Subexpansions,superexpansions and uniqueness properties in non-integer bases

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers 1<q<2 for which the β-expansion of 1 is the unique possible expansion with coefficients 0 or 1. Subsequently, the unique expansions were characterized in [9] and this characterization led to the determination (in [17]) of the smallest number q having this curious property. It is intimately related to the classical Thue-Morse sequence. Allouche and Cosnard recently proved that this q is transcendental. The purpose of this paper is to extend the previous results for expansions in arbitrary non-integer bases q>1. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section we answer a question concerning the existence of universal expansions, a notion introduced in [12]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Weighted polynomial inequalities

For the weights exp (?|x|^λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialP _n can decrease on [?1,1] ifP _n (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedL ^p spaces.     

Asymptotics of Christoffel functions on arcs and curves

For a system of smooth Jordan curves and arcs asymptotics for Christoffel functions is established. A separate new method is developed to handle the upper and lower estimates. In the course to the upper bound a theorem of Widom on the norm of Chebyshev polynomials is generalized.     

Electrochemically initiated tagging of thiols using an electrospray ionization based liquid microjunction surface sampling probe two‐electrode cell

This paper reports on the conversion of a liquid microjunction surface sampling probe (LMJ‐SSP) into a two‐electrode electrochemical cell using a conductive sample surface and the probe as the two electrodes with an appropriate battery powered circuit. With this LMJ‐SSP, two‐electrode cell arrangement, tagging of analyte thiol functionalities (in this case peptide cysteine residues) with hydroquinone tags was initiated electrochemically using a hydroquinone‐doped solution when the analyte either was initially in solution or was sampled from a surface. Efficient tagging (～90%), at flow rates of 5–10 µL/min, could be achieved for up to at least two cysteines on a peptide. The high tagging efficiency observed was explained with a simple kinetic model. In general, the incorporation of a two‐electrode electrochemical cell, or other multiple electrode arrangement, into the LMJ‐SSP is expected to add to the versatility of this approach for surface sampling and ionization coupled with mass spectrometric detection. Published in 2009 by John Wiley & Sons, Ltd.     

The norm of minimal polynomials on several intervals

Using works of Franz Peherstorfer, we examine how close the $n$th Chebyshev number for a set $E$ of finitely many intervals can get to the theoretical lower limit $2\mathrm{cap}{\left(E\right)}^n$.     

Univoque graphs for non-integer base expansions

Science China Mathematics - Unique expansions in non-integer bases have been investigated in many papers during the last thirty years. They are often conveniently generated by labeled directed...     

Experimental strategy for characterization of essential dynamical variables in oscillatory systems: effect of double-layer capacitance on the stability of electrochemical oscillators

An experimentally accessible algorithm for changing the time scale associated with a dynamical variable is proposed. In general, a differential controller can be applied to (a) identify the essential species in oscillatory systems and (b) explore their role in the feedback loops. Here, we report on classifying electrochemical oscillators by changing the time scale over which the electrode potential varies; the type of different electrochemical oscillators is identified based on whether the controlled modification of pseudo-capacitance induces or suppresses current oscillations.