Pub Date : 2023-10-19DOI: 10.1080/27684830.2023.2269657
Wei Ren
We propose reduced Collatz conjecture that is equivalent to Collatz conjecture, which states that every positive integer can return to an integer less than it, instead of 1. Reduced Collatz conjecture is easier to explore because certain structures must be presented in reduced dynamics, rather than in original dynamics (as original dynamics is a mixture of original dynamics). Reduced dynamics is a computation sequence from starting integer to the first integer less than it, in terms of “I” that represents (3×+1)/2 and “O” that represents x/2. We formally prove that all positive integers are partitioned into two halves and either present “I” or “O” in next ongoing computation. More specifically, (1) if any positive integer x that is i module 2t (i is an odd integer) is given, then the first t computations (each one is either “I” or “O” corresponding to whether current integer is odd or even) will be identical with that of i. (2) If current integer after t computations (in terms of “I” or “O”) is less than x, then reduced dynamics of x is available. Otherwise, the residue class of x (namely i module 2t) can be partitioned into two halves (namely i module 2t+1 and i+2t module 2t+1), and either half presents “I” or “O” in the intermediately forthcoming (t + 1)-th computation. This discovery will be helpful to the final proof of Collatz conjecture—if the union of residue classes who present reduced dynamics that become larger with the growth of residue module, equals all positive integers asymptotically, then reduced Collatz conjecture (or equivalently, Collatz conjecture) will be true.
{"title":"Collatz dynamics is partitioned by residue class regularly","authors":"Wei Ren","doi":"10.1080/27684830.2023.2269657","DOIUrl":"https://doi.org/10.1080/27684830.2023.2269657","url":null,"abstract":"We propose reduced Collatz conjecture that is equivalent to Collatz conjecture, which states that every positive integer can return to an integer less than it, instead of 1. Reduced Collatz conjecture is easier to explore because certain structures must be presented in reduced dynamics, rather than in original dynamics (as original dynamics is a mixture of original dynamics). Reduced dynamics is a computation sequence from starting integer to the first integer less than it, in terms of “I” that represents (3×+1)/2 and “O” that represents x/2. We formally prove that all positive integers are partitioned into two halves and either present “I” or “O” in next ongoing computation. More specifically, (1) if any positive integer x that is i module 2t (i is an odd integer) is given, then the first t computations (each one is either “I” or “O” corresponding to whether current integer is odd or even) will be identical with that of i. (2) If current integer after t computations (in terms of “I” or “O”) is less than x, then reduced dynamics of x is available. Otherwise, the residue class of x (namely i module 2t) can be partitioned into two halves (namely i module 2t+1 and i+2t module 2t+1), and either half presents “I” or “O” in the intermediately forthcoming (t + 1)-th computation. This discovery will be helpful to the final proof of Collatz conjecture—if the union of residue classes who present reduced dynamics that become larger with the growth of residue module, equals all positive integers asymptotically, then reduced Collatz conjecture (or equivalently, Collatz conjecture) will be true.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135779844","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 : 2023-10-18DOI: 10.1080/27684830.2023.2264581
Kbenesh W. Blayneh
Vertical transmissions of vector-borne diseases such as dengue virus, malaria, West Nile virus and Rift Valley fever are among challenging factors in disease control. In this paper, we used a system of differential equations to study the impacts of vertical transmission (in the vector) and horizontal (vector-to-host) transmissions cycle on the uniform persistence of a vertically transmitted disease. This is given analytically when the secondary reproduction number is greater than unity. Additional numerical results indicate that vertical transmission rates raise the epidemic level. However, we provide analytical results to reveal that vertical transmission of the disease in vectors alone without horizontal transmission, could not make the disease endemic. Furthermore, by transforming the system of differential equations to a nonlinear eigenvalue equation, the existence of a backward bifurcation is established. The directions of bifurcation is forward when the disease-induced death rate of hosts is reduced below a threshold. It is shown that the bifurcation is backward when the disease-induced death rate in the host is above a critical value. Additionally, increased disease transmission parameters also contribute to backward bifurcation. Numerical results are also provided to highlight that, depending on the initial conditions, the disease could be persistent even when the secondary reproduction number is less than unity.
