8,560 research outputs found
Geometric construction of spinors in orthogonal modular categories
A geometric construction of Z_2-graded orthogonal modular categories is
given. Their 0-graded parts coincide with categories previously obtained by
Blanchet and the author from the category of tangles modulo the Kauffman skein
relations. Quantum dimensions and twist coefficients for 1-graded simple
objects (spinors) are calculated. We show that our even orthogonal modular
categories admit cohomological refinements and the odd orthogonal ones lead to
spin refinements. The relation with the quantum group approach is discussed.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol3/agt-3-32.abs.htm
On the unification of quantum 3-manifold invariants
In 2006 Habiro initiated a construction of generating functions for
Witten-Reshetikhin-Turaev (WRT) invariants known as unified WRT invariants. In
a series of papers together with Irmgard Buehler and Christian Blanchet we
extended his construction to a larger class of 3-manifolds. The unified
invariants provide a strong tool to study properties of the whole collection of
WRT invariants, e.g. their integrality, and hence, their categorification. In
this paper we give a survey on ideas and techniques used in the construction of
the unified invariants.Comment: 18 page
Integrality of quantum 3-manifold invariants and rational surgery formula
We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an
arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a
rational surgery formula for the unified invariant dominating WRT SO(3)
invariants of rational homology 3-spheres at roots of unity of order co-prime
with the torsion. As an application, we compute the unified invariant for
Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this
invariant separates integral homology Seifert fibered spaces and can be used to
detect the unknot.Comment: 18 pages, Compositio Math. in pres
Unified quantum invariants and their refinements for homology 3-spheres with 2-torsion
For every rational homology 3-sphere with 2-torsion only we construct a
unified invariant (which takes values in a certain cyclotomic completion of a
polynomial ring), such that the evaluation of this invariant at any odd root of
unity provides the SO(3) Witten-Reshetikhin-Turaev invariant at this root and
at any even root of unity the SU(2) quantum invariant. Moreover, this unified
invariant splits into a sum of the refined unified invariants dominating spin
and cohomological refinements of quantum SU(2) invariants. New results on the
Ohtsuki series and the integrality of quantum invariants are the main
applications of our construction.Comment: 23 pages, results of math.QA/0510382 are include
Coupling between whistler waves and slow-mode solitary waves
The interplay between electron-scale and ion-scale phenomena is of general
interest for both laboratory and space plasma physics. In this paper we
investigate the linear coupling between whistler waves and slow magnetosonic
solitons through two-fluid numerical simulations. Whistler waves can be trapped
in the presence of inhomogeneous external fields such as a density hump or hole
where they can propagate for times much longer than their characteristic time
scale, as shown by laboratory experiments and space measurements. Space
measurements have detected whistler waves also in correspondence to magnetic
holes, i.e., to density humps with magnetic field minima extending on
ion-scales. This raises the interesting question of how ion-scale structures
can couple to whistler waves. Slow magnetosonic solitons share some of the main
features of a magnetic hole. Using the ducting properties of an inhomogeneous
plasma as a guide, we present a numerical study of whistler waves that are
trapped and transported inside propagating slow magnetosonic solitons.Comment: Submitted to Phys. of Plasma
The egalitarian sharing rule in provision of public projects
In this note we consider a society that partitions itself into disjoint
jurisdictions, each choosing a location of its public project and a taxation
scheme to finance it. The set of public project is multi-dimensional, and their
costs could vary from jurisdiction to jurisdiction. We impose two principles,
egalitarianism, that requires the equalization of the total cost for all agents
in the same jurisdiction, and efficiency, that implies the minimization of the
aggregate total cost within jurisdiction. We show that these two principles
always yield a core-stable partition but a Nash stable partition may fail to
exist.Comment: 7 page
Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case
We prove the existence of a complex valued -function on the unit circle,
a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class
, such that the perturbated operator does not belong to the
space of trace class operators. This resolves a problem of Peller
concerning the validity of the Koplienko-Neidhardt trace formula for unitaries
Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae
A formula for the norm of a bilinear Schur multiplier acting from the
Cartesian product of two copies of the
Hilbert-Schmidt classes into the trace class is established in
terms of linear Schur multipliers acting on the space of
all compact operators. Using this formula, we resolve Peller's problem on
Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice
continuously differentiable function with a bounded second derivative, a
self-adjoint (unbounded) operator and a self-adjoint operator such that
f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $
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