卷 24, 编号 3 (2024)
Mathematics
On semigroups of relations with the operation of the rectangular product
摘要
A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. The theory of algebras of relations is an essential part of modern algebraic logic and has numerous applications in semigroup theory. The following problems naturally arise when classes of algebras of relation are considered: find a system of axioms for these classes, and find a basis of of identities (quasi-identities) for the varieties (quasi-varieties) generated by these classes. In the paper, these problems are solved for the class of semigroups of relation with the binary associative operation of the rectangular product, the result of which is the Cartesian product of the first projection of the first relation on the second projection of the second one.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):320-329
320-329
Constructions of some secret sharing schemes based on linear codes
摘要
There are perfect and ideal threshold secret sharing schemes, for example, Shamir’s secret sharing scheme. For the case of general secret sharing schemes with an arbitrary access structure, it is possible to construct a perfect scheme for any access structure (for example, the Ito – Saito – Nishizeki scheme, the Benaloh – Leichter scheme), but in general, such a scheme will not be an ideal secret sharing scheme. In the paper, for some classes of access structures, the construction of perfect and ideal secret sharing schemes based on linear codes is given. We also give a construction of perfect verifiable secret sharing schemes for any access structure for which there is a line code that implements this structure.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):330-341
330-341
A new proof of the Krzyz conjecture for n = 3
摘要
The purpose of this paper is to solve the problem of the sharp estimation of the modulus of the third Taylor coefficient on the class of holomorphic bounded nonvanishing in the unit disk functions (hereafter referred to as the class of bounded nonvanishing functions). The problem of sharp estimation of the moduli of all Taylor coefficients depending on the coefficient number on this class is usually called the Krzyz problem. Consider the class of normalized holomorphic bounded functions in the unit disk (hereafter referred to as the class of bounded functions). The coefficient problem on this class is posed as follows: find the necessary and sufficient conditions to impose on a sequence of complex numbers so that the power series with coefficients from this sequence is the Taylor series of some function from the class of bounded functions. In this paper, by means of the solution of the coefficient problem for the class of bounded functions, we solve the problem of obtaining the sharp estimates of moduli of the first three Taylor coefficients on the class of bounded functions. It is pointed out that it is possible to visualize the first three coefficient bodies of the subclass of the class of bounded functions, consisting of functions with real coefficients. Next, we solve the problem of obtaining the sharp upper estimation of the modulus of the third Taylor coefficient on the class of bounded nonvanishing functions, by reducing it to the functional over the class of bounded functions. On the basis of the above-mentioned estimates on the class of bounded functions, it was possible to obtain a functional that majorizes the original one. The problem is then reduced to the problem of finding the conditional maximum of the function of three real variables with inequality-type constraints, which made it possible to apply standard methods of differential calculus to obtain this main result.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):342-350
342-350
Divergent series and generalized mixed problem for wave equation
摘要
Allowing the inversion of the operations of summation and integration for trigonometric Fourier series we present the solution by Fourier method of the generalized mixed problem for the homogeneous wave equation with zero initial velocity and fixed ends boundary conditions. The solution has the form of a series converging at an exponential rate. This series converges the classical solution if the latter equists. The results of the article reinforce the previously obtained results.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):351-358
351-358
Mechanics
The “Planning – Modelling – Prediction” methodology for preoperative planning in trauma orthopaedics
摘要
Preoperative planning of surgical treatment is an important stage of preparation for surgical treatment in traumatology and orthopaedics, which makes it possible to emphasise the peculiarities of the clinical case, prevent possible problems during surgery and reduce the risks of postoperative complications. The leading method of diagnostics for further planning of surgical treatment nowadays is radiological studies, primarily radiography and computed tomography. The results of radiological studies allow a sufficiently qualitative assessment of the zone of interest, planning of the required degree of correction and placement of fixing metal structures and endoprostheses. At the same time, when planning, the doctor relies mostly on the knowledge of the norms of anatomical relations and structures. And in the case of a multitude of possible treatment options, the doctor relies on his or her own medical experience to make a choice. This article presents a developed generalising methodology of preoperative planning in traumatology-orthopaedics, which includes biomechanical analysis and methods of accumulation and processing of quantitative data of clinical cases along with the usual methods of preoperative planning for doctors. The methodology brings together into a single system the criteria for evaluating the success of treatment by applying three classes of criteria: geometric (anatomical), biomechanical and clinical. The methodology allows the physician to perform biomechanical modelling of the proposed treatment options and quantitatively evaluate them on the basis of comparison of stress-strain states arising in the «bone-implant» system as a result of each of the planned options. The methodology allows to determine successful treatment options and to predict changes in the patient's quality of life after treatment. The presented methodology includes a mechanism for accumulation of quantitative data on clinical cases and quality control of the used biomechanical models.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):359-380
