Pions: Experimental Tests of Chiral Symmetry Breaking

Based on the spontaneous breaking of chiral symmetry, chiral perturbation theory (ChPT) is believed to approximate confinement scale QCD. Dedicated and increasingly accurate experiments and improving lattice calculations are confirming this belief, and we are entering a new era in which we can test confinement scale QCD in some well chosen reactions. This is demonstrated with an overview of low energy experimental tests of ChPT predictions of $\pi\pi$ scattering, pion properties, $\pi$N scattering and electromagnetic pion production. These predictions have been shown to be consistent with QCD in the meson sector by increasingly accurate lattice calculations. At present there is good agreement between experiment and ChPT calculations, including the $\pi\pi$ and $\pi$N s wave scattering lengths and the $\pi^{0}$ lifetime. Recent, accurate pionic atom data are in agreement with chiral calculations once isospin breaking effects due to the mass difference of the up and down quarks are taken into account, as was required to extract the $\pi\pi$ scattering lengths. In addition to tests of the theory, comparisons between $\pi\pi$ and $\pi$N interactions based on general chiral principles are discussed. Lattice calculations are now providing results for the fundamental, long and inconclusively studied, $\pi$N $\sigma$ term and the contribution of the strange quark to the mass of the proton. Increasingly accurate experiments in electromagnetic pion production experiments from the proton which test ChPT calculations (and their energy region of validity) are presented. These experiments are also beginning to measure the final state $\pi$N interaction. This paper is based on the concluding remarks made at the Chiral Dynamics Workshop CD12 held at Jefferson Lab in Aug. 2012.Comment: 13 pages, 8 fig

Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization

We study dynamic $(1+\epsilon)$-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected $n$-node $m$-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of $\tilde O(mn/\epsilon)$ and constant query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total update time of $O(mn^2)$ and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of $\tilde O(n^{5/2}/\epsilon)$ and constant query time that has an additive error of $2$ in addition to the $1+\epsilon$ multiplicative error. This beats the previous $\tilde O(mn/\epsilon)$ time when $m=\Omega(n^{3/2})$. Note that the additive error is unavoidable since, even in the static case, an $O(n^{3-\delta})$-time (a so-called truly subcubic) combinatorial algorithm with $1+\epsilon$ multiplicative error cannot have an additive error less than $2-\epsilon$, unless we make a major breakthrough for Boolean matrix multiplication [Dor et al. FOCS 1996] and many other long-standing problems [Vassilevska Williams and Williams FOCS 2010]. The algorithm can also be turned into a $(2+\epsilon)$-approximation algorithm (without an additive error) with the same time guarantees, improving the recent $(3+\epsilon)$-approximation algorithm with $\tilde O(n^{5/2+O(\sqrt{\log{(1/\epsilon)}/\log n})})$ running time of Bernstein and Roditty [SODA 2011] in terms of both approximation and time guarantees. (2) We present a deterministic algorithm with a total update time of $\tilde O(mn/\epsilon)$ and a query time of $O(\log\log n)$. The algorithm has a multiplicative error of $1+\epsilon$ and gives the first improved deterministic algorithm since 1981. It also answers an open question raised by Bernstein [STOC 2013].Comment: A preliminary version was presented at the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS 2013

Resampling images in Fourier domain

When simulating sky images, one often takes a galaxy image $F(x)$ defined by a set of pixelized samples and an interpolation kernel, and then wants to produce a new sampled image representing this galaxy as it would appear with a different point-spread function, a rotation, shearing, or magnification, and/or a different pixel scale. These operations are sometimes only possible, or most efficiently executed, as resamplings of the Fourier transform $\tilde F(u)$ of the image onto a $u$-space grid that differs from the one produced by a discrete Fourier transform (DFT) of the samples. In some applications it is essential that the resampled image be accurate to better than 1 part in $10^3$, so in this paper we first use standard Fourier techniques to show that Fourier-domain interpolation with a wrapped sinc function yields the exact value of $\tilde F(u)$ in terms of the input samples and kernel. This operation scales with image dimension as $N^4$ and can be prohibitively slow, so we next investigate the errors accrued from approximating the sinc function with a compact kernel. We show that these approximations produce a multiplicative error plus a pair of ghost images (in each dimension) in the simulated image. Standard Lanczos or cubic interpolators, when applied in Fourier domain, produce unacceptable artifacts. We find that errors $<1$ part in $10^3$ can be obtained by (1) 4-fold zero-padding of the original image before executing the $x\rightarrow u$ DFT, followed by (2) resampling to the desired $u$ grid using a 6-point, piecewise-quintic interpolant that we design expressly to minimize the ghosts, then (3) executing the DFT back to $x$ domain.Comment: Typographical and one algebraic correction, to appear in PASP March 201

