M. Nafikov, I. Zagirov, K. Karimov, Valery N. Permyakov, I. R. Akhmet’yanov
An experimental mechanism for cleaning grain mixtures with a drive from a linear asynchronous electric motor has been developed and manufactured in the Bashkir State Agrarian University. More favorable conditions for sifting the grain mixture are created as a result of the use of an asynchronous linear electric motor for its drives in comparison with traditionally used machines driven by rotation motors. Based on the Lagrange equations, a mathematical description of the movement of the links of the mechanism is carried out and the electromagnetic force required in accordance with the cleaning technology is found. It is created in the windings of a linear electric motor by an electronic switch. It is found that the shape of the pulse of the required electromagnetic force is close to sinusoidal. However, it is possible to replace the electromotive force varying according to the sinusoidal law by a force constant in modulus and equal in impulse. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.
{"title":"Determination of electromagnetic force of a linear electric motor in the drive of a grain cleaning machine by modeling","authors":"M. Nafikov, I. Zagirov, K. Karimov, Valery N. Permyakov, I. R. Akhmet’yanov","doi":"10.17223/19988621/76/11","DOIUrl":"https://doi.org/10.17223/19988621/76/11","url":null,"abstract":"An experimental mechanism for cleaning grain mixtures with a drive from a linear asynchronous electric motor has been developed and manufactured in the Bashkir State Agrarian University. More favorable conditions for sifting the grain mixture are created as a result of the use of an asynchronous linear electric motor for its drives in comparison with traditionally used machines driven by rotation motors. Based on the Lagrange equations, a mathematical description of the movement of the links of the mechanism is carried out and the electromagnetic force required in accordance with the cleaning technology is found. It is created in the windings of a linear electric motor by an electronic switch. It is found that the shape of the pulse of the required electromagnetic force is close to sinusoidal. However, it is possible to replace the electromotive force varying according to the sinusoidal law by a force constant in modulus and equal in impulse. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78435329","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}
Research on the dynamic compression of geological materials is important for understanding composition and physical condition of the deep interior of the Earth and other planets. It also provides some data on the interaction processes related to the formation and evolution of planets. Magnesium silicates dominate in Earth's mantle and, thus, are expected to become the major phases in rocky exoplanets. In particular, enstatite Mg2[Si2O6] and forsterite Mg2SiO4 are essential constituents of Earth's mantles. Strong emphasis is put on the phase transition possibility for magnesium silicates under study. A remarkable fact is the dissociation of Mg2SiO4 into the following oxides: MgO and SiO2 (stishovite). The experiments have been carried out at a pressure value of 33 GPa, which corresponds to that in Earth's mantle at a depth of 1000 km. In this paper, the results of modeling the shock-wave loading of enstatite and forsterite as the mixtures of quartz SiO2 and periclase MgO are presented. The proposed model assumes that the components of the mixture under shock-wave loading are in thermodynamic equilibrium. The components of the material under study are considered in a phase transition region as a mixture of low- and high-pressure phases. The model is also valid for a polymorphic phase transition region. The calculations of magnesium silicates are performed with account for the polymorphic phase transition of quartz and periclase. The results are validated using the data obtained in dynamic experiments.
{"title":"Magnesium silicates at high dynamic loading","authors":"Konstantin K. Maevskiy","doi":"10.17223/19988621/79/10","DOIUrl":"https://doi.org/10.17223/19988621/79/10","url":null,"abstract":"Research on the dynamic compression of geological materials is important for understanding composition and physical condition of the deep interior of the Earth and other planets. It also provides some data on the interaction processes related to the formation and evolution of planets. Magnesium silicates dominate in Earth's mantle and, thus, are expected to become the major phases in rocky exoplanets. In particular, enstatite Mg2[Si2O6] and forsterite Mg2SiO4 are essential constituents of Earth's mantles. Strong emphasis is put on the phase transition possibility for magnesium silicates under study. A remarkable fact is the dissociation of Mg2SiO4 into the following oxides: MgO and SiO2 (stishovite). The experiments have been carried out at a pressure value of 33 GPa, which corresponds to that in Earth's mantle at a depth of 1000 km. In this paper, the results of modeling the shock-wave loading of enstatite and forsterite as the mixtures of quartz SiO2 and periclase MgO are presented. The proposed model assumes that the components of the mixture under shock-wave loading are in thermodynamic equilibrium. The components of the material under study are considered in a phase transition region as a mixture of low- and high-pressure phases. The model is also valid for a polymorphic phase transition region. The calculations of magnesium silicates are performed with account for the polymorphic phase transition of quartz and periclase. The results are validated using the data obtained in dynamic experiments.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85094703","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}
In 2018, Abdolyusefi, Ashrafi, and Chen gave a definition of a 2-nil-good ring element in their work, generalizing the notion of a graceful ring element introduced two years earlier by Kalugeryan and Lam, as well as the definition of a 2-nil-good ring. In the same work, it was shown that the Morita context ring, i.e. a formal matrix ring of the second order is 2-nil-good if the rings over which it is considered are themselves 2-nil-good. In this paper, we generalize further, defining k-nil-good elements and k-nil-good rings, and state a condition under which a formal matrix ring of an arbitrary finite order is k-nil-good.
