Pub Date : 2019-04-26DOI: 10.33581/2520-6508-2019-1-18-27
B. Kalitine
The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.
{"title":"Stability of some differential equations of the fourth-order and fifth-order","authors":"B. Kalitine","doi":"10.33581/2520-6508-2019-1-18-27","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-18-27","url":null,"abstract":"The article is devoted to the study of the problem of stability of nonlinear ordinary differential equations by the method of semi-definite Lyapunov’s functions. The types of fourth-order and fifth-order scalar nonlinear differential equations of general form are singled out, for which the sign-constant auxiliary functions are defined. Sufficient conditions for stability in the large are obtained for such equations. The results coincide with the necessary and sufficient conditions in the corresponding linear case. Studies emphasize the advantages in using the semi-positive functions in comparison with the classical method of applying Lyapunov’s definite positive functions.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45343572","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 : 2019-04-26DOI: 10.33581/2520-6508-2019-1-28-32
Aliaksei Vaidzelevich
It is known that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order greater than two with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended real line. Moreover, for the spectra of upper frequencies of third-order equations this result was inverted under the assumption that the spectra contain zero. In the present paper we obtain an inversion of the above statement for equations of the fourth order and higher. Namely, for an arbitrary zero-containing Suslin subset S on the non-negative half-line of the extended real line and a positive integer number n greater than three a n order linear differential equation is constructed, which spectra of the upper Sergeev frequencies of zeros, signs, and roots coincide with the set S.
{"title":"On spectra of upper Sergeev frequencies of linear differential equations","authors":"Aliaksei Vaidzelevich","doi":"10.33581/2520-6508-2019-1-28-32","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-28-32","url":null,"abstract":"It is known that the spectra (ranges) of upper and lower Sergeev frequencies of zeros, signs, and roots of a linear differential equation of order greater than two with continuous coefficients belong to the class of Suslin sets on the nonnegative half-line of the extended real line. Moreover, for the spectra of upper frequencies of third-order equations this result was inverted under the assumption that the spectra contain zero. In the present paper we obtain an inversion of the above statement for equations of the fourth order and higher. Namely, for an arbitrary zero-containing Suslin subset S on the non-negative half-line of the extended real line and a positive integer number n greater than three a n order linear differential equation is constructed, which spectra of the upper Sergeev frequencies of zeros, signs, and roots coincide with the set S.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42312294","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 : 2019-04-26DOI: 10.33581/2520-6508-2019-1-69-76
V. Volkov, Alena Prakonina
Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an isolated anisotropic ring conductor a comparative efficiency analysis of some promising finite-difference schemes and iterative methods in terms of their compatibility with the fictitious domain method is carried out. On the basis of numerical experiments empirical estimates of the asymptotic dependence of the convergence rate of the biconjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners on the step size and the value of the small parameter determining the continuation of the conductivity coefficient in the fictitious domain method are obtained. It is shown, that for one of the considered schemes the Fourier – Jacobi preconditioner is spectrally optimal and allows to eliminate the asymptotical dependence of the iterations number to achieve a given accuracy both on the value of the step size and the value of the small parameter in the fictitious domain method.
{"title":"Iterative realization of finite difference schemes in the fictitious domain method for elliptic problems with mixed derivatives","authors":"V. Volkov, Alena Prakonina","doi":"10.33581/2520-6508-2019-1-69-76","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-69-76","url":null,"abstract":"Development of efficient finite difference schemes and iterative methods for solving anisotropic diffusion problems in an arbitrary geometry domain is considered. To simplify the formulation of the Neumann boundary conditions, the method of fictitious domains is used. On the example of a two-dimensional model problem of potential distribution in an isolated anisotropic ring conductor a comparative efficiency analysis of some promising finite-difference schemes and iterative methods in terms of their compatibility with the fictitious domain method is carried out. On the basis of numerical experiments empirical estimates of the asymptotic dependence of the convergence rate of the biconjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners on the step size and the value of the small parameter determining the continuation of the conductivity coefficient in the fictitious domain method are obtained. It is shown, that for one of the considered schemes the Fourier – Jacobi preconditioner is spectrally optimal and allows to eliminate the asymptotical dependence of the iterations number to achieve a given accuracy both on the value of the step size and the value of the small parameter in the fictitious domain method.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49067414","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 : 2019-04-26DOI: 10.33581/2520-6508-2019-1-35-45
T. Tu, Yuriy S. Kharin
The problem of sequential test for many simple hypotheses on parameters of time series with trend is considered. Two approaches, including M-ary sequential probability ratio test and matrix sequential probability ratio test are used for constructing the sequential test. The sufficient conditions of finite terminations of the test and the existence of finite moments of their stopping times are given. The upper bounds for the average numbers of observations are obtained. With the thresholds chosen suitably, these tests can belong to some specified classes of statistical tests. Numerical examples are presented.
