Pub Date : 2022-05-03DOI: 10.22363/2658-4670-2022-30-2-139-148
A. Zorin, M. Malykh, L. Sevastianov
One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value AA is associated with a pseudodifferential operator O^(A){hat{O}(A)}. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state ψ{psi} the quantity AA is equal to λ{lambda} if ψ{psi} is an eigenfunction of the operator O^(A){hat{O}(A)}. In this case, the variance O^((A-λ)2)ψ{hat{O}((A-lambda)2)psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkins mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-λ)2){hat{O}((A-lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)λ+λ2E^{hat{O}(A2)-2hat{O}(A)lambda + lambda 2 hat{E}} is studied by the methods of perturbation theory under the assumption of small variance D^(A)=O^(A2)-O^(A)2{hat{D}(A) = hat{O}(A2) - hat{O}(A) 2} of the observable AA. It is shown that in the neighborhood of the real eigenvalue λ{lambda} of the operator O^(A){hat{O}(A)}, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by ±i⟨D^⟩{pm i sqrt{langle hat{D} rangle}}.
量子力学的一个可能版本，被称为Kuryshkin-Wodkiewicz量子力学，被认为是。在这个版本中，量子分布函数是正的，但作为对此的报复，冯·诺依曼量化规则被一个更复杂的规则所取代，其中观测值AA与伪微分算子O^（a）｛hat｛O｝（a）｝相关联。这个版本是耗散量子系统的一个例子，因此，预计哈密顿量的本征值应该有虚部。然而，在这个理论中，类氢原子的哈密顿量的离散谱被证明是实值的。在本文中，我们对这个悖论提出以下解释。传统上假设，在某种状态下，如果ψ是算子O^（A）的本征函数，则量AA等于λ。在这种情况下，方差O^（（A-λ）2）ψ｛hat｛O｝（（A-lambda）2）psi｝在量子力学的标准版本中为零，但在Kuryshkins力学中为非零。因此，可以考虑这样一个值和与其对应的状态的范围，其中方差O^（（a-λ）2）｛hat｛O｝（（a-λ）2｝）｝为零。在小方差D^（A）=O^（A2）-O^（A}（A）}，算子笔有两个特征值，它们在微扰理论的一阶上相差±i⟨D^⟩｛pm i sqrt｛langlehat｛D｝rangle｝｝。
{"title":"Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics","authors":"A. Zorin, M. Malykh, L. Sevastianov","doi":"10.22363/2658-4670-2022-30-2-139-148","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-139-148","url":null,"abstract":"One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value AA is associated with a pseudodifferential operator O^(A){hat{O}(A)}. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state ψ{psi} the quantity AA is equal to λ{lambda} if ψ{psi} is an eigenfunction of the operator O^(A){hat{O}(A)}. In this case, the variance O^((A-λ)2)ψ{hat{O}((A-lambda)2)psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkins mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-λ)2){hat{O}((A-lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)λ+λ2E^{hat{O}(A2)-2hat{O}(A)lambda + lambda 2 hat{E}} is studied by the methods of perturbation theory under the assumption of small variance D^(A)=O^(A2)-O^(A)2{hat{D}(A) = hat{O}(A2) - hat{O}(A) 2} of the observable AA. It is shown that in the neighborhood of the real eigenvalue λ{lambda} of the operator O^(A){hat{O}(A)}, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by ±i⟨D^⟩{pm i sqrt{langle hat{D} rangle}}.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44963993","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}
Pub Date : 2022-05-03DOI: 10.22363/2658-4670-2022-30-2-149-159
A. L. Sevastyanov
The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.
{"title":"Investigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguides","authors":"A. L. Sevastyanov","doi":"10.22363/2658-4670-2022-30-2-149-159","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-149-159","url":null,"abstract":"The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49199231","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}
Pub Date : 2022-05-03DOI: 10.22363/2658-4670-2022-30-2-127-138
K. Lovetskiy, D. Kulyabov, Ali Weddeye Hissein
The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, inverse with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.
