Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition [frac{1}{T}int_{0}^{T}u(x,t)eta_1(t)dt=Phi_1(x), ;;;xin Bbb R^n] where $u$ is the unknown solution of the Cauchy problem, $eta_1$ and $Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.
{"title":"REGULAR SOLUTION OF THE INVERSE PROBLEM WITH INTEGRAL CONDITION FOR A TIME-FRACTIONAL EQUATION","authors":"H. Lopushanska, A. Lopushansky","doi":"10.31861/bmj2020.02.09","DOIUrl":"https://doi.org/10.31861/bmj2020.02.09","url":null,"abstract":"Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known.\u0000\u0000We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them.\u0000\u0000We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition\u0000[frac{1}{T}int_{0}^{T}u(x,t)eta_1(t)dt=Phi_1(x), ;;;xin Bbb R^n]\u0000where $u$ is the unknown solution of the Cauchy problem, $eta_1$ and $Phi_1$ are the given functions.\u0000\u0000Using the method of the Green's vector function,\u0000we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability.\u0000\u0000There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128618448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal control in a nonlocal boundary value problem with integral conditions for parabolic equation with degeneration","authors":"I. Pukal’skii, B. Yashan","doi":"10.31861/bmj2019.01.082","DOIUrl":"https://doi.org/10.31861/bmj2019.01.082","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116382797","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 problems for Eidelman type equations and systems of equations are considered in this paper. They were the large part of scientific interests for Prof. Ivasyshen S.D. The results of investigations of Cauchy problem, initial-boundary and the inverse problems for this type of equations in bounded or unbounded domains are given. The results are represented as the estimates of the solutions, the integral representations of solutions, theorems of the existence, uniqueness and stability of solutions.
{"title":"ON PROBLEMS FOR EIDELMAN TYPE EQUATIONS AND SYSTEM OF EQUATIONS","authors":"N. Protsakh, H. Ivasiuk, T. Fratavchan","doi":"10.31861/bmj2022.02.17","DOIUrl":"https://doi.org/10.31861/bmj2022.02.17","url":null,"abstract":"The problems for Eidelman type equations and systems of equations are considered in this paper. They were the large part of scientific interests for Prof. Ivasyshen S.D. The results of investigations of Cauchy problem, initial-boundary and the inverse problems for this type of equations in bounded or unbounded domains are given. The results are represented as the estimates of the solutions, the integral representations of solutions, theorems of the existence, uniqueness and stability of solutions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132980987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"WEAK R-SPACES AND UNIFORM LIMIT OF SEQUENCES OF THE FIRST BAIRE CLASS FUNCTIONS","authors":"Mykhaylo Lukan, O. Karlova","doi":"10.31861/bmj2019.02.039","DOIUrl":"https://doi.org/10.31861/bmj2019.02.039","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124015115","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}
