Distributive lattice structure on the set of perfect matchings of carbon nanotubes

2016 ◽  
Vol 54 (6) ◽  
pp. 1296-1305
Author(s):  
Niko Tratnik ◽  
Petra Žigert Pleteršek
Keyword(s):  
Carbon Nanotubes ◽  
Distributive Lattice ◽  
Lattice Structure ◽  
Perfect Matchings

Carbon-nanotubes on graphite: alignment of lattice structure

2003 ◽  
Vol 36 (7) ◽  
pp. 818-822 ◽  
Author(s):  
C Rettig ◽  
M B decker ◽  
H H vel
Keyword(s):  
Carbon Nanotubes ◽  
Lattice Structure

scholarly journals Partitions of an Integer into Powers

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
Matthieu Latapy
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Tree Structure ◽  
Dynamical Model ◽  
International Audience ◽  
Partitions Of Integers ◽  
Natural Way

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

Generalized intervals in partially ordered groups

1959 ◽  
Vol 55 (2) ◽  
pp. 165-171 ◽  
Author(s):  
D. C. J. Burgess
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Ordered Group ◽  
Partially Ordered ◽  
Partially Ordered Group ◽  
Ordered Groups ◽  
Normal Sense

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

Lattice Structure of D, T, and SR Fuzzy Flip-Flops Under Max-Min Logic

2005 ◽  
Vol 9 (6) ◽  
pp. 661-668 ◽  
Author(s):  
Shinichi Yoshida ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Logic ◽  
Distributive Lattice ◽  
System Design ◽  
Lattice Structure ◽  
Design Method ◽  
Boolean Lattice ◽  
Lattice Structures ◽  
Flip Flop ◽  
Different Types ◽  
Fuzzy Logical

Lattice structures of fuzzy flip-flops are described. A binary flip-flop (e.g. D, T, set-type SR, or reset-type SR flip-flop) can be extended to a fuzzy flip-flop in various ways. Under max-min fuzzy logic, there are 4 types of D fuzzy flip-flops extended from a binary D flip-flop, 136 types of SR fuzzy flip-flops extended from a binary SR flip-flop, and only one T fuzzy flip-flop. There is a lattice structure among different types of fuzzy flip-flops extended from a same binary flip-flop in terms of the order of ambiguity and the order of fuzzy logical value. These results show that fuzzy flip-flops under max-min fuzzy logic construct distributive lattice structures. Moreover D and T fuzzy flip-flops constructs Boolean lattice. And there exists a order monotone between two lattices of same fuzzy flip-flop under the order of ambiguity and the order of fuzzy logical value. Proposed analysis and results have potential to establish a fuzzy sequential system design method.

A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations

Order ◽  
2005 ◽  
Vol 22 (4) ◽  
pp. 311-328 ◽  
Author(s):  
Elena Barcucci ◽  
Antonio Bernini ◽  
Luca Ferrari ◽  
Maddalena Poneti
Keyword(s):  
Distributive Lattice ◽  
Lattice Structure ◽  
Noncrossing Partitions ◽  
Dyck Paths

scholarly journals Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

2017 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Stefan Felsner ◽  
Daniel Heldt
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Distributive Lattice ◽  
Maximum Degree ◽  
Lattice Structure ◽  
Mixing Time ◽  
Plane Graph ◽  
Negative Results ◽  
Slow Mixing ◽  
Constant Maximum

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.

Intercalated Fullerenes and Carbon Nanotubes

2003 ◽  
Author(s):  
Toshiaki Enoki ◽  
Morinobu Endo ◽  
Masatsugu Suzuki
Keyword(s):  
Carbon Nanotubes ◽  
Long Range ◽  
Lattice Structure ◽  
Arc Discharge ◽  
Van Der Waals ◽  
Graphite Electrodes ◽  
As Doping ◽  
Substitutional Doping ◽  
Guest Species ◽  
Host Materials

