Related papers: The arithmetic of simplices
A completely well-centered tetrahedral mesh is a triangulation of a three dimensional domain in which every tetrahedron and every triangle contains its circumcenter in its interior. Such meshes have applications in scientific computing and…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
The sample range of uniform random points $X_1, \dots , X_n$ chosen in a given convex set is the convex hull ${\rm conv}[X_1, \dots, X_n]$. It is shown that in dimension three the expected volume of the sample range is not monotone with…
Let $\mathcal{Z}$ be a spin $4$-manifold carrying a parallel spinor and $M\hookrightarrow \mathcal{Z}$ a hypersurface. The second fundamental form of the embedding induces a flat metric connection on $TM$. Such flat connections satisfy a…
We explore geometric properties of circular analogues of tractrices and pseudospheres in $\mathbb R^3$.
The connection between the commutativity of a family of $n\times n$ matrices and the generalized joint numerical ranges is studied. For instance, it is shown that ${\cal F}$ is a family of mutually commuting normal matrices if and only if…
We determine which simple algebraic groups of type $^3D_4$ over arbitrary fields of characteristic different from 2 admit outer automorphisms of order 3, and classify these automorphisms up to conjugation. The criterion is formulated in…
In this work we take a deep dive into the cone of copositive $3 \times 3$ matrices. In doing so we visualize the cone, make geometric observations about it, and prove them. We then use these observations to parametrize the set. In the…
It is well-known how to compute the structure of the second homotopy group of a space, $X$, as a module over the fundamental group, $\pi_1X$, using the homology of the universal cover and the Hurewicz isomorphism. We describe a new method…
We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4)…
This paper is divided into two parts. The first is a review, through categorical lenses, of the classical theory of regular-singular differential systems over $C((x))$ and $\mathbb P^1_C\smallsetminus\{0,\infty\}$, where $C$ is…
Polypolyhedra are edge-transitive compounds of polyhedra. In this paper we use group theory to determine the number of distinct polypolyhedra whose symmetry group is any given finite irreducible Coxeter group. We apply this result in order…
In this paper we prove two results, one semi-historical and the other new. The semi-historical result, which goes back to Thurston and Riley, is that the geometrization theorem implies that there is an algorithm for the homeomorphism…
The three-algebras used by Bagger and Lambert in N=6 theories of ABJM type are in one-to-one correspondence with a certain type of Lie superalgebras. We show that the description of three-algebras as generalized Jordan triple systems…
Orbit-finite models of computation generalise the standard models of computation, to allow computation over infinite objects that are finite up to symmetries on atoms, denoted by $\mathbb{A}$. Set theory with atoms is used to reason about…
We characterize all algebraic subsets of the tridisk that are Caratheodory sets, that is the intrinsic Caratheodory metric on the set equals the Caratheodory metric for the tridisk. We show that such sets are either retracts, or are…
It is known that the groups of Euclidean rotations in dimension 3 (isometries of $S^2$), general Lorentz transformations in dimension 4 (Hyperbolic isometries in dimension 3), and screw motions in dimension 3 can be represented by the…
The paper surveys recent progress in understanding geometric, topological and combinatorial properties of large simplicial complexes, focusing mainly on ampleness, connectivity and universality. In the first part of the paper we concentrate…
Extending previous results on a characterization of all equilateral triangle in space having vertices with integer coordinates ("in $\mathbb Z^3$"), we look at the problem of characterizing all regular polyhedra (Platonic Solids) with the…
An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space…