Dyadic rational (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Dyadic rational" in English language version.

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31doi.org

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27ams.org

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8arxiv.org

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3harvard.edu

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1normalesup.org

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ams.org

mathscinet.ams.org

Resnikoff, Howard L.; Wells, Raymond O. Jr. (1998), "2.2.1: Digital computers and measurement", Wavelet Analysis: The Scalable Structure of Information, New York: Springer-Verlag, pp. 17–18, doi:10.1007/978-1-4612-0593-7, ISBN 0-387-98383-X, MR 1712468
van der Hoeven, Joris (2006), "Computations with effective real numbers", Theoretical Computer Science, 351 (1): 52–60, doi:10.1016/j.tcs.2005.09.060, MR 2201092
Ko, Ker-I (1991), Complexity Theory of Real Functions, Progress in Theoretical Computer Science, Boston, Massachusetts: Birkhäuser Boston, Inc., pp. 41–43, doi:10.1007/978-1-4684-6802-1, ISBN 0-8176-3586-6, MR 1137517, S2CID 11758381
Zheng, Xizhong; Rettinger, Robert (2004), "Weak computability and representation of reals", Mathematical Logic Quarterly, 50 (4–5): 431–442, doi:10.1002/malq.200310110, MR 2090389, S2CID 15815720
Ambos-Spies, Klaus; Zheng, Xizhong (2019), "On the differences and sums of strongly computably enumerable real numbers", in Manea, Florin; Martin, Barnaby; Paulusma, Daniël; Primiero, Giuseppe (eds.), Computing with Foresight and Industry: 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11558, Cham: Springer, pp. 310–322, doi:10.1007/978-3-030-22996-2_27, MR 3981892, S2CID 195795492
Jerrum, Mark R.; Valiant, Leslie G.; Vazirani, Vijay V. (1986), "Random generation of combinatorial structures from a uniform distribution", Theoretical Computer Science, 43 (2–3): 169–188, doi:10.1016/0304-3975(86)90174-X, MR 0855970
Uiterwijk, Jos W. H. M.; Barton, Michael (2015), "New results for Domineering from combinatorial game theory endgame databases", Theoretical Computer Science, 592: 72–86, arXiv:1506.03949, doi:10.1016/j.tcs.2015.05.017, MR 3367582, S2CID 5899577
Equivalent formulas to these, written in the language of the Coq interactive theorem prover, are given by Krebbers, Robbert; Spitters, Bas (2013), "Type classes for efficient exact real arithmetic in Coq", Logical Methods in Computer Science, 9 (1): 1:01, 27, arXiv:1106.3448, doi:10.2168/LMCS-9(1:1)2013, MR 3029087, S2CID 218627153
O'Connor, Russell (2007), "A monadic, functional implementation of real numbers", Mathematical Structures in Computer Science, 17 (1): 129–159, arXiv:cs/0605058, doi:10.1017/S0960129506005871, MR 2311089, S2CID 221168970
Nilsson, Johan (2009), "On numbers badly approximable by dyadic rationals", Israel Journal of Mathematics, 171: 93–110, doi:10.1007/s11856-009-0042-9, MR 2520103
Kac, Mark (1959), Statistical Independence in Probability, Analysis and Number Theory, Carus Mathematical Monographs, vol. 12, New York: John Wiley & Sons for the Mathematical Association of America, pp. 2–3, MR 0110114
Pollen, David (1992), "Daubechies' scaling function on [0,3]", Wavelets, Wavelet Analysis and Its Applications, vol. 2, Boston, Massachusetts: Academic Press, pp. 3–13, MR 1161245
