In this paper we study geometric concepts in a general cubic structure. The well-known relationships on the cubic curve motivate us to introduce new concepts into a general cubic structure. We will define the concept of the tangential of a point in a general cubic structure and we will study tangentials of higher-order. The characterization of this concept will be also given by means of the associated totally symmetric quasigroup. We will introduce the concept of associated and corresponding points in a cubic structure, and discuss the number of mutually different corresponding points. The properties of the introduced geometric concepts will be investigated in a general cubic structure.
{"title":"Tangentials in cubic structures","authors":"V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper","doi":"10.3336/gm.55.2.10","DOIUrl":"https://doi.org/10.3336/gm.55.2.10","url":null,"abstract":"In this paper we study geometric concepts in a general cubic structure. The well-known relationships on the cubic curve motivate us to introduce new concepts into a general cubic structure. We will define the concept of the tangential of a point in a general cubic structure and we will study tangentials of higher-order. The characterization of this concept will be also given by means of the associated totally symmetric quasigroup. We will introduce the concept of associated and corresponding points in a cubic structure, and discuss the number of mutually different corresponding points. The properties of the introduced geometric concepts will be investigated in a general cubic structure.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78317232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we continue to investigate common properties of the products ac and bd in various categories under the assumption acd=dbd and dba=aca. These properties include generalized strongly Drazin invertibility and generalized Hirano invertibility in rings, abstract index of Fredholm elements and B-Fredholm elements in the Banach algebra context, complementability of kernels and ranges for bounded linear operators on Banach spaces.
{"title":"Further results on common properties of the products ac and bd","authors":"Qingping Zeng, Kai Yan, Zhenying Wu","doi":"10.3336/gm.55.2.07","DOIUrl":"https://doi.org/10.3336/gm.55.2.07","url":null,"abstract":"In this paper, we continue to investigate common properties of the products ac and bd in various categories under the assumption acd=dbd and dba=aca. These properties include generalized strongly Drazin invertibility and generalized Hirano invertibility in rings, abstract index of Fredholm elements and B-Fredholm elements in the Banach algebra context, complementability of kernels and ranges for bounded linear operators on Banach spaces.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79066070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We analyze two types of permutation orbifolds: (i) S2-orbifolds of the universal level k vertex operator algebra Vk(𝔰𝔩2) and of its simple quotient Lk(𝔰𝔩2), and (ii) the S3-orbifold of the level one simple vertex operator algebra L1(𝔰𝔩2). We determine their structures and discuss related W-algebras.
{"title":"Permutation orbifolds of 𝔰𝔩2 vertex operator algebras","authors":"A. Milas, M. Penn","doi":"10.3336/gm.55.2.08","DOIUrl":"https://doi.org/10.3336/gm.55.2.08","url":null,"abstract":"We analyze two types of permutation orbifolds: (i) S2-orbifolds of the universal level k vertex operator algebra Vk(𝔰𝔩2) and of its simple quotient Lk(𝔰𝔩2), and (ii) the S3-orbifold of the level one simple vertex operator algebra L1(𝔰𝔩2). We determine their structures and discuss related W-algebras.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75457403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
B. Zalar, Smetanova Maribor Slovenia architecture, B. Ferčec, Yilei Tang, M. Mencinger, Jadranska Ljubljana Slovenia mechanics
If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation z'=az2+bz+c as a particular case of 𝓐- Riccati equations z'=a · (z · z)+b · z+c where 𝓐=( ℝn,·) is a commutative, possibly nonassociative algebra, a,b,c∈𝓐 and z:I → 𝓐 is defined on some nontrivial real interval. In the case 𝓐=ℂ, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution * of ℂ. In the present paper we study the critical points of 𝓛(π)- Riccati equations, where 𝓛(π) is the limit case of the so-called family of planar Lyapunov algebras, which characterize 2-dimensional homogeneous systems of quadratic ODEs with stable origin. The number of possible critical points is 1, 3 or ∞, depending on coefficients. The nature of critical points is also completely described. Finally, simultaneous stability of the origin is considered for homogeneous quadratic part corresponding to algebras 𝓛(θ).
