Equilibrium triangulations of the complex projective plane

Starting with the well-known 7-vertex triangulation of the ordinary torus, we construct a 10-vertex triangulation of P² which fits the equilibrium decomposition of P² in the simplest possible way. By suitable positioning of the vertices, the full automorphism group of order 42 is realized by a discrete group of isometries in the Fubini-Study metric. A slight subdivision leads to an elementary proof of the theorem of Kuiper-Massey which says that P² modulo conjugation is PL homeomorphic to the standard 4-sphere. The branch locus of this identification is a 7-vertex triangulation P ₂ ⁷ of the real projective plane. We also determine all tight simplicial embeddings of P ₂ ¹⁰ and P ₂ ⁷ .     

Finding minimum areak-gons

Given a setP ofn points in the plane and a numberk, we want to find a polygon with vertices inP of minimum area that satisfies one of the following properties: (1) is a convexk-gon, (2) is an empty convexk-gon, or (3) is the convex hull of exactlyk points ofP. We give algorithms for solving each of these three problems in timeO(kn ³). The space complexity isO(n) fork=4 andO(kn ²) fork5. The algorithms are based on a dynamic programming approach. We generalize this approach to polygons with minimum perimeter, polygons with maximum perimeter or area, polygons containing the maximum or minimum number of points, polygons with minimum weight (for some weights added to vertices), etc., in similar time bounds.This paper includes work done while David Eppstein was at Columbia University, Department of Computer Science, and while Günter Rote and Gerhard Woeginger were at the Freie Universität Berlin, Fachbereich Mathematik, Institut für Informatik. Research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).     

On the Agmon-Miranda maximum principle for solutions of strongly elliptic equations in domains of ℝn with conical points

In this paper the Agmon-Miranda maximum principle for solutions of strongly elliptic differential equations Lu = 0 in a bounded domain G with a conical point is considered. Necessary and sufficient conditions for the validity of this principle are given both for smooth solutions of the equation Lu = 0 in G and for the generalized solution of the problem Lu = 0 in G, D _k ^v u = gk on G (k = 0,...,m-1). It will be shown that for every elliptic operator L of order 2m > 2 there exists such a cone in ⁿ (n4) that the Agmon-Miranda maximum principle fails in this cone.     

Uniformities for the convergence in law and in probability

The convergence (X _n, Y_n)0 is investigated and characterized for probability metrics which metrize convergence in distribution or in probability. Some related metrics are also considered.     

Sets of vectors with many orthogonal paris:

What is the most number of vectors inR ^d such that anyk+1 contain an orthogonal pair? The 24 positive roots of the root systemF ₄ inR ⁴ show that this number could exceeddk.     

Matroid Shellability, β-Systems,and Affine Hyperplane Arrangements

The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the -system of a matroid, nbc(M), whose cardinality is Crapo's -invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the -system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the -system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex , and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the -system labels its decreasing chains.Basic cycles can be carried over from The intersection poset of any (real or complex) afflnehyperplane arrangement is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by nbc(M), for the union of such an arrangement.     

On locally optimal nonparametric tests

Summary There is an abundancy of problems in which no parametric model realistically describes the situation and in which, accordingly, we have to resort to nonparametric methods. As the numerical problems connected with nonparametric tests are becoming less and less important, rank tests, permutation tests and the like are becoming more and more part of the standard armatory of applied statisticians. The lack of tabulated critical values, for instance, should no longer be a serious objection against the use of permutation tests in practice; cf. Edgington (1987).The rationale underlying permutation and rank tests has been outlined in quite a number of text books and papers; cf. Fraser (1957), Lehmann (1959), Hájek-Sidák (1967) or Witting (1970). Roughly speaking, permutation tests are constructibel if the data can be condensed by means of a sufficient and complete statistic allowing for the proper kind of conditioning. Rank tests arise if the underlying problem is invariant with respect to (w.r.t.) a large group of transformations which leads to a maximal invariant statistic consisting of (signed) ranks.Most practical nonparametric problems, however, are too complex to be tractable by just one of those approaches. Many of them, however, can be handled by a combination of both techniques. In this paper we outline the logic underlying that combined reduction method and apply it to construct locally most powerful tests. Moreover, we discuss what we label Hoeffding's transfer problem, i.e. the uniformity aspect of locally most powerful tests with respect to the starting point at the boundary. We are concentrating on the discussion of the nonparametric two-sample location and scale problem. Further important problems are mentioned in Section III.This is a written account of an invited lecture delivered by the third author on occasion of the 14th Symposium über Operations Research, Ulm, September 6–8, 1989.     

The complete matter sector in a three-generation compactification

We consider a Calabi-Yau compactification paradigm with three light generations and anR-symmetry. From a special form of the Tian-Yau manifold, we also construct a new three-generation model with markedly different phenomenology. Thecomplete spectrum of all light matter fields is obtained in a universal way and moreover in a physically suitable basis, allowing a straightforward analysis of all their couplings. Here we discuss all the renormalizable Yukawa couplings. This computation can equally well be repeated for all compactification models based on Calabi-Yau complete intersections in products of homogeneous spaces.     

Concrete quantum logics with covering properties

LetL be a concrete (=set-representable) quantum logic. Letn be a natural number (or, more generally, a cardinal). We say thatL admits intrinsic coverings of the ordern, and writeL C _n , if for any pairA, BL we can find a collection {C _i iI}, where cardI<n andC _i L for anyiI, such thatA B= _il C _i . Thus, in a certain sense, ifLC _n , then the rate of noncompatibility of an arbitrary pairA,BL is less than a given numbern. In this paper we first consider general and combinatorial properties of logics ofC _n and exhibit typical examples. In particular, for a givenn we construct examples ofL C _n+1\C _n . Further, we discuss the relation of the classesC _n to other classes of logics important within the quantum theories (e.g., we discover the interesting relation to the class of logics which have an abundance of Jauch-Piron states). We then consider conditions on which a class of concrete logics reduce to Boolean algebras. We conclude with some open questions.