U-statistics in danach spaces

U-statistics in Banach spaces are considered and thoroughly investigated. The martingale structure, estimates of moments, the law of large numbers, the central limit theorem, the invariance principle, estimates of the rate of convergence, and large deviations are established     

On Le’s and Bugeaud’s Papers about the Equation ax 2?+ b 2m?1 ?=?4 c p

 We correct an inaccuracy in the papers of Le, Mignotte, and Bugeaud on the generalized Ramanujan-Nagell equation. Received June 23, 2001     

Lie point symmetries of the Lane-Emden systems

It is shown that a Lie point symmetry of the Lane-Emden system is a Noether symmetry if and only if its parameters belong to the critical hyperbola.     

Covering group and graph of discretized volumes

We consider a discretized volume V consisting of finite, congruent and attached copies of a tile t. We find a group L _V the orbit of which, when applied to t, is just V. We show the connection between the structural matrixQ in the formal solution of a boundary value problem formulated for volume V and the so called auxiliary matrix of the graph Γ _v associated with V. We show boundary value problems to be isomorphic if the graphs associated with the volumes are isomorphic, or, if the covering groups are Sunada pairs.     

Interaction between Ti-gel and silver, copper, as well as nickel compounds under ultrasound irradiation

Ag-TiO₂, Cu-TiO₂ and Ni-TiO₂ were prepared by sonication. The interactions of Ti-gel with silver, copper and nickel ions under ultrasound irradiation are very different, although these ions can be dispersed very well in titania. The results of EDXA and XRD analysis indicate that Ti-gel does not react with AgCl and crystallizes unaffectedly to form rutile. M(OH)₂ (M = Cu and Ni), on the other hand, favors polycondensation with Ti-gel and affects the crystallization of Ti-gel. CuO favors stabilization of the anatase phase. Ni²⁺ ions tend to incorporate more easily into titania than Cu²⁺ ions do when the samples are calcined. TEM, and BET were also used to characterize the samples.     

Luminol Oxidation by Hydrogen Peroxide Catalyzed by Tobacco Anionic Peroxidase: Steady-State Luminescent and Transient Kinetic Studies

The properties of a newly isolated anionic tobacco peroxidase from transgenic tobacco plants overexpressing the enzyme have been studied with respect to the chemiluminescent reaction of luminol oxidation. These were compared to the properties of horseradish peroxidase in the cooxidation of luminol and p -iodophenol, the enhanced chemiluminescence reaction. The pH, luminol and hydrogen peroxide concentrations were optimized for maximum sensitivity using the tobacco enzyme. The detection limit for the latter under the optimal conditions (2.5 m M luminol, 2 m M hydrogen peroxide, 100 m M Naborate buffer, pH 9.3) was about 0.1 p M , which is at least five times lower than that for horseradish peroxidase in enhanced chemiluminescence with p -iodophenol. The rate constants for the elementary steps of the enzyme-catalyzed reaction have been determined: k ₁= 4.9 × 10⁶ M ⁻¹ s¹, k ₂= 7.3 × 10⁶ M ⁻¹ s⁻¹, k ³= 2.1 × 10⁶ M ⁻¹ s⁻¹ (pH 9.3). The similarity of these rate constants is unusual for plant peroxidases. The high catalytic activity of tobacco peroxidase in the luminescent reaction is explained by the high reactivity of its Compound II toward luminol and the high stability of the holoenzyme with respect to heme dissociation. This seems to be a unique property of this particular enzyme among other plant peroxidases.