Pub Date : 2022-08-01DOI: 10.33581/2520-6508-2022-2-34-46
V. I. Korzyuk, J. V. Rudzko
In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. This mixed problem models the propagation of displacement waves during a longitudinal impact on a bar, when the load remains in contact with the bar and the bar has a linear elastic element at the end. On the lower boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. The boundary condition, including the unknown function and its first and second order partial derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proven and the conditions are established under which a piecewise-smooth solution exists. The problem with matching conditions is considered.
{"title":"Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end","authors":"V. I. Korzyuk, J. V. Rudzko","doi":"10.33581/2520-6508-2022-2-34-46","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-34-46","url":null,"abstract":"In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. This mixed problem models the propagation of displacement waves during a longitudinal impact on a bar, when the load remains in contact with the bar and the bar has a linear elastic element at the end. On the lower boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. The boundary condition, including the unknown function and its first and second order partial derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proven and the conditions are established under which a piecewise-smooth solution exists. The problem with matching conditions is considered.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45538486","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 : 2022-08-01DOI: 10.33581/2520-6508-2022-2-57-69
E. Starovoitov, Alina V. Nesterovich
The statement of the boundary value problem on the deformation of a circular three-layer plate in its plane under the action of a non-axisymmetric load is herein presented. The materials of thin carrier layers obey the hypotheses of the theory of small elastoplastic deformations. The relatively thick filler is physically non-linearly elastic. A system of non-linear differential equilibrium equations in partial derivatives is obtained. A general technique for solving the problem in displacements based on the Fourier method and Ilyushin’s method of elastic solutions is proposed. The case of an external cosine load is considered. An iterative solution of a boundary value problem for a physically non-linear plate is obtained. The corresponding solution of the elastic problem is written out in the final form. The obtained solution is numerically tested.
{"title":"The non-axisymmetric loading of an elastoplastic three-layer plate in its plane","authors":"E. Starovoitov, Alina V. Nesterovich","doi":"10.33581/2520-6508-2022-2-57-69","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-57-69","url":null,"abstract":"The statement of the boundary value problem on the deformation of a circular three-layer plate in its plane under the action of a non-axisymmetric load is herein presented. The materials of thin carrier layers obey the hypotheses of the theory of small elastoplastic deformations. The relatively thick filler is physically non-linearly elastic. A system of non-linear differential equilibrium equations in partial derivatives is obtained. A general technique for solving the problem in displacements based on the Fourier method and Ilyushin’s method of elastic solutions is proposed. The case of an external cosine load is considered. An iterative solution of a boundary value problem for a physically non-linear plate is obtained. The corresponding solution of the elastic problem is written out in the final form. The obtained solution is numerically tested.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41901243","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 : 2022-07-27DOI: 10.33581/2520-6508-2022-2-15-22
E. V. Gromak
In this paper, we consider the generalised hierarchy of the first Painlevé equation which is a sequence of polynomial ordinary differential equations of even order that have a uniform differential-algebraic structure determined by the operator L~n. The first member of this hierarchy for n = 2 is the first Painlevé equation, and the subsequent equations of order 2n – 2 contain arbitrary parameters. They are named as higher analogues of the first Painlevé equation of 2n – 2 order. The article considers the analytical properties of solutions to the equations of the generalised hierarchy of the first Painlevé equation and the related linear equations. It is established that each hierarchy equation has one dominant term, and an arbitrary meromorphic solution of any hierarchy equation cannot have a finite number of poles. The character of the mobile poles of meromorphic solutions is determined. Using the Frobenius method, sufficient conditions are obtained for the meromorphicity of the general solution of the second-order linear equations with a linear potential defined by meromorphic solutions of the first three equations of the hierarchy.
