A generalized Beraha conjecture for non-planar graphs

We study the partition function _ZG(nk,k)(Q,v)$Z_{G(nk,k)}(Q,v)$ of the Q  -state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k)$G(nk,k)$. We study its zeros in the plane (Q,v)$(Q,v)$ for 1?k?7$1?k?7$. We also consider two specializations of _ZG(nk,k)$Z_{G(nk,k)}$, namely the chromatic polynomial _PG(nk,k)(Q)$P_{G(nk,k)}(Q)$ (corresponding to v=−1$v=-1$), and the flow polynomial _ΦG(nk,k)(Q)${\Phi }_{G(nk,k)}(Q)$ (corresponding to v=−Q$v=-Q$). In these two cases, we study their zeros in the complex Q  -plane for 1?k?7$1?k?7$. We pay special attention to the accumulation loci of the corresponding zeros when n→∞$n\to \infty$. We observe that the Berker–Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.     

A proof of Wahl's conjecture in the symplectic case

Let X denote a flag variety of type A or type C. We construct a canonical Frobenius splitting of X × X which vanishes with maximal multiplicity along the diagonal. This way we verify a conjecture by Lakshmibai, Mehta and Parameswaran [4] in type C, and obtain a new proof in type A. In particular, we obtain a proof of Wahl's conjecture in type C, and a new proof in type A. We also present certain cohomological consequences.     

Poly(ethylene-co-butylene) functionalized multi walled carbon nanotubes applied in polypropylene nanocomposites

A novel functionalized multi walled carbon nanotube (MWCNT) was prepared through grafting with α-azido-poly(ethylene-co-butylene) (PEB-N₃). The PEB-N₃ was prepared through a two step procedure and grafted onto an industrial grade multi walled carbon nanotube (MWCNT) through a highly efficient nitrene addition. This novel nano filler was melt mixed into polypropylene (PP) and the composite was characterized by FT-IR spectroscopy, Raman spectroscopy, Scanning Electron Microscopy (SEM), Rheology and Dielectric Relaxation Spectroscopy (DRS). The analyses showed that composites with the novel filler had a high degree of discharge from the surface and higher conductivity compared to the pristine filler, illustrating an efficient conductive network in the composites. The composites showed low percolation thresholds of 0.3 wt.% (0.15 vol.%) as well as improved stability at a range of temperatures from 25–135 °C.     

What can we learn from N2O isotope data? – Analytics,processes and modelling

The isotopic composition of nitrous oxide (N₂O) provides useful information for evaluating N₂O sources and budgets. Due to the co-occurrence of multiple N₂O transformation pathways, it is, however, challenging to use isotopic information to quantify the contribution of distinct processes across variable spatiotemporal scales. Here, we present an overview of recent progress in N₂O isotopic studies and provide suggestions for future research, mainly focusing on: analytical techniques; production and consumption processes; and interpretation and modelling approaches. Comparing isotope-ratio mass spectrometry (IRMS) with laser absorption spectroscopy (LAS), we conclude that IRMS is a precise technique for laboratory analysis of N₂O isotopes, while LAS is more suitable for in situ/inline studies and offers advantages for site-specific analyses. When reviewing the link between the N₂O isotopic composition and underlying mechanisms/processes, we find that, at the molecular scale, the specific enzymes and mechanisms involved determine isotopic fractionation effects. In contrast, at plot-to-global scales, mixing of N₂O derived from different processes and their isotopic variability must be considered. We also find that dual isotope plots are effective for semi-quantitative attribution of co-occurring N₂O production and reduction processes. More recently, process-based N₂O isotopic models have been developed for natural abundance and ¹⁵N-tracing studies, and have been shown to be effective, particularly for data with adequate temporal resolution. Despite the significant progress made over the last decade, there is still great need and potential for future work, including development of analytical techniques, reference materials and inter-laboratory comparisons, further exploration of N₂O formation and destruction mechanisms, more observations across scales, and design and validation of interpretation and modelling approaches. Synthesizing all these efforts, we are confident that the N₂O isotope community will continue to advance our understanding of N₂O transformation processes in all spheres of the Earth, and in turn to gain improved constraints on regional and global budgets.     

Two square‐pyramidal chromium(V)–nitride complexes: bis­(2‐methyl­quinolin‐8‐olato)nitridochromium(V) and nitridobis(2‐sulfidopyridine N‐oxide)chromium(V)

Two new chromium(V)–nitride complexes with a coordination sphere completed by bidentate ligands have been synthesized and structurally characterized. Bis(2‐methyl­quinolin‐8‐olato)nitridochromium(V), [Cr(C₁₀H₈NO)₂(N)], has the coordination sphere completed by an equatorial N₂O₂ set of ligators. The compound crystallizes with the five‐coordinate complexes at sites with twofold rotational symmetry and all Cr—N bond directions aligned with the crystallographic b axis. Nitridobis(2‐sulfidopyridine N‐oxide)chromium(V), [Cr(C₅H₄NOS)₂(N)], crystallizes with the mol­ecules on general positions and has an equatorial S₂O₂ coordination environment, which is unprecedented among nitride complexes of the first‐row transition metals. In both systems, Cr[triple‐bond]N bonds are short at ca 1.56 Å.     

Diastereoisomeric β‐ethyl aspartate–cobalt(III) complexes: Λ(+)578‐ and Δ(−)578‐bis(ethane‐1,2‐diamine)[β‐ethyl (S)‐aspartato]cobalt(III) bis(perchlorate) monohydrate

The structures of the diastereoisomers Λ(+)₅₇₈‐, (I), and Δ(−)₅₇₈‐bis(ethane‐1,2‐diamine)[β‐ethyl (S)‐aspartato‐κ²N,O¹]cobalt(III) bis(perchlorate) monohydrate, (II), both [Co(C₆H₁₀N₂O₄)(C₂H₈N₂)₂](ClO₄)₂·H₂O, are compared. In both structures, the ester group of the amino acid side chain is engaged only in intramolecular hydrogen bonding to coordinated amine groups. This interaction is stronger in (I) and correlates with previously observed diastereoisomeric equilibrium ratios for related metal complex systems in aqueous media. The two perchlorate anions of (II) are located on twofold axes. Both perchlorates in (I) and one of the perchlorates in (II) are affected by disorder. Both structures exhibit extensive three‐dimensional hydrogen‐bonding networks.     

Chemical structures assigned for the low molecular weight fractions from degradation of poly(styrene sulfides)

A non-destructive analysis is performed of thermally unstable sulfur-styrene reaction products, combining preparative size exclusion chromatography (P-SEC), ultraviolet-visible (UV-vis) and nuclear magnetic resonance (NMR). The crystallizing compounds are identified as be 2,4-diphenylthiophanes with short sulfur bridge (x ∼ 2) contrary to earlier suggestions which were based on destructive analysis. A new cyclic structure (styrene polysulfide x = 1 up to 8) was assigned to the amorphous species containing a single styrene repeating unit. Comparison with the amorphous fractions suggests that the rigid ring of styrene repeating units in adjacent sequences is the characteristic feature for the crystallisability. The melting and crystallization behaviour of this crystalline component was observed, by optical microscopy (OM) and differential scanning calorimetry (DSC), to be step-wise as well as broad, due to the variation in the length of the sulfur bridge.