359-380
On identification of two-dimensional density of an elastic inhomogeneous cylinder
摘要
Within the framework of the linear theory of elasticity, using the model of an isotropic body, the problem of steady-state vibrations of an inhomogeneous hollow cylinder is formulated. The vibrations of the cylinder are caused by a load applied to the side surface, and the conditions of sliding embedding are implemented at the ends. The variable material properties are described by the Lamé parameters and density, which change along the radial and longitudinal coordinates. The direct problem solution of the cylinder vibrations is constructed using the finite element method implemented in the FlexPDE package, its main advantages are noted. To study the influence of variable properties on the values of resonant frequencies and components of the displacement field, the laws of these properties are considered in the general form used in modern works for modeling functionally graded materials. On the basis of the performed numerical calculations, the degree of influence of the amplitude values of each of the Lamé parameters and density on the first resonant frequency and the displacement field are studied. Graphs are also presented that demonstrate the influence of the type of the law of density change on the values of the displacement field components. A new inverse coefficient problem is formulated to determine the density distribution function in the cylinder wall from the displacement field data measured at a finite set of points inside the considered area for a fixed frequency. The main difficulties in the implementation of the reconstruction procedure in practice are noted. To increase the accuracy of calculating the first and second derivatives of the two-dimensional functions calculated in the finite element package, which are used in solving the inverse problem, an approach based on the locally weighted regression algorithm is proposed. The results of computational experiments on solving the inverse problem are presented, which demonstrate the possibility of using the proposed method to restore various types of two-dimensional laws of density. Practical recommendations are given for the implementation of the most accurate reconstruction procedure.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):381-393
381-393
Hyperbolic boundary layer in the vicinity of the shear wave front in shells of revolution
摘要
Hyperbolic boundary layer equations in thin shells of revolution are constructed in small vicinities of the shear wave fronts (taking into account its geometry) at edge shock loading of the normal type. Special coordinate system is used for defining the small boundary layer region. In this system, the coordinate lines defined by the normal to the middle surface are replaced by lines forming the surface of the shear wave front. The asymptotic model of the geometry of such a wave front suggests that these lines are formed by rotated normal to the middle surface. Asymptotically main components of considered stress strain state are defined: the normal displacement and the shear stress. The governing equation of this boundary layer is the hyperbolic equation of the second order with the variable coefficients for the normal displacement.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):394-401
394-401
Control of the rolling of a dynamically symmetrical sphere on an inclined rotating plane
摘要
The work investigates the rolling dynamics of a dynamically symmetrical heavy sphere (or a heavy spherical shell) along an inclined rough plane (platform) rotating with constant or periodic angular velocity around an axis, which is perpendicular to the plane and passing through some fixed point of this plane. Nonholonomic and holonomic constraints are imposed at the point of contact of the sphere with the reference plane. The equations of motion of the sphere are constructed. In the case of the constant angular velocity of the plane at any slope and in the case of the periodic angular velocity of the plane located horizontally the boundedness of the velocities of the geometric center of the sphere is proved. Moreover, in the case of the constant angular velocity of the plane, solutions are found analytically. Based on numerical integration, it is shown that for the periodic angular velocity of the plane and for the nonzero slope the square of the velocity vector of the geometric center of the sphere increases indefinitely. Two controls for the slope of the plane proportional to the projections of the velocity vector of the sphere on the coordinate axes lying in the reference plane are introduced. In the case of the constant angular velocity of the plane, a qualitative analysis of the equations of motion has been carried out, the control parameters at which the square of the velocity vector of the geometric center of the sphere will be bounded and at which it will be unbounded have been analytically found. The results of this control are presented for the case of periodic angular velocity of the plane. It is shown that by controlling the slope of the plane, it is possible to achieve the boundedness of the square of the velocity vector of the geometric center of the sphere. The obtained results are illustrated, the trajectories of the contact point and graphs of the desired mechanical parameters are constructed.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):402-414
402-414
On renormalization of the approximate solution of the orbital coordinate system equations of orientation
摘要
