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Christou, Konstantinos P., Courtney Pollack, Jo Van Hoof, and Wim Van Dooren. "Natural number bias in arithmetic operations with missing numbers – A reaction time study." Journal of Numerical Cognition 6, no. 1 (2020): 22–49. http://dx.doi.org/10.5964/jnc.v6i1.228.
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When reasoning about numbers, students are susceptible to a natural number bias (NNB): When reasoning about non-natural numbers they use properties of natural numbers that do not apply. The present study examined the NNB when students are asked to evaluate the validity of algebraic equations involving multiplication and division, with an unknown, a given operand, and a given result; numbers were either small or large natural numbers, or decimal numbers (e.g., 3 × _ = 12, 6 × _ = 498, 6.1 × _ = 17.2). Equations varied on number congruency (unknown operands were either natural or rational numbers), and operation congruency (operations were either consistent – e.g., a product is larger than its operand – or inconsistent with natural number arithmetic). In a response-time paradigm, 77 adults viewed equations and determined whether a number could be found that would make the equation true. The results showed that the NNB affects evaluations in two main ways: a) the tendency to think that missing numbers are natural numbers; and b) the tendency to associate each operation with specific size of result, i.e., that multiplication makes bigger and division makes smaller. The effect was larger for items with small numbers, which is likely because these number combinations appear in the multiplication table, which is automatized through primary education. This suggests that students may count on the strategy of direct fact retrieval from memory when possible. Overall the findings suggest that the NNB led to decreased student performance on problems requiring rational number reasoning.
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Moss, Joan. "Research, Reflection, Practice: Introducing Percents in Linear Measurement to Foster an Understanding of Rational-Number Operations." Teaching Children Mathematics 9, no. 6 (2003): 335–39. http://dx.doi.org/10.5951/tcm.9.6.0335.
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How do we foster computational fluency with rational numbers when this topic is known to pose so many conceptual challenges for young students? How can we help students understand the operations of rational numbers when their grasp of the quantities involved in the rational-number system is often very limited? Traditional instruction in rational numbers focuses on rules and memorization. Teachers often give students instructions such as, “To add fractions, first find a common denominator, then add only the numerators” or “To add and subtract decimal numbers, line up the decimals, then do your calculations.”
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Schwarzweller, Christoph, and Artur Korniłowicz. "Characteristic of Rings. Prime Fields." Formalized Mathematics 23, no. 4 (2015): 333–49. http://dx.doi.org/10.1515/forma-2015-0027.
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Summary The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.
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Putra, Zetra Hainul. "Didactic Transposition of Rational Numbers: a Case From a Textbook Analysis and Prospective Elementary Teachers’ Mathematical and Didactic Knowledge." Revija za elementarno izobraževanje 13, no. 4 (2020): 365–94. http://dx.doi.org/10.18690/rei.13.4.365-394.2020.
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This study aims to present a detailed analysis of didactic transposition of rational numbers from knowledge to be taught into taught knowledge occurring in a teacher education institution. The knowledge to be taught of rational numbers is analysed from the mathematics textbook used by prospective elementary teachers in a mathematics education course. The analysis focuses on mathematical praxeology, especially the type of task and technique. Then, the taught knowledge is investigated from 32 prospective elementary teachers’ collaborative work on two hypothetical teacher tasks (HTT) related to operations with rational numbers.
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Nowlin, Douald. "Division with Fractions." Mathematics Teaching in the Middle School 2, no. 2 (1996): 116–19. http://dx.doi.org/10.5951/mtms.2.2.0116.
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Although operations with fractions have been in the elementary and middle school curriculum for many years, finding examples of practical problems that illustrate the usefulness of division with fractions and mixed numbers is not easy. Most real-world applications of rational numbers involve decimal numerals (Usiskin and Bell 1984), but examples of division with fractions and mixed numbers are often obviously contrived. In addition, many teachers and prospective teachers have difficulty constructing examples and concrete models for the operation of division with fractions (Borko 1992; Ball 1988).
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Gorbunov, Konstantin, and Vassily Lyubetsky. "Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions." Mathematics 8, no. 11 (2020): 2001. http://dx.doi.org/10.3390/math8112001.
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We propose a novel linear time algorithm which, given any directed weighted graphs a and b with vertex degrees 1 or 2, constructs a sequence of operations transforming a into b. The total cost of operations in this sequence is minimal among all possible ones or differs from the minimum by an additive constant that depends only on operation costs but not on the graphs themselves; this difference is small as compared to the operation costs and is explicitly computed. We assume that the double cut and join operations have identical costs, and costs of the deletion and insertion operations are arbitrary strictly positive rational numbers.
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Chung, Heewon, and Myungsun Kim. "Encoding of Rational Numbers and Their Homomorphic Computations for FHE-Based Applications." International Journal of Foundations of Computer Science 29, no. 06 (2018): 1023–44. http://dx.doi.org/10.1142/s0129054118500193.
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This work addresses a basic problem of security systems that operate on very sensitive information. Specifically, we are interested in the problem of privately handling numeric data represented by rational numbers (e.g., medical records). Fully homomorphic encryption (FHE) is one of the natural and powerful tools for ensuring privacy of sensitive data, while allowing complicated computations on the data. However, because the native plaintext domain of known FHE schemes is restricted to a set of quite small integers, it is not easy to obtain efficient algorithms for encrypted rational numbers in terms of space and computation costs. For example, the naïve decimal representation considerably restricts the choice of parameters in employing an FHE scheme, particularly the plaintext size. Our basic strategy is to alleviate this inefficiency by using a different representation of rational numbers instead of naïve expressions. In this work we express rational numbers as continued fractions. Because continued fractions enable us to represent rational numbers as a sequence of integers, we can use a plaintext space with a small size while preserving the same quality of precision. However, this encoding technique requires performing very complex arithmetic operations, such as division and modular reduction. Theoretically, FHE allows the evaluation of any function, including modular reduction at encrypted data, but it requires a Boolean circuit of very high degree to be constructed. Hence, the primary contribution of this work is developing an approach to solve this efficiency problem using homomorphic operations with small degrees.
