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.