Explicit left orders on free groups extending the lexicographic order on free monoids

For every finitely generated free group, we construct an explicit left order extending the lexicographic order on the free monoid generated by the positive letters. The order is defined by a left, free action on the orbit of 0 of a free group of piecewise linear homeomorphisms of the line. The membership in the positive cone is decidable in linear time in the length of the input word. The positive cone forms a context-free language closed under word reversal.     

A simple quantum statistical thermodynamics interpretation of an impressive phase diagram pressure shift upon (H/D) isotopic substitution in water + 3-methylpyridine

In a previous work (J. Phys. Chem. B 2003, 107, 9837), we reported liquid-liquid-phase splitting at negative pressures in mixtures of H2O + D2O + 3-methylpyridine (3-MP) at the limit of pure H2O as the solvent, thus extending for the first time the L-L phase diagrams to this metastable region. We showed that there is an intimate relation between pressure and solvent deuterium content. Isotopic substitution (H/D) in water provokes subtle entropic effects that, in turn, trigger a significant pressure shift, opening a pressure-wide miscibility window of as much as 1600 bar. Isotope effects are quantum in origin. Therefore, a model that is both pressure-dependent and considers quantization constitutes a necessary tool if one wishes to fully describe the p, T, x critical demixing in these systems. In the current work, the statistical-mechanical theory of isotope effects is combined with a compressible pressure-dependent model. This combination enabled us to predict successfully the overall L-L phase diagram via differences in the vibrational mode frequencies of water on its transfer from the pure state to that of dilution in 3-MP: each of the three librational modes undergo a calculated red-shift of -(250 +/- 30) cm(-1), while the overall internal frequencies contribution is estimated as a total +(400 +/- 25) cm(-1) blue-shift.     

Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions

Let X be a Lévy process in, , obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Lévy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Lévy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.     

Arsenopyrite mineral based electrochemical sensor for acid–base titrations in γ-butyrolactone and propylene carbonate

Abstract  

A novel acid–base sensor based on the natural mineral arsenopyrite for titrations in γ-butyrolactone and propylene carbonate is validated and studied. This sensor, which requires only small sample volumes, was employed for the titrations of some important organic acids (benzoic, anthranilic, and salicylic acid) with potassium hydroxide. A stable stationary potential was attained at the arsenopyrite electrode in γ-butyrolactone and propylene carbonate in less than 5 min. The change in the electrode potential from the acidic to basic region was 379 to −160 mV in γ-butyrolactone and 434 to −67 mV in propylene carbonate. The arsenopyrite electrode showed a relatively fast response time in the investigated solvents (11 s in γ-butyrolactone and 14 s in propylene carbonate). It can be used without any time limit or without considerable divergence in the potential. The investigated electrode showed a linear dynamic response for p-toluenesulfonic acid concentrations in the range 0.1–0.001 M, and a sub-Nernst dependence in γ-butyrolactone as well as in propylene carbonate. Unlike traditional pH titration, an end-point color indicator is unnecessary in this method, and real-time monitoring can be realized. The relative standard deviations for measurements of benzoic, anthranilic, and salicylic acids were 0.10–0.31%, which show that the repeatability and accuracy of measurements taken with the sensor are satisfactory.     

Dissipation anomaly and energy cascade in 3D incompressible flows

The purpose of this Note is to present a mathematical evidence of the dissipation anomaly in 3D turbulent flows within a general setting for the study of energy cascade in physical scales of 3D incompressible flows recently introduced by the authors.     

Some Aspects of Ionic Liquids as Diverse and Versatile Sustainable Solvents

In this mini review several particular cases of ionic liquid solution behavior are discussed, namely solutions of ionic liquids with aliphatic alcohols, polyalcohols, arenes, chloromethanes and poly(ethyleneglycol) as well as salting-out and salting-in effects. The presented cases clearly expose ionic liquids as being diverse and versatile sustainable solvents that exhibit flexible phase behavior and, consequently, variable phase diagrams. These can be tuned either by changing the length of the alkyl chain(s) of the ionic liquid??s cation (and/or sometimes the anion), or by introducing a different aliphatic nature to the other solution constituent (e.g., by varying polymer chain length or the number of carbon atoms in chloromethanes). Sometimes, the evolution of the phase diagrams is fine and continuous, showing several consecutive stages, revealing both qualitative and quantitative changes. Finally, the diversity and versatility of ionic liquids are viewed as important features that contribute to their efficiency as tunable solvents or salting media.     

Elastomer-Clay Nanocomposites: Effect of Elastomer Polarity

Dynamic mechanical properties of non-polar and polar elastomer-based nanocomposites with Montmorillonite clay are studied in order to ascertain differences caused by elastomer's polarity. Natural as non-polar and polychloroprene as polar rubber are used to experimentally verify predictions based on different nature of elastomer-clay interactions in the two cases. The properties foretold and elucidated in terms of van der Waals interactions in the non-polar case and dipole-dipole interactions in the polar case agree well with experiment, concurrently sustaining general depiction of these unusual nanocomposites.     

Damping optimization of parameter dependent mechanical systems by rational interpolation

We consider an optimization problem related to semi-active damping of vibrating systems. The main problem is to determine the best damping matrix able to minimize influence of the input on the output of the system. We use a minimization criteria based on the \(\mathcal {H}_{2}\) system norm. The objective function is non-convex and the associated optimization problem typically requires a large number of objective function evaluations. We propose an optimization approach that calculates ‘interpolatory’ reduced order models, allowing for significant acceleration of the optimization process. In our approach, we use parametric model reduction (PMOR) based on the Iterative Rational Krylov Algorithm, which ensures good approximations relative to the \(\mathcal {H}_{2}\) system norm, aligning well with the underlying damping design objectives. For the parameter sampling that occurs within each PMOR cycle, we consider approaches with predetermined sampling and approaches using adaptive sampling, and each of these approaches may be combined with three possible strategies for internal reduction. In order to preserve important system properties, we maintain second-order structure, which through the use of modal coordinates, allows for very efficient implementation. The methodology proposed here provides a significant acceleration of the optimization process; the gain in efficiency is illustrated in numerical experiments.     

Picosecond fluorescence spectroscopy with a CW laser and a radio — a simple technique

A simple picosecond-domain lifetime measurement technique is described, wherein ac null detection is performed with two modulated light beams, one from the sample, the other an optically delayed part of the excitation, both falling onto a single detector.