Pub Date : 2023-12-05DOI: 10.15330/cmp.15.2.482-494
H. A. Erdem, A. Uçum, K. Ilarslan, Ç. Camcı
In the theory of curves in Euclidean $3$-space, it is well known that a curve $beta $ is said to be a Bertrand curve if for another curve $beta^{star}$ there exists a one-to-one correspondence between $beta $ and $beta^{star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Minkowski $3$-space. Related examples that meet these conditions are given. Moreover, thanks to this new approach, timelike, spacelike and Cartan null Bertrand mates of a timelike general helix have been obtained.
{"title":"New approach to timelike Bertrand curves in 3-dimensional Minkowski space","authors":"H. A. Erdem, A. Uçum, K. Ilarslan, Ç. Camcı","doi":"10.15330/cmp.15.2.482-494","DOIUrl":"https://doi.org/10.15330/cmp.15.2.482-494","url":null,"abstract":"In the theory of curves in Euclidean $3$-space, it is well known that a curve $beta $ is said to be a Bertrand curve if for another curve $beta^{star}$ there exists a one-to-one correspondence between $beta $ and $beta^{star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Minkowski $3$-space. Related examples that meet these conditions are given. Moreover, thanks to this new approach, timelike, spacelike and Cartan null Bertrand mates of a timelike general helix have been obtained.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"81 19","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138600392","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 : 2023-11-21DOI: 10.15330/cmp.15.2.437-448
D. Bodnar, O.S. Bodnar, I. Bilanyk
Truncation error bounds for branched continued fractions of the special form are established. These fractions can be obtained by fixing the values of variables in branched continued fractions with independent variables, which is an effective tool for approximating complex functions of two variables. The main result is a two-dimensional analog of the theorem considered in [SCIAM J. Numer. Anal. 1983, 20 (3), 1187$-$1197] for van Vleck's continued fractions. For its proving, the $mathcal{C}$-figure convergence and estimates of the difference between approximants of fractions in an angular domain are significantly used. In comparison with the previously established results, the elements of a branched continued fraction of the special form can tend to zero at a certain rate. An example of the effectiveness of using a two-dimensional analog of van Vleck's theorem is considered.
建立了特殊形式支链续分数的截断误差界限。这些分数可以通过固定带自变量的支化连续分数中的变量值获得，这是逼近两变量复变函数的有效工具。主要结果是 [SCIAM J. Numer. Anal.为了证明该定理，大量使用了 $mathcal{C}$ 图收敛性和角域中分数近似值之间差值的估计。与之前建立的结果相比，特殊形式的分支连续分数的元素能以一定的速率趋于零。本论文以 van Vleck 定理的二维类比为例，说明了使用该定理的有效性。
{"title":"A truncation error bound for branched continued fractions of the special form on subsets of angular domains","authors":"D. Bodnar, O.S. Bodnar, I. Bilanyk","doi":"10.15330/cmp.15.2.437-448","DOIUrl":"https://doi.org/10.15330/cmp.15.2.437-448","url":null,"abstract":"Truncation error bounds for branched continued fractions of the special form are established. These fractions can be obtained by fixing the values of variables in branched continued fractions with independent variables, which is an effective tool for approximating complex functions of two variables. The main result is a two-dimensional analog of the theorem considered in [SCIAM J. Numer. Anal. 1983, 20 (3), 1187$-$1197] for van Vleck's continued fractions. For its proving, the $mathcal{C}$-figure convergence and estimates of the difference between approximants of fractions in an angular domain are significantly used. In comparison with the previously established results, the elements of a branched continued fraction of the special form can tend to zero at a certain rate. An example of the effectiveness of using a two-dimensional analog of van Vleck's theorem is considered.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"12 6","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139252894","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 : 2023-11-21DOI: 10.15330/cmp.15.2.449-467
W. Ramirez, D. Bedoya, A. Urieles, C. Cesarano, M. Ortega
In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomial matrices, which involves the generalized Pascal matrix.