{"title":"Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order","authors":"K. Lovetskiy, D. Kulyabov, Ali Weddeye Hissein","doi":"10.22363/2658-4670-2022-30-2-127-138","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-127-138","url":null,"abstract":"The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, inverse with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47444156","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}
Pub Date : 2022-05-03DOI: 10.22363/2658-4670-2022-30-2-105-114
A. Belov, N. Kalitkin
We consider Cauchy problem for ordinary differential equation with solution possessing a sequence of multiple poles. We propose the generalized reciprocal function method. It reduces calculation of a multiple pole to retrieval of a simple zero of accordingly chosen function. Advantages of this approach are illustrated by numerical examples. We propose two representative test problems which constitute interest for verification of other numerical methods for problems with poles.
{"title":"Numerical solution of Cauchy problems with multiple poles of integer order","authors":"A. Belov, N. Kalitkin","doi":"10.22363/2658-4670-2022-30-2-105-114","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-105-114","url":null,"abstract":"We consider Cauchy problem for ordinary differential equation with solution possessing a sequence of multiple poles. We propose the generalized reciprocal function method. It reduces calculation of a multiple pole to retrieval of a simple zero of accordingly chosen function. Advantages of this approach are illustrated by numerical examples. We propose two representative test problems which constitute interest for verification of other numerical methods for problems with poles.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43872419","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}
Pub Date : 2022-05-03DOI: 10.22363/2658-4670-2022-30-2-115-126
Zhanna O. Dombrovskaya
Mathematical statement of one-wavelength antireflective coating based on two-dimensional metamaterial is formulated for the first time. The constraints on geometric parameters of the structure are found. We propose a penalty function, which ensures the applicability of physical model and provides the uniqueness of the desired minimum. As an example, we consider the optimization of metasurface composed of PbTe spheres located on germanium substrate. It is shown that the accuracy of the minimization with properly chosen penalty term is the same as for the objective function without it.
{"title":"Optimization of an isotropic metasurface on a substrate","authors":"Zhanna O. Dombrovskaya","doi":"10.22363/2658-4670-2022-30-2-115-126","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-115-126","url":null,"abstract":"Mathematical statement of one-wavelength antireflective coating based on two-dimensional metamaterial is formulated for the first time. The constraints on geometric parameters of the structure are found. We propose a penalty function, which ensures the applicability of physical model and provides the uniqueness of the desired minimum. As an example, we consider the optimization of metasurface composed of PbTe spheres located on germanium substrate. It is shown that the accuracy of the minimization with properly chosen penalty term is the same as for the objective function without it.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47830511","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}
Pub Date : 2022-05-03DOI: 10.22363/2658-4670-2022-30-2-160-182
I. Zaryadov, Hilquias C. C. Viana, T. Milovanova
The paper presents a study of three queuing systems with a threshold renovation mechanism and an inverse service discipline. In the model of the first type, the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims). In the second model, the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In the model of the third type (generalizing the previous two models), two threshold values are used: one to activate the mechanism for dropping requests, the second - to set a safe zone in the queue. Based on the results obtained earlier, the main time-probabilistic characteristics of these models are presented. With the help of simulation modeling, the analysis and comparison of the behavior of the considered models were carried out.
