Let X be a homogeneous polynomial vector field of degree 2 on S^2. We show that if X has at least a non--hyperbolic singularity, then it has no limit cycles. We give necessary and sufficient conditions for determining if a singularity of X on S^2 is a center and we characterize the global phase portrait of X modulo limit cycles. We also study the Hopf bifurcation of X and we reduce the 16^{th} Hilbert's problem restricted to this class of polynomial vector fields to the study of two particular families. Moreover, we present two criteria for studying the nonexistence of periodic orbits for homogeneous polynomial vector fields on S^2 of degree n.
In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}= α_2(x),\quad\dot{y} = ky^2+β_1(x) y + γ_2(x), $$ with $(x,y)\in\mathbb{R}^2$, $γ_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\neq0$ (otherwise the system is a Lienard differential system), $β_ 1(x)$ a polynomial of degree at most $1$, $α_ 2(x)$ and $γ_ 2(x)$ polynomials of degree at most 2, and the maximum of the degrees of $ α_2(x)$ and $k y^2+β_1(x) y + γ_2(x)$ is 2. We give the complete description of their phase portraits in the Poincare disk
We study the phase portraits with positive probability of random planar homogeneous vector fields of degree n. In particular, for n=1,2,3, we give a complete solution of the problem and, moreover, either we give the exact value of each probability or we estimate it by using the Monte Carlo method. It is remarkable that all but two of these phase portraits are characterized by their index at the origin and by their number of invariant straight lines through it.
We compute the statistics of $SL_{d}(\mathbb{Z})$ matrices lying on level sets of an integral polynomial defined on $SL_{d}(\mathbb{R})$, a result that is a variant of the well known theorem proved by Linnik about the equidistribution of radially projected integral vectors from a large sphere into the unit sphere. Using the above result we generalize the work of Aka, Einsiedler and Shapira in various directions. For example, we compute the joint distribution of the residue classes modulo $q$ and the properly normalized orthogonal lattices of primitive integral vectors lying on the level set $-…
Partition regularity in imaginary quadratic rings of integers
Universidad de Chile（智利大学）；Centro de Modelamiento Matemático（数学建模中心）；Complutense University of Madrid（马德里康普顿斯大学）；Aristotle University of Thessaloniki（塞萨洛尼基亚里士多德大学）；Virginia Tech（弗吉尼亚理工大学）
