A note on the Stokes operator and its powers

The so-called Stokes operator is an important tool in the analysis of the solutions of the Navier-Stokes equations and their numerical approximation. The aim of this note is to clarify certain properties of the fractional powers of this operator which are sometimes misused.     

Density of rational points on del Pezzo surfaces of degree one

We state conditions under which the set S(k)$S(k)$ of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k   of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration S?P¹$S?{\mathbb{P}}^1$ induced by the anticanonical map has a nodal fiber over a k  -rational point of P¹${\mathbb{P}}^1$. It also suffices to require the existence of a point in S(k)$S(k)$ that does not lie on six exceptional curves of S   and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over R$\mathbb{R}$, the set of surfaces S   defined over Q$\mathbb{Q}$ for which the set S(Q)$S(\mathbb{Q})$ is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.     

Development and validation of a stability-indicative agar diffusion assay to determine the potency of linezolid in tablets in the presence of photodegradation products

Linezolid (LNZ) is one of the first commercially available (and most widely used) oxazolidinone antibiotics. This study describes the development and validation of a microbiological assay, applying the cylinder-plate method, for the determination of the antibiotic linezolid, as well as the evaluation of the ability of the method in determining the stability of linezolid in tablets. The validation method yielded good results and included linearity, precision, accuracy, robustness and selectivity. The assay is based on the inhibitory effect of LNZ upon the strain of Bacillus subtilis ATCC 9372 used as the test microorganism. The results of the assay were treated statistically by analysis of variance (ANOVA) and were found to be linear (r² = 0.9998) in the range of 20-80 μg mL⁻¹, precise (inter-assay: R.S.D. = 0.61) and accurate (R.S.D. = 1.7). The method developed and validated proved to be indicative of stability and capable of determining the decay of linezolid in the presence of photodegradation products. Comparison of bioassay and liquid chromatography by ANOVA showed no significant difference between methodologies. The results demonstrated the validity of the proposed bioassay, which is a simple and useful alternative methodology for LNZ determination in routine quality control.     

A Mathematical Characterization of Quantum Gaussian Stochastic Evolution Schemes

We give a common mathematical characterization of relevant stochastic evolution schemes built up in the literatute to attack the quantum measurement problem. This characterization is based on two hypotheses, namely, (i) the trace conservation with probability one and (ii) the existence of a complex phase determining a linear support for the stochastic process driving the random evolution.This paper is dedicated to Prof. Emilio Santos on the occasion of his 70th birthday.     

Numerical analysis of electric field formulations of the eddy current model

This paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current, subjected to boundary conditions involving only data easily available in applications. Two different cases are considered depending on the boundary data: input current intensities or differences of potential. Weak formulations in terms of the electric field are given in both cases. In the first one, the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit. In the second one, the electric potentials are imposed by means of Lagrange multipliers, which are proved to represent the input current intensities. Optimal error estimates are proved in both cases and implementation issues are discussed. Finally, numerical tests confirming the theoretical results are reported. Partially supported by Xunta de Galicia research grant PGIDIT02PXIC20701PN (Spain). Partially supported by FONDAP in Applied Mathematics (Chile). Partially supported by MCYT (Spain) project DPI2003-01316.     

Topological gravity from a transgression gauge field theory

It is shown that a topological action for gravity in even dimensions can be obtained from a gravity theory whose Lagrangian is given by a transgression form invariant under the Poincaré group.     

LC Assay for Ciprofloxacin Hydrochloride Ophthalmic Solution

The objective of the current study was to develop and subsequently validate a simple, sensitive and precise reversed-phase LC method for the determination of ciprofloxacin hydrochloride in ophthalmic solution form. The chromatographic separation of ciprofloxacin hydrochloride was achieved on a Symmetry Waters C₁₈ column using UV detection at 275 nm. The optimized mobile phase consisted of 2.5% acetic acid solution: methanol:acetonitrile (70:15:15, v/v/v). The proposed method provided linear responses within the concentration range 1.0–6.0 μg mL⁻¹ for ciprofloxacin hydrochloride. Correlation coefficient (r) for the ciprofloxacin hydrochloride was 0.9994. The precision of the method was demonstrated using intra- and inter-day assay RSD% values which were less than 5% in all instances. No interference from any components of pharmaceutical dosage forms was observed.

     

Dipyridylamide Donor Disposition and Isophthalate Substituent Steric Effect on the Dimensionality and Topology of Divalent Copper Coordination Polymers

Divalent copper coordination polymers containing an isophthalate ligand and a dipyridylamide ligand show different dimensionalities and topologies depending on pyridyl nitrogen donor disposition and the steric bulk of the substituent on the dicarboxylate aromatic ring. According to single‐crystal X‐ray diffraction, [Cu(ip)(3‐pna)]_n ( 1 , ip = isophthalate, 3‐pna = 3‐pyridylnicotinamide) shows a (4, 4) layered grid structure based on {Cu₂(OCO)₂} dimeric units. {[Cu(ip)(3‐pina)]·H₂O}_n ( 2 , 3‐pina = 3‐pyridylisonicotinamide) exhibits similar dimeric units, but in contrast to 1 these are connected into a non‐interpenetrated 3D 6⁵8 cds network. Both [Cu(mip)(3‐pina)]_n ( 3 , mip = 5‐methylisophthalate) and [Cu(meoip)(3‐pina)]_n ( 4 , mip = 5‐methoxyisophthalate) display dimer‐based 4¹²6³ pcu networks in contrast to 2 . Use of 5‐hydroxyisophthalate (H₂hip) as a precursor afforded a mixture of {[Cu₂(hip)₂(3‐pina)₄]·9.5H₂O}_n ( 5 ) and [Cu(hip)(3‐pina)]_n ( 6 ). Compound 5 shows a 2D interdigitated structure with [Cu(hip)]_n coordination polymer layers featuring {Cu₂(OCO)₂} dimeric units and pendant 3‐pina ligands, while 6 also showed a dimer‐based 4¹²6³ pcu network. Use of the very sterically bulky 5‐tert‐butylisophthalate (tbip) ligand afforded the 1D chain coordination polymer {[Cu(tbip)(3‐pina)₂(H₂O)]·H₂O}_n ( 7 ), which contains isolated copper ions in contrast to 1 – 6 , and has a curious “butterfly“ resemblance. Very weak antiferromagnetic coupling is seen within the {Cu₂(OCO)₂} dimeric units in 1 . Thermal decomposition behavior is also discussed.     

The Extended Cartan Homotopy Formula and a Subspace Separation Method for Chern–Simons Theory

In the context of Chern–Simons (CS) Theory, a subspace separation method for the Lagrangian is proposed. The method is based on the iterative use of the Extended Cartan Homotopy Formula, and allows one to (1) separate the action in bulk and boundary contributions, and (2) systematically split the Lagrangian in appropriate reflection of the subspace structure of the gauge algebra. In order to apply the method, one must regard CS forms as a particular case of more general objects known as transgression forms. Five-dimensional CS Supergravity is used as an example to illustrate the method.