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Geometry and Information Retrieval
"... With the arrival of the digital computer in the second half of the twentieth century, a vast amount of information has been stored and made available. The growing of accesible information has reached an exponential growing rate, and computer scientists have been worried about the problem of accessin ..."
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With the arrival of the digital computer in the second half of the twentieth century, a vast amount of information has been stored and made available. The growing of accesible information has reached an exponential growing rate, and computer scientists have been worried about the problem of accessing and searching this information accurately. The subfield of Computer Science that deals with the representation, automated storage and retrieval of information items is called information retrieval (IR) [10], [1], [12]. We denote these items as documents (unit of retrieval) which might be a paragraph, a section, a chapter, a web page, an article, or a whole book [1]. The two main views of an IR system are the following. The former, the indexing subsystem, which takes a set of documents and converts them to a suitable representation (what is called an index), and the latter (the most important one) retrieval subsystem which
Easy Proofs of Some Consequences of Gleason's Theorem
"... The famous Gleason's Theorem gives a characterization of measures on lattices of subspaces of Hilbert spaces. While the proof of Gleason's Theorem is higly advanced, some of its consequences, in particular the nonexistence of hidden variables (=two-valued states), can be proved relative ..."
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The famous Gleason's Theorem gives a characterization of measures on lattices of subspaces of Hilbert spaces. While the proof of Gleason's Theorem is higly advanced, some of its consequences, in particular the nonexistence of hidden variables (=two-valued states), can be proved relatively easily. Here we present some of such results.
Piron's and Bell's Geometrical Lemmas
, 2001
"... The famous Gleason's Theorem gives a characterization of measures on lattices of subspaces of Hilbert spaces. The attempts to simplify its proof lead to geometrical lemmas that possess also easy proofs of some consequences of Gleason's Theorem. We contribute to these results by solving ..."
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The famous Gleason's Theorem gives a characterization of measures on lattices of subspaces of Hilbert spaces. The attempts to simplify its proof lead to geometrical lemmas that possess also easy proofs of some consequences of Gleason's Theorem. We contribute to these results by solving two open problems formulated by Chevalier, Dvurecenskij and Svozil. Besides, our use of orthoideals provides a unified approach to finite and infinite measures.
m Mathematical Publications KOCHEN{SPECKER THEOREM: TWO GEOMETRIC PROOFS
"... ABSTRACT. We present two geometric proofs for Kochen{Specker's theorem. A quite similar argument has been used by Cooke, Keane, Moran, and by Kalmbach in her book to derive Gleason's theorem. ..."
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ABSTRACT. We present two geometric proofs for Kochen{Specker's theorem. A quite similar argument has been used by Cooke, Keane, Moran, and by Kalmbach in her book to derive Gleason's theorem.
Kochen–Specker Theorem: Two Geometric Proofs∗
"... We present two geometric proofs for KochenSpecker’s theorem [S. Kochen, E. P. Specker: The problem of hidden variables in quantum mechanics, J. Math. Mech. 17 (1967), 5987]. A quite similar argument has been used by Cooke, Keane, Moran [R. Cooke, M. Keane, W. Moran: An elementary proof of Gleason’s ..."
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We present two geometric proofs for KochenSpecker’s theorem [S. Kochen, E. P. Specker: The problem of hidden variables in quantum mechanics, J. Math. Mech. 17 (1967), 5987]. A quite similar argument has been used by Cooke, Keane, Moran [R. Cooke, M. Keane, W. Moran: An elementary proof of Gleason’s theorem, Math. Proc. Camb. Phil. Soc. 98 (1985), 117128], and by Kalmbach in her book to derive Gleason’s theorem.
Physical unknowables
, 2008
"... Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the n-body problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unk ..."
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Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the n-body problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unknowables include the random occurrence of single events, complementarity and value indefiniteness.
QUASI-STATE RIGIDITY FOR FINITE-DIMENSIONAL LIE ALGEBRAS
"... ABSTRACT. We say that a Lie algebra g quasi-state rigid if every Ad-invariant Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras Cnou(n), n ≥ 2, are rigid. For ..."
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ABSTRACT. We say that a Lie algebra g quasi-state rigid if every Ad-invariant Lie quasi-state on it is the directional derivative of a homogeneous quasimorphism. Extending work of Entov and Polterovich, we show that every reductive Lie algebra, as well as the algebras Cnou(n), n ≥ 2, are rigid. For solvable Lie algebras which split over a codimension one abelian ideal, we characterize rigidity in terms of spectral data. In particular we show that the Lie algebra of the (ax+b)-group is rigid, but the Lie algebras of the SOL group and the three-dimensional Heisenberg group are not. 1.
