Exact asymptotics of monomer-dimer model on rectangular semi-infinite lattices

By using the asymptotic theory of Pemantle and Wilson, exact asymptotic expansions of the free energy of the monomer-dimer model on rectangular $n \times \infty$ lattices in terms of dimer density are obtained for small values of $n$, at both high and low dimer density limits. In the high dimer density limit, the theoretical results confirm the dependence of the free energy on the parity of $n$, a result obtained previously by computational methods. In the low dimer density limit, the free energy on a cylinder $n \times \infty$ lattice strip has exactly the same first $n$ terms in the series expansion as that of infinite $\infty \times \infty$ lattice.Comment: 9 pages, 6 table

Karl Mannheim and Jean Floud: a false start for the sociology of education in Britain?

Arriving in the UK after exile from Nazi Germany, Karl Mannheim taught sociology at the London School of Economics and then also at the London Institute of Education, where he was awarded a chair just a year before his untimely death in 1947. In his later writings and teaching, Mannheim argued that the sociology of education could make a crucial contribution to the new type of society he regarded as essential if the problems of liberal democracy were to be overcome, and the slide towards totalitarianism avoided. And the period immediately after his death was a key phase in the development and establishment of the sociology of education in Britain. Jean Floud, who took over teaching the subject at the Institute of Education after Mannheim’s death, played a central role in this, but, while she had studied with him and served as his research assistant, she adopted a very different approach. This focused, in particular, on whether the existing structure and operation of educational institutions restricted social mobility. As a result of this change in focus, Mannheim’s work had a very marginal role in the subsequent history of British sociology of education. In this article, I compare Mannheim’s and Floud’s competing conceptions of the character and role of the subdiscipline, and how these fared in later developments within the field

Karl Mannheim and Jean Floud: a false start for the sociology of education in Britain?

Arriving in the UK after exile from Nazi Germany, Karl Mannheim taught sociology at the London School of Economics and then also at the London Institute of Education, where he was awarded a chair just a year before his untimely death in 1947. In his later writings and teaching, Mannheim argued that the sociology of education could make a crucial contribution to the new type of society he regarded as essential if the problems of liberal democracy were to be overcome, and the slide towards totalitarianism avoided. And the period immediately after his death was a key phase in the development and establishment of the sociology of education in Britain. Jean Floud, who took over teaching the subject at the Institute of Education after Mannheim’s death, played a central role in this, but, while she had studied with him and served as his research assistant, she adopted a very different approach. This focused, in particular, on whether the existing structure and operation of educational institutions restricted social mobility. As a result of this change in focus, Mannheim’s work had a very marginal role in the subsequent history of British sociology of education. In this article, I compare Mannheim’s and Floud’s competing conceptions of the character and role of the subdiscipline, and how these fared in later developments within the field

Old stellar Galactic disc in near-plane regions according to 2MASS: scales, cut-off, flare and warp

We have pursued two different methods to analyze the old stellar population near the Galactic plane, using data from the 2MASS survey. The first method is based on the isolation of the red clump giant population in the color-magnitude diagrams and the inversion of its star counts to obtain directly the density distribution along the line of sight. The second method fits the parameters of a disc model to the star counts in 820 regions. Results from both independent methods are consistent with each other. The qualitative conclusions are that the disc is well fitted by an exponential distribution in both the galactocentric distance and height. There is not an abrupt cut-off in the stellar disc (at least within R<15 kpc). There is a strong flare (i.e. an increase of scale-height towards the outer Galaxy) which begins well inside the solar circle, and hence there is a decrease of the scale-height towards the inner Galaxy. Another notable feature is the existence of a warp in the old stellar population whose amplitude is coincident with the amplitude of the gas warp. It is shown for low latitude stars (mean height: |z|~300 pc) in the outer disc (galactocentric radius R>6 kpc) that: the scale-height in the solar circle is h_z(R_sun)=3.6e-2 R_sun, the scale-length of the surface density is h_R=0.42 R_sun and the scale-length of the space density in the plane (i.e. including the effect of the flare) is H=0.25 R_sun. The variation of the scale-height due to the flare follows roughly a law h_z(R) =~ h_z(R_sun) exp [(R-R_\odot)/([12-0.6R(kpc)] kpc)] (for R<~15 kpc; R_sun=7.9 kpc). The warp moves the mean position of the disc to a height z_w=1.2e-3 R(kpc)^5.25 sin(phi+(5 deg.)) pc (for R<~13 kpc; R_sun=7.9 kpc).Comment: LaTEX, 20 pages, 23 figures, accepted to be published in A&