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Yakes, Christopher. "Rational Number Operations on the Number Line." Mathematics Enthusiast 14, no. 1-3 (2017): 309–24. http://dx.doi.org/10.54870/1551-3440.1400.
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Jay, Ar I. Laranang Edd, En Jay B. Santiago Prince, Louise Oca Danielle, and Vincent T. Bongon Patrick. "Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner's Excellence in Mathematics)." International Journal of Social Science and Education Research Studies 05, no. 05 (2025): 435–44. https://doi.org/10.5281/zenodo.15469834.
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Abstract : This action research explores the impact of an innovative game-based intervention entitled “Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner’s Excellence in Mathematics),” on the academic performance of grade 7 students at Mount Carmel College of Casiguran, Inc., academic year 2024-2025. The research employs a quasi-experimental design with purposive sampling consisting of an experimental and control group, wherein the experimental group undergoes the intervention. A validated 30-item pre-test and post-test was the main instruments for this study and administered to 52 participants (26 experimental group and 26 control group) to measure the academic performance of both group. This game-based intervention was designed to foster critical thinking and strengthen mathematical skills over 20 days. Results revealed significant differences among learners exposed to Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner’s Excellence in Mathematics). The experimental group’s post-test mean score increased to outstanding, compared to the control group’s Did Not Meet Expectations. Statistical analysis confirmed a significant difference between the two groups’ performances post-intervention, with a very large effect size indicated. Additionally, a dependent t-test for the experimental group showed a notable improvement from their pre-test mean score. The study’s findings highlighted the effectiveness of Math SCRABBLE (Strengthening Critical Thinking and Problem-Solving, Reinforcing Rational Numbers and Integers, Applying Basic Operations for Building Learner’s Excellence in Mathematics) as an innovative educational tool addressing mathematical challenges and enhancing foundational skills. It also offers valuable insights for educators, emphasizing the importance of engaging and interactive pedagogies in improving learner outcomes in mathematics.
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Carvalho, Alexia Bezerra de, Rita de Cássia da Costa Guimarães, William Vieira, Emanoel Fabiano Menezes Pereira, and Roberto Seidi Imafuku. "Sobre dificuldades de ingressantes no ensino médio na compreensão de números racionais." ForScience 9, no. 1 (2021): e00878. http://dx.doi.org/10.29069/forscience.2021v9n1.e878.
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Apresenta-se, neste artigo, uma análise da resolução de uma questão sobre números racionais, exposta por estudantes ingressantes no Ensino Médio. O objetivo foi identificar as principais dificuldades e erros apresentados pelos participantes sobre o conceito e operações de números racionais. Para isso, realizou-se uma análise de erros das resoluções apresentadas pelos participantes. A interação de aspectos algorítmicos, intuitivos e formais é o referencial teórico adotado nas análises. Verificou-se que os estudantes não possuem clareza sobre o conceito de número racional e apresentam, em decorrência, dificuldades no uso dos procedimentos algorítmicos necessários para a resolução da questão proposta. Palavras-chave: Números racionais. Análise de erros. Dificuldades. Interpretação. Difficulties of high school entrants in understanding rational numbers Abstract This paper presents an analysis of the resolution of a question about rational numbers, posed by students entering high school. The goal was to identify the main difficulties and errors presented by the participants about the concept and operations of rational numbers. To do so, an error analysis of the resolutions presented by the participants was performed. The interaction of algorithmic, intuitive and formal aspects is the theoretical framework adopted in the analysis. It was found that students do not have clarity about the concept of rational numbers and present, as a result, difficulties in the use of algorithmic procedures necessary to solve the proposed question. Keywords: Rational numbers. Error analysis. Difficulties. Interpretation.
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Bergstra, Jan A. "Most General Algebraic Specifications for an Abstract Datatype of Rational Numbers." Scientific Annals of Computer Science XXX, no. 1 (2020): 1–24. https://doi.org/10.7561/SACS.2020.1.1.
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The notion of a most general algebraic specification of an arithmetical datatype of characteristic zero is introduced.Three examples of such specifications are given. A preference is formulated for a specification by means of infinitely many equations which can be presented via a finite number of so-called schematic equations phrased in terms of an infinite signature. On the basis of the latter specification three topics are discussed: (i) fracterm decomposition operators and the numerator paradox, (ii) foundational specifications of arithmetical datatypes, and (iii) poly-infix operations.
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Azim, Diane. "Understanding Multiplication as One Operation." Mathematics Teaching in the Middle School 7, no. 8 (2002): 466–71. http://dx.doi.org/10.5951/mtms.7.8.0466.
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Some of the wonder of numbers becomes apparent when we use numbers to perform calculations. Understanding multiplication with positive rational numbers is not a simple process. It requires reconceptualizing the meanings of multiplication with whole numbers to include numbers that are less than 1 or are mixed numbers. (Numbers in this article are all non-negative numbers.) To learn how multiplication works with fractions is to experience a whole new meaning for multiplication.
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LINES, DANIEL. "KNOTS WITH UNKNOTTING NUMBER ONE AND GENERALISED CASSON INVARIANT." Journal of Knot Theory and Its Ramifications 05, no. 01 (1996): 87–100. http://dx.doi.org/10.1142/s0218216596000072.
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We extend the classical notion of unknotting operation to include operations on rational tangles. We recall the “classical” conditions (on the signature, linking form etc.) for a knot to have integral (respectively rational) unknotting number one. We show that the generalised Casson invariant of the twofold branched cover of the knot gives a further necessary condition. We apply these results to some Montesinos knots and to knots with less than nine crossings.
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Lin, Xi. "Difficulties in Learning and Teaching Numbers: A Literature Review on the Obstacles and Misconceptions of Learners and Instructors." Journal of Contemporary Educational Research 6, no. 6 (2022): 111–18. http://dx.doi.org/10.26689/jcer.v6i6.4073.
