Separation of Potential Fields Generated by Different Sources Based on Modified S-Approximations

Subject. This article focuses on the problems of solving systems of homogeneous linear algebraic equations in the pricing problems of an alternative financial model. Objectives. The article aims to develop a non-trivial approach to solving the problem of pricing from the scratch in the construction of a national alternative financial model, which boils down to solving large homogeneous systems of linear algebraic equations, complicated by the fuzzy setting of initial parameters (initial data) within the framework of creating a State model of a socially oriented economy, guaranteed to ensure economic, military and other types of integrated security of the Russian Federation. Methods. For the study, we used tensorial notations involving the use of elements of linear algebra, including operations with matrices and vectors, the apparatus of the fuzzy set theory (interval arithmetic), as well as methods and principles of scientific research and complex logical analysis and modeling of pricing and technological processes. Results. The article shows that an effective non-zero (different from zero, i.e. classical, trivial) solution of systems of homogeneous algebraic equations in pricing problems of an alternative financial model from the scratch is possible if the initial parameters are specified by fuzzy sets, which significantly simplifies the procedure of the computational process and the volume of arithmetic operations. Conclusions. The results obtained can be used in structures involved in the creation and testing of financial and economic models both at the federal and the Russian Federation constituent entity levels, as well as in governmental and non-governmental organizations dealing with the issues of pricing, planning and logistics support for business entities not only in the Russian Federation, but also in other countries experiencing problems in the field of financing.

Computation of anomalous gravity and magnetic fields generated by various models is a necessary step if techniques of curve‐matching are to be used for quantitative interpretation of potential field data. Recently developed methods show that anomalous magnetic and gravity fields are completely determined by the divergence of magnetization and the first vertical derivative of density, respectively. Using these methods, efficient algorithms can be developed for computing potential field anomalies caused by arbitrary distribution of magnetization and density in an irregularly shaped body. Automatic iterative procedures are normally employed in the space domain for estimating parameters of the selected model that yield a best‐fit anomaly curve for a set of discrete observed data. Examples of application of the Newton‐Raphson method, Marquardt method, and the Powell algorithm to the interpretation of magnetic data are presented and discussed. Amplitude and energy spectra of the anomalous fields are also used conveniently in many cases for systematic estimation of average and individual depths, horizontal and vertical extents, and density or magnetization contrasts of causative bodies in a bounded region. Some of these frequency‐domain approaches are found to have many useful applications.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

The basic theoretical foundations of the spectral analysis of gravity and magnetic data based on the approximation approach have been considered and corresponding expressions implementing various statements of the F-approximation method have been derived [Strakhov and Kerimov, 1999; 2000; 2001]. The author considered the algorithms and computer implementation of the F-approximation of anomalous gravity and magnetic fields presented in the results of testing with the use of a number of model examples and data on the accuracy of the model field approximation earlier on in [Kerimov, 2003]. In this paper, I consider methodological features of constructing the F-approximation from the detailed gravity and magnetic surveys, anomalous gravity and magnetic fields reconstructed with the use of the F -approximation, and examples of the field transforms obtained with the use of this approach.

One of the most important tasks of modern physical geodesy is to determine a high-frequency part of the Earth’s gravity field (EGF), which inevitably depends on the individual characteristics of a particular computational area. The author investigates the practical aspects of a relatively new theory for modeling the said part of the EGF in a local area, using a series of special spherical functions ensuring local orthogonality. With the global gravitational model EGM2008 as input information, we aim to construct a local analytical one for the high-frequency component of the EGF in the Republic of Burundi territory. To overcome the main technical difficulty, the problem of solving rank-deficient systems of linear algebraic equations (SLAE), special regularization techniques are employed. The developed algorithm generalizes the classical global spherical harmonic analysis (GSHA) and is treated as one of the preferred alternative methods of regional modeling. It enables representing fields with high spatial resolution using a much smaller number of parameters rather than the classical GSHA. Compared with other possible methods of regional analysis, the advantage of the LSGA is ultimate provision of a solution, not only analytical, but also meeting the Laplace differential equation, which, as we know, is fundamentally important in many problems of geophysics in general and physical geodesy in particular

The article deals with high-performance information technology (HPC) for the problems of stress-strain analysis at all stages of the life cycle of buildings and structures: construction, operation and reconstruction. The results of numerical simulation of high buildings using software as a processor component based on a new hybrid algorithm for solving systems of linear algebraic equations [1] with a symmetric positive-definite matrix that combines computation on multi-core processors and graphs. It has been found that to accelerate the calculations, hybrid systems that combine multi-core CPUs with accelerator coprocessors, including GPUs, are promising [5]. To test the effectiveness of the proposed parallel algorithm for solving systems of linear algebraic equations [1], numerical experiments were carried out at the most dangerous loads of a 27-story building. Results of numerical researches with use for preprocessor (input of initial data) and postprocessor (output of results of calculations) of processing of the LIRA-SAPR software complex are presented [2, 4, 6]. The results of numerical studies of the behavior of structures of high buildings have shown a multiple reduction in the time of solving systems of linear algebraic equations with symmetric matrices on multiprocessor (multi-core) computers with graphical accelerators using the proposed hybrid algorithms [1]. High-performance technologies based on parallel calculations give more effect than more complex processes: modeling of life cycle of high buildings, bridges, especially complex structures of NPPs, etc. for static and dynamic loads, including emergencies in normal and difficult geological conditions, which make up 70% of Ukraine's territories.

