On upper bounds on the smallest size of a saturating set in a projective plane

In a projective plane $\Pi _{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $\Pi _{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following upper bound on the smallest size $s(2,q)$ of a saturating set in $\Pi _{q}$ is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant $c\ge 1$ a random point set of size $k$ in $\Pi _{q}$ with $2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q$ is a saturating set with probability greater than $1-1/(q+1)^{2c^{2}-2}.$ Our probabilistic approach is also applied to multiple saturating sets. A point set $S\subset \Pi_{q}$ is $(1,\mu)$-saturating if for every point $Q$ of $\Pi _{q}\setminus S$ the number of secants of $S$ through $Q$ is at least $\mu$, counted with multiplicity. The multiplicity of a secant $\ell$ is computed as ${\binom{{\#(\ell \,\cap S)}}{{2}}}.$ The following upper bound on the smallest size $s_{\mu }(2,q)$ of a $(1,\mu)$-saturating set in $\Pi_{q}$ is proved: \begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a $(1,\mu)$-saturating set) in the projective space $PG(N,q)$ are obtained. All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and some references are adde

Towards Traditional Carbon Fillers: Biochar-Based Reinforced Plastic

The global market of carbon-reinforced plastic represents one of the largest economic platforms. This sector is dominated by carbon black (CB) produced from traditional oil industry. Recently, high technological fillers such as carbon fibres or nanostructured carbon (i.e. carbon nanotubes, graphene, graphene oxide) fillers have tried to exploit their potential but without economic success. So, in this chapter we are going to analyse the use of an unconventional carbon filler called biochar. Biochar is the solid residue of pyrolysis and can be a solid and sustainable replacement for traditional and expensive fillers. In this chapter, we will provide overview of the last advancement in the use of biochar as filler for the production of reinforced plastics

Introducing the Novel Mixed Gaussian-Lorentzian Lineshape in the Analysis of the Raman Signal of Biochar

In this research, an innovative procedure is proposed to elaborate Raman spectra obtained from nanostructured and disordered solids. As a challenging case study, biochar, a bio-derived carbon based material, was selected. The complex structure of biochar (i.e., channeled surface, inorganic content) represents a serious challenge for Raman characterization. As widely reported, the Raman spectra are closely linked to thermal treatments of carbon material. The individual contributions to the Raman spectra are difficult to identify due to the numerous peaks that contribute to the spectra. To tackle this problem, we propose a brand new approach based on the introduction, on sound theoretical grounds, of a mixed Gaussian-–Lorentzian lineshape. As per the experimental part, biochar samples were carbonized in an inert atmosphere at various temperatures and their respective spectra were successfully decomposed using the new lineshape. The evolution of the structure with carbonization temperature was investigated by Raman and XRD analysis. The results of the two techniques fairly well agree. Compared to other approaches commonly reported in the literature this method (i) gives a sounder basis to the lineshape used in disordered materials, and (ii) appears to reduce the number of components, leading to an easier understanding of their origin