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1.
Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS
3, connected sums of handlesS
1
× S
2
, euclidean and spherical space forms, and other examples with non-trivial fundamental group. 相似文献
2.
Abstract regular polytopes generalize the classical concept of a regular polytope and regular tessellation to more complicated combinatorial structures with a distinctive geometrical and topological flavour. In this paper the authors give an almost complete classification of the (universal) locally toroidal regular 4-polytopes of Schläfli types {4,4,3} and {4,4,4}. 相似文献
3.
Adam Stephanides 《Geometriae Dedicata》1994,53(3):263-270
Given a regular incidence (quasi-)polytopeP of type {a
1,a
2, ...,a
n–1} and a function on its directed edges satisfying certain conditions, we construct for everym 2 a regular incidence (quasi-)polytope of type {ma
1,a
2, ...,a
n–1} with the same vertex figure asP. 相似文献
4.
Many “highly symmetric” configurations of vectors in Cd, such as the vertices of the platonic solids and the regular complex polytopes, are equal-norm tight frames by virtue of being the orbit of the irreducible unitary action of their symmetry group. For nonabelian groups there are uncountably many such tight frames up to unitary equivalence. The aim of this paper is to single out those orbits which are particularly nice, such as those which are the vertices of a complex polytope. This is done by defining a finite class of tight frames of n vectors for Cd (n and d fixed) which we call the highly symmetric tight frames. We outline how these frames can be calculated from the representations of abstract groups using a computer algebra package. We give numerous examples, with a special emphasis on those obtained from the (Shephard–Todd) finite reflection groups. The interrelationships between these frames with complex polytopes, harmonic frames, equiangular tight frames, and Heisenberg frames (maximal sets of equiangular lines) are explored in detail. 相似文献
5.
Michael I. Hartley 《Combinatorica》2008,28(3):299-314
This article completes the classification of finite universal locally projective regular abstract polytopes, by summarising
(with careful references) previously published results on the topic, and resolving the few cases that do not appear in the
literature. In rank 4, all quotients of the locally projective polytopes are also noted. In addition, the article almost completes
the classification of the infinite universal locally projective polytopes, except for the {{5,3,3,},{3,3,5}15} and its dual. It is shown that this polytope cannot be finite, but its existence is not established. The most remarkable
feature of the classification is that a nondegenerate universal locally projective polytope is infinite if and only if the rank of is 5 and the facets of or its dual are the hemi-120-cell {5,3,3}15. 相似文献
6.
J. M. Wills 《Aequationes Mathematicae》1987,34(2-3):206-220
Summary We investigate polyhedral realizations of regular maps with self-intersections in E3, whose symmetry group is a subgroup of index 2 in their automorphism group. We show that there are exactly 5 such polyhedra. The polyhedral sets have been more or less known for about 100 years; but the fact that they are realizations of regular maps is new in at least one case, a self-dual icosahedron of genus 11. Our polyhedra are closely related to the 5 regular compounds, which can be interpreted as discontinuous polyhedral realizations of regular maps.The author was born on March 5, 1937; so exactly half a century after Otto Haupt.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday. 相似文献
7.
Andreas W. M. Dress 《Aequationes Mathematicae》1985,29(1):222-243
The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE
3 and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE
3, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE
n
is worked out in §9. 相似文献
8.
In ``Quotients of a Universal Locally Projective Polytope' (Math. Z. 247 663–674, DOI: 10.1007/s00209-003-0625-9), the authors analyse the a certain locally projective universal polytope, showing
it to be finite, and enumerating its quotients. The authors have since discovered some errors in the enumeration of the quotients.
This note corrects these errors.
The online version of the original article can be found at 相似文献
9.
Peter McMullen 《Aequationes Mathematicae》1989,37(1):38-56
Let be a finite regular incidence-polytope. A realization of is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group () of induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of forms, in a natural way, a closed convex cone, which is also denoted by The dimensionr of is the number of equivalence classes under () of diagonals of , and is also the number of unions of double cosets ** *–1* ( *), where * is the subgroup of () which fixes some given vertex of . The fine structure of corresponds to the irreducible orthogonal representations of (). IfG is such a representation, let its degree bed
G
, and let the subgroup ofG corresponding to * have a fixed space of dimensionw
G
. Then the relations
相似文献
10.
