On Grids in Point-Line Arrangements in the Plane

The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for point-line incidence. Let $\mathcal{L}_1$ and $\mathcal{L}_2$ be two sets of $t$ lines in the plane and let $P=\{\ell_1 \cap \ell_2 : \ell_1 \in \mathcal{L}_1, \ell_2 \in \mathcal{L}_2\}$ be the set of intersection points between $\mathcal{L}_1$ and $\mathcal{L}_2$. We say that $(P, \mathcal{L}_1 \cup \mathcal{L}_2)$ forms a \emph{natural $t\times t$ grid} if $|P| =t^2$, and $conv(P)$ does not contain the intersection point of some two lines in $\mathcal{L}_i,$ for $i = 1,2.$ For fixed $t > 1$, we show that any arrangement of $n$ points and $n$ lines in the plane that does not contain a natural $t\times t$ grid determines $O(n^{\frac{4}{3}- \varepsilon})$ incidences, where $\varepsilon = \varepsilon(t)$. We also provide a construction of $n$ points and $n$ lines in the plane that does not contain a natural $2 \times 2$ grid and determines at least $\Omega({n^{1+\frac{1}{14}}})$ incidences.Comment: 13 pages, 5 figure

Fluoride ionadsorptionontopalmstone: Optimizationthroughresponsesurface methodology,isotherm,andadsorbent characteristicsdata

In somepartoftheworld,groundwatersourcecanbecomeunsafe for drinkingduetothehighconcentrationof fluoride ions[1]. The low costandfacile-producedadsorbentlikepalmstonecould effectivelyremoved fluoride ionsthroughadsorptionprocess.In this dataset,theinfluence of fluoride ionconcentration,solution pH, adsorbentdosage,andcontacttimeon fluoride ionadsorption by palmstoneswastestedbycentralcompositedesign(CCD) under responsesurfacemethodology(RSM).Thedatastonecar- bonized adsorbentwaspreparedbyasimpleandfacilemethodat relativelylowtemperatureof250 °C during3h.Theadsorbenthad the mainfunctionalgroupsofO–H, –OH, Si–H, C¼O, N¼O, C–C, C– OR, C–H, andC–Br onitssurface.Attheoptimizedconditions obtained byRSM,about84.78%of fluoride ionwasremovedusing the adsorbent.TheLangmuirisothermwassuitableforcorrelation of equilibriumdata(maximumadsorptioncapacity¼ 3.95 mg/g). Overall,thedataofferafacileadsorbenttowaterandwastewater workswhichfacetohighlevelof fluoride water/wastewater content

Connections between additive combinatorics, graph theory, and incidence geometry

One of the Erd\H{o}s-like cornerstones in incidence geometry from which many other results follow is the celebrated Szemer\'edi-Trotter Theorem which states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this thesis, we study the effect of forbidding grids and short even cycles on the incidence graphs of point-line arrangements in the plane. Let $A$ and $B$ be two disjoint finite sets of points in the plane such that their union contains no three points on a line. We say that $A$ \emph{avoids} $B$ if no straight line determined by a pair of points in $A$ intersects the convex hull of $B.$ $A$ and $B$ arecalled mutually avoiding if $A$ avoids $B$ and $B$ avoids $A .$ Aronov et al. showed that any set of $n$ points in general positionin the plane contains a pair of mutually avoiding sets, each of size at least $\Omega(\sqrt{n})$. Moreover, they proved that any set of $n$ points in general position in $\mathbb{R}^{d}$ contains a pair of mutually avoiding sets, each of size at least $\Omega\left(n^{\frac{1}{d^{2}-d+1}}\right)$. In this thesis, we give a generalized version of mutually avoiding set theorem in the plane. Given an algebraic structure $R$ and a subset $A \subset R,$ define the sum set and the productset of $A$ to be $A+A=\{a+b: a, b \in A\}$ and $A \cdot A=\{a \cdot b: a, b \in A\}$ respectively.Showing under what conditions at least one of $|A+A|$ or $|A \cdot A|$ is large has a long history of study that continues to the present day. By employing recent developments on the energy of polynomials over finite fields, we give the best-known lowerbounds on $\max\{|A+A|, |f(A,A)|\}$, when $A$ is a small subset of $\mathbb{F}_p,$ and $f$ is a quadratic non-degenerate polynomial in $\mathbb{F}_p[x,y].

Connections between additive combinatorics, graph theory, and incidence geometry

One of the Erd\H{o}s-like cornerstones in incidence geometry from which many other results follow is the celebrated Szemer\'edi-Trotter Theorem which states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this thesis, we study the effect of forbidding grids and short even cycles on the incidence graphs of point-line arrangements in the plane. Let $A$ and $B$ be two disjoint finite sets of points in the plane such that their union contains no three points on a line. We say that $A$ \emph{avoids} $B$ if no straight line determined by a pair of points in $A$ intersects the convex hull of $B.$ $A$ and $B$ arecalled mutually avoiding if $A$ avoids $B$ and $B$ avoids $A .$ Aronov et al. showed that any set of $n$ points in general positionin the plane contains a pair of mutually avoiding sets, each of size at least $\Omega(\sqrt{n})$. Moreover, they proved that any set of $n$ points in general position in $\mathbb{R}^{d}$ contains a pair of mutually avoiding sets, each of size at least $\Omega\left(n^{\frac{1}{d^{2}-d+1}}\right)$. In this thesis, we give a generalized version of mutually avoiding set theorem in the plane. Given an algebraic structure $R$ and a subset $A \subset R,$ define the sum set and the productset of $A$ to be $A+A=\{a+b: a, b \in A\}$ and $A \cdot A=\{a \cdot b: a, b \in A\}$ respectively.Showing under what conditions at least one of $|A+A|$ or $|A \cdot A|$ is large has a long history of study that continues to the present day. By employing recent developments on the energy of polynomials over finite fields, we give the best-known lowerbounds on $\max\{|A+A|, |f(A,A)|\}$, when $A$ is a small subset of $\mathbb{F}_p,$ and $f$ is a quadratic non-degenerate polynomial in $\mathbb{F}_p[x,y].

Using Machine Learning to Estimate Nonorographic Gravity Wave Characteristics at Source Levels

International audienceMachine learning (ML) provides a powerful tool for investigating the relationship between the large-scale flow and unresolved processes, which need to be parameterized in climate models. The current work explores the performance of the random forest regressor (RF) as a nonparametric model in the reconstruction of nonorographic gravity waves (GWs) over midlatitude oceanic areas. The ERA5 dataset from the European Centre for Medium-Range Weather Forecasts (ECMWF) model outputs is employed in its full resolution to derive GW variations in the lower stratosphere. Coarse-grained variables in a column-based configuration of the atmosphere are used to reconstruct the GWs variability at the target level. The first important outcome is the relative success in reconstructing the GW signal (coefficient of determination R2 ≈ 0.85 for “E3” combination). The second outcome is that the most informative explanatory variable is the local background wind speed. This questions the traditional framework of gravity wave parameterizations, for which, at these heights, one would expect more sensitivity to sources below than to local flow. Finally, to test the efficiency of a relatively simple, parametric statistical model, the efficiency of linear regression was compared to that of random forests with a restricted set of only five explanatory variables. Results were poor. Increasing the number of input variables to 15 hardly changes the performance of the linear regression (R2 changes slightly from 0.18 to 0.21), while it leads to better results with the random forests (R2 increases from 0.29 to 0.37)