{"title":"On meromorphic solutions of the equations related to the first Painlevé equation","authors":"E. V. Gromak","doi":"10.33581/2520-6508-2022-2-15-22","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-15-22","url":null,"abstract":"In this paper, we consider the generalised hierarchy of the first Painlevé equation which is a sequence of polynomial ordinary differential equations of even order that have a uniform differential-algebraic structure determined by the operator L~n. The first member of this hierarchy for n = 2 is the first Painlevé equation, and the subsequent equations of order 2n – 2 contain arbitrary parameters. They are named as higher analogues of the first Painlevé equation of 2n – 2 order. The article considers the analytical properties of solutions to the equations of the generalised hierarchy of the first Painlevé equation and the related linear equations. It is established that each hierarchy equation has one dominant term, and an arbitrary meromorphic solution of any hierarchy equation cannot have a finite number of poles. The character of the mobile poles of meromorphic solutions is determined. Using the Frobenius method, sufficient conditions are obtained for the meromorphicity of the general solution of the second-order linear equations with a linear potential defined by meromorphic solutions of the first three equations of the hierarchy.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46371905","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 : 2022-07-27DOI: 10.33581/2520-6508-2022-2-47-56
V. Klimenok
We consider herein a single-server queueing system with a finite buffer and a batch Markovian arrival process. Customers staying in the buffer may have a lower or higher priority. Immediately after arrival each of the customer is assigned the lowest priority and a timer is set for it, which is defined as a random variable distributed according to the phase law and having two absorbing states. After the timer enters one of the absorbing states, the customer may leave the system forever (get lost) or change its priority to the highest. When the timer enters another absorbing state, the customer is lost with some probability and the timer is set again with an additional probability. If a customer enters a completely full system, it is lost. Systems of this type can serve as mathematical models of many real-life medical care systems, contact centers, perishable food storage systems, etc. The operation of the system is described in terms of a multidimensional Markov chain, the stationary distribution and a number of performance characteristics of the system are calculated.
{"title":"A queueing system with a batch Markovian arrival process and varying priorities","authors":"V. Klimenok","doi":"10.33581/2520-6508-2022-2-47-56","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-47-56","url":null,"abstract":"We consider herein a single-server queueing system with a finite buffer and a batch Markovian arrival process. Customers staying in the buffer may have a lower or higher priority. Immediately after arrival each of the customer is assigned the lowest priority and a timer is set for it, which is defined as a random variable distributed according to the phase law and having two absorbing states. After the timer enters one of the absorbing states, the customer may leave the system forever (get lost) or change its priority to the highest. When the timer enters another absorbing state, the customer is lost with some probability and the timer is set again with an additional probability. If a customer enters a completely full system, it is lost. Systems of this type can serve as mathematical models of many real-life medical care systems, contact centers, perishable food storage systems, etc. The operation of the system is described in terms of a multidimensional Markov chain, the stationary distribution and a number of performance characteristics of the system are calculated.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49573826","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 : 2022-07-20DOI: 10.33581/2520-6508-2022-2-82-93
D. Barovik, V. Taranchuk
The problem of computer modelling of the spread of surface forest fires in a two-dimensional formulation is herein considered. We describe the initial-boundary value problem in the form of a system of partial differential equations in the accepted approximation of the corresponding physical and chemical processes with refinements of the mutually agreed defining functions and the coefficients included in the equations. The Wolfram Mathematica computer algebra system is used as a platform for developing the computer model, performing calculations, and creating a database with the outcomes of computations. The results of numerical experiments investigating possible scenarios of how a fire zone spreads in different directions and its behaviour near the fuelbreaks are presented. Several qualitative features of the structure, the evolution of the temperature front, and the vector fields of the oxygen concentration gradient over the forest area are identified and illustrated with multidimensional graphics in the presence of areas of the low content of combustible materials of various shapes and sizes, including the demonstration of the influence of the equilibrium wind speed in the forest canopy. Possible variants of the fire front movement in the direction of the wind velocity and against it are identified and explained using representative examples.