{"title":"Analysis of queuing systems with threshold renovation mechanism and inverse service discipline","authors":"I. Zaryadov, Hilquias C. C. Viana, T. Milovanova","doi":"10.22363/2658-4670-2022-30-2-160-182","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-2-160-182","url":null,"abstract":"The paper presents a study of three queuing systems with a threshold renovation mechanism and an inverse service discipline. In the model of the first type, the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims). In the second model, the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In the model of the third type (generalizing the previous two models), two threshold values are used: one to activate the mechanism for dropping requests, the second - to set a safe zone in the queue. Based on the results obtained earlier, the main time-probabilistic characteristics of these models are presented. With the help of simulation modeling, the analysis and comparison of the behavior of the considered models were carried out.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42165348","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}
Pub Date : 2022-02-25DOI: 10.22363/2658-4670-2022-30-1-39-51
I. Belyaeva
The paper considers the class of Hamiltonian systems with two degrees of freedom. Based on the classical normal form, according to the rules of Born-Jordan and Weyl-MacCoy, its quantum analogs are constructed for which the eigenvalue problem is solved and approximate formulas for the energy spectrum are found. For particular values of the parameters of quantum normal forms using these formulas, numerical calculations of the lower energy levels were performed, and the obtained results were compared with the known data of other authors. It was found that the best and good agreement with the known results is obtained using the Weyl-MacCoy quantization rule. The procedure for normalizing the classical Hamilton function is an extremely time-consuming task, since it involves hundreds and even thousands of polynomials for the necessary transformations. Therefore, in the work, normalization is performed using the REDUCE computer algebra system. It is shown that the use of the Weyl-MacCoy and Born-Jordan correspondence rules leads to almost the same values for the energy spectrum, while their proximity increases for large quantities of quantum numbers, that is, for highly excited states. The canonical transformation is used in the work, the quantum analog of which allows us to construct eigenfunctions for the quantum normal form and thus obtain analytical formulas for the energy spectra of different Hamiltonian systems. So, it is shown that quantization of classical Hamiltonian systems, including those admitting the classical mode of motion, using the method of normal forms gives a very accurate prediction of energy levels.
{"title":"The quantization of the classical two-dimensional Hamiltonian systems","authors":"I. Belyaeva","doi":"10.22363/2658-4670-2022-30-1-39-51","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-1-39-51","url":null,"abstract":"The paper considers the class of Hamiltonian systems with two degrees of freedom. Based on the classical normal form, according to the rules of Born-Jordan and Weyl-MacCoy, its quantum analogs are constructed for which the eigenvalue problem is solved and approximate formulas for the energy spectrum are found. For particular values of the parameters of quantum normal forms using these formulas, numerical calculations of the lower energy levels were performed, and the obtained results were compared with the known data of other authors. It was found that the best and good agreement with the known results is obtained using the Weyl-MacCoy quantization rule. The procedure for normalizing the classical Hamilton function is an extremely time-consuming task, since it involves hundreds and even thousands of polynomials for the necessary transformations. Therefore, in the work, normalization is performed using the REDUCE computer algebra system. It is shown that the use of the Weyl-MacCoy and Born-Jordan correspondence rules leads to almost the same values for the energy spectrum, while their proximity increases for large quantities of quantum numbers, that is, for highly excited states. The canonical transformation is used in the work, the quantum analog of which allows us to construct eigenfunctions for the quantum normal form and thus obtain analytical formulas for the energy spectra of different Hamiltonian systems. So, it is shown that quantization of classical Hamiltonian systems, including those admitting the classical mode of motion, using the method of normal forms gives a very accurate prediction of energy levels.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43583067","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}
Pub Date : 2022-02-25DOI: 10.22363/2658-4670-2022-30-1-62-78
Serge Ndayisenga, L. Sevastianov, K. Lovetskiy
The paper discusses the formulation and analysis of methods for solving the one-dimensional Poisson equation based on finite-difference approximations - an important and very useful tool for the numerical study of differential equations. In fact, this is a classical approximation method based on the expansion of the solution in a Taylor series, based on which the recent progress of theoretical and practical studies allowed increasing the accuracy, stability, and convergence of methods for solving differential equations. Some of the features of this analysis include interesting extensions to classical numerical analysis of initial and boundary value problems. In the first part, a numerical method for solving the one-dimensional Poisson equation is presented, which reduces to solving a system of linear algebraic equations (SLAE) with a banded symmetric positive definite matrix. The well-known tridiagonal matrix algorithm, also known as the Thomas algorithm, is used to solve the SLAEs. The second part presents a solution method based on an analytical representation of the exact inverse matrix of a discretized version of the Poisson equation. Expressions for inverse matrices essentially depend on the types of boundary conditions in the original setting. Variants of inverse matrices for the Poisson equation with different boundary conditions at the ends of the interval under study are presented - the Dirichlet conditions at both ends of the interval, the Dirichlet conditions at one of the ends and Neumann conditions at the other. In all three cases, the coefficients of the inverse matrices are easily found and the algorithm for solving the problem is practically reduced to multiplying the matrix by the vector of the right-hand side.