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This paper attempts to summarize a number of research studies on numbers. The purpose of this study was to investigate and identify the obstacles encountered by students when they are dealing with number reasoning (whole numbers, integers, and rational numbers) and the difficulties faced by pre-service teachers in teaching arithmetic, including their misconceptions and weaknesses when they teach arithmetic and operations. There are two main sections in this paper: students’ cognitive obstacles for number reasoning, and pre-service teachers’ misconceptions of arithmetic. With the summarized misconceptions and obstacles of both, students and teachers, this paper provides efficient and effective thinking strategies that may help both, learners and instructors overcome obstacles, revise misconceptions, and strengthen their understanding, in order to develop proficiency in number reasoning and arithmetic operations.
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Egamnazarov, B. B. "ШОЛИЧИЛИК КЛАСТЕРИ БОШ МОДЕЛИ ВА МАШИНА ПАРКИ ТАРКИБИНИ АСОСЛАШ". Journal of Science and Innovative Development 6, № 3 (2023): 106–14. http://dx.doi.org/10.36522/2181-9637-2023-3-11.
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This article presents the results of a study on the development of a general model of the rice cluster and the determination of the composition, types and quantity of the required machine park. The concept of “General model of a rice-growing cluster” is introduced and its definition is given. The general model for sown area is substantiated by the following sequence: regional rice clusters are grouped according to the values of sown area; clusters with the largest sown areas were selected from the groups and the corresponding variational series of numbers was formed; the upper limit of the mathematical expectation of this series is taken as the rational sowing area of the general model. Using the standard coefficients (numbers) of technical equipment required per 1 000 hectares of rice land, the composition, brands and number of tractors and agricultural machines that perform technological operations on the sowing area of the rice-growing cluster are determined. In ricegrowing clusters with a rational technical park, all agrotechnical operations are carried out at the optimum time and agricultural products adapted for processing are grown.
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Fabulya, Zoltán. "Designing an Excel VBA function to recognize more important irrational numbers." Analecta Technica Szegedinensia 16, no. 1 (2022): 62–70. http://dx.doi.org/10.14232/analecta.2022.1.62-70.
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Calculations typically performed on a calculator or computer show the result as a decimal fraction if it is not an integer. It would be easier to interpret the result if a value could be expressed with integers and operations, such as the root subtraction operation. This article shows how this can be done with a developed algorithm in Microsoft Excel, which recognizes the most famous irrational numbers and displays them in text form together with the character of the operation sign. For example, “5√3/2” is given for 4.330127019. It is also useful to display irrational numbers with integers because only an infinite number of decimal places in a decimal fraction could show the exact value, and that is not possible. So, the developed algorithm can display a more interpretable and accurate form of the irrational number. In addition to the results that can be written as square roots, the algorithm is capable of displaying irrational numbers that can be expressed as the number Pi, using the π character. The Excel algorithm which was implemented in Visual Basic for Applications shows all rational numbers as the quotient of two integers that are relative primes.
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Ramírez, Juan. "A New Set Theory for Analysis." Axioms 8, no. 1 (2019): 31. http://dx.doi.org/10.3390/axioms8010031.
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We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.
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Siegler, Robert S., and Hugues Lortie-Forgues. "Hard Lessons: Why Rational Number Arithmetic Is So Difficult for So Many People." Current Directions in Psychological Science 26, no. 4 (2017): 346–51. http://dx.doi.org/10.1177/0963721417700129.
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Fraction and decimal arithmetic pose large difficulties for many children and adults. This is a serious problem because proficiency with these skills is crucial for learning more advanced mathematics and science and for success in many occupations. This review identifies two main classes of difficulties that underlie poor understanding of rational number arithmetic: inherent and culturally contingent. Inherent sources of difficulty are ones that are imposed by the task of learning rational number arithmetic, such as complex relations among fraction arithmetic operations. They are present for all learners. Culturally contingent sources of difficulty are ones that vary among cultures, such as teacher understanding of rational numbers. They lead to poorer learning among students in some places rather than others. We conclude by discussing interventions that can improve learning of rational number arithmetic.
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Abbasi, F., and T. Allahviranloo. "Fuzzy reliability estimation using the new operations of transmission average on Rational-linear patchy fuzzy numbers." Soft Computing 23, no. 10 (2018): 3383–96. http://dx.doi.org/10.1007/s00500-017-2996-6.
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Schneider, Gregory R. "Foliated compressing discs for Legendrian rational tangles." Journal of Knot Theory and Its Ramifications 25, no. 06 (2016): 1650029. http://dx.doi.org/10.1142/s0218216516500292.
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We establish a new framework for diagramming both Legendrian rational tangles in the standard contact structure on [Formula: see text] and the signed characteristic foliations of their associated compressing discs, as well as the technical means by which these diagrams can be used to study Legendrian isotopies of such tangles. We then establish a number of results that represent new progress in the ongoing effort to classify Legendrian rational tangles under a pair of operations known as Legendrian flypes. These operations, while topologically isotopies, are known to produce distinct Legendrian objects in many circumstances, a fact that has been of much interest throughout the study and classification of Legendrian knots.
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Zhou, Meijun, Jiayu Qin, Gang Mei, and John C. Tipper. "Simple and Robust Boolean Operations for Triangulated Surfaces." Mathematics 11, no. 12 (2023): 2713. http://dx.doi.org/10.3390/math11122713.
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Boolean operations on geometric models are important in numerical simulation and serve as essential tools in the fields of computer-aided design and computer graphics. The accuracy of these operations is heavily influenced by finite precision arithmetic, a commonly employed technique in geometric calculations, which introduces numerical approximations. To ensure robustness in Boolean operations, numerical methods relying on rational numbers or geometric predicates have been developed. These methods circumvent the accumulation of rounding errors during computation, thus preserving accuracy. Nonetheless, it is worth noting that these approaches often entail more intricate operation rules and data structures, consequently leading to longer computation times. In this paper, we present a straightforward and robust method for performing Boolean operations on both closed and open triangulated surfaces. Our approach aims to eliminate errors caused by floating-point operations by relying solely on entity indexing operations, without the need for coordinate computation. By doing so, we ensure the robustness required for Boolean operations. Our method consists of two main stages: (1) Firstly, candidate triangle intersection pairs are identified using an octree data structure, and then parallel algorithms are employed to compute the intersection lines for all pairs of triangles. (2) Secondly, closed or open intersection rings, sub-surfaces, and sub-blocks are formed, which is achieved entirely by cleaning and updating the mesh topology without geometric solid coordinate computation. Furthermore, we propose a novel method based on entity indexing to differentiate between the union, subtraction, and intersection of Boolean operation results, rather than relying on inner and outer classification. We validate the effectiveness of our method through various types of Boolean operations on triangulated surfaces.
