Abstract Sides and medians are both Jacobi coordinate magnitudes. These furthermore enter equably into the spherical coordinates on Kendall’s shape sphere and the Hopf coordinates. This motivates treating medians on the same footing as sides in the geometry of triangles and the ensuing Shape Theory. In this article, we consequently reformulate inequalities for the medians in terms of shape quantities, and proceed to find inequalities bounding the mass-weighted Jacobi coordinates. This work moreover identifies the 4/3 – powers of which occur frequently in the theory of medians – as the ratio of Jacobi masses.
One of the Hopf coordinates is tetra-area. Another is anisoscelesness, which parametrizes whether triangles are left-or-right leaning as bounded by isoscelesness itself. The third is ellipticity, which parametrizes tallness-or-flatness of triangles as bounded by regular triangles. Whereas tetra-area is clearly cluster-choice invariant, Jacobi coordinates, anisoscelesness and ellipticity are cluster-choice dependent. And yet they can be ‘democratized’ by averaging over all clusters. Democratized ellipticity moreover trivializes, due to ellipticity being the difference of base-side and median second moments, whose averages are equal to each others. Thus we introduce a distinct linear ellipticity’ quantifier of tallness-or-flatness of triangles whose democratization is nontrivial, and find inequalities bounding this.
Some of this article’s inequalities are shape-independent bounds. Others’ bounds depend on the isoperimetric ratio and arithmetic-to-geometric mean ratio of the sides: shape variables.
Abstract We systematically consider simple relational variables — relative variables, ratio variables and dilatational variables — for Graph Theory. We apply these to simplifying graph inequalities and elucidating a large number Graph-Theoretically significant probability-valued variables. This material has further use in developing network stucture quantifiers. It represents interaction between Similarity Geometry, and basic Shape Theory in the sense of Kendall, with Graph Theory.
Abstract Dirac based his theory of constrained systems on Linear Algebra foundations. It is a brackets-algebraic consistency procedure with multiple outcomes, including new constraints dropping out and redeclaring brackets becoming necessary (Dirac brackets). This procedure has not yet been edited, however, to caution about and remove scaffolding structures that turn out to not in general be brackets-algebraically consistent. We perform this task here. Our main innovation is moreover demonstration that substantial progress can be made from placing Dirac’s Algorithm on Linear Algebra and Order Theory foundations. For chains, lattices, posets, and digraphs abound therein For instance, in the simpler version of the algorithm, its iterations form a chain, of which its Dirac brackets updating steps are a subchain. Its consistent algebraic structures, meaningful notions of weak equality, of appended Hamiltonians and of observables form bounded lattices. Many key notions – such as Dirac’s extended Hamiltonian or Dirac observables – are identified as extrema of these lattices, cementing their permanence. Others are however revealed to be but simplest examples of middles. By this, e.g. Kuchar observables are in general to be replaced by a poset of algebraicially-consistent middling A-observables.
In the harder – path dependent – (previously called bifurcating or branching) version of the algorithm, moreover, iterations and Dirac brackets types become digraph-valued. What previously was a lattice of consistent constraint subalgebraic structures now becomes a competing lattice, described overall by a semi-lattice, with weak equalities, appended Hamiltonians and observables following suit. Order Theory conceptualization thus remains both lucid and under control within this harder case. Such Order-Theoretic considerations furthermore transcend to extended variants and to Temporal Relationalism implementing variants. And to the Generalized Lie Algorithm: a vast generalization of scope in which to apply Dirac’s insights from constrained dynamical systems to wherever Lie Theory is applicable.
Article dedicated to the memory of Niall o’Murchadha (see the Acknowledgments section if visiting here in this regard).
Based on material presented at the 2021 Summer school on Combinatorial and Topological Applications to Fundamental Physics.
Dirac algebroid formed by General Relativity’s constraints
Seminar 1) Interplay between foliations and algebroids Seminar 2) Combinatorial improvements in understanding of Dirac-type Algorithms for constrained systems. Seminar 3) Using differential geometric flow methods to better understand the subsequent Problem of Observables Seminar 4) how deriving spacetime from space rests on cohomology
This Summer School is on each Saturday for the next four Saturdays (starting August 21st 2021).
