R. Escobedo, P. Pellicer-Covarrubias, V. Sánchez-Gutiérrez
In this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. We show results of compactness, connectedness and local connectedness for this hyperspace. We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.
{"title":"The hyperspace of totally disconnected sets","authors":"R. Escobedo, P. Pellicer-Covarrubias, V. Sánchez-Gutiérrez","doi":"10.3336/gm.55.1.10","DOIUrl":"https://doi.org/10.3336/gm.55.1.10","url":null,"abstract":"In this paper we study the hyperspace of all nonempty closed totally disconnected subsets of a space, equipped with the Vietoris topology. We show results of compactness, connectedness and local connectedness for this hyperspace. We also include a study of path connectedness, particularly we prove that for a smooth dendroid this hyperspace is pathwise connected, and we present a general result which implies that for an Euclidean space this hyperspace has uncountably many arc components.","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":"89325759","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}
It is well known that various types of path problems in graphs can be treated together within a common algebraic framework. Thereby each type is characterized by a different “path algebra”, i.e., a different instance of the same abstract algebraic structure. This paper demonstrates that the common algebraic framework, although originally intended for conventional problem variants, can be extended to cover multiobjective and robust variants. Thus the paper is mainly concerned with constructing and justifying new path algebras that correspond to such more complex problem varieties. A consequence of the obtained algebraic formulation is that multi-objective or robust problem instances can be solved by well-known general algorithms designed to work over an arbitrary path algebra. The solutions obtained in this way comprise all paths that are efficient in the Pareto sense. The efficient paths are by default described only implicitly, as vectors of objective-function values. Still, it is shown in the paper that, with slightly extended versions of the involved algebras, the same paths can also be identified explicitly. Also, for robust problem instances it is possible to select only one “robustly optimal” path according to a generalized min-max or min-max regret criterion.
{"title":"An algebraic framework for multi-objective and robust variants of path problems","authors":"R. Manger","doi":"10.3336/gm.55.1.12","DOIUrl":"https://doi.org/10.3336/gm.55.1.12","url":null,"abstract":"It is well known that various types of path problems in graphs can be treated together within a common algebraic framework. Thereby each type is characterized by a different “path algebra”, i.e., a different instance of the same abstract algebraic structure. This paper demonstrates that the common algebraic framework, although originally intended for conventional problem variants, can be extended to cover multiobjective and robust variants. Thus the paper is mainly concerned with constructing and justifying new path algebras that correspond to such more complex problem varieties. A consequence of the obtained algebraic formulation is that multi-objective or robust problem instances can be solved by well-known general algorithms designed to work over an arbitrary path algebra. The solutions obtained in this way comprise all paths that are efficient in the Pareto sense. The efficient paths are by default described only implicitly, as vectors of objective-function values. Still, it is shown in the paper that, with slightly extended versions of the involved algebras, the same paths can also be identified explicitly. Also, for robust problem instances it is possible to select only one “robustly optimal” path according to a generalized min-max or min-max regret criterion.","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":"85510345","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}
A. Bazsó, A. Bérczes, Ondřej Kolouch, I. Pink, J. Šustek
We investigate the power and polynomial values of the polynomials Pn(X) = ∏ n k=0 ( X2·3 k − X3 k − 1 ) for n ∈ N. We prove various ineffective and effective finiteness results. In the case 0 ≤ n ≤ 3, we determine all pairs x, y of integers such that Pn(x) = y2 or Pn(x) = y3.
{"title":"Diophantine equations connected to the Komornik polynomials","authors":"A. Bazsó, A. Bérczes, Ondřej Kolouch, I. Pink, J. Šustek","doi":"10.3336/gm.55.1.02","DOIUrl":"https://doi.org/10.3336/gm.55.1.02","url":null,"abstract":"We investigate the power and polynomial values of the polynomials Pn(X) = ∏ n k=0 ( X2·3 k − X3 k − 1 ) for n ∈ N. We prove various ineffective and effective finiteness results. In the case 0 ≤ n ≤ 3, we determine all pairs x, y of integers such that Pn(x) = y2 or Pn(x) = y3.","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":"88424568","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}
A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.
{"title":"High rank elliptic curves induced by rational Diophantine triples","authors":"A. Dujella, J. C. Peral","doi":"10.3336/gm.55.2.05","DOIUrl":"https://doi.org/10.3336/gm.55.2.05","url":null,"abstract":"A rational Diophantine triple is a set of three nonzero rational a,b,c with the property that ab+1, ac+1, bc+1 are perfect squares. We say that the elliptic curve y2 = (ax+1)(bx+1)(cx+1) is induced by the triple {a,b,c}. In this paper, we describe a new method for construction of elliptic curves over ℚ with reasonably high rank based on a parametrization of rational Diophantine triples. In particular, we construct an elliptic curve induced by a rational Diophantine triple with rank equal to 12, and an infinite family of such curves with rank ≥ 7, which are both the current records for that kind of curves.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84870797","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 examine the shape of a triangle by means of a ternary operation which satisfies some properties. We prove that each system of the shapes of triangles can be obtained by means of the field with defined ternary operation. We give a geometric model of the shapes of triangles on the set of complex numbers which motivate us to introduce some geometric concepts. The concept of transfer is defined and some interesting properties are explored. By means of transfer the concept of a parallelogram is introduced.