Since the discovery of soccer-ball-shaped C60 (Kroto et al., 1985), fullerenes have been added to the family of allotropes of carbon element. During the fullerene formation by arc-discharge of graphite electrodes, carbon nanotubes were simultaneously grown as a deposit on the electrode (Iijima, 1991). The carbon nanotubes consist of single or multiple graphene sheets rolled in the form of a seamless cylinder, with the diameter of the hollow core being almost 10 Å (similar to that of fullerenes) or even as small as 4 Å. For these new forms of nanometer-sized carbon, so-called nanocarbons, basically similar concepts as GICs have been applied from the aspects of structures, electronic properties, and functionalities that can be controlled by doping or intercalation process. That is, the bonding force between nanoballs or nanotubes is governed by weak van der Waals forces, so that foreign species such as atoms or molecules can be intercalated (or doped) in the van der Waals gaps, similar to graphite. So, from applications and the basic science of these new carbon families, intercalation as well as doping to these hosts has been studied intensively in the last ten years. There are three kinds of doping reactions of guest species into these host materials, which are reflected in their specific structure. Guest atoms can be introduced by substituting the carbon atoms of the hosts. This process is generally called “doping,” as there is a similarity with the doping process in semiconductors, where, in general, there is no long-range periodicity in the guest arrangement against the host crystal. Guests can locate in the hollow cores of fullerenes or carbon nanotubes as well as on their outer surfaces. GICs establish the super-lattice structure between the host of the graphite lattice and the inserted guest species, where the long-range periodicity along the c-axis as well as on the a-b plane is formed. According to the original meaning of “intercalation,” periodic doping to the host materials is defined as intercalation. So, the three kinds of doping are: (1) endohedral doping into the hollow cage; (2) substitutional doping by replacing the carbon atoms on the cages; and (3) exohedral doping where the dopants are sited in the gaps between the cage molecules of a fullerene crystal or between carbon nanotubes in the array of the bundle form.

scholarly journals Lattice Structures from Planar Graphs

10.37236/1768 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Stefan Felsner
Keyword(s):  
Distributive Lattice ◽  
Planar Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Lattice Structure ◽  
General Theorem ◽  
Lattice Structures ◽  
Schnyder Woods

The set of all orientations of a planar graph with prescribed outdegrees carries the structure of a distributive lattice. This general theorem is proven in the first part of the paper. In the second part the theorem is applied to show that interesting combinatorial sets related to a planar graph have lattice structure: Eulerian orientations, spanning trees and Schnyder woods. For the Schnyder wood application some additional theory has to be developed. In particular it is shown that a Schnyder wood for a planar graph induces a Schnyder wood for the dual.

Optical Properties of Carbon Nanotubes Under Axial Strain

2008 ◽  
Vol 8 (1) ◽  
pp. 122-130 ◽  
Author(s):  
Rajay Kumar ◽  
Stephen B. Cronin
Keyword(s):  
Carbon Nanotubes ◽  
Raman Spectroscopy ◽  
Lattice Structure ◽  
Theoretical Calculations ◽  
Axial Strain ◽  
Resonance Raman Spectroscopy ◽  
High Strength ◽  
New Physics ◽  
Sensitive Material ◽  
Induced Changes

A review is given of Raman spectroscopy of carbon nanotubes under axial strain. Carbon nanotubes possess a high Young's modulus (1 TPa) and breaking strains of 5–15%. Resonance Raman spectroscopy reveals changes in the electronic energies and lattice structure of nanotubes under applied strain. Studies performed on composite materials, where nanotubes have been added to increase the material's strength, are reviewed. Measurements of individual nanotubes under strain are also presented. Emphasis is given to the important new physics revealed by observing the strain-induced changes in the Raman spectra of individual nanotubes. A brief review of theoretical calculations performed on nanotubes under strain is also presented. The implications for using carbon nanotubes as a high strength material and as a strain sensitive material are indicated.

Structural analysis of the photosynthetic membrane from Ectothiorhodospira halochloris

1983 ◽  
Vol 41 ◽  
pp. 740-741
Author(s):  
H. Engelhardt ◽  
R. Guckenberger ◽  
W. Baumeister
Keyword(s):  
Lattice Structure ◽  
Bacterial Species ◽  
Hexagonal Lattice ◽  
Reaction Centre ◽  
Photosynthetic Membrane ◽  
Rhodopseudomonas Viridis ◽  
Photosynthetic Membranes ◽  
Ectothiorhodospira Halochloris ◽  
Main Components

Bacterial photosynthetic membranes contain, apart from lipids and electron transport components, reaction centre (RC) and light harvesting (LH) polypeptides as the main components. The RC-LH complexes in Rhodopseudomonas viridis membranes are known since quite seme time to form a hexagonal lattice structure in vivo; hence this membrane attracted the particular attention of electron microscopists. Contrary to previous claims in the literature we found, however, that 2-D periodically organized photosynthetic membranes are not a unique feature of Rhodopseudomonas viridis. At least five bacterial species, all bacteriophyll b - containing, possess membranes with the RC-LH complexes regularly arrayed. All these membranes appear to have a similar lattice structure and fine-morphology. The lattice spacings of the Ectothiorhodospira haloohloris, Ectothiorhodospira abdelmalekii and Rhodopseudomonas viridis membranes are close to 13 nm, those of Thiocapsa pfennigii and Rhodopseudomonas sulfoviridis are slightly smaller (∼12.5 nm).

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