The chromatic number of random Borsuk graphs

We study a model of random graph where vertices are n i.i.d. uniform random points on the unit sphere S^d in , and a pair of vertices is connected if the Euclidean distance between them is at least 2??. We are interested in the chromatic number of this graph as n tends to infinity. It is not too hard to see that if ?>0 is small and fixed, then the chromatic number is d+2 with high probability. We show that this holds even if ?→0 slowly enough. We quantify the rate at which ? can tend to zero and still have the same chromatic number. The proof depends on combining topological methods (namely the Lyusternik–Schnirelman–Borsuk theorem) with geometric probability arguments. The rate we obtain is best possible, up to a constant factor—if ?→0 faster than this, we show that the graph is (d+1)‐colorable with high probability.25     

Experimental and Theoretical Studies on the Rearrangement of 2‐Oxoazepane α,α‐Amino Acids into 2′‐Oxopiperidine β2,3,3‐Amino Acids: An Example of Intramolecular Catalysis

Enantiopure β‐amino acids represent interesting scaffolds for peptidomimetics, foldamers and bioactive compounds. However, the synthesis of highly substituted analogues is still a major challenge. Herein, we describe the spontaneous rearrangement of 4‐carboxy‐2‐oxoazepane α,α‐amino acids to lead to 2′‐oxopiperidine‐containing β^2,3,3‐amino acids, upon basic or acid hydrolysis of the 2‐oxoazepane α,α‐amino acid ester. Under acidic conditions, a totally stereoselective synthetic route has been developed. The reordering process involved the spontaneous breakdown of an amide bond, which typically requires strong conditions, and the formation of a new bond leading to the six‐membered heterocycle. A quantum mechanical study was carried out to obtain insight into the remarkable ease of this rearrangement, which occurs at room temperature, either in solution or upon storage of the 4‐carboxylic acid substituted 2‐oxoazepane derivatives. This theoretical study suggests that the rearrangement process occurs through a concerted mechanism, in which the energy of the transition states can be lowered by the participation of a catalytic water molecule. Interestingly, it also suggested a role for the carboxylic acid at position 4 of the 2‐oxoazepane ring, which facilitates this rearrangement, participating directly in the intramolecular catalysis.     

Commuting <Emphasis Type="Italic">U</Emphasis>-operators and nondegenerate imbeddings of Jordan systems

Over an arbitrary ring of scalars, we build a Jordan algebra J having two elements x,y∈J such that x?y=0, but whose U-operators do not commute. This shows that nondegeneracy is a necessary condition in the main theorem of Commuting U-Operators in Jordan Algebras by J. A Anquela, T. Cortés and H. P. Petersson (Trans. Amer. Math. Soc. 366 (2014), 5877–5902). As a consequence, we obtain examples of Jordan systems over arbitrary rings of scalars that cannot be imbedded in nondegenerate systems.     

On the Structure and Chiral Aggregation of Liquid Crystalline Star-Shaped Triazines H-Bonded to Benzoic Acids

The ability of a star-shaped tris(triazolyl)triazine derivative to hierarchically build supramolecular chiral columnar organizations through the formation of H-bonded complexes with benzoic acids was studied from a theoretical and experimental point of view. The combined study has been done at three different levels including the study of the structure of the triazine core, the association with benzoic acids in stoichiometry 1:3, and the assembly of 1:3 complexes in helical aggregates. Although the star-shaped triazine core crystallizes in a non-C₃ conformation, the C₃-symmetric conformation is theoretically predicted to be more stable and gives rise to a favorable C₃ supramolecular 1:3 complex upon the interaction with three benzoic acids in their voids. In addition, calculations at different levels (DFT, PM7, and MM3) for the 1:3 host-guest complex predict the formation of large stable columnar helical aggregates stabilized by the compact packing of the interstitial acids by π–π and CH⋅⋅⋅π interactions. The acids restrict the movement of the the star-shaped triazine cores along the stacking axis causing a template effect in the self-assembly of the complex. Theoretical predictions correlate with experimental results, since the interaction with achiral or chiral 3,4,5-(4-alkoxybenzyloxy)benzoic acids gives rise to supramolecular complexes that organize in bulk hexagonal columnar mesophases stable at room temperature with intracolumnar order. The existence of supramolecular chirality in the mesophase was determined for complexes formed by acids derived from (S)-2-octanol. Chiral aggregation was also evidenced for complexes formed in dodecane.     

Controlling the Length of Cooperative Supramolecular Polymers with Chain Cappers

The design and the characterization of supramolecular additives to control the chain length of benzene-1,3,5-tricarboxamide (BTA) cooperative supramolecular polymers under thermodynamic equilibrium is unraveled. These additives act as chain cappers of supramolecular polymers and feature one face as reactive as the BTA discotic to interact strongly with the polymer end, whereas the other face is nonreactive and therefore impedes further polymerization. Such a design requires fine tuning of the conformational preorganization of the amides and the steric hindrance of the motif. The chain cappers studied are monotopic derivatives of BTA, modified by partial N-methylation of the amides or by positioning of a bulky cyclotriveratrylene cage on one face of the BTA unit. This study not only clarifies the interplay between structural variations and supramolecular interactions, but it also highlights the necessity to combine orthogonal characterization methods, spectroscopy and light scattering, to elucidate the structures and compositions of supramolecular systems.