On 3-stable number conditions in n-connected claw-free graphs

For a subgraph $X$ of $G$, let ${\alpha }_G^3\left(X\right)$ be the maximum number of vertices of $X$ that are pairwise distance at least three in $G$. In this paper, we prove three theorems. Let $n$ be a positive integer, and let $H$ be a subgraph of an $n$-connected claw-free graph $G$. We prove that if $n\ge 2$, then either $H$ can be covered by a cycle in $G$, or there exists a cycle $C$ in $G$ such that ${\alpha }_G^3(H?V\left(C\right))\le {\alpha }_G^3\left(H\right)?n$. This result generalizes the result of Broersma and Lu that $G$ has a cycle covering all the vertices of $H$ if ${\alpha }_G^3\left(H\right)\le n$. We also prove that if $n\ge 1$, then either $H$ can be covered by a path in $G$, or there exists a path $P$ in $G$ such that ${\alpha }_G^3(H?V\left(P\right))\le {\alpha }_G^3\left(H\right)?n?1$. By using the second result, we prove the third result. For a tree $T$, a vertex of $T$ with degree one is called a leaf of $T$. For an integer $k\ge 2$, a tree which has at most $k$ leaves is called a $k$-ended tree. We prove that if ${\alpha }_G^3\left(H\right)\le n+k?1$, then $G$ has a $k$-ended tree covering all the vertices of $H$. This result gives a positive answer to the conjecture proposed by Kano et al. (2012).     

A k-Tree Containing Specified Vertices

A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let ${S \subseteq V(G)}A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let S í V(G){S \subseteq V(G)} with κ(S) ≥ 1. Suppose that for every l > κ(S), there exists an integer t such that 1 ￡ t ￡ (k-1)l+2 - ?\fracl-1k ?{1 \le t \leq (k-1)l+2 - \lfloor \frac{l-1}{k} \rfloor} and the degree sum of any t independent vertices of S is at least n + tl − kl − 1. Then G has a k-tree containing S. We also show some new results on a spanning k-tree as corollaries of the above theorem.     

Dirac electrons in solids

Electronic band structures in solids sometimes have features similar to Dirac electrons in vacuum. Well-known examples are bismuth and graphite; 4×4 original Dirac matrix in three dimension (3d) in the former with strong spin–orbit interaction, while 2×2 massless Dirac in two dimension (2d) with weak inter-layer coupling described essentially by Weyl equation in the latter. Recently one layer of graphite, graphene, is realized and studied both extensively and intensively. Other recent examples include a molecular solid, α-(BEDT-TTF)₂I₃$\mathrm{\alpha }\mathrm{-}(\mathrm{BEDT}\mathrm{-}\mathrm{TTF}{)}_2{\mathrm{I}}_3$, which has a layered structure with electronic states described by tilted-Weyl equation, and Fe-pnictides. There is also a theoretical proposal that one of inverse perovskites, Ca₃PbO, can be a candidate in 3d with strong spin–orbit interaction similar to bismuth. The particular feature of Dirac electrons in solids is a small, or even vanishing, band gap and then thermodynamic or transport properties are affected by inter-band coupling of electronic states. Typical ones are responses to external magnetic field. Actually, it has long been known that orbital susceptibility of these Dirac electrons has very particular features resulting from inter-band effects of magnetic field. It is of interest to see such inter-band effects on Hall effects to be compared with orbital susceptibility, which will be introduced in this paper, together with possible consequences of mutual interaction between valleys triggered by tilting in molecular solids.     

Visible-light-mediated addition of α-aminoalkyl radicals generated from α-silylamines to α,β-unsaturated carbonyl compounds

Visible-light-mediated addition of α-aminoalkyl radicals generated from α-silylamines to α,β-unsaturated carbonyl compounds using a photoredox catalyst is developed. We also succeeded in the isolation of a silyl enol ether as a primary product of the photochemical reaction.     

Helicity inversion from left- to right-handed square planar Pd(II) complexes: synthesis of a diastereomer pair from a single chiral ligand and their structure dynamism

A diastereomer pair of left- and right-handed square planar Pd(II) complexes was synthesized from a single chiral ligand as kinetic and thermodynamic products. Helicity inversion between the diastereomers occurred rapidly under thermal and microwave irradiation conditions.     

Reversible structural switch in the nano-cavity of crystalline metallo-macrocycles with smooth ligand exchange by non-coordinating guest stimuli

A tetranuclear Ni(II)-macrocycle consisting of β-dipeptides demonstrates effects of water vapor on the structural switching of the nano-cavity by ligand exchange in the single-crystal state. The content of non-coordinating water molecules regulates the opening and closing of the cavity through cooperation of smooth ligand exchange and internal hydrogen-bond switching.     

Enantioselective alkylation of β-keto phosphonates by direct use of diaryl methanols as electrophiles

Enantioselective alkylation of β-keto phosphonates with diaryl methanols in the presence of catalytic amounts of copper(ii) trifluoromethanesulfonate and an optically active ligand gives the corresponding alkylated products in good to high yields with a high enantioselectivity.     

Vanadium isotopic composition of the sea squirt (Ciona savignyi)

Vanadium (V) in the sea squirt (Ciona savignyi) from Onagawa Bay, Miyagi, Japan, was isolated and purified through adsorption on a diamine resin and anion and cation exchanges after the dissolution of sea squirt samples with nitric acid and hydrogen peroxide. The (50)V/(51)V isotope ratio of V thus obtained was mass-spectrometrically determined to be from 2.51×10(-3) to 2.55×10(-3) with the average of 2.53×10(-3) by the thermal ionisation technique. This value agreed with those of vanadyl chloride and vanadyl nitrate both prepared from vanadyl sulphate (Wako Pure Chemical Industries, Ltd., Japan) and of V in coastal seawater (Shimokita Peninsula, Aomori, Japan) within experimental uncertainties (standard deviation of±0.04), which suggested that no appreciable V isotope fractionation occurs accompanying V uptake by the sea squirt from sea water.     

Quantum valence criticality as an origin of unconventional critical phenomena

It is shown that unconventional critical phenomena commonly observed in paramagnetic metals YbRh2Si2, YbRh2(Si0.95Ge0.05)2, and β-YbAlB4 are naturally explained by the quantum criticality of Yb-valence fluctuations. We construct the mode-coupling theory taking account of local correlation effects of f electrons and find that unconventional criticality is caused by the locality of the valence fluctuation mode. We show that measured low-temperature anomalies such as divergence of uniform spin susceptibility χ～T(-ζ) with ζ～0.6 giving rise to a huge enhancement of the Wilson ratio and the emergence of T-linear resistivity are explained in a unified way.