Pub Date : 2022-01-01DOI: 10.15330/cmp.14.2.543-563
Renato Fehlberg J'unior, I. Kaygorodov
We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [J. Algebra Appl. 2017, 16 (9), 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.
{"title":"On the Kantor product, II","authors":"Renato Fehlberg J'unior, I. Kaygorodov","doi":"10.15330/cmp.14.2.543-563","DOIUrl":"https://doi.org/10.15330/cmp.14.2.543-563","url":null,"abstract":"We describe the Kantor square (and Kantor product) of multiplications, extending the classification proposed in [J. Algebra Appl. 2017, 16 (9), 1750167]. Besides, we explicitly describe the Kantor square of some low dimensional algebras and give constructive methods for obtaining new transposed Poisson algebras and Poisson-Novikov algebras; and for classifying Poisson structures and commutative post-Lie structures on a given algebra.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90191251","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 : 2021-12-30DOI: 10.15330/cmp.13.3.851-861
S. Yanchenko, O. Radchenko
In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,theta}(mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,theta}B(mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,theta}(mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 leqslant p leqslant q leqslant infty$, $(p,q)neq {(1,1), (infty, infty)}$, $dgeq 1$. In the case $2
{"title":"Approximation characteristics of the isotropic Nikol'skii-Besov functional classes","authors":"S. Yanchenko, O. Radchenko","doi":"10.15330/cmp.13.3.851-861","DOIUrl":"https://doi.org/10.15330/cmp.13.3.851-861","url":null,"abstract":"In the paper, we investigates the isotropic Nikol'skii-Besov classes $B^r_{p,theta}(mathbb{R}^d)$ of non-periodic functions of several variables, which for $d = 1$ are identical to the classes of functions with a dominant mixed smoothness $S^{r}_{p,theta}B(mathbb{R})$. We establish the exact-order estimates for the approximation of functions from these classes $B^r_{p,theta}(mathbb{R}^d)$ in the metric of the Lebesgue space $L_q(mathbb{R}^d)$, by entire functions of exponential type with some restrictions for their spectrum in the case $1 leqslant p leqslant q leqslant infty$, $(p,q)neq {(1,1), (infty, infty)}$, $dgeq 1$. In the case $2<p=q<infty$, $d=1$, the established estimate is also new for the classes $S^{r}_{p,theta}B(mathbb{R})$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81355035","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 : 2021-12-30DOI: 10.15330/cmp.13.3.790-804
Jassim H.K, H. Ahmad, A. Shamaoon, C. Cesarano
In this paper, a hybrid technique called the homotopy analysis Sumudu transform method has been implemented solve fractional-order partial differential equations. This technique is the amalgamation of Sumudu transform method and the homotopy analysis method. Three examples are considered to validate and demonstrate the efficacy and accuracy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution which shows that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.
{"title":"An efficient hybrid technique for the solution of fractional-order partial differential equations","authors":"Jassim H.K, H. Ahmad, A. Shamaoon, C. Cesarano","doi":"10.15330/cmp.13.3.790-804","DOIUrl":"https://doi.org/10.15330/cmp.13.3.790-804","url":null,"abstract":"In this paper, a hybrid technique called the homotopy analysis Sumudu transform method has been implemented solve fractional-order partial differential equations. This technique is the amalgamation of Sumudu transform method and the homotopy analysis method. Three examples are considered to validate and demonstrate the efficacy and accuracy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the exact solution which shows that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"6 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87584826","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 : 2021-12-30DOI: 10.15330/cmp.13.3.805-817
D. Bushev, F. Abdullayev, I. Kal’chuk, M. Imashkyzy
In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.
