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Добірка наукової літератури з теми "Numbers, Rational"

Автор: Grafiati

Опубліковано: 4 червня 2021

Оновлено: 26 липня 2024

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Статті в журналах з теми "Numbers, Rational"

., Jyoti. "Rational Numbers." Journal of Advances and Scholarly Researches in Allied Education 15, no. 5 (July 1, 2018): 220–22. http://dx.doi.org/10.29070/15/57856.

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Scott Malcom, P. "Understanding Rational Numbers." Mathematics Teacher 80, no. 7 (October 1987): 518–21. http://dx.doi.org/10.5951/mt.80.7.0518.

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Understanding is sometimes an elusive goal in mathematics. Although we may believe we have a complete understanding of a concept, another approach to this same concept may bring us additional insight.

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Lennerstad, Håkan, and Lars Lundberg. "Decomposing rational numbers." Acta Arithmetica 145, no. 3 (2010): 213–20. http://dx.doi.org/10.4064/aa145-3-1.

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PEYTON JONES, SIMON. "12 Rational Numbers." Journal of Functional Programming 13, no. 1 (January 2003): 149–52. http://dx.doi.org/10.1017/s0956796803001412.

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Frougny, Christiane, and Karel Klouda. "Rational base number systems forp-adic numbers." RAIRO - Theoretical Informatics and Applications 46, no. 1 (August 22, 2011): 87–106. http://dx.doi.org/10.1051/ita/2011114.

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Xin Liu, Xin Liu, Xiaomeng Liu Xin Liu, Dan Luo Xiaomeng Liu, Gang Xu Dan Luo, and Xiu-Bo Chen Gang Xu. "Confidentially Compare Rational Numbers under the Malicious Model." 網際網路技術學刊 25, no. 3 (May 2024): 355–63. http://dx.doi.org/10.53106/160792642024052503002.

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<p>Secure multi-party computation is a hotspot in the cryptography field, and it is also a significant means to realize privacy computation. The Millionaires’ problem is the most fundamental problem among them, which is the basic module of secure multi-party computation protocols. Although there are many solutions to this problem, there are few anti-malicious adversarial protocols besides protocols based on Yao’s garbled circuit. Only a few solutions have low efficiency, and there is no protocol for rational numbers comparison under the malicious model, which restricts the solution of many secure multi-party computation problems. In this paper, the possible malicious behaviors are analyzed in the existing Millionaires’ problem protocols. These behaviors are discovered and taken precautions against through the triangle area formula, zero-knowledge proof, and cut-and-choose method, so the protocol of comparing confidentially rational numbers is proposed under the malicious model. And this paper adopts the real/ideal model paradigm to prove the security of the malicious model protocol. Efficiency analysis indicates that the proposed protocol is more effective than existing protocols. The protocol of rational numbers comparison under the malicious model is more suitable for the practical applications of secure multi-party computation, which has important theoretical and practical significance.</p> <p> </p>

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Roy, Damien, and Johannes Schleischitz. "Numbers with Almost all Convergents in a Cantor Set." Canadian Mathematical Bulletin 62, no. 4 (December 3, 2018): 869–75. http://dx.doi.org/10.4153/s0008439518000450.

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AbstractIn 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.

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Belin, Mervenur, and Gülseren Karagöz Akar. "Exploring Real Numbers as Rational Number Sequences With Prospective Mathematics Teachers." Mathematics Teacher Educator 9, no. 1 (September 1, 2020): 63–87. http://dx.doi.org/10.5951/mte.2020.9999.

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The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students’ understanding of real numbers. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real numbers as rational number sequences. We found that the instructional sequence enhanced prospective teachers’ understanding of real numbers by considering them as quantities and explaining them by using rational number sequences. In particular, results showed that prospective teachers reasoned about fractions and decimal representations of rational numbers using long division, the division algorithm, and diagrams. This further prompted their reasoning with decimal representations of rational and irrational numbers as rational number sequences, which leads to authentic construction of real numbers. Enacting the instructional sequence provides lenses for mathematics teacher educators to notice and eliminate difficulties of their students while developing relationships among multiple representations of real numbers.

