Graphon estimation via nearest‐neighbour algorithm and two‐dimensional fused‐lasso denoising

2021 ◽  
Author(s):  
Oscar Hernan Madrid Padilla ◽  
Yanzhen Chen
Keyword(s):  
Nearest Neighbour ◽  
Two Dimensional ◽  
Fused Lasso

Specific heat of ordered and isotopically disordered lattices

1978 ◽  
Vol 56 (10) ◽  
pp. 1390-1394
Author(s):  
K. P. Srivastava
Keyword(s):  
Specific Heat ◽  
Low Temperatures ◽  
Numerical Study ◽  
Three Dimensional ◽  
Constant Volume ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Appreciable Difference ◽  
Substantial Change ◽  
Neighbour Interaction

An extensive numerical study on specific heat at constant volume (Cv) for ordered and isotopically disordered lattices has been made. Cv at various temperatures for ordered and disordered linear and two-dimensional lattices have been compared and no appreciable difference in Cv between these two structures has been observed. Effect of concentration of light atoms on Cv for three-dimensional isotopically disordered lattices has also been shown.In spite of taking next-nearest-neighbour interaction into account, no substantial change in Cv between the ordered and isotopically disordered linear lattices has been found. It is shown that the low lying modes contribute substantially at low temperatures.

scholarly journals General up to next-nearest-neighbour elasticity of triangular lattices in three dimensions

2006 ◽  
Vol 462 (2073) ◽  
pp. 2695-2713 ◽  
Author(s):  
Cyril Dubus ◽  
Ken Sekimoto ◽  
Jean-Baptiste Fournier
Keyword(s):  
Blood Cell ◽  
Red Blood Cell ◽  
Triangular Lattice ◽  
Soft Matter ◽  
Rotational Symmetry ◽  
Three Dimensions ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Biological Physics ◽  
Crystalline System

We establish the most general form of the discrete elasticity of a two-dimensional triangular lattice embedded in three dimensions, taking into account up to next-nearest-neighbour interactions. Besides crystalline system, this is relevant to biological physics (e.g. red blood cell cytoskeleton) and soft matter (e.g. percolating gels, etc.). In order to correctly impose the rotational invariance of the bulk terms, it turns out to be necessary to take into account explicitly the elasticity associated with the vertices located at the edges of the lattice. We find that some terms that were suspected in the literature to violate rotational symmetry are, in fact, admissible.

The growth of crystals and the equilibrium structure of their surfaces

1951 ◽  
Vol 243 (866) ◽  
pp. 299-358 ◽  
Keyword(s):  
Activation Energy ◽  
Transition Temperature ◽  
Crystallographic Orientation ◽  
Crystal Surface ◽  
Equilibrium Structure ◽  
Surface Melting ◽  
Atomic Scale ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Rate Of Growth

Parts I and II deal with the theory of crystal growth, parts III and IV with the form (on the atomic scale) of a crystal surface in equilibrium with the vapour. In part I we calculate the rate of advance of monomolecular steps (i.e. the edges of incomplete monomolecular layers of the crystal) as a function of supersaturation in the vapour and the mean concentration of kinks in the steps. We show that in most cases of growth from the vapour the rate of advance of monomolecular steps will be independent of their crystallographic orientation, so that a growing closed step will be circular. We also find the rate of advance for parallel sequences of steps. In part II we find the resulting rate of growth and the steepness of the growth cones or growth pyramids when the persistence of steps is due to the presence of dislocations. The cases in which several or many dislocations are involved are analysed in some detail; it is shown that they will commonly differ little from the case of a single dislocation. The rate of growth of a surface containing dislocations is shown to be proportional to the square of the supersaturation for low values and to the first power for high values of the latter. Volmer & Schultze’s (1931) observations on the rate of growth of iodine crystals from the vapour can be explained in this way. The application of the same ideas to growth of crystals from solution is briefly discussed. Part III deals with the equilibrium structure of steps, especially the statistics of kinks in steps, as dependent on temperature, binding energy parameters, and crystallographic orientation. The shape and size of a two-dimensional nucleus (i.e. an ‘island* of new monolayer of crystal on a completed layer) in unstable equilibrium with a given supersaturation at a given temperature is obtained, whence a corrected activation energy for two-dimensional nucleation is evaluated. At moderately low supersaturations this is so large that a crystal would have no observable growth rate. For a crystal face containing two screw dislocations of opposite sense, joined by a step, the activation energy is still very large when their distance apart is less than the diameter of the corresponding critical nucleus; but for any greater separation it is zero. Part IV treats as a ‘co-operative phenomenon’ the temperature dependence of the structure of the surface of a perfect crystal, free from steps at absolute zero. It is shown that such a surface remains practically flat (save for single adsorbed molecules and vacant surface sites) until a transition temperature is reached, at which the roughness of the surface increases very rapidly (‘ surface melting ’). Assuming that the molecules in the surface are all in one or other of two levels, the results of Onsager (1944) for two-dimensional ferromagnets can be applied with little change. The transition temperature is of the order of, or higher than, the melting-point for crystal faces with nearest neighbour interactions in both directions (e.g. (100) faces of simple cubic or (111) or (100) faces of face-centred cubic crystals). When the interactions are of second nearest neighbour type in one direction (e.g. (110) faces of s.c. or f.c.c. crystals), the transition temperature is lower and corresponds to a surface melting of second nearest neighbour bonds. The error introduced by the assumed restriction to two available levels is investigated by a generalization of Bethe’s method (1935) to larger numbers of levels. This method gives an anomalous result for the two-level problem. The calculated transition temperature decreases substantially on going from two to three levels, but remains practically the same for larger numbers.

Statistical mechanics of the two-dimensional assembly

1950 ◽  
Vol 202 (1069) ◽  
pp. 202-207 ◽  
Keyword(s):  
Ising Model ◽  
Curie Temperature ◽  
Statistical Mechanics ◽  
Specific Heat ◽  
Considerable Effect ◽  
Square Lattice ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Triangular Lattices

The work of Kaufman & Onsager (1946) on the two-dimensional Ising model of a ferromagnet is extended from the plane square lattice to the plane honeycomb and triangular lattices. The specific heat anomaly, where it exists, turns out to be of the same type in all three lattices, an infinity in the specific heat at the Curie temperature. It is concluded that second-nearest neighbour interactions may have a considerable effect on the position of the Curie temperature.

Computer analysis of polarized spacing patterns with special reference to the larvae of the black fly, Simulium vittatum

1987 ◽  
Vol 65 (3) ◽  
pp. 602-604 ◽  
Author(s):  
M. Eymann ◽  
J. M. Schmidt ◽  
W. G. Friend
Keyword(s):  
Computer Analysis ◽  
Statistical Significance ◽  
Present Form ◽  
Nearest Neighbour ◽  
Two Dimensional ◽  
Neighbour Distance ◽  
Analysis Techniques ◽  
Simulium Vittatum ◽  
Spacing Pattern ◽  
Average Individual

A computer analysis of spacing patterns which makes fewer assumptions than either nearest-neighbour distance or plot frequency analysis techniques is described. This analysis determines if the spacing pattern is polarized in any direction, and can compensate for the possible effect of the size and shape of the organism on the spacing pattern. The output of the analysis is a two-dimensional frequency distribution which represents the likelihood of finding a neighbour near an average individual. This output can be represented by a two-dimensional map. The method requires a sample size greater than 50, and in its present form cannot assign statistical significance to the resulting probability distribution.

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