125 research outputs found
Quantum isometries and noncommutative spheres
We introduce and study two new examples of noncommutative spheres: the
half-liberated sphere, and the free sphere. Together with the usual sphere,
these two spheres have the property that the corresponding quantum isometry
group is "easy", in the representation theory sense. We present as well some
general comments on the axiomatization problem, and on the "untwisted" and
"non-easy" case.Comment: 16 page
Quantum Symmetries and Strong Haagerup Inequalities
In this paper, we consider families of operators in
a tracial C-probability space , whose joint
-distribution is invariant under free complexification and the action of
the hyperoctahedral quantum groups . We prove a strong
form of Haagerup's inequality for the non-self-adjoint operator algebra
generated by , which generalizes the
strong Haagerup inequalities for -free R-diagonal families obtained by
Kemp-Speicher \cite{KeSp}. As an application of our result, we show that
always has the metric approximation property (MAP). We also apply
our techniques to study the reduced C-algebra of the free unitary
quantum group . We show that the non-self-adjoint subalgebra generated by the matrix elements of the fundamental corepresentation of
has the MAP. Additionally, we prove a strong Haagerup inequality for
, which improves on the estimates given by Vergnioux's property
RD \cite{Ve}
Spectral analysis of the free orthogonal matrix
We compute the spectral measure of the standard generators of the
Wang algebra . We show in particular that this measure has support
, and that it has no atoms. The computation is
done by using various techniques, involving the general Wang algebra ,
a representation of due to Woronowicz, and several calculations with
orthogonal polynomials.Comment: 22 pages, 4 figure
A maximality result for orthogonal quantum groups
We prove that the quantum group inclusion is "maximal",
where is the usual orthogonal group and is the half-liberated
orthogonal quantum group, in the sense that there is no intermediate compact
quantum group . In order to prove this result, we
use: (1) the isomorphism of projective versions , (2) some
maximality results for classical groups, obtained by using Lie algebras and
some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.Comment: 10 page
Induction of Kanizsa Contours Requires Awareness of the Inducing Context
It remains unknown to what extent the human visual system interprets information about complex scenes without conscious analysis. Here we used visual masking techniques to assess whether illusory contours (Kanizsa shapes) are perceived when the inducing context creating this illusion does not reach awareness. In the first experiment we tested perception directly by having participants discriminate the orientation of an illusory contour. In the second experiment, we exploited the fact that the presence of an illusory contour enhances performance on a spatial localization task. Moreover, in the latter experiment we also used a different masking method to rule out the effect of stimulus duration. Our results suggest that participants do not perceive illusory contours when they are unaware of the inducing context. This is consistent with theories of a multistage, recurrent process of perceptual integration. Our findings thus challenge some reports, including those from neurophysiological experiments in anaesthetized animals. Furthermore, we discuss the importance to test the presence of the phenomenal percept directly with appropriate methods
Stability of the selfsimilar dynamics of a vortex filament
In this paper we continue our investigation about selfsimilar solutions of
the vortex filament equation, also known as the binormal flow (BF) or the
localized induction equation (LIE). Our main result is the stability of the
selfsimilar dynamics of small pertubations of a given selfsimilar solution. The
proof relies on finding precise asymptotics in space and time for the tangent
and the normal vectors of the perturbations. A main ingredient in the proof is
the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger
equation, connected to the binormal flow by Hasimoto's transform.Comment: revised version, 36 page
Quantum Isometries of the finite noncommutative geometry of the Standard Model
We compute the quantum isometry group of the finite noncommutative geometry F
describing the internal degrees of freedom in the Standard Model of particle
physics. We show that this provides genuine quantum symmetries of the spectral
triple corresponding to M x F where M is a compact spin manifold. We also prove
that the bosonic and fermionic part of the spectral action are preserved by
these symmetries.Comment: 29 pages, no figures v3: minor change
Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation
We consider the mass-critical focusing nonlinear Schrodinger equation in the
presence of an external potential, when the nonlinearity is inhomogeneous. We
show that if the inhomogeneous factor in front of the nonlinearity is
sufficiently flat at a critical point, then there exists a solution which blows
up in finite time with the maximal (unstable) rate at this point. In the case
where the critical point is a maximum, this solution has minimal mass among the
blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of
the mass-critical Schrodinger equation on some surfaces. The proof is based on
properties of the linearized operator around the ground state, and on a full
use of the invariances of the equation with an homogeneous nonlinearity and no
potential, via time-dependent modulations.Comment: 36 pages. More explanations, references updated, statement of Theorem
1.1 corrected. FInal versio
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