Detection of the old stellar component of the major Galactic bar

We present near-IR colour--magnitude diagrams and star counts for a number of regions along the Galactic plane. It is shown that along the l=27 b=0 line of sight there is a feature at 5.7 +-0.7kpc with a density of stars at least a factor two and probably more than a factor five times that of the disc at the same position. This feature forms a distinct clump on an H vs. J-H diagram and is seen at all longitudes from the bulge to about l=28, but at no longitude greater than this. The distance to the feature at l=20 is about 0.5kpc further than at l=27 and by l=10 it has merged with, or has become, the bulge. Given that at l=27 and l=21 there is also a clustering of very young stars, the only component that can reasonably explain what is seen is a bar with half length of around 4kpc and a position angle of about 43+-7.Comment: 5 pages, 5 figures accepted as a letter in MNRA

Monomer-dimer model in two-dimensional rectangular lattices with fixed dimer density

The classical monomer-dimer model in two-dimensional lattices has been shown to belong to the \emph{``#P-complete''} class, which indicates the problem is computationally ``intractable''. We use exact computational method to investigate the number of ways to arrange dimers on $m \times n$ two-dimensional rectangular lattice strips with fixed dimer density $\rho$. For any dimer density $0 < \rho < 1$, we find a logarithmic correction term in the finite-size correction of the free energy per lattice site. The coefficient of the logarithmic correction term is exactly -1/2. This logarithmic correction term is explained by the newly developed asymptotic theory of Pemantle and Wilson. The sequence of the free energy of lattice strips with cylinder boundary condition converges so fast that very accurate free energy $f_2(\rho)$ for large lattices can be obtained. For example, for a half-filled lattice, $f_2(1/2) = 0.633195588930$, while $f_2(1/4) = 0.4413453753046$ and $f_2(3/4) = 0.64039026$. For $\rho < 0.65$, $f_2(\rho)$ is accurate at least to 10 decimal digits. The function $f_2(\rho)$ reaches the maximum value $f_2(\rho^*) = 0.662798972834$ at $\rho^* = 0.6381231$, with 11 correct digits. This is also the \md constant for two-dimensional rectangular lattices. The asymptotic expressions of free energy near close packing are investigated for finite and infinite lattice widths. For lattices with finite width, dependence on the parity of the lattice width is found. For infinite lattices, the data support the functional form obtained previously through series expansions.Comment: 15 pages, 5 figures, 5 table

Inversion of stellar statistics equation for the Galactic Bulge

A method based on Lucy (1974, AJ 79, 745) iterative algorithm is developed to invert the equation of stellar statistics for the Galactic bulge and is then applied to the K-band star counts from the Two-Micron Galactic Survey in a number of off-plane regions (10 deg.>|b|>2 deg., |l|<15 deg.). The top end of the K-band luminosity function is derived and the morphology of the stellar density function is fitted to triaxial ellipsoids, assuming a non-variable luminosity function within the bulge. The results, which have already been outlined by Lopez-Corredoira et al.(1997, MNRAS 292, L15), are shown in this paper with a full explanation of the steps of the inversion: the luminosity function shows a sharp decrease brighter than M_K=-8.0 mag when compared with the disc population; the bulge fits triaxial ellipsoids with the major axis in the Galactic plane at an angle with the line of sight to the Galactic centre of 12 deg. in the first quadrant; the axial ratios are 1:0.54:0.33, and the distance of the Sun from the centre of the triaxial ellipsoid is 7860 pc. The major-minor axial ratio of the ellipsoids is found not to be constant. However, the interpretation of this is controversial. An eccentricity of the true density-ellipsoid gradient and a population gradient are two possible explanations. The best fit for the stellar density, for 1300 pc<t<3000 pc, are calculated for both cases, assuming an ellipsoidal distribution with constant axial ratios, and when K_z is allowed to vary. From these, the total number of bulge stars is ~ 3 10^{10} or ~ 4 10^{10}, respectively.Comment: 19 pages, 23 figures, accepted in MNRA

New Lower Bounds on the Self-Avoiding-Walk Connective Constant

We give an elementary new method for obtaining rigorous lower bounds on the connective constant for self-avoiding walks on the hypercubic lattice $Z^d$. The method is based on loop erasure and restoration, and does not require exact enumeration data. Our bounds are best for high $d$, and in fact agree with the first four terms of the $1/d$ expansion for the connective constant. The bounds are the best to date for dimensions $d \geq 3$, but do not produce good results in two dimensions. For $d=3,4,5,6$, respectively, our lower bound is within 2.4\%, 0.43\%, 0.12\%, 0.044\% of the value estimated by series extrapolation.Comment: 35 pages, 388480 bytes Postscript, NYU-TH-93/02/0