Orthocomplemented Posets with a Symmetric Difference

We endow orthocomplemented posets with a binary operation–an abstract symmetric difference of sets–and we study algebraic properties of this class, . Denoting its elements by ODP, we first investigate on the features related to compatibility in ODPs. We find, among others, that any ODP is orthomodular. This explicitly links with the theory of quantum logics. By analogy with Boolean algebras, we then ask if (when) an ODP is set representable. Though we find that general ODPs do not have to be set representable, many natural ODPs are shown to be. We characterize the set-representable ODPs in terms of two valued morphisms and prove that they form a quasivariety. This quasivariety contains the class of pseudocomplemented ODPs as we show afterwards. At the end we ask whether any orthomodular poset can be converted or, more generally, enlarged to an ODP. By countre-examples we answer these questions to the negative. The authors acknowledge the support of the research plan MSM 0021620839 that is financed by the Ministry of Education of the Czech Republic and the grant GAČR 201/07/1051 of the Czech Grant Agency.     

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

Synthesis, characterization, and the role of counterion in stabilizing trigonal pyramidal copper(I)/2,2′bipyridine complexes containing electron-poor methyl acrylate

Copper(I)/2,2′-bipyridine complexes, [Cu^I(bpy)(π-CH₂CHCOOCH₃)][A] have been synthesized and characterized. These complexes are used in copper(I) mediated cyclopropanation and aziridination reactions of methyl acrylate and represent the first class of trigonal pyramidal copper(I) complexes with π-coordinated electron poor olefins. In the case of 1 and 3, weak coordination of the counterion was observed. The counterion was noncoordinating in complex 2, which was dimeric in the solid state with the oxygen atoms of the carbonyl moieties in methyl acrylate bridging two copper(I) centers.     

UV-vis and fluorescence spectroscopic detection of anions by the conformational restriction of 2,2′-binaphthalene derivatives bearing thiourea groups through a methylene spacer

Novel fluorescence receptors, 2 and 3 based on 2,2′-binaphthalene possessing thiourea moieties via a methylene spacer have been synthesized. Hydrogen bonds of NH groups of thiourea moieties with acetate anion were confirmed by ¹H NMR study in MeCN-d₃. These receptors showed characteristic UV-vis spectral changes through isosbestic points on complexation with anions inspite of lacking conjugation between the chromophore and the binding sites in polar organic solvent such as acetonitrile. The UV-vis spectral changes arise from the conformational restriction of the 2,2′-binaphthyl skeleton on the complexation. The receptors exhibit high selectivities for AcO⁻ and F⁻. The fluorescence intensity of the receptors decreases with the increasing amount of the AcO⁻, however, addition of F⁻ induces a different change in its fluorescence spectrum, in which shorter emission of the receptors decreases with the increase in F⁻ concentration, while the longer emission of the receptors increases through an isoemissive point in MeCN. The results suggest that favorable dual-wavelength ratiometric fluorescence measurement can be conducted by the receptors for F⁻.     

Projective modules and prime submodules

In this paper, we use Zorn’s Lemma, multiplicatively closed subsets and saturated closed subsets for the following two topics (i) The existence of prime submodules in some cases (ii) The proof that submodules with a certain property satisfy the radical formula. We also give a partial characterization of a submodule of a projective module which satisfies the prime property.     

Spectral characteristics of 2-(4′-amino-3-pyridyl)benzimidazole: Effects of solvent and acid or base concentration

Spectral characteristics of 2-(4′-amino-3-pyridyl)benzimidazole (4-A3PyBI) have been studied in different solvents, as well as at different acid or base concentrations using absorption, fluorescence excitation and fluorescence spectroscopy. Excited singlet state (S₁) lifetimes for each species were measured using nanosecond time-dependent spectrofluorimeter. AM1 semi-empirical and density functional theoretical (DFT) calculations were performed on each species for the spectral assignment. From the above results it is concluded that 4-A3PyBI exists only in the amine form. First protonation occurs at pyridineN- atom and second protonation at the benzimidazole (BI)N- atom. When dication (DC) species is excited, two emission bands are observed, having the same fluorescence excitation spectra, suggesting the same ground state (S₀) precursor. Short wavelength (SW) emission band is assigned to the π-π* transition and long wavelength (LW) emission to the charge transfer transition. First deprotonation in S₀ state occurs from >N-H moiety, whereas in S₁ state it is from -NH₂ group. Monoanion (MA) so formed in S₁ state is non-fluorescent. Dianion (DA) is formed by further deprotonating >N-H moiety in S₁ state and it is fluorescent. pK_a values were determined and discussed.     

Silica gel-promoted synthesis of 3,4,5-triaryltetrahydro-1,4-thiazine derivatives from β,β′-dichloro sulfides and aromatic amines

A novel and efficient synthesis of 3,4,5-triaryltetrahydro-1,4-thiazine derivatives by treatment of β,β′-dichloro sulfides, 2:1 adducts of alkenes and sulfur dichloride, with aromatic amines in the presence of silica gel is described. The silica gel can be easily recovered and reused in up to four consecutive reactions without any significant decrease in product yield.     

On the complexity of the integral closure

The computation of the integral closure of an affine ring has been the focus of several modern algorithms. We will treat here one related problem: the number of generators the integral closure of an affine ring may require. This number, and the degrees of the generators in the graded case, are major measures of cost of the computation. We prove several polynomial type bounds for various kinds of algebras, and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus.

     

The Rings Characterized by Minimal Left Ideals

We study these rings with every minimal left ideal being a projective, direct summand and a p-injective module, respectively. Some characterizations of these rings are given, and the relations among them are obtained. With these rings, we characterize seinisiinple rings. Finally, we introduce MC2 rings, and give some characterizations of MC2 rings.