21,509 research outputs found
Sheaves that fail to represent matrix rings
There are two fundamental obstructions to representing noncommutative rings
via sheaves. First, there is no subcanonical coverage on the opposite of the
category of rings that includes all covering families in the big Zariski site.
Second, there is no contravariant functor F from the category of rings to the
category of ringed categories whose composite with the global sections functor
is naturally isomorphic to the identity, such that F restricts to the Zariski
spectrum functor Spec on the category of commutative rings (in a compatible way
with the natural isomorphism). Both of these no-go results are proved by
restricting attention to matrix rings.Comment: 13 pages; final versio
A prime ideal principle for two-sided ideals
Many classical ring-theoretic results state that an ideal that is maximal
with respect to satisfying a special property must be prime. We present a
"Prime Ideal Principle" that gives a uniform method of proving such facts,
generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam
and the author. Old and new "maximal implies prime" results are presented, with
results touching on annihilator ideals, polynomial identity rings, the
Artin-Rees property, Dedekind-finite rings, principal ideals generated by
normal elements, strongly noetherian algebras, and just infinite algebras.Comment: 22 page
On discretization of C*-algebras
The C*-algebra of bounded operators on the separable infinite-dimensional
Hilbert space cannot be mapped to a W*-algebra in such a way that each unital
commutative C*-subalgebra C(X) factors normally through .
Consequently, there is no faithful functor discretizing C*-algebras to
AW*-algebras, including von Neumann algebras, in this way.Comment: 5 pages. Please note that arXiv:1607.03376 supersedes this paper. It
significantly strengthens the main results and includes positive results on
discretization of C*-algebra
Row-Centric Lossless Compression of Markov Images
Motivated by the question of whether the recently introduced Reduced Cutset
Coding (RCC) offers rate-complexity performance benefits over conventional
context-based conditional coding for sources with two-dimensional Markov
structure, this paper compares several row-centric coding strategies that vary
in the amount of conditioning as well as whether a model or an empirical table
is used in the encoding of blocks of rows. The conclusion is that, at least for
sources exhibiting low-order correlations, 1-sided model-based conditional
coding is superior to the method of RCC for a given constraint on complexity,
and conventional context-based conditional coding is nearly as good as the
1-sided model-based coding.Comment: submitted to ISIT 201
Minimum Conditional Description Length Estimation for Markov Random Fields
In this paper we discuss a method, which we call Minimum Conditional
Description Length (MCDL), for estimating the parameters of a subset of sites
within a Markov random field. We assume that the edges are known for the entire
graph . Then, for a subset , we estimate the parameters
for nodes and edges in as well as for edges incident to a node in , by
finding the exponential parameter for that subset that yields the best
compression conditioned on the values on the boundary . Our
estimate is derived from a temporally stationary sequence of observations on
the set . We discuss how this method can also be applied to estimate a
spatially invariant parameter from a single configuration, and in so doing,
derive the Maximum Pseudo-Likelihood (MPL) estimate.Comment: Information Theory and Applications (ITA) workshop, February 201
Functions with Prescribed Best Linear Approximations
A common problem in applied mathematics is to find a function in a Hilbert
space with prescribed best approximations from a finite number of closed vector
subspaces. In the present paper we study the question of the existence of
solutions to such problems. A finite family of subspaces is said to satisfy the
\emph{Inverse Best Approximation Property (IBAP)} if there exists a point that
admits any selection of points from these subspaces as best approximations. We
provide various characterizations of the IBAP in terms of the geometry of the
subspaces. Connections between the IBAP and the linear convergence rate of the
periodic projection algorithm for solving the underlying affine feasibility
problem are also established. The results are applied to problems in harmonic
analysis, integral equations, signal theory, and wavelet frames
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