Semi-pointed partition posets and Species

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Fa{\`a} di Bruno Hopf algebra.Comment: 27 page

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

A new two--parameter family of isomonodromic deformations over the five punctured sphere

The object of this paper is to describe an explicit two--parameter family of logarithmic flat connections over the complex projective plane. These connections have dihedral monodromy and their polar locus is a prescribed quintic composed of a circle and three tangent lines. By restricting them to generic lines we get an algebraic family of isomonodromic deformations of the five--punctured sphere. This yields new algebraic solutions of a Garnier system. Finally, we use the associated Riccati one--forms to construct an interesting non--generic family of transversally projective Lotka--Volterra foliations.Comment: English text, 30 page

Manin's conjecture for a cubic surface with 2A_2+A_1 singularity type

We establish Manin's conjecture for a cubic surface split over Q and whose singularity type is 2A_2+A_1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath-Brown) and draws on the work of Deligne.Comment: 34 page

On shapes of Algebraic loci: wild-life in the forests of geometry

It is of great interest to determine the topological shape of the locus of solutions of any given set of polynomial equations. Answer to this geometrical question leads us to the interaction of topology, algebraic/analytic geometry and differential geometry

Very Local Interstellar Medium Revealed by a Complete Solar Cycle of Interstellar Neutral Helium Observations with IBEX

The IBEX-Lo instrument on board the Interstellar Boundary Explorer (IBEX) mission samples interstellar neutral (ISN) helium atoms penetrating the heliosphere from the very local interstellar medium (VLISM). In this study, we analyze the IBEX-Lo ISN helium observations covering a complete solar cycle, from 2009 through 2020 using a comprehensive uncertainty analysis including statistical and systematic sources. We employ the Warsaw Test Particle Model to simulate ISN helium fluxes at IBEX, which are subsequently compared with the observed count rate in the three lowest energy steps of IBEX-Lo. The χ2 analysis shows that the ISN helium flows from ecliptic \left(\lambda ,\beta \right)=(255\buildrel{\circ}\over{.} 59\pm 0\buildrel{\circ}\over{.} 23,5\buildrel{\circ}\over{.} 14\pm 0\buildrel{\circ}\over{.} 08), with speed vHP = 25.86 ± 0.21 km s−1 and temperature THP = 7450 ±140 K at the heliopause. Accounting for gravitational attraction and elastic collisions, the ISN helium speed and temperature in the pristine VLISM far from the heliopause are vVLISM = 25.9 km s−1 and TVLISM = 6150 K, respectively. The time evolution of the ISN helium fluxes at 1 au over 12 yr suggests significant changes in the IBEX-Lo detection efficiency, higher ionization rates of ISN helium atoms in the heliosphere than assumed in the model, or an additional unaccounted for signal source in the analyzed observations. Nevertheless, we do not find any indication of the evolution of the derived parameters of ISN helium over the period analyzed. Finally, we argue that the continued operation of IBEX-Lo to overlap with the Interstellar Mapping and Acceleration Probe will be pivotal in tracking possible physical changes in the VLISM