自由格的若干定理

格的一个等式类Ｋ上由一个弱偏格生成的自由格的定义可以仿照Ｋ上由偏格生成的自由格的定义来给出．本文给出了Ｋ上由弱偏格生成的自由格的存在定理，并对任一非序集生成的自由分配格的结构作了两点刻画．     

Isomorphous rare‐earth tris[bis(2,6‐diisopropylphenyl) phosphate] complexes and their catalytic properties in 1,3‐diene polymerization and in the inhibited oxidation of polydimethylsiloxane

Crystals of mononuclear tris[bis(2,6‐diisopropylphenyl) phosphato‐κO]pentakis(methanol‐κO)lanthanide methanol monosolvates of lanthanum, [La(C₂₄H₃₄O₄P)₃(CH₃OH)₅]·CH₃OH, ( 1 ), cerium, [Ce(C₂₄H₃₄O₄P)₃(CH₃OH)₅]·CH₃OH, ( 2 ), and neodymium, [Nd(C₂₄H₃₄O₄P)₃(CH₃OH)₅]·CH₃OH, ( 3 ), have been obtained by reactions between LnCl₃(H₂O)_n (n = 6 or 7) and lithium bis(2,6‐diisopropylphenyl) phosphate in a 1:3 molar ratio in methanol media. Compounds ( 1 )–( 3 ) crystallize in the monoclinic P2₁/c space group and have isomorphous crystal structures. All three bis(2,6‐diisopropylphenyl) phosphate ligands display a κO‐monodentate coordination mode. The coordination number of the metal atom is 8. Each [Ln{O₂P(O‐2,6‐ⁱPr₂C₆H₃)₂}₃(CH₃OH)₅] molecular unit exhibits four intramolecular O—H…O hydrogen bonds, forming six‐membered rings. The unit forms two intermolecular O—H…O hydrogen bonds with one noncoordinating methanol molecule. All six hydroxy H atoms are involved in hydrogen bonding within the [Ln{O₂P(O‐2,6‐ⁱPr₂C₆H₃)₂}₃(CH₃OH)₅]·CH₃OH unit. This, along with the high steric hindrance induced by the three bulky diaryl phosphate ligands, prevents the formation of a hydrogen‐bond network. Complexes ( 1 )–( 3 ) exhibit disorder of two of the isopropyl groups of the phosphate ligands. The cerium compound ( 2 ) demonstrates an essential catalytic inhibition in the thermal decomposition of polydimethylsiloxane in air at 573 K. Catalytic systems based on the neodymium complex tris[bis(2,6‐diisopropylphenyl) phosphato‐κO]neodymium, ( 3′ ), which was obtained as a dry powder of ( 3 ) upon removal of methanol, display a high catalytic activity in isoprene and butadiene polymerization.     

Hydrogen bonding and fluorous weak interactions in the non‐isomorphous {4,4′‐bis[(2,2,3,3‐tetrafluoropropoxy)methyl]‐2,2′‐bipyridine‐κ2N,N′}dibromidopalladium and ‐platinum complexes

The polyfluorinated title compounds, [MBr₂(C₁₈H₁₆F₈N₂O₂)] or [4,4′‐(HCF₂CF₂CH₂OCH₂)₂‐2,2′‐bpy]MBr₂, ( 1 ) (M = Pd and bpy is bipyridine) and ( 2 ) (M = Pt), have –CH_(α)2OCH_(β)2CF₂CF₂H side chains with methylene H‐atom donors at the α and β sites, and methine H‐atom donors at the terminal sites, in addition to aromatic H‐atom donors. In contrast to the original expectation of isomorphous structures, ( 1 ) crystallizes in the space group C2/c and ( 2 ) in P2₁/n, with similar unit‐cell volumes and Z = 4. The asymmetric unit of ( 1 ) is one half of the molecule, which resides on a crystallographic twofold axis. Both ( 1 ) and ( 2 ) display stacking of the molecules, indicating a planar (bpy)MBr₂ skeleton in each case. The structure of ( 1 ) exhibits columns with C—H_(β)…Br hydrogen bonds between consecutive layers which conforms to a static (β,β) linkage between layers. In the molecular plane, ( 1 ) shows double C—H_(α)…Br hydrogen bonds self‐repeating along the b axis, the planar molecules being connected into infinite belts. Compound ( 2 ) has no crystallographic symmetry and forms π‐dimer pairs as supermolecules, which then stack parallel to the a axis. The π‐dimer‐pair supermolecules exhibit (Pt—)Br…Br(—Pt) contacts [3.6937 (7) Å] to neighbouring π‐dimer pairs crosslinking the columns. The structure of ( 2 ) reveals many C—H…F(—C) interactions between F atoms and aromatic C—H groups, in addition to those between F atoms and methylene C—H groups.     

