Pub Date : 2022-06-07DOI: 10.21136/mb.2022.0128-21
M. M. Chems-Eddin
Let p ≡ 1 (mod 8) and q ≡ 3 (mod 8) be two prime integers and let l 6∈ {−1, p, q} be a positive odd square-free integer. Assuming that the fundamental unit of Q (√ 2p ) has a negative norm, we investigate the unit group of the fields Q (√ 2, √ p, √ q, √ −l ) .
{"title":"On units of some fields of the form $mathbb{Q}big(sqrt2, sqrt{p}, sqrt{q}, sqrt{-ell}big)$","authors":"M. M. Chems-Eddin","doi":"10.21136/mb.2022.0128-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0128-21","url":null,"abstract":"Let p ≡ 1 (mod 8) and q ≡ 3 (mod 8) be two prime integers and let l 6∈ {−1, p, q} be a positive odd square-free integer. Assuming that the fundamental unit of Q (√ 2p ) has a negative norm, we investigate the unit group of the fields Q (√ 2, √ p, √ q, √ −l ) .","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44744636","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 : 2022-06-06DOI: 10.21136/mb.2022.0197-20
Santosh Kumar, Johnson Allen Kessy
. The notions of compatible mappings play a crucial role in metrical fixed point theory. Partial metric spaces are a generalization of the notion of a metric space in the sense that distance of a point from itself is not necessarily zero. In this paper, we prove coincidence and fixed point theorems for a pair of single-valued and multi-valued weak compatible mappings on a complete partial metric space. Our main results generalize, in particular, the results of Kaneko and Sessa (1989), Pathak (1995) and Kessy, Kumar and Kakiko (2017). Examples that illustrate the generality of our results are also provided.
{"title":"Fixed point theorems for hybrid pair of weak compatible mappings\u0000 \u0000in partial metric spaces","authors":"Santosh Kumar, Johnson Allen Kessy","doi":"10.21136/mb.2022.0197-20","DOIUrl":"https://doi.org/10.21136/mb.2022.0197-20","url":null,"abstract":". The notions of compatible mappings play a crucial role in metrical fixed point theory. Partial metric spaces are a generalization of the notion of a metric space in the sense that distance of a point from itself is not necessarily zero. In this paper, we prove coincidence and fixed point theorems for a pair of single-valued and multi-valued weak compatible mappings on a complete partial metric space. Our main results generalize, in particular, the results of Kaneko and Sessa (1989), Pathak (1995) and Kessy, Kumar and Kakiko (2017). Examples that illustrate the generality of our results are also provided.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47492304","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 : 2022-06-02DOI: 10.21136/mb.2022.0100-21
Figen Takil Mutlu, A. Tercan, R. Yaşar
In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C3 properties and M/SocM has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak FI-extending, pseudo duo, C3 properties and ascending (or descending) chain condition on essential submodules then M = M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (or Artinian, respectively) submodule M2. Moreover, we show that a nonsingular weak CS (or weak C 11, or weak FI) module has a direct summand which essentially contains the socle of the module and is a CS (or C11, or FI-extending, respectively) module.
{"title":"Eventually semisimple weak $FI$-extending modules","authors":"Figen Takil Mutlu, A. Tercan, R. Yaşar","doi":"10.21136/mb.2022.0100-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0100-21","url":null,"abstract":"In this article, we study modules with the weak FI-extending property. We prove that if M satisfies weak FI-extending, pseudo duo, C3 properties and M/SocM has finite uniform dimension then M decomposes into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the weak FI-extending, pseudo duo, C3 properties and ascending (or descending) chain condition on essential submodules then M = M1 ⊕ M2 for some semisimple submodule M1 and Noetherian (or Artinian, respectively) submodule M2. Moreover, we show that a nonsingular weak CS (or weak C 11, or weak FI) module has a direct summand which essentially contains the socle of the module and is a CS (or C11, or FI-extending, respectively) module.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43808788","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 : 2022-05-04DOI: 10.21136/mb.2022.0137-21
M. M. Chems-Eddin, O. Ouzzaouit, A. Tamoussit
. Let D be an integral domain with the quotient field K , X an indeterminate over K and x an element of D . The Bhargava ring over D at x is defined to be B x ( D ) := { f ∈ K [ X ]: for all a ∈ D, f ( xX + a ) ∈ D [ X ] } . In fact, B x ( D ) is a subring of the ring of integer-valued polynomials over D . In this paper, we aim to investigate the behavior of B x ( D ) under localization. In particular, we prove that B x ( D ) behaves well under localization at prime ideals of D , when D is a locally finite intersection of localizations. We also attempt a classification of integral domains D such that B x ( D ) is locally free, or at least faithfully flat (or flat) as a D -module (or D [ X ]-module, respectively). Particularly, we are interested in domains that are (locally) essential. A particular attention is devoted to provide conditions under which B x ( D ) is trivial when dealing with essential domains. Finally, we calculate the Krull dimension of Bhargava rings over MZ-Jaffard domains. Interesting results are established with illustrating examples.
