Related papers: Formalizing dimensional analysis using the Lean th…
The Lagrangian formalism is used to derive covariant equations that are suitable for use in continuously distributed matter in curved spacetime. Special attention is given to theoretical representation, in which the Lagrangian and its…
Making meaning with math in physics requires blending physical conceptual knowledge with mathematical symbology. Students in introductory physics classes often struggle with this, but it is an essential component of learning how to think…
Chemical theory can be made more rigorous using the Lean theorem prover, an interactive theorem prover for complex mathematics. We formalize the Langmuir and BET theories of adsorption, making each scientific premise clear and every step of…
Roughly speaking, Buckingham's $\Pi$-Theorem provides a method to "guess" the structure of physical formulas simply by studying the dimensions (the physical units) of the involved quantities. Here we will prove a quantitative version of…
The Buckingham's $\pi$, theorem has been recently introduced in the context of Non destructive Testing \& Evaluation (NdT\&E) , giving a theoretical basis for developing simple but effective methods for multi-parameter estimation via…
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 physics community relies on index notation to effectively manipulate types of tensors. This paper introduces the first formally verified implementation of index notation in the interactive theorem prover Lean 4. By integrating index…
Given an ideal $I$ in a commutative ring $A$, a divided power structure on $I$ is a collection of maps $\{\gamma_n \colon I \to A\}_{n \in \mathbb{N}}$, subject to axioms that imply that it behaves like the family $\{x \mapsto…
Formalizing mathematical proofs using computerized verification languages like Lean 4 has the potential to significantly impact the field of mathematics, it offers prominent capabilities for advancing mathematical reasoning. However,…
Dimensional regularization of Euclidean momentum space integrals is a highly successful technique in renormalization of quantum field theories. While it yields a straightforward algorithmic method, with which to evaluate diagrams beyond…
Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a…
We report on our experience formalizing differential geometry with mathlib, the Lean mathematical library. Our account is geared towards geometers with no knowledge of type theory, but eager to learn more about the formalization of…
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…
When faced with a mathematical model, often the first step is to reduce the complexity of the model by turning variables and parameters into dimensionless quantities. This process is often performed by hand, relying on a skill practiced…
The numerical dimension is a numerical measure of the positivity of a pseudo-effective divisor $L$. There are several proposed definitions of the numerical dimension due to Nakayama (2004) and Boucksom et al. (2004). We prove the equality…
Traditional approaches for validating molecular simulations rely on making software open source and transparent, incorporating unit testing, and generally employing human oversight. We propose an approach that eliminates software errors…
Roughly speaking, functional analysis is the study of vector spaces of arbitrary dimension over the field of real or complex numbers, and the continuous linear mappings between such spaces. Naturally, the notion of continuity requires a…
We perform Dirac's canonical analysis for a four-dimensional $BF$ and for a generalized four-dimensional $BF$ theory depending on a connection valued in the Lie algebra of SO(3,1). This analysis is developed by considering the corresponding…
The traditional Pi-theorem tells us that for any dimensionally invariant relation there exists a full set of independent dimensionless "Pi groups" which can be used to nondimensionalise the relation. In this paper, we seek to understand…
Dimensional analysis, and in particular the Buckingham $\Pi$ theorem is widely used in fluid mechanics. In this article we obtain an expression for the impact parameter from Buckingham's theorem and we compare our result with Rutherford's…