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1.
We give a lower bound for the number of vertices of a generald-dimensional polytope with a given numberm ofi-faces for eachi = 0,..., d/2 – 1. The tightness of those bounds is proved using McMullen's conditions. Form greater than a small constant, those lower bounds are attained by simpliciali-neighbourly polytopes. 相似文献
2.
Deza M Dutour M Fowler PW 《Journal of chemical information and computer sciences》2004,44(4):1282-1293
Two connections between fullerene structures and alternating knots are established. Knots may appear in two ways: from zigzags, i.e., circuits (possibly self-intersecting) of edges running alternately left and right at successive vertices, and from railroads, i.e., circuits (possibly self-intersecting) of edge-sharing hexagonal faces, such that the shared edges occur in opposite pairs. A z-knot fullerene has only a single zigzag, doubly covering all edges: in the range investigated (n = 74) examples are found for C(34) and all C(n)() with n >/= 38, all chiral, belonging to groups C(1), C(2), C(3), D(3), or D(5). An r-knot fullerene has a railroad corresponding to the projection of a nontrivial knot: examples are found for C(52) (trefoil), C(54) (figure-of-eight or Flemish knot), and, with isolated pentagons, at C(96), C(104), C(108), C(112), C(114). Statistics on the occurrence of z-knots and of z-vectors of various kinds, z-uniform, z-transitive, and z-balanced, are presented for trivalent polyhedra, general fullerenes, and isolated-pentagon fullerenes, along with examples with self-intersecting railroads and r-knots. In a subset of z-knot fullerenes, so-called minimal knots, the unique zigzag defines a specific Kekulé structure in which double bonds lie on lines of longitude and single bonds on lines of latitude of the approximate sphere defined by the polyhedron vertices. 相似文献
3.
A theorem of Deza asserts that ifH
1, ...,H
m
ares-sets any pair of which intersects in exactlyd elements and ifm ≧s
2 −s+2, then theH
i
form aΔ-system, i.e.
. In other words, every large equidistant (0, 1)-code of constant weight is trivial. We give a (0, +1, −1) analogue of this
theorem. 相似文献
4.
We study facets of the cut coneC
n
, i.e., the cone of dimension 1/2n(n – 1) generated by the cuts of the complete graph onn vertices. Actually, the study of the facets of the cut cone is equivalent in some sense to the study of the facets of the cut polytope. We present several operations on facets and, in particular, a lifting procedure for constructing facets ofC
n+1 from given facets of the lower dimensional coneC
n
. After reviewing hypermetric valid inequalities, we describe the new class of cycle inequalities and prove the facet property for several subclasses. The new class of parachute facets is developed and other known facets and valid inequalities are presented. 相似文献
5.
The cut polytopeP n is the convex hull of the incidence vectors of the cuts (i.e. complete bipartite subgraphs) of the complete graph onn nodes. A well known class of facets ofP n arises from the triangle inequalities:x ij +x ik +x jk ≤ 2 andx ij -x ik -x jk ≤ 0 for 1 ≤i,j, k ≤n. Hence, the metric polytope Mn, defined as the solution set of the triangle inequalities, is a relaxation ofP n . We consider several properties of geometric type for Pn, in particular, concerning its position withinM n . Strengthening the known fact ([3]) thatP n has diameter 1, we show that any set ofk cuts,k ≤ log2 n, satisfying some additional assumption, determines a simplicial face ofM n and thus, also, ofP n . In particular, the collection of low dimension faces ofP n is contained in that ofM n . Among a large subclass of the facets ofP n , the triangle facets are the closest ones to the barycentrum of Pn and we conjecture that this result holds in general. The lattice generated by all even cuts (corresponding to bipartitions of the nodes into sets of even cardinality) is characterized and some additional questions on the links between general facets ofP n and its triangle facets are mentioned. 相似文献
6.
Julián I. Peña Rosselló Horacio S. Wio Roberto R. Deza Peter Hänggi 《The European Physical Journal B - Condensed Matter and Complex Systems》2017,90(2):34
The performance of a ring of linearly coupled, monostable nonlinear oscillators is optimized towards its goal of acting as energy harvester – through piezoelectric transduction – of mesoscopic fluctuations, which are modeled as Ornstein-Uhlenbeck noises. For a single oscillator, the maximum output voltage and overall efficiency are attained for a soft piecewise-linear potential (providing a weak attractive constant force) but they are still fairly large for a harmonic potential. When several harmonic springs are linearly and bidirectionally coupled to form a ring, it is found that counter-phase coupling can largely improve the performance while in-phase coupling worsens it. Moreover, it turns out that few (two or three) coupled units perform better than more. 相似文献
7.
