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1.
Let ?? be a set of n-dimensional polytopes. A set ?? of n-dimensional polytopes is said to be an element set for ?? if each polytope in ?? is the union of a finite number of polytopes in ?? identified along (n ? 1)-dimensional faces. The element number of the set ?? of polyhedra, denoted by e(??), is the minimum cardinality of the element sets for ??, where the minimum is taken over all possible element sets ${\Omega \in \mathcal{E}(\Sigma)}$ . It is proved in Theorem 1 that the element number of the convex regular 4-dimensional polytopes is 4, and in Theorem 2 that the element numbers of the convex regular n-dimensional polytopes is 3 for n ?? 5. The results in this paper together with our previous papers determine completely the element numbers of the convex regular n-dimensional polytopes for all n ?? 2. 相似文献
2.
Linking (n − 2)-dimensional panels inn-space II: (n − 2, 2)-frameworks and body and hinge structures
Tiong-Seng Tay 《Graphs and Combinatorics》1989,5(1):245-273
An (n – 1, 2)-framework inn-space is a structure consisting of a finite set of (n – 2)-dimensional panels and a set of rigid bars each joining a pair of panels using ball joints. A body and hinge (or (n + 1,n – 1)-) framework inn-space consists of a finite set ofn-dimensional bodies articulated by a set of (n – 2)-dimensional hinges, i.e., joints in 2-space, line hinges in 3-space, plane-hinges in 4-space, etc. In this paper we characterize the graphs of all rigid (n – 1, 2)- and (n + 1,n – 1)-frameworks inn-space. Rigidity here is statical rigidity or equivalently infinitesimal rigidity. 相似文献
3.
Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n
3/2
) . 相似文献
4.
Let Σ be a set of polyhedra. A set Ω of polyhedra is said to be an element set for Σ if each polyhedron in Σ is the union
of a finite number of polyhedra in Ω. We call each polyhedron of the element set Ω an element for Σ. In this paper, we determine
one element set for the set Π of the Platonic solids, and prove that this element set is, in fact, best possible; it achieves
the minimum in terms of cardinality among all the element sets for Π. We also introduce the notion of indecomposability of
a polyhedron and present a conjecture in Sect. 3. 相似文献
5.
Antonio Fernández Fernando Mayoral Francisco Naranjo Enrique A. Sánchez–Pérez 《Positivity》2009,13(1):129-143
Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L1(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L1(n) = L1(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation
of the space L1(m) associated to downward directed subsets of the set of all the representations.
This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02
(M.E.C. Spain) and FEDER is gratefully acknowledged. 相似文献
6.
In this paper, we prove the following theorem regarding the Wang–Yau quasi-local energy of a spacelike two-surface in a spacetime:
Let Σ be a boundary component of some compact, time-symmetric, spacelike hypersurface Ω in a time-oriented spacetime N satisfying the dominant energy condition. Suppose the induced metric on Σ has positive Gaussian curvature and all boundary
components of Ω have positive mean curvature. Suppose H ≤ H
0 where H is the mean curvature of Σ in Ω and H
0 is the mean curvature of Σ when isometrically embedded in
\mathbb R3{\mathbb R^3} . If Ω is not isometric to a domain in
\mathbb R3{\mathbb R^3}, then
1. |
the Brown–York mass of Σ in Ω is a strict local minimum of the Wang–Yau quasi-local energy of Σ. 相似文献
7.
W. Ishizuka C. Y. Wang 《Calculus of Variations and Partial Differential Equations》2008,32(3):387-405
For a bounded domain Ω ⊂ R
n
endowed with L
∞-metric g, and a C
5-Riemannian manifold (N, h) ⊂ R
k
without boundary, let u ∈ W
1,2(Ω, N) be a weakly harmonic map, we prove that (1) u ∈ C
α (Ω, N) for n = 2, and (2) for n ≥ 3, if, in additions, g ∈ VMO(Ω) and u satisfies the quasi-monotonicity inequality (1.5), then there exists a closed set Σ ⊂ Ω, with H
n-2(Σ) = 0, such that for some α ∈ (0, 1).
