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 $H$. The Lie algebras in this category are called Lie $H$-pseudoalgebras. The main result of this paper is the classification of all simple and all semisimple Lie $H$-pseudoalgebras which are finitely generated as $H$-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 $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $\Lambda$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |\Lambda, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|\Lambda, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $\Lambda$, and a sequence itself has a symbol which is a half-form, $\sigma$, on $\Lambda$. We prove estimates, as $k\to\infty$, of the norm squares $\langle \Lambda, k|\Lambda, k\rangle$ in terms of $\int_\Lambda \sigma\overline{\sigma}$. More generally, we show that if $\Lambda_1$ and $\Lambda_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langle\Lambda_1, k|\Lambda_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $\Lambda_1\cap\Lambda_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. 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 $\lambda \to \infty$) 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 $(M, g)$ with ergodic geodesic flow. If $\{\phi_{j_k} \}$ 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 $\bar{\partial}$ 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

Slow Diffeomorphisms of a Manifold with Two Dimensions Torus Action

The uniform norm of the differential of the n-th iteration of a diffeomorphism is called the growth sequence of the diffeomorphism. In this paper we show that there is no lower universal growth bound for volume preserving diffeomorphisms on manifolds with an effective two dimensions torus action by constructing a set of volume-preserving diffeomorphisms with arbitrarily slow growth.Comment: 12 p