Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erd?s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles.In the plane, our main result is an O(n44/19)=O(n^2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω(nlogn) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most ; there exist n-element point sets (for arbitrarily large n) that span (6/π²−o(1))n² minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω(nlogn) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds).In 3-space we prove an O(n17/7β(n))=O(n^2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is an old result of Erd?s and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n²+O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω(n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε>0. (iii) Every set of n points, not all on a line, determines at least Ω(n2/3/β(n)) triangles of distinct areas, which share a common side.     

Generic Fiber of Power Series Ring Extensions

Let D be a Noetherian domain containing a field, a ∈ D a nonzero nonunit, and z an indeterminate over D. We prove that the generic fiber of the extension D[[z]]? D[1/a][[z]] has dimension ≥ dim(D/aD).     

Reply to A Computational Evaluation of the Evidence for the Synthesis of 1,3‐Dimethylcyclobutadiene in Solid State and Aqueous Solution—Beyond the Experimental Reality

Following earlier reports on the photochemical synthesis of 1,3‐dimethylcyclobutadiene 8 , 10 in a protective host matrix, theoretical calculations for the formation of that adduct have been recently performed by Rzepa. 13 The author formulated criticisms based mainly on density functional theory calculations of ¹H NMR spectra. According to Rzepa the calculated spectra do not correspond with our measured spectra, which leads him to the conclusion that our interpretation is wrong, and that mainly cyclobutadiene has not been stabilized or even synthesized; we believe, however, that the initial model that Rzepa used for his calculations does not correspond to chemical reality or is at the very least a crude simplification of it, which implies that his calculations cannot match, in every point, our experimental spectra. Rzepa′s simplified models might be ‘reasonable’ from the theoretical point of view; however, in the case of assessment in the solid state, the theoretical setup does not force the system to preserve the confined stabilizing space defined by the crystalline matrix for encapsulated hosts in the solid state. Inversely, in the case of solution modeling, the theoretical setup is too rigid to properly assess the complex equilibria occurring in solution and to accurately determine the NMR spectra of exchanging species in solution. The inconsistency between our experimental results and the results of the theoretical models proposed by Rzepa is such that his conclusions are considered to be too far from experimental reality. Accurate modeling taking in account “reasonable” experimental details would be a worthwhile endeavor.     

Selectfluor and NFSI exo‐Glycal Fluorination Strategies Applied to the Enhancement of the Binding Affinity of Galactofuranosyltransferase GlfT2 Inhibitors

Two complementary methods for the synthesis of fluorinated exo‐glycals have been developed, for which previously no general reaction had been available. First, a Selectfluor‐mediated fluorination was optimized after detailed analysis of all the reaction parameters. A dramatic effect of molecular sieves on the course of the reaction was observed. The reaction was generalized with a set of biologically relevant furanosides and pyranosides. A second direct approach involving carbanionic chemistry and the use of N‐fluorobenzenesulfonimide (NFSI) was performed and this method gave better diastereoselectivities. Assignment of the Z/E configuration of all the fluorinated exo‐glycals was achieved based on the results of HOESY experiments. Furthermore, fluorinated exo‐glycal analogues of UDP‐galactofuranose were prepared and assayed against GlfT2, which is a key enzyme involved in the cell‐wall biosynthesis of major pathogens. The fluorinated exo‐glycals proved to be potent inhibitors as compared with a series of C‐glycosidic analogues of UDP‐Galf, thus demonstrating the double beneficial effect of the exocyclic enol ether functionality and the fluorine atom.     

The Role of Early-Career Chemists in European Policy-Making

Global societal challenges emphasize the importance of collaboration between scientists and policy-makers, while the participation of a diverse group of professionals, including early-career scientists, is critical towards a sustainable future. The European Young Chemists’ Network (EYCN) has been actively working with the European Chemical Society (EuChemS) to create a platform for early-career chemists in policy advice. This article comments on the possible roles of scientists in policy-making and provides an overview of relevant initiatives and platforms at the European level that could facilitate involvement. Opportunities for participation in policy advice from the perspective of early-career chemists are discussed and examples of impact are provided, hoping to stimulate further discussions and engagement in policy-making.     

Zuckerbestimmungen

Ohne Zusammenfassung     

Determination of sorbic acid with sodium chlorite

A new method for the determination of sorbic acid is proposed, using sodium chlorite in hydrochloric add medium as reagent. In these conditions the hydrochloric acid is oxidized to chlorine which adds on to the double bonds of the sorbic acid. The addition is rapid (30-90 sec) and for this reason the determination is more rapid then others similar to it. Sorbic acid has been determined in foods by this method, after a preliminary solvent extraction or steam-distillation.     

Polymerization of acetylenic derivatives. XXX. Isomers of polyphenylacetylene

Cis-transoidal (orange, soluble, and of low crystallinity) and cis-cisoidal (red, insoluble, and highly crystalline) polyphenylacetylenes (PPA) were prepared by Ziegler-Natta catalysts and trans-cisoidal (yellow, soluble, and amorphous) polyphenylacetylenes were prepared by using phosphine complexes, TiCl₄ and by thermal initiation. The cis-transoidal and cis-cisoidal structures isomerize thermally in the solid state above 100°C. In solution the cis-transoidal structure isomerizes above 80°C. The polymers obtained by thermal isomerization are soluble, amorphous, and have a trans-cisoidal structure. At temperatures higher than 120°C the cis–trans isomerization is accompanied by cyclization and by scission of the polymer chain. A method was developed for determination of cis content of cis-transoidal and cis-cisoidal polyphenylacetylenes.