The Ground State Energy of a Dilute Two-dimensional Bose Gas

The ground state energy per particle of a dilute, homogeneous, two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be $E_0/N = (2\pi \hbar^2\rho /m){|\ln (\rho a^2)|^{-1}}$, to leading order, with a relative error at most ${\rm O} (|\ln (\rho a^2)|^{-1/5})$. Here $N$ is the number of particles, $\rho =N/V$ is the particle density and $a$ is the scattering length of the two-body potential. We assume that the two-body potential is short range and nonnegative. The amusing feature of this result is that, in contrast to the three-dimensional case, the energy, $E_0$ is not simply $N(N-1)/2$ times the energy of two particles in a large box of volume (area, really) $V$. It is much larger

Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-ray CT

Iterative image reconstruction (IIR) with sparsity-exploiting methods, such as total variation (TV) minimization, investigated in compressive sensing (CS) claim potentially large reductions in sampling requirements. Quantifying this claim for computed tomography (CT) is non-trivial, because both full sampling in the discrete-to-discrete imaging model and the reduction in sampling admitted by sparsity-exploiting methods are ill-defined. The present article proposes definitions of full sampling by introducing four sufficient-sampling conditions (SSCs). The SSCs are based on the condition number of the system matrix of a linear imaging model and address invertibility and stability. In the example application of breast CT, the SSCs are used as reference points of full sampling for quantifying the undersampling admitted by reconstruction through TV-minimization. In numerical simulations, factors affecting admissible undersampling are studied. Differences between few-view and few-detector bin reconstruction as well as a relation between object sparsity and admitted undersampling are quantified.Comment: Revised version that was submitted to IEEE Transactions on Medical Imaging on 8/16/201

Patterned Irradiation of YBa_2Cu_3O_(7-x) Thin Films

We present a new experiment on YBa_2Cu_3O_{7-x} (YBCO) thin films using spatially resolved heavy ion irradiation. Structures consisting of a periodic array of strong and weak pinning channels were created with the help of metal masks. The channels formed an angle of +/-45 Deg with respect to the symmetry axis of the photolithographically patterned structures. Investigations of the anisotropic transport properties of these structures were performed. We found striking resemblance to guided vortex motion as it was observed in YBCO single crystals containing an array of unidirected twin boundaries. The use of two additional test bridges allowed to determine in parallel the resistivities of the irradiated and unirradiated parts as well as the respective current-voltage characteristics. These measurements provided the input parameters for a numerical simulation of the potential distribution of the Hall patterning. In contrast to the unidirected twin boundaries in our experiment both strong and weak pinning regions are spatially extended. The interfaces between unirradiated and irradiated regions therefore form a Bose-glass contact. The experimentally observed magnetic field dependence of the transverse voltage vanishes faster than expected from the numerical simulation and we interpret this as a hydrodynamical interaction between a Bose-glass phase and a vortex liquid.Comment: 7 pages, 8 Eps figures included. Submitted to PR

How biased are maximum entropy models?

Maximum entropy models have become popular statistical models in neuroscience and other areas in biology, and can be useful tools for obtaining estimates of mutual information in biological systems. However, maximum entropy models fit to small data sets can be subject to sampling bias; i.e. the true entropy of the data can be severely underestimated. Here we study the sampling properties of estimates of the entropy obtained from maximum entropy models. We show that if the data is generated by a distribution that lies in the model class, the bias is equal to the number of parameters divided by twice the number of observations. However, in practice, the true distribution is usually outside the model class, and we show here that this misspecification can lead to much larger bias. We provide a perturbative approximation of the maximally expected bias when the true model is out of model class, and we illustrate our results using numerical simulations of an Ising model; i.e. the second-order maximum entropy distribution on binary data.