Поточечное условие абсолютной непрерывности функции одной переменной и его применения

An absolutely continuous function in calculus is precisely such a function that, within the framework of Lebesgue integration, can be restored from its derivative, that is, the Newton--Leibniz theorem on the relationship between integration and differentiation is fulfilled for it. An equivalent definition is that the the sum of the moduli of the increments of the function with respect to arbitrary pair-wise disjoint intervals is less than any positive number if the sum of the lengths of the intervals is small enough. Certain sufficient conditions for absolute continuity are known, for example, the Banach--Zaretsky theorem. In this paper we prove a new sufficient condition for the absolute continuity of a function of one variable and give some of its applications to problems in the theory of function spaces. The proved condition makes it possible to significantly simplify the proof of the theorems on the pointwise description of functions of the Sobolev classes defined on Euclidean spaces and Сarnot groups.

The Multiple K -Riemann Integral

The aim of this paper is to extend the notion of K -Riemann integrability of functions defined over a , b to functions defined over a rectangular box of ℝ n . As a generalization of step functions, we introduce a notion of K -step functions which allows us to give an equivalent definition of the K -Riemann integrable functions.

Preservers of partial orders on the set of all variance-covariance matrices

Let H+n(R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L?wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H+ n (R). We describe the form of all surjective maps on H+ n (R), n > 1, that preserve the L?wner partial order in both directions. We present an equivalent definition of the minus partial order on H+ n (R) and also characterize all surjective, additive maps on H+ n (R), n ? 3, that preserve the minus partial order in both directions.

Remarks on some generalization of the notion of microscopic sets

We consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ-ideal of classic microscopic sets.

A Spectral Theory for Wright’s Inbreeding Coefficients and Related Quantities

Wright’s inbreeding coefficient, FST, is a fundamental measure in population genetics. Assuming a predefined population subdivision, this statistic is classically used to evaluate population structure at a given genomic locus. With large numbers of loci, unsupervised approaches such as principal component analysis (PCA) have, however, become prominent in recent analyses of population structure. In this study, we describe the relationships between Wright’s inbreeding coefficients and PCA for a model of K discrete populations. Our theory provides an equivalent definition of FST based on the decomposition of the genotype matrix into between and within-population matrices. Assuming that a separation condition is fulfilled, our main result states that the proportion of genetic variation explained by the first (K − 1) principal components can be accurately approximated by the average value of FST over all loci included in the genotype matrix. This equivalent definition of FST can be used to evaluate the fit of discrete population models to the data. It is also useful for computing inbreeding coefficients from surrogate genotypes, for example, obtained after correction of experimental artifacts or after removing genetic variation associated with environmental variables. The relationships between inbreeding coefficients and the spectrum of the genotype matrix not only allow interpretations of PCA results in terms of population genetic concepts but extend those concepts to population genetic analyses accounting for temporal, geographical and environmental contexts.

Characterizations for the n-strong Drazin invertibility in a ring

Recently, a new type of generalized inverse called the [Formula: see text]-strong Drazin inverse was introduced by Mosić in the setting of rings. Namely, let [Formula: see text] be a ring and [Formula: see text] be a positive integer, an element [Formula: see text] is called the [Formula: see text]-strong Drazin inverse of [Formula: see text] if it satisfies [Formula: see text], [Formula: see text] and [Formula: see text]. The main aim of this paper is to consider some equivalent characterizations for the [Formula: see text]-strong Drazin invertibility in a ring. Firstly, we give an equivalent definition of the [Formula: see text]-strong Drazin inverse, that is, [Formula: see text] is the [Formula: see text]-strong Drazin inverse of [Formula: see text] if and only if [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we obtain some existence criteria for this inverse by means of idempotents. In particular, the [Formula: see text]-strong Drazin invertibility of the product [Formula: see text] are investigated, where [Formula: see text] is regular and [Formula: see text] are arbitrary elements in a ring.

LOWER ASSOUAD-TYPE DIMENSIONS OF UNIFORMLY PERFECT SETS IN DOUBLING METRIC SPACES

In this paper, we are concerned with the relationship among the lower Assouad-type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as [Formula: see text] tends to 1 is equal to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as [Formula: see text] tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi-lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi-lower Assouad dimension.

Preservers of partial orders on the set of all variance-covariance matrices

Let H+n(R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L?wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H+n(R). We describe the form of all surjective maps on H+ n (R), n > 1, that preserve the L?wner partial order in both directions. We present an equivalent definition of the minus partial order on H+n(R) and also characterize all surjective, additive maps on H+ n (R), n ? 3, that preserve the minus partial order in both directions.

Signal Recovery Beyond Linear Estimates

This chapter explains why Σ‎0 1 formulas of Peano arithmetic (PA) capture all computably enumerable sets, as claimed by Alonzo Church's thesis from the previous chapter. This allows us to capture “computable analysis” in the language of PA, since computable sets and functions are definable in terms of computable enumerability. To justify the claim that Σ‎0 1 = “computably enumerable,” this chapter makes a thorough analysis of the concept of computation. It takes a precise, but intuitively natural, concept of computation and translates it into the language of PA. The chapter demonstrates that the translation is indeed Σ‎0 1, but with a slightly different (though equivalent) definition of Σ‎0 1.

Limit linear series for curves not of compact type

We introduce a notion of limit linear series for nodal curves which are not of compact type. We give a construction of a moduli space of limit linear series, which works also in smoothing families, and we prove a corresponding specialization result. For a more restricted class of curves which simultaneously generalizes two-component curves and curves of compact type, we give an equivalent definition of limit linear series, which is visibly a generalization of the Eisenbud–Harris definition. Finally, for the same class of curves, we prove a smoothing theorem which constitutes an improvement over known results even in the compact-type case.