PARAMETRIC METHOD OF SOLVING PROBLEMS OF MATHEMATICAL SAFE ON GRAPHS

We consider one method of solving the problem of mathematical safe on certain graphs called parametric. Its gist consist in denoting some variables, corresponding to graph vertices, by certain parameters. Other unknown variables are expressed through these parameters. Then unknown variables chosen in special way are compared and the mentioned parameters are found by solving additional system of equations for these parameters. Dimension of this system is equal to the number of parameters. Solution to the problem i.e. all unknown variables of the original system, are found by solving additional system of equations. In the paper this method is described on specially chosen examples. The method is demonstrated by solving the mathematical safe problem on the graphs of «chain», «ladder» and «window» types that showed its efficiency. Besides special attention is paid to special cases when solution does not exist. This occurs in the cases when the weighed sum of system equations is not divisable without remainder to its modulo. In such cases, to find solution the initial state of the vector b is corrected in such a way that the weighted sum of equations satisfies the above mentioned condition. Then solution of the problem is performed according to the general method scheme.

LEXICOGRAPHIC PROBLEMS OF CONVEX OPTIMIZATION: SOLVABILITY AND OPTIMALITY CONDITIONS, CUTTING PLANE METHOD

The lexicographic approach for solving multicriteria problems consists in the strict ordering of criteria concerning relative importance and allows to obtain optimization of more important criterion due to any losses of all another, to the criteria of less importance. Hence, a lot of problems including the ones of com­plex system optimization, of stochastic programming under risk, of dynamic character, etc. may be presented in the form of lexicographic problems of opti­mization. We have revealed conditions of existence and optimality of solutions of multicriteria problems of lexicographic optimization with an unbounded convex set of feasible solutions on the basis of applying properties of a recession cone of a convex feasible set, the cone which puts in order lexicographically a feasible set with respect to optimization criteria and local tent built at the boundary points of the feasible set. The properties of lexicographic optimal solutions are described. Received conditions and properties may be successfully used while developing algorithms for finding optimal solutions of mentioned problems of lexicographic optimization. A method of finding lexicographic of optimal solutions of convex lexicographic problems is built and grounded on the basis of ideas of method of linearization and Kelley cutting-plane method.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

DATABASE PROTECTION BASED ON WEB APPLICATION FIREWALL

Web applications are increasingly being used in activities such as reading news, paying bills, and shopping online. As these services grow, you can see an increase in the number and extent of attacks on them, such as: theft of personal information, bank data and other cases of cybercrime. All of the above is a consequence of the openness of information in the database. Web application security is highly dependent on database security. Client request data is usually retrieved by a set of requests that request the application user. If the data entered by the user is not scanned very carefully, you can collect a whole host of types of attacks that use web applications to create security threats to the database. Unfortunately, due to time constraints, web application programmers usually focus on the functionality of web applications, but only few worry about security. This article provides methods for detecting anomalies using a database firewall. The methods of penetration and types of hacks are investigated. A database firewall is proposed that can block known and unknown attacks on Web applications. This software can work in various ways depending on the configuration. There are almost no false positives, and the overhead of performance is relatively small. The developed database firewall is designed to protect against attacks on web application databases. It works as a proxy, which means that requests for SQL expressions received from the client will first be sent to the developed firewall, rather than to the database server itself. The firewall analyzes the request: requests that are considered strange are blocked by the firewall and an empty result is returned to the client.

ENHANCING ADEQUACY OF GRADING STUDY PROJECTS ON THE BASE OF PARAMETRIC RELAXATION OF PAIRWISE COMPARISONS

A problem of automated assessing of students’ study projects is regarded. A heuristic algorithm based on fuzzy estimating of projects and on pairwise comparisons among them is proposed. For improving adequacy and naturalness of grades, an approach based on introducing a parameter named relaxation parameter was suggested in the paper. This enables to reduce the spread between maximum and minimum values of projects in comparison with the one in the standard scale suggested by T. Saati. Reasonable values of this parameter were selected experimentally. For estimating the best alternative, a center of mass of a fuzzy max-min composition should be calculated. An estimation algorithm for a case of non-transitive preferences based on getting strongly connected components and on pairwise comparisons between them is also suggested. In this case, relaxation parameters should be chosen separately for each subtask. So the combined technique of evaluating alternatives proposed in the paper depends of the following parameters: relaxation parameters for pairwise comparisons matrices within each strongly connected components; relaxation parameter for pairwise comparisons matrices among strongly connected components; membership function for describing the best alternative.