In the notation of Estes and Ohm for rings that are both subrings of $\mathbb {Q}$ and overrings of $\mathbb {Z}$ , the dyadic rationals are the ring $\mathbb {Z} _{\{2\}}$ . See section 7 of Estes, Dennis; Ohm, Jack (1967), "Stable range in commutative rings" (PDF), Journal of Algebra, 7 (3): 343–362, doi:10.1016/0021-8693(67)90075-0, MR 0217052
Lucyshyn-Wright, Rory B. B. (2018), "Convex spaces, affine spaces, and commutants for algebraic theories", Applied Categorical Structures, 26 (2): 369–400, arXiv:1603.03351, doi:10.1007/s10485-017-9496-9, MR 3770912, S2CID 3743682
Manners, Freddie (2015), "A solution to the pyjama problem", Inventiones Mathematicae, 202 (1): 239–270, arXiv:1305.1514, Bibcode:2015InMat.202..239M, doi:10.1007/s00222-014-0571-7, MR 3402799, S2CID 119148680; see section 6.2.1, "A model case: ${\widehat {\mathbb {Z} [1/2]}}$ ", pp. 255–257.
Robert, Alain M. (2000), "5.4 Fractional and integral parts of $p$ -adic numbers", A Course in $p$ -adic Analysis, Graduate Texts in Mathematics, vol. 198, New York: Springer-Verlag, pp. 40–43, doi:10.1007/978-1-4757-3254-2, ISBN 0-387-98669-3, MR 1760253
de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang (2007), "On the isolated points in the space of groups" (PDF), Journal of Algebra, 307 (1): 254–277, arXiv:math/0511714, doi:10.1016/j.jalgebra.2006.02.012, MR 2278053, S2CID 11566447
Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1997), "Rational numbers", Notes on Infinite Permutation Groups, Texts and Readings in Mathematics, vol. 12, Berlin: Springer-Verlag, pp. 77–86, doi:10.1007/978-93-80250-91-5_9, ISBN 81-85931-13-5, MR 1632579
Girgensohn, Roland (1996), "Constructing singular functions via Farey fractions", Journal of Mathematical Analysis and Applications, 203 (1): 127–141, doi:10.1006/jmaa.1996.0370, MR 1412484
Cvitanović, Predrag; Gunaratne, Gemunu H.; Procaccia, Itamar (1988), "Topological and metric properties of Hénon-type strange attractors", Physical Review A, Third Series, 38 (3): 1503–1520, Bibcode:1988PhRvA..38.1503C, doi:10.1103/PhysRevA.38.1503, MR 0970237, PMID 9900529
Brin, Matthew G. (1999), "The ubiquity of Thompson's group $F$ in groups of piecewise linear homeomorphisms of the unit interval", Journal of the London Mathematical Society, Second Series, 60 (2): 449–460, arXiv:math/9705205, doi:10.1112/S0024610799007905, MR 1724861, S2CID 14490692
Cannon, J. W.; Floyd, W. J. (2011), "What is … Thompson's group?" (PDF), Notices of the American Mathematical Society, 58 (8): 1112–1113, MR 2856142
Fernandes, António M.; Ferreira, Fernando (2005), "Basic applications of weak König's lemma in feasible analysis" (PDF), Reverse Mathematics 2001, Lecture Notes in Logic, vol. 21, La Jolla, California: Association for Symbolic Logic, pp. 175–188, MR 2185433
Conway, J. H. (2001), On Numbers and Games (Second ed.), Natick, Massachusetts: A K Peters, ISBN 1-56881-127-6, MR 1803095; for the dyadic rationals, see "The numbers ${\tfrac {1}{4}}$ , ${\tfrac {3}{4}}$ , $1\,{\tfrac {1}{2}}$ , $3$ , and so on", pp. 10–12
Mauldon, J. G. (1978), "Num, a variant of Nim with no first-player win", The American Mathematical Monthly, 85 (7): 575–578, doi:10.2307/2320870, JSTOR 2320870, MR 0503877
Flanigan, J. A. (1982), "A complete analysis of black-white Hackendot", International Journal of Game Theory, 11 (1): 21–25, doi:10.1007/BF01771244, MR 0665515, S2CID 119964871