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Korhonen, Risto. "Approximation of real numbers with rational number sequences." Proceedings of the American Mathematical Society 137, no. 01 (August 14, 2008): 107–13. http://dx.doi.org/10.1090/s0002-9939-08-09479-3.

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Gong, Linming, Bo Yang, Tao Xue, Jinguang Chen, and Wei Wang. "Secure rational numbers equivalence test based on threshold cryptosystem with rational numbers." Information Sciences 466 (October 2018): 44–54. http://dx.doi.org/10.1016/j.ins.2018.07.046.

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Дисертації з теми "Numbers, Rational"

Ketkar, Pallavi S. (Pallavi Subhash). "Primitive Substitutive Numbers are Closed under Rational Multiplication." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278637/.

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Lehr (1991) proved that, if M(q, r) denotes the set of real numbers whose expansion in base-r is q-automatic i.e., is recognized by an automaton A = (Aq, Ar, ao, δ, φ) (or is the image under a letter to letter morphism of a fixed point of a substitution of constant length q) then M(q, r) is closed under addition and rational multiplication. Similarly if we let M(r) denote the set of real numbers α whose base-r digit expansion is ultimately primitive substitutive, i.e., contains a tail which is the image (under a letter to letter morphism) of a fixed point of a primitive substitution then in an attempt to generalize Lehr's result we show that the set M(r) is closed under multiplication by rational numbers. We also show that M(r) is not closed under addition.

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Coward, Daniel R. "Sums of two rational cubes." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.320587.

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Brown, Bruce John Lindsay. "The initial grounding of rational numbers : an investigation." Thesis, Rhodes University, 2007. http://hdl.handle.net/10962/d1006351.

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This small scale exploratory research project investigated the grounding of rational number concepts in informal, everyday life situations. A qualitative approach was taken to allow for the identification and then in depth investigation, of issues of importance for such a grounding of rational number understanding. The methodology followed could be seen as a combination of grounded theory and developmental research. And the data was generated through in-depth and clinical interviews structured around a number of grounded tasks related to rational numbers. The research comprised three cycles of interviews that were transcribed and then analysed in detail, interspersed with periods of reading and reflection. The pilot cycle involved a single grade three teacher, the second cycle involved 2 grade three teachers and the third cycle involved 2 grade three children. The research identified a number of different perspectives that were all important for the development of a fundamental intuitive understanding that could be considered personally meaningful to the individual concerned and relevant to the development of rational number concepts. Firstly in order to motivate and engage the child on a personal level the grounding situation needed to be seen as personally significant by the child. Secondly, coordinating operations provided a means of developing a fundamental intuitive understanding, through coordination with affording structures of the situation that are relevant to rational numbers. Finally, goal directed actions that are deliberately structured to achieve explicit goals in a situation are important for the development of more explicit concepts and skills fundamental for rational number understanding. Different explicit structures give rise to different interpretations of rational numbers in grounding situations. In addition to these perspectives, it became evident that building and learning representations was important for developing a more particularly mathematical understanding, based on the fundamental understanding derived from the child's grounded experience. The conclusion drawn in this research as a result of this complexity, is that to achieve a comprehensive and meaningful grounding, children's learning of rational numbers will not follow a simple linear trajectory. Rather this process forms a web of learning, threading coordinating operations for intuitive development, interpretations for explicit grounding and representations to develop more formal mathematical conceptions.

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Shaughnessy, John F. "Finding Zeros of Rational Quadratic Forms." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/cmc_theses/849.

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In this thesis, we introduce the notion of quadratic forms and provide motivation for their study. We begin by discussing Diophantine equations, the field of p-adic numbers, and the Hasse-Minkowski Theorem that allows us to use p-adic analysis determine whether a quadratic form has a rational root. We then discuss search bounds and state Cassels' Theorem for small-height zeros of rational quadratic forms. We end with a proof of Cassels' Theorem and suggestions for further reading.

