No Arabic abstract
The mesochronic velocity is the average of the velocity field along trajectories generated by the same velocity field over a time interval of finite duration. In this paper we classify initial conditions of trajectories evolving in incompressible vector fields according to the character of motion of material around the trajectory. In particular, we provide calculations that can be used to determine the number of expanding directions and the presence of rotation from the characteristic polynomial of the Jacobian matrix of mesochronic velocity. In doing so, we show that (a) the mesochronic velocity can be used to characterize dynamical deformation of three-dimensional volumes, (b) the resulting mesochronic analysis is a finite-time extension of the Okubo--Weiss--Chong analysis of incompressible velocity fields, (c) the two-dimensional mesochronic analysis from Mezic et al. emph{A New Mixing Diagnostic and Gulf Oil Spill Movement}, Science 330, (2010), 486-489, extends to three-dimensional state spaces. Theoretical considerations are further supported by numerical computations performed for a dynamical system arising in fluid mechanics, the unsteady Arnold--Beltrami--Childress (ABC) flow.
In this short note, we study the distribution of spreads in a point set $mathcal{P} subseteq mathbb{F}_q^d$, which are analogous to angles in Euclidean space. More precisely, we prove that, for any $varepsilon > 0$, if $|mathcal{P}| geq (1+varepsilon) q^{lceil d/2 rceil}$, then $mathcal{P}$ generates a positive proportion of all spreads. We show that these results are tight, in the sense that there exist sets $mathcal{P} subset mathbb{F}_q^d$ of size $|mathcal{P}| = q^{lceil d/2 rceil}$ that determine at most one spread.
We prove that a sufficiently large subset of the $d$-dimensional vector space over a finite field with $q$ elements, $ {Bbb F}_q^d$, contains a copy of every $k$-simplex. Fourier analytic methods, Kloosterman sums, and bootstrapping play an important role.
The dynamics in three-dimensional billiards leads, using a Poincare section, to a four-dimensional map which is challenging to visualize. By means of the recently introduced 3D phase-space slices an intuitive representation of the organization of the mixed phase space with regular and chaotic dynamics is obtained. Of particular interest for applications are constraints to classical transport between different regions of phase space which manifest in the statistics of Poincare recurrence times. For a 3D paraboloid billiard we observe a slow power-law decay caused by long-trapped trajectories which we analyze in phase space and in frequency space. Consistent with previous results for 4D maps we find that: (i) Trapping takes place close to regular structures outside the Arnold web. (ii) Trapping is not due to a generalized island-around-island hierarchy. (iii) The dynamics of sticky orbits is governed by resonance channels which extend far into the chaotic sea. We find clear signatures of partial transport barriers. Moreover, we visualize the geometry of stochastic layers in resonance channels explored by sticky orbits.
In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of dimension n over a finite field of characteristic l, for l at least 5, which can be derived from work of Kantor: G is either reducible, symplectically imprimitive or it contains Sp(n, l). This result is for instance useful for proving big image results for symplectic Galois representations.
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${mathbb F}_q$, the finite field with q elements, by $A cdot A+... +A cdot A$, where A is a subset ${mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.