{"title":"Uniform persistence and backward bifurcation of vertically transmitted vector-borne diseases","authors":"Kbenesh W. Blayneh","doi":"10.1080/27684830.2023.2264581","DOIUrl":"https://doi.org/10.1080/27684830.2023.2264581","url":null,"abstract":"Vertical transmissions of vector-borne diseases such as dengue virus, malaria, West Nile virus and Rift Valley fever are among challenging factors in disease control. In this paper, we used a system of differential equations to study the impacts of vertical transmission (in the vector) and horizontal (vector-to-host) transmissions cycle on the uniform persistence of a vertically transmitted disease. This is given analytically when the secondary reproduction number is greater than unity. Additional numerical results indicate that vertical transmission rates raise the epidemic level. However, we provide analytical results to reveal that vertical transmission of the disease in vectors alone without horizontal transmission, could not make the disease endemic. Furthermore, by transforming the system of differential equations to a nonlinear eigenvalue equation, the existence of a backward bifurcation is established. The directions of bifurcation is forward when the disease-induced death rate of hosts is reduced below a threshold. It is shown that the bifurcation is backward when the disease-induced death rate in the host is above a critical value. Additionally, increased disease transmission parameters also contribute to backward bifurcation. Numerical results are also provided to highlight that, depending on the initial conditions, the disease could be persistent even when the secondary reproduction number is less than unity.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135884164","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 : 2023-10-17DOI: 10.1080/27684830.2023.2266902
Wondwosen Zemene Norahun
In this paper, we study the concept of O-fuzzy ideals in distributive lattices with the least element 0 as a generalization of δ-fuzzy ideals. We investigate the properties of O-fuzzy ideals and establishes their relationships with other types of fuzzy ideals. We observe that χ{0} may not be an O-fuzzy ideal. Hence, we derive a necessary and sufficient condition for χ{0} to become an O-fuzzy ideal. It is proved that the class of O-fuzzy is a distributive lattice. Moreover, we establish a necessary and sufficient condition for the class of O-fuzzy ideals to be a complete distributive lattice.
{"title":"O-Fuzzy ideals in distributive lattices","authors":"Wondwosen Zemene Norahun","doi":"10.1080/27684830.2023.2266902","DOIUrl":"https://doi.org/10.1080/27684830.2023.2266902","url":null,"abstract":"In this paper, we study the concept of O-fuzzy ideals in distributive lattices with the least element 0 as a generalization of δ-fuzzy ideals. We investigate the properties of O-fuzzy ideals and establishes their relationships with other types of fuzzy ideals. We observe that χ{0} may not be an O-fuzzy ideal. Hence, we derive a necessary and sufficient condition for χ{0} to become an O-fuzzy ideal. It is proved that the class of O-fuzzy is a distributive lattice. Moreover, we establish a necessary and sufficient condition for the class of O-fuzzy ideals to be a complete distributive lattice.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135993032","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 : 2023-10-17DOI: 10.1080/14794802.2023.2260356
Taro Fujita
Click to increase image sizeClick to decrease image size Disclosure statementNo potential conflict of interest was reported by the author(s).
点击放大图片点击缩小图片披露声明作者未发现潜在的利益冲突。
{"title":"BSRLM day conference proceedings","authors":"Taro Fujita","doi":"10.1080/14794802.2023.2260356","DOIUrl":"https://doi.org/10.1080/14794802.2023.2260356","url":null,"abstract":"Click to increase image sizeClick to decrease image size Disclosure statementNo potential conflict of interest was reported by the author(s).","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135993397","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 : 2023-10-12DOI: 10.1080/14794802.2023.2261410
Esther Brunner
Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsEsther BrunnerEsther Brunner is a Professor of Mathematics Education at the Thurgau University of Teacher Education in Kreuzlingen in Switzerland, where she directs the Mathematics Department. Her research aims to understand mathematics teaching processes, the quality of mathematics instructions from a content-specific perspective, mathematical reasoning and proof, and early mathematics education. She combines her research with a reference to practice and makes results usable for schools.
{"title":"Helping Students Become Powerful Mathematical Thinkers. Case Studies of Teaching for Robust Understanding“ <b>Helping Students Become Powerful Mathematical Thinkers. Case Studies of Teaching for Robust Understanding</b> ” by Alan Schoenfeld, Heather Fink, and Sandra Zuñiga-Ruiz, with Siqi Huang, Xinyu Wei, and Brantina Chirinda, New York/London, Routledge, 2023, 255 pp., ISBN: 978-1-03-244168-9“ <b>Mathematics Teaching on Target. A Guide to Teaching for Robust Understanding at All Grade Levels…","authors":"Esther Brunner","doi":"10.1080/14794802.2023.2261410","DOIUrl":"https://doi.org/10.1080/14794802.2023.2261410","url":null,"abstract":"Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsEsther BrunnerEsther Brunner is a Professor of Mathematics Education at the Thurgau University of Teacher Education in Kreuzlingen in Switzerland, where she directs the Mathematics Department. Her research aims to understand mathematics teaching processes, the quality of mathematics instructions from a content-specific perspective, mathematical reasoning and proof, and early mathematics education. She combines her research with a reference to practice and makes results usable for schools.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136013541","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 paper, we study the concept of fuzzy ℓ-ideals and fuzzy ℓ-filters of an ordered group. We provide several characterizations for fuzzy ℓ-ideals and fuzzy ℓ-filters in different perspective. We observe that for any fuzzy convex sub ℓ-group we can find a fuzzy ℓ-ideal and a fuzzy ℓ-filter containing it. Finally, for any fuzzy convex sub ℓ-group can be expressed as the intersection of some fuzzy ℓ-ideals and fuzzy ℓ-filters containing it.