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Li, Wan Qing, Zhe Zhao, and Ling Yu Meng. "Approach for Setting Project Buffer with Engineering Materials Based on the Theory of Unascertained Rational Number." Advanced Materials Research 459 (January 2012): 364–67. http://dx.doi.org/10.4028/www.scientific.net/amr.459.364.
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The paper proposes an approach for setting the size of project buffer based on the theory of unascertained rational numbers and critical chain management. This method applies the optimistic time into programming schedule plan and using the addition operation of unascertained rational numbers to obtain the unascertained schedule plan, to finish and generate cumulative reliability. This technique is applied to calculate the buffer time by using the optimistic construction period and total project period when finishing reliability is 90%. This feasibility of the model has been proved by an example
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Trümper, Manfred. "The Collatz Problem in the Light of an Infinite Free Semigroup." Chinese Journal of Mathematics 2014 (April 30, 2014): 1–21. http://dx.doi.org/10.1155/2014/756917.
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The Collatz (or 3m+1) problem is examined in terms of a free semigroup on which suitable diophantine and rational functions are defined. The elements of the semigroup, called T-words, comprise the information about the Collatz operations which relate an odd start number to an odd end number, the group operation being the concatenation of T-words. This view puts the concept of encoding vectors, first introduced in 1976 by Terras, in the proper mathematical context. A method is described which allows to determine a one-parameter family of start numbers compatible with any given T-word. The result brings to light an intimate relationship between the Collatz 3m+1 problem and the 3m-1 problem. Also, criteria for the rise or fall of a Collatz sequence are derived and the important notion of anomalous T-words is established. Furthermore, the concept of T-words is used to elucidate the question what kind of cycles—trivial, nontrivial, rational—can be found in the Collatz 3m+1 problem and also in the 3m-1 problem. Furthermore, the notion of the length of a Collatz sequence is discussed and applied to average sequences. Finally, a number of conjectures are proposed.
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Kärki, Tomi, Jake McMullen, and Erno Lehtinen. "Improving rational number knowledge using the NanoRoboMath digital game." Educational Studies in Mathematics 110, no. 1 (2021): 101–23. http://dx.doi.org/10.1007/s10649-021-10120-6.
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AbstractRational number knowledge is a crucial feature of primary school mathematics that predicts students’ later mathematics achievement. Many students struggle with the transition from natural number to rational number reasoning, so novel pedagogical approaches to support the development of rational number knowledge are valuable to mathematics educators worldwide. Digital game-based learning environments may support a wide range of mathematics skills. NanoRoboMath, a digital game-based learning environment, was developed to enhance students’ conceptual and adaptive rational number knowledge. In this paper, we tested the effectiveness of a preliminary version of the game with fifth and sixth grade primary school students (N = 195) using a quasi-experimental design. A small positive effect of playing the NanoRoboMath game on students’ rational number conceptual knowledge was observed. Students’ overall game performance was related to learning outcomes concerning their adaptive rational number knowledge and understanding of rational number representations and operations.
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Picciarelli, V., M. Di Gennaro, A. Loconsole, and R. Stella. "Development in mastering of rational number concepts and rational number operations: the role of gender, cognitive style and intellectual skills." International Journal of Mathematical Education in Science and Technology 26, no. 3 (1995): 407–16. http://dx.doi.org/10.1080/0020739950260309.
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Talambutsa, Alexey Leonidovich, and Tobias Hartnick. "Efficient computations with counting functions on free groups and free monoids." Sbornik: Mathematics 214, no. 10 (2023): 1458–99. http://dx.doi.org/10.4213/sm9683e.
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We present efficient algorithms to decide whether two given counting functions on nonabelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on nonabelian free groups are cohomologous. We work in the multi-tape Turing machine model with nonconstant-time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, that is, it is the same as in addition of rational numbers (implemented using the Harvey-van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions. Bibliography: 20 titles.
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Rahayuningsih, Sri, Ade Eka Anggriani, Mardhatillah, et al. "Examining natural number bias processes in elementary school students: A case study in Indonesia." Multidisciplinary Science Journal 7, no. 8 (2025): 2025381. https://doi.org/10.31893/multiscience.2025381.
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Elementary, middle, and college students often encounter difficulties in grasping rational numbers and their properties. This qualitative study utilized a case study approach to examine the cognitive processes of elementary students affected by natural number bias, contributing to the advancement of cognitive theories in mathematics instruction. Non-random purposive sampling was employed to select participants, with 171 fifth-grade students in Malang undergoing a paper-and-pencil exam. After screening, only seven students qualified as research participants. Data collection involved mathematical tests and video-audio interviews, with responses recorded and analyzed through coding and interpretation. Investigator triangulation ensured reliability. The findings reveal that students predominantly use rule-based strategies for fractional operations and number sense strategies for fractional comparisons, demonstrating a clear distinction in their approaches. Furthermore, a pattern of systematic errors was identified: students relying on rule-based strategies tended to make errors in conceptual tasks, while those using number sense strategies struggled with procedural tasks. These results emphasize the importance of balancing rule-based procedural skills and number sense understanding in teaching. The study recommends developing comprehensive mathematics learning materials that promote diagram representation, problem-solving strategies, conceptual understanding of fractions, and mathematical reasoning to support students' deeper understanding of rational numbers.
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Feliu, Elisenda. "A chain morphism for Adams operations on rational algebraic K-theory." Journal of K-Theory 5, no. 2 (2009): 349–402. http://dx.doi.org/10.1017/is009010016jkt085.