ams.org

Cannon, J. W.; Floyd, W. J. (2011), "What is … Thompson's group?" (PDF), Notices of the American Mathematical Society, 58 (8): 1112–1113, MR 2856142

arxiv.org

Uiterwijk, Jos W. H. M.; Barton, Michael (2015), "New results for Domineering from combinatorial game theory endgame databases", Theoretical Computer Science, 592: 72–86, arXiv:1506.03949, doi:10.1016/j.tcs.2015.05.017, MR 3367582, S2CID 5899577
Equivalent formulas to these, written in the language of the Coq interactive theorem prover, are given by Krebbers, Robbert; Spitters, Bas (2013), "Type classes for efficient exact real arithmetic in Coq", Logical Methods in Computer Science, 9 (1): 1:01, 27, arXiv:1106.3448, doi:10.2168/LMCS-9(1:1)2013, MR 3029087, S2CID 218627153
O'Connor, Russell (2007), "A monadic, functional implementation of real numbers", Mathematical Structures in Computer Science, 17 (1): 129–159, arXiv:cs/0605058, doi:10.1017/S0960129506005871, MR 2311089, S2CID 221168970
Lucyshyn-Wright, Rory B. B. (2018), "Convex spaces, affine spaces, and commutants for algebraic theories", Applied Categorical Structures, 26 (2): 369–400, arXiv:1603.03351, doi:10.1007/s10485-017-9496-9, MR 3770912, S2CID 3743682
Manners, Freddie (2015), "A solution to the pyjama problem", Inventiones Mathematicae, 202 (1): 239–270, arXiv:1305.1514, Bibcode:2015InMat.202..239M, doi:10.1007/s00222-014-0571-7, MR 3402799, S2CID 119148680; see section 6.2.1, "A model case: ${\widehat {\mathbb {Z} [1/2]}}$ ", pp. 255–257.
de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang (2007), "On the isolated points in the space of groups" (PDF), Journal of Algebra, 307 (1): 254–277, arXiv:math/0511714, doi:10.1016/j.jalgebra.2006.02.012, MR 2278053, S2CID 11566447
Brin, Matthew G. (1999), "The ubiquity of Thompson's group $F$ in groups of piecewise linear homeomorphisms of the unit interval", Journal of the London Mathematical Society, Second Series, 60 (2): 449–460, arXiv:math/9705205, doi:10.1112/S0024610799007905, MR 1724861, S2CID 14490692
Erickson, Jeff; Nivasch, Gabriel; Xu, Junyan (June 2021), "Fusible numbers and Peano arithmetic", Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), IEEE, pp. 1–13, arXiv:2003.14342, doi:10.1109/lics52264.2021.9470703, S2CID 214727767

books.google.com

Rudman, Peter S. (2009), How Mathematics Happened: The First 50,000 Years, Prometheus Books, p. 148, ISBN 978-1-61592-176-8
Barnes, John (2016), Nice Numbers, Springer International Publishing, doi:10.1007/978-3-319-46831-0, ISBN 978-3-319-46830-3, Note that binary measures (2, 4, 8, 16) are very common indeed. This is particularly obvious with volumes.
Miller, Heather M.-L. (2013), "Weighty matters: evidence for unity and regional diversity from the Indus civilization weights", in Abraham, Shinu Anna; Gullapalli, Praveena; Raczek, Teresa P.; Rizvi, Uzma Z. (eds.), Connections and Complexity: New Approaches to the Archaeology of South Asia, Left Coast Press, pp. 161–177, doi:10.4324/9781315431857, ISBN 978-1-59874-686-0; see in particular p. 166
Resnikoff, Howard L.; Wells, Raymond O. Jr. (1998), "2.2.1: Digital computers and measurement", Wavelet Analysis: The Scalable Structure of Information, New York: Springer-Verlag, pp. 17–18, doi:10.1007/978-1-4612-0593-7, ISBN 0-387-98383-X, MR 1712468
Kirk, David B.; Hwu, Wen-mei W. (2013), "7.2 Representable numbers", Programming Massively Parallel Processors: A Hands-on Approach (2nd ed.), Morgan Kaufmann, pp. 155–159, ISBN 978-0-12-391418-7
Ko, Ker-I (1991), Complexity Theory of Real Functions, Progress in Theoretical Computer Science, Boston, Massachusetts: Birkhäuser Boston, Inc., pp. 41–43, doi:10.1007/978-1-4684-6802-1, ISBN 0-8176-3586-6, MR 1137517, S2CID 11758381
Libbey, Theodore (2006), "Time signature", The NPR Listener's Encyclopedia of Classical Music, Workman Publishing, p. 873, ISBN 978-0-7611-2072-8
Yanakiev, Ivan K. (2020), "Mathematical devices in aid of music theory, composition, and performance", in Bozhikova, Milena (ed.), Music between Ontology and Ideology, Cambridge Scholars Publishing, pp. 35–62, ISBN 978-1-5275-4758-2; see in particular p. 37.
Wells, David Graham (2015), Motivating Mathematics: Engaging Teachers And Engaged Students, World Scientific, pp. 32–33, ISBN 978-1-78326-755-2
Sabin, Malcolm (2010), Analysis and Design of Univariate Subdivision Schemes, Geometry and Computing, vol. 6, Springer, p. 51, ISBN 9783642136481

