Related papers: Beyond polarities: Generalizing Maxwell's reciproc…
A natural way to represent on the plane both a planar graph and its dual is to follow the definition of the dual, thus, to place vertices inside their corresponding primal faces, and to draw the dual edges so that they only cross their…
The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by…
The existence of an interpolating master action does not guarantee the same spectrum for the interpolated dual theories. In the specific case of a generalized self-dual (GSD) model defined as the addition of the Maxwell term to the…
We study parallel surfaces and dual surfaces of cuspidal edges. We give concrete forms of principal curvature and principal direction for cuspidal edges. Moreover, we define ridge points for cuspidal edges by using those. We clarify…
We investigate some connections between two different ways of defining Poincar\'e Duality, and relate them geometrically to the level curve mapping.
We expand on the recent derivation of 3d dualities using bosonization. We present in some detail a general class of Abelian duals.
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
We recall and define various kinds of supersingular $\ell$-isogeny graphs and precise graph isomorphism with a corresponding quaternion $\ell$-ideal graph. In particular, we introduce the notion of double-orientations on supersingular…
We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers.…
The two postulates of special relativity plus the postulates of conserved charges, both electric and magnetic, and a resulting linear system are sufficient for the derivation of the generalized vacuum Maxwell equations with both charges.…
Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm…
We give a combinatorial characterization of upward planar graphs in terms of upward planar orders, which are special linear extensions of edge posets.
A conjecture regarding the structure of expander graphs is discussed.
This is mostly* a non-technical exposition of the joint work arXiv:1212.0373 with Caporaso and Payne. Topics include: Moduli of Riemann surfaces / algebraic curves; Deligne-Mumford compactification; Dual graphs and the combinatorics of the…
We provide a new perspective on the divisor theory of graphs, using additive combinatorics. As a test case for this perspective, we compute the gonality of certain families of outerplanar graphs, specifically the strip graphs. The Jacobians…
The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may…
We formalise the teleparallel version of extended geometry (including gravity) by the introduction of a complex, the differential of which provides the linearised dynamics. The main point is the natural replacement of the two-derivative…
Structures of commuting semigroups of isometries under certain additional assumptions like double commutativity or dual double commutativity are found.
Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected…
The underlying algebra for a noncommutative geometry is taken to be a matrix algebra, and the set of derivatives the adjoint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of…