In the quaternion formulation, the problem of mathematical modeling of the spacecraft motion in an elliptical orbit is considered. The control is a modulo-limited acceleration vector from the jet thrust, directed orthogonally to the plane of the spacecraft orbit. The motion of the spacecraft center of mass is described using the quaternion differential equation of the orientation of the orbital coordinate system. An approximate analytical solution of the quaternion differential equation of the orientation of the orbital coordinate system is constructed in the form of a uniformly suitable asymptotic expansion by degrees of eccentricity of the spacecraft orbit (small parameter). To eliminate the secular terms in this expansion, the renormalization method was applied. Taking into account the known solution of the equation of orientation of the orbital coordinate system for the case when the spacecraft orbit is circular, allowed to simplify the form of the above expansion. The nonlinear oscillation frequencies of each component of the desired quaternion were found. Analytical transformations were performed using the SymPy symbolic algebra package. To carry out numerical simulation of the spacecraft motion, a program was written in Python. Calculations based on analytical formulas obtained in the paper (in the absence of secular terms) and previously obtained results in the presence of secular terms are compared. An example of modeling the controlled motion of a spacecraft is given for the case when the initial orientation of the orbital coordinate system corresponds to the orientation of the orbit of one of the satellites of the GLONASS orbital grouping. Graphs were built to show error in the module (and components) of the quaternion describing the orientation of the orbital coordinate system. It is shown that the elimination of secular terms using the renormalization method made it possible to reduce the error in determining this module with an increase in the number of spacecraft revolutions around the Earth. The analysis of the obtained approximate analytical solution is carried out. The features and regularities of the spacecraft motion in an elliptical orbit are established.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):415-422
415-422
Computer Sciences
Methods for obtaining information for biomedical monitoring of the level of oxygenation and blood pressure using built-in sensors of smartphone technology
摘要
The article is devoted to the actual problem of non-invasive self-monitoring of oxygenation and blood pressure indicators by patients. The article provides an overview of the available promising approaches for monitoring the biomarkers under consideration. Also, it demonstrates the main problems associated with applying the approaches under consideration and those caused by the test sample itself.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):423-431
423-431
Extraction of features in images based on integral transformations in solving problems of classification of fragments of photographs
摘要
The article proposes a method for calculating features in an image to form a training data set for solving various problems of video image classification. This method involves the use of well-known integral transformations — the Radon transform and the Steklov function. The proposed method is compared with convolutional neural networks both in terms of the percentage of correct prediction and in terms of its execution time. As a test task, the problem of finding a fragment of a photograph containing an image of a car is considered.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):432-441
432-441
The quality improvement method for detecting attacks on web applications using pre-trained natural language models
摘要
This paper explores the use of deep learning techniques to improve the performance of web application firewalls (WAFs), describes a specific method for improving the performance of web application firewalls, and presents the results of its testing on publicly available CSIC 2010 data. Most web application firewalls work on the basis of rules that have been compiled by experts. When running, firewalls inspect HTTP requests exchanged between client and server to detect attacks and block potential threats. Manual drafting of rules requires experts' time, and distributed ready-made rule sets do not take into account the specifics of particular user applications, therefore they allow many false positives and miss many network attacks. In recent years, the use of pretrained language models has led to significant improvements in a diverse set of natural language processing tasks as they are able to perform knowledge transfer. The article describes the adaptation of these approaches to the field of information security, i.e. the use of a pretrained language model as a feature extractor to match an HTTP request with a feature vector. These vectors are then used to train the classifier. We offer a solution that consists of two stages. In the first step, we create a deep pre-trained language model based on normal HTTP requests to the web application. In the second step, we use this model as a feature extractor and train a one-class classifier. Both steps are performed for each application. The experimental results show that the proposed approach significantly outperforms the classical Mod-Security approaches based on rules configured using OWASP CRS and does not require the involvement of a security expert to define trigger rules.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):442-451
442-451
Detection of sources of network attacks based on the data sampling
摘要
This article defines the rules for finding the threshold values for the main network variables used to detect network intrusions under conditions of limited data sampling. The sFlow technology operates with a limited sample of packets, and one packet out of 50 can be analyzed, but this value can reach 5000. The main conclusion is that the product of the threshold value and sample resolution remains a constant value. The article defines the size of the maximum resolution, at which an attack with a given threshold can be detected. Based on the experimental data, this hypothesis was tested; considering the experimental error, it was verified.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):452-462
452-462
Appendices
In memory of Yakov G. Sapunkov
摘要
The article is devoted to the outstanding mechanical scientist and pedagogue Yakov Grigoryevich Sapunkov (04.11.1938–23.10.2020), whose important scientific results range from hypersonic aerodynamics, fluid mechanics and computational methods to the theory of optimal control, mechanics of space flight and the theory of strapdown inertial navigation systems (SINS). An outline of his scientific activity and list of his most significant scientific works are given.
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics. 2024;24(3):463-471
463-471