On null geodesically complete spacetimes under NEC and NGC: Is the Gao-Wald “time dilation” a topological effect?

We review a theorem of Gao-Wald on a kind of a gravitational “time delay” effect in null geodesically complete spacetimes under NEC and NGC, and we observe that it is not valid anymore throughout its statement, as well as a conclusion that there is a class of cosmological models where particle horizons are absent, if one substituted the manifold topology with a finer (spacetime) topology. Since topologies of the Zeeman-Göbel class incorporate the causal, differential, and conformal structure of a spacetime and there are serious mathematical arguments in favour of such topologies and against the manifold topology, there is a strong evidence that “time dilation” theorems of this kind are topological in nature rather than having a particular physical meaning.     

Equifacetted 4-polytopes

Many polyhedra (convex 3-polytopes) are known which occur as facets (3-faces) of convex 4-polytopes. The purpose of this paper is to determine the combinatorial types of infinitely many more polyhedra with this property. This is achieved by determining the facets of polar bicyclic polytopes.     

Vertex-faithful regular polyhedra

We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a “vertex-faithful” polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.     

Lattice triangulations of $\mathbb{E}^3 $ and of the 3-torus

This paper gives answers to a few questions concerning tilings of Euclidean spaces where the tiles are topological simplices with curvilinear edges. We investigate lattice triangulations of Euclidean 3-space in the sense that the vertices form a lattice of rank 3 and such that the triangulation is invariant under all translations of that lattice. This is the dual concept of a primitive lattice tiling where the tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation by Euclidean tetrahedra (and with straight edges) but there are infinitely many non-standard lattice triangulations where the tetrahedra necessarily have certain curvilinear edges. From the view-point of Discrete Differential Geometry this tells us that there are such triangulations of 3-space which do not carry any flat discrete metric which is equivariant under the lattice. Furthermore, we investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number n ≥ 15 of vertices. The unique one with 15 vertices is neighborly, i.e., any two vertices are joined by an edge. It turns out that for any odd n ≥ 17 there is an n-vertex neighborly triangulation of the 3-torus as a quotient of a certain non-standard lattice triangulation. Combinatorially, one can obtain these neighborly 3-tori as slight modifications of the boundary complexes of the cyclic 4-polytopes. As a kind of combinatorial surgery, this is an interesting construction by itself.     

Reprint of: Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

Soft Null Hypotheses: A Case Study of Image Enhancement Detection in Brain Lesions

This work is motivated by a study of a population of multiple sclerosis (MS) patients using dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) to identify active brain lesions. At each visit, a contrast agent is administered intravenously to a subject and a series of images are acquired to reveal the location and activity of MS lesions within the brain. Our goal is to identify the enhancing lesion locations at the subject level and lesion enhancement patterns at the population level. We analyze a total of 20 subjects scanned at 63 visits (～30Gb), the largest population of such clinical brain images. After addressing the computational challenges, we propose possible solutions to the difficult problem of transforming a qualitative scientific null hypothesis, such as “this voxel does not enhance,” to a well-defined and numerically testable null hypothesis based on the existing data. We call such procedure “soft null” hypothesis testing as opposed to the standard “hard null” hypothesis testing. This problem is fundamentally different from: (1) finding testing statistics when a quantitative null hypothesis is given; (2) clustering using a mixture distribution; or (3) setting a reasonable threshold with a parametric null assumption. Supplementary materials are available online.     

Reguläre Inzidenzkomplexe I

The concept of regular incidence-complexes generalizes the notion of regular polyhedra in a combinatorial sense. A regular incidence-complex is a partially ordered set with regularity defined by certain transitivity properties of its automorphism group. The concept includes all regular d-polytopes and all regular complex d-polytopes as well as many geometries and well-known configurations.     

