A qq-Queens Problem. II. The Square Board

We apply to the $n\times n$ chessboard the counting theory from Part I for nonattacking placements of chess pieces with unbounded straight-line moves, such as the queen. Part I showed that the number of ways to place $q$ identical nonattacking pieces is given by a quasipolynomial function of $n$ of degree $2q$, whose coefficients are (essentially) polynomials in $q$ that depend cyclically on $n$. Here we study the periods of the quasipolynomial and its coefficients, which are bounded by functions, not well understood, of the piece's move directions, and we develop exact formulas for the very highest coefficients. The coefficients of the three highest powers of $n$ do not vary with $n$. On the other hand, we present simple pieces for which the fourth coefficient varies periodically. We develop detailed properties of counting quasipolynomials that will be applied in sequels to partial queens, whose moves are subsets of those of the queen, and the nightrider, whose moves are extended knight's moves. We conclude with the first, though strange, formula for the classical $n$-Queens Problem and with several conjectures and open problems.Comment: 23 pp., 1 figure, submitted. This = second half of 1303.1879v1 with great improvements. V2 has a new proposition, better definitions, and corrected conjectures. V3 has results et al. renumbered to correspond with published version, and expands dictionary's cryptic abbreviation

Fast spectral source integration in black hole perturbation calculations

This paper presents a new technique for achieving spectral accuracy and fast computational performance in a class of black hole perturbation and gravitational self-force calculations involving extreme mass ratios and generic orbits. Called \emph{spectral source integration} (SSI), this method should see widespread future use in problems that entail (i) point-particle description of the small compact object, (ii) frequency domain decomposition, and (iii) use of the background eccentric geodesic motion. Frequency domain approaches are widely used in both perturbation theory flux-balance calculations and in local gravitational self-force calculations. Recent self-force calculations in Lorenz gauge, using the frequency domain and method of extended homogeneous solutions, have been able to accurately reach eccentricities as high as $e \simeq 0.7$. We show here SSI successfully applied to Lorenz gauge. In a double precision Lorenz gauge code, SSI enhances the accuracy of results and makes a factor of three improvement in the overall speed. The primary initial application of SSI--for us its \emph{raison d'\^{e}tre}--is in an arbitrary precision \emph{Mathematica} code that computes perturbations of eccentric orbits in the Regge-Wheeler gauge to extraordinarily high accuracy (e.g., 200 decimal places). These high accuracy eccentric orbit calculations would not be possible without the exponential convergence of SSI. We believe the method will extend to work for inspirals on Kerr, and will be the subject of a later publication. SSI borrows concepts from discrete-time signal processing and is used to calculate the mode normalization coefficients in perturbation theory via sums over modest numbers of points around an orbit. A variant of the idea is used to obtain spectral accuracy in solution of the geodesic orbital motion.Comment: 15 pages, 7 figure

Quantum Hopfield neural network

Quantum computing allows for the potential of significant advancements in both the speed and the capacity of widely used machine learning techniques. Here we employ quantum algorithms for the Hopfield network, which can be used for pattern recognition, reconstruction, and optimization as a realization of a content-addressable memory system. We show that an exponentially large network can be stored in a polynomial number of quantum bits by encoding the network into the amplitudes of quantum states. By introducing a classical technique for operating the Hopfield network, we can leverage quantum algorithms to obtain a quantum computational complexity that is logarithmic in the dimension of the data. We also present an application of our method as a genetic sequence recognizer.Comment: 13 pages, 3 figures, final versio

Continuous-variable quantum neural networks

We introduce a general method for building neural networks on quantum computers. The quantum neural network is a variational quantum circuit built in the continuous-variable (CV) architecture, which encodes quantum information in continuous degrees of freedom such as the amplitudes of the electromagnetic field. This circuit contains a layered structure of continuously parameterized gates which is universal for CV quantum computation. Affine transformations and nonlinear activation functions, two key elements in neural networks, are enacted in the quantum network using Gaussian and non-Gaussian gates, respectively. The non-Gaussian gates provide both the nonlinearity and the universality of the model. Due to the structure of the CV model, the CV quantum neural network can encode highly nonlinear transformations while remaining completely unitary. We show how a classical network can be embedded into the quantum formalism and propose quantum versions of various specialized model such as convolutional, recurrent, and residual networks. Finally, we present numerous modeling experiments built with the Strawberry Fields software library. These experiments, including a classifier for fraud detection, a network which generates Tetris images, and a hybrid classical-quantum autoencoder, demonstrate the capability and adaptability of CV quantum neural networks

Intra and Inter-Neighborhood Income Inequality and Crime

Author's manuscript made available in accordance with the publisher's policy.One important factor in many macro-level theories of crime is income inequality. Although research generally shows that low levels of neighborhood income are associated with crime, research studies have been less clear on whether income inequality is a robust, independent predictor of crime, particularly in small area studies, and few studies have explicitly considered income inequality between neighborhoods, and those that do typically focus on homicide. The current study examines whether within and between neighborhood income inequality is associated with variation in violent and property crime. We employ geocoded Uniform Crime Report data from the Indianapolis police department and economic and demographic characteristics of the population from the American Community Survey for 2005–2009. Consistent with prior research, lower levels of income were associated with higher violent and property crime counts. Within-tract income inequality was also associated with higher Uniform Crime Reports violent and property crimes in most models. Results also showed that the ratio of tract income levels to neighboring tracts is associated with variation in crime. Thus, both local and nearby income inequality affect crime. Implications for theory and policy are discussed

The Effect of Foreclosures on Crime in Indianapolis, 2003-2008

Author's manuscript made available in accordance with the publisher's policy.Objective Until recently, few studies have examined the relationship between home foreclosures and crime. Foreclosures are one major source of neighborhood instability and can be expected to affect crime from several theoretical perspectives. Some recent research has found conflicting results on whether foreclosures cause crime. Method This study examines whether foreclosures are a robust predictor of crime and whether the effect of foreclosures on crime varies across neighborhood contexts. We estimate fixed-effects negative binomial models using geocoded Indianapolis foreclosure and crime data for 2003–2008 to predict crime counts in 1,000 feet × 1,000 feet square grid cells. Result Foreclosures exhibit consistent positive effects on indices of overall, property, and violent UCR-reported (where UCR is Uniform Crime Report) offenses in a cell and rape, aggravated assault, and burglary counts. In addition, foreclosures had greater effects on reported UCR crimes in stable neighborhoods, especially those with more owner-occupied homes. Conclusion Foreclosures were a robust predictor of crime in the current study