2,638 research outputs found
A note on Freiman's theorem in vector spaces
We show that if A is a subset of F_2^n and |A+A| < K|A| then A is contained
in a subspace of size at most 2^{O(K^{3/2}log K)}|A|. This improves on the
previous best of 2^{O(K^2)}.Comment: 9 pp. Corrected typos. Updated references
Adjointness Relations as a Criterion for Choosing an Inner Product
This is a contribution to the forthcoming book "Canonical Gravity: {}From
Classical to Quantum" edited by J. Ehlers and H. Friedrich. Ashtekar's
criterion for choosing an inner product in the quantisation of constrained
systems is discussed. An erroneous claim in a previous paper is corrected and a
cautionary example is presented.Comment: 6 pages, MPA-AR-94-
On the tensor convolution and the quantum separability problem
We consider the problem of separability: decide whether a Hermitian operator
on a finite dimensional Hilbert tensor product is separable or entangled. We
show that the tensor convolution defined for certain mappings on an almost
arbitrary locally compact abelian group, give rise to formulation of an
equivalent problem to the separability one.Comment: 13 pages, two sections adde
Power of unentangled measurements on two antiparallel spins
We consider a pair of antiparallel spins polarized in a random direction to
encode quantum information. We wish to extract as much information as possible
on the polarization direction attainable by an unentangled measurement, i.e.,
by a measurement, whose outcomes are associated with product states. We develop
analytically the upper bound 0.7935 bits to the Shannon mutual information
obtainable by an unentangled measurement, which is definitely less than the
value 0.8664 bits attained by an entangled measurement. This proves our main
result, that not every ensemble of product states can be optimally
distinguished by an unentangled measurement, if the measure of
distinguishability is defined in the sense of Shannon. We also present results
from numerical calculations and discuss briefly the case of parallel spins.Comment: Latex file, 18 pages, 1 figure; published versio
Density of states of a two-dimensional electron gas in a non-quantizing magnetic field
We study local density of electron states of a two-dimentional conductor with
a smooth disorder potential in a non-quantizing magnetic field, which does not
cause the standart de Haas-van Alphen oscillations. It is found, that despite
the influence of such ``classical'' magnetic field on the average electron
density of states (DOS) is negligibly small, it does produce a significant
effect on the DOS correlations. The corresponding correlation function exhibits
oscillations with the characteristic period of cyclotron quantum
.Comment: 7 pages, including 3 figure
A Robust Iterative Unfolding Method for Signal Processing
There is a well-known series expansion (Neumann series) in functional
analysis for perturbative inversion of specific operators on Banach spaces.
However, operators that appear in signal processing (e.g. folding and
convolution of probability density functions), in general, do not satisfy the
usual convergence condition of that series expansion. This article provides
some theorems on the convergence criteria of a similar series expansion for
this more general case, which is not covered yet by the literature.
The main result is that a series expansion provides a robust unbiased
unfolding and deconvolution method. For the case of the deconvolution, such a
series expansion can always be applied, and the method always recovers the
maximum possible information about the initial probability density function,
thus the method is optimal in this sense. A very significant advantage of the
presented method is that one does not have to introduce ad hoc frequency
regulations etc., as in the case of usual naive deconvolution methods. For the
case of general unfolding problems, we present a computer-testable sufficient
condition for the convergence of the series expansion in question.
Some test examples and physics applications are also given. The most
important physics example shall be (which originally motivated our survey on
this topic) the case of pi^0 --> gamma+gamma particle decay: we show that one
can recover the initial pi^0 momentum density function form the measured single
gamma momentum density function by our series expansion.Comment: 23 pages, 9 figure
Quantum graphs as holonomic constraints
We consider the dynamics on a quantum graph as the limit of the dynamics
generated by a one-particle Hamiltonian in R^2 with a potential having a deep
strict minimum on the graph, when the width of the well shrinks to zero. For a
generic graph we prove convergence outside the vertices to the free dynamics on
the edges. For a simple model of a graph with two edges and one vertex, we
prove convergence of the dynamics to the one generated by the Laplacian with
Dirichlet boundary conditions in the vertex.Comment: 28 pages, 3 figure
Algebras generated by two bounded holomorphic functions
We study the closure in the Hardy space or the disk algebra of algebras
generated by two bounded functions, of which one is a finite Blaschke product.
We give necessary and sufficient conditions for density or finite codimension
of such algebras. The conditions are expressed in terms of the inner part of a
function which is explicitly derived from each pair of generators. Our results
are based on identifying z-invariant subspaces included in the closure of the
algebra. Versions of these results for the case of the disk algebra are given.Comment: 22 pages ; a number of minor mistakes have been corrected, and some
points clarified. Conditionally accepted by Journal d'Analyse Mathematiqu
Sampling, Intervention, Prediction, Aggregation: A Generalized Framework for Model-Agnostic Interpretations
Model-agnostic interpretation techniques allow us to explain the behavior of
any predictive model. Due to different notations and terminology, it is
difficult to see how they are related. A unified view on these methods has been
missing. We present the generalized SIPA (sampling, intervention, prediction,
aggregation) framework of work stages for model-agnostic interpretations and
demonstrate how several prominent methods for feature effects can be embedded
into the proposed framework. Furthermore, we extend the framework to feature
importance computations by pointing out how variance-based and
performance-based importance measures are based on the same work stages. The
SIPA framework reduces the diverse set of model-agnostic techniques to a single
methodology and establishes a common terminology to discuss them in future
work
Hidden Symmetry of the Differential Calculus on the Quantum Matrix Space
A standard bicovariant differential calculus on a quantum matrix space is considered. The principal result of this work is in observing
that the is in fact a
-module differential algebra.Comment: 5 page
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