Existence of Prograde Double-Double Orbits in the Equal-Mass Four-Body Problem

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle {\theta\in(0,\pi/7]} in the equal-mass four-body problem. A new geometric argument is introduced to show that for any {\theta\in(0,\pi/2)}, the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.

A note on the generalization of bivariate normal quadrant probabilities to bivariate elliptical distributions

I present a geometric argument to show that the quadrant probability for the bivariate normal distribution can be generalized to the case of all elliptical distributions.

Geometric classification of semidirect products in the maximal parabolic subgroup of Sp(2, ℝ)

We classify up to conjugation by [Formula: see text] (more precisely, block diagonal symplectic matrices) all the semidirect products inside the maximal parabolic of [Formula: see text] by means of an essentially geometric argument. This classification has already been established in [G. S. Alberti, L. Balletti, F. De Mari and E. De Vito, Reproducing subgroups of [Formula: see text]. Part I: Algebraic classification, J. Fourier Anal. Appl. 9(4) (2013) 651–682] without geometry, under a stricter notion of equivalence, namely, conjugation by arbitrary symplectic matrices. The present approach might be useful in higher dimensions and provides some insight.

THE VECTOR-VALUED TENT SPACES AND

Tent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to $H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases $p=1$ and $p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case $p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.

Some Sampling Properties of Common Phase Estimators

The instantaneous phase of neural rhythms is important to many neuroscience-related studies. In this letter, we show that the statistical sampling properties of three instantaneous phase estimators commonly employed to analyze neuroscience data share common features, allowing an analytical investigation into their behavior. These three phase estimators—the Hilbert, complex Morlet, and discrete Fourier transform—are each shown to maximize the likelihood of the data, assuming the observation of different neural signals. This connection, explored with the use of a geometric argument, is used to describe the bias and variance properties of each of the phase estimators, their temporal dependence, and the effect of model misspecification. This analysis suggests how prior knowledge about a rhythmic signal can be used to improve the accuracy of phase estimates.

Computing nilpotent and unipotent canonical forms: a symmetric approach

Let k be an algebraically closed field of any characteristic except 2, and let G = GLn(k) be the general linear group, regarded as an algebraic group over k. Using an algebro-geometric argument and Dynkin–Kostant theory for G we begin by obtaining a canonical form for nilpotent Ad(G)-orbits in (k) which is symmetric with respect to the non-main diagonal (i.e. it is fixed by the map f : (xi,j) ↦ (xn+1−j,n+1−i)), with entries in {0,1}. We then show how to modify this form slightly in order to satisfy a non-degenerate symmetric or skew-symmetric bilinear form, assuming that the orbit does not vanish in the presence of such a form. Replacing G by any simple classical algebraic group we thus obtain a unified approach to computing representatives for nilpotent orbits of all classical Lie algebras. By applying Springer morphisms, this also yields representatives for the corresponding unipotent classes in G. As a corollary we obtain a complete set of generic canonical representatives for the unipotent classes in finite general unitary groups GUn(q) for all prime powers q.

GRID DIAGRAMS OF LORENZ LINKS

In paper [1] Birman and Kofman prove the coincidence of the class of Lorenz links and the class of twisted links. The proof in that work is algebraic. We will identify this class in terms of grid diagrams and provide a transparent geometric argument for Birman–Kofman's result.

A NEW PROOF THAT ALTERNATING LINKS ARE NON-TRIVIAL

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. Unlike most other proofs of this result, this proof uses only classic results in topology and combinatorial group theory.