STOCHASTIC DIFFERENTIAL GAMES IN DISTRIBUTED SYSTEMS WITH DELAY

We study a differential game of approach in a delay stochastic system. The evolution of the system is described by Ito`s linear stochastic differential equation in Hilbert space. The considered Hilbert spaces are assumed to be real and separable. The Wiener process takes values in a Hilbert space and has a nuclear symmetric positive covariance operator. The pursuer and evader controls are non-anticipating random processes, taking on values, generally, in different Hilbert spaces. The operator multiplying the system state is the generator of an analytic semigroup. Solutions of the equation are represented with the help of a formula of variation of constants by the initial data and the control block. The delay effect is taken into account by summing shift type operators. To study the differential game, the method of resolving functions is extended to case of delay stochastic systems in Hilbert spaces. The technique of set-valued mappings and their selectors is used. We consider the application of obtained results in abstract Hilbert spaces to systems described by stochastic partial differential equations with time delay. By taking into account a random external influence and time delay, we study the heat propagation process with controlled distributed heat source and leak.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

INVESTIGATION OF THE PROBLEM OF OPTIMIZATION OF THE HIERARCHICAL STRUCTURE OF A SPARSE AND DENSE COMMUNICATION NETWORK

The article is devoted to the study of the optimization problem for the hierarchical structure of a multicommodity communication network with discrete flows and parameters when its density changes (the ratio of the number of network arcs to the maximum possible number of arcs for a given number of nodes in the network). The network has three levels of hierarchy — a backbone, a zonal and a internal and four types of nodes — backbone nodes of the first, second and third types, forming the backbone and zonal levels of the network, and nodes of the fourth type, which subordinate to each backbone node and form the internal levels of the network. The types of nodes different from one another in functionality. The main task of the study is to establish how the structure of the backbone network changes (the number and location of backbone nodes of the first, second and third types), the scheme of processing and distribution of flows and technical and economic indicators of the network's functioning for various degrees of its density. The principles of organizing the sorting and distribution of flows in a three-level network and its mathematical model are given. A mathematical model of the problem of optimizing the structure of the backbone network and the schemes of sorting and distribution of flows is formulated. The algorithms for solving the problem are based on the discrete analogue of the local descent method proposed earlier by the authors, when the neighborhoods of the metric space of possible solutions are chosen from heuristic considerations, taking into account the peculiarities of the problem being solved. Computer modeling of the problem on a homogeneous network containing 100 nodes with a change in the degree of nodes from 2 to 99 is carried out. The modeling was carried out on the example of a road transport network for the transportation of cargo using a computer program, which is part of the instrumental software of the Information and Analytical Decision Support System (IA DSS), which is being developed at the Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine. An experimental study of the solution to the problem showed that the best technical, economic and operational indicators of its functioning are achieved with a degree nodes from 9 to 14, when high network connectivity is ensured, and is significantly reduced the average and maximum time of delivery of cargos to recipients. The proposed computer technology for solving problem with changing network density allows you to interactively modeling various options of a network , changing the topology, hierarchical structure, flows, parameters and constraints of the model and from the family of the resived results choose the best option, taking into account the selected a goal function and the accepted constraints; calculate preliminary technical and economic indicators of the network's functioning, estimate the cost of additional resources and plan the amount of investment required for the modernization and construction of its structural elements, which ultimately makes it possible to increase the efficiency functioning of the network by optimizing use of its resources and reducing the operating costs for the processing and transportation of flows.

SYSTEM OF AUTOMATIC SEGMENTATION OF PAUSES IN PHONOGRAMS ON THE BASIS OF NEURON NETWORKS OF THE DEEP LEARNING

The use of neuron networks of the deep learning for the construction of tool for realization of examinations of materials and apparatus of the digital audio recording allows to solve the «frigging» problem of such examination — problem of exposure of tracks of editing in digital phonograms. These networks provide high probability of exposure of such tracks in the pauses of speech information writtenin on a phonogram. Before man-hunting of tracks of editing in the investigated phonogram it is necessary to distinguish pauses (to perform its segmentation), and tool built on the basis of neuron networks of the deep learning, requires its work to be done in automatic mode. The basic requirement of automatic segmentation is high efficiency of selection of pauses in the conditions of permanent change of level of noises in phonograms. It is determined by probability of errors of І and ІІ kinds. It is offered on the basis of neuron networks of the deep learning to create CAS of segmentation of phonograms, possessing high efficiency of selection of pauses in speech information. Thus the system must be independent of level of noises in every concrete pause, and also language, context and announcer, whose speech is fixed in a phonogram. It is suggested to examine pauses as one of the types of voice information, which characteristics differ from characteristics of speech information fixed in a phonogram. For educating of such network it was required to create the primary base of these sounds and pauses. On its basis three arrays of the data, intended for learning, testing and determination of the crooked errors of І and ІІ kinds, are created. After learning and testing the system passed verification on the real phonograms. As a result taking into account some features of speech on the neuron networks of deep learning there has been built the system providing effective segmentation of pauses in phonograms in the automatics mode. The obtained results suit examination that is conformed by given curves over of errors of І and ІІ kinds.

ON APPROXIMATE OPTIMAL CONTROL FOR THE REACTION-DIFFUSION PROCESS IN MICROINHOMOGENEOUS MEDIUM

The problem of constructing an approximate optimal control for controlled processes of chemical kinetics in microinhomogeneous medium is considered. Such processes are described by semilinear parabolic equations of the reaction-diffusion type with coefficients of the form . The preference of an approximate control as the optimal control in the problem with averaged coefficients is justified. An example of the construction of such a control is discussed and its efficiency is demonstrated.