Contact dr.e.anderson.maths.physics *at* protonmail.com to join , subject to approval and to abiding by the Summer School’s rules.
ABSTRACT: We consider global issues with Relationalism: a first piece of Background Independence that resolves various Problem of Time facets and is locally implemented by Lie derivatives. Spacetime Relationalism and Configurational Relationalism are quite similar, though the second of these requires supplementing with a heterogeneous Temporal Relationalism. Avoiding both zeroes and Killing vectors is involved, as are some fibre bundle effects, including bibundles, monopoles on configuration space and the Gribov effect. Most of Relationalism’s globality follows from quotient configuration spaces being stratified manifolds. Nine distinct strategies for dealing with stratified manifolds are compared. Compactness and metric-space guarantees for nice (in particular Hausdorff) stratified manifolds are provided. Fibre bundles do not suffice for stratified manifolds, so general bundles, differential spaces, presheaves and sheaves are brought in instead. Hausdorff paracompact (HP) spaces continue to afford simplifications, with some support when local compactness and second-coutability apply. Relational Particle Mechanics, Gauge Theory and GR examples of these various global issues are provided.
[79 pages, including 25 figures. Second of a series of six Fundamental Physics and Applied Topology Articles.]
ABSTRACT: The Problem of Time is due to conceptual gaps between General Relativity and the other observationally-confirmed theories of Physics; it is a major foundational issue in Quantum Gravity. The Problem of Time’s multiple facets were mostly pointed out over 50 years ago by Wheeler, DeWitt and Dirac. These facets were subsequently classified by Kuchar and Isham. They argued that the lion’s share of the problem consists of interferences between facets. They also posed the question of in which order the facets should be approached. By further considering the nature of each facet at the local classical level, the Author showed the facets to be two copies of Lie Theory — spacetime and canonical — with a Wheelerian 2-way route therebetween. This solves the facet ordering question. The resulting mathematical framework turns out moreover to be consistent enough to smooth out all local classical facet interferences as well.
It would furthermore be preferable if all of the Background Independence aspects, resultant Problem of Time facets, and strategies to resolve these, were treated in a globally well-defined manner. The current article begins to address this by classifying what is meant by `global’. Be this at the level of mathematical structure (Topology, Differential Geometry, Lie Theory, PDEs, Functional Analysis). At the level of which spaces the modelling actually requires (space, spacetime, configuration space, phase space, space of spacetimes…). Or at the level of each aspect of time, space or Background Independence more generally. We also include globalization strategies and justification of A Local Resolution of the Problem of Time being possible in the first place. This is based on Hausdorff paracompact spaces, which for now support a Shrinking Lemma.
[50 pages including 14 figures. First Article in a series of six on fundamental Physics and Applied Topology.]
The Problem of Time is due to conceptual gaps between General Relativity and the other observationally-confirmed theories of Physics. It is a fundamental issue in Quantum Gravity. A key point in resolving the Problem of Time turns out to be that Algebra rapidly takes centre stage. The first algebraic aspect encountered is that constraints must close, as must spacetime generators. This Closure aspect is assessed by the generalized Lie Algorithm. Dirac’s Algorithm is the constrained canonical perspective’s subcase of this. Such algorithms have the capacity to shut down trial sets of generators for being inconsistent. Thus they constitute a type of selection principle. Those sets of generators which survive form Lie algebras or Lie algebroids. Examples include the Lie algebra of spacetime diffeomorphisms and the Dirac algebroid of constraints in GR. Around 3/4 of Problem of Time aspects revolve around brackets algebraic structures. Observables and Constructability join Closure in this regard, while Relationalism is distinct. Moreover, Closure’s centrality has been under-represented in the literature to date. The current series justifies this centrality both algebraically and graph-theoretically. We furthermore proceed to compensate for previous literature’s under-representation of this point.