arXiv subjects
Publications and source records attributed to M. Shapiro.
We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real provided that the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable.
We introduce the notion of rationality for hyperholomorphic functions (functions in the kernel of the Cauchy-Fueter operator). Following the case of one complex variable, we give three equivalent definitions: the first in terms of Cauchy-Kovalevskaya quotients of polynomials, the second in terms of realizations and the third in terms of backward-shift invariance. Also introduced and studied are the counterparts of the Arveson space and Blaschke factors.
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.
We discuss a functional model for multi--diagonal selfadjoint operators with almost periodic coefficients that generalizes the well known model for finite band Jacobi matrices. It give us an opportunity to construct examples of almost periodic operators with different spectral properties. Main result deals with an exact condition for the uniqueness of the model of the given type.
We study the distribution of the number of permutations with a given periodic up-down sequence w.r.t. the last entry, find exponential generating functions and prove asymptotic formulas for this distribution.
The possibility of accelerating molecular reactions by lasers has attracted considerable theoretical and experimental interest. A particular example of laser-modified reaction dynamics is laser catalysis, a process in which the tunneling through a potential barrier is enhanced by transient excitation to a bound electronic state. We have performed detailed calculations of pulsed laser catalysis on one- and two-dimensional potentials, as a function of the reactants' collision energy and the laser's central frequency. In agreement with previous CW results, the reactive lineshapes are Fano-type curves, resulting from interference between nonradiative tunneling and the optically assisted pathway. In contrast to the CW process, the power requirements of pulsed laser catalysis are well within the reach of commonly used pulsed laser sources, making an experimental realization possible. The laser catalysis scenario is shown to be equivalent in the ``dressed'' state picture, to resonant tunneling through a double-barrier potential, admitting perfect transmission when the incident energy matches a quasibound state of the well within the barriers. Possible applications for atom optics, solid-state devices, and scanning tunneling microscopy, are discussed.
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
We prove that the number of connected components in the intersection of two open opposite Schubert cells in the variety of complete real n-dimensional flags equals 3*2^{n-1} for n>5.
In this short note we give an explicit presentation of the algebra A_n generated by the curvature 2-forms of the standard Hermitiam line bundles over SL_n/B as the quotient of the polynomial ring. The difference between A_n and H^*(SL_n/B) reflects the fact that SL_n/B is not a symmetric space. Possible applications of A_n lie in the field of arithmetic intersection theory on flag varieties.
The group of real 4 by 4 upper triangular matrices with 1s on the diagonal has a left-invariant subRiemannian (or Carnot-Caratheodory) structure whose underlying distribution corresponds to the superdiagonal. We prove that the associated subRiemannian geodesic flow is not completely integrable. This provides the first example of a Carnot group (graded nilpotent Lie group with an invariant subRiemannian structure supported on the generating subspace) with a non-integrable geodesic flow. We apply this result to prove that the centralizer for the corresponding quadratic ``quantum'' Hamiltonian in the universal enveloping algebra for this group is ``as small as possible''.
We define a class of groups based on parallel computations by pushdown automata. This class generalizes automatic groups. It includes the fundamental groups of all 3-manifolds which obey Thurston' s geometrization conjecture. It also includes nilpotent groups of arbitrary class and polynomial degree isoperimetric inequality. It is closed under wreath product.
In this paper we prove the complete integrability of Toda flows on generic coadjoint orbits in simple Lie algebras.
In this paper we reduce the problem of counting the number of connected components in the intersection of two opposite open Schubert cells in the variety of real complete flags to a purely combinatorial question of counting the number of orbits of a certain intriguing group action in the space of upper triangular matrices with {0,1}-valued entries. The crucial step of our reduction uses the parametrization of the space of real unipotent totally positive upper triangular matrices introduced by Lusztig and Berenstein, Fomin, Zelevinski.