因子上非线性混合三重积保持映射

设$\mathcal{A}$, $\mathcal{B}$是两个因子且$\dim\mathcal{A}>4$.本文证明了双射$\phi:\mathcal{A}\rightarrow\mathcal{B}$ 满足对所有的$A,B,C\in\mathcal A$有$\phi([A,B]\bullet C)=[\phi(A),\phi(B)]\bullet\phi(C)$当且仅当$\phi$是线性*-同构, 共轭线性*- 同构,负的线性*-同构, 负的共轭线性*-同构.     

Isomorphism of lattices of recursively enumerable sets

Let , and for , let be the lattice of subsets of which are recursively enumerable relative to the ``oracle' . Let be , where is the ideal of finite subsets of . It is established that for any , is effectively isomorphic to if and only if , where is the Turing jump of . A consequence is that if , then . A second consequence is that can be effectively embedded into preserving least and greatest elements if and only if .

     

Upperbound procedures for the identification of similar three-dimensional chemical structures

Summary This paper describes techniques for calculating the degree of similarity between an input query molecule and each of the molecules in a database of 3-D chemical structures. The inter-molecular similarity measure used is the number of atoms in the 3-D common substructure (CS) between the two molecules which are being compared. The identification of 3-D CSs is very demanding of computational resources, even when an efficient clique detection algorithm is used for this purpose. Two types of upperbound calculation are described which allow reductions in the number of exact CS searches which need to be carried out to identify those molecules from a database which are similar to a 3-D target molecule.     

Isomorphism of inclusion compounds built with composite host lattices of thiadiazole and V-shaped molecules with the existence of tetrabutylammonium guest template

By using 4,4′-dihydroxydiphenylsulfone (BPS) and 4,4′-dihydroxydiphenylsulfide (BTS) react with amidinethiourea and tetrabutylammonium respectively, two isomorphs of inclusion compounds of (n-C₄H₉)₄N⁺·C₁₂H₉O₄S^?·C₂H₄N₄S (1) and (n-C₄H₉)₄N⁺·C₁₂H₉O₂S^?·C₂H₄N₄S (2) were prepared and characterized by Single crystal X-ray diffraction. Although two varied types of V-shaped molecules were selected to react in the actual experimental process, two isomorphs with similar packing modes were finally obtained. Interestingly, amidinothiourea molecule occurred in situ oxidation reaction to generate a new compound of 3,5-dimido-1,2,4,-thiadiazole during the experiments, and these two inclusion compounds here are the pioneer examples of thiadiazole and tetraalkylammonium and can be regarded as two isomorphs due to their similar crystallization modes. In these two isomorphs, thiadiazole molecules derived from amidinothiourea link with various V-type molecules to develop the similar dumbbell-type hydrogen-bonded ribbons by the alike hydrogen bonding modes, then the ribbons interpenetrate with tetrabutylammonium to result in two stable isomorphism structures.     

Groups with the same non-commuting graph

The non-commuting graph Γ_G of a non-abelian group G is defined as follows. The vertex set of Γ_G is G−Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy≠yx. It has been conjectured that if G and H are two non-abelian finite groups such that Γ_G≅Γ_H, then |G|=|H| and moreover in the case that H is a simple group this implies G≅H. In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H. Then for certain simple groups H, we show that the graph isomorphism Γ_G≅Γ_H implies G≅H.     

关于GL(3,2)的两个反例

指出GL(3,2)的两个不共轭的{2,3}-Hall子群都和S_4同构.由此,进一步给出两个不同构的S_4-模提供相同2-Brauer特征标的例子.     

On graphs whose Laplacian matrix’s multipartite separability is invariant under graph isomorphism

Normalized Laplacian matrices of graphs have recently been studied in the context of quantum mechanics as density matrices of quantum systems. Of particular interest is the relationship between quantum physical properties of the density matrix and the graph theoretical properties of the underlying graph. One important aspect of density matrices is their entanglement properties, which are responsible for many nonintuitive physical phenomena. The entanglement property of normalized Laplacian matrices is in general not invariant under graph isomorphism. In recent papers, graphs were identified whose entanglement and separability properties are invariant under isomorphism. The purpose of this note is to completely characterize the set of graphs whose separability is invariant under graph isomorphism. In particular, we show that this set consists of K_2,2 and its complement, all complete graphs and no other graphs.