Narrow Results

To describe the "slow" motions of n interacting mass points, we give the most general four-dimensional (4D) noninstantaneous, nonparticle symmetric Galilei-invariant variational principle. It involves two-body invariants constructed from particle 4-positions and 4-velocities of the proper orthochronous inhomogeneous Galilei group. The resulting 4D equations of motion and multiple-time conserved quantities involve integrals over the worldlines of the other n-1 interacting particles. For a particular time-asymmetric retarded (advanced) interaction, we show the vanishing of all integrals over worldlines in the ten standard 4D multiple-time conserved quantities, thus yielding a Newtonian-like initial value problem. This interaction gives 3D noninstantaneous, nonparticle symmetric, coupled nonlinear second-order delay-differential equations of motion that involve only algebraic combinations of nonsimultaneous particle positions, velocities, and accelerations. The ten 3D noninstantaneous, nonparticle symmetric conserved quantities involve only algebraic combinations of nonsimultaneous particle positions and velocities. A two-body example with a generalized Newtonian gravity is provided. We suggest that this formalism might be useful as an alternative slow-motion mechanics for astrophysical applications.

In this paper we prove that the (non-trivial) irreducible complex unipotent characters of a finite classical group G of type B_m or C_m in odd characteristic are reducible over any proper subgroup of G, apart from very few notable exceptions.

In this paper, we determine the finite classical simple groups of dimension n = 3, 5 which are (2, 3)-generated (the cases n = 2, 4 are known). If n = 3, they are PSL₃(q), q ≠ 4, and PSU₃(q²), q² ≠ 9, 25. If n = 5 they are PSL₅(q), for all q, and PSU₅(q²), q² ≥ 9. Also, the soluble group PSU₃(4) is not (2, 3)-generated. We give explicit (2, 3)-generators of the linear preimages, in the special linear groups, of the (2, 3)-generated simple groups.

Based on the general strategy described by Borel and Serre and the Voronoi algorithm for computing unit groups of orders we present an algorithm for finding presentations of $S$-unit groups of orders. The algorithm is then used for some investigations concerning the congruence subgroup property.

In this paper, we construct a Cartesian authentication code from subspaces of orthogonal space of odd characteristic and compute its parameters. Assuming that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the probabilities of successful impersonation attack and substitution attack are also computed.

In this paper, we construct a Cartesian authentication code from subspaces of singular symplectic space and compute its parameters. Assuming that the encoding rules of the transmitter and the receiver are chosen according to a uniform probability distribution, the probabilities of successful impersonation attack and substitution attack are also computed.

The purpose of this paper is to put forward the claim that Hurwitz’s paper [Über die Erzeugung der invarianten durch integration, Nachr. Ges. Wiss. Göttingen1897 (1897) 71–90.] should be regarded as the origin of random matrix theory in mathematics. Here Hurwitz introduced and developed the notion of an invariant measure for the matrix groups $SO(N)$ and $U(N)$. He also specified a calculus from which the explicit form of these measures could be computed in terms of an appropriate parametrization — Hurwitz chose to use Euler angles. This enabled him to define and compute invariant group integrals over $SO(N)$ and $U(N)$. His main result can be interpreted probabilistically: the Euler angles of a uniformly distributed matrix are independent with beta distributions (and conversely). We use this interpretation to give some new probability results. How Hurwitz’s ideas and methods show themselves in the subsequent work of Weyl, Dyson and others on foundational studies in random matrix theory is detailed.

Let F be a field and K a subfield of F. We classify the overgroups in GL(n, F) of an Sp(n, K, f_K), SU(n, K, f_K) or Ω(n, K, Q_K) provided that the index [F : K] is not much bigger than the Witt index ν(f_k) or ν(Q_K).

In this paper, we first present the basic ideas of the method of determining reducibility or irreducibility of parabolically induced representations of classical p-adic groups using Jacquet modules. After that we explain the construction of irreducible square integrable representations by considering characteristic examples. We end with a brief presentation of the classification of irreducible square integrable representations of these groups modulo cuspidal data, which was obtained jointly with C. Mœglin.