Crystal structure of β-cyclodextrin-4-chlorobenzoic acid complex:Unusual C-Cl…π interaction between 4-chlorobenzoic acids in β-cyclodextrin dimer

The inclusion complex ofβ-cyclodextrin（β-CD） with 4-chlorobenzoic acid was synthesized and its crystal structure was determined by single crystal X-ray diffraction under cryogenic condition.The complex contains twoβ-CDs,two 4-chlorobenzoic acids and twenty-nine water molecules in the asymmetric unit,and can be formulated as 2(C₄₂H₇₀O₃₅)·2（C₇H₅O₂Cl）·29（H₂O）.In the crystal lattice,twoβ-CDs form a face-to-face dimer jointed together through hydrogen bonding between the secondary hydroxyl groups ofβ-CDs.Two 4-chlorobenzoic acid molecules which contact by C-Cl…πinteractions are included in theβ-CD dimer cavity.Theβ-CD dimers are arranged in brickwork-like pattern along the crystallographic a-axis.The effect of guest molecule length on the inclusion geometry was discussed by comparative study of someβ-CD complexes containing similar guests.     

Hopf π-代数与π-子代数

主要讨论局部有限维的Hopf π-代数的π-子代数与Hopf π-子代数,分别得到了Hopf π-代数的π-子代数和Hopf π-子代数的一些充分必要条件.     

Hopf π-子模

设H为有限型Hopfπ-代数,研究Hopfπ-代数H上的Hopfπ-模与Hopf π-余代数H *上的Hopfπ-余模之间的对偶关系,得出了Hopfπ-子模与Hopfπ-子 余模之间的充分必要条件,推广了Hopf代数中的相关结论.     

Upper and Lower Bounds of Table Sums

For a group G,we produce upper and lower bounds for the sum of the entries of the Brauer character table of G and the projective indecomposable character table of G.When G is a π-separable group,we show that the sum of the entries in the table of Isaacs' partial characters is a real number,and we obtain upper and lower bounds for this sum.     

Synthesis and Crystal Structure of [Zn(FcCOO)2(2,2′-bipy)(H2O)]2·H2O

A novel complex [Zn(FcCOO)2(2,2′-bipy)(H2O)]2·H2O (Fc = (η5-C5H5)Fe(η5-C5H4), 2,2′-bipy = 2,2′-bipyridine) has been synthesized and characterized by elemental analysis, IR and X-ray diffraction. It crystallizes in monoclinic system, space group P21/c with a = 6.8187(4), b = 21.7155(13), c = 19.7411(12) (A), α = 90, β = 97.7420(10), γ = 90°, C64H58Fe4Zn2N4O11, Mr = 1413.28, V = 2896.4(3) (A)3, Dc = 1.620 g/cm3, Z = 2, F(000) = 1444, μ(MoKα) = 1.857 mm-1, R = 0.0523 and wR = 0.0982 for 3219 observed reflections (I ＞ 2σ(I)). Structural analysis shows that the zinc atom is coordinated with three oxygen atoms from two ferrocenemonocarboxylates and one water molecule together with two nitrogen atoms from 2,2′-bipyridine, giving a distorted square-pyramidal coordination geometry. The complex molecules are linked to form an infinite one-dimensional chain by both intermolecular hydrogen bonds and π-π stacking interactions of the bipyridine rings.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

弱Hopf群余代数Kegel定理

本文研究了余三角弱Hopfπ-余代数H的左弱π-H-余模代数.通过构造左弱π-H-余模代数的导出π-σ-李代数,得到了弱Hopf π-余代数Kegel定理,推广了文献[4]的结果.     

Categorical Abstract Algebraic Logic: Subdirect Representation of Pofunctors

Pa?asińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in the theory of categorical abstract algebraic logic. In this note, an order subdirect representation theorem for partially ordered algebraic systems is proven. This is an analog of the Order Subdirect Representation Theorem of Pa?asińska and Pigozzi, which, in turn, generalizes the well-known Subdirect Representation Theorem of Universal Algebra.     

Coquasitriangular Hopf Group Algebras and Drinfel'd Co-Doubles

A ring R is called “semicommutative” if any right annihilator over R is an ideal of R. We show that special subrings of upper triangular matrix rings over a reduced ring are maximal semicommutative. Consequently, new families of semicommutative rings are presented.     

On Dedekind Modules

In this article the authors give the relation between a finitely-generated torsionfree Dedekind module M over a domain R and prime submodules of the 𝒪(M)-module M and the ring 𝒪(M). They also prove that M is a finitely-generated torsionfree Dedekind module over a domain R if and only if every semi-maximal submodule of R-module M is invertible.