565,521 research outputs found
All the lowest order PDE for spectral gaps of Gaussian matrices
Tracy-Widom (TW) equations for one-matrix unitary ensembles (UE) (equivalent
to a particular case of Schlesinger equations for isomonodromic deformations)
are rewritten in a general form which allows one to derive all the lowest order
equations (PDE) for spectral gap probabilities of UE without intermediate
higher-order PDE. This is demonstrated on the example of Gaussian ensemble
(GUE) for which all the third order PDE for gap probabilities are obtained
explicitly. Moreover, there is a {\it second order} PDE for GUE probabilities
in the case of more than one spectral endpoint.
This approach allows to derive all PDE at once where possible, while in the
method based on Hirota bilinear identities and Virasoro constraints starting
with different bilinear identities leads to different subsets of the full set
of equations.Comment: 22 pages, references corrected, remark adde
Generating dynamic higher-order Markov models in web usage mining
Markov models have been widely used for modelling users’ web navigation behaviour. In previous work we have presented a dynamic clustering-based Markov model that accurately represents second-order transition probabilities given by a collection of navigation sessions. Herein, we propose a generalisation of the method that takes into account higher-order conditional probabilities. The method makes use of the state cloning concept together with a clustering technique to separate the navigation paths that reveal differences in the conditional probabilities. We report on experiments conducted with three real world data sets. The results show that some pages require a long history to understand the users choice of link, while others require only a short history. We also show that the number of additional states induced by the method can be controlled through a probability threshold parameter
Testing Born's Rule in Quantum Mechanics with a Triple Slit Experiment
In Mod. Phys. Lett. A 9, 3119 (1994), one of us (R.D.S) investigated a
formulation of quantum mechanics as a generalized measure theory. Quantum
mechanics computes probabilities from the absolute squares of complex
amplitudes, and the resulting interference violates the (Kolmogorov) sum rule
expressing the additivity of probabilities of mutually exclusive events.
However, there is a higher order sum rule that quantum mechanics does obey,
involving the probabilities of three mutually exclusive possibilities. We could
imagine a yet more general theory by assuming that it violates the next higher
sum rule. In this paper, we report results from an ongoing experiment that sets
out to test the validity of this second sum rule by measuring the interference
patterns produced by three slits and all the possible combinations of those
slits being open or closed. We use attenuated laser light combined with single
photon counting to confirm the particle character of the measured light.Comment: Submitted to the proceedings of Foundations of Probability and
Physics-5, Vaxjo, Sweden, August 2008. 8 pages, 8 figure
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