Response to Comment on ‘Preconditioning of matrices partitioned in 2 × 2 block form: Eigenvalue estimates and Schwarz DD for mixed FEM’

As has been pointed out in a study by Cao, there is an erroneous omission in the study by Axelsson, Corollary 3.2 and Proposition 3.2, and the authors thank Zhi‐hao Cao for pointing this out. We mention, however, that these are side results and do not influence the major result of the paper ‘Preconditioning of matrices partitioned in 2 by 2 block form: Eigenvalue estimates and Schwarz DD for mixed FEM’. In this reply, we give a short background to the relations between inverses of saddle point matrices, thereby providing justified references to previous publications of the topic and present a corrected and extended version of Proposition 3.2. This gives an alternative presentation of the statements in the study by Cao, which might be useful for readers. Copyright © 2012 John Wiley & Sons, Ltd.     

Polypyrrole-Coated and Polysulfate-Modified CE Capillaries

Fused-silica capillaries covalently coated with polypyrrole offer some interesting properties for capillary electrophoresis. The polymeric, surface-bonded coating is chemically stable and reduces the adsorptive properties of the silica surface. At the same time, the weak positive surface charges offer the possibility of creating a strongly anionic surface when poly-sulfates are used as counter-ions. This affords an electroosmotic flow which is largely independent of pH. With sodium dodecyl sulfate in the background electrolyte, this effect is further enhanced. The potential of the approach is demonstrated with the separation of herbicides by micellar electrokinetic chromatography. Repeatability of electroosmotic mobility of less than 1% RSD is achieved.

     

Two Simple Derivations of Universal Bounds for the C.B.S. Inequality Constant

Universal bounds for the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for piecewise linear-linear and piecewise quadratic-linear finite element spaces in 2 space dimensions are derived. The bounds hold for arbitrary shaped triangles, or equivalently, arbitrary matrix coefficients for both the scalar diffusion problems and the elasticity theory equations.     

Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems

Two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied. Both techniques are based on spectral equivalence approach. The first technique consists in an incomplete factorization of the separate displacement component part of the stiffness matrix. The second technique uses an incomplete factorization of the isotropic approximation to the stiffness matrix. Results concerning existence, stability and efficiency of these preconditioning techniques are presented. The efficiency and robustness of the described techniques are illustrated by numerical experiments.     

Residuated Lattices of Size ≤ 12

We present the numbers of all non-isomorphic residuated lattices with up to 12 elements and a link to a database of these lattices. In addition, we explore various characteristics of these lattices such as the width, length, and various properties considered in the literature and provide the corresponding statistics. We also present algorithms for computing finite residuated lattices including a fast heuristic test of non-isomorphism.     

Polypyrrole-Coated and Polysulfate-Modified CE Capillaries

Fused-silica capillaries covalently coated with polypyrrole offer some interesting properties for capillary electrophoresis. The polymeric, surface-bonded coating is chemically stable and reduces the adsorptive properties of the silica surface. At the same time, the weak positive surface charges offer the possibility of creating a strongly anionic surface when poly-sulfates are used as counter-ions. This affords an electroosmotic flow which is largely independent of pH. With sodium dodecyl sulfate in the background electrolyte, this effect is further enhanced. The potential of the approach is demonstrated with the separation of herbicides by micellar electrokinetic chromatography. Repeatability of electroosmotic mobility of less than 1% RSD is achieved.     

Universal measurement apparatus controlled by quantum software

We propose a quantum device that can approximate any projective measurement on a qubit-a quantum "multimeter." The desired measurement basis is selected by the quantum state of a "program register." Two different kinds of programs are considered and in both cases the device is optimized with respect to maximal average fidelity (assuming uniform distribution of measurement bases). Quantum multimeters exhibiting the covariance property are introduced and an optimal covariant multimeter with a single-qubit program register is found. Possible experimental realization of the simplest proposed device is presented.     

How quantum correlations enhance prediction of complementary measurements

If there are correlations between two qubits, then the results of the measurement on one of them can help to predict measurement results on the other one. It is an interesting question as to what can be predicted about the results of two complementary projective measurements on the first qubit. To quantify these predictions the complementary knowledge excesses are used. A nontrivial constraint restricting them is derived. For any mixed state and for arbitrary measurements the knowledge excesses are bounded by a factor depending only on the maximal violation of Bell's inequalities. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.     

Some Properties of Residuated Lattices

We investigate some (universal algebraic) properties of residuated lattices--algebras which play the role of structures of truth values of various systems of fuzzy logic.     

Effects of the limited analyte solubility on its mobility and zone shape: electrophoretic behavior of sanguinarine and chelerythrine around pH 7

Electrophoretic mobilities and shapes of zones of sanguinarine and chelerythrine in aqueous media around pH 7 are affected by limited solubility of their uncharged forms and by the pH-dependent chemical equilibrium between cationic and uncharged forms of these alkaloids. The sanguinarine solubility in sodium MOPS of pH 7.4 was estimated at 50 micromol x L(-1). Sanguinarine zones in this buffer have the shape of tailed peak with concentration-independent mobility if the injected sanguinarine concentration exceeds this solubility limit only slightly. The chelerythrine solubility is higher because of lower dissociation constants of its cations. Precipitation of sanguinarine and chelerythrine with the phosphate anions decelerates their electrophoretic transport in phosphate buffer. Sanguinarine solubility is 5 micromol x L(-1) at the most in 13 mmol x L(-1) sodium phosphate buffer of pH 7.4. Acidifying of the sample up to pH 3 decreases the tailing of the peaks of sanguinarine and chelerythrine and contributes to the rise of sharp maxima of their migrating zones. Any capillary coating deteriorates the peak shape.