Let $f$ be an entire function with $f(0)=1$, $(lambda_n)_{ninmathbb N}$ be the sequence of its zeros, $n(t)=sum_{|lambda_n|le t}1$, $N(r)=int_0^r t^{-1}n(t), dt$, $r>0$, $h(varphi)$ be the indicator of $f$, and $F(z)=zf'(z)/f(z)$, $z=re^{ivarphi}$. An entire function $f$ is called a function of improved regular growth if for some $rhoin (0,+infty)$ and $rho_1in (0,rho)$, and a $2pi$-periodic $rho$-trigonometrically convex function $h(varphi)notequiv -infty$ there exists a set $Usubsetmathbb C$ contained in the union of disks with finite sum of radii and such that begin{equation*} log |{f(z)}|=|z|^rho h(varphi)+o(|z|^{rho_1}),quad Unotni z=re^{ivarphi}toinfty. end{equation*} In this paper, we prove that an entire function $f$ of order $rhoin (0,+infty)$ with zeros on a finite system of rays ${z: arg z=psi_{j}}$, $jin{1,ldots,m}$, $0lepsi_1
{"title":"ASYMPTOTIC BEHAVIOR OF THE LOGARITHMIC DERIVATIVE OF ENTIRE FUNCTION OF IMPROVED REGULAR GROWTH IN THE METRIC OF $L^q[0,2pi]$","authors":"R. Khats","doi":"10.31861/bmj2021.01.04","DOIUrl":"https://doi.org/10.31861/bmj2021.01.04","url":null,"abstract":"Let $f$ be an entire function with $f(0)=1$, $(lambda_n)_{ninmathbb N}$ be the sequence of its zeros, $n(t)=sum_{|lambda_n|le t}1$, $N(r)=int_0^r t^{-1}n(t), dt$, $r>0$, $h(varphi)$ be the indicator of $f$, and $F(z)=zf'(z)/f(z)$, $z=re^{ivarphi}$. An entire function $f$ is called a function of improved regular growth if for some $rhoin (0,+infty)$ and $rho_1in (0,rho)$, and a $2pi$-periodic $rho$-trigonometrically convex function $h(varphi)notequiv -infty$ there exists a set $Usubsetmathbb C$ contained in the union of disks with finite sum of radii and such that\u0000begin{equation*}\u0000log |{f(z)}|=|z|^rho h(varphi)+o(|z|^{rho_1}),quad Unotni z=re^{ivarphi}toinfty.\u0000end{equation*}\u0000In this paper, we prove that an entire function $f$ of order $rhoin (0,+infty)$ with zeros on a finite system of rays ${z: arg z=psi_{j}}$, $jin{1,ldots,m}$, $0lepsi_1<psi_2<ldots<psi_m<2pi$, is a function of improved regular growth if and only if for some $rho_3in (0,rho)$\u0000begin{equation*}\u0000N(r)=c_0r^rho+o(r^{rho_3}),quad rto +infty,quad c_0in [0,+infty),\u0000end{equation*}\u0000and for some $rho_2in (0,rho)$ and any $qin [1,+infty)$, one has\u0000begin{equation*}\u0000left{frac{1}{2pi}int_0^{2pi}left|frac{Im F(re^{ivarphi})}{r^rho}+h'(varphi)right|^q, dvarphiright}^{1/q}=o(r^{rho_2-rho}),quad rto +infty.\u0000end{equation*}","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128971888","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}
Parabolicity in the sense of both Petrosky and Shilov has a scalar character. It is not able to take into account the specificity of the heterogeneity of the environment. In this regard, in the early 70-s, S.D. Eidelman proposed the so-called $vec{2b}$-parabolicity, which is a natural generalization of the Petrovsky parabolicity for the case of an anisotropic medium. A detailed study of the Cauchy problem for equations with such parabolicity was carried out in the works of S.D. Eidelman, S.D. Ivasishena, M.I. Matiichuk and their students. An extension of parabolicity according to Shilov for the case of anisotropic media is ${vec{p},vec h}$-parabolicity. The class of equations with such parabolicity is quite broad, it includes the classes of Eidelman, Petrovskii, and Shilov and allows unifying the classical theory of the Cauchy problem for parabolic equations. In this work, for inhomogeneous ${vec{p},vec h}$-parabolic equations with vector positive genus, the conditions under which the Cauchy problem in the class of generalized initial functions of the type of Gelfand and Shilov distributions will be correctly solvable are investigated. At the same time, the inhomogeneities of the equations are continuous functions of finite smoothness with respect to the set of variables, which decrease with respect to the spatial variable, and are unbounded with the integrable feature with respect to the time variable.
