Economical tight examples for the biased Erd?s-Selfridge theorem

A positional game is essentially a generalization of tic-tac-toe 0;played on a hypergraph (V,F). A pivotal result in the study of positional games is the Erd?s-Selfridge 0;theorem, which gives simple criteria for the existence of a Breaker's winning strategy on a hypergraph F. It has been shown that the Erd?s-Selfridge 0;theorem can be tight and that numerous extremal systems exist for that theorem. We focus on a generalization of the Erd?s-Selfridge 0;theorem proven by Beck for biased (p:q) 0;games, which we call the (p:q)-Erd?s-Selfridge 0;theorem. We show that for pn-uniform hypergraphs there is a unique extremal system for the (p:q)-Erd?s-Selfridge 0;theorem (q?2) when Maker 0;must win in exactly n turns (i.e., as quickly as possible).     

A New Supramolecular Assembly Formed by Melamine and Sulfuric Acid

Polymeric melaminium sulfate [(LH₂)₂(SO₄)₂]_n has been synthesized by reaction of melamine L with sulfuric acid in aqueous solution. The compound was characterized by elemental analysis, ¹H NMR, ESI MS and a single crystal X‐ray diffraction analysis. The architecture of the assembly formed is based on hydrogen bonded dimers of diprotonated melaminium cations (LH₂)²⁺ which are linked by a hydrogen bonded network with sulfate ions forming 2D sheets. A 3D polymeric structure results from the presence of mutual hydrogen bonds between sulfate ions and melaminium cations in different sheets. Significant π‐π stacking is also present between the aromatic cations in this supramolecular arrangement.     

Self‐assembly of π‐conjugated bis(terpyridine) ligands with zinc(II) ions: New metallosupramolecular materials for optoelectronic applications

A series of main‐chain metallopolymers ( P1–P10 ) was prepared by the self‐assembly of rigid‐linear π‐conjugated bis(terpyridine) monomers ( 1–10 ) with Zn^II ions and was fully characterized. The polymerization was additionally confirmed by UV/vis titration experiments. A strong increase in viscosities (around 1.6 times) relative to those of the monomer solutions was found. The thermal stability of P1–P10 compared with that of 1–10 was enhanced as a result of the metallopolymerization. The electro‐optical properties of the materials were investigated in detail. Tuning of the electrochemical and photophysical properties was enabled; thus, bright purple to green photoluminescent (PL) emission (PL quantum yields of 0.12–0.81) for P1–P10 was observed in solution with the emission color strongly depending on the nature of the π‐conjugated bis(terpyridine) system. Thin homogeneous films of P6 were prepared by solution processing, that is, spin‐coating and inkjet‐printing, and exhibited intense yellow PL emission in the solid state. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 4083–4098, 2009     

Fourier transform Raman and DFT study of three annulated oligothiophenes with different molecular shapes.

Herein, we study the conjugation properties of three different thienoacenes, each of which has three or four fused thiophene rings, by means of Fourier transform Raman spectroscopy. The B3LYP/6-31G** vibrational analysis of all of the collected spectroscopic data evidences that the selective enhancement of a limited number of Raman scatterings is related to the occurrence in the three thienoacenes of a vibronic coupling between the lowest unoccupied frontier molecular orbital (LUMO) and some Raman-active skeletal nu(C==C) stretching modes of 1600-1300 cm(-1).     

Variations on a theme by Mark Kac

Mark Kac's theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations, most of which are formulated in terms of 0013;1 processes. Whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0, this condition is dropped in part of the variations; two others refer to the variance of the recurrence time and two to the Poincaré cycle of a dynamical system. Most variations consist in inequalities and formal identities for the mean first-arrival time and subsequent recurrence times for the given state.     

A Large π‐Extended Carbon Nanoring Based on Nanographene Units: Bottom‐Up Synthesis,Photophysical Properties,and Selective Complexation with Fullerene C70

Herein we report the organoplatinum‐mediated bottom‐up synthesis, characterization, and properties of a novel large π‐extended carbon nanoring based on a nanographene hexa‐peri ‐hexabenzocoronene (HBC) building unit. This tubular structure can be considered as an example of the longitudinal extension of the cycloparaphenylene scaffold to form a large π‐extended carbon nanotube (CNT) segment. The cyclic tetramer of a tetramesityl HBC ([4]CHBC) was synthesized by the reaction of a 2,11‐diborylated hexa‐peri ‐hexabenzocoronene with a platinum complex, followed by reductive elimination. The structure of this tubular molecule was further confirmed by physical characterization. Theoretical calculations indicate that the strain energy of this nanoring is as high as 49.18 kcal mol⁻¹. The selective supramolecular host–guest interaction between [4]CHBC and C₇₀ was also investigated.     

A Near‐Infrared Dye That Undergoes Multiple Interconversions through Acid–Base Equilibrium and Reversible Redox Processes

A near‐infrared (NIR) polymethine dye ( 1 ), consisting of a cyclohepta[1,2‐b ;4,3‐b′ ]dithiophene and two phenol moieties, was synthesized. This dye exhibited pH‐responsive changes in its photophysical properties due to a two‐step acid–base equilibrium that produced a protonated cation ( 1H⁺ ) and an anion ( 1⁻ ). While 1H⁺ showed an intense fluorescence in the red region of the visible spectrum, 1⁻ exhibited a strong absorption in the NIR region. The tropylium ion character in 1H⁺ induces high pK _a1 and pK _a2 values for 1 . Moreover, a stable radical ( 1^. ) was prepared, which showed a NIR absorption band with a maximum at circa 1600 nm. The cyclic voltammogram of 1^. revealed a two‐step reversible redox process that produced 1⁻ and the cation 1⁺ , which is different from 1H⁺ . These redox processes accompany drastic electrochromic changes in the vis–NIR region. Overall, 1 is susceptible to multiple interconversions between five forms, due to the multifaceted character of the cycloheptadithiophene skeleton.