12,639 research outputs found
Quantum D-modules, elliptic braid groups, and double affine Hecke algebras
We build representations of the elliptic braid group from the data of a
quantum D-module M over a ribbon Hopf algebra U. The construction is modelled
on, and generalizes, similar constructions by Lyubashenko and Majid, and also
certain geometric constructions of Calaque, Enriquez, and Etingof concerning
trigonometric Cherednik algebras. In this context, the former construction is
the special case where M is the basic representation, while the latter
construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t
limits 1. In the latter case, we produce representations of the double affine
Hecke algebra of type A_{n-1}, for each n
A Modified Crank-Nicolson Method
In order to obtain a numerical solution to the heat equation using finite differences, either implicit or explicit equations are used to formulate a solution. The advantage in an explicit formulation is its simplicity and minimal computer storage requirements while its disadvantage is its instability. The opposite is true for an implicit formulation such as the Crank-Nicolson method; although it is stable it is more difficult to implement and requires a much larger memory capacity. In this paper we examine the accuracy and stability of a hybrid approach, a modified Crank-Nicolson formulation, that combines the advantageous features of both the implicit and explicit formulations. This hybrid approach results in a 20% reduction in the amount of work required compared to the standard Crank-Nicolson solution if both methods use a special tridiagonal system solver. If Gaussian elimination is used, the modified Crank-Nicolson approach reduces the amount of work by 87%. Regardless of the linear system solver used, the modified Crank-Nicolson approach reduces by 50% the memory requirement of the standard Crank-Nicolson method
Fourier transform for quantum -modules via the punctured torus mapping class group
We construct a certain cross product of two copies of the braided dual
of a quasitriangular Hopf algebra , which we call the elliptic
double , and which we use to construct representations of the punctured
elliptic braid group extending the well-known representations of the planar
braid group attached to . We show that the elliptic double is the universal
source of such representations. We recover the representations of the punctured
torus braid group obtained in arXiv:0805.2766, and hence construct a
homomorphism to the Heisenberg double , which is an isomorphism if is
factorizable.
The universal property of endows it with an action by algebra
automorphisms of the mapping class group of the
punctured torus. One such automorphism we call the quantum Fourier transform;
we show that when , the quantum Fourier transform
degenerates to the classical Fourier transform on as .Comment: 12 pages, 1 figure. Final version, to appear in Quantum Topolog
Syntactic Topic Models
The syntactic topic model (STM) is a Bayesian nonparametric model of language
that discovers latent distributions of words (topics) that are both
semantically and syntactically coherent. The STM models dependency parsed
corpora where sentences are grouped into documents. It assumes that each word
is drawn from a latent topic chosen by combining document-level features and
the local syntactic context. Each document has a distribution over latent
topics, as in topic models, which provides the semantic consistency. Each
element in the dependency parse tree also has a distribution over the topics of
its children, as in latent-state syntax models, which provides the syntactic
consistency. These distributions are convolved so that the topic of each word
is likely under both its document and syntactic context. We derive a fast
posterior inference algorithm based on variational methods. We report
qualitative and quantitative studies on both synthetic data and hand-parsed
documents. We show that the STM is a more predictive model of language than
current models based only on syntax or only on topics
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