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AbstractFor any regular noetherian scheme X and every k ≥ 1, we define a chain morphism ψk between two chain complexes whose homology with rational coefficients is isomorphic to the algebraic K-groups of X tensored by ℚ. It is shown that the morphisms ψk induce in homology the Adams operations defined by Gillet and Soulé or the ones defined by Grayson.
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Masáková, Z., J. Patera, and E. Pelantová. "Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals." Canadian Journal of Physics 79, no. 2-3 (2001): 687–96. http://dx.doi.org/10.1139/p01-003.
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There are only three irrationalities directly related to experimentally observed quasicrystals, namely, those which appear in extensions of rational numbers by Ö5, Ö2, Ö3. In this article, we demonstrate that the algebraically defined aperiodic point sets with precisely these three irrational numbers play an exceptional role. The exceptional role stems from the possibility of equivalent characterization of these point sets using one binary operation. PACS Nos.: 61.90+d, 61.50-f
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Belinska, A., O. Varankina, O. Bliznjuk, N. Masalitina, and L. Krichkovska. "CONTROL AND OPERATIONS MANAGEMENT OF EXTRACTION PROCESS IN INDUSTRIAL BIOTECHNOLOGY OF BETA-CAROTINE FROM BLAKESLEA TRISPORA." Integrated Technologies and Energy Saving, no. 3 (November 9, 2021): 46–56. http://dx.doi.org/10.20998/2078-5364.2021.3.05.
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The technological parameters, namely temperature and duration of β-carotene extraction process from the biomass of filamentous fungus Blakeslea trispora with vegetable oils of various fatty acid compositions and with various contents of natural antioxidants (refined deodorized sunflower, viso-oleic sunflower, corn and sesame) have been investigated. Statistical models of dependences of β-carotene, as well as analytical numbers, characterizing the content of free fatty acids (acid number) and primary products of lipid oxidation (peroxide number) content, in oil extracts of the specified refined deodorized oils, from temperature and extraction duration have been built.
 Rational parameters of β-carotene extraction from Blakeslea trispora biomass with selected extractants (refined deodorized sunflower, high oleic sunflower, corn and sesame oils) have been determined for the extracts technological properties control. It has been proven that the use of these refined deodorized oils as extractants practically does not affect the content of the target product in oil extracts of biomass, but it does affect the analytical numbers of extracts characterizing the content of free fatty acids, peroxides and hydroperoxides. The highest content of free fatty acids in β-carotene containing biomass extraction with sunflower oil has been observed. The minimum content of free fatty acids in extracts with corn and sesame oils using has been be achieved. The highest content of primary products of lipid oxidation (peroxides and hydroperoxides) during β-carotene containing biomass extraction by sunflower oil has been observed. The minimum content of free fatty acids in extracts with sesame oil using has been achieved.
 It is possible to predict β-carotene content, as well as acid and peroxide numbers of oil extracts from biomass in specified refined deodorized oils, depending on temperature and extraction process duration using the obtained approximation dependences.
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Norton, Anderson, and Jesse L. M. Wilkins. "The Splitting Group." Journal for Research in Mathematics Education 43, no. 5 (2012): 557–83. http://dx.doi.org/10.5951/jresematheduc.43.5.0557.
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Piagetian theory describes mathematical development as the construction and organization of mental operations within psychological structures. Research on student learning has identified the vital roles of two particular operations–splitting and units coordination–play in students' development of advanced fractions knowledge. Whereas Steffe and colleagues (e.g., Steffe, 2001; Steffe & Olive, 2010) describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students' psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students' development with a structure that is isomorphic to the positive rational numbers under multiplication–the splitting group. We use a quantitative analysis of written assessments from 58 eighth grade students to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students' constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students' constructions of 3 levels of units coordination.
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Asadi, Mohammadali, Alexander Brandt, Robert H. C. Moir, and Marc Moreno Maza. "Algorithms and Data Structures for Sparse Polynomial Arithmetic." Mathematics 7, no. 5 (2019): 441. http://dx.doi.org/10.3390/math7050441.
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We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, respectively, where the divisor set is assumed to form a triangular set. Our operations make use of two data structures for sparse distributed polynomials and sparse recursively viewed polynomials, with a keen focus on locality and memory usage for optimized performance on modern memory hierarchies. Experimentation shows that these new implementations compare favorably against competing implementations, performing between a factor of 3 better (for multiplication over the integers) to more than 4 orders of magnitude better (for pseudo-division with respect to a triangular set).
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Tang, Shujue. "A TODIM Method Based on 2- Tuple Linguistic Neutrosophic Numbers and Its Application." Journal of Innovation and Development 2, no. 3 (2023): 78–83. http://dx.doi.org/10.54097/jid.v2i3.7280.
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This paper proposes a TODIM method based on 2-Tuple Linguistic Neutrosophic Numbers for the decision-making problem of bounded rational online shopping consumers. Firstly, the definition of 2- Tuple Linguistic Neutrosophic sets, normalized Hamming distance and the operation method of traditional TODIM method are introduced. Then the paper puts forward the operation steps of TODIM method based on 2-Tuple Linguistic Neutrosophic Numbers. Finally, an actual example analysis application is carried out to illustrate the operability and rationality of the method. Suggestions on recommendation function of shopping website and online business operators are put forward.
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Matczak, K., A. Mućka, and A. B. Romanowska. "Duality for dyadic intervals." International Journal of Algebra and Computation 29, no. 01 (2019): 41–60. http://dx.doi.org/10.1142/s0218196718500625.
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In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.
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Улитенко, Ю. А., та О. В. Кислов. "МЕТОД ОБҐРУНТУВАННЯ СХЕМИ ТА ВИБОРУ ПАРАМЕТРІВ СИЛОВОЇ УСТАНОВКИ ЛІТАЛЬНОГО АПАРАТУ ДЛЯ ШВИДКОСТЕЙ ПОЛЬОТУ МП = 0…5". Aerospace Technic and Technology, № 7 (7 липня 2017): 5–9. http://dx.doi.org/10.32620/aktt.2017.7.01.
