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Observability of Lattice Graphs [article]

Fangqiu Han, Subhash Suri, Xifeng Yan
2015 arXiv   pre-print
This shows an interesting dependence of graph observability on the ratio between the dimension of the lattice and that of the walk.  ...  All of our results extend easily to non-square lattices: given a lattice graph of size N = n_1 * n_2 * ... * n_d, the number of colors for t-observability is Theta (N^(1/t)).  ...  We prove the results for subgraphs of square lattices, but the bounds are easily extended to rectangular lattices: given a lattice graph of size N = n 1 × n 2 × · · · × n d , the number of colors for t-observability  ... 
arXiv:1505.02224v1 fatcat:uittvvigbneiviugnhz6hxx7uu

Diagonalisation of quantum observables on regular lattices and general graphs

Niall Moran, Graham Kells, Jiri Vala
2011 Computer Physics Communications  
This code is capable of constructing and diagonalising the observables for spin 1 2 and spinless fermionic particles with many body interactions on arbitrary graphs using massively parallel distributed  ...  In the study of quantum mechanical systems, exact diagonalisation (ED) methods play an extremely important role. We have developed an ED code named DoQO (Diagonalisation of Quantum Observables).  ...  project nmphy002a "Topological Phases in Quantum Lattice Systems".  ... 
doi:10.1016/j.cpc.2010.12.051 fatcat:2xokqjdxcjbpzmrtw3h4dwxwn4

Minimax and Adaptive Estimation of Covariance Operator for Random Variables Observed on a Lattice Graph

T. Tony Cai, Ming Yuan
2015 Figshare  
In a range of applications including imaging analysis and fMRI studies, random variables are observed on a lattice graph.  ...  The results show that the dimension of the lattice graph significantly affects the optimal rates convergence, often much more so than the dimension of the random variables.  ...  S.5 Proof of Lemma 6 Observe that E   ∑ B∈B d Σ B − Σ B   4 ≤ p 6 E   ∑ B∈B d Σ B − Σ B 4   ≤ p 6 E   ∑ B∈B d ( ∥S B − Σ B ∥ + λ 0 √ s d (B) + log p n ) 4   ≤ Cp 6   E ∑ B∈B d ∥S B − Σ B  ... 
doi:10.6084/m9.figshare.1298112 fatcat:qhnrk5dodbh57na4wd35wuyw5q

Minimax and Adaptive Estimation of Covariance Operator for Random Variables Observed on a Lattice Graph

T. Tony Cai, Ming Yuan
2016 Journal of the American Statistical Association  
In a range of applications including imaging analysis and fMRI studies, random variables are observed on a lattice graph.  ...  The results show that the dimension of the lattice graph significantly affects the optimal rates convergence, often much more so than the dimension of the random variables.  ...  lattice graph.  ... 
doi:10.1080/01621459.2014.1001067 fatcat:ednwsuqvnbe5bhlbdkn4ipmp4e

A Graph Lattice Approach to Maintaining Dense Collections of Subgraphs as Image Features

Eric Saund
2011 2011 International Conference on Document Analysis and Recognition  
The graph lattice enables methods for adaptively growing a feature space of subgraphs tailored to observed document genres.  ...  This paper shows how large families of complex features can be built out of simpler ones through construction of a graph lattice-a hierarchy of related subgraphs linked in a lattice.  ...  Thank you to Fang Liu for discussions and for building a preliminary implementation of the graph lattice machinery.  ... 
doi:10.1109/icdar.2011.216 dblp:conf/icdar/Saund11 fatcat:favkvvory5b5xphnqav5dvcrf4

A Graph Lattice Approach to Maintaining and Learning Dense Collections of Subgraphs as Image Features

Eric Saund
2013 IEEE Transactions on Pattern Analysis and Machine Intelligence  
Additionally, the graph lattice enables methods for adaptively growing a feature space of subgraphs tailored to observed data.  ...  Efficiency in exact matching is gained by exploitation of the graph lattice data structure.  ...  Appreciation is also due to Fang Liu for discussions and for building a preliminary implementation of the graph lattice machinery.  ... 
doi:10.1109/tpami.2012.267 pmid:23267200 fatcat:2324ywfd5fgx7lmaylfcsoexji

The relation graphs of finite lattices

Pramada Ramachandran, Teena Liza John
2020 Malaya Journal of Matematik  
We investigate the properties of relation graphs and characterize those lattices whose relation graphs are complete. The association between the relation graphs of isomorphic lattices is studied.  ...  The relation graph of a finite lattice L is defined to be a simple graph with the elements of L as vertices and two distinct vertices are adjacent if and only if they are comparable in L.  ...  Fig. 1 : 1 Lattice and its associated relation graph G(L) Observation 3.2. {0, 1} is the only lattice whose diagraph coincides with its relation graph.  ... 
doi:10.26637/mjm0804/0147 fatcat:vbdlxhw5mzadxpvqc2kee2y3ka

