In the article, we perform a classification of algebras with dimensions $\leq$ 3 and with the property that each element is colinear with its square. The classification is complete up to properties of the ground field.

Using electron microscopy, we studied the morphology of secretory granules in rat pars intermedia cells. We found figures of apparent intergranule fusion, characterized by a tight association of two granules. The fusion was detected in around 2% of all measured granules, indicating a low occurrence of intergranule fusion. To study whether intergranule fusion affects the distribution of granule diameters a simple probabilistic model was developed. It is based on the theory that larger granules are formed by fusion of two or more spherical granules of fixed size, and that the surface of a newly formed granule is equal to the sum of fused granule membranes. The model accounts for the bias on granule diameter measurements due to sectioning of granules. Although the electron microscopy data strongly indicates the existence of intergranule fusion in rat melanotrophs, this process as modelled in the present work does not contribute to the granule diameter distribution significantly. It is likely that in addition to the fusion of larger granules, other processes, such as fusion of microvesicles, may affect the distribution of granule diameters.

We introduced a new bivariate distribution for a random vector Z = [X, Y]T, where X and Y are positive continuous variables. The distribution is a generalization of the Weibull distribution, it has 7 parameters: r and s are power parameters, m and n are moment parameters, a and b are scaling parameters, and p is the linking parameter. The ML estimation turns out to be a very difficult task. In this paper we present the estimation procedure. It was tested on simulated data, which we generated using the acceptance-rejection method. The results are very satisfactory.

We propose a new bivariate distribution with five shape parameters and two scale parameters. It allows great flexibility for the bivariate situation and successfully replaces the assumption of the bivariate normal distribution when the ratio of two positive bounded variables is in question

To enable correct statistical inference, the knowledge about the existence of moments is crucial. The objective of this paper is to study the existence of the moments for the ratio \(Z = X/Y\) , where \(X\) and \(Y\) are arbitrary random variables with the additional assumption \(P(Y = 0) = 0\). We present three existence theorems showing that specific behaviour of the distribution of \(Y\) in the neighbourhood of zero is essential. Simple consequences of these theorems give evidence to the existence of the moments for particular random variables; some of these results are well known from standard probability theory. However, we obtain them in a simple way.

This paper presents the interval estimate for specific points in polynomial regression: zero of a linear regression, abscissa of the extreme of a quadratic regression, abscissa of the inflection point of a cubic regression. Two different approaches are under study. An application of these two approaches based on quadratic regression in presented: interval estimate for the plant density giving optimal yield of maize is under consideration.

We prove that the rounded expected value of the rounded random variable differs from the expected value of the original random variable by less then 2δ, where δ is the distance between two neighbouring rounded numbers. The same conclusion holds for standard deviation. Hence, in practice the influence of rounding to expected value and standard deviation is almost negligible.

We derive the probability density of the ratio of components of the bivariate normal distribution with arbitrary parameters. The density is a product of two factors, the first is a Cauchy density, the second a very complicated function. We show that the distribution under study does not possess an expected value or other moments of higher order. Our particular interest is focused on the shape of the density. We introduce a shape parameter and show that according to its sign the densities are classified into three main groups. As an example, we derive the distribution of the ratio Z = − Bm−1 /(mBm ) for a polynomial regression of order m. For m=1, Z is the estimator for the zero of a linear regression, for m = 2 , an estimator for the abscissa of the extreme of a quadratic regression, and for m = 3 , an estimator for the abscissa of the inflection point of a cubic regression.

Minotov zakon, po katerem je relativna stopnja rasti padajoča na vsem območju, kjer je mikrobna rastna krivulja naraščajoča, razširimo še na območje, kjer opazovana kultura umira. Pokažemo, da je rastna krivulja, za katero velja ta zakon, logaritmično konkavna in da ima zato vselej obliko , kjer je padajoča funkcija. Konec lag-faze definiramo kot začetek logaritmične konkavnosti rastne krivulje. V nadaljevanju naštejemo ostale splošne lastnosti takih rastnih krivulj, utemeljimo osnovni princip aproksimacije konkretnih podatkov in predlagamo preprost model.

The objective of this paper is to quantify and compare the loss functions of the standard two-stage design and its composite sample alternative in the context of multivariate soil sampling. The loss function is defined (conceptually) as the ratio of cost over information and measures design inefficiency. The efficiency of the design is the reciprocal of the loss function. The focus of this paper is twofold: (a) we define a measure of multivariate information using the Kullback–Leibler distance, and (b) we derive the variance-covariance structure for two soil sampling designs: a standard two-stage design and its composite sample counterpart. Randomness in the mass of soil samples is taken into account in both designs. A pilot study in Slovenia is used to demonstrate the calculations of the loss function and to compare the efficiency of the two designs. The results show that the composite sample design is more efficient than the two-stage design. The efficiency ratio is 1.3 for pH, 2.0 for C, 2.1 for N, and 2.5 for CEC. The multivariate efficiency ratio is 2.3. These ratios primarily reflect cost ratios; influence of the information is small.

In statistics, Fisher was the first to introduce the measure of the amount of information supplied by the data about the unknown parameter. We analyze the disadvantages of Fisher's information measure for optimization of sampling designs. To overcome this problem, we modify Fisher's information measure and we upgrade it to the multivariate setting. On a case study of soil we demonstrate the evaluation of different information measures derived from Fisher's information measure. The variables under study were the concentrations of several chemical compounds in soil (such as pH, N, C, Zn, etc.).

Let B be a real or complex complete normed quadratic algebra. All homomorphisms from arbitrary (possibly non associative) complete normed algebras into B are continuous if and only if B has no non-zero element with zero square.

The objective of a long-term soil survey is to determine the mean concentrations of several chemical parameters for the pre-defined soil layers and to compare them with the corresponding values in the past. A two-stage random sampling procedure is used to achieve this goal. In the first step, n subplots are selected from N subplots by simple random sampling without replacement; in the second step, m sampling sites are chosen within each of the n selected subplots. Thus n · m soil samples are collected for each soil layer. The idea of the composite sample design comes from the challenge of reducing very expensive laboratory analyses: m laboratory samples from one subplot and one soil layer are physically mixed to form a composite sample. From each of the n selected subplots, one composite sample per soil layer is analyzed in the laboratory, thus n per soil layer in total. In this paper we show that the cost is reduced by the factor m — 1 when instead of the two-stage sampling its composite sample alternative is used; however, the variance of the composite sample mean is increased. In the case of positive intraclass correlation the increase is less than 12.5%; in the case of negative intraclass correlation the increase depends on the properties of the variable as well. For the univariate case we derive the optimal number of subplots and sampling sites. A case study is discussed at the end.

In statistics, Fisher was the first to introduce the measure of the amount of information supplied by the data about the unknown parameter. We analyze the disadvantages of Fisher information measure for optimization of sampling designs. To overcome this problem, we modify Fisher information measure and we upgrade it to the multivariate setting. It turns out that a reasonable modification of Fisher information measure leads to a special case of Kullback information measure, both in the univariate and multivariate setting. Using Shannon’s and Wiener’s concept of information we also show a simple derivation of Kullback information measure for a special case when the prior distribution of the parameter is uniform and the posterior distribution is truncated normal.

Minot's law, that the relative growth rate is decreasing everywhere on the domain of increasing microbial growth function, we extend also to the domain where the observed culture decays. We show that the growth function which fulfils this law is logarithmically concave and is always of the form      ⋅ = ∫ t a dx