Related papers: The Hilbert 16-th problem and an estimate for cycl…

I discuss some recent work linking certain aspects of the second part of Hilbert's 16th problem to the theory of \hbox{o-minimality}. These notes are adapted from a lecture I gave in the Jour fixe seminar series at the Zukunfts\-kolleg of…

Let $\xi$ be a real analytic vector field with an elementary isolated singularity at $0\in \mathbb{R}^3$ and eigenvalues $\pm bi,c$ with $b,c\in \mathbb{R}$ and $b\neq 0$. We prove that all cycles of $\xi$ in a sufficiently small…

Let $\{v_{\alpha}\}$ be a system of polynomial solutions of the parabolic equation $a_{hk}\partial_{x_{h}x_{k}}u - \partial_t u =0$ in a bounded $C^1$-cylinder $\Omega_{T}$ contained in $\mathbb{R}^{n+1}$. Here $a_{hk}\partial_{x_{h}x_{k}}$…

We study Hamiltonicity and pancyclicity in the graph obtained as the union of a deterministic $n$-vertex graph $H$ with $\delta(H)\geq\alpha n$ and a random $d$-regular graph $G$, for $d\in\{1,2\}$. When $G$ is a random $2$-regular graph,…

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this…

The paper deals with a complex polynomial $H$ in two variables having - a generic highest homogeneous part (without multiple zero lines), - nonconstant lower terms. In particular, under these conditions the polynomial $H$ has at least two…

Motivated by the classical Hilbert's Sixteenth Problem, we extend some main developments obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context. Specifically, we study the growth of the maximum number of…

Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer…

Consider a semiclassical Hamiltonian \begin{equation*} H_{V, h} := h^{2} \Delta + V - E \end{equation*} where $h > 0$ is a semiclassical parameter, $\Delta$ is the positive Laplacian on $\mathbb{R}^{d}$, $V$ is a smooth, compactly supported…

The Kervaire Invariant 1 Problem until recently was an open problem in algebraic topology. Hill-Hopkins-Ravenel theorem clams a negative solution of the problem for all dimensions $n=2^l-2$, $l \ge 8$. We prove the statement of…

The Hilbert functions and the regularity of the graded components of local cohomology of a bigraded algebra are considered. Explicit bounds for these invariants are obtained for bigraded hypersurface rings.

For any $k\ge 3$ and $\ell \in [k-1]$ such that $(k,\ell) \ne (3,1)$, we show that any sufficiently large $k$-graph $G$ must contain a Hamilton $\ell$-cycle provided that it has no isolated vertices and every set of $k-1$ vertices contained…

This paper contains two parts. In the first part, we shall study the Abelian integrals for Zoladek's example [13], in which it is claimed the existence integrals of 11 small-amplitude limit cycles around a singular point in a particular…

In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth…

We study the number of limit cycles that a planar polynomial vector field can have as a function of its number $m$ of monomials. We prove that the number of limit cycles increases at least quadratically with $m$ and we provide good lower…

Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining…

For $n\ge 5$ and $k\ge 4$, we show that any minimizing biharmonic map from $\Omega\subset R^n$ to $S^k$ is smooth off a closed set whose Hausdorff dimension is at most $n-5$. When $n=5$ and $k=4$, for a parameter $\lambda\in [0,1]$ we…

The description of many dynamical problems like the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of hyperelliptic integrals of all three kinds. The result of the inversion…

The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable $(C_M, C_N)$-decomposition of $K_v$ into $\alpha$ $C_M$-factors and $\beta$ $C_N$-factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by…

This paper presents a numerical study on multigrid algorithms of $V$-cycle type for problems posed in the Hilbert space $H(\mathbf{curl})$ in three dimensions. The multigrid methods are designed for discrete problems originated from the…