{"title":"Tools for the analysis and visualisation of distributions and vector fields in surface forest fires modelling","authors":"D. Barovik, V. Taranchuk","doi":"10.33581/2520-6508-2022-2-82-93","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-82-93","url":null,"abstract":"The problem of computer modelling of the spread of surface forest fires in a two-dimensional formulation is herein considered. We describe the initial-boundary value problem in the form of a system of partial differential equations in the accepted approximation of the corresponding physical and chemical processes with refinements of the mutually agreed defining functions and the coefficients included in the equations. The Wolfram Mathematica computer algebra system is used as a platform for developing the computer model, performing calculations, and creating a database with the outcomes of computations. The results of numerical experiments investigating possible scenarios of how a fire zone spreads in different directions and its behaviour near the fuelbreaks are presented. Several qualitative features of the structure, the evolution of the temperature front, and the vector fields of the oxygen concentration gradient over the forest area are identified and illustrated with multidimensional graphics in the presence of areas of the low content of combustible materials of various shapes and sizes, including the demonstration of the influence of the equilibrium wind speed in the forest canopy. Possible variants of the fire front movement in the direction of the wind velocity and against it are identified and explained using representative examples.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47025543","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 : 2022-07-20DOI: 10.33581/2520-6508-2022-2-6-14
Maksim M. Vaskovskii
In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented.
{"title":"On the uniqueness of higher order Gubinelli derivatives and an analogue of the Doob – Meyer theorem for rough paths of the arbitrary positive Holder index","authors":"Maksim M. Vaskovskii","doi":"10.33581/2520-6508-2022-2-6-14","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-6-14","url":null,"abstract":"In this paper, we investigate the features of higher order Gubinelli derivatives of controlled rough paths having an arbitrary positive Holder index. There is used a notion of the (α, β)-rough map on the basis of which the sufficient conditions are given for the higher order Gubinelli derivatives uniqueness. Using the theorem on the uniqueness of higher order Gubinelli derivatives an analogue of the Doob – Meyer theorem for rough paths with an arbitrary positive Holder index is proved. In the final section of the paper, we prove that the law of the local iterated logarithm for fractional Brownian motion allows using all the main results of this paper for integration over the multidimensional fractional Brownian motions of the arbitrary Hurst index. The examples demonstrating the connection between the rough path integrals and the Ito and Stratonovich integrals are represented.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41819369","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 : 2022-07-15DOI: 10.33581/2520-6508-2022-2-107-114
M. Dymkov, Siarhey M. Dymkou
In this paper, we consider the optimal control problem described by a system of ordinary differential equations in the presence of state constraints. Theoretical results are obtained concerning the approximation of this problem by a sequence of new optimal control problems with a modified right-hand side of the control system and no state constraints. The issues of the approximation of continuous control systems by their discrete versions are also discussed.
{"title":"A method for relaxing state constraints in nonsmooth optimal control problems","authors":"M. Dymkov, Siarhey M. Dymkou","doi":"10.33581/2520-6508-2022-2-107-114","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-107-114","url":null,"abstract":"In this paper, we consider the optimal control problem described by a system of ordinary differential equations in the presence of state constraints. Theoretical results are obtained concerning the approximation of this problem by a sequence of new optimal control problems with a modified right-hand side of the control system and no state constraints. The issues of the approximation of continuous control systems by their discrete versions are also discussed.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44408255","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 : 2022-06-24DOI: 10.33581/2520-6508-2022-2-23-33
A. I. Kalinin, Leonid I. Lavrinovich, Darya Y. Prudnikova
We consider an optimisation problem for the transient process in a quasi-linear dynamical system (contains a small parameter at non-linearities) with a performance index that is a linear combination of energy costs and the duration of the process. An algorithm for constructing asymptotic approximations of a given order to the solution of this problem is proposed. The algorithm is based on the asymptotic decomposition by integer powers of a small parameter of the initial values of adjoint variables and the duration of the process that are finite-dimensional elements, according to which the solution of the problem is easily restored. The computational procedure of the algorithm includes solving the problem of optimising the transient process in a linear dynamical system, integrating systems of linear differential equations, and finding the roots of non-degenerate linear algebraic systems. We also show how the constructed asymptotic approximations can be used to construct optimal control in the problem under consideration for a given value of a small parameter.
