In the present paper, based on the ideas of Algerian physicist M.E. Hassani, the generalizedHassani spatial-temporal transformations in real Hilbert space are introduced. The originaltransformations, introduced by M.E. Hassani, are the particular cases of the transformations,introduced in this paper. It is proven that the classes of generalized Hassani transforms donot form a group of operators in the general case. Further, using these generalized Hassanitransformations as well as the theory of changeable sets and universal kinematics, the mathematicallystrict models of Hassani kinematics are constructed and the performance of the relativityprinciple in these models is discussed.
{"title":"On some properties of Hassani transforms","authors":"Y. Grushka","doi":"10.30970/ms.57.1.79-91","DOIUrl":"https://doi.org/10.30970/ms.57.1.79-91","url":null,"abstract":"In the present paper, based on the ideas of Algerian physicist M.E. Hassani, the generalizedHassani spatial-temporal transformations in real Hilbert space are introduced. The originaltransformations, introduced by M.E. Hassani, are the particular cases of the transformations,introduced in this paper. It is proven that the classes of generalized Hassani transforms donot form a group of operators in the general case. Further, using these generalized Hassanitransformations as well as the theory of changeable sets and universal kinematics, the mathematicallystrict models of Hassani kinematics are constructed and the performance of the relativityprinciple in these models is discussed.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42964574","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}
When Loday and Ronco studied ternary planar trees, they introduced types of algebras,called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associativeoperations satisfying additional eight axioms relating these operations, while a trialgebra is justa linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notionof a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are ageneralization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids andtrialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraictopology. One of the most useful concepts in algebra is the free object. Every variety containsfree algebras and free objects in any variety of algebras are important in the study of thatvariety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, thefree rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid wereconstructed and the least dimonoid congruences as well as the least semigroup congruence onthe first four free algebras were characterized. However, just mentioned congruences on freeleft (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, wecharacterize the least dimonoid congruences and the least semigroup congruence on free left(right) n-trinilpotent trioids and free abelian trioids.
{"title":"The least dimonoid congruences on relatively free trioids","authors":"A. Zhuchok","doi":"10.30970/ms.57.1.23-31","DOIUrl":"https://doi.org/10.30970/ms.57.1.23-31","url":null,"abstract":"When Loday and Ronco studied ternary planar trees, they introduced types of algebras,called trioids and trialgebras. A trioid is a nonempty set equipped with three binary associativeoperations satisfying additional eight axioms relating these operations, while a trialgebra is justa linear analog of a trioid. If all operations of a trioid (trialgebra) coincide, we obtain the notionof a semigroup (associative algebra), and if two concrete operations of a trioid (trialgebra)coincide, we obtain the notion of a dimonoid (dialgebra) and so, trioids (trialgebras) are ageneralization of semigroups (associative algebras) and dimonoids (dialgebras). Trioids andtrialgebras have close relationships with the Hopf algebras, the Leibniz 3-algebras, the Rota-Baxter operators, and the post-Jordan algebras. Originally, these structures arose in algebraictopology. One of the most useful concepts in algebra is the free object. Every variety containsfree algebras and free objects in any variety of algebras are important in the study of thatvariety. Loday and Ronco constructed the free trioid of rank 1 and the free trialgebra. Recently,the free trioid of an arbitrary rank, the free commutative trioid, the free n-nilpotent trioid, thefree rectangular triband, the free left n-trinilpotent trioid and the free abelian trioid wereconstructed and the least dimonoid congruences as well as the least semigroup congruence onthe first four free algebras were characterized. However, just mentioned congruences on freeleft (right) n-trinilpotent trioids and free abelian trioids were not considered. In this paper, wecharacterize the least dimonoid congruences and the least semigroup congruence on free left(right) n-trinilpotent trioids and free abelian trioids.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42084488","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 study's primary purpose is to investigate the $mathscr{A}/mathscr{T}$ structure of a quotient ring, where $mathscr{A}$ is an arbitrary ring and $mathscr{T}$ is a semi-prime ideal of $mathscr{A}$. In more details, we look at the differential identities in a semi-prime ideal of an arbitrary ring using $mathscr{T}$-commuting generalized derivation. The article proves a number of statements. A characteristic representative of these assertions is, for example, the following Theorem 3: Let $mathscr{A}$ be a ring with $mathscr{T}$ a semi-prime ideal and $mathscr{I}$ an ideal of $mathscr{A}.$ If $(lambda, psi)$ is a non-zero generalized derivation of $mathscr{A}$ and the derivation satisfies any one of the conditions:1) $lambda([a, b])pm[a, psi(b)]in mathscr{T}$, 2) $lambda(acirc b)pm acirc psi(b)in mathscr{T}$,$forall$ $a, bin mathscr{I},$ then $psi$ is $mathscr{T}$-commuting on $mathscr{I}.$ Furthermore, examples are provided to demonstrate that the constraints placed on the hypothesis of the various theorems were not unnecessary.