core.ac.uk

In the notation of Estes and Ohm for rings that are both subrings of $\mathbb {Q}$ and overrings of $\mathbb {Z}$ , the dyadic rationals are the ring $\mathbb {Z} _{\{2\}}$ . See section 7 of Estes, Dennis; Ohm, Jack (1967), "Stable range in commutative rings" (PDF), Journal of Algebra, 7 (3): 343–362, doi:10.1016/0021-8693(67)90075-0, MR 0217052

doi.org

Barnes, John (2016), Nice Numbers, Springer International Publishing, doi:10.1007/978-3-319-46831-0, ISBN 978-3-319-46830-3, Note that binary measures (2, 4, 8, 16) are very common indeed. This is particularly obvious with volumes.
Curtis, Lorenzo J. (1978), "Concept of the exponential law prior to 1900", American Journal of Physics, 46 (9): 896–906, Bibcode:1978AmJPh..46..896C, doi:10.1119/1.11512
Miller, Heather M.-L. (2013), "Weighty matters: evidence for unity and regional diversity from the Indus civilization weights", in Abraham, Shinu Anna; Gullapalli, Praveena; Raczek, Teresa P.; Rizvi, Uzma Z. (eds.), Connections and Complexity: New Approaches to the Archaeology of South Asia, Left Coast Press, pp. 161–177, doi:10.4324/9781315431857, ISBN 978-1-59874-686-0; see in particular p. 166
Resnikoff, Howard L.; Wells, Raymond O. Jr. (1998), "2.2.1: Digital computers and measurement", Wavelet Analysis: The Scalable Structure of Information, New York: Springer-Verlag, pp. 17–18, doi:10.1007/978-1-4612-0593-7, ISBN 0-387-98383-X, MR 1712468
Kneusel, Ronald T. (2017), "Chapter 6: Fixed-point numbers", Numbers and Computers (2nd ed.), Springer International Publishing, pp. 183–214, doi:10.1007/978-3-319-50508-4_6
van der Hoeven, Joris (2006), "Computations with effective real numbers", Theoretical Computer Science, 351 (1): 52–60, doi:10.1016/j.tcs.2005.09.060, MR 2201092
Ko, Ker-I (1991), Complexity Theory of Real Functions, Progress in Theoretical Computer Science, Boston, Massachusetts: Birkhäuser Boston, Inc., pp. 41–43, doi:10.1007/978-1-4684-6802-1, ISBN 0-8176-3586-6, MR 1137517, S2CID 11758381
Zheng, Xizhong; Rettinger, Robert (2004), "Weak computability and representation of reals", Mathematical Logic Quarterly, 50 (4–5): 431–442, doi:10.1002/malq.200310110, MR 2090389, S2CID 15815720
Ambos-Spies, Klaus; Zheng, Xizhong (2019), "On the differences and sums of strongly computably enumerable real numbers", in Manea, Florin; Martin, Barnaby; Paulusma, Daniël; Primiero, Giuseppe (eds.), Computing with Foresight and Industry: 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11558, Cham: Springer, pp. 310–322, doi:10.1007/978-3-030-22996-2_27, MR 3981892, S2CID 195795492
Jerrum, Mark R.; Valiant, Leslie G.; Vazirani, Vijay V. (1986), "Random generation of combinatorial structures from a uniform distribution", Theoretical Computer Science, 43 (2–3): 169–188, doi:10.1016/0304-3975(86)90174-X, MR 0855970
Jones, Shelly M.; Pearson, Dunn (May 2013), "Music: highly engaged students connect music to math", General Music Today, 27 (1): 18–23, doi:10.1177/1048371313486478, S2CID 220604326
Hiebert, James; Tonnessen, Lowell H. (November 1978), "Development of the fraction concept in two physical contexts: an exploratory investigation", Journal for Research in Mathematics Education, 9 (5): 374–378, doi:10.2307/748774, JSTOR 748774
Pothier, Yvonne; Sawada, Daiyo (November 1983), "Partitioning: the emergence of rational number ideas in young children", Journal for Research in Mathematics Education, 14 (5): 307–317, doi:10.2307/748675, JSTOR 748675
Uiterwijk, Jos W. H. M.; Barton, Michael (2015), "New results for Domineering from combinatorial game theory endgame databases", Theoretical Computer Science, 592: 72–86, arXiv:1506.03949, doi:10.1016/j.tcs.2015.05.017, MR 3367582, S2CID 5899577
