On the Poincare Polynomial of the full Linear Group

Generic-case approach to algorithmic problems was suggested by A. Miasnikov, I. Kapovich, P. Schupp and V. Shpilrain in 2003. This approach studies behavior of an algo-rithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we prove that the subset sum problems for the monoid of integer positive unimodular matrices of the second order, the special linear group of the second order, and the modular group are generically solvable in polynomial time.

Download Full-text

Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

Download Full-text

Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09495-y ◽

2021 ◽

Author(s):

Adrien Laurent ◽

Gilles Vilmart

Keyword(s):

Invariant Measure ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Linear Group ◽

Langevin Equation ◽

Numerical Experiments ◽

Special Linear Group ◽

Order Conditions ◽

High Order ◽

Differential Equations On Manifolds

AbstractWe derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.

Download Full-text

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

Forum Mathematicum ◽

10.1515/forum-2020-0054 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Ivan Matić

Keyword(s):

Linear Group ◽

Local Field ◽

General Linear Group ◽

Discrete Series ◽

Series Representation ◽

General Linear ◽

Composition Series ◽

Induced Representations ◽

Discrete Series Representation

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

Download Full-text

TOPOLOGICAL QUANTIZATION OF THE MAGNETIC FLUX

Modern Physics Letters A ◽

10.1142/s021773230000181x ◽

2000 ◽

Vol 15 (24) ◽

pp. 1491-1495 ◽

Cited By ~ 1

Author(s):

DANIEL WISNIVESKY

Keyword(s):

Quantum Theory ◽

Charged Particle ◽

Magnetic Flux ◽

Linear Group ◽

Connected Region ◽

Magnetic Tube ◽

Dirac Monopole ◽

Multiply Connected ◽

Quantum Problem ◽

Multiply Connected Region

We discuss the quantum problem of a charged particle in a multiply connected region encircling a magnetic tube, using a theory in which space and internal coordinates are derived from the parameters of a linear group of transformations (group space quantum theory). Based only on symmetry considerations, we show that, the magnetic flux in the tube must be quantized in multiples of the Dirac monopole charge.

Download Full-text