2,207 research outputs found
he geometry of statistical efficiency
We will place certain parts of the theory of statistical efficiency into the author’s
operator trigonometry (1967), thereby providing new geometrical understanding
of statistical efficiency. Important earlier results of Bloomfield
and Watson, Durbin and Kendall, Rao and Rao, will be so interpreted. For
example, worse case relative least squares efficiency corresponds to and is
achieved by the maximal turning antieigenvectors of the covariance matrix.
Some little-known historical perspectives will also be exposed. The overall
view will be emphasized
Rank-1 Tensor Approximation Methods and Application to Deflation
Because of the attractiveness of the canonical polyadic (CP) tensor
decomposition in various applications, several algorithms have been designed to
compute it, but efficient ones are still lacking. Iterative deflation
algorithms based on successive rank-1 approximations can be used to perform
this task, since the latter are rather easy to compute. We first present an
algebraic rank-1 approximation method that performs better than the standard
higher-order singular value decomposition (HOSVD) for three-way tensors.
Second, we propose a new iterative rank-1 approximation algorithm that improves
any other rank-1 approximation method. Third, we describe a probabilistic
framework allowing to study the convergence of deflation CP decomposition
(DCPD) algorithms based on successive rank-1 approximations. A set of computer
experiments then validates theoretical results and demonstrates the efficiency
of DCPD algorithms compared to other ones
Bicomplexes and Integrable Models
We associate bicomplexes with several integrable models in such a way that
conserved currents are obtained by a simple iterative construction. Gauge
transformations and dressings are discussed in this framework and several
examples are presented, including the nonlinear Schrodinger and sine-Gordon
equations, and some discrete models.Comment: 17 pages, LaTeX, uses amssymb.sty and diagrams.st
Shock wave structure in highly rarefied flows
The Boltzmann equation is written in terms of two functions associated
with the gain and loss of a certain type of molecule due to collisions.
Its integral form is then applied to the problem of normal shock structure,
and an iteration technique is used to determine the solution. The first
approximation to the velocity distribution function of the Chapman-Enskog
sequence, which leads to the Navier-Stokes equations, is used to initiate
the iteration scheme. Expressions for the distribution function and the
flow parameters pertinent to the first iteration are derived and show that
the B-G-K model results can be obtained as a special case. This model is
found to be valid in the continuum regime only, and is consequently limited
to the study of strong shocks. In the present treatment the iteration is
carried out on the distribution function and the analysis indicates that
the method is equally valid for variations in both Mach and Knudsen numbers.
Finally, the results of the first approximation are simplified, and expressed
in a form suitable for numerical computation, and the range of their validity
is discussed. The method should be equally suitable for other flow problems
of linear or nonlinear nature
A convexity of functions on convex metric spaces of Takahashi and applications
We quickly review and make some comments on the concept of convexity in
metric spaces due to Takahashi. Then we introduce a concept of convex structure
based convexity to functions on these spaces and refer to it as convexity.
convex functions generalize convex functions on linear spaces. We discuss
illustrative examples of (strict) convex functions and dedicate the major
part of this paper to proving a variety of properties that make them fit in
very well with the classical theory of convex analysis. Finally, we apply some
of our results to the metric projection problem and fixed point theory
How Listing's Law May Emerge from Neural Control of Reactive Saccades
We hypothesize that Listing's Law emerges as a result of two key properties of the saccadic sensory-motor system: 1) The visual sensory apparatus has a 2-D topology and 2) motor synergists are synchronized. The theory is tested by showing that eye attitudes that obey Listing's Law are achieved in a 3-D saccadic control system that translates visual eccentricity into synchronized motor commands via a 2-D spatial gradient. Simulations of this system demonstrate that attitudes assumed by the eye upon accurate foveation tend to obey Listing's Law.Office of Naval Research (N00014-92-J-1309, N00014-95-1-1409); Air Force Office of Scientific Research (90-0083
Two-loop renormalization of vector, axial-vector and tensor fermion bilinears on the lattice
We compute the two-loop renormalization functions, in the RI' scheme, of
local bilinear quark operators , where
corresponds to the Vector, Axial-Vector and Tensor Dirac operators, in the
lattice formulation of QCD. We consider both the flavor nonsinglet and singlet
operators.
We use the clover action for fermions and the Wilson action for gluons. Our
results are given as a polynomial in , in terms of both the
renormalized and bare coupling constant, in the renormalized Feynman gauge.
Finally, we present our results in the MSbar scheme, for easier comparison
with calculations in the continuum. The corresponding results, for fermions in
an arbitrary representation, together with some special features of
superficially divergent integrals, are included in the Appendices.Comment: 42 pages, 10 figures. Version accepted in PRD. Added comments and
references, provided per diagram numerical values; final results and
conclusions left unchanged. This paper is a sequel to arXiv:0707.2906 (Phys.
Rev. D76 (2007) 094514), which regards the scalar and pseudoscalar cases
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a
matrix with columns that form an orthonormal basis for a subspace \X, and
a Hermitian matrix , the eigenvalues of are called Ritz values of
with respect to \X. If the subspace \X is -invariant then the Ritz
values are some of the eigenvalues of . If the -invariant subspace \X
is perturbed to give rise to another subspace \Y, then the vector of absolute
values of changes in Ritz values of represents the absolute eigenvalue
approximation error using \Y. We bound the error in terms of principal angles
between \X and \Y. We capitalize on ideas from a recent paper [DOI:
10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute
values of differences between Ritz values for subspaces \X and \Y was
weakly (sub-)majorized by a constant times the sine of the vector of principal
angles between \X and \Y, the constant being the spread of the spectrum of
. In that result no assumption was made on either subspace being
-invariant. It was conjectured there that if one of the trial subspaces is
-invariant then an analogous weak majorization bound should only involve
terms of the order of sine squared. Here we confirm this conjecture.
Specifically we prove that the absolute eigenvalue error is weakly majorized by
a constant times the sine squared of the vector of principal angles between the
subspaces \X and \Y, where the constant is proportional to the spread of
the spectrum of . For many practical cases we show that the proportionality
factor is simply one, and that this bound is sharp. For the general case we can
only prove the result with a slightly larger constant, which we believe is
artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and
Applications (SIMAX
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