A New Route to Crystalline Salts of the Hydrolytic Dimer of Chromium(III)

Crystalline salts of the hydrolytic dimer of Cr(III), [Cr₂(μ-OH)₂(H₂O)₈]X₄·n H₂O (X = p-toluenesulfonate (tos) or mesitylene-2-sulfonate (dmtos)) have been prepared in good yields via a simple two-step procedure: H⁺ oxidation of Cr metal to give Cr²⁺ (T ≈? 70°) followed by O₂ oxidation, of Cr²⁺ to give the dimer (T ≈? 25°). The mechanism of conversion of Cr²⁺ into the dimer is discussed.     

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On the occupation times of cones by Brownian motion

Summary We study some features concerning the occupation timeA _t of a d-dimensional coneC by Brownian motion. In particular, in the case whereC is convex, we investigate the asymptotic behaviour ofP(A₁u0, when the Brownian motion starts at the vertex ofC. We also give the precise integral test, which decides whether a.s., lim inf _t A _t/(tf(t))=0 or for a decreasing functionf.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Applications of capillary electrophoresis in element speciation analysis of plant and food extracts

The performance of a new separation technique for metal-speciation — capillary electrophoresis (CE) — is presented. Polyhydroxy compounds such as polyphenols and organic tanning agents can be determined in the g/L range. Certain complexes of these compounds with iron (III) ions may also be directly or indirectly verified. The work has shown that it is possible to determine certain complexes existing in beverages (tea and coffee) by decomplexation with chelating agents (CDTA). The ligands released can be separated by CE and identified by their migration time and UV/VIS absorption spectra.     

Quantitative determination of chloride by means of flow injection analysis with spectrophotometric detection in the UV/VIS range

Chloride is determined indirectly by Spectrophotometric flow injection analysis. Two systems are compared, both based on the principle of ion exchange of easily detectable anions versus chloride from suitable mercury salts. The first method is based on the exchange of chloride with chloranilate which is detected at 332 nm or at 306 nm in neutral or in acidic medium respectively. In the second case, chloride reacts with Hg(SCN)₂. The liberated thiocyanate forms a strongly coloured complex with Fe(III) in acidic solution with an absorption maximum at 460 nm. Both methods have a detection limit of about 5 mol Cl^–/l (175 ng/ml). In the case of the thiocyanate method, the relative standard deviation is about 2% (7 measurements) in the range of 5 to 150 mol/l and decreases significantly to a value of approximately 0.2% at higher concentrations; for the chloranilate method it is 10% for lower and about 1% for higher concentrations respectively.     

An invariant trace formula for the universal covering group of SL(2,ℝ)

Automorphic forms of arbitrary real weight can be considered as functions on the universal covering group of SL(2, ). In this situation, we prove an invariant form of the Selberg trace formula for Hecke operators. For this purpose, the Fourier transforms of weightet orbital integrals, obtained by J. Arthur, R. Herb and P. Sally, jr., are explicitly calculated. Our formula does not follow from Arthur's invariant trace formula, since the group has infinite centre, and vector-valued automorphic forms with respect to non-congruence lattices are considered.     

The finite section method for Moore-Penrose inversion of Toeplitz operators

It is shown that for certain classes of infinite block Toeplitz matricesT(a)=[a _j-k ] ₀ the Moore-Penrose inverses of the finite sectionT _n (a)=[a _j-k ] ₀ ^n–1 converge to the Moore-Penrose inverse ofT(a). Furthermore the convergence for modified finite section methods and the finite section method for Wiener-Hopf integral and related operators are studied.     

On the integrated density of states for crystals with randomly distributed impurities

In the present paper, we discuss spectral properties of a periodic Schrödinger operator which is perturbed by randomly distributed impurities; such operators occur as simple models for crystals (or semi-conductors) with impurities. While the spectrum itself is independent of the concentrationp of impurities, for 0<p<1, we focus our attention on the limiting behavior of the integrated density of states _p of the random Schrödinger operator, inside a spectral gap of the periodic operator, asp0. Denoting byU ₀ the set of eigenvalues (in the gap) of the reference problem having precisely one impurity (located at the origin, say), we show that the integrated density of states concentrates around the points ofU ₀, in the sense that _p (U ) is of orderp, for any fixed -neighborhoodU ofU ₀, while _p (K)C·p ², for any compact subsetK of the gap which does not intersectU .Research partially supported by Deutsche Forschungsgemeinschaft