Related papers: Formalizing dimensional analysis using the Lean th…
In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…
In our previous article Phys. Rev. Lett. 127 (2021) 271601, we announced a novel 'democratic' Lagrangian formulation of general nonlinear electrodynamics in four dimensions that features electric and magnetic potentials on equal footing.…
The goal of this contribution is to introduce the Hamiltonian formalism of theoretical mechanics for analysing motion in generic linear and non-linear dynamical systems, including particle accelerators. This framework allows the derivation…
This is a self-contained purely algebraic treatment of desingularization of fields of fractions $\mathbf{L}:=Q(\mathbf{A})$ of $d$-dimensional domains of the form \[\mathbf{A}:=\bar{\mathbf{F}}[\underline{x}]/\langle…
We develop dimension theory for a large class of structures called espaliers, consisting of a set $L$ equipped with a partial order $\leq$, an orthogonality relation $\perp$, and an equivalence relation $\sim$, subject to certain axioms.…
We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that…
A ridge function is a function of several variables that is constant along certain directions in its domain. Using classical dimensional analysis, we show that many physical laws are ridge functions; this fact yields insight into the…
An exponential interaction is constructed so that one-dimensional atoms and chains of atoms mimic the general behavior of their three-dimensional counterparts. Relative to the more commonly used soft-Coulomb interaction, the exponential…
Mechanics studies the relationships between space, time, and matter, which can be expressed in terms of the dimensions of length $\mathcal{L}$, time $\mathcal{T}$, and mass $\mathcal{M}$. Each dimension broadens the scope of mechanics, from…
Many models require integrals of high-dimensional functions: for instance, to obtain marginal likelihoods. Such integrals may be intractable, or too expensive to compute numerically. Instead, we can use the Laplace approximation (LA). The…
One can perform equational reasoning about computational effects with a purely functional programming language thanks to monads. Even though equational reasoning for effectful programs is desirable, it is not yet mainstream. This is partly…
Numerical characteristics of identities of finite-dimensional nonassociative algebras are studied. The main result is the construction of a four-dimensional simple unitary algebra with fractional PI-exponent strictly less than its…
A variational principle is applied to 4D Euclidean space provided with a tensor refractive index, defining what can be seen as 4-dimensional optics (4DO). The geometry of such space is analysed, making no physical assumptions of any kind.…
We employ the wavelet formalism of quantum field theory to study field theories in the nonperturbative Hamiltonian framework. Specifically, we make use of Daubechies wavelets in momentum space. These basis elements are characterised by a…
We generalize the concept of a field by allowing addition to be a partial operation. We show that elements of such a "partially additive field" share many similarities with physical quantities. In particular, they form subsets of mutually…
The numerical evolution of Einstein's field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modelling black hole production in TeV…
We find the minimal number of independent preparations and measurements certifying the dimension of a classical or quantum system limited to $d$ states, optionally reduced to the real subspace. As a dimension certificate, we use the linear…
The evolution equation is used as the fundamental equation of field theory, which is described entirely by the geometry of the four-dimensional space. The evolution kernel determines the covariant action of physical fields by the proper…
We introduce three measures of complexity for families of sets. Each of the three measures, that we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family: for upper…