A cluster expansion approach to exponential random graph models

The exponential family of random graphs is among the most widely-studied network models. We show that any exponential random graph model may alternatively be viewed as a lattice gas model with a finite Banach space norm. The system may then be treated by cluster expansion methods from statistical mechanics. In particular, we derive a convergent power series expansion for the limiting free energy in the case of small parameters. Since the free energy is the generating function for the expectations of other random variables, this characterizes the structure and behavior of the limiting network in this parameter region.Comment: 15 pages, 1 figur

Modelling quasicrystals at positive temperature

We consider a two-dimensional lattice model of equilibrium statistical mechanics, using nearest neighbor interactions based on the matching conditions for an aperiodic set of 16 Wang tiles. This model has uncountably many ground state configurations, all of which are nonperiodic. The question addressed in this paper is whether nonperiodicity persists at low but positive temperature. We present arguments, mostly numerical, that this is indeed the case. In particular, we define an appropriate order parameter, prove that it is identically zero at high temperatures, and show by Monte Carlo simulation that it is nonzero at low temperatures

Two-point correlation properties of stochastic "cloud processes''

We study how the two-point density correlation properties of a point particle distribution are modified when each particle is divided, by a stochastic process, into an equal number of identical "daughter" particles. We consider generically that there may be non-trivial correlations in the displacement fields describing the positions of the different daughters of the same "mother" particle, and then treat separately the cases in which there are, or are not, correlations also between the displacements of daughters belonging to different mothers. For both cases exact formulae are derived relating the structure factor (power spectrum) of the daughter distribution to that of the mother. These results can be considered as a generalization of the analogous equations obtained in ref. [1] (cond-mat/0409594) for the case of stochastic displacement fields applied to particle distributions. An application of the present results is that they give explicit algorithms for generating, starting from regular lattice arrays, stochastic particle distributions with an arbitrarily high degree of large-scale uniformity.Comment: 14 pages, 3 figure

Preparation of 6‐ 3 H glucocerebroside

Glucocerebroside (1–0‐β‐glucosyl ceramide) can be labeled with 3 H‐borohydride at the 6‐position of the glucose moiety. The 6‐trityl ether of cerebroside is formed first, the remaining hydroxyl groups are acetylated, the trityl group is removed, and the free 6‐hydroxyl group is oxidized to an aldehyde. The carbonyl group is then reduced with borohydride and the acetyl groups are removed, regenerating the original glycolipid.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90217/1/2580130309_ftp.pd

Stimulation of liver growth and DNA synthesis by glucosylceramide

The nature of the growth‐stimulating effect of glucosylceramide was studied. Mice were injected intraperitoneally with emulsified glucosylceramide and conduritol B epoxide, an inhibitor of cerebroside glucosidase. Within one or two days, the liver grew 18–24%, as reported. Two enzymes involved in DNA synthesis also increased more than the weight. The total liver activity of thymidine kinase increased 46–73%, and the total activity of ornithine decarboxylase increased as much as 101%. It is suggested that elevated liver levels of glucocerebroside stimulate cell proliferation through a relatively direct mechanism.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141613/1/lipd0508.pd

Stimulation in vitro of galactocerebroside galactosidase by N‐decanoyl 2‐amino‐2‐methylpropanol

Amides resembling ceramide (fatty acyl sphingosine) were synthesized and tested in vitro for their effects on the rat brain β‐galactosidase which hydrolyzes galactosyl ceramide. The N‐decanoyl derivative of 2‐amino‐2‐methyl‐1‐propanol was most effective, giving a 34% stimulation at 0.15 mM concentration and a 60% stimulation at maximal levels. Addition of a hydroxyl group in the 3 position reduced the degree of stimulation, as did increasing or decreasing the length of the fatty acid portion. Omission of the branched methyl group resulted in inhibition instead of stimulation. Kinetic analysis indicates that the stimulator does not affect the binding of substrate to enzyme, but does speed the rate of hydrolytic action. Stimulation was also observed with the cerebrosidase in spleen and kidney. It is suggested that the stimulators act on an enzyme site other than the substrate‐active site.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/141447/1/lipd0056.pd

Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra

The paper investigates how correlations can completely specify a uniformly discrete point process. The setting is that of uniformly discrete point sets in real space for which the corresponding dynamical hull is ergodic. The first result is that all of the essential physical information in such a system is derivable from its $n$-point correlations, $n= 2, 3, >...$. If the system is pure point diffractive an upper bound on the number of correlations required can be derived from the cycle structure of a graph formed from the dynamical and Bragg spectra. In particular, if the diffraction has no extinctions, then the 2 and 3 point correlations contain all the relevant information.Comment: 16 page

Local Complexity of Delone Sets and Crystallinity

This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to X being an ideal crystal. Explicitly, for N(T), if R is the covering radius of X then either N(T) is bounded or N(T) >= T/2R for all T>0. The constant 1/2R in this bound is best possible in all dimensions. For M(T), either M(T) is bounded or M(T) >= T/3 for all T>0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M(T) >= c(n)T for all T>0, for a certain constant c(n) which depends on the dimension n of X and is greater than 1/3 when n > 1.Comment: 26 pages. Uses latexsym and amsfonts package

Crystalline ground states for classical particles

Pair interactions whose Fourier transform is nonnegative and vanishes above a wave number K_0 are shown to give rise to periodic and aperiodic infinite volume ground state configurations (GSCs) in any dimension d. A typical three dimensional example is an interaction of asymptotic form cos(K_0 r)/r^4. The result is obtained for densities rho >= rho_d where rho_1=K_0/2pi, rho_2=(sqrt{3}/8)(K_0/pi)^2 and rho_3=(1/8sqrt{2})(K_0/pi)^3. At rho_d there is a unique periodic GSC which is the uniform chain, the triangular lattice and the bcc lattice for d=1,2,3, respectively. For rho>rho_d the GSC is nonunique and the degeneracy is continuous: Any periodic configuration of density rho with all reciprocal lattice vectors not smaller than K_0, and any union of such configurations, is a GSC. The fcc lattice is a GSC only for rho>=(1/6 sqrt{3})(K_0/pi)^3.Comment: final versio