485 research outputs found
The Maslov Gerbe
Let Lag(E) be the grassmannian of lagrangian subspaces of a complex
symplectic vector space E. We construct a Maslov class which generates the
second integral cohomology of Lag(E), and we show that its mod 2 reduction is
the characteristic class of a flat gerbe with structure group Z_2. We explain
the relation of this gerbe to the well-known flat Maslov line bundle with
structure group Z_4 over the real lagrangian grassmannian, whose characteristic
class is the mod 4 reduction of the real Maslov class.Comment: 8 page
On estimation of covariance function for functional data with detection limits
In many studies on disease progression, biomarkers are restricted by detection limits, hence informatively missing. Current approaches ignore the problem by just filling in the value of the detection limit for the missing observations for the estimation of the mean and covariance function, which yield inaccurate estimation. Inspired by our recent work [Liu and Houwing-Duistermaat (2022), ‘Fast Estimators for the Mean Function for Functional Data with Detection Limits’, Stat, e467.] in which novel estimators for mean function for data subject to detection limit are proposed, in this paper, we will propose a novel estimator for the covariance function for sparse and dense data subject to a detection limit. We will derive the asymptotic properties of the estimator. We will compare our method to the standard method, which ignores the detection limit, via simulations. We will illustrate the new approach by analysing biomarker data subject to a detection limit. In contrast to the standard method, our method appeared to provide more accurate estimates of the covariance. Moreover its computation time is small
A Cantor set of tori with monodromy near a focus-focus singularity
We write down an asymptotic expression for action coordinates in an
integrable Hamiltonian system with a focus-focus equilibrium. From the
singularity in the actions we deduce that the Arnol'd determinant grows
infinitely large near the pinched torus. Moreover, we prove that it is possible
to globally parametrise the Liouville tori by their frequencies. If one
perturbs this integrable system, then the KAM tori form a Whitney smooth
family: they can be smoothly interpolated by a torus bundle that is
diffeomorphic to the bundle of Liouville tori of the unperturbed integrable
system. As is well-known, this bundle of Liouville tori is not trivial. Our
result implies that the KAM tori have monodromy. In semi-classical quantum
mechanics, quantisation rules select sequences of KAM tori that correspond to
quantum levels. Hence a global labeling of quantum levels by two quantum
numbers is not possible.Comment: 11 pages, 2 figure
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Evolution of spectral properties along the O(6)-U(5) transition in the interacting boson model. II. Classical trajectories
This article continues our previous study of level dynamics in the
[O(6)-U(5)]O(5) transition of the interacting boson model
[nucl-th/0504016] using the semiclassical theory of spectral fluctuations. We
find classical monodromy, related to a singular bundle of orbits with infinite
period at energy E=0, and bifurcations of numerous periodic orbits for E>0. The
spectrum of allowed ratios of periods associated with beta- and
gamma-vibrations exhibits an abrupt change around zero energy. These findings
explain anomalous bunching of quantum states in the E0 region, which
is responsible for the redistribution of levels between O(6) and U(5)
multiplets.Comment: 11 pages, 7 figures; continuation of nucl-th/050401
Vanishing Twist near Focus-Focus Points
We show that near a focus-focus point in a Liouville integrable Hamiltonian
system with two degrees of freedom lines of locally constant rotation number in
the image of the energy-momentum map are spirals determined by the eigenvalue
of the equilibrium. From this representation of the rotation number we derive
that the twist condition for the isoenergetic KAM condition vanishes on a curve
in the image of the energy-momentum map that is transversal to the line of
constant energy. In contrast to this we also show that the frequency map is
non-degenerate for every point in a neighborhood of a focus-focus point.Comment: 13 page
Gauge fixing and equivariant cohomology
The supersymmetric model developed by Witten to study the equivariant
cohomology of a manifold with an isometric circle action is derived from the
BRST quantization of a simple classical model. The gauge-fixing process is
carefully analysed, and demonstrates that different choices of gauge-fixing
fermion can lead to different quantum theories.Comment: 18 pages LaTe
Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems
A new construction using finite dimensional dual grassmannians is developed
to study rational and soliton solutions of the KP hierarchy. In the rational
case, properties of the tau function which are equivalent to bispectrality of
the associated wave function are identified. In particular, it is shown that
there exists a bound on the degree of all time variables in tau if and only if
the wave function is rank one and bispectral. The action of the bispectral
involution, beta, in the generic rational case is determined explicitly in
terms of dual grassmannian parameters. Using the correspondence between
rational solutions and particle systems, it is demonstrated that beta is a
linearizing map of the Calogero-Moser particle system and is essentially the
map sigma introduced by Airault, McKean and Moser in 1977.Comment: LaTeX, 24 page
Semitoric integrable systems on symplectic 4-manifolds
Let M be a symplectic 4-manifold. A semitoric integrable system on M is a
pair of real-valued smooth functions J, H on M for which J generates a
Hamiltonian S^1-action and the Poisson brackets {J,H} vanish. We shall
introduce new global symplectic invariants for these systems; some of these
invariants encode topological or geometric aspects, while others encode
analytical information about the singularities and how they stand with respect
to the system. Our goal is to prove that a semitoric system is completely
determined by the invariants we introduce
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