{"title":"Some results on fuzzy ℓ-ideals and fuzzy ℓ-filters in ℓ-group","authors":"Taye Sisay Tadesse, Kishore Parimi Radhakrishna, Melesse Mulugeta Habte","doi":"10.1080/27684830.2023.2259687","DOIUrl":"https://doi.org/10.1080/27684830.2023.2259687","url":null,"abstract":"In this paper, we study the concept of fuzzy ℓ-ideals and fuzzy ℓ-filters of an ordered group. We provide several characterizations for fuzzy ℓ-ideals and fuzzy ℓ-filters in different perspective. We observe that for any fuzzy convex sub ℓ-group we can find a fuzzy ℓ-ideal and a fuzzy ℓ-filter containing it. Finally, for any fuzzy convex sub ℓ-group can be expressed as the intersection of some fuzzy ℓ-ideals and fuzzy ℓ-filters containing it.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135538860","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 : 2023-09-24DOI: 10.1080/27684830.2023.2259178
I. Jahan, N. Ajmal
{"title":"The join problem of subnormal <i>L</i> -subgroups of an <i>L</i> -group","authors":"I. Jahan, N. Ajmal","doi":"10.1080/27684830.2023.2259178","DOIUrl":"https://doi.org/10.1080/27684830.2023.2259178","url":null,"abstract":"","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135924317","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 paper, we introduce the concept of γ−fuzzy filters in a pseudo-complemented distributive lattices. We study the special class of fuzzy filters called γ−fuzzy filters, which is isomorphic to the set of all fuzzy ideals of the lattice of boosters. We observe that every γ−fuzzy filters is the intersection of all prime γ−fuzzy filters containing it. We also topologize the set of all prime γ−fuzzy filters of a pseudo-completed distributive lattice. Properties of the space also studied. We show that there is a one to one correspondence between the class of γ−fuzzy filters and the lattice of all open sets in Pγ. It is proved that the space Pγ is a T0 space.
{"title":"<i>γ</i> -fuzzy filters and space of <i>γ</i> -- fuzzy filters in P-algebra","authors":"Wondwosen Zemene Norahun, Teferi Getachew Alemayehu","doi":"10.1080/27684830.2023.2258591","DOIUrl":"https://doi.org/10.1080/27684830.2023.2258591","url":null,"abstract":"In this paper, we introduce the concept of γ−fuzzy filters in a pseudo-complemented distributive lattices. We study the special class of fuzzy filters called γ−fuzzy filters, which is isomorphic to the set of all fuzzy ideals of the lattice of boosters. We observe that every γ−fuzzy filters is the intersection of all prime γ−fuzzy filters containing it. We also topologize the set of all prime γ−fuzzy filters of a pseudo-completed distributive lattice. Properties of the space also studied. We show that there is a one to one correspondence between the class of γ−fuzzy filters and the lattice of all open sets in Pγ. It is proved that the space Pγ is a T0 space.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236964","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 : 2023-09-14DOI: 10.1080/27684830.2023.2230705
Carlos Morton Barrera
There have been derivations for the Sums of Powers published since the sixteenth century. All techniques have used recursive processes, producing the following formula in the series. I present a new method that calculates the Sums of Powers and Harmonic Numbers. Starting with a novel relationship between Pascal’s Numbers and Stirling’s Numbers of the First Kind, the Sums of Powers is developed. This formula, published previously using a different methodology, is in terms of Pascal Numbers multiplied by constant coefficients. However, a further step is introduced. A recursive relationship is discovered among the coefficients of these formulae. A double sigma master formula is developed, allowing one to calculate all formulae for Sums of Powers without needing Bernoulli Numbers. Finally, from the Sums of Powers master formula, I derive a formula to calculate the Bernoulli Numbers. I further develop a summation formula for the Harmonic Numbers using the same relationships.
{"title":"Sums of Powers and Harmonic Numbers: A new approach","authors":"Carlos Morton Barrera","doi":"10.1080/27684830.2023.2230705","DOIUrl":"https://doi.org/10.1080/27684830.2023.2230705","url":null,"abstract":"There have been derivations for the Sums of Powers published since the sixteenth century. All techniques have used recursive processes, producing the following formula in the series. I present a new method that calculates the Sums of Powers and Harmonic Numbers. Starting with a novel relationship between Pascal’s Numbers and Stirling’s Numbers of the First Kind, the Sums of Powers is developed. This formula, published previously using a different methodology, is in terms of Pascal Numbers multiplied by constant coefficients. However, a further step is introduced. A recursive relationship is discovered among the coefficients of these formulae. A double sigma master formula is developed, allowing one to calculate all formulae for Sums of Powers without needing Bernoulli Numbers. Finally, from the Sums of Powers master formula, I derive a formula to calculate the Bernoulli Numbers. I further develop a summation formula for the Harmonic Numbers using the same relationships.","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911503","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 : 2023-09-12DOI: 10.1080/14794802.2023.2243261
Birke-Johanna Weber, Jana Breuer, Anke Lindmeier
{"title":"How do school-related mathematical problems become relevant for prospective teachers in mathematics courses at university? A qualitative interview study","authors":"Birke-Johanna Weber, Jana Breuer, Anke Lindmeier","doi":"10.1080/14794802.2023.2243261","DOIUrl":"https://doi.org/10.1080/14794802.2023.2243261","url":null,"abstract":"","PeriodicalId":45396,"journal":{"name":"Research in Mathematics Education","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135877996","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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