A subset R of a product set X×Y is called a relation on X to Y . A relation U on the power set P (X) to Y is called a super relation on X to Y . The relation R can be identified, to some extent, with the set-valued function φR defined by φR (x) = R (x) = { y ∈ Y : (x, y) ∈ R } for all x ∈ X, and the union-preserving super relation R . defined by R (A) = R [A ] = ⋃ a∈A R (a) for all A ⊆ X. By using the relation R , we also define two super relations lbR and clR on Y to X such that lbR (B) = { x ∈ X : {x}×B ⊆ R } and clR (B) = { x ∈ X : R (x) ∩ B 6= ∅ } for all B ⊆ X . By using complement and inverse relations, we prove that lbR = cl c Rc and clR (B) = R−1 [B ] . We also consider the dual super relations ubR = lbR−1 and intR = cl c R ◦ CY . If U is a super relation on X to Y and V is a super relation on Y to X, then having in mind Galois connections and residuated mappings, we say that U is V –normal if, for all A ⊆ X and B ⊆ Y , we have U (A) ⊆ B if and only if A ⊆ V (B) . Thus, if U is V –normal, then by defining Φ = V ◦ U and following Pataki’s ideas, we see that U is Φ–regular in the sense that, for all A1 , A2 ⊆ X, we have U (A1) ⊆ U (A2) if and only if A1 ⊆ Φ (A2) . In this paper, by considering a relator (family of relations) R on X to Y , we investigate normality properties of the more general super relations lbR = ⋃ R∈R lbR and clR = ⋂ R∈R clR , and their duals ubR = lbR−1 and intR = cl c R ◦ CY . However, as some applicable results of the paper, we only prove that if R is a relation on X to Y , then the following assertions hold : (1) clR−1 is intR – normal, or equivalently clR is intR−1 – normal ; (2) ub c R is lbR ◦ CY – normal, or equivalently lb c R is ubR ◦ CX – normal ; (3) R is a function of X to Y if and only if clR−1 is clR – normal, or equivalently intR is intR−1 – normal . The closure-interior and the upper-lower-bound Galois connections, established in assertions (1) and (2), are applied in the calculus of relations and the completion of posets, respectively. Some of the implications in assertion (3) require that Y 6= ∅ .
乘积集X×Y的子集R称为X到Y的关系。幂集P (X)到Y上的关系U称为X到Y上的超关系。关系R在一定程度上可以用φR (x) = R (x) = {y∈y: (x, y)∈R}定义的集值函数φR和保并超关系R来标识。由R (A) = R [A] = ` ` A∈A R (A)定义，对于所有A的X，我们还利用关系R在Y到X上定义了两个超关系lbR和clR，使得对于所有B的X, lbR (B) = {X∈X: {X}×B任任R}， clR (B) = {X∈X: R (X)∩B 6=∅}。利用补和逆关系，证明了lbR = cl c Rc和clR (B) = R−1 [B]。我们还考虑了对偶超关系ubR = lbR−1和intR = cl c R◦CY。如果U是X到Y上的超关系，V是Y到X上的超关系，那么考虑到伽罗瓦连接和剩余映射，我们说U是V -正规的，当且仅当，对于所有的a, X和B，我们有U (a)，它是B。因此，如果U为V -法线，则通过定义Φ = V◦U并遵循Pataki的思想，我们可以看到U为Φ-regular，即对于所有A1、A2的任一个X，当且仅当A1≥Φ (A2)时，我们有U (A1)≥U (A2)。本文通过考虑X到Y上的一个关系族R，研究了更一般的超关系lbR =∈R lbR和clR = R∈R clR及其对偶ubR = lbR−1和intR = cl c R◦CY的正态性性质。然而，作为本文的一些适用结果，我们只证明了如果R是X到Y上的关系，则下列断言成立:(1)clR−1是intR -正规的，或者等价地clR是intR−1 -正规的;(2) b c R为lbR◦CY -正常，或b c R为ubR◦CX -正常;(3) R是X到Y的函数当且仅当clR−1是clR -正规的，或者等价地，intR是intR−1 -正规的。在断言(1)和断言(2)中建立的闭包内连接和上界下界伽罗瓦连接分别应用于关系演算和偏序集补全。断言(3)中的某些含意要求y6 =∅。
{"title":"Galois and Pataki Connections for Ordinary Functions and Super Relations","authors":"Santanu Acharjee, M. Rassias, Á. Száz","doi":"10.47443/ejm.2022.017","DOIUrl":"https://doi.org/10.47443/ejm.2022.017","url":null,"abstract":"A subset R of a product set X×Y is called a relation on X to Y . A relation U on the power set P (X) to Y is called a super relation on X to Y . The relation R can be identified, to some extent, with the set-valued function φR defined by φR (x) = R (x) = { y ∈ Y : (x, y) ∈ R } for all x ∈ X, and the union-preserving super relation R . defined by R (A) = R [A ] = ⋃ a∈A R (a) for all A ⊆ X. By using the relation R , we also define two super relations lbR and clR on Y to X such that lbR (B) = { x ∈ X : {x}×B ⊆ R } and clR (B) = { x ∈ X : R (x) ∩ B 6= ∅ } for all B ⊆ X . By using complement and inverse relations, we prove that lbR = cl c Rc and clR (B) = R−1 [B ] . We also consider the dual super relations ubR = lbR−1 and intR = cl c R ◦ CY . If U is a super relation on X to Y and V is a super relation on Y to X, then having in mind Galois connections and residuated mappings, we say that U is V –normal if, for all A ⊆ X and B ⊆ Y , we have U (A) ⊆ B if and only if A ⊆ V (B) . Thus, if U is V –normal, then by defining Φ = V ◦ U and following Pataki’s ideas, we see that U is Φ–regular in the sense that, for all A1 , A2 ⊆ X, we have U (A1) ⊆ U (A2) if and only if A1 ⊆ Φ (A2) . In this paper, by considering a relator (family of relations) R on X to Y , we investigate normality properties of the more general super relations lbR = ⋃ R∈R lbR and clR = ⋂ R∈R clR , and their duals ubR = lbR−1 and intR = cl c R ◦ CY . However, as some applicable results of the paper, we only prove that if R is a relation on X to Y , then the following assertions hold : (1) clR−1 is intR – normal, or equivalently clR is intR−1 – normal ; (2) ub c R is lbR ◦ CY – normal, or equivalently lb c R is ubR ◦ CX – normal ; (3) R is a function of X to Y if and only if clR−1 is clR – normal, or equivalently intR is intR−1 – normal . The closure-interior and the upper-lower-bound Galois connections, established in assertions (1) and (2), are applied in the calculus of relations and the completion of posets, respectively. Some of the implications in assertion (3) require that Y 6= ∅ .","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87061791","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}
