English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We work with the following expression for the entropy (density) of a dimergas on an infinite r-regular lattice lambda(p) = 1/2 [pln(r)-ln(p)-2(1-p)ln(1-p)-p ]+sum_{k=2}(d_k)(p^k) where the indicated sumconverges for density, p, small enough. Pernici has computed the coefficientsd_k for k < 13. He found these d_k to be polynomials in certain interesting"geometric quantites" arising in the work of Wanless. Each of these quantitiesis the number density of isomorphic mappings of some graph into the lattice(graph). So for a bipartite lattice d_2 = c_2 d_3 = c_3 d_4 = c_4 + c_5hat{G}_1 d_5 = c_6 + c_7 hat{G}_1. The c_i depend only on r. Here hat{G}_1 isthe density of mapping classes of the four loop graph into the lattice. Thelimit of 1/V times the number of such mapping classes into a lattice of volumeV as V goes to infinity. The infinite volume limit. There is a simple linearrelation that yields the kth virial coefficient from the value of d_k! We feelthis expression gives the deepest insight into the virial coefficients so farobtained.  What we show in this paper is that such polynomial relations for the d_k inthese geometric quantities holds for the d_k for k < 28. Of course we expect itto hold for all k. We use the same computation procedure as Pernici. We notethis procedure is not rigorously established. So far a procedure for thephysicist, perhaps not the mathematician (their loss). It is a worthy challengefor the mathematical physicist to supply the needed rigor.

On the Pernici-Wanless Expansion for the Entropy ( and Virial Coefficients ) of a Dimer Gas on an Infinite Regular Lattice