{"title":"New $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomials and their matrices","authors":"W. Ramirez, D. Bedoya, A. Urieles, C. Cesarano, M. Ortega","doi":"10.15330/cmp.15.2.449-467","DOIUrl":"https://doi.org/10.15330/cmp.15.2.449-467","url":null,"abstract":"In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomial matrices, which involves the generalized Pascal matrix.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"16 3","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139253502","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 : 2023-11-14DOI: 10.15330/cmp.15.2.396-410
A. Bougoutaia, A. Belacel, R. Macedo, H. Hamdi
In this article, we establish new relationships involving the class of Cohen positive strongly $p$-summing multilinear operators. Furthermore, we introduce a new class of multilinear operators on Banach lattices, called positive Cohen weakly nuclear multilinear operators. We establish a Pietsch domination-type theorem for this new class of multilinear operators. As an application, we show that every positive Cohen weakly $p$-nuclear multilinear operator is positive Dimant strongly $p$-summing and Cohen positive strongly $p$-summing. We conclude with a tensor representation of our class.
{"title":"On positive Cohen weakly nuclear multilinear operators","authors":"A. Bougoutaia, A. Belacel, R. Macedo, H. Hamdi","doi":"10.15330/cmp.15.2.396-410","DOIUrl":"https://doi.org/10.15330/cmp.15.2.396-410","url":null,"abstract":"In this article, we establish new relationships involving the class of Cohen positive strongly $p$-summing multilinear operators. Furthermore, we introduce a new class of multilinear operators on Banach lattices, called positive Cohen weakly nuclear multilinear operators. We establish a Pietsch domination-type theorem for this new class of multilinear operators. As an application, we show that every positive Cohen weakly $p$-nuclear multilinear operator is positive Dimant strongly $p$-summing and Cohen positive strongly $p$-summing. We conclude with a tensor representation of our class.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"46 43","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134901898","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 work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the $p$th power of the absolute value is Lebesgue integrable, where $pin[1,+infty),$ and Lebesgue measurable essentially bounded functions on $[0,1].$ We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras.
{"title":"Algebras of weakly symmetric functions on spaces of Lebesgue measurable functions","authors":"I.V. Burtnyak, Yu.Yu. Chopyuk, S.I. Vasylyshyn, T.V. Vasylyshyn","doi":"10.15330/cmp.15.2.411-419","DOIUrl":"https://doi.org/10.15330/cmp.15.2.411-419","url":null,"abstract":"In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the $p$th power of the absolute value is Lebesgue integrable, where $pin[1,+infty),$ and Lebesgue measurable essentially bounded functions on $[0,1].$ We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"46 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134902994","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 : 2023-10-29DOI: 10.15330/cmp.15.2.388-395
N. Elsharkawy, C. Cesarano, R. Dmytryshyn, A. Elsharkawy
In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermore, we give certain characterizations of equiform spherical curves in 3-dimensional Minkowski space, which are timelike with an equiform spacelike principal normal vector.
{"title":"Timelike spherical curves according to equiform Bishop frame in 3-dimensional Minkowski space","authors":"N. Elsharkawy, C. Cesarano, R. Dmytryshyn, A. Elsharkawy","doi":"10.15330/cmp.15.2.388-395","DOIUrl":"https://doi.org/10.15330/cmp.15.2.388-395","url":null,"abstract":"In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermore, we give certain characterizations of equiform spherical curves in 3-dimensional Minkowski space, which are timelike with an equiform spacelike principal normal vector.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"3 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136135287","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 : 2023-10-19DOI: 10.15330/cmp.15.2.381-387
M.M. Osypchuk
In the paper, the transition probability density of an isotropic $alpha$-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo-differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols $|lambda|^varkappa$ and $lambda|lambda|^{varkappa-1}$, where $varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.
