Lattice Green functions in all dimensions

We give a systematic treatment of lattice Green functions (LGF) on the $d$-dimensional diamond, simple cubic, body-centred cubic and face-centred cubic lattices for arbitrary dimensionality $d \ge 2$ for the first three lattices, and for $2 \le d \le 5$ for the hyper-fcc lattice. We show that there is a close connection between the LGF of the $d$-dimensional hypercubic lattice and that of the $(d-1)$-dimensional diamond lattice. We give constant-term formulations of LGFs for all lattices and dimensions. Through a still under-developed connection with Mahler measures, we point out an unexpected connection between the coefficients of the s.c., b.c.c. and diamond LGFs and some Ramanujan-type formulae for $1/\pi.$Comment: 30 page

Room-temperature photon-number-resolved detection using a two-mode squeezer

We study the average intensity-intensity correlations signal at the output of a two-mode squeezing device with |N)⊠- |α) as the two input modes. We show that the input photon number can be resolved from the average intensity-intensity correlations. In particular, we show jumps in the average intensity-intensity correlations signal as a function of input photon number N. Therefore, we propose that such a device may be deployed as a photon-number-resolving detector at room temperature with high efficiency

Exact results for some Madelung type constants in the finite-size scaling theory

A general formula is obtained from which the madelung type constant: $$C(d|\nu)=\int_0^\infty dx x^{d/2-\nu-1}[(\sum_{l=-\infty}^\infty e^{-xl^2})^d-1-(\frac\pi x)^{d/2}]$$ extensively used in the finite-size scaling theory is computed analytically for some particular cases of the parameters $d$ and $\nu$. By adjusting these parameters one can obtain different physical situations corresponding to different geometries and magnitudes of the interparticle interaction.Comment: IOP- macros, 5 pages, replaced with amended version (1 ref. added

Smooth Paths on Three Dimensional Lattice

A particular class of random walks with a spin factor on a three dimensional cubic lattice is studied. This three dimensional random walk model is a simple generalization of random walk for the two dimensional Ising model. All critical diffusion constants and associated critical exponents are calculated. Continuum field theories such as Klein-Gordon, Dirac and massive Chern-Simons theories are constructed near several critical points.Comment: 7 pages,NUP-A-94-

Uniform tiling with electrical resistors

The electric resistance between two arbitrary nodes on any infinite lattice structure of resistors that is a periodic tiling of space is obtained. Our general approach is based on the lattice Green's function of the Laplacian matrix associated with the network. We present several non-trivial examples to show how efficient our method is. Deriving explicit resistance formulas it is shown that the Kagom\'e, the diced and the decorated lattice can be mapped to the triangular and square lattice of resistors. Our work can be extended to the random walk problem or to electron dynamics in condensed matter physics.Comment: 22 pages, 14 figure

Spanning tree generating functions and Mahler measures

We define the notion of a spanning tree generating function (STGF) $\sum a_n z^n$, which gives the spanning tree constant when evaluated at $z=1,$ and gives the lattice Green function (LGF) when differentiated. By making use of known results for logarithmic Mahler measures of certain Laurent polynomials, and proving new results, we express the STGFs as hypergeometric functions for all regular two and three dimensional lattices (and one higher-dimensional lattice). This gives closed form expressions for the spanning tree constants for all such lattices, which were previously largely unknown in all but one three-dimensional case. We show for all lattices that these can also be represented as Dirichlet $L$-series. Making the connection between spanning tree generating functions and lattice Green functions produces integral identities and hypergeometric connections, some of which appear to be new.Comment: 26 pages. Dedicated to F Y Wu on the occasion of his 80th birthday. This version has additional references, additional calculations, and minor correction

Informatics and data mining tools and strategies for the Human Connectome Project

The Human Connectome Project (HCP) is a major endeavor that will acquire and analyze connectivity data plus other neuroimaging, behavioral, and genetic data from 1,200 healthy adults. It will serve as a key resource for the neuroscience research community, enabling discoveries of how the brain is wired and how it functions in different individuals. To fulfill its potential, the HCP consortium is developing an informatics platform that will handle: 1) storage of primary and processed data, 2) systematic processing and analysis of the data, 3) open access data sharing, and 4) mining and exploration of the data. This informatics platform will include two primary components. ConnectomeDB will provide database services for storing and distributing the data, as well as data analysis pipelines. Connectome Workbench will provide visualization and exploration capabilities. The platform will be based on standard data formats and provide an open set of application programming interfaces (APIs) that will facilitate broad utilization of the data and integration of HCP services into a variety of external applications. Primary and processed data generated by the HCP will be openly shared with the scientific community, and the informatics platform will be available under an open source license. This paper describes the HCP informatics platform as currently envisioned and places it into the context of the overall HCP vision and agenda