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Lozier, Stephane. "On simultaneous approximation to a real number and its cube by rational numbers." Thesis, University of Ottawa (Canada), 2010. http://hdl.handle.net/10393/28701.

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One of the fundamental problems in Diophantine approximation is approximation to real numbers by algebraic numbers of bounded degree. In 1969, H. Davenport and W. M. Schmidt developed a new method to approach the problem. This method combines a result on simultaneous approximation to successive powers of a real number xi with geometry of numbers. For now, the only case where the estimates are optimal is the case of two consecutive powers. Davenport and Schmidt show that if a real number xi is such that 1, xi, xi² are linearly independent over Q , then the exponent of simultaneous approximation to xi and xi² by rational numbers with the same denominator is at most ( 5 - 1}/2 = 0.618..., the inverse of the Golden ratio. In this thesis, we consider the case of a number and its cube. Our main result is that if a real number xi is such that 1, xi, xi³ are linearly independent over Q , then the exponent of simultaneous approximation to xi and xi³ by rational numbers with the same denominator is at most 5/7 = 0.714.... As corollaries, we deduce a result on approximation by algebraic numbers and a version of Gel'fond's lemma for polynomials of the form a + bT + cT³.

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Millsaps, Gayle M. "Interrelationships between teachers' content knowledge of rational number, their instructional practice, and students' emergent conceptual knowledge of rational number." Connect to resource, 2005. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1124225634.

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Thesis (Ph. D.)--Ohio State University, 2005.
Title from first page of PDF file. Document formatted into pages; contains xviii, 339 p.; also includes graphics (some col.). Includes bibliographical references (p. 296-306). Available online via OhioLINK's ETD Center

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Carbone, Rose Elaine. "Elementary Teacher Candidates’ Understanding of Rational Numbers: An International Perspective." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-79565.

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This paper combines data from two different international research studies that used problem posing in analyzing elementary teacher candidates’ understanding of rational numbers. In 2007, a mathematics educator from the United States and a mathematician from Northern Ireland collaborated to investigate their respective elementary teacher candidates’ understanding of addition and division of fractions. A year later, the same US mathematics educator collaborated with a mathematics educator from South Africa on a similar research project that focused solely on the addition of fractions. The results of both studies show that elementary teacher candidates from the three different continents share similar misconceptions regarding the addition of fractions. The misconceptions that emerged were analyzed and used in designing teaching strategies intended to improve elementary teacher candidates’ understanding of rational numbers. The research also suggests that problem posing may improve their understanding of addition of fractions.

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Clark, David Alan. "The Euclidean algorithm for Galois extensions of the rational numbers." Thesis, McGill University, 1992. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=39408.

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Let K be a totally real, quartic, Galois extension of $ doubq$ whose ring of integers R is a principal ideal domain. If there is a prime ideal p of R such that the unit group maps onto $(R/{ bf p} sp2$)*, then R is a Euclidean domain. This criterion is generalized to arbitrary Galois extensions.
Let E be an elliptic curve over a number field F. Suppose ($F: doubq rbrack le 4$ and $F(E lbrack q rbrack ) not subseteq F$ for all primes q such that F contains a primitive $q sp{ rm th}$ root of unity, then the reduced elliptic curve $ tilde{E}(F sb{ bf p})$ is cyclic infinitely often. In general, if $ Gamma$ a subgroup of $E(F)$ with the range of $ Gamma$ sufficiently large, there are infinitely many prime ideals p of F such that the reduced curve $ tilde{E}(F sb{ bf p}) = Gamma sb{ bf p}$, where $ Gamma sb{ bf p}$ is the reduction modulo p of $ Gamma$.

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Bruyns, P. "Aspects of the group of homeomorphisms of the rational numbers." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375224.