Geometric properties of the three-dimensional Ising andXYmodels

Frank Winter, Wolfhard Janke, Adriaan M. J. Schakel
2008 Physical Review E  
The fractal structure of high-temperature graphs of the three-dimensional Ising and XY models is investigated by simulating these graphs directly on a cubic lattice and analyzing them with the help of  ...  percolation observables.  ...  Observables The HT graphs are analyzed with the help of standard percolation observables ͓10͔. An important characteristic is whether a graph spans the lattice or not.  ... 
doi:10.1103/physreve.77.061108 pmid:18643218 fatcat:di52byywiff4jn4byiu4nphhzu

Protein Evolution within a Structural Space

Eric J. Deeds, Nikolay V. Dokholyan, Eugene I. Shakhnovich
2003 Biophysical Journal  
We develop a divergent evolution model based on the lattice space which produces graphs that, within certain parameter regimes, recapitulate the scale-free behavior observed in similar graphs of real protein  ...  We use this measure of structural relatedness to create a graph of lattice structures and compare this graph (in which nodes are lattice structures and edges are defined using structural similarity) to  ...  Given that such distributions can also be observed in random graphs (Dokholyan et al., 2002) , we desired to identify corresponding ''families'' of lattice structures.  ... 
doi:10.1016/s0006-3495(03)74716-x pmid:14581198 pmcid:PMC1303574 fatcat:vrvh5jjr7jfzbffqklyk323rfe

Distributive Lattice Polymorphism on Reflexive Graphs [article]

Mark Siggers
2014 arXiv   pre-print
In this paper we give two characterisations of the class of reflexive graphs admitting distributive lattice polymorphisms and use these characterisations to address the problem of recognition: for a reflexive  ...  graph G in which no two vertices have the same neighbourhood, we find a polynomial time algorithm to decide if G admits a distributive lattice polymorphism.  ...  PI Characterisation of DL Graphs We start with some observations relating lattice homomorphisms and graph homomorphisms. The first is clear.  ... 
arXiv:1411.7879v1 fatcat:cztb2gkfife3rb5w5qgcxqtxki

General Fifth M- Zagreb Polynomials of the TUC4C8(R)[p, q] 2D-Lattice and its Derived Graphs

2020 Letters in Applied NanoBioScience  
A molecular graph or a chemical graph is a graph related to the structure of a chemical compound.  ...  , its subdivision, and para-line graphs.  ...  The authors would also like to thank the referees for their thoughtful suggestions for the improvement of the article. Conflicts of Interest The authors declare no conflict of interest.  ... 
doi:10.33263/lianbs101.17381747 fatcat:lpjb2736o5a55lcn4jdqxnvwna

The Z_6-symmetric model partition function on triangular lattice

Nor Sakinah Mohd Manshur, Siti Fatimah Zakaria, Nasir Ganikhodjaev
2020 Malaysian Journal of Fundamental and Applied Sciences  
Q = 6 on a triangular lattice. Mainly, the model is defined on the triangular lattice graph with the nearest neighbour interaction.  ...  There is a study on a square lattice that can predict the existence of multiple phase transitions on a complex plane.  ...  From our result on the zeros distribution, this behaviour is already observed. To support our claim and observation, we present the graph of the specific heat for our model.  ... 
doi:10.11113/mjfas.v16n3.1722 fatcat:swh7hspusze6xjx2fbff2i2fvm

Numerical evidence against a conjecture on the cover time of planar graphs

J. Ricardo G. Mendonça
2011 Physical Review E  
We investigate a conjecture on the cover times of planar graphs by means of large Monte Carlo simulations.  ...  The conjecture states that the cover time τ(G_N) of a planar graph G_N of N vertices and maximal degree d is lower bounded by τ(G_N) ≥ C_d N( N)^2 with C_d = (d/4π) tan (π/d), with equality holding for  ...  Other lattices observe the bound, with the square lattice being "borderline."  ... 
doi:10.1103/physreve.84.022103 pmid:21929045 fatcat:vmqnm2n3mjhfzik4wu622d6doy

ダイヤモンドはなぜ美しい?
Why do diamonds look so beautiful?

Toshikazu Sunada
2006 Expositions of Current Mathematics  
The lattice R0 will be said to be the maximal lattice of X. (2) Try to define the d-dimensional diamond lattice, and observe that the hexagonal lattice is the 2-dimensional diamond lattice. 4 Crystal  ...  We may also define the notion of "inner symmetry in terms of automorphisms of the diamond lattice as an abstract graph.  ... 
doi:10.11429/emath1996.2006.spring-meeting_26 fatcat:3ltl7u7z7nertae3mc4hd73dhq

On centrally symmetric graphs

Manfred Stern
1996 Mathematica Bohemica  
I am grateful to the CNRS of France for inviting me to the Laboratoire de Mathematiques Discretes in Marseille, Luminy, during February/March 1994. In particular I thank Professor G.  ...  If the covering graph of a lattice is a CS-graph or an S-graph, then the lattice will be called a CS-lattice or S-lattice, respectively.  ...  Let us observe that a lattice L which is graded and whose dual lattice is strong need not be upper semimodular: just take for L the dual of a nonmodular geometric lattice.  ... 
doi:10.21136/mb.1996.125944 fatcat:ttbb7m624ncnzk2mqv7jhitqui
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