Related papers: Spherical Geometry and the Least Symmetric Triangl…
This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are…
We study the problem of computing the minimum area triangle that circumscribes a given $n$-sided convex polygon touching edge-to-edge. In other words, we compute the minimum area triangle that is the intersection of 3 half-planes out of $n$…
A common criterion in the design of finite Hilbert space frames is minimal coherence, as this leads to error reduction in various signal processing applications. Frames that achieve minimal coherence relative to all unit-norm frames are…
We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic…
We propose a geometric formulation of effective field theories via nonlinear supersymmetry. Non-supersymmetric particles are embedded in constrained superfields governed by a nonlinear sigma model, and operators are collected into…
In this paper we develop new Newton and conjugate gradient algorithms on the Grassmann and Stiefel manifolds. These manifolds represent the constraints that arise in such areas as the symmetric eigenvalue problem, nonlinear eigenvalue…
A general geometrical scheme is presented for the construction of novel classical gravity theories whose solutions obey two-sided bounds on the sectional curvatures along certain subvarieties of the Grassmannian of two-planes. The…
The discrete isoperimetric inequality states that among all n -gons with a fixed area, the regular n -gon has the least perimeter. We prove analogues of the discrete isoperimetric inequality (involving circumradius or inradius) for cyclic…
This is an innovative treatise on triangles, resting upon 1) 3-body problem techniques including mass-weighted relative Jacobi coordinates. 2) Part I's detailed layer-by-layer topological and geometrical study of Kendall-type shape spaces -…
Following a strictly geometric approach we construct globally supersymmetric scalar field theories on the supersphere, defined as the quotient space $S^{2|2} = UOSp(1|2)/\mathcal{U}(1)$. We analyze the superspace geometry of the…
Geometrical analysis of a new type of Unified Field Theoretical models follow the guidelines of previous works of the authors is presented. These new unified theoretical models are characterized by an underlying hypercomplex structure, zero…
The study of extremal problems on triangle areas was initiated in a series of papers by Erd\H{o}s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that…
We introduce a general framework for training flow matching models on Riemannian symmetric spaces, a large class of manifolds that includes the sphere, hyperbolic space and Grassmannians. We exploit their algebraic structure to reformulate…
In this note, we use a result of Osserman and Schiffer \cite{OS} to give a variational characterization of the catenoid. Namely, we show that subsets of the catenoid minimize area within a geometrically natural class of minimal annuli. To…
We first review some topics in the classical computational geometry of lines, in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of the set of lines in R^3 interacting with $n$ objects of fixed description…
We survey the classification of the Riemannian metrics on spheres with respect to which all equators are minimal hypersurfaces, and discuss problems related to these geometries.
Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their…
Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach…
We exhibit the duality between best Lipschitz (infinity harmonic) maps and least gradient maps in the case of maps from surfaces to the circle. We show that given a homotopy class of a map from a surface to the circle the infinity harmonic…
A geometric approach to understanding recursion relations for scattering amplitudes is developed. We achieve this by studying intersection numbers of triangulated accordiohedra presented as hyperplane arrangements. The cancellation of…