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].     

Secondary and Tertiary Structure of Polysaccharides in Solutions and Gels

Many polysaccharide chains can adopt ordered helical and ribbon-like secondary structures. It seems however that these chains are often so stiff and extended that the cooperative interactions necessary for stability in the solvent environment can only be achieved when inter-chain as well as intra-chain interactions are favorable. Hence we commonly find two-or more-stranded associations of helices, of ribbons, or of helices with ribbons. These can be regarded as tertiary and higher levels of structure. The ordered secondary structure characteristically requires a regular repeating sequence of sugar residues, and the termination of this sequence by insertion of a residue of different type may also terminate the secondary structure and hence the association in which it is involved. This is the mechanism by which native polysaccharides may link up to form three dimensional networks, or gels, in which state they perform their natural roles in maintaining the hydration and integrity of biological tissues. For several polysaccharides there is evidence that the mechanism of biological control over the fine topology and properties of the gel network is mediated by enzymes which modify sugar residues at the polymer level to change the pattern of “interrupting” sugar residues.     

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.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Subsolidus phase relations and crystal structure of compounds in the PrOx-CaO-CuO system

The subsolidus phase relations of the PrO_x-CaO-CuO pseudo-ternary system sintered at 950-1000°C have been investigated by X-ray powder diffraction. In this system, there exist one compound Ca₁₀Pr₄Cu₂₄O₄₁, one Ca₂Pr₂Cu₅O₁₀-based solid solution, seven three-phase regions and two two-phase regions. The crystal structures of Ca₁₀Pr₄Cu₂₄O₄₁ and Ca₂Pr₂Cu₅O₁₀-based solid solution have been determined. Compound Ca₁₀Pr₄Cu₂₄O₄₁ crystallizes in an orthorhombic cell with space group D_2h²⁰−Cccm, Z=4. Its lattice parameters are a=11.278(2) Å, b=12.448(3) Å and c=27.486(8) Å. The crystal structure of Ca₂Pr₂Cu₅O₁₀-based solid solution is an incommensurate phase based on the orthorhombic NaCuO₂ type subcell. The lattice parameters of the subcell of the Ca_2.4Pr_1.6Cu₅O₁₀ are a₀=2.8246(7) Å, b₀=6.3693(5) Å, c₀=10.679(1) Å, and those of the orthorhombic superstructure are with a=5a₀, b=b₀, c=5c₀. The Ca_2.4Pr_1.6Cu₅O₁₀ structure can also be determined by using a monoclinic supercell with space group C_2h⁵−P2₁/c, Z=4, a=5a₀, b=b₀, and β=104.79(1)° or 136.60(1)°, V=5a₀b₀c₀.