{"title":"About fc-nil-good formal matrix rings","authors":"Tsyrendorzhi D. Norbosambuev, E. A. Timoshenko","doi":"10.17223/19988621/77/2","DOIUrl":"https://doi.org/10.17223/19988621/77/2","url":null,"abstract":"In 2018, Abdolyusefi, Ashrafi, and Chen gave a definition of a 2-nil-good ring element in their work, generalizing the notion of a graceful ring element introduced two years earlier by Kalugeryan and Lam, as well as the definition of a 2-nil-good ring. In the same work, it was shown that the Morita context ring, i.e. a formal matrix ring of the second order is 2-nil-good if the rings over which it is considered are themselves 2-nil-good. In this paper, we generalize further, defining k-nil-good elements and k-nil-good rings, and state a condition under which a formal matrix ring of an arbitrary finite order is k-nil-good.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76392522","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}
In this paper, a steady laminar non-isothermal flow of a power-law fluid in an axisymmetric sudden pipe expansion is numerically simulated. The rheological behavior of the fluid is described by the modified Ostwald-de Waele law; the apparent viscosity is an exponential function of temperature. The equations are written in terms of dimensionless stream function - vortex - temperature. No-slip conditions and zero temperature are used on the solid wall. At the inlet boundary, the velocity and temperature profiles correspond to a one-dimensional steady non-isothermal flow of the considered fluid. “Soft” boundary conditions are assigned at the outlet boundary. The formulated problem is solved using the finite-difference method. The structure of the flow through a sudden pipe expansion is shown to include one- and two-dimensional flow zones with a recirculation region occurring in the inner corner vicinity. The variation in the two-dimensional flow zone length is analyzed with respect to a power-law index and dimensionless criteria of the problem. Distributions of the velocity, temperature, and apparent viscosity are presented at various Peclet and Reynolds numbers for dilatant and pseudoplastic fluids.
{"title":"A numerical study of the effect of nonisothermality on the power-law fluid flow characteristics in a sudden pipe expansion","authors":"Dilara A. Mamazova, K. Ryltseva, G. Shrager","doi":"10.17223/19988621/79/11","DOIUrl":"https://doi.org/10.17223/19988621/79/11","url":null,"abstract":"In this paper, a steady laminar non-isothermal flow of a power-law fluid in an axisymmetric sudden pipe expansion is numerically simulated. The rheological behavior of the fluid is described by the modified Ostwald-de Waele law; the apparent viscosity is an exponential function of temperature. The equations are written in terms of dimensionless stream function - vortex - temperature. No-slip conditions and zero temperature are used on the solid wall. At the inlet boundary, the velocity and temperature profiles correspond to a one-dimensional steady non-isothermal flow of the considered fluid. “Soft” boundary conditions are assigned at the outlet boundary. The formulated problem is solved using the finite-difference method. The structure of the flow through a sudden pipe expansion is shown to include one- and two-dimensional flow zones with a recirculation region occurring in the inner corner vicinity. The variation in the two-dimensional flow zone length is analyzed with respect to a power-law index and dimensionless criteria of the problem. Distributions of the velocity, temperature, and apparent viscosity are presented at various Peclet and Reynolds numbers for dilatant and pseudoplastic fluids.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87768789","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}
The article is devoted to bistochastic quadratic operators. The concept of a bistochastic quadratic operator is introduced and the property of such operators is studied. A necessary and sufficient condition for bistochasticity is given. An analogue of Birkhoff s theorem is studied for the class of bistochastic quadratic operators. A sufficient condition for the extremity of bistochastic quadratic operators is obtained, and a necessary condition is also obtained for small dimensions. The concept of bistochastic quadratic operators is generalized, and the concept of a bistochastic operator of arbitrary order is introduced. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.