{"title":"Sequential probability ratio test for many simple hypotheses on parameters of time series with trend","authors":"T. Tu, Yuriy S. Kharin","doi":"10.33581/2520-6508-2019-1-35-45","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-35-45","url":null,"abstract":"The problem of sequential test for many simple hypotheses on parameters of time series with trend is considered. Two approaches, including M-ary sequential probability ratio test and matrix sequential probability ratio test are used for constructing the sequential test. The sufficient conditions of finite terminations of the test and the existence of finite moments of their stopping times are given. The upper bounds for the average numbers of observations are obtained. With the thresholds chosen suitably, these tests can belong to some specified classes of statistical tests. Numerical examples are presented.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43187845","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 : 2019-04-26DOI: 10.33581/2520-6508-2019-1-12-17
E. V. Zubei
A finite non-nilpotent group G is called a B-group if every proper subgroup of the quotient group G/Φ(G) is nilpotent. We establish the r-solvability of the group in which some Sylow r-subgroup permutes with the derived subgroups of 2-nilpotent (or 2-closed) B-subgroups of even order and the solvability of the group in which the derived subgroups of 2-closed and 2-nilpotent B-subgroups of even order are permutable.
{"title":"On the permutability of Sylow subgroups with derived subgroups of B-subgroups","authors":"E. V. Zubei","doi":"10.33581/2520-6508-2019-1-12-17","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-12-17","url":null,"abstract":"A finite non-nilpotent group G is called a B-group if every proper subgroup of the quotient group G/Φ(G) is nilpotent. We establish the r-solvability of the group in which some Sylow r-subgroup permutes with the derived subgroups of 2-nilpotent (or 2-closed) B-subgroups of even order and the solvability of the group in which the derived subgroups of 2-closed and 2-nilpotent B-subgroups of even order are permutable.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46431974","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 : 2019-04-26DOI: 10.33581/2520-6508-2019-1-46-57
S. Bosiakov, G. Mikhasev
Pathological changes in the oscillating system of the middle ear can lead to a decrease of the susceptibility threshold of the auditory analyser to sound vibrations and, consequently, to partial or complete hearing loss. Cartilage implants are most often used for the reconstruction of the tympanic membrane, since they help to avoid complications after treatment. Evaluation of the dynamic characteristics (eigenmodes and eigenfrequencies) of the reconstructed middle ear is the most important problem for analysing the quality of operations that improve the auditory conductivity and develop further recommendations for optimal prosthetics. The aim of this study is to estimate the eigenfrequencies of the middle ear free oscillations after prosthetics on the basis of a mathematical model involving transverse vibrations of the cartilage graft and movement of the prosthesis connecting the reconstructed tympanic membrane and the base of the foot stapes plate. The values of natural frequencies are evaluated for different positions of the nodal lines, averaged geometrical parameters and elastic properties of the tympanic membrane, as well as the foot plate of stapes and prosthesis.
{"title":"Free oscillations of the middle ear after total tympanoplasty and ossiculoplasty with functional mobility of the foot plate of stapes","authors":"S. Bosiakov, G. Mikhasev","doi":"10.33581/2520-6508-2019-1-46-57","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-46-57","url":null,"abstract":"Pathological changes in the oscillating system of the middle ear can lead to a decrease of the susceptibility threshold of the auditory analyser to sound vibrations and, consequently, to partial or complete hearing loss. Cartilage implants are most often used for the reconstruction of the tympanic membrane, since they help to avoid complications after treatment. Evaluation of the dynamic characteristics (eigenmodes and eigenfrequencies) of the reconstructed middle ear is the most important problem for analysing the quality of operations that improve the auditory conductivity and develop further recommendations for optimal prosthetics. The aim of this study is to estimate the eigenfrequencies of the middle ear free oscillations after prosthetics on the basis of a mathematical model involving transverse vibrations of the cartilage graft and movement of the prosthesis connecting the reconstructed tympanic membrane and the base of the foot stapes plate. The values of natural frequencies are evaluated for different positions of the nodal lines, averaged geometrical parameters and elastic properties of the tympanic membrane, as well as the foot plate of stapes and prosthesis.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42111859","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 : 2019-04-08DOI: 10.33581/2520-6508-2019-1-58-68
Галина Алексеевна Расолько
In the paper, computational schemes for solving the Cauchy problem for the singular integro-differential Prandtl equation with a singular integral over a segment of the real axis, understood in the sense of the Cauchy principal value, are constructed and justified. This equation is reduced to equivalent Fredholm equations of the second kind by inversion of the singular integral in three classes of Muskhelishvili functions and applying spectral relations for the singular integral. At the same time, we investigate the conditions for the solvability of integral Fredholm equations of the second kind with a logarithmic kernel of a special form and are approximately solved. The new computational schemes are based on applying the spectral relations for the singular integral to the integral entering into the equivalent equation. Uniform estimates of the errors of approximate solutions are obtained.
{"title":"To the numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials","authors":"Галина Алексеевна Расолько","doi":"10.33581/2520-6508-2019-1-58-68","DOIUrl":"https://doi.org/10.33581/2520-6508-2019-1-58-68","url":null,"abstract":"In the paper, computational schemes for solving the Cauchy problem for the singular integro-differential Prandtl equation with a singular integral over a segment of the real axis, understood in the sense of the Cauchy principal value, are constructed and justified. This equation is reduced to equivalent Fredholm equations of the second kind by inversion of the singular integral in three classes of Muskhelishvili functions and applying spectral relations for the singular integral. At the same time, we investigate the conditions for the solvability of integral Fredholm equations of the second kind with a logarithmic kernel of a special form and are approximately solved. The new computational schemes are based on applying the spectral relations for the singular integral to the integral entering into the equivalent equation. Uniform estimates of the errors of approximate solutions are obtained.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":"45 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41263989","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：