Lattices generated by orbits of subspaces under finite singular pseudo-symplectic groups I

Let be the (2ν+1+l)-dimensional vector space over the finite field . In the paper we assume that is a finite field of characteristic 2, and the singular pseudo-symplectic groups of degree 2ν+1+l over . Let be any orbit of subspaces under . Denote by the set of subspaces which are intersections of subspaces in and the intersection of the empty set of subspaces of is assumed to be . By ordering by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice , and the characteristic polynomial of .     

Adjoint orbits of the odd real symplectic group

In this paper we classify the adjoint orbits of the odd symplectic group over the field of real numbers. We need a noneigenvalue modulus to classify certain orbits.      

Large element orders and the characteristic of finite simple symplectic groups

The characteristic of a simple group of Lie type is the characteristic of the field over which this group is defined. Let G = Sp_2n (q), where q = 2 ^k . It is shown that every finite group of Lie type with the same two largest element orders as G has characteristic 2.     

On adjacencies of orbits of a symplectic group action on the manifold of complete flags in a symplectic space

Translated from Itogi Nauki i Tekhniki, Seriya Sovremenaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 20, Topologia-3, 1994.     

Preservation and destruction of periodic orbits by symplectic integrators

We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.     

On the determination of a measure by the orbits generated by its logarithmic potential

We say that a logarithmic potential generates a curve in the plane if a unit mass traces the curve under the action of the potential. We consider the following problem: A one-parameter family of plane curves is given. We assume that these curves lie in the complement of a compact set . Find all measures supported in whose potentials generate each of the given curves. We solve this problem when is the unit circle in three specific cases: (a) when the given curves are straight lines through the origin, (b) when the curves are straight lines through a point on the unit circle, and (c) when the curves are circles centered at the origin. The solution involves the Poisson integral and its boundary behavior.

     

On the common part of the spectrum of finite difference operators generated by systems of polynomials

In this paper, we study the spectra of finite difference operators generated by systems of multiple orthogonal polynomials and the corresponding systems of measures of Stieltjes type. We show that the common support of the orthogonality measures coincides with the intersection of the spectra of the family of finite difference operators with common collection of Weyl functions.     

Integral geometry under the action of the first symplectic group

We show a Poincaré formula of real surfaces in the quaternion numbers H, which we regard as the Euclidean space of dimension four, under the action of the semidirect product of Sp (1) and the parallel translations of H.     

On an iterated construction of irreducible polynomials over finite fields of even characteristic by Kyuregyan

We deal with the construction of sequences of irreducible polynomials with coefficients in finite fields of even characteristic. We rely upon a transformation used by Kyuregyan in 2002, which generalizes the Q-transform employed previously by Varshamov and Garakov (1969) as well as by Meyn (1990) for the synthesis of irreducible polynomials. While in the iterative procedure described by Kyuregyan the coefficients of the initial polynomial of the sequence have to satisfy certain hypotheses, in the present paper these conditions are removed. We construct infinite sequences of irreducible polynomials of nondecreasing degree starting from any irreducible polynomial.     

Finding representatives for the orbits under the automorphism group of a bounded abelian group

Given any bounded abelian p-group A for some prime p, we determine a system of representatives for the orbits of A under its automorphism group. This is used to determine the number of orbits. As a consequence, certain types of groups are characterized by this number.