{"title":"Infinite distohastic square operators in li","authors":"A. Imomov, A. I. Eshniyazov","doi":"10.17223/19988621/76/2","DOIUrl":"https://doi.org/10.17223/19988621/76/2","url":null,"abstract":"The article is devoted to bistochastic quadratic operators. The concept of a bistochastic quadratic operator is introduced and the property of such operators is studied. A necessary and sufficient condition for bistochasticity is given. An analogue of Birkhoff s theorem is studied for the class of bistochastic quadratic operators. A sufficient condition for the extremity of bistochastic quadratic operators is obtained, and a necessary condition is also obtained for small dimensions. The concept of bistochastic quadratic operators is generalized, and the concept of a bistochastic operator of arbitrary order is introduced. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88914634","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}
This paper presents the results of modeling of unsteady fluid filtration in the formation penetrated by a well, which intersects a vertical hydraulic fracture throughout the entire thickness of the formation. The model of a bilinear fluid flow in a formation vertical hydraulic fracture system is considered in the case when the horizontal extent of the formation is considered to be infinite, and the fracture is of a finite length. The bilinearity of the flow means that in the formation - fracture system there are two mutually perpendicular fluid flows: from the formation to the fracture and along the fracture to the well. The analytical solution to a system of equations describing fluid filtration in the formation and fracture is obtained using the Laplace transform method. Analyzing the derived solution, main characteristic features of the filtration in the formation - fracture system are determined.
{"title":"Modeling of unsteady filtration in a formation - hydraulic fracture system","authors":"I. L. Khabibullin, Artur A. Khisamov","doi":"10.17223/19988621/77/12","DOIUrl":"https://doi.org/10.17223/19988621/77/12","url":null,"abstract":"This paper presents the results of modeling of unsteady fluid filtration in the formation penetrated by a well, which intersects a vertical hydraulic fracture throughout the entire thickness of the formation. The model of a bilinear fluid flow in a formation vertical hydraulic fracture system is considered in the case when the horizontal extent of the formation is considered to be infinite, and the fracture is of a finite length. The bilinearity of the flow means that in the formation - fracture system there are two mutually perpendicular fluid flows: from the formation to the fracture and along the fracture to the well. The analytical solution to a system of equations describing fluid filtration in the formation and fracture is obtained using the Laplace transform method. Analyzing the derived solution, main characteristic features of the filtration in the formation - fracture system are determined.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87424396","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}
In this paper, we study various functionals of convex hulls generated by the realization of a homogeneous Poisson point process in a polygon on a plane. The convex hull is a generalization of the extreme elements of the sample when estimating the carrier of the distribution; and in the multidimensional case, as an estimate of the carrier of the distribution, it retains many properties of one-dimensional estimates, such as consistency, asymptotic unbiasedness, and sufficiency. Works on the study of random convex hulls in polygons and various functionals of them are usually referred to the field of probabilistic geometry. It should be noted that studying the properties of even the simplest functionals of convex hulls, such as the number of vertices or the area, is not an easy problem (see, for example, [1-4]). This also explains the fact that before the appearance of the work of P. Groeneboom [6], the main progress in this field was achieved only in the study of the properties of the mean values of such functionals. In [6], he succeeded in proving the central limit theorem for the number of vertices of a convex hull in the case when the support of the original uniform distribution is either a convex polygon or an ellipse. The main result of this paper consists in proving that the difference between the perimeters of the distribution carrier and the convex hull converges in probability to a random variable that has a distribution different from normal, and it is asymptotically independent of the number of vertices and the area of the convex hull.
{"title":"The limit distribution of the perimeter of a convex hull generated by a Poisson point process in a convex polygon","authors":"I. Khamdamov, Z. S. Chay, L. Sharipova","doi":"10.17223/19988621/79/4","DOIUrl":"https://doi.org/10.17223/19988621/79/4","url":null,"abstract":"In this paper, we study various functionals of convex hulls generated by the realization of a homogeneous Poisson point process in a polygon on a plane. The convex hull is a generalization of the extreme elements of the sample when estimating the carrier of the distribution; and in the multidimensional case, as an estimate of the carrier of the distribution, it retains many properties of one-dimensional estimates, such as consistency, asymptotic unbiasedness, and sufficiency. Works on the study of random convex hulls in polygons and various functionals of them are usually referred to the field of probabilistic geometry. It should be noted that studying the properties of even the simplest functionals of convex hulls, such as the number of vertices or the area, is not an easy problem (see, for example, [1-4]). This also explains the fact that before the appearance of the work of P. Groeneboom [6], the main progress in this field was achieved only in the study of the properties of the mean values of such functionals. In [6], he succeeded in proving the central limit theorem for the number of vertices of a convex hull in the case when the support of the original uniform distribution is either a convex polygon or an ellipse. The main result of this paper consists in proving that the difference between the perimeters of the distribution carrier and the convex hull converges in probability to a random variable that has a distribution different from normal, and it is asymptotically independent of the number of vertices and the area of the convex hull.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89651639","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}
This work is related to one of the methods of medical imaging - electrical impedance tomography (EIT). A feature of the considered two-dimensional mathematical formulation for the direct EIT problem is the use of an elliptic type equation with piecewise constant coefficients and a special integro-differential boundary condition at the contact boundary of the electrodes. For an approximate solution of the problem under consideration, a numerical method was developed using unstructured grids, the finite volume method for barycentric cells, and the Gaussian elimination method with the choice of the main element. The validation of the method was performed on a known analytical solution.
