Related papers: Formalizing dimensional analysis using the Lean th…
In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables and parameters, the Buckingham Pi theorem provides a procedure…
We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem,…
The languages of mathematical physics and modelling are endowed with a rich ``grammar of dimensions'' that common abstractions of programming languages fail to represent. We propose a dependently typed domain-specific language (embedded in…
This paper introduces dimensional analysis in Non-Destructive Testing & Evaluation (NDT&E) problems. This is the first time that this approach is adopted in the framework of NDT&E, and the paper opens to the development of probes and…
A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…
We present a formalization, in the theorem prover Lean, of the classification of solvable Lie algebras of dimension at most three over arbitrary fields. Lie algebras are algebraic objects which encode infinitesimal symmetries, and as such…
We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
Classical dimensional analysis is one of the cornerstones of qualitative physics and is also used in the analysis of engineering systems, for example in engineering design. The basic power product relationship in dimensional analysis is…
Dimensional analysis provides many simple and useful tools for various situations in science. The objective of this paper is to investigate its relations to functions, i.e., the dimensions for functions that yield physical quantities and…
Lurking variables represent hidden information, and preclude a full understanding of phenomena of interest. Detection is usually based on serendipity -- visual detection of unexplained, systematic variation. However, these approaches are…
Physical units are fundamental to scientific computing. However, many finite element frameworks lack built-in support for dimensional analysis. In this work, we present a systematic framework for integrating physical units into the Unified…
A group of dimensionless numbers, termed DLV (Density-Length-Velocity) system, is put forward to represent the scaled behavior of structures under impact loads. It is obtained by means of the Buckingham Pi theorem with an alternative basis.…
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…
This study addresses the often underestimated importance of physical dimensions and units in the formal reconstruction of physical theories, focusing on structuralist approaches that use the concept of ``species of structure" as a…
This paper introduces and formally verifies a novel geometric framework for first-order stochastic dominance (FSD) in $N$ dimensions using the Lean 4 theorem prover. Traditional analytical approaches to multi-dimensional stochastic…
On the verge of the centenary of dimensional analysis (DA), we present a generalisation of the theory and a methodology for the discovery of empirical laws from observational data. It is well known that DA: a) reduces the number of free…
We present an innovative approach to dimensional analysis, based on a general representation theorem for complete quantity functions admitting a covariant scalar representation; this theorem is in turn grounded in a purely algebraic theory…
Dimensional analysis is one of the most fundamental tools for understanding physical systems. However, the construction of dimensionless variables, as guided by the Buckingham-$\pi$ theorem, is not uniquely determined. Here, we introduce…
This paper explores formalizing Geometric (or Clifford) algebras into the Lean 3 theorem prover, building upon the substantial body of work that is the Lean mathematics library, mathlib. As we use Lean source code to demonstrate many of our…
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…