{"title":"Bilateral estimates of some pseudo-derivatives of the transition probability density of an isotropic $alpha$-stable stochastic process","authors":"M.M. Osypchuk","doi":"10.15330/cmp.15.2.381-387","DOIUrl":"https://doi.org/10.15330/cmp.15.2.381-387","url":null,"abstract":"In the paper, the transition probability density of an isotropic $alpha$-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo-differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols $|lambda|^varkappa$ and $lambda|lambda|^{varkappa-1}$, where $varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135779533","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 : 2023-09-10DOI: 10.15330/cmp.15.2.377-380
E. Zikkos
Recently D.T. Stoeva proved that if two Bessel sequences in a separable Hilbert space $mathcal H$ are biorthogonal and one of them is complete in $mathcal H$, then both sequences are Riesz bases for $mathcal H$. This improves a well known result where completeness is assumed on both sequences.
In this note we present an alternative proof of Stoeva's result which is quite short and elementary, based on the notion of Riesz-Fischer sequences.
{"title":"Characterizing Riesz bases via biorthogonal Bessel sequences","authors":"E. Zikkos","doi":"10.15330/cmp.15.2.377-380","DOIUrl":"https://doi.org/10.15330/cmp.15.2.377-380","url":null,"abstract":"Recently D.T. Stoeva proved that if two Bessel sequences in a separable Hilbert space $mathcal H$ are biorthogonal and one of them is complete in $mathcal H$, then both sequences are Riesz bases for $mathcal H$. This improves a well known result where completeness is assumed on both sequences.
 In this note we present an alternative proof of Stoeva's result which is quite short and elementary, based on the notion of Riesz-Fischer sequences.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136071673","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 : 2023-09-07DOI: 10.15330/cmp.15.2.356-376
K.N. Soltanov
In this article, the existence of the spectrum (the eigenvalues) for the nonlinear continuous operators acting in the Banach spaces is investigated. For the study this question it is used a different approach that allows the studying of all eigenvalues of a nonlinear operator relative to another nonlinear operator. Here we show that in nonlinear operators case it is necessary to seek the spectrum of the given nonlinear operator relative to another nonlinear operator satisfying certain conditions. The different examples, for which eigenvalues can be found, are provided. Moreover, the nonlinear problems including parameters are studied.
{"title":"Some remarks on spectrum of nonlinear continuous operators","authors":"K.N. Soltanov","doi":"10.15330/cmp.15.2.356-376","DOIUrl":"https://doi.org/10.15330/cmp.15.2.356-376","url":null,"abstract":"In this article, the existence of the spectrum (the eigenvalues) for the nonlinear continuous operators acting in the Banach spaces is investigated. For the study this question it is used a different approach that allows the studying of all eigenvalues of a nonlinear operator relative to another nonlinear operator. Here we show that in nonlinear operators case it is necessary to seek the spectrum of the given nonlinear operator relative to another nonlinear operator satisfying certain conditions. The different examples, for which eigenvalues can be found, are provided. Moreover, the nonlinear problems including parameters are studied.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135096980","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 : 2023-08-03DOI: 10.15330/cmp.15.2.321-330
S. Nykorovych, O. Nykyforchyn
In two ways we introduce metrics on the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric on a fixed set, and prove that the obtained metrics are compact and topologically equivalent. To achieve this, we give a characterization of the sets being the hypographs of the mentioned pseudoultrametrics, and apply Hausdorff metric to their family. It is proved that the uniform convergence metric is a limit case of metrics defined via hypographs. It is shown that the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric, with the induced topology is a Lawson compact Hausdorff upper semilattice.
{"title":"Metric and topology on the poset of compact pseudoultrametrics","authors":"S. Nykorovych, O. Nykyforchyn","doi":"10.15330/cmp.15.2.321-330","DOIUrl":"https://doi.org/10.15330/cmp.15.2.321-330","url":null,"abstract":"In two ways we introduce metrics on the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric on a fixed set, and prove that the obtained metrics are compact and topologically equivalent. To achieve this, we give a characterization of the sets being the hypographs of the mentioned pseudoultrametrics, and apply Hausdorff metric to their family. It is proved that the uniform convergence metric is a limit case of metrics defined via hypographs. It is shown that the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric, with the induced topology is a Lawson compact Hausdorff upper semilattice.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88867723","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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