Zhang–Zhang Polynomials of Multiple Zigzag Chains Revisited: A Connection with the John–Sachs Theorem

Multiple zigzag chains Zm,n of length n and width m constitute an important class of regular graphene flakes of rectangular shape. The physical and chemical properties of these basic pericondensed benzenoids can be related to their various topological invariants, conveniently encoded as the coefficients of a combinatorial polynomial, usually referred to as the ZZ polynomial of multiple zigzag chains Zm,n. The current study reports a novel method for determination of these ZZ polynomials based on a hypothesized extension to John–Sachs theorem, used previously to enumerate Kekulé structures of various benzenoid hydrocarbons. We show that the ZZ polynomial of the Zm,n multiple zigzag chain can be conveniently expressed as a determinant of a Toeplitz (or almost Toeplitz) matrix of size m2×m2 consisting of simple hypergeometric polynomials. The presented analysis can be extended to generalized multiple zigzag chains Zkm,n, i.e., derivatives of Zm,n with a single attached polyacene chain of length k. All presented formulas are accompanied by formal proofs. The developed theoretical machinery is applied for predicting aromaticity distribution patterns in large and infinite multiple zigzag chains Zm,n and for computing the distribution of spin densities in biradical states of finite multiple zigzag chains Zm,n.     

Oxygen- and Nitrogen-Embedded Zigzag Hydrocarbon Belts

Despite the aesthetically appealing structures and tantalizing physical and chemical properties, zigzag hydrocarbon belts and their heteroatom-embedded analogues remain challenging synthetic targets. We report herein the synthesis of diverse O/N-doped zigzag hydrocarbon belts based on selective bridging of the fjords of resorcin[4]arene derivatives through intramolecular S_NAr and palladium-catalyzed intermolecular C−N bond formation reactions. Preorganized conformations of mono-macrocyclic, half-belt and quasi-belt compounds were revealed to facilitate cyclization reactions to construct heteroatom-linked octahydrobelt[8]arenes. The acquired products had strained square-prism-shaped belt structures in which all six-membered heterocyclic rings adopted an unusual boat conformation with equatorially configured alkyl groups. The unprecedented heteroatom-bearing belts also exhibited different photophysical and redox properties to those of octahydrobelt[8]arene analogues.     

Formation of Stone–Wales edge: Multistep reconstruction and growth mechanisms of zigzag nanographene

Although the existence of Stone–Wales (5‐7) defect at graphene edge has been clarified experimentally, theoretical study on the formation mechanism is still imperfect. In particular, the regioselectivity of multistep reactions at edge (self‐reconstruction and growth with foreign carbon feedstock) is essential to understand the kinetic behavior of reactive boundaries but investigations are still lacking. Herein, by using finite‐sized models, multistep reconstructions and carbon dimer additions of a bared zigzag edge are introduced using density functional theory calculations. The zigzag to 5‐7 transformation is proved as a site‐selective process to generate alternating 5‐7 pairs sequentially and the first step with largest barrier is suggested as the rate‐determining step. Conversely, successive C₂ insertions on the active edge are calculated to elucidate the formation of 5‐7 edge during graphene growth. A metastable intermediate with a triple sequentially fused pentagon fragment is proved as the key structure for 5‐7 edge formation. © 2017 Wiley Periodicals, Inc.     

一维聚合物{[Co(N3)2(dps)2]·0.5H2O}n的合成，结构和磁性质

A coordination polymer {[Co(N₃)₂(dps)₂]·0.5H₂O}_n (1) (dps=dipyridin-4-ylsulfane) with one dimensional chains has been synthesized and the structure and magnetic properties were characterized. Complex 1 crystallizes belongs to monoclinic system and has space group P2₁/c. The Co^Ⅱ ion is six-coordinated and located at a compressed octahedral environment due to the Jahn-Teller effect. The ligand of dps bridges between Co^Ⅱ ions producing the infinite chains of Co…Co along a direction and, interestingly, four pyridyl groups of dps ligands form a propeller-shaped coordination geometry around Co^Ⅱ ion with the same spiral within the chain. These 1D chains are further linked through the hydrogen bonds to generate a 3D supermolecular network. Magnetic studies indicated that complex 1 shows very weak antiferromagnetic coupling between Co^Ⅱ ions in low temperature. CCDC: 769596.     