Introduction: Natural earth’s magnetic field (EMF) is promoter of mother cell division into two daughter cells. Cancer is a cell that divides unlimited and destroys the body. Aim: Measurements presented in this paper show that malignant diseases are formed in anomalous magnetic fields (AMF) in patient’s beds. Removal of patients from AMF explains the etiological uncertainties in the medical literature. Methods: Measuring of AMF in patients’ beds was performed by proton magnetometer. Measuring points are on square net, 10 × 10 cm. The results are shown with contour lines on drawings. The health status of residence in the Earth’s natural magnetic field (EMF) was monitored. The etiologic data from the literature are explained. Results: Research lasted 25 years. AMF were measured in 1500 patients’ beds. There was a correlation between AMF and location of the diseased organ. AMF surfaces were shown in beds where the diseased organ is, on the contrary, there were natural values of EMF in the rest of the bed surface. Furthermore, the results found that three cases of breast cancer with metastases survived after surgical interventions and subsequent removal into natural EMF in contrast to 26 cases with the same diagnosis and surgical interventions who died after short period and staying in AMF. Another example was brain malignancy, which was decreased by living in natural EMF for one year. Conclusion: Based on the results of measurement of AMF, the medical condition of the patient when staying in a natural EMF, and explanation of uncertainties etiologies, the main conclusion was: AMF is the main cause of the malignant diseases.

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.

Radial Basis Functions (RBF) are popular for interpolating scattered data. In this context, the solution of the system of linear algebraic equations (SLAE) is the most time-consuming operation. Techniques fail with large point sets consisting of more than several thousands of points when direct methods and global support are used. In this paper we demostrate that the solution of the SLAE in the wavelet domain is suitable for the problem of image interpolation by means of Compactly-Supported Radial Basis Functions (CSRBF). The iterative solution of SLAE with highly irregular matrices cannot be accelerated by wavelet transformation and subsequent sparcification if the transformed matrix is still highly irregular. To solve the SLAE in the wavelet domain, the ordering of the samples defines the spacial relationship and the energy of the coefficients in the low frequency domain. Two sorting algorithms for the wavelet domain solution are tested and compared with the spacial solution of the SLAE. Examples of image interpolation by means of CSRBF demostrate the superiority of the solution in the wavelet domain using GMRES iterative method.

Conditionality in the problem of the minimum of a quadratic functional

According to statistical research, a violation of license agreements annually causes huge losses to software companies. On the one hand, illegal copies of the software product are created, on the other hand, some fragments of the programs are used by third parties unauthorized. Another important problem is the violation of the program integrity, for example, in terms of blocking functions of the license key checking. In this regard, the task of construction methods for protecting intellectual property in software applications is highly relevant. Previously known methods solve this problem by means of fragile digital watermarks. Below is presented a method for constructing a fragile digital watermark used in executable files. A model of a developers team creating software product protected by DWM is considered. The application of this method will allow to reveal the fact of the container integrity violation, on the one hand, and, on the other hand, will allow the author, if it is necessary, to confirm his participation in the development and embedding of the DWM. In this method we use mathematical properties of systems of linear algebraic equations, digital signature and cryptographic hash functions. The scheme is based on the Kronecker – Capelli theorem. To find the group password the co-author who is in the group finds one root of the consistent system of linear algebraic equations. The indicated system consists of n equations and contains n + 1 variables. For an outsider who is not in the group, the system of equations does not have a unique solution. The co-author of the group is able to calculate one variable in system based on their passport data. Therefore, the system of equations for such co-author has a unique solution. Various attacks on a protected by the new method are analyzed, and it is shown its efficiency. The constructed method can be applied in companies with a large team of developers.

Geophysical Inverse Theory and Regularization Problems

Research subject. The deep structure of the Yugan-Koltogor zone located in West Siberia was investigated with the purpose of detecting prospective oil and gas bearing areas.Materials and methods. The methods of lineament extraction and the computer modelling of rock density were employed. Lineament extraction was conducted on the basis of geophysical data, including detailed (1 : 200 000) maps of anomalous magnetic and gravity fields. In order to detect faults, telemetering methods were used, along with the results of studies conducted to investigate core materials from wells in the region. The modelling of the deep structure of the pre-Jurassic basement was performed on the basis of its geological map by solving a direct problem while fitting geological body densities.Results. Six largest faults of the Yugan-Koltogor zone identified on the geological map by a special sign “deep faults and regional schistosity zones” are of particular interest as possible oil bearing areas.Conclusions. The modelling of the deep structure of the pre-Jurassic basement of the West Siberian Platform has shown the granite decompaction areas of the Yugan zone to be highly promising in terms of oil and gas deposits.