Detectability of CMB tensor B modes via delensing with weak lensing galaxy surveys

We analyze the possibility of delensing CMB polarization maps using foreground weak lensing (WL) information. We build an estimator of the CMB lensing potential out of optimally combined projected potential estimators to different source redshift bins. Our estimator is most sensitive to the redshift depth of the WL survey, less so to the shape noise level. Estimators built using galaxy surveys like LSST and SNAP yield a 30-50% reduction in the lensing B-mode power. We illustrate the potential advantages of a 21-cm survey by considering a fiducial WL survey for which we take the redshift depth zmax and the effective angular concentration of sources n as free parameters. For a noise level of 1 muK arcmin in the polarization map itself, as projected for a CMBPol experiment, and a beam with FWHM=10 arcmin, we find that going to zmax=20 at n=100 gal/sqarcmin yields a delensing performance similar to that of a quadratic lensing potential estimator applied to small-scale CMB maps: the lensing B-mode contamination is reduced by almost an order of magnitude. In this case, there is also a reduction by a factor of ~4 in the detectability threshold of the tensor B-mode power. At this CMB noise level, there is little gain from sources with zmax>20. The delensing gains are lost if the CMB beam exceeds ~20 arcmin. The delensing efficiency and useful zmax depend acutely on the CMB map noise level, but beam sizes below 10 arcmin do not help. Delensing via foreground sources does not require arcminute-resolution CMB observations, a substantial practical advantage over the use of CMB observables for delensing.Comment: 10 pages, 5 figures; accepted for publication in Physical Review

The Amorphous-Crystal Interface in Silicon: a Tight-Binding Simulation

The structural features of the interface between the cystalline and amorphous phases of Si solid are studied in simulations based on a combination of empirical interatomic potentials and a nonorthogonal tight-binding model. The tight-binding Hamiltonian was created and tested for the types of structures and distortions anticipated to occur at this interface. The simulations indicate the presence of a number of interesting features near the interface. The features that may lead to crystallization upon heating include chains with some defects, most prominently dimers similar to those on the Si(001) 2x1 reconstructed free surface. Within the amorphous region order is lost over very short distances. By examining six different samples with two interfaces each, we find the energy of the amorphous-crystal interface to be 0.49 +/- 0.05 J/m^2Comment: Submitted to Phys. Rev.

Corporate Taxes and Incentives and the Structure of Production: A Selected Survey

In this paper we develop a general intertemporal model of production, emphasizing the role of present and expected future corporate income taxes, credits and allowances along with costly adjustment and variable utilization of the quasi-fixed factors. Three specific issues are considered: 1) the direct and indirect effects of taxes operating through factor prices on the long-run input substitution, thus altering the structure of the production process; 2) the effects of tax policy changes on the rate and direction of technological change; and 3) the effects of tax policy on the inter- temporal pattern of substitutions and complementarities among the inputs that arise due to presence of quasi-fixity of some inputs. The rates of utilization of the quasi-fixed factors are determined in the short-run in conjunction with the demands for the variable factors of production. Hence, utilization rates depend on product and factor prices and therefore on tax policy. We specialize the general model in order to highlight each of the three themes and their interaction with tax policy. We also discuss the various ways in which empirical implementation of the theoretical models and a brief summary of the empirical results in the literature is also provided. Lastly, we discuss some policy implications which emerge from the analysis and empirical results.