Law of the iterated logarithm for random graphs

A milestone in probability theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables with mean 0 and variance 1 In this paper we prove that LIL holds for various functionals of random graphs and hypergraphs models. We first prove LIL for the number of copies of a fixed subgraph H. Two harder results concern the number of global objects: perfect matchings and Hamiltonian cycles. The main new ingredient in these results is a large deviation bound, which may be of independent interest. For random k‐uniform hypergraphs, we obtain the Central Limit Theorem and LIL for the number of Hamilton cycles.     

Optimal packings of two to four equal circles on any flat torus

We find explicit formulas for the radii and locations of the circles in all the optimally dense packings of two, three or four equal circles on any flat torus, defined to be the quotient of the Euclidean plane by the lattice generated by two independent vectors. We prove the optimality of the arrangements using techniques from rigidity theory and topological graph theory.     

Synthesis of Cyclic Compounds via Photoinduced Radical Cyclization Cascade of C=C bonds

Cyclic compounds constitute a great important class of substances in the science of medicine and biology, which renders the research on facile and efficient construction of such complex scaffolds from simple starting materials to be hot and appealing. Recently, the radical cascade reaction involving multiple bond formation/cleavage has emerged as an ideal and powerful route to give high‐value cyclic products, along with diminished cost and waste. As a simple and benign methodology, photoredox catalysis offers a readily available access to the generation of radical species. Alkenes have been recognized as one of the most valuable building blocks for the reason they allow installation of different functional groups simultaneously through addition to the C=C bonds. This account summarizes the recent advances in photoinduced radical cascade cyclization to the synthesis of cyclic compounds with C=C bonds working as the initial radical acceptors, and emphasis is put on the related reaction mechanisms.     

Synthesis and Electronic Properties of Novel 5,7-Diazapentacene Derivatives

A route to the synthesis of novel 5,7-diazapentacenes and some preliminary studies on their properties is reported. A single crystal X-ray diffraction study of the dihexyl derivative showed it had formed a dimer during the analysis. The materials possess lower lying frontier orbitals than pentacene and may have potential applications in organic electronic devices. This synthetic method may be applicable to the synthesis of other azaacenes.     

Synthesis,Structures, and Near‐IR Absorption of Heterole‐Fused Earring Porphyrins

A reaction sequence of regioselective peripheral bromination, Suzuki–Miyaura coupling with 2‐borylated thiophene or pyrrole, and oxidative ring‐closure with FeCl₃ allowed the synthesis of heterole‐fused earring porphyrins 4Pd and 9Pd from the parent earring porphyrin 1 . Differently pyrrole‐fused porphyrins 5H and 6H and their Pd^II complexes 5Pd and 6Pd were also synthesized. The structures of 4Pd , 5H, 6Pd , and 8Pd have been revealed by X‐ray analysis to be slightly twisted owing to constraints imposed by heterole‐fused structures. 5Pd exhibits an intensified band at 1505 nm, while 4Pd and 9Pd display small but remarkably red‐shifted absorption bands reaching around 2200 nm.     

Multifold factorizations of cyclic groups into subsets

Let $(G,+)$ be an abelian group. A finite multiset A over G is said to give a λ-fold factorization of G if there exists a multiset B over G such that each element of G occurs λ times in the multiset $A+B:=\{a+b:a\in A,b\in B\}$. In this article, restricting G to a cyclic group, we will provide sufficient conditions on a given multiset A under which the exact value or an upper bound of the minimum multiplicity λ of a factorization of G can be given by introducing a concept of ‘lcm-closure’. Furthermore, a couple of properties on a given factor A will be shown when A has a prime or prime power order (cardinality). A relation to multifold factorizations of the set of integers will be also glanced at a general perspective.     

Dimerisation of Dipiperidinoacetylene: Convenient Access to Tetraamino-1,3-Cyclobutadiene and Tetraamino-1,2-Cyclobutadiene Metal Complexes

The reaction of 1,2-dipiperidinoacetylene ( 1 ) with 0.5 equivalents of SnCl₂ or GeCl₂⋅dioxane afforded the 1,2,3,4-tetrapiperidino-1,3-cyclobutadiene tin and germanium dichloride complexes 2 a and 2 b , respectively. A competing redox reaction was observed with excess amounts of SnCl₂, which produced a tetrapiperidinocyclobutadiene dication with two trichlorostannate(II) counterions. Heating neat 1 to 110 °C for 16 h cleanly produced the dimer 1,3,4,4-tetrapiperidino-3-buten-1-yne ( 3 ); its reaction with stoichiometric amounts of SnCl₂ or GeCl₂⋅dioxane furnished the 1,3,4,4-tetrapiperidino-1,2-cyclobutadiene tin and germanium dichloride complexes 4 a and 4 b , respectively. Transition-metal complexes containing this novel four-membered cyclic bent allene (CBA) ligand were prepared by reaction of 3 with [(tht)AuCl], [RhCl(CO)₂]₂, and [(Me₃N)W(CO)₅] to form [(CBA)AuCl] ( 5 ), [(CBA)RhCl(CO)₂] ( 6 ), and [(CBA)W(CO)₅] ( 7 ). The molecular structures of all compounds 2 – 7 were determined by X-ray diffraction analyses, and density functional theory (DFT) calculations were carried out to rationalise the formation of 3 and 4 a .     

Rapid,Traceless, AgI‐Promoted Macrocyclization of Peptides Possessing an N‐Terminal Thioamide

Peptide macrocyclization is often a slow process, plagued by epimerization and cyclodimerization. Herein, we describe a new method for peptide macrocyclization employing the Ag^I‐promoted transformation of peptide thioamides. The Ag^I has a dual function: chemoselectively activating the thioamide and tethering the N‐terminal thioamide to the C‐terminal carboxylate. Extrusion of Ag₂S generates an isoimide intermediate, which undergoes acyl transfer to generate the native cyclic peptide, resulting in a rapid, traceless macrocylization process. Cyclic peptides are furnished in high yields within 1 hour, free of epimerization and cyclodimerization.