into the theoretical foundations of artificial and natural general intelligence, which has been expressed in a series of books and papers and used to guide a series of practical and research-prototype software…
An informal guide to the history of Heisenberg's matrix mechanics. It is designed for mathematicians with only a minimal background in either physics or geometry, and it is based upon Heisenberg's original arguments.