Related papers: Theory and Application of Augmented Dimensional An…
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…
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…
It is time to renew old ways of thinking about dimensional analysis. Specifically, more than $n-r$ invariants and more than one functional relation between invariants need to be considered simultaneously. Thus generalized, dimensional…
In this paper, we develop a systematic theory for high dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new \emph{U}~type…
Dimensional analysis (DA) pays attention to fundamental physical dimensions such as length and mass when modelling scientific and engineering systems. It goes back at least a century to Buckingham's Pi theorem, which characterizes a…
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.
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…
Dimensional analysis is fundamental to the formulation and validation of physical laws, ensuring that equations are dimensionally homogeneous and scientifically meaningful. In this work, we use Lean 4 to formalize the mathematics of…
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…
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…
The fundamental purpose of the present research article is to introduce the basic principles of Dimensional Analysis in the context of the neoclassical economic theory, in order to apply such principles to the fundamental relations that…
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…
We explain how the representation theory associated with supersymmetry in diverse dimensions is encoded within the representation theory of supersymmetry in one time-like dimension. This is enabled by algebraic criteria, derived, exhibited,…
A new, coordinate-free (geometric) approach to multivariate statistical analysis. General multivariate linear models and linear hypotheses are defined in geometric form. A method of constructing statistical criteria is defined for linear…
The advent of modern technology, permitting the measurement of thousands of characteristics simultaneously, has given rise to floods of data characterized by many large or even huge datasets. This new paradigm presents extraordinary…
We present new findings in regard to data analysis in very high dimensional spaces. We use dimensionalities up to around one million. A particular benefit of Correspondence Analysis is its suitability for carrying out an orthonormal…
We present a new method for enhancing symbolic regression for differential equations via dimensional analysis, specifically Ipsen's and Buckingham pi methods. Since symbolic regression often suffers from high computational costs and…
High-dimensional big data appears in many research fields such as image recognition, biology and collaborative filtering. Often, the exploration of such data by classic algorithms is encountered with difficulties due to `curse of…
The growing number of dimensionality reduction methods available for data visualization has recently inspired the development of quality assessment measures, in order to evaluate the resulting low-dimensional representation independently…
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…