Related papers: Types, equations, dimensions and the Pi theorem
Formal definitions of quantities, quantity spaces, dimensions and dimension groups are introduced. Based on these concepts, a theoretical framework and a practical algorithm for dimensional analysis are developed, and examples of…
Programming requires much more than just writing code in a programming language. It is usually done in the context of a stateful environment, by interacting with a system through a graphical user interface. Yet, this wide space of…
In scientific and engineering applications, physical quantities embodied as units of measurement (UoM) are frequently used. The loss of the Mars climate orbiter, attributed to a confusion between the metric and imperial unit systems,…
A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the…
Embedding spaces contain interpretable dimensions indicating gender, formality in style, or even object properties. This has been observed multiple times. Such interpretable dimensions are becoming valuable tools in different areas of…
Mathematics is a highly specialized domain with its own unique set of challenges that has seen limited study in natural language processing. However, mathematics is used in a wide variety of fields and multidisciplinary research in many…
This thesis embarks on a comprehensive exploration of formal computational models that underlie typed programming languages. We focus on programming calculi, both functional (sequential) and concurrent, as they provide a compelling rigorous…
Advances in computational power and hardware efficiency have enabled tackling increasingly complex, high-dimensional problems. While artificial intelligence (AI) achieves remarkable results, the interpretability of high-dimensional…
This note states and proves a representation theorem for regular quantity functions, based on the theory of quantity spaces, thereby giving a new perspective on dimensional analysis and the classical $\pi$ theorem.
Theorem: Let X and Y be two Banach spaces, Phi: X to Y a continuous, linear, surjective operator, and Psi: X to Y a completely continuous operator with bounded range. Then, one has dim{x in X : Phi(x)=Psi(x)} >= dim(Phi^{-1}(0)). Here dim…
The theory of physical dimensions and units in physics is outlined. This includes a discussion of the universal applicability and superiority of quantity equations. The International System of Units (SI) is one example thereof. By analyzing…
Dependently typed programming languages have become increasingly relevant in recent years. They have been adopted in industrial strength programming languages and have been extremely successful as the basis for theorem provers. There are…
Physical quantities and physical dimensions are among the first concepts encountered by students in their undergraduate career. In this pedagogical review, I will start from these concepts and, using the powerful tool of dimensional…
Building on recent work in statistical science, the paper presents a theory for modelling natural phenomena that unifies physical and statistical paradigms based on the underlying principle that a model must be nondimensionalizable. After…
The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
Types are an important part of any modern programming language, but we often forget that the concept of type we understand nowadays is not the same it was perceived in the sixties. Moreover, we conflate the concept of "type" in programming…
Many formal languages of contemporary mathematical music theory -- particularly those employing category theory -- are powerful but cumbersome: ideas that are conceptually simple frequently require expression through elaborate categorical…
Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust…
Dependent types provide a lightweight and modular means to integrate programming and formal program verification. In particular, the types of programs written in dependently typed programming languages (Agda, Idris, F*, etc.) can be used to…