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A two-dimensional framework (G,p) is
a graph G = (V,E) together with a map p: V → ℝ2. We view (G,p) as a straight line realization of G in ℝ2. Two realizations of G are equivalent if the corresponding edges in the two frameworks have the same length.
A pair of vertices {u,v} is globally linked in G if %and for all equivalent frameworks (G,q), the distance between the points
corresponding to u and v is the same
in all pairs of equivalent generic realizations of G. The graph G is globally rigid
if all of its pairs of vertices are globally linked. We extend the characterization of globally rigid graphs given by the
first two authors [13] by characterizing globally linked pairs in M-connected graphs, an important family of rigid graphs.
As a byproduct we simplify the proof of a result of Connelly [6] which is a key step in the characterization of globally rigid
graphs. We also determine the number of distinct realizations of an M-connected graph, each of which is equivalent to a given
generic realization. Bounds on this number for minimally rigid graphs were obtained by Borcea and Streinu in [3]. 相似文献
5.
C. Uranga C. Connell G.M. Borstad L.R. Zink M. Jackson 《Applied physics. B, Lasers and optics》2007,88(4):503-505
A three-laser heterodyne system was used to measure the frequencies of twelve previously observed far-infrared laser emissions
from the partially deuterated methanol isotopologues 13CD3OH and CHD2OH. Two laser emissions, a 53.773 μm line from 13CD3OH and a 74.939 μm line from CHD2OH, have also been discovered and frequency measured. The CO2 pump laser offset frequency was measured with respect to its center frequency for twenty-four FIR laser emissions from CH3OH, 13CD3OH and CHD2OH.
PACS 07.57.Hm; 42.55.Lt; 42.62.Eh 相似文献
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Let G = (V,E) be a graph or digraph and r : V → Z+. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν1,…,νr(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003 相似文献
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9.
S. David Morley Raymond J. Abraham Ian S. Haworth David E. Jackson Martin R. Saunders Jeremy G. Vinter 《Journal of computer-aided molecular design》1991,5(5):475-496
Summary Four modifications to the COSMIC molecular mechanics force field are described, which greatly increase both its versatility and the accuracy of calculated conformational energies. The Hill non-bonded van der Waals potential function has been replaced by a two-parameter Morse curve and a new H-H potential, similar to that in MM3, incorporated. Hydrocarbon energies in particular are much improved.A simple iterative Hückel pi-electron molecular orbital calculation allows modelling of conjugated systems. Calculated bond lengths and rotational barriers for a series of conjugated hydrocarbons and nitrogen heterocycles are shown to be as accurate as those determined by the MM2 SCF method.Explicit hydrogen-bonding potentials for H-bond acceptor-donor atom pairs have been included to give better hydrogen bond energies and lengths. The van der Waals radii of protonic hydrogens are reduced to 0.5 Å and the energy well depth is increased to 1.0 kcal mol-1.Two new general atom types, N+
sp
2
and O-
sp
3
, have been introduced which allow a wide variety of charged conjugated systems to be studied. A minimum of parameterisation is required, as the new types are easily included in the Hückel scheme which automatically adjusts bond and torsional parameters according to the defined bond-order relationships. 相似文献
10.
Brad M. Rosen Virgil Percec 《Journal of polymer science. Part A, Polymer chemistry》2007,45(21):4950-4964
Atom transfer radical polymerization (ATRP) and single electron‐transfer living radical polymerization (SET‐LRP) both utilize copper complexes of various oxidation states with N‐ligands to perform their respective activation and deactivation steps. Herein, we utilize DFT (B3YLP) methods to determine the preferred ligand‐binding geometries for Cu/N‐ligand complexes related to ATRP and SET‐LRP. We find that those ligands capable of achieving tetrahedral complexes with CuI and trigonal bipyramidal with axial halide complexes with [CuIIX]+ have higher energies of stabilization. We were able to correlate calculated preferential stabilization of [CuIIX]+ with those ligands that perform best in SET‐LRP. A crude calculation of energy of disproportionation revealed that the same preferential binding of [CuIIX]+ results in increased propensity for disproportionation. Finally, by examining the relative energies of the basic steps of ATRP and SET‐LRP, we were able to rationalize the transition from the ATRP mechanism to the SET‐LRP mechanism as we transition from typical nonpolar ATRP solvents to polar SET‐LRP solvents. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 4950–4964, 2007 相似文献