{"title":"The use of the isometry of function spaces with different numbers of variables in the theory of approximation of functions","authors":"D. Bushev, F. Abdullayev, I. Kal’chuk, M. Imashkyzy","doi":"10.15330/cmp.13.3.805-817","DOIUrl":"https://doi.org/10.15330/cmp.13.3.805-817","url":null,"abstract":"In the work, we found integral representations for function spaces that are isometric to spaces of entire functions of exponential type, which are necessary for giving examples of equality of approximation characteristics in function spaces isometric to spaces of non-periodic functions. Sufficient conditions are obtained for the nonnegativity of the delta-like kernels used to construct isometric function spaces with various numbers of variables.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"30 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83048346","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 : 2021-12-30DOI: 10.15330/cmp.13.3.838-850
O. Fedunyk-Yaremchuk, S. Hembars’ka
In this paper, we continue the study of approximation characteristics of the classes $B^{Omega}_{p,theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{Omega}_{p,theta}$ in the space $L_{q},$ $1leq p
在本文中，我们继续研究了若干变量周期函数$B^{Omega}_{p,theta}$类的逼近特性，这些函数的连续性混合模的主模同时包含指数乘子和对数乘子。利用满足一定条件的线性算子，我们得到了在$L_{q},$$1leq p
{"title":"Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity","authors":"O. Fedunyk-Yaremchuk, S. Hembars’ka","doi":"10.15330/cmp.13.3.838-850","DOIUrl":"https://doi.org/10.15330/cmp.13.3.838-850","url":null,"abstract":"In this paper, we continue the study of approximation characteristics of the classes $B^{Omega}_{p,theta}$ of periodic functions of several variables whose majorant of the mixed moduli of continuity contains both exponential and logarithmic multipliers. We obtain the exact-order estimates of the orthoprojective widths of the classes $B^{Omega}_{p,theta}$ in the space $L_{q},$ $1leq p<q<infty,$ and also establish the exact-order estimates of approximation for these classes of functions in the space $L_{q}$ by using linear operators satisfying certain conditions.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"30 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85710156","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 : 2021-12-30DOI: 10.15330/cmp.13.3.831-837
N. Parfinovych
Let $S_{h,m}$, $h>0$, $min {mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $kin {mathbb Z}$. We obtain exact values of the best $(alpha, beta)$-approximations by spaces $S_{h,m}cap L_1({mathbb R})$ in the space $L_1({mathbb R})$ for the classes $W^r_{1,1}({mathbb R})$, $rin {mathbb N}$, of functions, defined on the whole real line, integrable on ${mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({mathbb R})$. These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({mathbb R})$ by splines from $S_{h,m}cap L_1({mathbb R})$ (case $alpha=beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({mathbb T})$ of $2pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({mathbb T})$ by periodic polynomial splines of minimal deficiency. As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({mathbb T})$.
{"title":"Non-symmetric approximations of functional classes by splines on the real line","authors":"N. Parfinovych","doi":"10.15330/cmp.13.3.831-837","DOIUrl":"https://doi.org/10.15330/cmp.13.3.831-837","url":null,"abstract":"Let $S_{h,m}$, $h>0$, $min {mathbb N}$, be the spaces of polynomial splines of order $m$ of deficiency 1 with nodes at the points $kh$, $kin {mathbb Z}$. \u0000We obtain exact values of the best $(alpha, beta)$-approximations by spaces $S_{h,m}cap L_1({mathbb R})$ in the space $L_1({mathbb R})$ for the classes $W^r_{1,1}({mathbb R})$, $rin {mathbb N}$, of functions, defined on the whole real line, integrable on ${mathbb R}$ and such that their $r$th derivatives belong to the unit ball of $L_1({mathbb R})$. \u0000These results generalize the well-known G.G. Magaril-Ilyaev's and V.M. Tikhomirov's results on the exact values of the best approximations of classes $W^r_{1,1}({mathbb R})$ by splines from $S_{h,m}cap L_1({mathbb R})$ (case $alpha=beta=1$), as well as are non-periodic analogs of the V.F. Babenko's result on the best non-symmetric approximations of classes $W^r_1({mathbb T})$ of $2pi$-periodic functions with $r$th derivative belonging to the unit ball of $L_1({mathbb T})$ by periodic polynomial splines of minimal deficiency. \u0000As a corollary of the main result, we obtain exact values of the best one-sided approximations of classes $W^r_1$ by polynomial splines from $S_{h,m}({mathbb T})$. This result is a periodic analogue of the results of A.A. Ligun and V.G. Doronin on the best one-sided approximations of classes $W^r_1$ by spaces $S_{h,m}({mathbb T})$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89731888","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 : 2021-12-30DOI: 10.15330/cmp.13.3.818-830
M. Qasim, A. Khan, Z. Abbas, M. Mursaleen
In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.