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In this article, we briefly review the main aspects of the method of justifying the choice of the circuit and the parameters of the power plant of an aircraft with flight speeds from 0 to 5 Mach numbers in the early stages of design. The analysis of existing methods is carried out. The algorithm for performing operations for determining the optimum composition and parameters of the power plant, as well as the size of the aircraft depending on the mass of the payload, is stated and justified. Application of the obtained results will allow to shorten the terms of creation of competitive engines for high-speed aircraft due to a purposeful search for their rational thermodynamic and constructive-geometric appearance.
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Campit, Marvin Nonoy B., Cristine A. Enomis, Desiree T. Castillo, and Jane C. Tuan. "Addressing Learning Loss in Mathematics: A Research-Based Intervention." Jurnal Pendidikan Indonesia Gemilang 4, no. 1 (2024): 75–88. https://doi.org/10.53889/jpig.v4i1.348.
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This study aimed to investigate the "learning loss" resulting from recent academic disruptions. This study used a classroom action research, utilizing peer tutoring as the primary intervention amidst various psychological factors. The pretest identifies ten least-mastered mathematics skills, including simplifying numerical expressions, operations on integers, converting fractions to decimals, ordering rational numbers, and solving word problems. Challenges are also noted in approximating volume and area, understanding angles, and interpreting graphical data. Focus group discussions highlight the positive feedback for the intervention and factors contributing to learning loss, such as limited study time, connectivity issues, social media distractions, poor study habits, and math difficulties. Implementing tutorial intervention before the post-test effectively addresses learning loss and significantly reduces learning gaps.
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Marei, Sa'ida Tawfiq, and Khalid Muhammad Abo Loum. "The Effect of the Realistic Mathematics Education Approach (RME) in Improving the Conceptual and Operational Level of Proportional Thinking among the Classroom Teacher’s Students." Jordanian Educational Journal 9, no. 3 (2024): 403–27. https://doi.org/10.46515/jaes.v9i3.901.
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This study aimed to reveal the effect of the Realistic Mathematics Education approach (RME) in improving the conceptual and operational level in proportional thinking among the students of the classroom teacher’s students at the Hashemite University. Intentionally choosing two groups who registered the subject of numbers and operations on it, and one of the two groups was randomly assigned as an experimental group, and their number was (39) who were taught using the (RME), and (39) as a control group who were taught through using traditional way. To achieve the objectives of the study, the researchers built two tests of the conceptual and operational level in proportional thinking in the unit of rational numbers, the two study instruments were applied after checking the validity and reliability of the two tests. The results of the study showed that there were statistically significant differences at the conceptual level (analysis synthesis, and transfer) in favor of the experimental group, and there were no statistically significant differences at the operational level, and for both levels (accounts, applications).
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Cherchi, Gianmarco, Fabio Pellacini, Marco Attene, and Marco Livesu. "Interactive and Robust Mesh Booleans." ACM Transactions on Graphics 41, no. 6 (2022): 1–14. http://dx.doi.org/10.1145/3550454.3555460.
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Boolean operations are among the most used paradigms to create and edit digital shapes. Despite being conceptually simple, the computation of mesh Booleans is notoriously challenging. Main issues come from numerical approximations that make the detection and processing of intersection points inconsistent and unreliable, exposing implementations based on floating point arithmetic to many kinds of degeneracy and failure. Numerical methods based on rational numbers or exact geometric predicates have the needed robustness guarantees, that are achieved at the cost of increased computation times that, as of today, has always restricted the use of robust mesh Booleans to offline applications. We introduce an algorithm for Boolean operations with robustness guarantees that is capable of operating at interactive frame rates on meshes with up to 200K triangles. We evaluate our tool thoroughly, considering not only interactive applications but also batch processing of large collections of meshes, processing of huge meshes containing millions of elements and variadic Booleans of hundreds of shapes altogether. In all these experiments, we consistently outperform prior robust floating point methods by at least one order of magnitude.
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Cheng, Harry H. "Handling of Complex Numbers in the CHProgramming Language." Scientific Programming 2, no. 3 (1993): 77–106. http://dx.doi.org/10.1155/1993/427160.
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The handling of complex numbers in the CHprogramming language will be described in this paper. Complex is a built-in data type in CH. The I/O, arithmetic and relational operations, and built-in mathematical functions are defined for both regular complex numbers and complex metanumbers of ComplexZero, Complexlnf, and ComplexNaN. Due to polymorphism, the syntax of complex arithmetic and relational operations and built-in mathematical functions are the same as those for real numbers. Besides polymorphism, the built-in mathematical functions are implemented with a variable number of arguments that greatly simplify computations of different branches of multiple-valued complex functions. The valid lvalues related to complex numbers are defined. Rationales for the design of complex features in CHare discussed from language design, implementation, and application points of views. Sample CHprograms show that a computer language that does not distinguish the sign of zeros in complex numbers can also handle the branch cuts of multiple-valued complex functions effectively so long as it is appropriately designed and implemented.
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ZHANG, WEI, MARK YEARY, J. Q. TRELEWICZ, and MONTE TULL. "EFFICIENT COMPUTATION OF MULTIPLIERLESS FILTERS IN EMBEDDED SYSTEMS EMPLOYING AN OPTIMAL APPROXIMATION METHOD." International Journal of Computational Methods 03, no. 02 (2006): 177–204. http://dx.doi.org/10.1142/s0219876206000801.
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An integerization technique for creating fixed integer transforms with computationally optimal representations is presented, and the improved performance in embedded systems by employing these integerized implementations is explored. This technique uses an optimal approximation algorithm that finds the lowest-length fractional representation of the rational numbers. The integer transform approximation allows multiplication to be replaced by shift-and-add operations in hardware systems; where multiplication can take several cycles, shifts and adds take one or fewer cycles each. The multiplierless implementation furthermore benefits from employing the proposed method to represent the floating-point coefficients in very high precision requirement areas, like decimation filter design. This paper is strongly oriented around the design of coefficients with hardware constraints in mind, such as minimizing the number of required adds/subtracts and shifts required for some engineering algorithms.