Equivalent formulas to these, written in the language of the Coq interactive theorem prover, are given by Krebbers, Robbert; Spitters, Bas (2013), "Type classes for efficient exact real arithmetic in Coq", Logical Methods in Computer Science, 9 (1): 1:01, 27, arXiv:1106.3448, doi:10.2168/LMCS-9(1:1)2013, MR 3029087, S2CID 218627153
O'Connor, Russell (2007), "A monadic, functional implementation of real numbers", Mathematical Structures in Computer Science, 17 (1): 129–159, arXiv:cs/0605058, doi:10.1017/S0960129506005871, MR 2311089, S2CID 221168970
Nilsson, Johan (2009), "On numbers badly approximable by dyadic rationals", Israel Journal of Mathematics, 171: 93–110, doi:10.1007/s11856-009-0042-9, MR 2520103
Bajnok, Béla (2013), An Invitation to Abstract Mathematics, Undergraduate Texts in Mathematics, New York: Springer, p. 186, doi:10.1007/978-1-4614-6636-9, ISBN 978-1-4614-6635-2
In the notation of Estes and Ohm for rings that are both subrings of $\mathbb {Q}$ and overrings of $\mathbb {Z}$ , the dyadic rationals are the ring $\mathbb {Z} _{\{2\}}$ . See section 7 of Estes, Dennis; Ohm, Jack (1967), "Stable range in commutative rings" (PDF), Journal of Algebra, 7 (3): 343–362, doi:10.1016/0021-8693(67)90075-0, MR 0217052
Lucyshyn-Wright, Rory B. B. (2018), "Convex spaces, affine spaces, and commutants for algebraic theories", Applied Categorical Structures, 26 (2): 369–400, arXiv:1603.03351, doi:10.1007/s10485-017-9496-9, MR 3770912, S2CID 3743682
Manners, Freddie (2015), "A solution to the pyjama problem", Inventiones Mathematicae, 202 (1): 239–270, arXiv:1305.1514, Bibcode:2015InMat.202..239M, doi:10.1007/s00222-014-0571-7, MR 3402799, S2CID 119148680; see section 6.2.1, "A model case: ${\widehat {\mathbb {Z} [1/2]}}$ ", pp. 255–257.
Robert, Alain M. (2000), "5.4 Fractional and integral parts of $p$ -adic numbers", A Course in $p$ -adic Analysis, Graduate Texts in Mathematics, vol. 198, New York: Springer-Verlag, pp. 40–43, doi:10.1007/978-1-4757-3254-2, ISBN 0-387-98669-3, MR 1760253
de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang (2007), "On the isolated points in the space of groups" (PDF), Journal of Algebra, 307 (1): 254–277, arXiv:math/0511714, doi:10.1016/j.jalgebra.2006.02.012, MR 2278053, S2CID 11566447
Nadler, S. B. Jr. (1973), "The indecomposability of the dyadic solenoid", The American Mathematical Monthly, 80 (6): 677–679, doi:10.2307/2319174, JSTOR 2319174
Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1997), "Rational numbers", Notes on Infinite Permutation Groups, Texts and Readings in Mathematics, vol. 12, Berlin: Springer-Verlag, pp. 77–86, doi:10.1007/978-93-80250-91-5_9, ISBN 81-85931-13-5, MR 1632579
Girgensohn, Roland (1996), "Constructing singular functions via Farey fractions", Journal of Mathematical Analysis and Applications, 203 (1): 127–141, doi:10.1006/jmaa.1996.0370, MR 1412484
Cvitanović, Predrag; Gunaratne, Gemunu H.; Procaccia, Itamar (1988), "Topological and metric properties of Hénon-type strange attractors", Physical Review A, Third Series, 38 (3): 1503–1520, Bibcode:1988PhRvA..38.1503C, doi:10.1103/PhysRevA.38.1503, MR 0970237, PMID 9900529
Brin, Matthew G. (1999), "The ubiquity of Thompson's group $F$ in groups of piecewise linear homeomorphisms of the unit interval", Journal of the London Mathematical Society, Second Series, 60 (2): 449–460, arXiv:math/9705205, doi:10.1112/S0024610799007905, MR 1724861, S2CID 14490692
Mauldon, J. G. (1978), "Num, a variant of Nim with no first-player win", The American Mathematical Monthly, 85 (7): 575–578, doi:10.2307/2320870, JSTOR 2320870, MR 0503877
Flanigan, J. A. (1982), "A complete analysis of black-white Hackendot", International Journal of Game Theory, 11 (1): 21–25, doi:10.1007/BF01771244, MR 0665515, S2CID 119964871
Erickson, Jeff; Nivasch, Gabriel; Xu, Junyan (June 2021), "Fusible numbers and Peano arithmetic", Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), IEEE, pp. 1–13, arXiv:2003.14342, doi:10.1109/lics52264.2021.9470703, S2CID 214727767

