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Dan Jonsson

Publications and source records attributed to Dan Jonsson.

14 recordsLinked to original sources
arXiv2026

Periods of N-body Systems Determined Through Dimensional Analysis

A generalization of classical dimensional analysis, presented in a separate article, makes it possible to derive Kepler's third law for the period of a two-body system, up to a multiplicative constant, without solving the equations of motion. Here we show how to derive generalizations of Kepler's third law to n-body systems by the same technique. Our results agree with conjectures by Sun on the period of a classical n-body system and by Semay and Sun on the quantum-theoretical counterpart of the period of a classical n-body system.

math-ph↗
arXiv2025

Differential Geometry on Pointwise Affine Spaces

We introduce an alternative formalization of curved spaces in which the concept of a pointwise affine space, as defined here, replaces that of a manifold. New or modified definitions of familiar notions from differential geometry such as connections, torsion, Riemann curvature, vector fields and differentiation of vector fields are presented, and examples of applications of the resulting framework are given. In general, the notions and results considered receive simpler and more transparent forms than in conventional approaches.

math.DG↗
arXiv2022

Theory and Application of Augmented Dimensional Analysis

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, grounded in a purely algebraic theory of quantity spaces, allows the classical \pi theorem to be restated in an explicit and precise form and its prerequisites to be clarified and relaxed. Augmented dimensional analysis, in contrast to classical dimensional analysis, is guaranteed to take into account all relations among the quantities involved. Several examples are given to show that the information thus gained, together with symmetry assumptions, can lead to new or stronger results. We also explore the connection between dimensional analysis and matroid theory, elucidating combinatorial aspects of dimensional analysis. It is emphasized that dimensional analysis rests on a principle of covariance.

math-ph↗
arXiv2021

Magnitudes, Scalable Monoids and Quantity Spaces

In ancient Greek mathematics, magnitudes such as lengths were strictly distinguished from numbers. In modern quantity calculus, a distinction is made between quantities and scalars that serve as measures of quantities. It can be argued that quantities should play a more prominent, independent role in modern mathematics, as magnitudes earlier. The introduction includes a sketch of the development and structure of the pre-modern theory of magnitudes and numbers. Then, a scalable monoid over a ring is defined and its basic properties are described. Congruence relations on scalable monoids, direct and tensor products of scalable monoids, subalgebras and homomorphic images of scalable monoids, and unit elements of scalable monoids are also defined and analyzed. A quantity space is defined as a commutative scalable monoid over a field, admitting a finite basis similar to a basis for a free abelian group. The mathematical theory of quantity spaces forms the basis of a rigorous quantity calculus and is developed with a view to applications in metrology and foundations of physics.

math.RA↗
arXiv2020

An Algebraic Foundation of Amended Dimensional Analysis

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 of quantity spaces. Examples of dimensional analysis based on this approach are given, showing that it allows results obtained by traditional dimensional analysis to be strengthened. For example, the orbital period of a two-body system can be derived without use of equations of motion.

math-ph↗
arXiv2020

Magnitudes Reborn: Quantity Spaces as Scalable Monoids

This article includes a survey of the historical development and theoretical structure of the pre-modern theory of magnitudes and numbers. In Part 1, work, insights and controversies related to quantity calculus from Euler onward are reviewed. In Parts 2 and 3, we define scalable monoids and, as a special case, quantity spaces; both can be regarded as universal algebras. Scalable monoids are related to rings and modules, and quantity spaces are to scalable monoids as vector spaces are to modules. Subalgebras and homomorphic images of scalable monoids can be formed, and tensor products of scalable monoids can be constructed as well. We also define and investigate congruence relations on scalable monoids, unit elements of scalable monoids, basis-like substructures of scalable monoids and quantity spaces, and scalar representations of elements of quantity spaces. The mathematical theory of quantity spaces is presented with a view to metrological applications. This article supersedes arXiv:1503.00564 and complements arXiv:1408.5024.

math.RA↗
arXiv2019

On Scalable Monoids

This brief exposition presents some basic properties of scalable monoids and quantity spaces, introduced in arXiv:1408.5024

math.RA↗
arXiv2019

On Group-Like Magmoids

A magmoid is a non-empty set with a partial binary operation; group-like magmoids generalize group-like magmas such as semigroups, monoids and groups. In this article, we first consider the many ways in which the notions of associative multiplication, identities and inverses can be generalized when the total binary operation is replaced by a partial binary operation. Poloids, groupoids, skew-poloids, skew-groupoids, prepoloids, pregroupoids, skew-prepoloids and skew-pregroupoids are then defined in terms of generalized associativity, generalized identities and generalized inverses. Some basic results about these magmoids are derived, and connections between poloid-like and prepoloid-like magmoids, in particular semigroups, are described. Notably, analogues of the Ehresmann-Schein-Nampooribad theorem are proved.

math.GR↗
arXiv2015

Quantities, Dimensions and Dimensional Analysis

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 are given.

math.HO↗
arXiv2014

Dimensional Analysis: A Centenary Update

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 analysis can yield more information than previously recognized.

math.HO↗
arXiv2014

Interpretations and Representations of Classical Tensors

Classical tensors, the familiar mathematical objects denoted by symbols such as $t_{i}$, $t^{ij}$ and $t_{k}^{ij}$, are usually interpreted either as 'coordinatizable objects' with coordinates changing in a specific way under a change of coordinate system or as elements of tensor spaces of the form $V^{\otimes n}\otimes\left(V^{*}\right)^{\otimes m}$. An alternative interpretation of classical tensors as linear tensor maps of the form $V^{\otimes m}\rightarrow V^{\otimes n}$ is presented here. In this interpretation, tensor multiplication is seen as generalized function composition. Representations of classical tensors by means of arrays are also considered.

math.HO↗