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Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 26 (9 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Moments of twovariable functions and the uniqueness of graph limits
, 2009
"... For a symmetric bounded measurable function W on [0, 1] 2 and a simple graph F, the homomorphism density t(F, W) = W (xi, xj) dx. [0,1] V (F) ij∈E(F) can be thought of as a “moment ” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the ..."
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Cited by 22 (2 self)
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For a symmetric bounded measurable function W on [0, 1] 2 and a simple graph F, the homomorphism density t(F, W) = W (xi, xj) dx. [0,1] V (F) ij∈E(F) can be thought of as a “moment ” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (Gn) of dense graphs is said to be convergent if the probability, t(F, Gn), that a random map from V (F) into V (Gn) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric
The BellKochenSpecker Theorem
"... Meyer, Kent and Clifton (MKC) claim to have nullified the BellKochenSpecker (BellKS) theorem. It is true that they invalidate KS’s account of the theorem’s physical implications. However, they do not invalidate Bell’s point, that quantum mechanics is inconsistent with the classical assumption, th ..."
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Cited by 6 (0 self)
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Meyer, Kent and Clifton (MKC) claim to have nullified the BellKochenSpecker (BellKS) theorem. It is true that they invalidate KS’s account of the theorem’s physical implications. However, they do not invalidate Bell’s point, that quantum mechanics is inconsistent with the classical assumption, that a measurement tells us about a property previously possessed by the system. This failure of classical ideas about measurement is, perhaps, the single most important implication of quantum mechanics. In a conventional colouring there are some remaining patches of white. MKC fill in these patches, but only at the price of introducing patches where the colouring becomes “pathologically” discontinuous. The discontinuities mean that the colours in these patches are empirically unknowable. We prove a general theorem which shows that their extent is at least as great as the patches of white in a conventional approach. The theorem applies, not only to the MKC colourings, but also to any other such attempt to circumvent the BellKS theorem (Pitowsky’s colourings, for example). We go on to discuss the implications. MKC do not nullify the BellKS theorem. They do, however, show that we did not, hitherto, properly understand the theorem. For that reason their results (and Pitowsky’s earlier results) are of major importance. 1 1.
On Rboundedness of unions of sets of operators
"... It is shown that the union of a sequence T1, T2,... of Rbounded sets of operators from X into Y with Rbounds τ1, τ2,..., respectively, is Rbounded if X is a Banach space of cotype q, Y a Banach space of type p, and ∑∞ k=1 τ r k < ∞, where r = pq/(q−p) if q < ∞ and r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ ..."
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Cited by 1 (0 self)
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It is shown that the union of a sequence T1, T2,... of Rbounded sets of operators from X into Y with Rbounds τ1, τ2,..., respectively, is Rbounded if X is a Banach space of cotype q, Y a Banach space of type p, and ∑∞ k=1 τ r k < ∞, where r = pq/(q−p) if q < ∞ and r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and p ̸ = q. The power r is sharp. Each Banach space that contains an isomorphic copy of c0 admits operators T1, T2,... such that ‖Tk ‖ = 1/k, k ∈ N, and {T1, T2,...} is not Rbounded. Further it is shown that the set of positive linear contractions in an infinite dimensional Lp is Rbounded only if p = 2.
Estimating betamixing coefficients
"... The literature on statistical learning for time series assumes the asymptotic independence or “mixing ” of the datagenerating process. These mixing assumptions are never tested, and there are no methods for estimating mixing rates from data. We give an estimator for the betamixing rate based on a ..."
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The literature on statistical learning for time series assumes the asymptotic independence or “mixing ” of the datagenerating process. These mixing assumptions are never tested, and there are no methods for estimating mixing rates from data. We give an estimator for the betamixing rate based on a single stationary sample path and show it is L1risk consistent. 1
TABLE OF CONTENTS
"... study, in the Department of Mathematics. It is intended as a supplement to the bulletin ..."
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study, in the Department of Mathematics. It is intended as a supplement to the bulletin
the Continuity Equation organized by Philippe Clément and Onno van Gaans at Leiden
, 2006
"... 1 Probability measures on metric spaces 1 ..."