Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this set. Our main result provides an essentially sharp uniform bound, depending only on $N$, for $L^2$ operator norm dimensions 3 and higher. The ingredients proof consist polynomial partitioning tools from incidence geometry almost-ort...

Despite a search, no chaotic driven complex-variable oscillators of the form z+f(z)=e(iOmegat) or z+f(z)=e(iOmegat) are found, where f is a polynomial with real coefficients. It is shown that, for analytic functions f(z), driven complex-variable oscillators of the form z+f(z)=e(iOmegat) cannot have chaotic solutions. Seven simple driven chaotic oscillators of the form z+f(z,z)=e(iOmegat) with p...

A “double star” is a tree with two internal vertices. It known that the Gyárfás–Sumner conjecture holds for double stars, is, every star H $H$ , there function f ${f}_{H}$ such if G $G$ does not contain as an induced subgraph then ? ( ) ? ? $\chi (G)\le {f}_{H}(\omega (G))$ (where ,\omega $ are chromatic number and clique of ). Here we prove can be chosen to polynomial.

In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing , if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic comp...

The $L^2$-orthogonal projection $\Pi_h:L^2(\Omega)\rightarrow\mathbb{V}_h$ onto a finite element (FE) space $\mathbb{V}_h$ is called $H^1$-stable iff $\|\nabla\Pi_h u\|_{L^2(\Omega)}\leq C\|u\|_{H^1(\Omega)}$, for any $u\in H^1(\Omega)$ with positive constant $C\neq C(h)$ independent of the mesh size $h>0$. In this work, we discuss local criteria $H^1$-stability adaptively refined meshes. We sh...

The first measurements of O2 broadening of NO lines is reported. For the NO fundamental vibration-rotation band, polynomial representations for this broadening at 299 K are given for Omega = 32 and Omega = (1/2) e and f transitions, respectively: gamma3/2(|m|) = 0.06463 - 0.00262|m| + 0.000147|m|2 - 0.00000327|m|3, gammaf1/2 (|m|) = 0.06515 - 0.00381|m| + 0.000271|m|2 - 0.00000665|m|3, and gamm...

Rhombellanes are mathematical structures existing in various environments, in crystal or quasicrystal networks, or even in their homeomorphs, further possible becoming real molecules. Rhombellanes originate in the K2.3 complete bipartite graph, a tile found in the linear polymeric staffanes. In close analogy, a rod-like polymer derived from hexahydroxy-cyclohexane was imagined. Further, the ide...

We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of polynomial growth Cayley graph $\Gamma$. prove that for each $k \in \mathbb{N}$, $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally $1/|B|^{\frac{1}{d-1}}$, where $d$ represents rate The method consists in associating manifold $M$ $\Gamma$ and bounded domain $N \subset M$ $(\Omega, B)$ find upper bounds spectrum $N$ tr...