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LORIO, MARCELO NASCIMENTO. "APPROXIMATIONS OF REAL NUMBERS BY RATIONAL NUMBERS: WHY THE CONTINUED FRACTIONS CONVERGING PROVIDE THE BEST APPROXIMATIONS?" PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2014. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=23981@1.

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PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO
COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR
Frações Contínuas são representações de números reais que independem da base de numeração escolhida. Quando se trata de aproximar números reais por frações, a escolha da base dez oculta, frequentemente, aproximações mais eficientes do que as exibe. Integrar conceitos de aproximações de números reais por frações contínuas com aspectos geométricos traz ao assunto uma abordagem diferenciada e bastante esclarecedora. O algoritmo de Euclides, por exemplo, ao ganhar significado geométrico, se torna um poderoso argumento para a visualização dessas aproximações. Os teoremas de Dirichlet, de Hurwitz-Markov e de Lagrange comprovam, definitivamente, que as melhores aproximações de números reais veem das frações contínuas, estimando seus erros com elegância técnica matemática incontestável.
Continued fractions are representations of real numbers that are independent of the choice of the numerical basis. The choice of basis ten frequently hides more than shows efficient approximations of real numbers by rational ones. Integrating approximations of real numbers by continued fractions with geometrical interpretations clarify the subject. The study of geometrical aspects of Euclids algorithm, for example, is a powerful method for the visualization of continued fractions approximations. Theorems of Dirichlet, Hurwitz-Markov and Lagrange show that, definitely, the best approximations of real numbers come from continued fractions, and the errors are estimated with elegant mathematical technique.

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Книги з теми "Numbers, Rational"

Rational numbers: Poems. [Kirksville, Mo.]: Truman State University Press, 2000.

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H, Salzmann, ed. The classical fields: Structural features of the real and rational numbers. Cambridge: Cambridge University Press, 2007.

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Bellos, Alex. Here's Looking at Euclid: A Surprising Excursion through the Astonishing World of Math. New York: Free Press, 2010.

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Bellos, Alex. Here's looking at Euclid: A surprising excursion through the astonishing world of math. New York: Free Press, 2010.

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Hertzberg, Hendrik. One million. New York: Times Books, 1993.

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Hertzberg, Hendrik. One million. New York: Abrams Image, 2009.

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S, Bezuk Nadine, ed. Understanding rational numbers and proportions. Reston, Va: National Council of Teachers of Mathematics, 1994.

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P, Carpenter Thomas, Fennema Elizabeth, and Romberg Thomas A, eds. Rational numbers: An integration of research. Hillsdale, N.J: Lawrence Erlbaum Associates, 1992.

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Mary, Stroh, and Sopris West Inc, eds. TransMath: Making sense of rational numbers. Longmont, Colo: Cambium Learning/Sopris West, 2010.

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Lappan, Glenda. Bits and pieces I: Understanding rational numbers. Palo Alto, CA: Dale Seymour Publications, 1998.

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Частини книг з теми "Numbers, Rational"

Eriksson, Kenneth, Donald Estep, and Claes Johnson. "Rational Numbers." In Applied Mathematics: Body and Soul, 71–86. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-05796-4_7.

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Bhattacharjee, Meenaxi, Rögnvaldur G. Möller, Dugald Macpherson, and Peter M. Neumann. "Rational Numbers." In Notes on Infinite Permutation Groups, 77–86. Gurgaon: Hindustan Book Agency, 1997. http://dx.doi.org/10.1007/978-93-80250-91-5_9.

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Bhattacharjee, Meenaxi, Dugald Macpherson, Rögnvaldur G. Möller, and Peter M. Neumann. "Rational numbers." In Lecture Notes in Mathematics, 77–86. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/bfb0092559.

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Shah, Nita H., and Vishnuprasad D. Thakkar. "Rational Numbers." In Journey from Natural Numbers to Complex Numbers, 47–60. Boca Raton : CRC Press, 2021. | Series: Advances in mathematics and engineering: CRC Press, 2020. http://dx.doi.org/10.1201/9781003105244-3.