{"title":"$mathscr{T}$-Commuting Generalized Derivations on Ideals and Semi-Prime Ideal-II","authors":"N. Rehman, Hafedh M. Alnoghashi","doi":"10.30970/ms.57.1.98-110","DOIUrl":"https://doi.org/10.30970/ms.57.1.98-110","url":null,"abstract":"The study's primary purpose is to investigate the $mathscr{A}/mathscr{T}$ structure of a quotient ring, where $mathscr{A}$ is an arbitrary ring and $mathscr{T}$ is a semi-prime ideal of $mathscr{A}$. In more details, we look at the differential identities in a semi-prime ideal of an arbitrary ring using $mathscr{T}$-commuting generalized derivation. The article proves a number of statements. A characteristic representative of these assertions is, for example, the following Theorem 3: Let $mathscr{A}$ be a ring with $mathscr{T}$ a semi-prime ideal and $mathscr{I}$ an ideal of $mathscr{A}.$ If $(lambda, psi)$ is a non-zero generalized derivation of $mathscr{A}$ and the derivation satisfies any one of the conditions:1) $lambda([a, b])pm[a, psi(b)]in mathscr{T}$, 2) $lambda(acirc b)pm acirc psi(b)in mathscr{T}$,$forall$ $a, bin mathscr{I},$ then $psi$ is $mathscr{T}$-commuting on $mathscr{I}.$ \u0000Furthermore, examples are provided to demonstrate that the constraints placed on the hypothesis of the various theorems were not unnecessary.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41623577","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 deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$ $R$ is a ring of stable range 1; $2.$ $R$ is a ring of von Neumann regular range 1. The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).
{"title":"Stable range conditions for abelian and duo rings","authors":"A. Dmytruk, A. Gatalevych, M. Kuchma","doi":"10.30970/ms.57.1.92-97","DOIUrl":"https://doi.org/10.30970/ms.57.1.92-97","url":null,"abstract":"The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$ $R$ is a ring of stable range 1; $2.$ $R$ is a ring of von Neumann regular range 1. \u0000The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41418255","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}
Denote by $mathfrak{D}_0$ a class of absolutely convergent in half-plane $Pi_0={scolon text{Re},s<0}$ Dirichlet series$F(s)=e^{sh}-sum_{k=1}^{infty}f_kexp{s(lambda_k+h)},, s=sigma+it$, where $h> 0$, $h0$.For $0lealphafrac{alpha}{h}$,and belongs to the class $mathfrak{DG}_h(l,alpha)$ if and only if$text{Re}{e^{-hs}((1-l)F'(s)+frac{l}{h}F''(s))}>alpha$ for all $sin Pi_0$. It is provedthat $Fin mathfrak{DF}_h(l,alpha)$ if and only if $ sum_{k=1}^{infty}(h+llambda_k)f_kle h-alpha$, and$Fin mathfrak{DG}_h(l,alpha)$ if and only if $sum_{k=1}^{infty}(h+llambda_k)(lambda_k+h)f_kle h(h-alpha)$. If $F_jin mathfrak{DF}_h(l_j,alpha_j)$, $j=1, 2$, where $l_jge0$ and $0le alpha_j0$ and $delta>0$ the neighborhood of the function $Fin mathfrak{D}_0$ is defined as follows$O_{j,delta}(F)={G(s)=e^{s}-sum_{k=1}^{infty}g_kexp{slambda_k}in mathfrak{D}_0colon sum_{k=1}^{infty}lambda^j_k|g_k-f_k|ledelta}$. It is described the neighborhoods of functions from classes $mathfrak{DF}_h(l,alpha)$ and $mathfrak{DG}_h(l,alpha)$. Conditions on real parameters $gamma_0,,gamma_1,,gamma_2,,a_1$ and $a_2$ of the differential equation $w''+(gamma_0e^{2hs}+gamma_1e^{hs}+gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solutioneither in $mathfrak{DF}_h(l,alpha)$ or in $mathfrak{DG}_h(l,alpha)$.