{"title":"Numerical solution of the direct problem of electroimpedance tomography in a complete electrode formulation","authors":"Anna A. Afanas’eva, A. Starchenko","doi":"10.17223/19988621/78/1","DOIUrl":"https://doi.org/10.17223/19988621/78/1","url":null,"abstract":"This work is related to one of the methods of medical imaging - electrical impedance tomography (EIT). A feature of the considered two-dimensional mathematical formulation for the direct EIT problem is the use of an elliptic type equation with piecewise constant coefficients and a special integro-differential boundary condition at the contact boundary of the electrodes. For an approximate solution of the problem under consideration, a numerical method was developed using unstructured grids, the finite volume method for barycentric cells, and the Gaussian elimination method with the choice of the main element. The validation of the method was performed on a known analytical solution.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84169478","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}
Left-invariant para-complex structures on six-dimensional nilpotent Lie groups are considered. A complete list of six-dimensional nilpotent Lie groups that admit para-Kähler structures is obtained, explicit expressions for para-complex structures are found, and curvature properties of associated para-Kähler metrics are investigated. It is shown that paracomplex structures are nilpotent and the corresponding para-Kähler metrics are Ricci-flat.
{"title":"Left-invariant para-Kahler structures on six-dimensional nilpotent Lie groups","authors":"N. K. Smolentsev","doi":"10.17223/19988621/78/3","DOIUrl":"https://doi.org/10.17223/19988621/78/3","url":null,"abstract":"Left-invariant para-complex structures on six-dimensional nilpotent Lie groups are considered. A complete list of six-dimensional nilpotent Lie groups that admit para-Kähler structures is obtained, explicit expressions for para-complex structures are found, and curvature properties of associated para-Kähler metrics are investigated. It is shown that paracomplex structures are nilpotent and the corresponding para-Kähler metrics are Ricci-flat.","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81708306","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}
The purpose of the work is to show the usefulness and features of experimental mathematics. Two number theory problems are solved using Wolfram Mathematica. The solution to the first problem has already been published. Congruencies of the form F(A(p)) ≡ εF(S) (mod p) by prime modulo p are proved, whenever A(p) is a polynomial respect p with integer coefficients. Here, F(n) is an nth Fibonacci number, ε is 1 or -1, and S is a simple expression which contains only coefficients of the polynomial A(p). The second problem examines the behavior of prime gaps. It is proved that if G is the set of all prime gaps whose length is a multiple of 6, the asymptotic density of G is ½. The first study is mentioned to compare the role of experimentation for these two tasks. In the first study, experiments were necessary - they helped, starting with known facts, to formulate chains of reliable guesses which turned out to be easy to prove. In the second study, it was not certain that the calculations being done could lead to anything. It was possible to arrive at the formulation of a theorem on the value of ½ for the limit without experimental calculations. Only a conjecture about the formulation of the theorem is required. However, the experiments additionally led to a hypothesis on how the passage to the limit is implemented for the first 80 million primes. AMS Mathematical Subject Classification: MSC 11A41, 11A07, 11B39
{"title":"Experimental mathematics and its use in number theory","authors":"V. M. Zyuz'kov","doi":"10.17223/19988621/75/2","DOIUrl":"https://doi.org/10.17223/19988621/75/2","url":null,"abstract":"The purpose of the work is to show the usefulness and features of experimental mathematics. Two number theory problems are solved using Wolfram Mathematica. The solution to the first problem has already been published. Congruencies of the form F(A(p)) ≡ εF(S) (mod p) by prime modulo p are proved, whenever A(p) is a polynomial respect p with integer coefficients. Here, F(n) is an nth Fibonacci number, ε is 1 or -1, and S is a simple expression which contains only coefficients of the polynomial A(p). The second problem examines the behavior of prime gaps. It is proved that if G is the set of all prime gaps whose length is a multiple of 6, the asymptotic density of G is ½. The first study is mentioned to compare the role of experimentation for these two tasks. In the first study, experiments were necessary - they helped, starting with known facts, to formulate chains of reliable guesses which turned out to be easy to prove. In the second study, it was not certain that the calculations being done could lead to anything. It was possible to arrive at the formulation of a theorem on the value of ½ for the limit without experimental calculations. Only a conjecture about the formulation of the theorem is required. However, the experiments additionally led to a hypothesis on how the passage to the limit is implemented for the first 80 million primes. AMS Mathematical Subject Classification: MSC 11A41, 11A07, 11B39","PeriodicalId":43729,"journal":{"name":"Vestnik Tomskogo Gosudarstvennogo Universiteta-Matematika i Mekhanika-Tomsk State University Journal of Mathematics and Mechanics","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80560714","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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