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Confidence intervals are constructed for an unknown correlation coefficient and mathematical expectation using 3s 1 and 2s 1 rules. The intervals make it possible to solve the problem of statistical analysis of the dependence between different random variables. The results obtained are used to analyze the correlation between the content of homocysteine and various clinical indices.

The properties of a mathematical model of a nuclear pore complex are studied using a random cap process on a two-dimensional Euclidean sphere of unit radius. The results obtained make it possible to calculate stereometric characteristics of a nuclear pore complexes from electron-diffraction patterns.

Spline regression is used to analyze the influence of radiation on the cancer probability in a group of participants of Chernobyl accident mitigation, depending on the exposure dose. A new method is proposed to approximate modified polygons by linear splines with two nodes. An algorithm for the identification of a transition point is outlined.

A new method of stratification analysis of general populations is proposed that is based on the approximation of inverse empirical distribution functions. With the help of the methods developed, it is shown that the number of modal classes in a heterogeneous population equals the number of segments of a linear regression spline. The proposed method is used for the investigation of populations of cancer cells that are sensitive or resistant to cisplatin.

We propose a new measure of proximity of samples based on confidence limits for the bulk of a population constructed using order statistics. For this measure of proximity, we compute approximate confidence limits corresponding to a given significance level in the cases where the null hypothesis on the equality of hypothetical distribution functions may or may not be true. We compare this measure of proximity with the Kolmogorov–Smirnov and Wilcoxon statistics for samples from various populations. On the basis of the proposed measure of proximity, we construct a statistical test for testing the hypothesis on the equality of hypothetical distribution functions.

The object of the investigation reported in this paper was to study, from the point of view of statistical and geometric theory of pattern recognition, the DNA optical density distribution peculiarities in the interphase nuclei of buccal epithelium present in the pathology of the thyroid and mammary glands. Two new indices to characterize this distribution (ratio of modal class volumes and relief index) are proposed. It is shown that in malignant neoplasms of the thyroid and mammary glands the changes in the nuclei of buccal epithelium are characterized by an increase in the optical density of DNA over a range from 0.15 to 0.30 in conventional units of measure, as compared with its values in benign pathological processes. The sensitivity of the proposed criterion for diseases of the thyroid gland is equal to 76.2% and the specificity is equal to 85.8%. For diseases of the mammary gland (excluding IDLC) we have discovered that the sensitivity of the method is equal to 94.29% and its specificity equal to 90.91%. In diseases of the mammary gland (including IDLC) we have discovered that the sensitivity of the method is equal to 71.42% and its specificity is equal to 90.91%.

The problem of pattern recognition with the help of spherical and elliptic discriminant functions is studied; in so doing the pattern of an object is assumed to be a vector of its characters from a finite-dimensional Euclidean space. Using a conformal mapping of a punctured sphere onto the plane as well as the inversion transformation, a criterion for the error-free recognition of two sets containing a finite number of points of training samples is obtained with the help of spherical discriminant functions. An algorithm for solving approximately a problem of construction of an ellipsoid of the minimal volume containing a given finite set of points is described. Bibliography: 5 titles.

A class of quadratic estimates is constructed for the second-order moment and variance of a random variable. These estimates are found on the basis of sample values obtained by simple sampling. The best quadratic estimates are found for the second-order moment and variance in the case of known mathematical expectation. The exactness of biased and unbiased estimates of variance is investigated in the case of unknown mathematical expectation.

To summarize, we can state that, in the present paper, the new metric ρ(1) has been constructed in a rather wide class of plane figures. This metric is a topologically majorant continuation of the metric of the space of continuously differentiable functions. Such metrics are called smooth. It is clear that in the smooth metric ρ(1) the proximity of figures from the classS implies the proximity of their perimeters. Note that there are many other examples of metrization of spaces of plane figures. In [3], a metric ρ3 is introduced that generalizes the Hausdorff metric and takes into account the behavior of the boundaries of figures, but it is not smooth. One can easily convince oneself by the following example.

We continue the investigation of the problem of construction of a minimum-area ellipse for a given convex polygon (this problem is solved for a rectangle and a trapezoid). For an arbitrary polygon, we prove that, in the case where the boundary of the minimum-area ellipse has exactly four or five common points with the polygon, this ellipse is the minimum-area ellipse for the quadrangles and pentagons formed by these common points.

We introduce a new concept of generalized solution of operator equations with closed linear operator in a Banach space as an element of the completion of the space in certain locally convex topology. We prove a theorem on the existence and uniqueness of a generalized solution and give examples of finding the generalized solution for infinite systems of the linear algebraic equations.

The independent trials scheme is applied to develop two tests for the hypothesis of identical distributions of two populations. Consistency of the tests is proved under certain conditions on the distribution functions.

We study the structure of the critical function of an optimal statistical criterion for testing an arbitrary finite set of simple alternative hypotheses.

The purpose of Chap. 2 is to give a natural definition of generalized solution of non-solvable linear operator equation in Banach spaces. In Sects. 2.1–2.4 two concepts of generalized solutions are proposed: strong generalized solution and weak generalized solution based on the duality. In Sect. 2.5 the theorem on existence of a weak generalized solution is proved. The relation between these two approaches is studied in Sect. 2.6 in the case of equations with linear injective operator.