C. Y. Wang Partially supported by NSF. 相似文献
8.
Isabel M. C. Salavessa 《Bulletin of the Brazilian Mathematical Society》2010,41(4):495-530
On a Riemannian manifold $
\bar M^{m + n}
$
\bar M^{m + n}
with an (m + 1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space ℝH ⋇ TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the case described
by Barbosa, do Carmo and Eschenburg, when n = 1 and Ω is the volume element of $
\bar M
$
\bar M
. To the second variation we associate an Ω-Jacobi operator and define Ω-stability. Under natural conditions, we show that
the Euclidean m-spheres are the unique Ω-stable submanifolds of ℝ
m+n
. We study the Ω-stability of geodesic m-spheres of a fibred space form M
m+n
with totally geodesic (m + 1)-dimensional fibres. 相似文献
9.
Consider an arrangement of n hyperplanes in \real
d
. Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper.
We aim to bound their maximum combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes
are cells of the arrangement. Situations where the polytopes are pairwise openly disjoint have also been considered in the
past. However, no nontrivial bound was known for the general case where the polytopes may have overlapping interiors, for
d>2 . We analyze families of polytopes that do not share vertices. In \real
3
we show an O(k
1/3
n
2
) bound on the number of faces of k such polytopes. We also discuss worst-case lower bounds and higher-dimensional versions of the problem. Among other results,
we show that the maximum number of facets of k pairwise vertex-disjoint polytopes in \real
d
is Ω(k
1/2
n
d/2
) which is a factor of away from the best known upper bound in the range n
d-2
≤ k ≤ n
d
. The case where 1≤ k ≤ n
d-2
is completely resolved as a known Θ(kn) bound for cells applies here.
Received September 20, 1999, and in revised form March 10, 2000. Online publication September 22, 2000. 相似文献
10.
Let (Ω,Σ,μ) be a measure space and letP be an operator onL
2(Ω,Σ,μ) with ‖P‖≦1,Pf≧0 a.e. wheneverf≧0. If the subspaceK is defined byK={x| ||P
n
x||=||P
*n
x||=||x||,n=1,2,...} thenK=L
2(Ω,Σ1,μ), where Σ1 ⊂ Σ and onK the operatorP is “essentially” a measure preserving transformation. Thus the eigenvalues ofP of modulus one, form a group under multiplication.
This last result was proved by Rota for finiteμ here finiteness is not assumed) and is a generalization of a theorem of Frobenius and Perron on positive matrices.
The research reported in this document has been sponsored in part by Air Force Office of Scientific Research, OAR through
the European Office, Aerospace Research, United States Air Force. 相似文献
11.
《Discrete and Computational Geometry》2008,28(4):577-583
Abstract. Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f(n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f(n)=Ω(n
3/2
) . 相似文献
12.
The computational complexity of the partition problem , which concerns the partitioning of a set of n vectors in d -space into p parts so as to maximize an objective function which is convex on the sum of vectors in each part, is determined by the number
of vertices of the corresponding p-partition polytope defined to be the convex hull in (d\times p) -space of all solutions. In this article, introducing and using the class of Momentopes , we establish the lower bound v
p,d
(n)=Ω(n^ \lfloor (d-1)/2 \rfloor p ) on the maximum number of vertices of any p -partition polytope of a set of n points in d -space, which is quite compatible with the recent upper bound v
p,d
(n)=O(n
d(p-1)-1
) , implying the same bound on the complexity of the partition problem. We also discuss related problems on the realizability
of Davenport—Schinzel sequences and describe some further properties of Momentopes.
Received April 4, 2001, and in revised form October 3, 2001. Online publication February 28, 2002. 相似文献
13.