{"title":"Partial qualitative analysis of planar 𝓐Q-Riccati equations","authors":"B. Zalar, Smetanova Maribor Slovenia architecture, B. Ferčec, Yilei Tang, M. Mencinger, Jadranska Ljubljana Slovenia mechanics","doi":"10.3336/gm.55.2.11","DOIUrl":"https://doi.org/10.3336/gm.55.2.11","url":null,"abstract":"If we view the field of complex numbers as a 2-dimensional commutative real algebra, we can consider the differential equation z'=az2+bz+c as a particular case of 𝓐- Riccati equations z'=a · (z · z)+b · z+c where 𝓐=( ℝn,·) is a commutative, possibly nonassociative algebra, a,b,c∈𝓐 and z:I → 𝓐 is defined on some nontrivial real interval. In the case 𝓐=ℂ, the nature of (at most two) critical points can be described using purely algebraic conditions involving involution * of ℂ. In the present paper we study the critical points of 𝓛(π)- Riccati equations, where 𝓛(π) is the limit case of the so-called family of planar Lyapunov algebras, which characterize 2-dimensional homogeneous systems of quadratic ODEs with stable origin. The number of possible critical points is 1, 3 or ∞, depending on coefficients. The nature of critical points is also completely described. Finally, simultaneous stability of the origin is considered for homogeneous quadratic part corresponding to algebras 𝓛(θ).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80158386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that bounded solutions of the quasilinear elliptic equation (Delta_{p(x)} u=g+div(textbf{F})) are locally Hölder continuous provided that the functions (g) and (textbf{F}) are in suitable Lebesgue spaces.
{"title":"Hölder continuity for the solutions of the p(x)-Laplace equation\u0000with general right-hand side","authors":"A. Lyaghfouri","doi":"10.3336/gm.57.1.03","DOIUrl":"https://doi.org/10.3336/gm.57.1.03","url":null,"abstract":"We show that bounded solutions of the quasilinear elliptic equation\u0000(Delta_{p(x)} u=g+div(textbf{F})) are locally Hölder continuous\u0000provided that the functions (g) and (textbf{F}) are in suitable\u0000Lebesgue spaces.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76702888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We characterize the solutions of the Markoff-Rosenberger equation ax 2 + by 2 + cz 2 = dxyz with a,b,c,d ∈ Z , gcd( a,b ) = gcd( a,c ) = gcd( b,c ) = 1 and a,b,c | d , for which ( x,y,z ) = ( F i ,F j ,F k ) , where F n denotes the n -th Fibonacci number for any integer n ≥ 0 .
{"title":"Markoff-Rosenberger triples with Fibonacci components","authors":"Szabolcs Tengelys","doi":"10.3336/gm.55.1.03","DOIUrl":"https://doi.org/10.3336/gm.55.1.03","url":null,"abstract":". We characterize the solutions of the Markoff-Rosenberger equation ax 2 + by 2 + cz 2 = dxyz with a,b,c,d ∈ Z , gcd( a,b ) = gcd( a,c ) = gcd( b,c ) = 1 and a,b,c | d , for which ( x,y,z ) = ( F i ,F j ,F k ) , where F n denotes the n -th Fibonacci number for any integer n ≥ 0 .","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83273193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a new approach for dealing with scalar conservation laws with the flux discontinuous with respect to the space variable and merely continuous with respect to the state variable which employs a variant of the kinetic formulation. We use it to improve results about the existence of solutions for non-degenerate scalar conservation laws with Caratheodory flux under a variant of non-degeneracy conditions.
