5,236 research outputs found
Superintegrable Hamiltonian systems with noncompact invariant submanifolds. Kepler system
The Mishchenko-Fomenko theorem on superintegrable Hamiltonian systems is
generalized to superintegrable Hamiltonian systems with noncompact invariant
submanifolds. It is formulated in the case of globally superintegrable
Hamiltonian systems which admit global generalized action-angle coordinates.
The well known Kepler system falls into two different globally superintegrable
systems with compact and noncompact invariant submanifolds.Comment: 23 page
Semiclassical almost isometry
Let M be a complex projective manifold, and L an Hermitian ample line bundle
on it. A fundamental theorem of Gang Tian, reproved and strengthened by
Zelditch, implies that the Khaeler form of L can be recovered from the
asymptotics of the projective embeddings associated to large tensor powers of
L. More precisely, with the natural choice of metrics the projective embeddings
associated to the full linear series |kL| are asymptotically symplectic, in the
appropriate rescaled sense. In this article, we ask whether and how this result
extends to the semiclassical setting. Specifically, we relate the Weinstein
symplectic structure on a given isodrastic leaf of half-weighted
Bohr-Sommerfeld Lagrangian submanifolds of M to the asymptotics of the the
pull-back of the Fubini-Study form under the semiclassical projective maps
constructed by Borthwick, Paul and Uribe.Comment: exposition improve
Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebras
We describe the group of continuous automorphisms of all simple
infinite-dimensional linearly compact Lie superalgebras and use it in order to
classify F-forms of these superalgebras over any field F of characteristic
zero.Comment: 24 page
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
Theory of Finite Pseudoalgebras
Conformal algebras, recently introduced by Kac, encode an axiomatic
description of the singular part of the operator product expansion in conformal
field theory. The objective of this paper is to develop the theory of
``multi-dimensional'' analogues of conformal algebras. They are defined as Lie
algebras in a certain ``pseudotensor'' category instead of the category of
vector spaces. A pseudotensor category (as introduced by Lambek, and by
Beilinson and Drinfeld) is a category equipped with ``polylinear maps'' and a
way to compose them. This allows for the definition of Lie algebras,
representations, cohomology, etc. An instance of such a category can be
constructed starting from any cocommutative (or more generally,
quasitriangular) Hopf algebra . The Lie algebras in this category are called
Lie -pseudoalgebras.
The main result of this paper is the classification of all simple and all
semisimple Lie -pseudoalgebras which are finitely generated as -modules.
We also start developing the representation theory of Lie pseudoalgebras; in
particular, we prove analogues of the Lie, Engel, and Cartan-Jacobson Theorems.
We show that the cohomology theory of Lie pseudoalgebras describes extensions
and deformations and is closely related to Gelfand-Fuchs cohomology. Lie
pseudoalgebras are closely related to solutions of the classical Yang-Baxter
equation, to differential Lie algebras (introduced by Ritt), and to Hamiltonian
formalism in the theory of nonlinear evolution equations. As an application of
our results, we derive a classification of simple and semisimple linear Poisson
brackets in any finite number of indeterminates.Comment: 102 pages, 7 figures, AMS late
Legendrian Distributions with Applications to Poincar\'e Series
Let be a compact Kahler manifold and a quantizing holomorphic
Hermitian line bundle. To immersed Lagrangian submanifolds of
satisfying a Bohr-Sommerfeld condition we associate sequences , where is a
holomorphic section of . The terms in each sequence concentrate
on , and a sequence itself has a symbol which is a half-form,
, on . We prove estimates, as , of the norm
squares in terms of . More generally, we show that if and
are two Bohr-Sommerfeld Lagrangian submanifolds intersecting
cleanly, the inner products have an
asymptotic expansion as , the leading coefficient being an integral
over the intersection . Our construction is a
quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of . We prove
that the Poincar\'e series on hyperbolic surfaces are a particular case, and
therefore obtain estimates of their norms and inner products.Comment: 41 pages, LaTe
Global action-angle coordinates for completely integrable systems with noncompact invariant submanifolds
The obstruction to the existence of global action-angle coordinates of
Abelian and noncommutative (non-Abelian) completely integrable systems with
compact invariant submanifolds has been studied. We extend this analysis to the
case of noncompact invariant submanifolds.Comment: 13 pages, to be published in J. Math. Phys. (2007
Resorption of Natural Calcium Carbonate by Avian Osteoclasts In Vitro
Osteoclasts isolated from the endosteum of 2.5 to 3-week chick tibia were cultured on glass coverslips or natural CaC03 (Tridacna) wafers for 2 and 4 days. The cells were exposed to the pH-dependent dye, acridine orange, and fluorescence was measured by a light microscope photometer. Fluorescence intensity values were higher in cells adherent to Tridacna wafers than in those incubated on glass after 2 and 4 days of culture (three and two-fold, respectively). Moreover, osteoclasts on Tridacna wafers were more flattened and were found to produce resorption pits. Acid production by osteoclasts cultured on Tridacna wafers was stimulated with 10-8 M parathyroid hormone and inhibited with 10-7 M acetazolamide or 10-7 M hydroxybenezoyl thiophene sulfonamide, as shown by changes in intensity of acridine orange fluorescence after 30, 60 and 120 minutes of treatment. These results indicate that osteoclasts cultured on natural CaC03 wafers mimic the behavior of osteoclasts cultured on other substrates. Further, the capacity to acidify was enhanced in cells cultured on CaC03 wafers. These results indicate that natural CaC03 Tridacna wafers provide a suitable substrate for osteoclasts in culture and demonstrate that carbonic anhydrase plays a role in carbonated substrate resorption
Complex zeros of real ergodic eigenfunctions
We determine the limit distribution (as ) of complex
zeros for holomorphic continuations \phi_{\lambda}^{\C} to Grauert tubes of
real eigenfunctions of the Laplacian on a real analytic compact Riemannian
manifold with ergodic geodesic flow. If is an
ergodic sequence of eigenfunctions, we prove the weak limit formula
\frac{1}{\lambda_j} [Z_{\phi_{j_k}^{\C}}] \to \frac{i}{\pi} \bar{\partial}
{\partial} |\xi|_g, where [Z_{\phi_{j_k}^{\C}}] is the current of
integration over the complex zeros and where is with respect
to the adapted complex structure of Lempert-Sz\"oke and Guillemin-Stenzel.Comment: Added some examples and references. Also added a new Corollary, and
corrected some typo
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