{"title":"On certain subclass of Dirichlet series absolutely convergent in half-plane","authors":"M. Sheremeta","doi":"10.30970/ms.57.1.32-44","DOIUrl":"https://doi.org/10.30970/ms.57.1.32-44","url":null,"abstract":"Denote by $mathfrak{D}_0$ a class of absolutely convergent in half-plane $Pi_0={scolon text{Re},s<0}$ Dirichlet series$F(s)=e^{sh}-sum_{k=1}^{infty}f_kexp{s(lambda_k+h)},, s=sigma+it$, where $h> 0$, $h<lambda_kuparrow+infty$ and $f_k>0$.For $0lealpha<h$ and $lge 0$ we say that $F$ belongs to the class $mathfrak{DF}_h(l,alpha)$ if and only if$text{Re}{e^{-hs}((1-l)F(s)+frac{l}{h}F'(s))}>frac{alpha}{h}$,and belongs to the class $mathfrak{DG}_h(l,alpha)$ if and only if$text{Re}{e^{-hs}((1-l)F'(s)+frac{l}{h}F''(s))}>alpha$ for all $sin Pi_0$. It is provedthat $Fin mathfrak{DF}_h(l,alpha)$ if and only if $ sum_{k=1}^{infty}(h+llambda_k)f_kle h-alpha$, and$Fin mathfrak{DG}_h(l,alpha)$ if and only if $sum_{k=1}^{infty}(h+llambda_k)(lambda_k+h)f_kle h(h-alpha)$. \u0000If $F_jin mathfrak{DF}_h(l_j,alpha_j)$, $j=1, 2$, where $l_jge0$ and $0le alpha_j<h$, then Hadamard composition$(F_1*F_2)in mathfrak{D}F_h(l,alpha)$, where $l=min{l_1,l_2}$ and$alpha=h-frac{(h-alpha_1)(h-alpha_2)}{h+llambda_1}$. Similar statement is correct for the class $F_jin mathfrak{DG}_h(l,alpha)$. \u0000For $j>0$ and $delta>0$ the neighborhood of the function $Fin mathfrak{D}_0$ is defined as follows$O_{j,delta}(F)={G(s)=e^{s}-sum_{k=1}^{infty}g_kexp{slambda_k}in mathfrak{D}_0colon sum_{k=1}^{infty}lambda^j_k|g_k-f_k|ledelta}$. It is described the neighborhoods of functions from classes $mathfrak{DF}_h(l,alpha)$ and $mathfrak{DG}_h(l,alpha)$. \u0000Conditions on real parameters $gamma_0,,gamma_1,,gamma_2,,a_1$ and $a_2$ of the differential equation $w''+(gamma_0e^{2hs}+gamma_1e^{hs}+gamma_2) w=a_1e^{hs}+a_2e^{2hs}$ are found, under which this equation has a solutioneither in $mathfrak{DF}_h(l,alpha)$ or in $mathfrak{DG}_h(l,alpha)$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42605731","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 $A$ and $B$ be given linear operators on Banach spaces $X$ and $Y$, we denote by $M_C$ the operator defined on $X oplus Y$ by $M_{C}=begin{pmatrix}A & C 0 & B%end{pmatrix}.$In this paper, we study an abstract boundaryvalue matrix problems with a spectral parameter described by Drazin invertibile operators of the form $$begin{cases}U_L=lambda M_{C}w+F, & Gamma w=Phi, & end{cases}%$$where $U_L , M_C$ are upper triangular operators matrices $(2times 2)$ acting in Banach spaces, $Gamma$ is boundary operator, $F$ and $Phi $ are given vectors and $lambda $ is a complex spectral parameter.We introduce theconcept of initial boundary operators adapted to the Drazin invertibility andwe present a spectral approach for solving the problem. It can be shown thatthe considered boundary value problems are uniquely solvable and that theirsolutions are explicitly calculated. As an application we give an example to illustrate our results.
{"title":"Boundary value matrix problems and Drazin invertible operators","authors":"K. Miloud Hocine","doi":"10.30970/ms.57.1.16-22","DOIUrl":"https://doi.org/10.30970/ms.57.1.16-22","url":null,"abstract":"Let $A$ and $B$ be given linear operators on Banach spaces $X$ and $Y$, we denote by $M_C$ the operator defined on $X oplus Y$ by $M_{C}=begin{pmatrix}A & C 0 & B%end{pmatrix}.$In this paper, we study an abstract boundaryvalue matrix problems with a spectral parameter described by Drazin invertibile operators of the form $$begin{cases}U_L=lambda M_{C}w+F, & Gamma w=Phi, & end{cases}%$$where $U_L , M_C$ are upper triangular operators matrices $(2times 2)$ acting in Banach spaces, $Gamma$ is boundary operator, $F$ and $Phi $ are given vectors and $lambda $ is a complex spectral parameter.We introduce theconcept of initial boundary operators adapted to the Drazin invertibility andwe present a spectral approach for solving the problem. It can be shown thatthe considered boundary value problems are uniquely solvable and that theirsolutions are explicitly calculated. As an application we give an example to illustrate our results.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41708523","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 introduce universally prestarlikegeneralized functions of order $vartheta $ with $vartheta leq 1$ associated with shell like domain, and we getcoefficient bounds and the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ forsuch functions.
