Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

Left dihedral codes over finite chain rings

Let R be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size 2n and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when $\gcd (char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over R and also over $R\times S$, where S is an isomorphic copy of R. As a particular result, we give the structure of cyclic codes of length 2 over R. In the case where $R={\mathbb{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\mathbb{F}}_{p^m}$, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.     

Shimalactone Biosynthesis Involves Spontaneous Double Bicyclo-Ring Formation with 8π-6π Electrocyclization

Shimalactones A and B are neuritogenic polyketides possessing characteristic oxabicyclo[2.2.1]heptane and bicyclo[4.2.0]octadiene ring systems that are produced by the marine fungus Emericella variecolor GF10. We identified a candidate biosynthetic gene cluster and conducted heterologous expression analysis. Expression of ShmA polyketide synthase in Aspergillus oryzae resulted in the production of preshimalactone. Aspergillus oryzae and Saccharomyces cerevisiae transformants expressing ShmA and ShmB produced shimalactones A and B, thus suggesting that the double bicyclo-ring formation reactions proceed non-enzymatically from preshimalactone epoxide. DFT calculations strongly support the idea that oxabicyclo-ring formation and 8π-6π electrocyclization proceed spontaneously after opening of the preshimalactone epoxide ring through protonation. We confirmed the formation of preshimalactone epoxide in vitro, followed by its non-enzymatic conversion to shimalactones in the dark.     

Dimerisation of Dipiperidinoacetylene: Convenient Access to Tetraamino-1,3-Cyclobutadiene and Tetraamino-1,2-Cyclobutadiene Metal Complexes

The reaction of 1,2-dipiperidinoacetylene ( 1 ) with 0.5 equivalents of SnCl₂ or GeCl₂⋅dioxane afforded the 1,2,3,4-tetrapiperidino-1,3-cyclobutadiene tin and germanium dichloride complexes 2 a and 2 b , respectively. A competing redox reaction was observed with excess amounts of SnCl₂, which produced a tetrapiperidinocyclobutadiene dication with two trichlorostannate(II) counterions. Heating neat 1 to 110 °C for 16 h cleanly produced the dimer 1,3,4,4-tetrapiperidino-3-buten-1-yne ( 3 ); its reaction with stoichiometric amounts of SnCl₂ or GeCl₂⋅dioxane furnished the 1,3,4,4-tetrapiperidino-1,2-cyclobutadiene tin and germanium dichloride complexes 4 a and 4 b , respectively. Transition-metal complexes containing this novel four-membered cyclic bent allene (CBA) ligand were prepared by reaction of 3 with [(tht)AuCl], [RhCl(CO)₂]₂, and [(Me₃N)W(CO)₅] to form [(CBA)AuCl] ( 5 ), [(CBA)RhCl(CO)₂] ( 6 ), and [(CBA)W(CO)₅] ( 7 ). The molecular structures of all compounds 2 – 7 were determined by X-ray diffraction analyses, and density functional theory (DFT) calculations were carried out to rationalise the formation of 3 and 4 a .     

Na原子链表面等离激元特性的含时密度泛函研究

本文用含时密度泛函理论研究了线性Na原子链的表面等离激元机理.主要在原子尺度下模拟计算了体系随着原子数增加及原子间距变化的集体激发过程.研究发现线性原子链有一个普遍的特性——存在一个纵模和两个横模.两个横模一般在实验上很难被观测到.纵模随着原子链长度增加,能量红移的同时,该纵模主峰的强度呈线性增长.随着原子个数的增加,端点模式(TE)开始蓝移,能量和偶极强度都逐渐趋向饱和.横模能量被劈裂的原因概括如下:(一)每个位置的电子受到的势不同,在两端的电子受到的势要比在中间的电子受到的势要高,因此两端的电荷积累也比中间多;(二)端点存在悬挂键,所以中间的电子-电子间相互作用与端点的不一样,这两方面又都与原子间距d有关.