{"title":"INHOMOGENEOUS DIFFERENTIAL EQUATIONS OF VECTOR ORDER WITH DISSIPATIVE PARABOLICITY AND POSITIVE GENUS","authors":"V. Litovchenko, M. Gorbatenko","doi":"10.31861/bmj2022.02.10","DOIUrl":"https://doi.org/10.31861/bmj2022.02.10","url":null,"abstract":"Parabolicity in the sense of both Petrosky and Shilov has a scalar character. It is not able to take into account the specificity of the heterogeneity of the environment. In this regard, in the early 70-s, S.D. Eidelman proposed the so-called $vec{2b}$-parabolicity, which is a natural generalization of the Petrovsky parabolicity for the case of an anisotropic medium. A detailed study of the Cauchy problem for equations with such parabolicity was carried out in the works of S.D. Eidelman, S.D. Ivasishena, M.I. Matiichuk and their students.\u0000\u0000An extension of parabolicity according to Shilov for the case of anisotropic media is ${vec{p},vec h}$-parabolicity. The class of equations with such parabolicity is quite broad, it includes the classes of Eidelman, Petrovskii, and Shilov and allows unifying the classical theory of the Cauchy problem for parabolic equations.\u0000\u0000In this work, for inhomogeneous ${vec{p},vec h}$-parabolic equations with vector positive genus, the conditions under which the Cauchy problem in the class of generalized initial functions of the type of Gelfand and Shilov distributions will be correctly solvable are investigated. At the same time, the inhomogeneities of the equations are continuous functions of finite smoothness with respect to the set of variables, which decrease with respect to the spatial variable, and are unbounded with the integrable feature with respect to the time variable.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126628241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On some properties of solutions of linear differential equations according to given sequences","authors":"O. Shavala","doi":"10.31861/bmj2019.01.121","DOIUrl":"https://doi.org/10.31861/bmj2019.01.121","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132837587","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}
Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have the form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete logistic equation and Ricker's equation are most often used in practice. In the given paper, these equations are considered width taking into account an effect of harvesting, that is, the equations of the form below are studied $N_{t+1}=r N_t (1- N_t) - c$ and $N_{t+1}= N_t exp (r(1 - N_t / K )) - c$, where the parameters $r$, $K>0$, $c>0$ are harvesting intensity. Positive equilibrium points and conditions for their stability for these equations were found. These kinds of states are often realized in nature. For practice, periodic solutions are also important, especially with periods $T=2 (N_{t+2} = N_t)$ and $T=3 (N_{t+3} = N_t)$, since, with their existence, by Sharkovskii's theorem, one can do conclusions about the existence of periodic solutions of other periods. For the discrete logistic equation in analytical form, the values that make up the periodic solution with period $T=2$ were found. We used numerical methods in order to find solutions with period $T=3$. For Ricker's model, the question of the existence of periodic solutions can be investigated by computer analysis only. In the paper, a number of computer experiments were conducted in which periodic solutions were found and their stability was studied. For Ricker's model with harvesting, chaotic solutions were also found. As we can see, the study of difference equations gives many unexpected results.
{"title":"MODELING HARVESTING PROCESSES FOR POPULATIONS WITH NON-OVERLAPPING GENERATIONS","authors":"V. Matsenko","doi":"10.31861/bmj2022.02.12","DOIUrl":"https://doi.org/10.31861/bmj2022.02.12","url":null,"abstract":"Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time.\u0000\u0000In the simplest case such equations have the form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function.\u0000\u0000Among such equations the discrete logistic equation and Ricker's equation are most often used in practice.\u0000\u0000In the given paper, these equations are considered width taking into account an effect of harvesting, that is, the equations of the form below are studied $N_{t+1}=r N_t (1- N_t) - c$ and $N_{t+1}= N_t exp (r(1 - N_t / K )) - c$, where the parameters $r$, $K>0$, $c>0$ are harvesting intensity.\u0000\u0000Positive equilibrium points and conditions for their stability for these equations were found. These kinds of states are often realized in nature.\u0000\u0000For practice, periodic solutions are also important, especially with periods $T=2 (N_{t+2} = N_t)$ and $T=3 (N_{t+3} = N_t)$, since, with their existence, by Sharkovskii's theorem, one can do conclusions about the existence of periodic solutions of other periods.\u0000\u0000For the discrete logistic equation in analytical form, the values that make up the periodic solution with period $T=2$ were found. We used numerical methods in order to find solutions with period $T=3$. For Ricker's model, the question of the existence of periodic solutions can be investigated by computer analysis only.\u0000\u0000In the paper, a number of computer experiments were conducted in which periodic solutions were found and their stability was studied. For Ricker's model with harvesting, chaotic solutions were also found.\u0000\u0000As we can see, the study of difference equations gives many unexpected results.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132131958","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}
It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood later, that the question of existence of such trajectories is closely related to existence of invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.