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Potgieter, Petrus H. "Computability of sets in Euclidean space." Pure Mathematics and Applications 30, no. 3 (2022): 65–77. http://dx.doi.org/10.2478/puma-2022-0025.
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Abstract We consider several concepts of computability (recursiveness) for sets in Euclidean space. A list of four ideal properties for such sets is proposed and it is shown in a very elementary way that no notion can satisfy all four desiderata. Most notions introduced here are essentially based on separability of ℝ n and this is natural when thinking about operations on an actual digital computer where, in fact, rational numbers are the basis of everything. We enumerate some properties of some naïve but practical notions of recursive sets and contrast these with others, including the widely used and accepted notion of computable set developed by Weihrauch, Brattka and others which is based on the “Polish school” notion of a computable real function. We also offer a conjecture about the Mandelbrot set.
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Ural, Alattin. "A Classification of Mathematical Modeling Problems of Prospective Mathematics Teachers." Journal of Educational Issues 6, no. 1 (2020): 98. http://dx.doi.org/10.5296/jei.v6i1.16566.
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The purpose of this research is to classify the mathematical modelling problems produced by pre-service mathematics teachers in terms of the number of variables and to determine the mathematical modelling skills and mathematical skills used in solving the problems in each class. The current study is a qualitative research and the data was analyzed using descriptive analysis. The data of the study was obtained from the mathematical modelling problem written by 59 senior mathematics teachers. They were given a 1-week period to write the problems and solutions. The participants took mathematical modelling course for one semester period prior to the research. The problems are the original problems that the participants themselves produced. The mathematical modelling problems produced are categorically as follows: “Which option is more economical” problems, “Profit-making” problems, “Future prediction” problems and “Relationship between two quantities” problems. The mathematical modelling skills used are as follows: to be able to collect appropriate data, organize the data, write dependent and independent variables, write fixed values, visualize the real situation mathematically or geometrically, use mathematical concepts. The mathematical skills used are generally; to be able to do four operations with rational numbers, draw distribution and column graph, write algebraic expression, do arithmetic operation in algebraic rational expressions, write/solve equation and inequality in 1 or 2 variables, write an appropriate mathematical function explaining the data related to the data, solve 1st degree equations in 1 variable, establish proportion, use trigonometric ratios in right triangle, use basic geometry information, draw and interpret a 1st degree inequality in 2 variables.
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Shiu, Daniel, та Peter Shiu. "A poor person's approximation to π". Mathematical Gazette 96, № 537 (2012): 408–14. http://dx.doi.org/10.1017/s0025557200005027.
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Suppose that we have a computing machine that can only deal with the operations of addition, subtraction and multiplication, but not division, of integers. Can such a machine be used to find the decimal expansion of a given real number α to an arbitrary length? The answer is ‘yes’, at least in the sense that α is the limit of a sequence of rational numbers, and one can obtain the decimal expansion of a positive rational number a/b without division. For example, for any positive k, we try all non-negative c < b and d < 10ka, and see if 10ka = db + c. There will be success because all we are doing is reducing 10ka modulo b in a particularly moronic way. This guarantees the existence and uniqueness of d, and d/10k is then the desired expansion. Take, for example, = and k = 12; then, after a tedious search, we should find that, for c = 3, there is d = 230 769 230 769 because 3 × 1012 = 230769230769 × 13 + 3, so that = 0.230769230769 + × 10-12. Such a procedure of searching for c, d is hopelessly inefficient, of course, and a more efficient method is given in the next section.
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Okada, Isamu. "Evolution of cooperative study." Impact 2020, no. 8 (2020): 76–78. http://dx.doi.org/10.21820/23987073.2020.8.76.
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Associate Professor Isamu Okada is based at the Department of Business Administration, Faculty of Business Administration, Soka University in Japan, as well as a visiting professor of Department of Information Systems and Operations, Vienna University of Economics and Business in Austria. Okada has dedicated his career to understanding more about the evolution of cooperation which is a strand of thought that falls under evolutionary biology. Academics around the world have long considered the issues relating to the evolution of cooperation. In these studies, cooperation is taken to mean providing benefits to others by paying some kind of cost, whether that be money, time, effort, etc. One of the most fascinating aspects of the theory is that rational thought holds a person has no incentive to cooperate. Indeed, despite decades of research and huge numbers of studies, a rational reason for cooperating has still not been cultivated properly. One of the mechanisms that lie behind cooperation are known as reciprocity and there are many different types. Three specific types have been studied in great detail; direct reciprocity, indirect reciprocity and network reciprocity. Okada's team has conducted investigations that shine new light on indirect reciprocity which could open up new directions for the field of evolutionary biology.
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Volodymyr, Kudin, Onotskyi Viacheslav, Al-Ammouri Ali, and Shkvarchuk Lyudmyla. "ADVANCEMENT OF A LONG ARITHMETIC TECHNOLOGY IN THE CONSTRUCTION OF ALGORITHMS FOR STUDYING LINEAR SYSTEMS." Eastern-European Journal of Enterprise Technologies 1, no. 4 (97) (2019): 14–22. https://doi.org/10.15587/1729-4061.2019.157521.
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We have advanced the application of algorithms within a method of basic matrices, which are equipped with the technology of long arithmetic to improve the precision of performing the basic operations in the course of studying the ill-conditioned linear systems, specifically, the systems of linear algebraic equations (SLAE). Identification of the fact of ill-conditionality of a system is a rather time-consuming computational procedure. The possibility to control computations entering the state of incorrectness and the impossibility of accumulating calculation errors, which is a desirable property of the methods and algorithms for solving practical problems, were introduced. Modern computers typically use the standard types of integers whose size does not exceed 64 bytes. This hardware limitation was resolved using software, specifically, by developing a proprietary type of data in the form of a special Longnum library in the C++ language (using the STL (Standard Template Library)). Software implementation was aimed at carrying out computations for methods of basic matrices (MBM) and Gauss matrices, that is, long arithmetic for models with rational elements was used. We have proposed the algorithms and computer realization of the Gauss type methods and methods of artificial basic matrices (a variant of the method of basic matrices) in MatLAB environment and Visual C++ environment using precise computation of the methods' elements, first of all, for the ill-conditioned systems of varying dimensionality. The Longnum library with the types of long integers (longint3) and rational numbers (longrat3) with the numerator and denominator of the longint3 type was developed. Arithmetic operations on long integers were performed based on the modern methods, including the Strassen multiplication method. We give the results from the computational experiment employing the mentioned methods, in which test models of the systems were generated, specifically, based on the Gilbert matrices of different dimensionality
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Hill, Heather C. "The Nature and Predictors of Elementary Teachers' Mathematical Knowledge for Teaching." Journal for Research in Mathematics Education 41, no. 5 (2010): 513–45. http://dx.doi.org/10.5951/jresematheduc.41.5.0513.
