The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schr\"odinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.Comment: 36 pages, 6 figures. Revised in light of referees' comments, and further references adde

Circular No. 59 - Control of Stinking Smut of Wheat with Copper Carbonate

Stinking smut or bunt of wheat is an ever-present and destructive disease in the wheat fields of Utah. During the past season (1925) this disease was especially prevalent, causing losses in certain fields of from 25 to 50 per cent, not counting the loss to the grower in reduced grade of grain. In the threshing of smutty wheat there is also the risk of loss from smut explosion. Almost every season cases of this sort are reported. In addition of all of the wheat tested by the U. S. Grain Inspector at Logan for Northern Utah and Southern Idaho 30 per cent showed smut infection in 1925. The average reduction for smut is near ten cents a bushel with a variation from five to twenty cents. The cost of producing a smutted crop may equal or even exceed the cost of producing a clean crop. Loss occurring from this disease, since it is preventable, can hardly be considered attached to the total gross returns; it is a subtraction from the net profit. Effective methods for the prevention of these losses by smut are now available to every grain grower

Atmospheric and Oceanographic Information Processing System (AOIPS) system description

The development of hardware and software for an interactive, minicomputer based processing and display system for atmospheric and oceanographic information extraction and image data analysis is described. The major applications of the system are discussed as well as enhancements planned for the future

Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.Comment: LaTeX, 22 pages, 2 figure

Deep Bilevel Learning

We present a novel regularization approach to train neural networks that enjoys better generalization and test error than standard stochastic gradient descent. Our approach is based on the principles of cross-validation, where a validation set is used to limit the model overfitting. We formulate such principles as a bilevel optimization problem. This formulation allows us to define the optimization of a cost on the validation set subject to another optimization on the training set. The overfitting is controlled by introducing weights on each mini-batch in the training set and by choosing their values so that they minimize the error on the validation set. In practice, these weights define mini-batch learning rates in a gradient descent update equation that favor gradients with better generalization capabilities. Because of its simplicity, this approach can be integrated with other regularization methods and training schemes. We evaluate extensively our proposed algorithm on several neural network architectures and datasets, and find that it consistently improves the generalization of the model, especially when labels are noisy.Comment: ECCV 201

Hamiltonians for the Quantum Hall Effect on Spaces with Non-Constant Metrics

The problem of studying the quantum Hall effect on manifolds with nonconstant metric is addressed. The Hamiltonian on a space with hyperbolic metric is determined, and the spectrum and eigenfunctions are calculated in closed form. The hyperbolic disk is also considered and some other applications of this approach are discussed as well.Comment: 16 page

Eigenvalus of Casimir Invariants for Type-I Quantum Superalgebras

We present the eigenvalues of the Casimir invariants for the type I quantum superalgebras on any irreducible highest weight module.Comment: 13 pages, AmsTex file; to appear in Lett. Math. Phy

Noncommutative Geometry Framework and The Feynman's Proof of Maxwell Equations

The main focus of the present work is to study the Feynman's proof of the Maxwell equations using the NC geometry framework. To accomplish this task, we consider two kinds of noncommutativity formulations going along the same lines as Feynman's approach. This allows us to go beyond the standard case and discover non-trivial results. In fact, while the first formulation gives rise to the static Maxwell equations, the second formulation is based on the following assumption $m[x_{j},\dot{x_{k}}]=i\hbar \delta_{jk}+im\theta_{jk}f.$ The results extracted from the second formulation are more significant since they are associated to a non trivial $\theta$-extension of the Bianchi-set of Maxwell equations. We find $div_{\theta}B=\eta_{\theta}$ and $\frac{\partial B_{s}}{\partial t}+\epsilon_{kjs}\frac{\partial E_{j}}{\partial x_{k}}=A_{1}\frac{d^{2}f}{dt^{2}}+A_{2}\frac{df}{dt}+A_{3},$ where $\eta_{\theta}$, $A_{1}$, $A_{2}$ and $A_{3}$ are local functions depending on the NC $\theta$-parameter. The novelty of this proof in the NC space is revealed notably at the level of the corrections brought to the previous Maxwell equations. These corrections correspond essentially to the possibility of existence of magnetic charges sources that we can associate to the magnetic monopole since $div_{\theta}B=\eta_{\theta}$ is not vanishing in general.Comment: LaTeX file, 16 page

On the solution of a supersymmetric model of correlated electrons

We consider the exact solution of a model of correlated electrons based on the superalgebra $Osp(2|2)$. The corresponding Bethe ansatz equations have an interesting form. We derive an expression for the ground state energy at half filling. We also present the eigenvalue of the transfer matrix commuting with the Hamiltonian.Comment: Palin latex , 8 page

Multivortex Solutions of the Weierstrass Representation

The connection between the complex Sine and Sinh-Gordon equations on the complex plane associated with a Weierstrass type system and the possibility of construction of several classes of multivortex solutions is discussed in detail. We perform the Painlev\'e test and analyse the possibility of deriving the B\"acklund transformation from the singularity analysis of the complex Sine-Gordon equation. We make use of the analysis using the known relations for the Painlev\'{e} equations to construct explicit formulae in terms of the Umemura polynomials which are $\tau$-functions for rational solutions of the third Painlev\'{e} equation. New classes of multivortex solutions of a Weierstrass system are obtained through the use of this proposed procedure. Some physical applications are mentioned in the area of the vortex Higgs model when the complex Sine-Gordon equation is reduced to coupled Riccati equations.Comment: 27 pages LaTeX2e, 1 encapsulated Postscript figur