Mathematics and Mathematical Modeling最新文献

{"title":"Selection of the Optimal Parameters of a Pneumatic Three-line Direct-acting Pressure Reducing Valve","authors":"K. Trukhanov, K. D. Efremova","doi":"10.24108/mathm.0222.0000303","DOIUrl":"https://doi.org/10.24108/mathm.0222.0000303","url":null,"abstract":"Reducing pneumatic valves, which are part of modern pneumatic control systems, are widely used in the automation of production and technological processes in various fields of technology. The efficiency of using pressure reducing valves depends on their parameters and characteristics. To select the optimal parameters of pressure reducing valves, various methods are used based on the search for the minimum of the target (target or criteria) function.The object of study is a three-line direct-acting pressure reducing valve, which ensures the determination of its optimal parameters. To calculate the optimal parameters of the pressure reducing valve, the method of orthogonal experimental design was applied. The main advantage of this method is that in this case all variables are varied simultaneously.The article presents the design features of a three-line direct-acting pressure reducing valve.The developed mathematical model of a pressure reducing valve is presented, which is a system of equations for a three-line direct-acting pressure reducing valve, which should be used to select the optimal parameters of the pressure reducing valve.To calculate the optimal parameters of the pressure reducing valve, the method of orthogonal experimental design was used. As an optimality criterion, the integral criterion of the squared error of the transient process, the change in the pressure level in the outlet line of the valve, was chosen.The optimal parameters of the pressure reducing valve are found, and transient processes of pressure change in the outlet line of the pressure reducing valve are calculated. The main requirements for optimization and selection of the optimal parameters of pressure reducing valves are formulated. A comparison of the transient processes of the outlet pressure before and after optimization is given.Based on the results of the work, the following conclusions were drawn:From the obtained comparative results of the transient processes of the level of outlet pressure in the outlet of the pressure reducing valve before and after optimization, it was found that after optimization the pressure reducing valve has a faster response.The speed has been increased from 0.35 s to 0.27 s, that is, the speed has increased by 1.30 times.The transient overshoot is less than 5%, and the static error is reduced by 1.9% and is completely eliminated.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132796726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On Laplacian Discrete Spectrum Behavior with Two Distant Perturbations on the Plane in the case of a double limiting eigenvalue","authors":"A. Golovina","doi":"10.24108/mathm.0222.0000301","DOIUrl":"https://doi.org/10.24108/mathm.0222.0000301","url":null,"abstract":"We consider the Laplacian with two distant perturbations on the plane. Perturbations are a real finite continuous potentials. The investigation is aimed at a behavior of the perturbed operator eigenvalues when the distance between the potentials tends to infinity. The study concerns an existence of the perturbed eigenvalues in the case of a double limiting eigenvalue (a double eigenvalue of the Laplacian with the first finite potential).The paper aim is to construct the first terms of the asymptotic expansions of the perturbed eigenvalues and the corresponding eigenfunctions in the case of a double limiting eigenvalue.The technique used to obtain the results is also applicable to the construction of complete asymptotic expansions of perturbed eigenvalues аnd their corresponding eigenfunctions. The finiteness of the distant potentials made it possible to reveal the complex exponential-power structure of the asymptotics obtained.The main study results include  the following:the first terms of the asymptotic expansions of the perturbed eigenvalues and their corresponding eigenfunctions;the first corrections of the asymptotics of the perturbed eigenvalues being equal to zero;exponential-power structure of the asymptotics of perturbed eigenvalues and their corresponding eigenfunctions.