6,041 research outputs found
Applying Hallgrenâs algorithm for solving Pellâs equation to finding the irrational slope of the launch of a billiard ball
This thesis is an exploration of Quantum Computing applied to Pellâs equation in an attempt to find solutions to the Billiard Ball Problem. Pellâs equation is a Diophantine equation in the form of x2 â ny2 = 1, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. We will be applying Hallgrenâs algorithm for finding irrational periods in functions, in the context of billiard balls and their movement on a friction-less unit square billiard table. Our central research question has been the following: Given the cutting sequence of the billiard ballâs movement, can you find the irrational slope value in which the billiard ball was put in motion
Cyclic cohomology for graded -algebras and its pairings with van Daele -theory
We consider cycles for graded -algebras (Real -algebras)
which are compatible with the -structure and the real structure. Their
characters are cyclic cocycles. We define a Connes type pairing between such
characters and elements of the van Daele -groups of the -algebra
and its real subalgebra. This pairing vanishes on elements of finite order. We
define a second type of pairing between characters and -group elements which
is derived from a unital inclusion of -algebras. It is potentially
non-trivial on elements of order two and torsion valued. Such torsion valued
pairings yield topological invariants for insulators. The two-dimensional
Kane-Mele and the three-dimensional Fu-Kane-Mele strong invariant are special
cases of torsion valued pairings. We compute the pairings for a simple class of
periodic models and establish structural results for two dimensional aperiodic
models with odd time reversal invariance.Comment: 57 page
Boundary algebras and Kac modules for logarithmic minimal models
Virasoro Kac modules were initially introduced indirectly as representations
whose characters arise in the continuum scaling limits of certain transfer
matrices in logarithmic minimal models, described using Temperley-Lieb
algebras. The lattice transfer operators include seams on the boundary that use
Wenzl-Jones projectors. If the projectors are singular, the original
prescription is to select a subspace of the Temperley-Lieb modules on which the
action of the transfer operators is non-singular. However, this prescription
does not, in general, yield representations of the Temperley-Lieb algebras and
the Virasoro Kac modules have remained largely unidentified. Here, we introduce
the appropriate algebraic framework for the lattice analysis as a quotient of
the one-boundary Temperley-Lieb algebra. The corresponding standard modules are
introduced and examined using invariant bilinear forms and their Gram
determinants. The structures of the Virasoro Kac modules are inferred from
these results and are found to be given by finitely generated submodules of
Feigin-Fuchs modules. Additional evidence for this identification is obtained
by comparing the formalism of lattice fusion with the fusion rules of the
Virasoro Kac modules. These are obtained, at the character level, in complete
generality by applying a Verlinde-like formula and, at the module level, in
many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.Comment: 71 pages. v3: version published in Nucl. Phys.
Noncommutative Spheres and Instantons
We report on some recent work on deformation of spaces, notably deformation
of spheres, describing two classes of examples. The first class of examples
consists of noncommutative manifolds associated with the so called
-deformations which were introduced out of a simple analysis in terms
of cycles in the -complex of cyclic homology. These examples have
non-trivial global features and can be endowed with a structure of
noncommutative manifolds, in terms of a spectral triple (\ca, \ch, D). In
particular, noncommutative spheres are isospectral
deformations of usual spherical geometries. For the corresponding spectral
triple (\cinf(S^{N}_\theta), \ch, D), both the Hilbert space of spinors \ch=
L^2(S^{N},\cs) and the Dirac operator are the usual ones on the
commutative -dimensional sphere and only the algebra and its action
on are deformed. The second class of examples is made of the so called
quantum spheres which are homogeneous spaces of quantum orthogonal
and quantum unitary groups. For these spheres, there is a complete description
of -theory, in terms of nontrivial self-adjoint idempotents (projections)
and unitaries, and of the -homology, in term of nontrivial Fredholm modules,
as well as of the corresponding Chern characters in cyclic homology and
cohomology.Comment: Minor changes, list of references expanded and updated. These notes
are based on invited lectures given at the ``International Workshop on
Quantum Field Theory and Noncommutative Geometry'', November 26-30 2002,
Tohoku University, Sendai, Japan. To be published in the workshop proceedings
by Springer-Verlag as Lecture Notes in Physic
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Spectacular narratives: Twister, independence day, and frontier mythology
Big-screen spectacle has become increasingly important to Hollywood in recent decades. It formed a central part of a post-war strategy aimed at tempting lost audiences back to the cinema in the face of demographic changes and the development of television and other domestic leisure activities. More recently, in an age in which the big Hollywood studios have become parts of giant conglomerates, the prevalence of spectacle and special effects has been boosted by a demand to engineer products that can be further exploited in multimedia forms such as computer games and theme-park rides, secondary outlets that can sometimes generate more profits than the films on which they are based. These and other developments have led some commentators to announce, or predict, the imminent demise of narrative as a central component of Hollywood cinema. But the case has been considerably overstated. Narrative is far from being eclipsed, even in the most spectacular and effects-oriented of todayâs blockbuster attractions. These films still tend to tell reasonably coherent stories, even if they may sometimes be looser and less well integrated than classical models. More important for my argument, contemporary spectaculars also continue to manifest the kinds of underlying thematic oppositions and reconciliations associated with a broadly âstructuralistâ analysis of narrative. This very important dimension of narrative has been largely ignored by those who identify, celebrate or more often bemoan a weakening of plot or character development in many spectacular features
Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
This monograph offers an overview on the topological invariants in fermionic
topological insulators from the complex classes. Tools from K-theory and
non-commutative geometry are used to define bulk and boundary invariants, to
establish the bulk-boundary correspondence and to link the invariants to
physical observables.Comment: Monograph in Springer Series in Mathematical Physics Studies, see
ISBN below. Correction of a few remaining typos. ISBN 978-3-319-29350-9,
eBook ISBN 978-3-319-29351-6, (Springer, 2016
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