Chapter 4 is the largest one. It is in a sense “barycenter” of the book. Here we consider typical examples of applications of generalized solutions in various branches of pure and applied analysis. In Sect. 4.1, 4.3, and 4.4 we prove theorems on generalized solvability of equations with Hilbert–Schmidt operators and Volterra equations of the first kind and describe their applications in the random processes estimation theory. In Sect. 4.2 the generalized solvability of linear operator equations in classic spaces of sequences is studied. In Sect. 4.5, 4.6, and 4.7 the theorems on generalized solvability of boundary value problems for parabolic and generalized wave equations are proved. Section 4.7 is devoted to discussion of theorems of Lax-Milgram kind in Banach and locally convex spaces.

In Chap. 6 we propose the general approach to development of the theory of generalized solvability of linear operator equations. In Sect. 6.1 we prove the theorems on existence and uniqueness of generalized solutions of linear operator equations in locally convex linear spaces. In Sect. 6.2 we show that the proposed construction generalizes the definitions of generalized solutions given in Chaps. 2 and 3. We study the a priory inequalities method for proving theorems on existence of generalized solutions in abstract cases.

The purpose of Chap. 7 is to give a natural definition of generalized solution of non-solvable non-linear equation. In Sect. 7.1 we give the definition of generalized solution of equation with operator which acts in metric spaces. In Sect. 7.2 we give the definition of new concept of near-solution of non-linear operator equation. In Sect. 7.3 and 7.4 the existence, uniqueness, and correctness of generalized solution are proved. In Sect. 7.5 and 7.6 we investigate issues of embedding of metric spaces and extension of operators which are important for the theory of generalized solutions. Sections 7.7 and 7.8 contain examples of operators and describe numerical aspects of computation of generalized solutions. Section 7.9 is devoted to the development of the theory of generalized solutions of equations in uniform spaces and proximity spaces.

Chapter is devoted to the concept of generalized solution of extreme problems. In Sect. 8.1 we give necessary motivations and the definition of generalized extreme element of bounded continuous functional defined of a bounded closed subset of a Banach space. In Sect. 8.2 we prove the theorem on existence of the generalized extreme element of a linear continuous functional. In Sect. 8.3 we study an interest issue concerning the possibility of compact dense embedding of linear normed space in Banach one. The purpose of Sects. 8.4 and 8.5 is to develop the theory of generalized solvability of convex minimization problems in infinite-dimensional Banach spaces.

In Chap. 3 the operator equations with continuous linear injective operator which acts in linear normed spaces are considered in greater detail. The spaces of strong and weak generalized solutions defined in previous chapter do not allow any constructive description in general case. In Sects. 3.1 and 3.2 we develop a variant of the theory, using the duality techniques, that is more useful for concrete applications. In Sects. 3.3 and 3.4 we study the alternative constructive approach to general solvability, based on the duality techniques.

In Chap. 1 we give major definitions, concepts and auxiliary facts concerning the theory of generalized solutions, namely: linear vector space, dual pair, linear functional, continuous functional, conjugate space, algebraically conjugate space, second conjugate space, bilinear form, linear operator, injective operator, bijective operator, surjective operator, adjoint operator, algebraically adjoint operator, intermediate space, topological space, linear normed space, embedding, dense embedding, natural embedding, strong topology, weak topology, bounded set, subdifferential, Banach space, Sobolev space, neighborhood.

Chapter 5 is devoted to numerical aspects of developed theory of generalized solvability. In Sect. 5.2–5.6 we discuss conceptual aspects of reliability and effectiveness of precise and iterative method for solving system of linear algebraic equations obtained as a result of finite-dimensional approximation of initial operator equations. In Sect. 5.7 we prove the criteria of classic solvability of linear operator equations in Banach and Hilbert spaces associated with the Neumann series.

Continuity and monotonicity of the inverse operator is investigated for the case when the original operator is a bijective mapping acting in topological manifolds or in Banach spaces. For finite-dimensional topological manifolds, the inverse operator is always continuous; for Banach spaces, continuity of the inverse operator is guaranteed only if the space is finite-dimensional. For an arbitrary differentiable bijective mapping acting in the Euclidean space, the Jacobian preserves its sign.

A number of new statistical tests for homogeneity of samples are proposed. The properties of the tests are examined. The tests use information provided by learning samples.

We examine the structure of a finite collection of nonintersecting homothetic centrally symmetric convex sets in Rn that form a system of linearly separable sets.

The frequency of occurrences of events in a modified Bernoulli model under the assumption Fx(u) ≡ Fy(u) is studied. A homogeneity test for two samples based on properties of this frequency and its properties are investigated.

We propose consistent tests of statistical hypotheses based on training samples. Unbiased consistent estimators of the probability of no decision are obtained.

A study of the proposed mathematical model of the mechanism of operative memory indicates that formation of circulating excitations is accounted for by short-term trace effects of synaptic transmission, well known in neurophysiology (posttetanic potentiation, temporary facilitation, and path formation). These results give added support to the recirculation theory, which maintains that fixation and storage of incoming information occurs via a stream of nervous impulses circulating in closed chains. The circulation of a stream, in turn, leads to stable morphological or chemical alterations in the structure of the synaptic apparatus, which constitutes the basis of long-term memory.