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Noël, Marie-Pascale, and Giannis Karagiannakis. "Rational numbers." In Effective Teaching Strategies for Dyscalculia and Learning Difficulties in Mathematics, 236–94. London: Routledge, 2022. http://dx.doi.org/10.4324/b22795-6.

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Ovchinnikov, Sergei. "Rational Numbers." In Real Analysis: Foundations, 1–30. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64701-8_1.

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Stillwell, John. "Rational Points." In Numbers and Geometry, 111–42. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0687-3_4.

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Kramer, Jürg, and Anna-Maria von Pippich. "The Rational Numbers." In Springer Undergraduate Mathematics Series, 93–139. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-69429-0_3.

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Stillwell, John. "The Rational Numbers." In Elements of Algebra, 18–37. New York, NY: Springer New York, 1994. http://dx.doi.org/10.1007/978-1-4757-3976-3_2.

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Kay, Anthony. "Rational Numbers, ℚ." In Number Systems, 107–48. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9780429059353-6.

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Тези доповідей конференцій з теми "Numbers, Rational"

Vălcan, Teodor-Dumitru. "Structures of Fields of Rational Numbers, Isomorphic Between Them." In 10th International Conference Education, Reflection, Development. European Publisher, 2023. http://dx.doi.org/10.15405/epes.23056.8.

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Pion, Sylvain, and Chee K. Yap. "Constructive root bound for k-ary rational input numbers." In the nineteenth conference. New York, New York, USA: ACM Press, 2003. http://dx.doi.org/10.1145/777792.777831.

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Cheng, Howard, and Eugene Zima. "On accelerated methods to evaluate sums of products of rational numbers." In the 2000 international symposium. New York, New York, USA: ACM Press, 2000. http://dx.doi.org/10.1145/345542.345581.

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May, John P., B. David Saunders, and David Harlan Wood. "Numerical techniques for computing the inertia of products of matrices of rational numbers." In ISSAC07: International Symposium on Symbolic and Algebraic Computation. New York, NY, USA: ACM, 2007. http://dx.doi.org/10.1145/1277500.1277520.

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Daghigh, Hassan, Somayeh Didari, and Ruholla Khodakaramian Gilan. "A deterministic algorithm for discrete logarithm on some special elliptic curves over rational numbers." In 2015 12th International Iranian Society of Cryptology Conference on Information Security and Cryptology (ISCISC). IEEE, 2015. http://dx.doi.org/10.1109/iscisc.2015.7387912.

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Pinto, Hélia. "THE GALLERY WALK AS A WAY TO TRAIN PRESERVICE TEACHERS FOR TEACHING RATIONAL NUMBERS." In 16th International Conference on Education and New Learning Technologies. IATED, 2024. http://dx.doi.org/10.21125/edulearn.2024.1370.

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Ge, Q. J., and Donglai Kang. "Rational Bézier and B-Spline Ruled Surface Patches." In ASME 1996 Design Engineering Technical Conferences and Computers in Engineering Conference. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/96-detc/dac-1495.

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Abstract This paper presents a geometric method for constructing bounded rational Bézier and B-spline ruled surfaces directly from line-segments. Oriented line-segments in a Euclidean three-space are represented by vectors with four homogeneous components over the ring of dual numbers. Projective algorithms for rational Bézier and B-spline curves are dualized to yield algorithms for rational Bézier and B-spline ruled surfaces.

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Pomrehn, Leonard P., and Panos Y. Papalambros. "Optimal Approximation of Real Values Using Rational Numbers With Application to the Kinematic Design of Gearboxes." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0384.