{"title":"Scalar conservation laws with Charatheodory flux revisited","authors":"Nikola Konatar","doi":"10.3336/gm.55.1.09","DOIUrl":"https://doi.org/10.3336/gm.55.1.09","url":null,"abstract":"We introduce a new approach for dealing with scalar conservation laws with the flux discontinuous with respect to the space variable and merely continuous with respect to the state variable which employs a variant of the kinetic formulation. We use it to improve results about the existence of solutions for non-degenerate scalar conservation laws with Caratheodory flux under a variant of non-degeneracy conditions.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88770999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 2012, V. Fedorchuk, using m-pairs and n-partitions, introduced the notion of the (m, n)-dimension of a space. It generalizes covering dimension. Here we are going to look at this concept in the setting of approximate inverse systems of compact metric spaces. We give a characterization of (m,n)-dimX, where X is the limit of an approximate inverse system, strictly in terms of the given system.
{"title":"Approximate inverse limits and (m,n)-dimensions","authors":"M. Lynam, L. Rubin","doi":"10.3336/gm.55.1.11","DOIUrl":"https://doi.org/10.3336/gm.55.1.11","url":null,"abstract":"In 2012, V. Fedorchuk, using m-pairs and n-partitions, introduced the notion of the (m, n)-dimension of a space. It generalizes covering dimension. Here we are going to look at this concept in the setting of approximate inverse systems of compact metric spaces. We give a characterization of (m,n)-dimX, where X is the limit of an approximate inverse system, strictly in terms of the given system.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87948929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In 1972, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i. e., if the functions Φj are continuous in C and Φ0(ζ(s), . . . , ζ(N−1)(s)) + · · ·+ sΦn(ζ(s), . . . , ζ(N−1)(s)) ≡ 0, then Φj ≡ 0 for j = 0, . . . , n. The problem goes back to Hilbert who obtained the algebraic-differential independence of ζ(s). In the paper, the functional independence of compositions F (ζ(s)) for some classes of operators F in the space of analytic functions is proved. For example, as a particular case, the functional independence of the function cos ζ(s) follows.
{"title":"Extension of the functional independence of the Riemann zeta-function","authors":"A. Laurinčikas","doi":"10.3336/gm.55.1.05","DOIUrl":"https://doi.org/10.3336/gm.55.1.05","url":null,"abstract":"In 1972, Voronin proved the functional independence of the Riemann zeta-function ζ(s), i. e., if the functions Φj are continuous in C and Φ0(ζ(s), . . . , ζ(N−1)(s)) + · · ·+ sΦn(ζ(s), . . . , ζ(N−1)(s)) ≡ 0, then Φj ≡ 0 for j = 0, . . . , n. The problem goes back to Hilbert who obtained the algebraic-differential independence of ζ(s). In the paper, the functional independence of compositions F (ζ(s)) for some classes of operators F in the space of analytic functions is proved. For example, as a particular case, the functional independence of the function cos ζ(s) follows.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80879086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bernadette Faye-Fall, F. Luca, Unam Morelia Mexico Centro de Ciencias Matemáticas
. In this paper, we show that if ( X k ,Y k ) is the k th solution of the Pell equation X 2 − dY 2 = 1 for some non–square integer d > 1, then the equation Y k = 2 n − 1 has at most two positive integer solutions ( k,n ).
． 本文证明了对于非平方整数d > 1，如果(X k, Y k)是Pell方程x2−dY 2 = 1的第k个解，则方程Y k = 2n−1最多有两个正整数解(k,n)。
{"title":"On Y-coordinates of Pell equations which are base 2 rep-digits","authors":"Bernadette Faye-Fall, F. Luca, Unam Morelia Mexico Centro de Ciencias Matemáticas","doi":"10.3336/gm.55.1.01","DOIUrl":"https://doi.org/10.3336/gm.55.1.01","url":null,"abstract":". In this paper, we show that if ( X k ,Y k ) is the k th solution of the Pell equation X 2 − dY 2 = 1 for some non–square integer d > 1, then the equation Y k = 2 n − 1 has at most two positive integer solutions ( k,n ).","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88724889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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