### Article

## Statistical testing of segment homogeneity in classification of piecewise-regular objects

The paper is focused on the problem of multi-class classification of composite (piecewise-regular) objects (e.g., speech signals, complex images, etc.). We propose a mathematical model of composite object representation as a sequence of independent segments. Each segment is represented as a random sample of independent identically distributed feature vectors. Based on this model and statistical approach we reduce the task to a problem of composite hypothesis testing of segment homogeneity. Several nearest-neighbor criteria are implemented, for some of them the well-known special cases (e.g., the Kullback-Leibler minimum information discrimination principle, the probabilistic neural network) are highlighted. It is experimentally shown that the proposed approach improves the accuracy when compared with contemporary classifiers.

Low-cost gaze tracking systems are in great demand due to their wide range of application. Commonly, extra devices are needed (for instance, head mounted cameras); however, in this investigation gaze tracking is performed in real-time based on the video stream from an infrared video camera. A comparative analysis of the existing analogues was executed and the main features of gaze tracking systems were highlighted and prioritized. These features are price, tracking accuracy, angle error, flexibility, and usability.

A methodology was developed which allows to calculate a gaze direction vector according to the relative position of eye center and corneal reflection from an infrared diode. The centers of an eye and reflection are estimated using the vector field of image gradients and additional weighting. CUDA technology is used to accelerate the developed algorithms.

The main advantage of the developed algorithm is the ability to detect and continuously track pupils’ centers, regardless of the head position, which significantly extends the scope of the gaze tracking system under consideration.

Problems of identification of plane unclosed curves are considered. Methods are proposed that allow one to classify graphic objects invariantly to affine transformations. An answer is given to the question on the types and the quantity of features that are needed to construct a mathematical description of curves for the recognition of an unclosed contour of an object. Metrics are introduced on the basis of which one can identify unclosed curves. The quality of identification on the basis of the metrics introduced is analyzed.

The parallel computing algorithms are explored to improve the efficiency of image recognition with large database. The novel parallel version of the directed enumeration method (DEM) is proposed. The experimental study results in face recognition problem with FERET and Essex datasets are presented. We compare the performance of our parallel DEM with the original DEM and parallel implementations of the nearest neighbor rule and conventional Best Bin First (BBF) k-d tree. It is shown that the proposed method is characterized by increased computing efficiency (2-10 times in comparison with exhaustive search and the BBF) and lower error rate than the original DEM.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.

It is well-known that the class of sets that can be computed by polynomial size circuits is equal to the class of sets that are polynomial time reducible to a sparse set. It is widely believed, but unfortunately up to now unproven, that there are sets in EXPNP, or even in EXP that are not computable by polynomial size circuits and hence are not reducible to a sparse set. In this paper we study this question in a more restricted setting: what is the computational complexity of sparse sets that are *selfreducible*? It follows from earlier work of Lozano and Torán (in: Mathematical systems theory, 1991) that EXPNP does not have sparse selfreducible hard sets. We define a natural version of selfreduction, tree-selfreducibility, and show that NEXP does not have sparse tree-selfreducible hard sets. We also construct an oracle relative to which all of EXP is reducible to a sparse tree-selfreducible set. These lower bounds are corollaries of more general results about the computational complexity of sparse sets that are selfreducible, and can be interpreted as super-polynomial circuit lower bounds for NEXP.