8,202 research outputs found
Codimension and pseudometric in co-Heyting algebras
In this paper we introduce a notion of dimension and codimension for every
element of a distributive bounded lattice . These notions prove to have a
good behavior when is a co-Heyting algebra. In this case the codimension
gives rise to a pseudometric on which satisfies the ultrametric triangle
inequality. We prove that the Hausdorff completion of with respect to this
pseudometric is precisely the projective limit of all its finite dimensional
quotients. This completion has some familiar metric properties, such as the
convergence of every monotonic sequence in a compact subset. It coincides with
the profinite completion of if and only if it is compact or equivalently if
every finite dimensional quotient of is finite. In this case we say that
is precompact. If is precompact and Hausdorff, it inherits many of the
remarkable properties of its completion, specially those regarding the
join/meet irreducible elements. Since every finitely presented co-Heyting
algebra is precompact Hausdorff, all the results we prove on the algebraic
structure of the latter apply in particular to the former. As an application,
we obtain the existence for every positive integers of a term
such that in every co-Heyting algebra generated by an -tuple ,
is precisely the maximal element of codimension .Comment: 34 page
The physical interpretation of daseinisation
We provide a conceptual discussion and physical interpretation of some of the
quite abstract constructions in the topos approach to physics. In particular,
the daseinisation process for projection operators and for self-adjoint
operators is motivated and explained from a physical point of view.
Daseinisation provides the bridge between the standard Hilbert space formalism
of quantum theory and the new topos-based approach to quantum theory. As an
illustration, we will show all constructions explicitly for a three-dimensional
Hilbert space and the spin-z operator of a spin-1 particle. This article is a
companion to the article by Isham in the same volume.Comment: 39 pages; to appear in "Deep Beauty", ed. Hans Halvorson, Cambridge
University Press (2010
Effective dimension of finite semigroups
In this paper we discuss various aspects of the problem of determining the
minimal dimension of an injective linear representation of a finite semigroup
over a field. We outline some general techniques and results, and apply them to
numerous examples.Comment: To appear in J. Pure Appl. Al
Time Delay Estimation from Low Rate Samples: A Union of Subspaces Approach
Time delay estimation arises in many applications in which a multipath medium
has to be identified from pulses transmitted through the channel. Various
approaches have been proposed in the literature to identify time delays
introduced by multipath environments. However, these methods either operate on
the analog received signal, or require high sampling rates in order to achieve
reasonable time resolution. In this paper, our goal is to develop a unified
approach to time delay estimation from low rate samples of the output of a
multipath channel. Our methods result in perfect recovery of the multipath
delays from samples of the channel output at the lowest possible rate, even in
the presence of overlapping transmitted pulses. This rate depends only on the
number of multipath components and the transmission rate, but not on the
bandwidth of the probing signal. In addition, our development allows for a
variety of different sampling methods. By properly manipulating the low-rate
samples, we show that the time delays can be recovered using the well-known
ESPRIT algorithm. Combining results from sampling theory with those obtained in
the context of direction of arrival estimation methods, we develop necessary
and sufficient conditions on the transmitted pulse and the sampling functions
in order to ensure perfect recovery of the channel parameters at the minimal
possible rate. Our results can be viewed in a broader context, as a sampling
theorem for analog signals defined over an infinite union of subspaces
A Royal Road to Quantum Theory (or Thereabouts)
This paper fails to derive quantum mechanics from a few simple postulates.
But it gets very close --- and it does so without much exertion. More exactly,
I obtain a representation of finite-dimensional probabilistic systems in terms
of euclidean Jordan algebras, in a strikingly easy way, from simple
assumptions. This provides a framework within which real, complex and
quaternionic QM can play happily together, and allows some --- but not too much
--- room for more exotic alternatives. (This is a leisurely summary, based on
recent lectures, of material from the papers arXiv:1206:2897 and
arXiv:1507.06278, the latter joint work with Howard Barnum and Matthew Graydon.
Some further ideas are also explored.)Comment: 33 pages, 3 figures. An expanded and somewhat informal account of
material from arXiv:1206:2897, plus some new results. A number of typos and
other minor errors are corrected in version
Normal forms and entanglement measures for multipartite quantum states
A general mathematical framework is presented to describe local equivalence
classes of multipartite quantum states under the action of local unitary and
local filtering operations. This yields multipartite generalizations of the
singular value decomposition. The analysis naturally leads to the introduction
of entanglement measures quantifying the multipartite entanglement (as
generalizations of the concurrence and the 3-tangle), and the optimal local
filtering operations maximizing these entanglement monotones are obtained.
Moreover a natural extension of the definition of GHZ-states to e.g. systems is obtained.Comment: Proof of uniqueness of normal form adde
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