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This article explores elementary school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teacher characteristics. The Learning Mathematics for Teaching project administered a multiple-choice assessment covering topics in number and operation to a nationally representative sample of teachers (n = 625) and at the same time collected information on teacher and student characteristics. Performance did not vary according to mathematical topic (e.g., whole numbers or rational numbers), and items categorized as requiring specialized knowledge of mathematics proved more difficult for this sample of teachers. There were few substantively significant relationships between mathematical knowledge for teaching and teacher characteristics, including leadership activities and self-reported college-level mathematics preparation. Implications for current policies aimed at improving teacher quality are addressed.
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Frič, Roman, and Martin Papčo. "Upgrading Probability via Fractions of Events." Communications in Mathematics 24, no. 1 (2016): 29–41. http://dx.doi.org/10.1515/cm-2016-0004.
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Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.
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Gaibyldaev, Zhanybek, Zhamalbek Ashimov, Damirbek Abibillaev, and Fuat Kocyigit. "Survival analysis of renal patients underwent transplantation in Kyrgyz Republic and various countries by 10 years follow-up." Heart, Vessels and Transplantation 3, Issue 4 (2019): 243. http://dx.doi.org/10.24969/hvt.2019.168.
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In our study we conducted survival analysis of 204 patients visited Scientific-Research Institute of Heart Surgery and Organs transplantation and who underwent renal transplantation in Kyrgyzstan and other Eurasian countries between 2005 and 2016 years (age range: 9-71 years, mean: 38.21 (12.74) years, median: 34.0 (0.89) years; gender: 142 male (69.6%)). During follow-up period, mortality event was observed in 16 (7.84%) patients. Survival function probabilities of patients and rational risk factors of survival functions were evaluated by Kaplan-Meier and Cox regression analyses, respectively. According to Kaplan-Meier results survival probabilities calculated for 1st year: 0.96 (0.014), for 3rd year: 0.94 (0.018), for 5th year: 0.86 (0.04), for 7th year: 0.75 (0.10). Among age groups 28-39 age ranges prevailed by 11 patients. Nevertheless, that difference did not show statistical significance: p˃0.322. The intensity of transplantation also analyzed according to years, which revealed increasing in numbers of operations by time. For instance, when in 2006 only two cases were registered in our center, but numbers of transplanted patients reached up to 48 in 2015. The association of mortality states and years of transplantation found significantly by Kaplan-Meier test (Breslow p˂0.001). The survival analysis was compared according to countries and revealed significant results (Breslow p˂0.05). From other factors influencing mortality, sex did not show strong impact on survival by Kaplan-Meier analysis, but significant association was found by Cox regression analysis.
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Moxness, Vidar Weidemann, Knut Gåseidnes, and Harald Asheim. "Skimmer Capacity for Viscous Oil." SPE Journal 16, no. 01 (2010): 155–61. http://dx.doi.org/10.2118/118701-pa.
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Summary When a skimmer removes oil locally, oil floating further away will flow toward it. The maximum flow rate toward the skimmer defines its natural capacity. Traditional skimmer-capacity modeling considers flow driven by height potential and resisted by inertial forces but neglects viscosity. On the basis of theory and experiments, this paper claims that high oil viscosity may govern the skimmer capacity. It is shown that viscous resistance relates to a dimensionless quantity called the Goose number, representing the ratio of inertial to viscous forces. At high Goose numbers, viscosity may be neglected. At sufficiently low Goose numbers, viscous resistance dominates. A numerical solution applicable to all Goose numbers has been developed. Analytical formulas for skimmer capacity at high and low Goose numbers are provided. A scaled laboratory facility was built to investigate the skimming of viscous oil. The measured rates were consistent with the numerical predictions and with the formula for low Goose number. With decreasing viscosity, predictions by the current model converged to traditional formulas that neglect viscosity. The model quantifies how skimming capacity is affected by size and by properties of the oil spill and skimmer geometry and submergence. This may enable more rational skimmer design and operation, and even optimization.
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Strauss, D. F. M. "Philosophical tendencies in the genesis of our understanding of physical nature." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 25, no. 2 (2006): 93–110. http://dx.doi.org/10.4102/satnt.v25i2.150.
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The rise of a long-standing legacy of natural scientific thought is found in ancient Greece – the well-spring of Western civilization and the source of articulated rational reflection. The earliest phase of Greek culture already gave birth to theoretical thinking about the universe. The Pythagoreans are first of all famous for their emphasis on number as a mode of explanation. However, in their thesis that everything is number they solely acknowledged rational numbers (fractions) and this approach eventually stranded on the discovery of irrational numbers that led to the geometrization of Greek mathematics. This transition generated at once also a powerful space metaphysics overarching the entire medieval period. It was only during the early modern period that the predecessors and successors of Galileo contemplated an appreciation for motion as a new principle of explanation (compare the classical mechanistic world view of the universe as a mechanism of material particles in motion). But also this mechanistic reduction (through which all physical processes were reduced to the motion of charged or uncharged mass-points) eventually failed because it was unable to account for the irreversibility of physical processes. As a result it was only 20th century physics that managed to acknowledge the decisive qualifying role of energy-operation (thus of the physical aspect) in the existence of material things and processes. This article is concluded with an explanation of the significance of the preceding considerations for a theoretical approximation of the mysterious nature of matter.