Random projections of regular simplices

Precise asymptotic formulae are obtained for the expected number ofk-faces of the orthogonal projection of a regularn-simplex inn-space onto a randomly chosen isotropic subspace of fixed dimension or codimension, as the dimensionn tends to infinity.F. Affentranger was supported by a grant from the Swiss National Foundation.     

Large regular simplices contained in a hypercube

We prove that the n-dimensional unit hypercube contains an n-dimensional regular simplex of edge length c√n, where c > 0 is a constant independent of n. Supported by MEXT Grant-in-Aid for Scientific Research (B) 20340022.     

Gaussian samples, regular simplices, and exchangeability

The interchangeable Goodman—Pollack-type random finite sets are characterized. Received June 14, 1995, and in revised form October 16, 1995.     

Subdivision of simplices relative to a cutting plane and finite concave minimization

In this article, our primary concern is the classical problem of minimizing globally a concave function over a compact polyhedron (Problem (P)). We present a new simplicial branch and bound approach, which combines triangulations of intersections of simplices with halfspaces and ideas from outer approximation in such a way, that a class of finite algorithms for solving (P) results. For arbitrary compact convex feasible sets one obtains a not necessarily finite but convergent algorithm. Theoretical investigations include determination of the number of simplices in each applied triangulation step and bounds on the number of iterations in the resulting algorithms. Preliminary numerical results are given, and additional applications are sketched.     

New regular embeddings of n‐cubes Qn

We investigate highly symmetrical embeddings of the n‐dimensional cube Q_n into orientable compact surfaces which we call regular embeddings of Q_n. We derive some general results and construct some new families of regular embeddings of Q_n. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Q_n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004     

Orthogonal polynomials and cubature formulae on balls,simplices, and spheres

We report on recent developments on orthogonal polynomials and cubature formulae on the unit ball B^d, the standard simplex T^d, and the unit sphere S^d. The main result shows that orthogonal structures and cubature formulae for these three regions are closely related. This provides a way to study the structure of orthogonal polynomials; for example, it allows us to use the theory of h-harmonics to study orthogonal polynomials on B^d and on T^d. It also provides a way to construct new cubature formulae on these regions.     

A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP². The automorphism group of X is isomorphic to S ₄ × S ₃. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP² admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision $ \bar X $ \bar X of the triangulation X such that the classical moment mapping μ: ℂP² → Δ² is a simplicial mapping of the triangulation $ \bar X $ \bar X onto the barycentric subdivision of the triangle Δ². We study the relationship of the triangulation X with complex crystallographic groups.     

The largest projections of regular polytopes

The projections of the regularn-dimensional simplex and crosspolytope intoR ^k with the largestk-volume are determined here for the casesk=2,n≧2 andk=3, 4≦n≦6. The proofs involve a combination of exterior algebra and computer gradient methods.     

An extremal characterization of regular simplices via the addition theorem for Gegenbauer polynomials

Let p ₁, ..., p _N+2 be unit vectors in E ^N+1. Then the sum of the squares of all the (N+2, 2) inner products which they determine is minimal if and only if ± signs can be chosen so that the ±p _i are the vertices of a regular simplex. The proof involves a special case of the addition theorem for Gegenbauer polynomials.AMS(MOS) subject classifications (1970). Primary 52A25, 52A40.     

Generalized Sperner lemma and subdivisions into simplices of equal volume

A generalization of the well-known Sperner lemma is suggested, which covers the case of arbitrary subdivisions of (convex) polyhedra into (convex) polyhedra. It is used for giving a new proof of the Thomas-Monsky-Mead theorem saying that the n-cube can be subdivided into N simplices of equal volume if and only if N is divisible by n!. Some new related results are announced. Bibliography: 6 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 245–254.     

Polytopes which are orthogonal projections of regular simplexes

We consider the polytopes which are certain orthogonal projections of k-dimensional regular simplexes in k-dimensional Euclidean space R ^k . We call such polytopes -polytopes. Every sufficiently symmetric polytope, such as a regular polytope, a quasi-regular polyhedron, etc., belongs to this class. We denote by P _m,n all n-dimensional -polytopes with m vertices. We show that there is a one-to-one correspondence between the elements of P _m,n and those of P _m,m–n–1 and that this correspondence preserves the symmetry of -polytopes. Using this duality, we determine some of the P _m,n 's. We also show that a -polytope is an orthogonal projection of a cross polytope if and only if it has central symmetry.     

Decompositions of Km,n into cubes

For a complete bipartite graph to be decomposable into isomorphic cubes, certain conditions on the number of cube and bipartition vertices must hold. We prove these necessary conditions sufficient in some cases. For cubes of fixed dimension d (indeed for d-regular bipartite graphs in general) we show that proving sufficiency can be reduced to decomposing a finite number of complete bipartite graphs. When t = 2^d−1 and r is the remainder on dividing t by d, we show K_t,t is decomposable into d-cubes and an r-factor, where if r > 0 this r-factor itself is decomposable into r-cubes. © 1996 John Wiley & Sons, Inc.     

Equisingularity classes of birational projections of normal singularities to a plane

Given a birational normal extension O of a two-dimensional local regular ring (R,m), we describe all the equisingularity types of the complete m-primary ideals J in R whose blowing-up X=BlJ(R) has some point Q whose local ring OX,Q is analytically isomorphic to O.