Mathematical analysis of sideband instabilities with application to rayleigh-bénard convection

Summary We introduce a new method for the analysis of sideband instabilities which are important for periodic patterns appearing in systems close to the instability threshold. The method relies on a two-fold application of the Liapunov-Schmidt reduction procedure, a first application to the nonlinear bifurcation problem and a second application to the linear spectral problem. We obtain rigorous results on the spectrum of the associated linearization in spaces allowing for general sideband perturbations by treating the sideband vector and the spectral parameter as small bifurcation parameters. We apply the theory to the small roll solutions in the Rayleigh-Bénard convection and derive domains in Rayleigh, Prandtl, and wave number space where the rolls are unstable. We recover the Eckhaus, zigzag, and skew-varicose instabilities obtained earlier by formal methods. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.     

How a Zigzag Carbon Nanotube Grows

Owing to the unique structure of zigzag (ZZ) carbon nanotubes (CNTs), their ring‐by‐ring growth behavior is different from that of chiral or armchair (AC) CNTs, on the rims of which kinks serve as active sites for carbon attachment. Through first‐principle calculations, we found that, because of the high energy barrier of initiating a new carbon ring at the rim of a ZZ CNT, the growth rate of a ZZ CNT is proportional to its diameter and significantly (10–1000 times) slower than that of other CNTs. This study successfully explained the broad experimental observation of the lacking of ZZ CNTs in CNT samples and completed the theory of CNT growth.     

压荷载下类岩石材料中的锯齿形裂纹分析

针对类岩石材料的复杂性和非均质性，特别是材料中一些强度相对较弱的晶界的存在等，对形状较为复杂的两种锯齿形裂纹：内张型滑动裂纹和单翼滑动裂纹的应力强度因子进行了分析．结合经典的滑动裂纹模型，讨论了裂纹起裂角、侧压以及翼裂长度等对应力强度因子的影响．分析的结果为进一步研究类岩石材料的断裂损伤机理提供了理论基础．     

Quasi-hereditary covers of higher zigzag algebras of type A

In this paper we define and study some quasi-hereditary covers for higher zigzag algebras of type A. We show how these algebras satisfy three different Koszul properties: they are Koszul in the classical sense, standard Koszul and Koszul with respect to the standard module Δ, according to the definition given in [24]. This last property gives rise to a well defined duality and we compute the Δ-Koszul dual as the path algebra of a quiver with relations.     

Renormalization phenomena in Ba-diluted ferroelastic lead phosphate, (Pb1-x Ba x )3(PO4)2

Abstract

In the improper ferroelastic palmierite-type lead phosphate order parameter coupling with a defect induced conjugate field leads to the renormalization of the two different critical temperatures of three order parameter components {Q ₃} and {Q ₁ Q ₂}. The influence of the lead dilution by barium on the ferroelastic domain pattern, the critical temperature of the ferroelastic transformation R m–C2/c and the development of the intermediate regime in (Pb_1–x Ba _x )₃(PO₄)₂ is studied using optical birefringence measurements, Raman and infrared spectroscopy. At the ferroelastic transition temperature T _c the orientational contribution of the three-states Potts model becomes critical. T _c is reduced from 453 K (x = 0) to zero K at x? 0.12. Modifications of the shape of zigzag needle domains as well as the angle between the monoclinic binary axis and the W walls along [031] with temperature and increasing Ba-content are reported. Above the ferroelastic transition point the component Q ₃, which corresponds to the displacive part in the Gibbs free energy, leads to dynamic short-range monoclinic deformation in the trigonal matrix. The temperature where Q ₃ shows critical behaviour is renormalized to 720 K (x = 0.12) as compared with 563 K in pure lead phosphate. For x>0.13 no monoclinic precursor clusters were found.     

Multiplicative degree-Kirchhoff index and number of spanning trees of a zigzag polyhex nanotube TUHC[2n, 2]

Let L_n denote the linear hexagonal chain containing n hexagons. Then identifying the opposite lateral edges of L_n in ordered way yields TUHC[2n, 2] , the zigzag polyhex nanotube, whereas identifying those of L_n in reversed way yields M_n, the hexagonal Möbius chain. In this article, we first obtain the explicit formulae of the multiplicative degree-Kirchhoff index, the Kemeny's invariant, the total number of spanning trees of TUHC[2n, 2] , respectively. Then we show that the multiplicative degree-Kirchhoff index of TUHC[2n, 2] is approximately one-third of its Gutman index. Based on these obtained results we can at last get the corresponding results for M_n.