Bellarosa L Fowler PW Lijnen E Deza M 《Journal of chemical information and computer sciences》2004,44(4):1314-1323
The problem of predicting stoichiometries and patterns of chemical addition to a carbon framework, subject solely to the restriction that each addend excludes neighboring sites up to some distance d, is equivalent to determination of d-codes of a graph, and for d = 2 to determination of maximum independent sets. Sizes, symmetries, and numbers of d-codes are found for the all-heptagon Klein graph (prototype for "plumber's nightmare" carbon) and for three related graphs. The independence number of the Klein graph is 23, which increases to 24 for a related, but sterically relaxed, all-heptagon network with the same number of vertices and modified adjacencies. Expansion of the Klein graph and its relaxed analogue by insertion of hexagonal faces to form leapfrog graphs also allows all heptagons to achieve their maximum of 3 addends. Consideration of the pi system that is the complement of the addition pattern imposes a closed-shell requirement on the adjacency spectrum, which typically reduces the size of acceptable independent sets. The closed-shell independence numbers of the Klein graph and its relaxed analogue are 18 and 20, respectively. 相似文献
8.
Given a connected surface \({\mathbb {F}}^2\) with Euler characteristic \(\chi \) and three integers \(b>a\ge 1<k\), an \((\{a,b\};k)\)-\({\mathbb {F}}^2\) is a \({\mathbb {F}}^2\)-embedded graph, having vertices of degree only k and only a- and b-gonal faces. The main case are (geometric) fullerenes (5, 6; 3)-\({\mathbb {S}}^2\). By \(p_a\), \(p_b\) we denote the number of a-gonal, b-gonal faces. Call an \((\{a,b\};k)\)-map lego-admissible if either \(\frac{p_b}{p_a}\), or \(\frac{p_a}{p_b}\) is integer. Call it lego-like if it is either \(ab^f\)-lego map, or \(a^fb\)-lego map, i.e., the face-set is partitioned into \(\min (p_a,p_b)\) isomorphic clusters, legos, consisting either one a-gon and \(f=\frac{p_b}{p_a}\,b\)-gons, or, respectively, \(f=\frac{p_a}{p_b}\,a\)-gons and one b-gon; the case \(f=1\) we denote also by ab. Call a \((\{a,b\};k)\)-map elliptic, parabolic or hyperbolic if the curvature \(\kappa _b=1+\frac{b}{k}-\frac{b}{2}\) of b-gons is positive, zero or negative, respectively. There are 14 lego-like elliptic \((\{a,b\};k)\)-\({\mathbb {S}}^2\) with \((a,b)\ne (1,2)\). No \((\{1,3\};6)\)-\({\mathbb {S}}^2\) is lego-admissible. For other 7 families of parabolic \((\{a,b\};k)\)-\({\mathbb {S}}^2\), each lego-admissible sphere with \(p_a\le p_b\) is \(a^fb\) and an infinity (by Goldberg–Coxeter operation) of \(ab^f\)-spheres exist. The number of hyperbolic \(ab^f\,(\{a,b\};k)\)-\({\mathbb {S}}^2\) with \((a,b)\ne (1,3)\) is finite. Such \(a^f b\)-spheres with \(a\ge 3\) have \((a,k)=(3,4),(3,5),(4,3),(5,3)\) or (3, 3); their number is finite for each b, but infinite for each of 5 cases (a, k). Any lego-admissible \((\{a,b\};k)\)-\({\mathbb {S}}^2\) with \(p_b=2\le a\) is \(a^f b\). We list, explicitly or by parameters, lego-admissible \((\{a,b\};k)\)-maps among: hyperbolic spheres, spheres with \(a\in \{1,2\}\), spheres with \(p_b\in \{2,\frac{p_a}{2}\}\), Goldberg–Coxeter’s spheres and \((\{a,b\};k)\)-tori. We present extensive computer search of lego-like spheres: 7 parabolic (\(p_b\)-dependent) families, basic examples of all 5 hyperbolic \(a^fb\) (b-dependent) families with \(a\ge 3\), and lego-like \((\{a,b\};3)\)-tori. 相似文献
9.
Mangioni SE Deza RR Toral R Wio HS 《Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics》2000,61(1):223-232
A recently introduced lattice model, describing an extended system which exhibits a reentrant (symmetry-breaking, second-order) noise-induced nonequilibrium phase transition, is studied under the assumption that the multiplicative noise leading to the transition is colored. Within an effective Markovian approximation and a mean-field scheme it is found that when the self-correlation time tau of the noise is different from zero, the transition is also reentrant with respect to the spatial coupling D. In other words, at variance with what one expects for equilibrium phase transitions, a large enough value of D favors disorder. Moreover, except for a small region in the parameter subspace determined by the noise intensity sigma and D, an increase in tau usually prevents the formation of an ordered state. These effects are supported by numerical simulations. 相似文献
10.
Applying the negative-type inequalities for square Euclidean distance, we present (1) a parallelatope theorem (a generalization of the parallelogram theorem), (2) a short proof of Rankin’s theorem for the maximum number of dispersed points on a sphere, and (3) a proof of impossibility of a certain geometric embedding for some graphs. 相似文献