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Abstract This article proposes a method for optimally approximating real values with rational numbers. Such requirements arise in the design of various types of gear sets, where integer numbers of gear teeth force individual stage ratios to assume rational values. The kinematic design of an 18-speed gearbox, taken from the literature, is analyzed and solved using the proposed method. The method, called sequential exhaustion, sequentially considers each stage of the gearbox design, exhaustively examining each stage. Examination of 94 solutions leads to a pareto-optimal set containing 11 solutions. Further, although the layout of the gearbox is predefined for the kinematic design problem, certain solutions of the problem exhibit “non-reducing” gear pairs, revealing previously unforeseen changes in the gearbox layout.

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Aliyev, Yagub N. "The 3x+1 Problem For Rational Numbers : Invariance of Periodic Sequences in 3x+1 Problem." In 2020 IEEE 14th International Conference on Application of Information and Communication Technologies (AICT). IEEE, 2020. http://dx.doi.org/10.1109/aict50176.2020.9368585.

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Annathurai, K., Z. Zamzamir, S. Shafie, F. Rahmat, R. Masri, and N. Hasan. "Development of InterFrac Matching Kit integrates game-based learning in the form 1 rational numbers topic." In INTERNATIONAL CONFERENCE ON INNOVATION IN MECHANICAL AND CIVIL ENGINEERING (i-MACE 2022). AIP Publishing, 2023. http://dx.doi.org/10.1063/5.0149564.

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Звіти організацій з теми "Numbers, Rational"

Lu, Chao. A Computational Library Using P-adic Arithmetic for Exact Computation With Rational Numbers in Quantum Computing. Fort Belvoir, VA: Defense Technical Information Center, November 2005. http://dx.doi.org/10.21236/ada456488.

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Lutz, Carsten. Adding Numbers to the SHIQ Description Logic - First Results. Aachen University of Technology, 2001. http://dx.doi.org/10.25368/2022.117.

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Recently, the Description Logic (DL) SHIQ has found a large number of applications. This success is due to the fact that SHIQ combines a rich expressivity with efficient reasoning, as is demonstrated by its implementation in DL systems such as FaCT and RACER. One weakness of SHIQ, however, limits its usability in several application areas: numerical knowledge such as knowledge about the age, weight, or temperature of real-world entities cannot be adequately represented. In this paper, we propose an extension of SHIQ that aims at closing this gap. The new Description Logic Q-SHIQ, which augments SHIQ by additional, 'concrete domain' style concept constructors, allows to refer to rational numbers in concept descriptions, and also to define concepts based on the comparison of numbers via predicates such as < or =. We argue that this kind of expressivity is needed in many application areas such as reasoning about the semantic web. We prove reasoning with Q-SHIQ to be EXPTIME-complete (thus not harder than reasoning with SHIQ) by devising an automata-based decision procedure.

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Gonzales, Lorenzo. Ir-Rational Number Institute Report 2017-2018. Office of Scientific and Technical Information (OSTI), June 2018. http://dx.doi.org/10.2172/1440467.

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Rosenfeld. L51741 Development of a Model for Fatigue Rating Shallow Unrestrained Dents. Chantilly, Virginia: Pipeline Research Council International, Inc. (PRCI), September 1997. http://dx.doi.org/10.55274/r0010337.

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The ability to fatigue-rate dents in line pipe is important for three reasons. First, field failures give evidence that fatigue crack growth occurs in dents under certain circumstances in spite of the fact that plain dents are generally thought to have little effect on the integrity of natural gas pipelines based on the results of single-cycle burst tests. Second, many operators are discovering large numbers of dents on the bottom quadrants of their pipe associated with rocks and backfill loads. Guidance is needed for discerning dents for which excavation and inspection is economically wasteful and counterproductive to pipeline safety from those dents for which further action would be beneficial. Third, fatigue life may be a more rational basis for rating the severity of a dent than present criteria which rely solely on dent depth with a maximum depth of 6 percent of the pipe diameter as a generally accepted limit. In some cases deeper dents might be permitted to remain in service, while in other cases, shallower dents should be repaired. The goal of this project was to develop guidelines for pipeline operators to assess the severity of dents on the basis of their fatigue life in-service. The assessment uses pipeline operating pressures and simple geometric measurements of the dent.