{"title":"ON APPROXIMATION OF ALMOST-PERIODIC SOLUTIONS FOR A NON-LINEAR COUNTABLE SYSTEM OF DIFFERENTIAL EQUATIONS BY QUASI-PERIODIC SOLUTIONS FOR SOME LINEAR SYSTEM","authors":"Yuri V Teplinsky","doi":"10.31861/bmj2021.02.09","DOIUrl":"https://doi.org/10.31861/bmj2021.02.09","url":null,"abstract":"It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood\u0000later, that the question of existence of such trajectories is closely related to existence of \u0000invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131637504","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}
From an intuitive point of view universal kinematics are collections (sets) of changing objects, which evolve, being in a certain spatial-geometric environment, and evolution of whi- ch can be observed from many different frames of reference. Moreover, the definition of uni- versal kinematics impose the existence of some (preassigned) universal coordinate transform between every two reference frames of such kinematics. Transferable self-consistently translati- onal reference frames (in vector universal kinematics) are interesting because for such reference frames it is possible to give a clear and unambiguous definition of displacement of a moving reference frame relative to a fixed one, which does not depend on the choice of a fixed point in the moving frame of reference. In the present paper it is shown that an arbitrary reference frame m is transferable self-consistently translational relatively to a reference frame l (in some vector uni- versal kinematics F) if and only if the coordinate transform operator from the reference frame m to the reference frame l is transferable self-consistently translational. Therefore transferable self-consistently translational coordinate transform operators describe the conversion of coordi- nates from the moving and transferable self-consistently translational frame of reference to the (given) fixed frame in vector universal kinematics. Also in the paper it is described the structure of transferable self-consistently translational coordinate transform operators (this is the main result of the article). Using this result it have been obtained the necessary and sufficient conditi- on for transferable self-consistently translationality of one reference frame relatively to another in vector universal kinematics.
{"title":"THE CRITERION FOR TRANSFERABLE SELF-CONSISTENTLY TRANSLATIONALITY OF COORDINATE TRANSFORM OPERATORS AND REFERENCE FRAMES IN UNIVERSAL KINEMATICS","authors":"Y. Grushka","doi":"10.31861/bmj2021.01.10","DOIUrl":"https://doi.org/10.31861/bmj2021.01.10","url":null,"abstract":"From an intuitive point of view universal kinematics are collections (sets) of changing\u0000objects, which evolve, being in a certain spatial-geometric environment, and evolution of whi-\u0000ch can be observed from many different frames of reference. Moreover, the definition of uni-\u0000versal kinematics impose the existence of some (preassigned) universal coordinate transform\u0000between every two reference frames of such kinematics. Transferable self-consistently translati-\u0000onal reference frames (in vector universal kinematics) are interesting because for such reference\u0000frames it is possible to give a clear and unambiguous definition of displacement of a moving\u0000reference frame relative to a fixed one, which does not depend on the choice of a fixed point in the\u0000moving frame of reference. In the present paper it is shown that an arbitrary reference frame m\u0000is transferable self-consistently translational relatively to a reference frame l (in some vector uni-\u0000versal kinematics F) if and only if the coordinate transform operator from the reference frame\u0000m to the reference frame l is transferable self-consistently translational. Therefore transferable\u0000self-consistently translational coordinate transform operators describe the conversion of coordi-\u0000nates from the moving and transferable self-consistently translational frame of reference to the\u0000(given) fixed frame in vector universal kinematics. Also in the paper it is described the structure\u0000of transferable self-consistently translational coordinate transform operators (this is the main\u0000result of the article). Using this result it have been obtained the necessary and sufficient conditi-\u0000on for transferable self-consistently translationality of one reference frame relatively to another\u0000in vector universal kinematics.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133403438","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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