Uniform Spacing of Zeros of Orthogonal Polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ _m,k , θ _m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ _m,k −θ _m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.     

Smooth equilibrium measures and approximation

A necessary and sufficient condition is given for approximation with weighted expressions of the form wnPn, where w is a given continuous weight function and Pn are polynomials of degree . The condition is that the extremal measure that solves an associated equilibrium problem is smooth (asymptotically optimal doubling). As corollaries we get all previous (positive and negative) results for approximation, as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.     

Asymptotics for Christoffel Functions with Varying Weights

We prove sharp asymptotics for Christoffel functions λ_n(w²ⁿ, · ) with respect to varying weights w²ⁿ. The result has an interpretation in the theory of statistical-mechanical models of quantum systems, and it is sufficiently strong to yield asymptotics for Christoffel functions for weights defined on several intervals.     

Some problems of A. Kroó on multiple Chebyshev polynomials

Three problems of A. Kroó on multiple Chebyshev polynomials are solved using the Borsuk–Ulam antipodal theorem.     

Minimizing analyte electrolysis in electrospray ionization mass spectrometry using a redox buffer coated emitter electrode

An emitter electrode with an electroactive poly(pyrrole) (PPy) polymer film coating was constructed for use in electrospray ionization mass spectrometry (ESI‐MS). The PPy film acted as a surface‐attached redox buffer limiting the interfacial potential of the emitter electrode. While extensive oxidation of selected analytes (reserpine and amodiaquine) was observed in positive ion mode ESI using a bare metal (gold) emitter electrode, the oxidation was suppressed for these same analytes when using the PPy‐coated electrode. A semi‐quantitative relationship between the rate of oxidation observed and the interfacial potential of the emitter electrode was shown. The redox buffer capacity, and therefore the lifetime of the redox buffering effect, correlated with the oxidation potential of the analyte and with the magnitude of the film charge capacity. Online reduction of the PPy polymer layer using negative ion mode ESI between analyte injections was shown to successfully restore the redox buffering capacity of the polymer film to its initial state. Published in 2010 by John Wiley & Sons, Ltd.     

Kinetics of the photoaquation of Hexacyanoferrate(II) ion

Kinetics of the photoaquation of hexacyanoferrate(II) ion in aqueous solution were studied potentiometrically and spectrophotometrically. Supposing the simplest mechanism (see Fig. 3. in text), the photoaquation in alkaline medium can be well described. The value of the constants at pH = ll.0 are: ø = 0.8-1.0, k₆ = (3.0 ± 0.5) × 10^?8 s^?1 and k_?6 = 1.5 ± 0.2 mol^?1 dm₃ s^?1. To describe the photoaquation in neutral medium t was extended (k′ = 3.33 x 10² mol^?1 dm³s^?1). The quantum yield in acidic medium can be calculated by combination of ø values of different protonated complexes. The reversibility of photoaquation in alkaline medium is also explained by the scheme.