{"title":"Finite-difference methods for solving 1D Poisson problem","authors":"Serge Ndayisenga, L. Sevastianov, K. Lovetskiy","doi":"10.22363/2658-4670-2022-30-1-62-78","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-1-62-78","url":null,"abstract":"The paper discusses the formulation and analysis of methods for solving the one-dimensional Poisson equation based on finite-difference approximations - an important and very useful tool for the numerical study of differential equations. In fact, this is a classical approximation method based on the expansion of the solution in a Taylor series, based on which the recent progress of theoretical and practical studies allowed increasing the accuracy, stability, and convergence of methods for solving differential equations. Some of the features of this analysis include interesting extensions to classical numerical analysis of initial and boundary value problems. In the first part, a numerical method for solving the one-dimensional Poisson equation is presented, which reduces to solving a system of linear algebraic equations (SLAE) with a banded symmetric positive definite matrix. The well-known tridiagonal matrix algorithm, also known as the Thomas algorithm, is used to solve the SLAEs. The second part presents a solution method based on an analytical representation of the exact inverse matrix of a discretized version of the Poisson equation. Expressions for inverse matrices essentially depend on the types of boundary conditions in the original setting. Variants of inverse matrices for the Poisson equation with different boundary conditions at the ends of the interval under study are presented - the Dirichlet conditions at both ends of the interval, the Dirichlet conditions at one of the ends and Neumann conditions at the other. In all three cases, the coefficients of the inverse matrices are easily found and the algorithm for solving the problem is practically reduced to multiplying the matrix by the vector of the right-hand side.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42424511","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}
Pub Date : 2022-02-25DOI: 10.22363/2658-4670-2022-30-1-79-87
E. Shchetinin
The paper proposes a trading strategy for investing in the cryptocurrency market that uses instant market entries based on additional sources of information in the form of a developed dataset. The task of predicting the moment of entering the market is formulated as the task of classifying the trend in the value of cryptocurrencies. To solve it, ensemble models and deep neural networks were used in the present paper, which made it possible to obtain a forecast with high accuracy. Computer analysis of various investment strategies has shown a significant advantage of the proposed investment model over traditional machine learning methods.
{"title":"On methods of building the trading strategies in the cryptocurrency markets","authors":"E. Shchetinin","doi":"10.22363/2658-4670-2022-30-1-79-87","DOIUrl":"https://doi.org/10.22363/2658-4670-2022-30-1-79-87","url":null,"abstract":"The paper proposes a trading strategy for investing in the cryptocurrency market that uses instant market entries based on additional sources of information in the form of a developed dataset. The task of predicting the moment of entering the market is formulated as the task of classifying the trend in the value of cryptocurrencies. To solve it, ensemble models and deep neural networks were used in the present paper, which made it possible to obtain a forecast with high accuracy. Computer analysis of various investment strategies has shown a significant advantage of the proposed investment model over traditional machine learning methods.","PeriodicalId":34192,"journal":{"name":"Discrete and Continuous Models and Applied Computational Science","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45600630","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}
Pub Date : 2022-02-25DOI: 10.22363/2658-4670-2022-30-1-52-61
M. M. Gambaryan, M. Malykh
The classical problem of the interaction of charged particles is considered in the framework of the concept of short-range interaction. Difficulties in the mathematical description of short-range interaction are discussed, for which it is necessary to combine two models, a nonlinear dynamic system describing the motion of particles in a field, and a boundary value problem for a hyperbolic equation or Maxwells equations describing the field. Attention is paid to the averaging procedure, that is, the transition from the positions of particles and their velocities to the charge and current densities. The problem is shown to contain several parameters; when they tend to zero in a strictly defined order, the model turns into the classical many-body problem. According to the Galerkin method, the problem is reduced to a dynamic system in which the equations describing the dynamics of particles, are added to the equations describing the oscillations of a field in a box. This problem is a simplification, different from that leading to classical mechanics. It is proposed to be considered as the simplest mathematical model describing the many-body problem with short-range interaction. This model consists of the equations of motion for particles, supplemented with equations that describe the natural oscillations of the field in the box. The results of the first computer experiments with this short-range interaction model are presented. It is shown that this model is rich in conservation laws.
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