{"title":"Misconceptions and resulting errors displayed by in service teachers in the learning of linear independence","authors":"L. Mutambara, S. Bansilal","doi":"10.29333/iejme/12483","DOIUrl":"https://doi.org/10.29333/iejme/12483","url":null,"abstract":"","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73289406","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}
{"title":"Comparison of the learning outcomes in online and in-class environments in the divisibility lessons","authors":"Dina Kamber Hamzić, Daniela Zubović, Lamija Šćeta","doi":"10.29333/iejme/12473","DOIUrl":"https://doi.org/10.29333/iejme/12473","url":null,"abstract":"","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87815886","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}
{"title":"The didactical phenomenology in learning the circle equation","authors":"Clement Ayarebilla Ali","doi":"10.29333/iejme/12472","DOIUrl":"https://doi.org/10.29333/iejme/12472","url":null,"abstract":"ABSTRACT","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73761655","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}
{"title":"Trends in learning and teaching of geometry: The case of the Geometry and its Applications Meeting","authors":"Paola Castro, P. Gómez, M. Cañadas","doi":"10.29333/iejme/12474","DOIUrl":"https://doi.org/10.29333/iejme/12474","url":null,"abstract":": The","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79083738","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}
{"title":"Configuring the landscape of research on problem-solving in mathematics teacher education","authors":"Anette de Ron, I. Christiansen, Kicki Skog","doi":"10.29333/iejme/12457","DOIUrl":"https://doi.org/10.29333/iejme/12457","url":null,"abstract":"","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74629627","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}
For n ∈ { 1 , 2 , . . . } , let Ω n = π n/ 2 / Γ( n 2 + 1) be the volume of the unit ball in R n . In this paper, we give a new asymptotic expansion for Ω n . Based on the obtained result, we also establish a sharp double inequality for Ω n .
{"title":"A new asymptotic expansion and sharp inequality for the volume of the unit ball in R^n","authors":"Xiao Zhang, Chao-Ping Chen","doi":"10.47443/ejm.2022.032","DOIUrl":"https://doi.org/10.47443/ejm.2022.032","url":null,"abstract":"For n ∈ { 1 , 2 , . . . } , let Ω n = π n/ 2 / Γ( n 2 + 1) be the volume of the unit ball in R n . In this paper, we give a new asymptotic expansion for Ω n . Based on the obtained result, we also establish a sharp double inequality for Ω n .","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76425562","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}
We evaluate in closed form a number of power series where the coefficients are products of Stirling numbers of the second kind and other special numbers or polynomials. The results include harmonic, hyperharmonic, derangement, Cauchy, Catalan numbers, zeta values, and also Bernoulli, Euler, and Laguerre polynomials.
{"title":"The Hadamard product of series with Stirling numbers of the second kind and other special numbers","authors":"Khristo N. Boyadzhiev, R. Frontczak","doi":"10.47443/ejm.2022.024","DOIUrl":"https://doi.org/10.47443/ejm.2022.024","url":null,"abstract":"We evaluate in closed form a number of power series where the coefficients are products of Stirling numbers of the second kind and other special numbers or polynomials. The results include harmonic, hyperharmonic, derangement, Cauchy, Catalan numbers, zeta values, and also Bernoulli, Euler, and Laguerre polynomials.","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85396205","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}
{"title":"Aspects of attitudes towards mathematics in modeling activities: Usefulness, interest, and social roles of mathematics","authors":"A. P. C. Lopes","doi":"10.29333/iejme/12394","DOIUrl":"https://doi.org/10.29333/iejme/12394","url":null,"abstract":"","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74808887","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}
By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.
{"title":"Fibonacci and Lucas Identities Derived via the Golden Ratio","authors":"K. Adegoke","doi":"10.47443/ejm.2022.018","DOIUrl":"https://doi.org/10.47443/ejm.2022.018","url":null,"abstract":"By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.","PeriodicalId":29770,"journal":{"name":"International Electronic Journal of Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90313799","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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