{"title":"Convergence properties of generalized Lupaş-Kantorovich operators","authors":"M. Qasim, A. Khan, Z. Abbas, M. Mursaleen","doi":"10.15330/cmp.13.3.818-830","DOIUrl":"https://doi.org/10.15330/cmp.13.3.818-830","url":null,"abstract":"In the present paper, we consider the Kantorovich modification of generalized Lupaş operators, whose construction depends on a continuously differentiable, increasing and unbounded function $rho$. For these new operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"84 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83463642","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 : 2021-12-30DOI: 10.15330/cmp.13.3.775-789
K. Bozkurt, M. L. Limmam, A. Aral
Difference of exponential type Szász and Szász-Kantorovich operators is obtained. Similar estimates are given for higher order $mu$-derivatives of the Szász operators and the Szász-Kantorovich type operators acting on the same order $mu$-derivative of the function. These differences are given in quantitative form using first modulus of continuity. Convergence in variation of the operators in the space of functions with bounded variation with respect to the variation seminorm is obtained. The results propose a general framework covering the results provided by previous literature.
{"title":"Generalization of Szász operators: quantitative estimate and bounded variation","authors":"K. Bozkurt, M. L. Limmam, A. Aral","doi":"10.15330/cmp.13.3.775-789","DOIUrl":"https://doi.org/10.15330/cmp.13.3.775-789","url":null,"abstract":"Difference of exponential type Szász and Szász-Kantorovich operators is obtained. Similar estimates are given for higher order $mu$-derivatives of the Szász operators and the Szász-Kantorovich type operators acting on the same order $mu$-derivative of the function. These differences are given in quantitative form using first modulus of continuity. Convergence in variation of the operators in the space of functions with bounded variation with respect to the variation seminorm is obtained. The results propose a general framework covering the results provided by previous literature.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"50 9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91003961","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 : 2021-12-29DOI: 10.15330/cmp.13.3.764-774
H. Afshari, E. Karapınar
In this paper, we study the existence of solutions for the following differential equations by using a fixed point theorems [ begin{cases} D^{mu}_{c}w(varsigma)pm D^{nu}_{c}w(varsigma)=h(varsigma,w(varsigma)),& varsigmain J, 0
{"title":"A solution of the fractional differential equations in the setting of $b$-metric space","authors":"H. Afshari, E. Karapınar","doi":"10.15330/cmp.13.3.764-774","DOIUrl":"https://doi.org/10.15330/cmp.13.3.764-774","url":null,"abstract":"In this paper, we study the existence of solutions for the following differential equations by using a fixed point theorems [ begin{cases} D^{mu}_{c}w(varsigma)pm D^{nu}_{c}w(varsigma)=h(varsigma,w(varsigma)),& varsigmain J, 0<nu<mu<1, w(0)=w_0,& end{cases} ] where $D^{mu}$, $D^{nu}$ is the Caputo derivative of order $mu$, $nu$, respectively and $h:Jtimes mathbb{R}rightarrow mathbb{R}$ is continuous. The results are well demonstrated with the aid of exciting examples.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"69 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82554712","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 : 2021-12-29DOI: 10.15330/cmp.13.3.750-763
Z. Çakir, C. Aykol, V. Guliyev, A. Serbetci
In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${mathcal{M}}_{p(cdot),lambda(cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${mathcal{M}}_{p(cdot),lambda(cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${mathcal{widetilde{M}}}_{p(cdot),lambda(cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${mathcal{M}}_{p(cdot),lambda(cdot)}(I_{0},w)$.
{"title":"Approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces","authors":"Z. Çakir, C. Aykol, V. Guliyev, A. Serbetci","doi":"10.15330/cmp.13.3.750-763","DOIUrl":"https://doi.org/10.15330/cmp.13.3.750-763","url":null,"abstract":"In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${mathcal{M}}_{p(cdot),lambda(cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${mathcal{M}}_{p(cdot),lambda(cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${mathcal{widetilde{M}}}_{p(cdot),lambda(cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${mathcal{M}}_{p(cdot),lambda(cdot)}(I_{0},w)$.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"5 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91239065","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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