3,862 research outputs found
Nonsquare Spectral Factorization for Nonlinear Control Systems
This paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time. More specifically, we obtain a parametrization of nonsquare spectral factors in terms of invariant Lagrangian submanifolds and associated solutions of Hamilton–Jacobi inequalities. This inequality is a nonlinear analogue of the bounded real lemma and the control algebraic Riccati inequality. By way of an application, we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system.
From Monge to Higgs: a survey of distance computations in noncommutative geometry
This is a review of explicit computations of Connes distance in
noncommutative geometry, covering finite dimensional spectral triples,
almost-commutative geometries, and spectral triples on the algebra of compact
operators. Several applications to physics are covered, like the metric
interpretation of the Higgs field, and the comparison of Connes distance with
the minimal length that emerges in various models of quantum spacetime. Links
with other areas of mathematics are studied, in particular the horizontal
distance in sub-Riemannian geometry. The interpretation of Connes distance as a
noncommutative version of the Monge-Kantorovich metric in optimal transport is
also discussed.Comment: Proceedings of the workshop "Noncommutative Geometry and Optimal
Transport", Besan\c{c}on november 201
Classification of Finite Spectral Triples
It is known that the spin structure on a Riemannian manifold can be extended
to noncommutative geometry using the notion of a spectral triple. For finite
geometries, the corresponding finite spectral triples are completely described
in terms of matrices and classified using diagrams. When tensorized with the
ordinary space-time geometry, finite spectral triples give rise to Yang-Mills
theories with spontaneous symmetry breaking, whose characteristic features are
given within the diagrammatic approach: vertices of the diagram correspond to
gauge multiplets of chiral fermions and links to Yukawa couplings.Comment: Latex, 29 pages with 2 figures, reference adde
Spectral triples and the super-Virasoro algebra
We construct infinite dimensional spectral triples associated with
representations of the super-Virasoro algebra. In particular the irreducible,
unitary positive energy representation of the Ramond algebra with central
charge c and minimal lowest weight h=c/24 is graded and gives rise to a net of
even theta-summable spectral triples with non-zero Fredholm index. The
irreducible unitary positive energy representations of the Neveu-Schwarz
algebra give rise to nets of even theta-summable generalised spectral triples
where there is no Dirac operator but only a superderivation.Comment: 27 pages; v2: a comment concerning the difficulty in defining cyclic
cocycles in the NS case have been adde
Flows of constant mean curvature tori in the 3-sphere: The equivariant case
We present a deformation for constant mean curvature tori in the 3-sphere. We
show that the moduli space of equivariant constant mean curvature tori in the
3-sphere is connected, and we classify the minimal, the embedded, and the
Alexandrov embedded tori therein. We conclude with an instability result.Comment: v2: 33 pages, 9 figures. Instability result adde
Geometry of Quantum Spheres
Spectral triples on the q-deformed spheres of dimension two and three are
reviewed.Comment: 23 pages, revie
On orthogonal tensors and best rank-one approximation ratio
As is well known, the smallest possible ratio between the spectral norm and
the Frobenius norm of an matrix with is and
is (up to scalar scaling) attained only by matrices having pairwise orthonormal
rows. In the present paper, the smallest possible ratio between spectral and
Frobenius norms of tensors of order , also
called the best rank-one approximation ratio in the literature, is
investigated. The exact value is not known for most configurations of . Using a natural definition of orthogonal tensors over the real
field (resp., unitary tensors over the complex field), it is shown that the
obvious lower bound is attained if and only if a
tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal
or unitary tensors exist depends on the dimensions and the
field. A connection between the (non)existence of real orthogonal tensors of
order three and the classical Hurwitz problem on composition algebras can be
established: existence of orthogonal tensors of size
is equivalent to the admissibility of the triple to the Hurwitz
problem. Some implications for higher-order tensors are then given. For
instance, real orthogonal tensors of order
do exist, but only when . In the complex case, the situation is
more drastic: unitary tensors of size with exist only when . Finally, some numerical illustrations
for spectral norm computation are presented
A walk in the noncommutative garden
This text is written for the volume of the school/conference "Noncommutative
Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in
noncommutative geometry, based on a discussion of significant examples of
noncommutative spaces in geometry, number theory, and physics. The paper also
contains an outline (the ``Tehran program'') of ongoing joint work with Consani
on the noncommutative geometry of the adeles class space and its relation to
number theoretic questions.Comment: 106 pages, LaTeX, 23 figure
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