Hamel bases and measurability

This is a note - set in the background of some historic comments - discussing the relationship between measurability and Hamel bases for R over Q. We explicitly note that such a basis must necessarily fail to be Borel measurable (or even 'analytic' in the sense of descriptive set theory). We also discuss some constructions in the literature which yield Hamel bases which even fail to be Lebesgue measurable, and discussan elementary construction of a Hamel basis which is Lebesgue measurable

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.Comment: 13 pages, LaTe

Diagonalizing operators over continuous fields of C*-algebras

It is well known that in the commutative case, i.e. for $A=C(X)$ being a commutative C*-algebra, compact selfadjoint operators acting on the Hilbert C*-module $H_A$ (= continuous families of such operators $K(x)$, $x\in X$) can be diagonalized if we pass to a bigger W*-algebra $L^\infty(X)={\bf A} \supset A$ which can be obtained from $A$ by completing it with respect to the weak topology. Unlike the "eigenvectors", which have coordinates from $\bf A$, the "eigenvalues" are continuous, i.e. lie in the C*-algebra $A$. We discuss here the non-commutative analog of this well-known fact. Here the "eigenvalues" are defined not uniquely but in some cases they can also be taken from the initial C*-algebra instead of the bigger W*-algebra. We prove here that such is the case for some continuous fields of real rank zero C*-algebras over a one-dimensional manifold and give an example of a C*-algebra $A$ for which the "eigenvalues" cannot be chosen from $A$, i.e. are discontinuous. The main point of the proof is connected with a problem on almost commuting operators. We prove that for some C*-algebras if $h\in A$ is a selfadjoint, $u\in A$ is a unitary and if the norm of their commutant $[u,h]$ is small enough then one can connect $u$ with the unity by a path $u(t)$ so that the norm of $[u(t),h]$ would be also small along this path.Comment: 21 pages, LaTeX 2.09, no figure

Finite difference lattice Boltzmann model with flux limiters for liquid-vapor systems

In this paper we apply a finite difference lattice Boltzmann model to study the phase separation in a two-dimensional liquid-vapor system. Spurious numerical effects in macroscopic equations are discussed and an appropriate numerical scheme involving flux limiter techniques is proposed to minimize them and guarantee a better numerical stability at very low viscosity. The phase separation kinetics is investigated and we find evidence of two different growth regimes depending on the value of the fluid viscosity as well as on the liquid-vapor ratio.Comment: 10 pages, 10 figures, to be published in Phys. Rev.

The Bond-Algebraic Approach to Dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix representation. Dualities like exact dimensional reduction, emergent, and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the (\mathbb{Z}_2) Higgs model is dual to the extended toric code model {\it in any number of dimensions}. Non-local dual variables and Jordan-Wigner dictionaries are derived from the local mappings of bond algebras. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.Comment: 131 pages, 22 figures. Submitted to Advances in Physics. Second version including a new section on the eight-vertex model and the correction of several typo

N 3-[(E)-Morpholin-4-yl­methyl­idene]-1-phenyl-1H-1,2,4-triazole-3,5-diamine monohydrate

In the title compound, C13H16N6O·H2O, the mean planes of the benzene and 1,2,4-triazole rings form a dihedral angle of 54.80 (5)°. The N atom of the amino group adopts a trigonal–pyramidal configuration. Conjugation in the amidine N=C—N fragment results in sufficient shortening of the formal single bond. In the crystal, inter­molecular N—H⋯O and O—H⋯N hydrogen bonds link mol­ecules into double layers parallel to the bc plane

Quality indicators in a mycobacteriology laboratory supporting clinical trials for pulmonary tuberculosis

AbstractBackgroundDocumentation of structured quality indicators for mycobacteriology laboratories supporting exclusively controlled clinical trials in pulmonary tuberculosis (PTB) is lacking.ObjectiveTo document laboratory indicators for a solid (Lowenstein–Jensen medium) culture system in a mycobacteriology laboratory for a period of 4years (2007–2010).MethodsThe sputum samples, collected from PTB suspects/patients enrolled in clinical trials, were subjected to fluorescence microscopy, culture and drug sensitivity testing (DST). Data was retrospectively collected from TB laboratory registers and computed using pre-formulated Microsoft Office Excel. Laboratory indicators were calculated and analyzed.ResultsThe number of samples processed in a calendar year varied from 6261 to 10,710. Of the samples processed in a calendar year, specimen contamination (4.8–6.9%), culture positives (78.4–85.1%) among smear positives, smear positives (71.8–79.0%) among culture positive samples, smear negatives among culture negative samples (95.2–96.7%), and average time to report DST results (76–97days) varied as shown in parentheses.ConclusionValues of quality indicators in mycobacteriology laboratories supporting exclusively clinical trials of PTB have to be defined and used for meaningful monitoring of laboratories

Dendritic Core-Shell Macromolecules Soluble in Supercritical Carbon Dioxide

International audienceSupercritical carbon dioxide has found strong interest as a reaction medium recently.1,2 As an alternative to organic solvents, compressed carbon dioxide is toxicologically harmless, nonflammable, inexpensive, and environmentally benign.3 Its accessible critical temperature and pressure (Tc ) 31 °C, Pc ) 7.38 MPa, Fc ) 0.468 g cm-3)4 and the possibility of tuning the solvent-specific properties between the ones of liquid and gas are very attractive