Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

Characterization of the normal distribution by constant regression of quadratic statistics on linear statistics

This paper contains applications of theorems of [1] for quadratic statistics which have constant regression on linear statistics. Two theorems are proved. The first is a sufficient condition which assumes that the characteristic function of a sample is an entire function. The second gives a new characterization of the normal distribution.     

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].     

Copper—A “Modern” Bioelement

Copper is a bioessential element in biology with truly unique chemical characteristics in its two relevant oxidation states +I and +II. Significant progress has been made in recent years in the elucidation of the frequently surprising biochemistry of this trace element. Those advances were especially furthered through mutual stimulation involving results from biochemistry, molecular biology, and medicine on one hand and the synthesis as well as the structural and spectroscopic characterization of low molecular weight model complexes on the other. The most notable features of protein-bound active copper are its almost exclusive function in the metabolism of O₂ or N/O compounds (NO, N₂O) and its frequent association with oxidizing organic and inorganic radicals such as tyrosyl, semiquinones, superoxide, or nitrosyl. This unique biological role of copper can be rationalized given its chemical and assumed evolutionary background.     

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.     

Preparation and characterization of Mg(II)-, Ca(II) and Cd(II) complexes of 1,2-ethanediol and water

Mixed ligand complexes of different compositions were prepared with water, sulfate ion and 1,2-ethanediol as ligand. IR spectra and the thermoanalytical curves of the complexes were recorded. Oxygen atoms bound by one or two coordinate bonds to the metal ion, or by hydrogen-bonds in the crystal, were observed. As for the water molecule, 1,2-ethanediol molecules of crystal and monohydrate type were found, depending on the type of binding of the oxygen atoms.This revised version was published online in November 2005 with corrections to the Cover Date.     

Linear combination of concomitants of order statistics with application to testing and estimation

Summary Let (X ₁,Y ₁), (X ₂,Y ₂),…, (X _n,Y _n) be i.i.d. as (X, Y). TheY-variate paired with therth orderedX-variateX _rn is denoted byY _rn and terms the concomitant of therth order statistic. Statistics of the form are considered. The asymptotic normality ofT _n is established. The asymptotic results are used to test univariate and bivariate normality, to test independence and linearity ofX andY, and to estimate regression coefficient based on complete and censored samples.     

On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.