On noncommutative weighted local ergodic theorems on <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces

In the present paper we consider a von Neumann algebra M with a faithful normal semi-finite trace τ, and {α _t }, a strongly continuous extension to L ^p (M, τ) of a semigroup of absolute contractions on L ¹(M, τ). By means of a non-commutative Banach Principle we prove for a Besicovitch function b and x ∊ L ^p (M, τ), that the averages 1/T ∫₀ ^T b(t)α _t (x)dt converge bilateral almost uniformly in L ^p (M, τ) as T → 0. Communicated by Dénes Petz     

Von Neumann algebras generated by translation-invariant Gibbs states of the Ising model on a Bethe lattice

The Ising model on a Bethe lattice of orderk≥2 is considered. For maximum or minimum translation-invariant Gibbs states of this model, the relations between the von Neumann algebras generated by these states for the Gelfand-Neimark-Segal representation are found. These algebras can be of types III_λ, λ∈(0, 1), and III₁. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 1, pp. 88–93, April, 2000.     

Benzazoles 5*. Synthesis and arylsulfonylation of 1-hydroxymethylbenzimidazole

1-Hydroxymethylbenzimidazole was synthesized by the reaction of benzimidazole with formaldehyde. Arylsulfonylation of the former in the presence of triethylamine occurred anomalously with deformylation to give 1-arylsulfonylbenzimidazoles in place of the expected 1-arylsulfonyloxymethyl-benzimidazoles.     

Recurrence equations over trees in a non-Archimedean context

In the present paper we study recurrence equations over k-ary trees. Namely, each equation is assigned to a vertex of the tree, and they are generated by contractive functions defined on an arbitrary non-Archimedean algebra. The main result of this paper states that the given equations have at most one solution. Moreover, we also provide the existence of unique solution of the equations. We should stress that the non-Archimedeanity of the algebra is essentially used, therefore, the methods applied in the present paper are not valid in the Archimedean setting.     

Benzazoles: II. Synthesis and arenesulfonylation of 1-methyl-1H-benzimidazol-2-amine

The reaction of 1-methyl-1H-benzimidazol-2-amine with arenesulfonyl chlorides in the presence of triethylamine afforded 1-arenesulfonyl-3-methyl-2,3-dihydro-1H-benzimidazol-2-imines instead of expected N-(1-methyl-1H-benzimidazol 2-yl)arenesulfonamides.     

Complexes of acetic acid α-(N-benzoxazolin-2-one) with Zn(II), Cu(II), and Co(II): Syntheses and crystal structures

The results of syntheses and X-ray diffraction analyses of mononuclear complexes [ML₂(H₂O)₄] (M = Co²⁺(I), Cu²⁺(II), and Zn²⁺(III)) containing water molecules and anions of acetic acid α-(N-benzoxazolin-2-one) (L = C₉H₆O₄) are presented. The crystals of complexes I–III are isostructural (space group P2₁/n, Z = 2) and are built of discrete neutral complex molecules. The crystallographic data are as follows: for complex I, a = 6.1470(5), b = 5.3310(3), c = 30.5894(17) Å, β = 95.056(6)°, V = 998.50(11) ų; for complex II, a = 5.9661(6) Å, b = 5.1414(4) Å, c = 32.672(2) Å, β = 92.395(6)°, V = 1001.33(14) ų; and for complex III, a = 6.1404(3) Å, b = 5.3476(2) Å, c = 30.5865(12) Å, β = 94.708(4)°, V = 1000.96(7) ų. The metal atoms (M) of the complexing agents are localized in the crystallographic symmetry centers and have a distorted octahedral environment due to two oxygen atoms of the carboxy groups of two monodentate ligands (L) and four water molecules. The M-O(1w)(H₂O) and M-O(2w)(H₂O) bond lengths for the indicated complexes are 2.088(3) and 2.118(3), 2.446(3) and 1.971(3), and 2.113(4) and 2.093(3) Å for M = Co²⁺, Cu²⁺, and Zn²⁺, respectively. The crystal structures are formed due to packing of chains built of inter-molecular hydrogen bonds O-H…O.     

Rheological properties of polycomplex gels of carboxymethyl cellulose with urea-formaldehyde oligomers

Structural and rheological properties of polycomplex gels of sodium carboxymethyl cellulose (Na-CMC) with urea-formaldehyde oligomers were studied at various component ratios at 298–343 K. The polycomplexes were examined by IR spectroscopy. Mechanical characteristics of solutions of Na-CMC and its complexes with urea-formaldehyde oligomers and the size of structural units of these solutions were determined.     

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.     

Benzazoles. 1. Acylation of Benzothiazolin-2-ones by Aromatic Acid Chlorides Using Small Amounts of Zinc Chloride

We show that the yield of 6-aroylbenzothiazolin-2-ones, products of acylation of benzothiazolin-2-ones by aromatic acid chlorides in the presence of ZnCl₂, depends on the degree of nucleophilicity of benzothiazolin-2-ones.