Stability in the Erdős–Gallai Theorem on cycles and paths,II

The Erd?s–Gallai Theorem states that for $k\ge 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k?1)(n?1)$ edges. A stronger version of the Erd?s–Gallai Theorem was given by Kopylov: If $G$ is a 2-connected $n$-vertex graph with no cycle of length at least $k$, then $e\left(G\right)\le max\{h(n,k,2),h(n,k,?\frac{k?1}{2}?)\}$, where $h(n,k,a)?\left(\genfrac{}{}{0ex}{}{k?a}{2}\right)+a(n?k+a)$. Furthermore, Kopylov presented the two possible extremal graphs, one with $h(n,k,2)$ edges and one with $h(n,k,?\frac{k?1}{2}?)$ edges.In this paper, we complete a stability theorem which strengthens Kopylov’s result. In particular, we show that for $k\ge 3$ odd and all $n\ge k$, every $n$-vertex 2-connected graph $G$ with no cycle of length at least $k$ is a subgraph of one of the two extremal graphs or $e\left(G\right)\le max\{h(n,k,3),h(n,k,\frac{k?3}{2})\}$. The upper bound for $e\left(G\right)$ here is tight.     

Scattering Below Critical Energy for the Radial 4D Yang-Mills Equation and for the 2D Corotational Wave Map System

Given g and f  =  gg′, we consider solutions to the following non linear wave equation :

Solution processable diketopyrrolopyrrole semiconductor: towards bio-electronic applications

In this paper, the possibility to use diketopyrrolopyrrole (DPP) for the construction of electrical devices designed to interact with animal cells was studied. For this purpose, the biocompatibility and electrical properties of the selected DPP derivative (3,6-bis(5-(benzofuran-2-yl)thiophen-2-yl)-2,5-bis(2-ethyl-hexyl)pyrrolo[3,4-c]pyrrole-1,4-dione) [referred as DPP(TBFu)₂] were researched. The electrical properties were studied using model organic field-effect transistors. Mainly investigated was under what conditions maximum charge carrier mobility can be achieved. Using the cumulative effect of self-assembled monolayers on dielectrics and electrodes and detailed thermal analysis of the DPP, a higher charge carrier mobility was achieved than has been previously reported (5.5?×?10^?3 cm² V^?1 s^?1). The biocompatibility was studied based on a culture of 3T3 fibroblasts. This research revealed that DPP(TBFu)₂ can be used in applications involving direct contact with living animal cells. The conclusions found with these model devices can be applied to components suitable for biosensing applications, e.g., water- or electrolyte-gated organic field-effect transistors.     

Structural characterization of oxidized lumiflavin hydrochloride hydrate: Comparison with the hydrogen bonding at the cofactor of flavoproteins

In order to examine the possibility of hydrogen bonding around the flavin ring, an X-ray diffraction study of the title compound was undertaken. Crystals of lumiflavin hydrochloride hydrate [2(C₁₃H₁₂N₄O₂) 4 HCl 7 H₂O] are triclinic, space group ,Z=2,a=11.064(1),b=17.903(1),c=9.891(1)Å, =111.9(6), -96.4(8), =91.6(7)°. We present the crystal structure of this compound and compare it with other flavin derivatives and with the structure of the active site of some flavoproteins.     

A quantitative prediction of the electronic spectra of thiocarbonyl chromophores: TD-DFT versus SAC-CI

In this contribution, we set up a SAC-CI methodology to evaluate the n → π* and π → π* vertical transition energies of a series of thiocarbonyl derivatives. We show that Frozen-Core SAC-CI provides accurate vertical excitations energies. Nevertheless, in order to obtain converged results, the R2S2 unliked integrals have to be taken into account in L3-SAC-CI calculations. In addition, we present the comparative performances of three computational procedures, INDO/S, TD-DFT, and SAC-CI, for the calculation of valence excited states energies and it turns out that: (1) no tuning of the exact exchange (α) included in TD-DFT allows to consistently reproduce the SAC-CI results; (2) SAC-CI and TD-B3LYP both evaluate the n → π*, as well as the π → π* transition energies, with a similar accuracy. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Homophymine A, an anti-HIV cyclodepsipeptide from the sponge Homophymia sp

A new anti-HIV cyclodepsipeptide, homophymine A, was isolated from a New Caledonian collection of the marine sponge Homophymia sp. The structure of homophymine A was determined by interpretation of spectroscopic data, acid hydrolysis, and LC-MS analysis. Homophymine A contains 11 amino acid residues and an amide-linked 3-hydroxy-2,4,6-trimethyloctanoic acid moiety. Along with four D-, two L-, and one N-methyl amino acids, it also contains four unusual amino acid residues: (2S,3S,4R)-3,4-diMe-Gln, (2R,3R,4S)-4-amino-2,3-dihydroxy-1,7-heptandioic acid, L-ThrOMe, and (2R,3R,4R)-2-amino-3-hydroxy-4,5-dimethylhexanoic acid. In a cell-based XTT assay, homophymine A exhibited cytoprotective activity against HIV-1 infection with a IC50 of 75 nM.     

Fast and reliable theoretical determination of pKa* for photoacids

In this Letter a fast and reliable computational protocol for the calculation of pKa* of photoacids is presented. The results obtained for several coumarins indicate that, beyond the obtained numerical accuracy, it is indeed possible to develop a reliable computational procedure for pKa* calculation by combining, in a judicious way, a reliable model for the excited states (TD-DFT) with a fast solvent model (PCM). The characteristics of the different components of the model will allow for routine applications to medium and large chemical systems, so that the proposed protocol could nicely integrate experimental analyses.