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.     

A Novel Synthetic Approach to Phosphorylated Peptidomimetics

It has been shown that the derivatives of diethyl 5‐amino‐2‐phthalimidoalkyl‐1,3‐oxazol‐4‐ylphosphonates can be employed in the synthesis of phosphorylated peptidomimetics containing the phosphonoglycine residue. The reaction of diethyl 5‐alkylamino‐2‐aminoalkyl‐1,3‐oxazol‐4‐ylphosphonates with unsaturated azlactones was utilized to obtain phosphorylated peptidomimetics with dehydroamino acid moieties. The double bond in the latter was reduced with zinc in acetic acid to provide the corresponding saturated peptidomimetics containing a diethoxyphosphoryl group in the side chain.     

Thermal expansion of Sn2P2S6 crystals

We present the results for thermal expansion coefficients of Sn₂P₂S₆ crystals determined both in the crystallographic system and the system based on eigenvectors of thermal expansion tensor. Peculiarities of temperature evolution of the indicative surface of thermal expansion tensor for Sn₂P₂S₆ are discussed, including the region of their ferroelectric phase transition.     

Evolution of Oxygen–Metal Electron Transfer and Metal Electronic States During Manganese Oxide Catalyzed Water Oxidation Revealed with In Situ Soft X‐Ray Spectroscopy

Manganese oxide (MnO_x) electrocatalysts are examined herein by in situ soft X‐ray absorption spectroscopy (XAS) and resonant inelastic X‐ray scattering (RIXS) during the oxidation of water buffered by borate (pH 9.2) at potentials from 0.75 to 2.25 V vs. the reversible hydrogen electrode. Correlation of L‐edge XAS data with previous mechanistic studies indicates Mn^IV is the highest oxidation state involved in the catalytic mechanism. MnO_x is transformed into birnessite at 1.45 V and does not undergo further structural phase changes. At potentials beyond this transformation, RIXS spectra show progressive enhancement of charge transfer transitions from oxygen to manganese. Theoretical analysis of these data indicates increased hybridization of the Mn?O orbitals and withdrawal of electron density from the O ligand shell. In situ XAS experiments at the O K‐edge provide complementary evidence for such a transition. This step is crucial for the formation of O₂ from water.     

Synthesis of Polysubstituted Iodoarenes Enabled by Iterative Iodine‐Directed para and ortho C−H Functionalization

Among halogenated aromatics, iodoarenes are unique in their ability to produce the bench‐stable halogen(III) form. Earlier, such iodine(III) centers were shown to enable C?H functionalization ortho to iodine via halogen‐centered rearrangement. The broader implications of this phenomenon are explored by testing the extent of an unusual iodane‐directed para C?H benzylation, as well as by developing an efficient C?H coupling with sulfonyl‐substituted allylic silanes. Through the combination of the one‐shot nature of the coupling event and the iodine retention, multisubstituted arenes can be prepared by sequentially engaging up to three aromatic C?H sites. This type of iodine‐based iterative synthesis will serve as a tool for the formation of value‐added aromatic cores.     

Nucleophilic catalysis of phosphorus trichloride oxygen oxidation

The effect of nucleophilic catalysis on phosphorus trichloride oxygen oxidation is discovered. A number of nucleophilic catalysts for the phosphorus trichloride oxidation are found. Triethylamine, 4‐dimethylaminopyridine (DMAP), and tetramethylurea are amongst them. © 2008 Wiley Periodicals, Inc. Heteroatom Chem 19:408–411, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20439     

Charge Transfer from Lead Sulfide Quantum Dots to MoS2 Nanoplatelets

Optics and Spectroscopy - The research interest in the transition metal dichalcogenides (TMD) has been reborn a few years ago. This had happened due to the remarkable properties of monolayered TMD...