Finite combinations of Baire numbers

Letκ be a regular cardinal. Consider the Bair numbers of the spaces (2^θ)κ for variousθ≥κ. Letl be the number of such different Baire numbers. Models of set theory withl=1 orl=2 are known and it is also known thatl is finite. We show here that ifκ>ω, thenl could be any given finite number.     

Sol-Gel Physical and Covalent Entrapment of Three Methyl Red Indicators: A Comparative Study

We report the results of a comparative study of the properties of three azo dyes from the methyl red indicator family, which were coentrapped with the surfactant cetyltrimethylammonium bromide within tetramethoxysilane-derived silica sol-gel matrices. Characteristic parameters studied were spectral shifts, shifts in indicating ranges and in pKi values, isosbestic-point behavior, changes in the titration curves and stability to leaching. Physical entrapment of methyl red and para-methyl red (pMR) is compared to the covalent entrapment of a trialkoxysilane derivative of methyl red. High stability for leaching was found for all three entrapped indicators. The pKi change in entrapped pMR is interpreted in terms of intracage motion induced by pH changes.     

High-order harmonic generation of pulses with multiple timescales: selection rules,carrier envelope phase and cutoff energy

ABSTRACT

High harmonic generation (HHG) is sensitive to the carrier envelope phase (CEP) of its driving laser field if it is a sufficiently short pulse (several-cycle pulse). Here we show that strong CEP effects can also be found in HHG from long duration multi-cycle pulses (up to 200?fs at 800?nm central wavelength). We find that HHG from multi-cycle pulses may be CEP dependent when the driving pulse exhibits two distinct timescales (multi-timescale pulse): (i) a short timescale associated with the average frequency, and (ii) a long timescale associated with the pulse’s temporal periodicity. The interplay of these timescales results in significant changes to both the cutoff frequency, and the appearance of symmetry allowed harmonics in the spectrum as function of CEP, similar to HHG from several-cycle pulses. We relate this effect to the multi-timescale intensity variations in the driving pulse, and construct an analytical condition to access the phenomenon. Lastly, we numerically demonstrate reconstruction of the CEP through HHG from long duration multi-timescale pulses. Our work may be useful in several areas of strong-field physics and attosecond science, for example, allowing spectroscopy of multi-timescale processes (e.g. HHG from vibrationally active media), and paving the way towards CEP characterisation using long pulses.     

Nonlinear variational inequalities and differential games with stopping times

Differential games with stopping times are introduced. The upper value satisfies a variational inequality of the first order. Next, stochastic differential games with stopping times are introduced and studied using nonlinear parabolic variational inequalities. The behavior of the value is studied as the coefficient of the diffusion matrix goes to zero.     

Synthesis and characterization of a new tetradentate ligand for Cu(II) metal ions

The synthesis and characterization of a tailor-made ligand 1 is described. Self-assembly of 1 on a GaAs surface was studied by FT-IR spectroscopy. The IR spectrum of the thin film points to the conclusion that the monolayer adsorbs to the surface via both carboxylates of the malonic acid derivative. The distinction between the possible binding modes is discussed.