The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p = c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real‐world graphs. Our model includes as special cases many models previously studied. We show that, under one very weak assumption (that the expected number of edges is “what it should be”), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is “stable”: for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

Self-Dual Regular 4-Polytopes and Their Petrie-Coxeter-Polyhedra

Coxeter’s regular skew polyhedra in euclidean 4-space IE⁴ are intimately related to the self-dual regular 4-polytopes. The same holds for two of the three Petrie-Coxeter-polyhedra and the (self-dual) cubical tessellation in IE³. In this paper we discuss the combinatorial Petrie-Coxeter-polyhedra associated with the self-dual regular 4-incidence-polytopes.     

On the Description of the Content-Equality by a Relation

In the geometry of polyhedra we understand by an elementary content-functional a real valued, non-negative, finite additive measure on the set of polyhedra which is invariant under isometries. There are close relations between the content-measurement and the relation of equidecomposability. Two polyhedra are called equidecomposable if they are decomposed into pairwise congruent pieces. For an example we consider the set of all polygons in the euclidean plane. It is well known that planar polygons have the same area if and only if they are equidecomposable. In the three-dimensional euclidean space one also can describe the content-equality of polyhedra by a relation. Two polyhedra have the same volume if they are equidecomposable with respect to equiaffine mappings (see [3]). In [4] the concept of an invariant content of polyhedra in a topological Klein space is introduced. Each regular closed quasicompact set ot the space is called polyhedron. Under this supposition two polyhedra have equal contents if they are equivalent by decomposition. The relation “equivalent by decomposition” is closely related to the relation “equidecomposable”.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.     

On regular polyhedra with hidden symmetries

We consider polyhedral realizations of oriented regular maps with or without self-intersections in E³ whose symmetry group is a subsgroup of small index in their. automorphism group. The four classical kepler-poinsot polyhedra are the only ones of index 1. There are exactly five of Index 2, all with icosahedral symmetry group [W2] as the Kepler-poinsot polyhedra. In this paper we show that there are no such polyhedral realizations with octahedral (tetrahedral) symmetry group and index 2 or 3 (2,3,4,5), which is best possible in the octahedral case.     

A universal result for consecutive random subdivision of polygons

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable ξ. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non‐degenerateness conditions on ξ, the shapes of these polygons eventually become “flat” The convergence rate to flatness is also investigated; in particular, in the case of triangles (d = 3), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of [11] which are achieved mostly by using ad hoc methods. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 341–371, 2017     

Monohedral Idemvalent Polyhedra that are Toroids

In this article we describe polyhedra of genus one that satisfy the following two properties: They have all faces congruent (monohedral) and they have the same number of faces at each vertex (idemvalent). In particular, the construction given yields polyhedra with six congruent triangles at each vertex. The construction given is a generalization of one given half a century ago by Alaoglu and Giese, and leads to many interesting toroidal polyhedra. We end by showing how the construction can yield a monohedral idemvalent toroidal polyhedron that is knotted.     

Regular polyhedra—old and new

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.This work was supported in part by National Science Foundation Grant MPS74-07547 A01.     

On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

In this paper, we establish new characterizations of totally geodesic spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime, which is supposed to obey the null convergence condition. As applications, we get nonparametric results concerning to entire maximal vertical graphs in a such ambient spacetime. Proceeding, we obtain a lower estimate of the index of relative nullity of complete r-maximal spacelike hypersurfaces immersed in Robertson–Walker spacetimes of constant sectional curvature. In particular, we prove a sort of weak extension of the classical Calabi–Bernstein theorem.     

On Hamilton decompositions of prisms over simple 3-polytopes

In this paper it is shown that certain families of simple 4-polytopes have a Hamilton decomposition, that is, the edges of these polytopes can be partitioned into two Hamilton cycles.This research was partially supported by N.S.E.R.C. under Grant A-4792.     

Connectivity threshold of Bluetooth graphs

We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc” wireless networks. The graphs are obtained as “irrigation subgraphs” of the well‐known random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors” of each vertex and the number of edges c that each vertex is allowed to send to these. The randomness comes from the underlying distribution of vertices in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 45–66, 2014