Counterexamples in optimal quadrature

It is widely believed that order of exactness is a good measure of the quality of an algorithm for numerical quadrature. We show that this is not the case, by exhibiting a situation in which the optimal algorithm does not even integrate constants exactly. We also show that there are situations in which the penalty for using equidistant nodes is unbounded. Finally, we show that the complexity of obtaining an -approximation can be an arbitrary function of , i.e., there is no hardest quadrature problem.     

Quartic X-splines

Summary The quartic periodic and nonperiodic X-spline are separated from the class of all piecewise-quartic interpolatory polynomials and their orders of convergence, smoothness and complexity of construction are examined. In particular, error estimates of interpolation of smooth functions at uniformly spaced knots by eight quartic X-splines of special interest are presented. The results are illustrated by a numerical example.     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].     

Direct Electrochemistry of Multi‐Copper Oxidases at Carbon Nanotubes Noncovalently Functionalized with Cellulose Derivatives

This study describes the direct electron transfer of multi‐copper oxidases, i.e., laccase (from Trametes versicolor) and bilirubin oxidase (BOD, from Myrothecium verrucaria) at multiwalled carbon nanotubes (MWNTs) noncovalently functionalized with biopolymers of cellulose derivatives, i.e., hydroxyethyl cellulose (HEC), methyl cellulose (MC), and carboxymethyl cellulose (CMC). The functionalization of the MWNTs with the cellulose derivatives is found to substantially solubilize the MWNTs into aqueous media and to avoid their aggregation on electrode surface. Under anaerobic conditions, the redox properties of laccase and BOD are difficult to be defined with cyclic voltammetry at either laccase/MWNT‐modified or BOD/MWNT‐modified electrodes. The direct electron transfer properties of laccase and BOD are thus studied in terms of the bioelectrocatalytic activities of the laccase/MWNT‐modified and BOD/MWNT‐modified electrodes toward the reduction of oxygen and found to be facilitated at the functionalized MWNTs. The possible application of the laccase‐catalyzed O₂ reduction at the laccase/MWNT‐modified electrode is illustrated by constructing a CNT‐based ascorbate/O₂ biofuel cell with the MWNT‐modified electrode as the anode for the oxidation of ascorbate biofuel.     

Numerical implementation of coherence for the example of the “Swiss Cross”

Summary We consider the two-dimensional Helmholtz equation u+u=0 inD with the boundary conditionsu=0 on D. D is the Swiss Cross — a region consisting of five unit squares. A method based on the concept of Coherence is utilized to determine an approximation for the first eigenvalue= ₁ more accurate than calculated by classical difference methods. The numerical result is used to illustrate isoperimetric upper and lower bounds for ₁, and to test some conjectures on its relations with torsional rigidity.Dedicated to the memory of Professor Lathar Collatz     

Functional inequality characterizing nonnegative concave functions in (0, ∞) k

Summary In the present note we prove that every functionf: (0, ) [0, ) satisfying the inequalityaf(s) + bf(t) f(as + bt), s, t > 0, for somea andb such that 0 <a < 1 <a + b must be of the formf(t) = f(1)t, (t > 0). This improves our recent result in [2], where the inequality is assumed to hold for alls, t 0, and gives a positive answer to the question raised there.An analogue for functions of several real variables of the above result characterizes concave functions. Conjugate functions and some relations to Hölder's and Minkowski's inequalities are mentioned.     

Problème inverse de la théorie du potentiel logarithmique pour les lemniscates

The paper is related to the question of uniqueness in the inverse logarithmic potential problem. This question is to find the conditions on which two domains D ₁ and D ₂ producing the same external potential must coincide. Assuming the general hypothesis of regularity and an additional condition of connectivity of (D₁D₂)_c, we prove a theorem of uniqueness in the case when one of the domains is a lemniscate. The main tool is one lemma for Cauchy's potential due to M. Sakai. We give a simple proof of its extension to Newtonian potential.

     

On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

Factorial regular representation of groups in complete graphs

A necessary and sufficient condition for the group of isomorphisms involved in a factorization of a complete graph into isomorphic factors is established.     

Within the Ischemic Penumbra,Sub-Cellular Compartmentalization of Heat Shock Protein 70 Overlaps with Autophagy Proteins and Fails to Merge with Lysosomes

The brain area which surrounds the frankly ischemic region is named the area penumbra. In this area, most cells are spared although their oxidative metabolism is impaired. area penumbra is routinely detected by immunostaining of a molecule named Heat Shock Protein 70 (HSP70). Within the area penumbra, autophagy-related proteins also increase. Therefore, in the present study, the autophagy-related microtubule-associated protein I/II-Light Chain 3 (LC3) was investigated within the area penumbra along with HSP70. In C57 black mice, ischemia was induced by permanent occlusion of the distal part of the middle cerebral artery. Immunofluorescence and electron microscopy show that LC3 and HSP70 are overexpressed and co-localize within the area penumbra in the same cells and within similar subcellular compartments. In the area penumbra, marked loss of co-localization of HSP70 and LC3-positive autophagy vacuoles, with lysosomal-associated membrane protein 1 (LAMP1) or cathepsin-D-positive lysosome vacuoles occurs. This study indicates that, within the area penumbra, a failure of autophagolysosomes depends on defective compartmentalization of LC3, LAMP1 and cathepsin-D and a defect in merging between autophagosomes and lysosomes. Such a deleterious effect is likely to induce a depletion of autophagolysosomes and cell clearing systems, which needs to be rescued in the process of improving neuronal survival.