Csaba D. Tóth 《Periodica Mathematica Hungarica》2008,57(2):217-225
It is shown that for every subdivision of the d-dimensional Euclidean space, d ≥ 2, into n convex cells, there is a straight line that stabs at least Ω((log n/log log n)1/(d−1)) cells. In other words, if a convex subdivision of d-space has the property that any line stabs at most k cells, then the subdivision has at most exp(O(k
d−1 log k)) cells. This bound is best possible apart from a constant factor. It was previously known only in the case d = 2.
Supported in part by NSERC grant RGPIN 35586. 相似文献
14.
We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R
N
, N=2,3, surrounded by a thin layer Σ
ε
, along a part Γ2 of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ
ε
with Reynolds’ number of order 1/ε in Σ
ε
. Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier
law involving a matrix of Borel measures having the same support contained in the interface Γ2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider
an optimal control problem within this context. 相似文献
15.
Gábor Hetyei 《Discrete and Computational Geometry》2009,42(4):705-721
We construct an n-dimensional polytope whose boundary complex is compressed and whose face numbers for any pulling triangulation are the coefficients
of the powers of (x−1)/2 in the nth Legendre polynomial. We show that the non-central Delannoy numbers count all faces in the lexicographic pulling triangulation
that contain a point in a given open generalized orthant. We thus provide a geometric interpretation of a relation between
the central Delannoy numbers and Legendre polynomials, observed over 50 years ago (Good in Proc. Camb. Philos. Soc. 54:39–42,
1958; Lawden in Math. Gaz. 36:193–196, 1952; Moser and Zayachkowski in Scr. Math. 26:223–229, 1963). The polytopes we construct are closely related to the root polytopes introduced by Gelfand et al. (Arnold–Gelfand mathematical
seminars: geometry and singularity theory, pp. 205–221. Birkhauser, Boston, 1996). 相似文献
16.
For a positive integer n and a subset S⊆[n−1], the descent polytope DP
S
is the set of points (x
1,…,x
n
) in the n-dimensional unit cube [0,1]
n
such that x
i
≥x
i+1 if i∈S and x
i
≤x
i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F
S
(t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes. 相似文献
17.
Carl W. Lee 《Israel Journal of Mathematics》1984,47(4):261-269
Letf(P
s
d
) be the set of allf-vectors of simpliciald-polytopes. ForP a simplicial 2d-polytope let Σ(P) denote the boundary complex ofP. We show that for eachf ∈f(P
s
d
) there is a simpliciald-polytopeP withf(P)=f such that the 11 02 simplicial diameter of Σ(P) is no more thanf
0(P)−d+1 (one greater than the conjectured Hirsch bound) and thatP admits a subdivision into a simpliciald-ball with no new vertices that satisfies the Hirsch property. Further, we demonstrate that the number of bistellar operations
required to obtain Σ(P) from the boundary of ad-simplex is minimum over the class of all simplicial polytopes with the samef-vector. This polytopeP will be the one constructed to prove the sufficiency of McMullen's conditions forf-vectors of simplicial polytopes. 相似文献
18.
Ding-hua YANG College of Mathematics Software Sciences Sichuan Normal University Chengdu China Chengdu Institute of Computer Applications Chinese Academy of Sciences Chengdu China 《中国科学A辑(英文版)》2007,50(3):423-438
In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought. 相似文献
19.
Yves Benoist 《Inventiones Mathematicae》2006,164(2):249-278
Divisible convex sets IV: Boundary structure in dimension 3
Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient
M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex:
The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M.
Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these
triangles is dense in the boundary of Ω (see Figs. 1 to 4).
Moreover, we construct examples of such divisible convex open sets Ω.
相似文献
20.
Hillel Gauchman 《Israel Journal of Mathematics》1979,33(1):37-51
It is shown that if ann-dimensional (n≧3) Riemannian manifold admitsr≧2 locally symmetric vector fields (LSVF's), then it is aV(k)-space. In particular, ifr=n−1 then the manifold is a space of constant curvature. In the case of a 3-dimensional Riemannian manifold a close connection
between LSVF's and eigenvectors of the Ricci tensor is found. 相似文献
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