{"title":"The small parameter method in the optimisation of a quasi-linear dynamical system problem","authors":"A. I. Kalinin, Leonid I. Lavrinovich, Darya Y. Prudnikova","doi":"10.33581/2520-6508-2022-2-23-33","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-2-23-33","url":null,"abstract":"We consider an optimisation problem for the transient process in a quasi-linear dynamical system (contains a small parameter at non-linearities) with a performance index that is a linear combination of energy costs and the duration of the process. An algorithm for constructing asymptotic approximations of a given order to the solution of this problem is proposed. The algorithm is based on the asymptotic decomposition by integer powers of a small parameter of the initial values of adjoint variables and the duration of the process that are finite-dimensional elements, according to which the solution of the problem is easily restored. The computational procedure of the algorithm includes solving the problem of optimising the transient process in a linear dynamical system, integrating systems of linear differential equations, and finding the roots of non-degenerate linear algebraic systems. We also show how the constructed asymptotic approximations can be used to construct optimal control in the problem under consideration for a given value of a small parameter.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44454730","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 : 2022-04-15DOI: 10.33581/2520-6508-2022-1-66-74
S. Agievich
We provide an upper bound on binomial coefficients that holds over the entire parameter range an whose form repeats the form of the de Moivre – Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the number of continuations of a given Boolean function to bent functions, investigate dependencies into the Walsh – Hadamard spectra, obtain restrictions on the number of representations as sums of squares of integers bounded in magnitude.
{"title":"An upper bound on binomial coefficients in the de Moivre – Laplace form","authors":"S. Agievich","doi":"10.33581/2520-6508-2022-1-66-74","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-66-74","url":null,"abstract":"We provide an upper bound on binomial coefficients that holds over the entire parameter range an whose form repeats the form of the de Moivre – Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the number of continuations of a given Boolean function to bent functions, investigate dependencies into the Walsh – Hadamard spectra, obtain restrictions on the number of representations as sums of squares of integers bounded in magnitude.","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42867249","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 : 2022-04-14DOI: 10.33581/2520-6508-2022-1-6-13
Anastasia I. Zhuk, Helena N. Zashchuk
Herein, we investigate systems of non-autonomous differential equations with generalised coefficients using the algebra of new generalised functions. We consider a system of non-autonomous differential equations with generalised coefficients as a system of equations in differentials in the algebra of new generalised functions. The solution of such a system is a new generalised function. It is shown that the different interpretations of the solutions of the given systems can be described by a unique approach of the algebra of new generalised functions. In this paper, for the first time in the literature, we describe associated solutions of the system of non-autonomous differential equations with generalised coefficients in the Lebesgue spaces Lp(T).
{"title":"On associated solutions of the system of non-autonomous differential equations in the Lebesgue spaces","authors":"Anastasia I. Zhuk, Helena N. Zashchuk","doi":"10.33581/2520-6508-2022-1-6-13","DOIUrl":"https://doi.org/10.33581/2520-6508-2022-1-6-13","url":null,"abstract":"Herein, we investigate systems of non-autonomous differential equations with generalised coefficients using the algebra of new generalised functions. We consider a system of non-autonomous differential equations with generalised coefficients as a system of equations in differentials in the algebra of new generalised functions. The solution of such a system is a new generalised function. It is shown that the different interpretations of the solutions of the given systems can be described by a unique approach of the algebra of new generalised functions. In this paper, for the first time in the literature, we describe associated solutions of the system of non-autonomous differential equations with generalised coefficients in the Lebesgue spaces Lp(T).","PeriodicalId":36323,"journal":{"name":"Zhurnal Belorusskogo Gosudarstvennogo Universiteta. Matematika. Informatika","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49629563","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}
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