{"title":"A complete system of the shapes of triangles","authors":"V. Volenec, Z. Kolar-Begović, R. Kolar-Šuper","doi":"10.3336/gm.54.2.07","DOIUrl":"https://doi.org/10.3336/gm.54.2.07","url":null,"abstract":"In this paper we examine the shape of a triangle by means of a ternary operation which satisfies some properties. We prove that each system of the shapes of triangles can be obtained by means of the field with defined ternary operation. We give a geometric model of the shapes of triangles on the set of complex numbers which motivate us to introduce some geometric concepts. The concept of transfer is defined and some interesting properties are explored. By means of transfer the concept of a parallelogram is introduced.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72440419","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 deal with the Brocard-Ramanujan-type equations An1An2 · · ·Ank ± 1 = Am or Gm or G 2 m where {An}n≥0 and {Gm}m≥0 are either balancing-like sequences or associated balancing-like sequences.
{"title":"Diophantine equations with balancing-like sequences associated to Brocard-Ramanujan-type problem","authors":"M. K. Sahukar, G. Panda","doi":"10.3336/gm.54.2.01","DOIUrl":"https://doi.org/10.3336/gm.54.2.01","url":null,"abstract":"In this paper, we deal with the Brocard-Ramanujan-type equations An1An2 · · ·Ank ± 1 = Am or Gm or G 2 m where {An}n≥0 and {Gm}m≥0 are either balancing-like sequences or associated balancing-like sequences.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88759783","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}
For any continuous single-valued functions f, g : [0, 1] → [0, 1] we define upper semicontinuous set-valued functions F,G : [0, 1] ⊸ [0, 1] by their graphs as the unions of the diagonal ∆ and the graphs of setvalued inverses of f and g respectively. We introduce when two functions are ∆-related and show that if f and g are ∆-related, then the inverse limits lim − ⊸ F and lim − ⊸ G are homeomorphic. We also give conditions under which lim − ⊸ G is a quotient space of lim − ⊸ F .
{"title":"Δ-related functions and generalized inverse limits","authors":"Tina Sovič","doi":"10.3336/gm.54.2.09","DOIUrl":"https://doi.org/10.3336/gm.54.2.09","url":null,"abstract":"For any continuous single-valued functions f, g : [0, 1] → [0, 1] we define upper semicontinuous set-valued functions F,G : [0, 1] ⊸ [0, 1] by their graphs as the unions of the diagonal ∆ and the graphs of setvalued inverses of f and g respectively. We introduce when two functions are ∆-related and show that if f and g are ∆-related, then the inverse limits lim − ⊸ F and lim − ⊸ G are homeomorphic. We also give conditions under which lim − ⊸ G is a quotient space of lim − ⊸ F .","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80187353","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 determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations of the odd orthogonal group defined over a local nonarchimedean field of characteristic zero.
我们确定了在特征为零的局部非阿基米德域上定义的奇正交群的所有不可约一般表示在所有层次上的提拉。
{"title":"Theta lifts of generic representations: the case of odd orthogonal groups","authors":"Petar Bakić","doi":"10.3336/gm.54.2.08","DOIUrl":"https://doi.org/10.3336/gm.54.2.08","url":null,"abstract":"We determine the occurrence and explicitly describe the theta lifts on all levels of all the irreducible generic representations of the odd orthogonal group defined over a local nonarchimedean field of characteristic zero.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90906139","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 give a Maschke-type theorem for a Hom-crossed product on a finite dimensional monoidal Hom-Hopf algebra, and investigate a sufficient and necessary condition for two Hom-crossed products to be equivalent. Furthermore, we construct an equivalent Homcrossed system based on a same Hom-crossed product by using lazy Hom2-cocyle.
{"title":"Equivalent crossed products of monoidal Hom-Hopf algebras","authors":"Zhong-wei Wang, Liangyun Zhang, Huihui Zheng","doi":"10.3336/gm.54.2.05","DOIUrl":"https://doi.org/10.3336/gm.54.2.05","url":null,"abstract":"In this paper, we give a Maschke-type theorem for a Hom-crossed product on a finite dimensional monoidal Hom-Hopf algebra, and investigate a sufficient and necessary condition for two Hom-crossed products to be equivalent. Furthermore, we construct an equivalent Homcrossed system based on a same Hom-crossed product by using lazy Hom2-cocyle.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81804589","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}
Let m > 31 be an even integer with gcd(m, 31) = 1. In this paper, using some elementary methods, we prove that the equation (m2 −312)x +(62m) = (m2 +312)z has only the positive integer solution (x, y, z) = (2, 2, 2). This result resolves an open problem raised by T. Miyazaki (Acta Arith. 186 (2018), 1–36) about Jeśmanowicz’ conjecture concerning primitive Pythagorean triples.
{"title":"An open problem on Jeśmanowicz' conjecture concerning primitive Pythagorean triples","authors":"Hai Yang, Ruiqin Fu","doi":"10.3336/gm.54.2.02","DOIUrl":"https://doi.org/10.3336/gm.54.2.02","url":null,"abstract":"Let m > 31 be an even integer with gcd(m, 31) = 1. In this paper, using some elementary methods, we prove that the equation (m2 −312)x +(62m) = (m2 +312)z has only the positive integer solution (x, y, z) = (2, 2, 2). This result resolves an open problem raised by T. Miyazaki (Acta Arith. 186 (2018), 1–36) about Jeśmanowicz’ conjecture concerning primitive Pythagorean triples.","PeriodicalId":55601,"journal":{"name":"Glasnik Matematicki","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86042003","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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