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134506594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Grid Method for Solving an Initial-Boundary-value Problem for a General Multidimensional Equation of Parabolic Type","authors":"Z. Beshtokova, V. Vodakhova","doi":"10.24108/mathm.0222.0000284","DOIUrl":"https://doi.org/10.24108/mathm.0222.0000284","url":null,"abstract":"The problems concerning the physical process studies that lead to mathematical models based on a parabolic type equation are of great practical importance.When solving the parabolic equation, a transition from the one-dimensional equation to the multidimensional one gives a rise to some difficulties. The trouble is that amount of computations significantly increases in the transition from the one-dimensional problems to the multidimensional ones In this regard a task to build the economical difference schemes for a numerical solution of multidimensional problems is a challenge.A difference scheme that approximates the problem over time is economical, if it is unconditionally stable and the desirable number of arithmetic operations in the transition from layer to layer is proportional to the number of nodes in each time layer.The paper dwells on the construction of a locally one-dimensional (economical) difference scheme for the approximate solution of a parabolic type equation of a general form in a multidimensional domain, the main idea of which is to reduce a complex problem to a sequential solution of boundary value problems of a simpler structure. At the same time, there is a construction of the economical, unconditionally stable difference scheme for each of the intermediate problems. To have a numerical solution of the problem, there is a construction of the locally one-dimensional difference scheme of A.A. Samarsky. Using a method of energy inequalities allows us to obtain a priori estimates in the differential and difference interpretations, whence it follow that there are the uniqueness, stability, as well as the convergence of the solution of the locally one-dimensional difference scheme to the solution of the original differential problem. For a bi-dimensional problem, a numerical solution algorithm is constructed.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134121975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regarding Classification of n-sharing of Multivariate Mappings over Finite Fields and One NSUCrypto'2019 Olympiad Problem","authors":"A. Chilikov","doi":"10.24108/mathm.0122.0000262","DOIUrl":"https://doi.org/10.24108/mathm.0122.0000262","url":null,"abstract":"A problem of great importance that arises in designing and implementation of a cryptosystem is countering side channel attacks. Often an appropriate mathematical algorithm, implemented on a specific physical device to work in the physical environment,   becomes vulnerable to such attacks.The “function sharing” technique is a prospective and efficient way to avoid this problem. In the paper we investigate “non-complete sharing” of Boolean functions and mappings, and functions and mappings over finite fields and provide a complete description of the set of functions with n variables, which have sharing.The main findings are the following: introducing and investigating a new concept of “weak” non-complete n-sharing, establishing its connection with “weak” and “classical” n-sharing, and substantiating its advantages from the algebraic point-of-view as well as establishing and proving a criterion for the existence of weak non-complete n-sharing for an arbitrary function. The results also include an explicit description of a set of functions which have weak sharing in terms of algebraic normal form, obtaining the precise and simple descriptions for the boundary (“border”) cases: n = 2, n=m and binary fields.  Applying these results to the AES S-box allows complete solving the problem,  i.e. a complete answer to the question of a representability of the S-box of the AES cipher as a sharing is available. We believe that the same way can be successful for other cryptographic algorithms.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132004620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysis of the Mathematical Model of Cancer Dynamics","authors":"O. A. Podderegin","doi":"10.24108/mathm.0122.0000299","DOIUrl":"https://doi.org/10.24108/mathm.0122.0000299","url":null,"abstract":"Qualitative analysis of dynamic systems is a powerful research tool used to solve a great deal of application problems. In mathematical models described by ordinary differential equations, the study of equilibrium positions has actually become the standard: their number and location depending on the parameters, as well as issues of local stability. In mathematical models in various fields of chemistry and biology, it is also important that the natural requirements for the possible values ​​of phase variables are not violated in the process of dynamics (as a rule, this is the non-negativity condition). More sophisticated study orientation is global sustainability.The need for a qualitative analysis of a dynamic system arises because, as a rule, differential equations are not integrated and it is impossible to define the general properties of the system trajectories by integration. However, even if the system of differential equations is integrated, the formulas can be so cumbersome that their use in the analysis of the system dynamics turns out to be a challenge.In this paper, we study an 8-dimensional system of ordinary differential equations that describes the dynamics of cancer during chemotherapy and immunotherapy. The phase space of the system is a non-negative orthant.The paper studies the equilibrium positions of the system. There is always one equilibrium in a system, but there may be two more equilibria. There are conditions for the system parameters found, under which the equilibrium position, being always present is asymptotically stable. Moreover, there is a provement that the asymptotic stability is global.