{"title":"On Bhargava rings","authors":"M. M. Chems-Eddin, O. Ouzzaouit, A. Tamoussit","doi":"10.21136/mb.2022.0137-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0137-21","url":null,"abstract":". Let D be an integral domain with the quotient field K , X an indeterminate over K and x an element of D . The Bhargava ring over D at x is defined to be B x ( D ) := { f ∈ K [ X ]: for all a ∈ D, f ( xX + a ) ∈ D [ X ] } . In fact, B x ( D ) is a subring of the ring of integer-valued polynomials over D . In this paper, we aim to investigate the behavior of B x ( D ) under localization. In particular, we prove that B x ( D ) behaves well under localization at prime ideals of D , when D is a locally finite intersection of localizations. We also attempt a classification of integral domains D such that B x ( D ) is locally free, or at least faithfully flat (or flat) as a D -module (or D [ X ]-module, respectively). Particularly, we are interested in domains that are (locally) essential. A particular attention is devoted to provide conditions under which B x ( D ) is trivial when dealing with essential domains. Finally, we calculate the Krull dimension of Bhargava rings over MZ-Jaffard domains. Interesting results are established with illustrating examples.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48839216","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 : 2022-05-02DOI: 10.21136/mb.2022.0047-21
W. Kota, R. M. El-Ashwah
. Using the operator D mq,̺ ( λ, l ), we introduce the subclasses Y ∗ mq,̺ ( l, λ, γ ) and K ∗ mq,̺ ( l, λ, γ ) of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.
{"title":"Some applications of subordination theorems associated with fractional $q$-calculus operator","authors":"W. Kota, R. M. El-Ashwah","doi":"10.21136/mb.2022.0047-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0047-21","url":null,"abstract":". Using the operator D mq,̺ ( λ, l ), we introduce the subclasses Y ∗ mq,̺ ( l, λ, γ ) and K ∗ mq,̺ ( l, λ, γ ) of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47186350","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 : 2022-04-28DOI: 10.21136/mb.2022.0007-21
T. Madaras, Mária Surimová
. A proper vertex k -colouring of a graph G is called l -homogeneous if the number of colours in the neigbourhood of each vertex of G equals l . We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs.
{"title":"Homogeneous colourings of graphs","authors":"T. Madaras, Mária Surimová","doi":"10.21136/mb.2022.0007-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0007-21","url":null,"abstract":". A proper vertex k -colouring of a graph G is called l -homogeneous if the number of colours in the neigbourhood of each vertex of G equals l . We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48358411","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 : 2022-04-28DOI: 10.21136/mb.2022.0142-21
M. Sahmoudi, M. Charkani
. Let L = K ( α ) be an extension of a number field K , where α satisfies the monic irreducible polynomial P ( X ) = X p − β of prime degree belonging to o K [ X ] ( o K is the ring of integers of K ). The purpose of this paper is to study the monogenity of L over K by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field L with a pure cubic subfield, which is not necessarily a composite extension of two cubic subfields. We obtain a slightly simpler computation of the discriminant d L/ Q .
． 设L = K (α)是数域K的一个扩展，其中α满足素数次的一元不可约多项式P (X) = X P−β，属于o K [X] (o K是K的整数环)。本文的目的是利用高斯值和指标理想，用一个简单实用的Dedekind判据来研究任意Dedekind环上幂积分基存在性的L / K的单调性。作为一个例子，我们确定了具有纯三次子域的纯非子域L的一个积分基，它不一定是两个三次子域的复合扩展。我们得到了一个稍微简单的判别d L/ Q的计算。
{"title":"On relative pure cyclic fields with power integral bases","authors":"M. Sahmoudi, M. Charkani","doi":"10.21136/mb.2022.0142-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0142-21","url":null,"abstract":". Let L = K ( α ) be an extension of a number field K , where α satisfies the monic irreducible polynomial P ( X ) = X p − β of prime degree belonging to o K [ X ] ( o K is the ring of integers of K ). The purpose of this paper is to study the monogenity of L over K by a simple and practical version of Dedekind’s criterion characterizing the existence of power integral bases over an arbitrary Dedekind ring by using the Gauss valuation and the index ideal. As an illustration, we determine an integral basis of a pure nonic field L with a pure cubic subfield, which is not necessarily a composite extension of two cubic subfields. We obtain a slightly simpler computation of the discriminant d L/ Q .","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47184667","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 : 2022-04-28DOI: 10.21136/mb.2022.0072-21
Henri Mühle
{"title":"Meet-distributive lattices have the intersection property","authors":"Henri Mühle","doi":"10.21136/mb.2022.0072-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0072-21","url":null,"abstract":"","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68443461","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 : 2022-03-30DOI: 10.21136/mb.2022.0186-20
S. Majumder, L. Mahato
. The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation where P d ( z, f ) is a difference-differential polynomial in f ( z ) of degree d 6 n − 1 with small functions of f ( z ) as its coefficients, p 1 , p 2 are nonzero rational functions and α 1 , α 2 are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.
{"title":"On the meromorphic solutions of a certain type of nonlinear difference-differential equation","authors":"S. Majumder, L. Mahato","doi":"10.21136/mb.2022.0186-20","DOIUrl":"https://doi.org/10.21136/mb.2022.0186-20","url":null,"abstract":". The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation where P d ( z, f ) is a difference-differential polynomial in f ( z ) of degree d 6 n − 1 with small functions of f ( z ) as its coefficients, p 1 , p 2 are nonzero rational functions and α 1 , α 2 are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48088617","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 : 2022-03-16DOI: 10.21136/mb.2022.0064-21
S. M. Tabatabaie, A. B. Salec
{"title":"On the inclusions of $X^Phi$ spaces","authors":"S. M. Tabatabaie, A. B. Salec","doi":"10.21136/mb.2022.0064-21","DOIUrl":"https://doi.org/10.21136/mb.2022.0064-21","url":null,"abstract":"","PeriodicalId":45392,"journal":{"name":"Mathematica Bohemica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46140487","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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