The rectilinear local crossing number of KnK_n

We determine ${\bar{\rm{lcr}}}(K_n)$, the rectilinear local crossing number of the complete graph $K_n$ for every $n$. More precisely, for every $n \notin \{8, 14 \},$ $${\bar{\rm{lcr}}}(K_n)=\left\lceil \frac{1}{2} \left( n-3-\left\lceil \frac{n-3}{3} \right\rceil \right) \left\lceil \frac{n-3}{3} \right\rceil \right\rceil,$$ ${\bar{\rm{lcr}}}(K_8)=4$, and ${\bar{\rm{lcr}}}(K_{14})=15$.Comment: 6 Figures. Changes from v1: Added keywords, MSC2010 codes, a single formula to consider all cases together, and the resolution of the case n=14 that remained as a conjecture on the previous version. Changes from v2: A minor error in Lemma 2 was corrected. Some typos were fixed. Figure 1 was eliminated and Figures 2 and 5 were improved slightly. The last section was split into two section

Book crossing numbers of the complete graph and small local convex crossing numbers

A $k$-page book drawing of a graph $G$ is a drawing of $G$ on $k$ halfplanes with common boundary $l$, a line, where the vertices are on $l$ and the edges cannot cross $l$. The $k$-page book crossing number of the graph $G$, denoted by $\nu_k(G)$, is the minimum number of edge-crossings over all $k$-page book drawings of $G$. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $\nu_k(K_n)$ for all $k\geq 14$ and determine $\nu_k(K_n)$ whenever $2 < n/k \leq 3$. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $\ell$ can have for $\ell\leq 4$.Comment: Version 2 extends our result on the maximum number of edges on a convex graph with local crossing number at most c for c=4. In This in turn improves most of the bounds in Version

Environmental Impact of Different Business Models: An {LCA} Study of Fresh Tomato Production in Italy

This paper aims to define and measure the environmental impacts associated with three distinct industry techniques used in the production of fresh tomatoes, from planting to harvesting. A comparative Life Cycle Assessment (LCA) is conducted to evaluate the environmental performance of producing one kilogram of fresh processing tomatoes. The study includes three Italian companies representing different Business Models: standard production, integrated production, and precision farming. The impact evaluation method used is ReCiPe, at both midpoint and endpoint levels. The results reveal that the adoption of advanced technology, such as precision farming, in managing the agricultural production of fresh tomatoes leads to a significant reduction in environmental impacts across all categories analyzed. Interestingly, the largest contribution to climate-changing gases comes from the use of fertilizers. Precision farming demonstrates a remarkable reduction of CO2 emissions by 52% compared to integrated production and 67% compared to standard production. This analysis provides valuable insights for managers and policymakers in developing strategies for large-scale, sustainable production using innovative techniques. The main contribution of this study lies in comparing the performance of three different Business Models employed in Italian fresh tomato production

Gold-Catalyzed Reaction of propargyl Esters and alkynylsilanes: Synthesis of vinylallene derivatives thorugh a Twofold 1,2-Rearrangement

The reaction of propargyl esters with alkynylsilanes under gold catalysis provides vinylallene derivatives through consecutive [1,2]-acyloxy/[1,2]-silyl rearrangements. Good yields, full atom-economy, broad substrate scope, easy scale-up and low catalyst loadings are salient features of this novel transformation. Density Functional Theory (DFT) calculations suggest a reaction mechanism involving initial [1,2]-acyloxy rearrangement to generate a gold vinylcarbene intermediate which upon regioselective attack of the alkynylsilane affords a vinyl cation which undergoes a type II-dyotropic rearrangement involving the silyl group and the metal fragment. Preliminary results on the enantioselective version of this transformation are also disclose

Geometric drawings of Kn with few crossings

AbstractWe give a new upper bound for the rectilinear crossing number cr¯(n) of the complete geometric graph Kn. We prove that cr¯(n)⩽0.380559(n4)+Θ(n3) by means of a new construction based on an iterative duplication strategy starting with a set having a certain structure of halving lines