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115287223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic Behavior of the Eigenvalues of a Periodic Operator with Two Distant Perturbations on the Axis","authors":"A. Golovina","doi":"10.24108/mathm.0122.0000300","DOIUrl":"https://doi.org/10.24108/mathm.0122.0000300","url":null,"abstract":"We consider a second-order periodic operator with two distant perturbations on the real axis. Perturbations are real finite continuous potentials. The objective is to investigate a behavior of the eigenvalues of the perturbed operator when the distance between the potentials tends to infinity.  The study issue is an existence of perturbed eigenvalues in the case of a double limiting eigenvalue (the simple and isolated eigenvalue of a periodic operator with first potential + the simple and isolated eigenvalue of a periodic operator with а second potential).The paper aim is to construct the first terms of the asymptotic expansions of the perturbed eigenvalues and the corresponding eigenfunctions in the case of a double limiting eigenvalue.The technique to obtain the results can also find application when constructing complete asymptotic expansions of perturbed eigenvalues and their corresponding eigenfunctions. The finiteness of the distant potentials allowed us to reveal the complex exponential structure of the asymptotics obtained.The main results include the following:the first terms of the asymptotic expansions of the perturbed eigenvalues and their corresponding eigenfunctions;symmetry of the first corrections of the asymptotics of the perturbed eigenvalues with respect to zero;exponential structure of the asymptotics of perturbed eigenvalues and their corresponding eigenfunctions.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132189797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Efficiency Analysis of OLAP-data Hypercube Decomposition for Exponential Computational Complexity Methods","authors":"A. Nosov, A. A. Akhrem, V. Rakhmankulov","doi":"10.24108/mathm.0321.0000258","DOIUrl":"https://doi.org/10.24108/mathm.0321.0000258","url":null,"abstract":"The paper studies problems of reduction (decomposition) of OLAP-hypercube multidimensional data models. When decomposing large hyper-cubes of multidimensional data into sub-cube components the goal is to increase the computational performance of analytical OLAP systems, which is related to decreasing computational complexity of reduction methods for solving OLAP-data analysis problems with respect to the computational complexity of non-reduction methods, applied to data directly all over the hypercube. The paper formalizes the concepts of reduction and non-reduction methods and gives a definition of the upper bound for the change in the computational complexity of reduction methods in the decomposition of the problem of analyzing multidimensional OLAP-data in comparison with non-reduction methods in the class of exponential degree of computational complexity.The exact values of the upper bound for changing computational complexity are obtained for the hypercube decomposition into two sub-cubes on sets consisting of an even and an odd number of sub-cube structures, and its main properties are given, which are used to determine the decomposition efficiency. A formula for the efficiency of decomposition into two sub-cube structures for reduction of OLAP data analysis problems is obtained, and it is shown that with an increase in the dimension “n” of the lattice specifying the number of sub-cubes in the hypercube data structure, the efficiency of such a decomposition obeys an exponential law with an exponent “n/2”, regardless of the parity “n”. The examples show the possibility to use the values (found) of the upper bound for the change in computational complexity to establish the effectiveness criteria for reduction methods and the expediency of decomposition in specific cases.The paper results can be used in processing and analysis of information arrays of hypercube structures of analytical OLAP systems belonging to the Big-Data or super-large computer systems of multidimensional data.