If Δ is a polytope in real affine space, each edge of Δ determines a reflection in the perpendicular bisector of the edge.
The exchange groupW (Δ) is the group generated by these reflections, and Δ is a (Coxeter) matroid polytope if this group is finite. This simple
concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The Gelfand-Serganova Theorem and
the structure of the exchange group both give us information about the matroid polytope. We then specialize this information
to the case of ordinary matroids; the matroid polytope by our definition in this case turns out to be a facet of the classical
matroid polytope familiar to matroid theorists.
This work was supported in part by NSA grant MDA904-95-1-1056. 相似文献
11.
An abstract polytope is called regular if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n-polytopes with the smallest number of flags are found, for every rank n>1. With a few small exceptions, the smallest regular n-polytopes come from a family of ‘tight’ polytopes with 2⋅4n−1 flags, one for each n, with Schläfli symbol {4∣4∣?∣4}. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram. 相似文献
12.
Let P be a simple lattice polytope. We define an action of the Hecke operators on E(P), the Ehrhart polynomial of P, and describe their effect on the coefficients of E(P). We also describe how the Brion–Vergne formula for E(P) transforms under the Hecke operators for nonsingular lattice polytopes P.
相似文献
13.
In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants. 相似文献
14.
We construct examples of Gromov hyperbolic
Coxeter groups of arbitrarily large dimension.
We also extend Vinbergs theorem to show that if a Gromov
hyperbolic Coxeter group is a virtual Poincaré duality group
of dimension n,
then n 61.Coxeter groups acting on their associated complexes have been extremely
useful source of examples and insight into nonpositively curved spaces
over last several years. Negatively curved (or Gromov hyperbolic)
Coxeter groups were much more elusive. In particular their existence in
high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on
the virtual cohomological dimension of any Gromov hyperbolic Coxeter group.
This question was also raised by Misha Gromov [G] (who thought that perhaps
any construction of high dimensional negatively curved spaces requires
nontrivial number theory in the guise of arithmetic groups in an essential
way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter
groups do exist, and we construct them by geometric or group theoretic but
not arithmetic means. 相似文献
15.
C. Palencia 《Numerische Mathematik》1999,82(3):471-490
We consider semidiscretizations in time, based on the backward Euler method, of an abstract, non-autonomous parabolic initial
value problem where , , is a family of sectorial operators in a Banach space X. The domains are allowed to depend on t. Our hypotheses are fulfilled for classical parabolic problems in the , , norms. We prove that the semidiscretization is stable in a suitable sense. We get optimal estimates for the error even when
non-homogeneous boundary values are considered. In particular, the results are applicable to the analysis of the semidiscretizations
of time-dependent parabolic problems under non-homogeneous Neumann boundary conditions.
Received October 17, 1997 / Revised version received April 17, 1998 相似文献
16.
Michael I. Hartley 《Annals of Combinatorics》2006,10(3):343-352
The main results of this article facilitate the search for quotients of regular abstract polytopes. A common approach in the
study of abstract polytopes is to construct polytopes with specified facets and vertex figures. Any nonregular polytope
may be constructed as a quotient of a regular polytope
by a (so-called) semisparse subgroup of its automorphism group W (which will be a string C-group). It becomes important, therefore, to be able to identify whether or not a given subgroup N of a string C-group W is semisparse. This article proves a number of properties of semisparse subgroups. These properties may be used to test for
semisparseness in a way which is computationally more efficient than previous methods. The methods are used to find an example
of a section regular polytope of type {6, 3, 3} whose facets are Klein bottles.
Received February 15, 2005 相似文献
17.
We give new weighted decompositions for simple polytopes, generalizing previous results of Lawrence-Varchenko and Brianchon-Gram. We start with Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment map in the context of toric varieties to obtain a decomposition for Delzant polytopes. Then, by a purely combinatorial argument, we show that this formula holds for any simple polytope. As an application, we study Euler-Maclaurin formulas. 相似文献
18.
We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n?2d+1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill. 相似文献
19.
We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812. 相似文献
20.
Andreas W. M. Dress 《Aequationes Mathematicae》1981,23(1):252-265
The new regular polyhedra, defined and investigated by Branko Grünbaum in [4], and theirn-dimensional generalizations are classified in terms of their symmetry group. 相似文献
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