{"title":"Universally prestarlike functions associated with shell like domain","authors":"K. Vijaya, G. Murugusundaramoorthy, S. Yalçın","doi":"10.30970/ms.57.1.53-61","DOIUrl":"https://doi.org/10.30970/ms.57.1.53-61","url":null,"abstract":"In this paper, we introduce universally prestarlikegeneralized functions of order $vartheta $ with $vartheta leq 1$ associated with shell like domain, and we getcoefficient bounds and the second Hankel determinant $|a_{2}a_{4}-a_{3}^{2}|$ forsuch functions.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44862242","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 : 2021-12-27DOI: 10.30970/ms.56.2.176-184
S. Bak, G. Kovtonyuk
We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_Theta$. We suppose that the weight $Theta$ satisfies some regularity assumption. Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2left((-T, T); l^2)$ belongs to $C^2left((-T, T); l^2_Thetaright)$. And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.
{"title":"Well-posedness of the Cauchy problem for system of oscillators on 2D–lattice in weighted $l^2$-spaces","authors":"S. Bak, G. Kovtonyuk","doi":"10.30970/ms.56.2.176-184","DOIUrl":"https://doi.org/10.30970/ms.56.2.176-184","url":null,"abstract":"We consider an infinite system of ordinary differential equations that describes the dynamics of an infinite system of \u0000linearly coupled nonlinear oscillators on a two dimensional integer-valued lattice. It is assumed that each oscillator \u0000interacts linearly with its four nearest neighbors and the oscillators are at the rest at infinity. We study the initial value problem (the Cauchy problem) for such system. This system naturally can be considered as an operator-differential equation \u0000in the Hilbert, or even Banach, spaces of sequences. We note that $l^2$ is the simplest choice of such spaces. With this choice of the configuration space, the phase space is $l^2times l^2$, and the equation can be written in the Hamiltonian form with the Hamiltonian $H$. Recall that from a physical point of view the Hamiltonian represents the full energy of the system, i.e., the sum of kinetic and potential energy. Note that the Hamiltonian $H$ is a conserved quantity, i.e., for any solution of equation the Hamiltonian is constant. For this space, there are some results on the global solvability of the corresponding Cauchy problem. In the present paper, results on the $l^2$-well-posedness are extended to weighted $l^2$-spaces $l^2_Theta$. We suppose that the weight $Theta$ satisfies some regularity assumption. \u0000Under some assumptions for nonlinearity and coefficients of the equation, we prove that every solution of the Cauchy problem from $C^2left((-T, T); l^2)$ belongs to $C^2left((-T, T); l^2_Thetaright)$. \u0000And we obtain the results on existence of a unique global solutions of the Cauchy problem for system of oscillators on a two-dimensional lattice in a wide class of weighted $l^2$-spaces. These results can be applied to discrete sine-Gordon type equations and discrete Klein-Gordon type equations on a two-dimensional lattice. In particular, the Cauchy problems for these equations are globally well-posed in every weighted $l^2$-space with a regular weight.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43175815","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 : 2021-12-27DOI: 10.30970/ms.56.2.215-217
A. Ravsky
We found a solution of Henry E. Dudeney’s star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which was claimed impossible by the puzzle author. Generalizing this result to other board sizes, we obtained bounds on minimal number of moves in a board filling queen path with given source and destination.
{"title":"A counterexample to Henry E. Dudeney’s star puzzle","authors":"A. Ravsky","doi":"10.30970/ms.56.2.215-217","DOIUrl":"https://doi.org/10.30970/ms.56.2.215-217","url":null,"abstract":"We found a solution of Henry E. Dudeney’s star puzzle (a path on a chessboard from c5 to d4 in 14 straight strokes) in 14 queen moves, which was claimed impossible by the puzzle author. Generalizing this result to other board sizes, we obtained bounds on minimal number of moves in a board filling queen path with given source and destination.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42011114","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 : 2021-12-27DOI: 10.30970/ms.56.2.185-192
Y. Baranetskij, I. Demkiv, M. Kopach, A. Solomko
Some approaches to the construction of interpolation rational integral approximations with arbitrary multiplicity of nodes are analyzed. An integral rational Hermitian-type interpolant of the third order on a continual set of nodes, which is the ratio of a functional polynomial of the first degree to a functional polynomial of the second degree, is constructed and investigated. The resulting interpolant is one that holds any rational functional of the resulting form. Проаналізовано ряд підходів до побудови інтерполяційних раціональних інтегральних наближень з довільною кратністю вузлів. Будується та досліджується інтегральний раціональний інтерполянт типу Ерміта третього порядку на континуальній множині вузлів, який є відношенням функціонального полінома першого степеня до функціонального полінома другого степеня. Одержаний інтерполянт є таким, що зберігає будь який раціональний функціонал одержаного вигляду.
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