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114603691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Mathematical Modelling of the Elliptical Vortex Ring in a Viscous Fluid with the Vortex Filament Method","authors":"O. Kotsur","doi":"10.24108/mathm.0321.0000263","DOIUrl":"https://doi.org/10.24108/mathm.0321.0000263","url":null,"abstract":"The article deals with modelling an elliptical vortex ring in a viscous fluid using the Lagrangian vortex filament method. The novelty is that earlier only inviscid flows restricted vortex filament method application. The proposed viscosity model uses an analogue of the diffusion rate method, which is widely applied to simulate plane-parallel and axisymmetric flows of viscous fluid. A transfer of the formula of a diffusion rate from two-dimensional flows to the model of spatial vortex filament is due to assumption that swirling of vortex lines (helicity of vorticity) is unavailable. Despite the laxity of the diffusion rate model for general spatial flows, its application enables taking into account the effect of viscous diffusion of vorticity, which provides expansion of vortex tubes in space. The paper formulates the vortex filament method in which the filaments are broken into the vortex segments. Such discretization enables turning from the equation of vorticity evolution in partial derivatives to a system of ordinary differential equations with respect to the parameters of the segments. Formulas to calculate a filament system-induced flow rate as well as formulas to perform approximate calculation of an analogue of the diffusion rate are given.The objective is to propose the viscosity model as an application to the vortex filament method by the example of modelling the evolution of an elliptical vortex ring in viscous fluid. The calculation results obtained by the vortex method are compared with the existing experiment and with the calculation performed by the grid method in the OpenFOAM package. A feature of the problem is that there are zones of nonzero helicity of vorticity where the proposed model of viscosity, strictly speaking, is not correct. It is shown that the results of calculations are in good agreement with each other and are in complete agreement with experiment. This allows saying that the effects of swirling vortex lines do not significantly affect the results of modelling a specific example of the spatial flow of viscous fluid by the proposed modification of the vortex filament method.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128922053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Analysing Efficiency Methods of Polynomial Complexity Degree in Multidimensional OLAP Cube Data Decomposition","authors":"A. A. Akhrem, A. Nosov, V. Rakhmankulov","doi":"10.24108/MATHM.0121.0000244","DOIUrl":"https://doi.org/10.24108/MATHM.0121.0000244","url":null,"abstract":"The article investigates the problems of reduction (decomposition) of multidimensional data models in terms of hypercube OLAP-structures. Describes the case when a data structure is defined by the array that slices and dices the hypercube into the odd number of subcubes, and this set of subcube structures becomes decomposed. Defines an exact upper bound for increasing a computational performance of methods to analyze OLAP-data on subcubes, which determines the decomposition approach efficiency in comparison with the OLAP-data analysis on a complete unreduced hypercube. A compared efficiency of the hypercube decomposition into two subcubes on the sets consisting of the even and odd number of subcube structures has shown that with considerable data partitioning for methods of a polynomial complexity degree the decomposition efficiency essentially is independent on this factor and rises with increasing complexity degree of methods applied.When using the mathematical methods to study decomposition (reduction) of large hyper-cubes of multidimensional data of analytical OLAP systems into subcube components, there is a need to find conditions for minimising the computational complexity of methods to solve the problems of the OLAP hyper-cube analysis during data decomposition in comparison with using these methods for analyzing large amounts of information that is accumulated directly in the hyper-cubes of multidimensional OLAP-data to establish the criteria for decreasing or increasing computational performance when applying methods on the subcube components (reduction methods) as compared to applying these methods on a hypercube (non-reduction or traditional methods), depending on one or another degree of complexity of complex methods.The article provides an accurate quantitative estimate of decreasing computational complexity of reduction methods for analyzing OLAP-cubes as compared to the non-reduction methods in the case when said methods have the polynomial complexity and the original hypercube array of data comprises the odd number of subcubes.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129525118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems","authors":"V. Roitenberg","doi":"10.24108/mathm.0120.0000213","DOIUrl":"https://doi.org/10.24108/mathm.0120.0000213","url":null,"abstract":"There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied  the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.","PeriodicalId":124696,"journal":{"name":"Mathematics and Mathematical Modeling","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124208174","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}