Statistical Methods in Topological Data Analysis for Complex, High-Dimensional Data

The utilization of statistical methods an their applications within the new field of study known as Topological Data Analysis has has tremendous potential for broadening our exploration and understanding of complex, high-dimensional data spaces. This paper provides an introductory overview of the mathematical underpinnings of Topological Data Analysis, the workflow to convert samples of data to topological summary statistics, and some of the statistical methods developed for performing inference on these topological summary statistics. The intention of this non-technical overview is to motivate statisticians who are interested in learning more about the subject.Comment: 15 pages, 7 Figures, 27th Annual Conference on Applied Statistics in Agricultur

On the relevance of numerical simulations to booming sand

We have performed a simulation study of 3D cohesionless granular flows down an inclined chute. We find that the oscillations observed in [L.E. Silbert, Phys. Rev. Lett., 94, 098002 (2005)] near the angle of repose are harmonic vibrations of the lowest normal mode. Their frequencies depend on the contact stiffness as well as on the depth of the flow. Could these oscillations account for the phenomena of "booming sand"? We estimate an effective contact stiffness from the Hertz law, but this leads to frequencies several times higher than observed. However, the Hertz law also predicts interpenetrations of a few nanometers, indicating that the oscillations frequencies are governed by the surface stiffness, which can be much lower than the bulk one. This is in agreement with previous studies ascribing the ability to sing to the presence of a soft coating on the grain surface.Comment: accepted for publication in Physical Review E http://pre.aps.org; Physical Review E (2012) to be publishe

U(1) Gauge Theory of the Hubbard Model : Spin Liquid States and Possible Application to k-(BEDT-TTF)_2 Cu_2 (CN)_3

We formulate a U(1) gauge theory of the Hubbard model in the slave-rotor representation. From this formalism it is argued that spin liquid phases may exist near the Mott transition in the Hubbard model on triangular and honeycomb lattices at half filling. The organic compound k-(BEDT-TTF)_2 Cu_2 (CN)_3 is a good candidate for the spin liquid state on a triangular lattice. We predict a highly unusual temperature dependence for the thermal conductivity of this material.Comment: 5 pages, 2 figures; paper shortened and the phase diagram of anisotropic triangular lattice correcte

Random quantum codes from Gaussian ensembles and an uncertainty relation

Using random Gaussian vectors and an information-uncertainty relation, we give a proof that the coherent information is an achievable rate for entanglement transmission through a noisy quantum channel. The codes are random subspaces selected according to the Haar measure, but distorted as a function of the sender's input density operator. Using large deviations techniques, we show that classical data transmitted in either of two Fourier-conjugate bases for the coding subspace can be decoded with low probability of error. A recently discovered information-uncertainty relation then implies that the quantum mutual information for entanglement encoded into the subspace and transmitted through the channel will be high. The monogamy of quantum correlations finally implies that the environment of the channel cannot be significantly coupled to the entanglement, and concluding, which ensures the existence of a decoding by the receiver.Comment: 9 pages, two-column style. This paper is a companion to quant-ph/0702005 and quant-ph/070200

Deriving the Hirshfeld partitioning using distance metrics

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