Selective Demethylations in 2, 3, 4-Trimethoxyaryl Carbonyl Compounds

2, 3, 4-Trimethoxybenzaldehyde, -acetophenone and -benzoic acid give the corresponding 2, 3-dihydroxy-4-methoxy compounds in good yield on treatment with BCl₃.     

The Cartan,Choquet and Kellogg properties for the fine topology on metric spaces

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.     

Harnack's inequality for a nonlinear eigenvalue problem on metric spaces

We prove Harnack's inequality for first eigenfunctions of the p-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (1,p)-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.     

Hyperbolic Harmonic Functions: Weak Approach with Applications in Function Spaces

Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular, we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among else Bloch spaces, Bergman-spaces, Besov-spaces, and Q_p-spaces. The second author was supported by the Finnish Cultural Foundation.     

Continuity of weak solutions of elliptic partial differential equations

The continuity of weak solutions of elliptic partial differential equations

Optimization of the gravimetric determination method of nickel as dimethylglyoximate for nickel raw materials

The method parameters of an almost one century old method for analyzing nickel as dimethylglyoximate were critically examined for the analysis of nickel raw materials and many of the method parameters were observed to have a significant effect on the Ni recovery. Thus, because the Ni precipitation method parameters vary a lot in analytical literature and also in practice, the obtained Ni results by different methods are not comparable. During this study it was found that the double precipitation worked out perfectly in eliminating the effects of impurity elements. The residual Ni content in the filtrates should also be measured to obtain accurate and precise Ni results. In complexing the impurity elements, tartaric acid, stabilized by acetic acid, turned out to be effective, and when added to the sample solution before ammonium addition, the best pH conditions for homogenous Ni precipitation with dimethylglyoxime were obtained. The optimized Ni determination method described was found to be accurate and highly reproducible when tested with Ni concentrates and standard reference materials containing high Ni concentrations.     

A strong Lebesgue point property for Sobolev functions

We show that first-order Sobolev functions fulfill a Wiener integral type Lebesgue point property outside a set of Sobolev capacity zero. Our condition is stronger than the standard Lebesgue point property, but the exceptional set is slightly larger.

     

Sobolev embeddings in metric measure spaces with variable dimension

In this article we study metric measure spaces with variable dimension. We consider Lebesgue spaces on these sets, and embeddings of the Riesz potential in these spaces. We also investigate Hajłasz-type Sobolev spaces, and prove Sobolev and Trudinger inequalities with optimal exponents. All of these questions lead naturally to function spaces with variable exponents. Supported the Research Council of Norway, Project 160192/V30.     

Optimization of the gravimetric determination method of nickel as dimethylglyoximate for nickel raw materials

The method parameters of an almost one century old method for analyzing nickel as dimethylglyoximate were critically examined for the analysis of nickel raw materials and many of the method parameters were observed to have a significant effect on the Ni recovery. Thus, because the Ni precipitation method parameters vary a lot in analytical literature and also in practice, the obtained Ni results by different methods are not comparable. During this study it was found that the double precipitation worked out perfectly in eliminating the effects of impurity elements. The residual Ni content in the filtrates should also be measured to obtain accurate and precise Ni results. In complexing the impurity elements, tartaric acid, stabilized by acetic acid, turned out to be effective, and when added to the sample solution before ammonium addition, the best pH conditions for homogenous Ni precipitation with dimethylglyoxime were obtained. The optimized Ni determination method described was found to be accurate and highly reproducible when tested with Ni concentrates and standard reference materials containing high Ni concentrations. Received: 26 December 1998 / Revised: 12 April 1999 / Accepted: 14 April 1999     

A Theorem on Fine Connectedness

We prove that E is a p-fine domain whenever R ⁿ is a p-fine domain, E R ⁿ is p-polar, and 1 < p n. By a p-fine domain we understand an open connected set in the p-fine topology, i.e. in the coarsest topology making all p-superharmonic functions continuous. As an application of our main result, we establish a general version of minimum principle.