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ADA JOINT PROGRAM OFFICE ARLINGTON VA. Ada (Tradename) Compiler Validation Summary Report: Certificate Number: 880815W1.09143 Rational VAX-VMS, Version 2.0.45 Rational R1000 Series 200 Model 20 and VAX-11/750 (Host) and (Target). Fort Belvoir, VA: Defense Technical Information Center, August 1988. http://dx.doi.org/10.21236/ada205908.

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Kaiser, Frederick M. Interagency Collaborative Arrangements and Activities: Types, Rationales, Considerations (Interagency Paper, Number 5, June 2011). Fort Belvoir, VA: Defense Technical Information Center, June 2011. http://dx.doi.org/10.21236/ada551190.

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Xiong, Wei. Rational Optimization of Microbial Processing for High Yield CO2-to-Isopropanol Conversion: Cooperative Research and Development Final Report, CRADA Number CRD-20-17114. Office of Scientific and Technical Information (OSTI), January 2024. http://dx.doi.org/10.2172/2283521.

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Munoz, Laura, Giulia Mascagni, Wilson Prichard, and Fabrizio Santoro. Should Governments Tax Digital Financial Services? A Research Agenda to Understand Sector-Specific Taxes on DFS. Institute of Development Studies (IDS), February 2022. http://dx.doi.org/10.19088/ictd.2022.002.

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Digital financial services (DFS) have rapidly expanded across Africa and other low-income countries. At the same time, low-income countries face strong pressures to increase domestic resource mobilisation, and major challenges in taxing the digital economy. A growing number are therefore advancing or considering new taxes on DFS. These have generated much debate and there are significant disagreements over the rationale for the taxes and their likely impacts. This paper examines three key questions that could help governments and other stakeholders to better understand the rationale for, and impacts of, different decisions around taxing DFS – and to arrive at policies that best meet competing needs. First, what is the rationale for imposing specific taxes on money transfers or mobile money in particular? Second, and most importantly, what is the likely impact of DFS taxes? Third, how do the policy processes through which taxes on DFS and money transfers are introduced function in practice? The paper looks at the core principles of good taxation and presents the existing debate around whether taxes on DFS observe them. It explains why understanding the landscape of financial services is essential to designing suitable tax policies and lays out a framework for developing the necessary analysis of the impacts of taxes on DFS. It also highlights the importance of better understanding the processes that give rise to these taxes.

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VISTA RESEARCH CORP TUCSON AZ. Ada Compiler Validation Summary Report: Certificate Number: 940630W1. 11369 Rational Software Corporation VADS Sun4 => PowerPC, Product Number 2100- 01444, Version 6.2 Sun 4 Model SPARCcenter 2000 under Solaris 2.3 => Motorola MVME160 (PowerPC 601 Bare Machine). Fort Belvoir, VA: Defense Technical Information Center, July 1994. http://dx.doi.org/10.21236/ada285107.

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Emmerson, Stephen. Modulations through time. Norges Musikkhøgskole, August 2018. http://dx.doi.org/10.22501/nmh-ar.530427.

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This article explores the rationale behind a performance given by the authors at the Unfolding the Process symposium held in Oslo in November 2015. For this occasion, the authors devised a new version of Bach’s Goldberg Variations that builds upon Emmerson’s arrangement of the work for two pianos in 2012. A shortened version of the work (c.30 minutes) was designed that aimed nonetheless to maintain the original work’s sense of structural balance and coherence. This version involved the transposition of a number of variations into different keys to explore the possibility of adding a satisfying tonal structure to our experience of the work, in a context where both performers see potential communicative value in 'playing with' dimensions of original masterworks with a view to giving fresh perspective to the listener experience. The article is written from the alternating perspectives of the authors; one of which is primarily concerned with the rationale and process of devising the arrangement while the other reflects upon the performative aspects and implications arising from it.

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