harvard.edu

ui.adsabs.harvard.edu

Curtis, Lorenzo J. (1978), "Concept of the exponential law prior to 1900", American Journal of Physics, 46 (9): 896–906, Bibcode:1978AmJPh..46..896C, doi:10.1119/1.11512
Manners, Freddie (2015), "A solution to the pyjama problem", Inventiones Mathematicae, 202 (1): 239–270, arXiv:1305.1514, Bibcode:2015InMat.202..239M, doi:10.1007/s00222-014-0571-7, MR 3402799, S2CID 119148680; see section 6.2.1, "A model case: ${\widehat {\mathbb {Z} [1/2]}}$ ", pp. 255–257.
Cvitanović, Predrag; Gunaratne, Gemunu H.; Procaccia, Itamar (1988), "Topological and metric properties of Hénon-type strange attractors", Physical Review A, Third Series, 38 (3): 1503–1520, Bibcode:1988PhRvA..38.1503C, doi:10.1103/PhysRevA.38.1503, MR 0970237, PMID 9900529

illinois.edu

jeffe.cs.illinois.edu

Erickson, Jeff; Nivasch, Gabriel; Xu, Junyan (June 2021), "Fusible numbers and Peano arithmetic", Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), IEEE, pp. 1–13, arXiv:2003.14342, doi:10.1109/lics52264.2021.9470703, S2CID 214727767

jstor.org

Hiebert, James; Tonnessen, Lowell H. (November 1978), "Development of the fraction concept in two physical contexts: an exploratory investigation", Journal for Research in Mathematics Education, 9 (5): 374–378, doi:10.2307/748774, JSTOR 748774
Pothier, Yvonne; Sawada, Daiyo (November 1983), "Partitioning: the emergence of rational number ideas in young children", Journal for Research in Mathematics Education, 14 (5): 307–317, doi:10.2307/748675, JSTOR 748675
Nadler, S. B. Jr. (1973), "The indecomposability of the dyadic solenoid", The American Mathematical Monthly, 80 (6): 677–679, doi:10.2307/2319174, JSTOR 2319174
Mauldon, J. G. (1978), "Num, a variant of Nim with no first-player win", The American Mathematical Monthly, 85 (7): 575–578, doi:10.2307/2320870, JSTOR 2320870, MR 0503877

nih.gov

pubmed.ncbi.nlm.nih.gov

Cvitanović, Predrag; Gunaratne, Gemunu H.; Procaccia, Itamar (1988), "Topological and metric properties of Hénon-type strange attractors", Physical Review A, Third Series, 38 (3): 1503–1520, Bibcode:1988PhRvA..38.1503C, doi:10.1103/PhysRevA.38.1503, MR 0970237, PMID 9900529

normalesup.org

de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang (2007), "On the isolated points in the space of groups" (PDF), Journal of Algebra, 307 (1): 254–277, arXiv:math/0511714, doi:10.1016/j.jalgebra.2006.02.012, MR 2278053, S2CID 11566447

oeis.org

Sloane, N. J. A. (ed.), "Sequence A188545", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation

