1,735 research outputs found
Conservation laws for invariant functionals containing compositions
The study of problems of the calculus of variations with compositions is a
quite recent subject with origin in dynamical systems governed by chaotic maps.
Available results are reduced to a generalized Euler-Lagrange equation that
contains a new term involving inverse images of the minimizing trajectories. In
this work we prove a generalization of the necessary optimality condition of
DuBois-Reymond for variational problems with compositions. With the help of the
new obtained condition, a Noether-type theorem is proved. An application of our
main result is given to a problem appearing in the chaotic setting when one
consider maps that are ergodic.Comment: Accepted for an oral presentation at the 7th IFAC Symposium on
Nonlinear Control Systems (NOLCOS 2007), to be held in Pretoria, South
Africa, 22-24 August, 200
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
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
Group Theory and Quasiprobability Integrals of Wigner Functions
The integral of the Wigner function of a quantum mechanical system over a
region or its boundary in the classical phase plane, is called a
quasiprobability integral. Unlike a true probability integral, its value may
lie outside the interval [0,1]. It is characterized by a corresponding
selfadjoint operator, to be called a region or contour operator as appropriate,
which is determined by the characteristic function of that region or contour.
The spectral problem is studied for commuting families of region and contour
operators associated with concentric disks and circles of given radius a. Their
respective eigenvalues are determined as functions of a, in terms of the
Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in
Hilbert space carrying the positive discrete series representations of the
algebra su(1,1)or so(2,1). The explicit relation between the spectra of
operators associated with disks and circles with proportional radii, is given
in terms of the dicrete variable Meixner polynomials.Comment: 11 pages, latex fil
Links between different analytic descriptions of constant mean curvature surfaces
Transformations between different analytic descriptions of constant mean
curvature (CMC) surfaces are established. In particular, it is demonstrated
that the system descriptive of CMC surfaces within the
framework of the generalized Weierstrass representation, decouples into a
direct sum of the elliptic Sh-Gordon and Laplace equations. Connections of this
system with the sigma model equations are established. It is pointed out, that
the instanton solutions correspond to different Weierstrass parametrizations of
the standard sphere
Covariant spinor representation of and quantization of the spinning relativistic particle
A covariant spinor representation of is constructed for the
quantization of the spinning relativistic particle. It is found that, with
appropriately defined wavefunctions, this representation can be identified with
the state space arising from the canonical extended BFV-BRST quantization of
the spinning particle with admissible gauge fixing conditions after a
contraction procedure. For this model, the cohomological determination of
physical states can thus be obtained purely from the representation theory of
the algebra.Comment: Updated version with references included and covariant form of
equation 1. 23 pages, no figure
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
Solutions to the Quantum Yang-Baxter Equation with Extra Non-Additive Parameters
We present a systematic technique to construct solutions to the Yang-Baxter
equation which depend not only on a spectral parameter but in addition on
further continuous parameters. These extra parameters enter the Yang-Baxter
equation in a similar way to the spectral parameter but in a non-additive form.
We exploit the fact that quantum non-compact algebras such as
and type-I quantum superalgebras such as and are
known to admit non-trivial one-parameter families of infinite-dimensional and
finite dimensional irreps, respectively, even for generic . We develop a
technique for constructing the corresponding spectral-dependent R-matrices. As
examples we work out the the -matrices for the three quantum algebras
mentioned above in certain representations.Comment: 13 page
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
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