semanticscholar.org

api.semanticscholar.org

Ko, Ker-I (1991), Complexity Theory of Real Functions, Progress in Theoretical Computer Science, Boston, Massachusetts: Birkhäuser Boston, Inc., pp. 41–43, doi:10.1007/978-1-4684-6802-1, ISBN 0-8176-3586-6, MR 1137517, S2CID 11758381
Zheng, Xizhong; Rettinger, Robert (2004), "Weak computability and representation of reals", Mathematical Logic Quarterly, 50 (4–5): 431–442, doi:10.1002/malq.200310110, MR 2090389, S2CID 15815720
Ambos-Spies, Klaus; Zheng, Xizhong (2019), "On the differences and sums of strongly computably enumerable real numbers", in Manea, Florin; Martin, Barnaby; Paulusma, Daniël; Primiero, Giuseppe (eds.), Computing with Foresight and Industry: 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings, Lecture Notes in Computer Science, vol. 11558, Cham: Springer, pp. 310–322, doi:10.1007/978-3-030-22996-2_27, MR 3981892, S2CID 195795492
Jones, Shelly M.; Pearson, Dunn (May 2013), "Music: highly engaged students connect music to math", General Music Today, 27 (1): 18–23, doi:10.1177/1048371313486478, S2CID 220604326
Uiterwijk, Jos W. H. M.; Barton, Michael (2015), "New results for Domineering from combinatorial game theory endgame databases", Theoretical Computer Science, 592: 72–86, arXiv:1506.03949, doi:10.1016/j.tcs.2015.05.017, MR 3367582, S2CID 5899577
Equivalent formulas to these, written in the language of the Coq interactive theorem prover, are given by Krebbers, Robbert; Spitters, Bas (2013), "Type classes for efficient exact real arithmetic in Coq", Logical Methods in Computer Science, 9 (1): 1:01, 27, arXiv:1106.3448, doi:10.2168/LMCS-9(1:1)2013, MR 3029087, S2CID 218627153
O'Connor, Russell (2007), "A monadic, functional implementation of real numbers", Mathematical Structures in Computer Science, 17 (1): 129–159, arXiv:cs/0605058, doi:10.1017/S0960129506005871, MR 2311089, S2CID 221168970
Lucyshyn-Wright, Rory B. B. (2018), "Convex spaces, affine spaces, and commutants for algebraic theories", Applied Categorical Structures, 26 (2): 369–400, arXiv:1603.03351, doi:10.1007/s10485-017-9496-9, MR 3770912, S2CID 3743682
Manners, Freddie (2015), "A solution to the pyjama problem", Inventiones Mathematicae, 202 (1): 239–270, arXiv:1305.1514, Bibcode:2015InMat.202..239M, doi:10.1007/s00222-014-0571-7, MR 3402799, S2CID 119148680; see section 6.2.1, "A model case: ${\widehat {\mathbb {Z} [1/2]}}$ ", pp. 255–257.
de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang (2007), "On the isolated points in the space of groups" (PDF), Journal of Algebra, 307 (1): 254–277, arXiv:math/0511714, doi:10.1016/j.jalgebra.2006.02.012, MR 2278053, S2CID 11566447
Brin, Matthew G. (1999), "The ubiquity of Thompson's group $F$ in groups of piecewise linear homeomorphisms of the unit interval", Journal of the London Mathematical Society, Second Series, 60 (2): 449–460, arXiv:math/9705205, doi:10.1112/S0024610799007905, MR 1724861, S2CID 14490692
Flanigan, J. A. (1982), "A complete analysis of black-white Hackendot", International Journal of Game Theory, 11 (1): 21–25, doi:10.1007/BF01771244, MR 0665515, S2CID 119964871
Erickson, Jeff; Nivasch, Gabriel; Xu, Junyan (June 2021), "Fusible numbers and Peano arithmetic", Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021), IEEE, pp. 1–13, arXiv:2003.14342, doi:10.1109/lics52264.2021.9470703, S2CID 214727767

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Fernandes, António M.; Ferreira, Fernando (2005), "Basic applications of weak König's lemma in feasible analysis" (PDF), Reverse Mathematics 2001, Lecture Notes in Logic, vol. 21, La Jolla, California: Association for Symbolic Logic, pp. 175–188, MR 2185433