Relevant bibliographies by topics / Numbers and Geometry

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Numbers and Geometry'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 February 2022

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Journal articles on the topic "Numbers and Geometry"

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Kock, Anders. "Differential Calculus and Nilpotent Real Numbers." Bulletin of Symbolic Logic 9, no. 2 (June 2003): 225–30. http://dx.doi.org/10.2178/bsl/1052669291.

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Do there exist real numbers d with d2 = 0 (besides d = 0, of course)? The question is formulated provocatively, to stress a formalist view about existence: existence is consistency, or better, coherence.Also, the provocation is meant to challenge the monopoly which the number system, invented by Dedekind et al., is claiming for itself as THE model of the geometric line. The Dedekind approach may be termed “arithmetization of geometry”.We know that one may construct a number system out of synthetic geometry, as Euclid and followers did (completed in Hilbert's Grundlagen der Geometrie, [2, Chapter 3]): “geometrization of arithmetic”. (Picking two distinct points on the geometric line, geometric constructions in an ambient Euclidean plane provide structure of a commutative ring on the line, with the two chosen points as 0 and 1).Starting from the geometric side, nilpotent elements are somewhat reasonable, although Euclid excluded them. The sophist Protagoras presented a picture of a circle and a tangent line; the apparent little line segment D which tangent and circle have in common, are, by Pythagoras' Theorem, precisely the points, whose abscissae d (measured along the tangent) have d2 = 0. Protagoras wanted to use this argument for destructive reasons: to refute the science of geometry.A couple of millenia later, the Danish geometer Hjelmslev revived the Protagoras picture. His aim was more positive: he wanted to describe Nature as it was. According to him (or extrapolating his position), the Real Line, the Line of Sensual Reality, had many nilpotent infinitesimals, which we can see with our naked eyes.
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Bradley, C. J., and Liang-shin Hahn. "Complex Numbers and Geometry." Mathematical Gazette 79, no. 484 (March 1995): 237. http://dx.doi.org/10.2307/3620117.

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Hoare, Graham, C. D. Olds, Anneli Lax, and Giuliana P. Davidoff. "The Geometry of Numbers." Mathematical Gazette 86, no. 506 (July 2002): 368. http://dx.doi.org/10.2307/3621910.

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Kannan, R. "Algorithmic Geometry of Numbers." Annual Review of Computer Science 2, no. 1 (June 1987): 231–67. http://dx.doi.org/10.1146/annurev.cs.02.060187.001311.

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Siriwardene, Nihal. "Double numbers in geometry." Journal of the National Science Foundation of Sri Lanka 15, no. 2 (June 30, 1987): 209. http://dx.doi.org/10.4038/jnsfsr.v15i2.8295.

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Schlickewei, Hans Peter, and Wolfgang M. Schmidt. "Quadratic geometry of numbers." Transactions of the American Mathematical Society 301, no. 2 (February 1, 1987): 679. http://dx.doi.org/10.1090/s0002-9947-1987-0882710-3.

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Fei, S. M., H. Y. Guo, and Y. Yu. "Symplectic geometry and geometric quantization on supermanifold withU numbers." Zeitschrift für Physik C Particles and Fields 45, no. 2 (June 1989): 339–44. http://dx.doi.org/10.1007/bf01674466.

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Cleary, Joan, Sidney A. Morris, and David Yost. "Numerical Geometry-Numbers for Shapes." American Mathematical Monthly 93, no. 4 (April 1986): 260. http://dx.doi.org/10.2307/2323675.

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Cleary, Joan, Sidney A. Morris, and David Yost. "Numerical Geometry-Numbers for Shapes." American Mathematical Monthly 93, no. 4 (April 1986): 260–75. http://dx.doi.org/10.1080/00029890.1986.11971802.

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Harkin, Anthony A., and Joseph B. Harkin. "Geometry of Generalized Complex Numbers." Mathematics Magazine 77, no. 2 (April 2004): 118–29. http://dx.doi.org/10.1080/0025570x.2004.11953236.

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Dissertations / Theses on the topic "Numbers and Geometry"

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Rivard-Cooke, Martin. "Parametric Geometry of Numbers." Thesis, Université d'Ottawa / University of Ottawa, 2019. http://hdl.handle.net/10393/38871.

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This thesis is primarily concerned in studying the relationship between different exponents of Diophantine approximation, which are quantities arising naturally in the study of rational approximation to a fixed n-tuple of real irrational numbers. As Khinchin observed, these exponents are not independent of each other, spurring interest in the study of the spectrum of a given family of exponents, which is the set of all possible values that can be taken by said family of exponents. Introduced in 2009-2013 by Schmidt and Summerer and completed by Roy in 2015, the parametric geometry of numbers provides strong tools with regards to the study of exponents of Diophantine approximation and their associated spectra by the introduction of combinatorial objects called n-systems. Roy proved the very surprising result that the study of spectra of exponents is equivalent to the study of certain quantities attached to n-systems. Thus, the study of rational approximation can be replaced by the study of n-systems when attempting to determine such spectra. Recently, Roy proved two new results for the case n=3, the first being that spectra are semi-algebraic sets, and the second being that spectra are stable under the minimum with respect to the product ordering. In this thesis, it is shown that both of these results do not hold in general for n>3, and examples are given. This thesis also provides non-trivial examples for n=4 where the spectra is stable under the minimum. An alternate and much simpler proof of a recent result of Marnat-Moshchevitin proving an important conjecture of Schmidt-Summerer is also given, relying only on the parametric geometry of numbers instead. Further, a conjecture which generalizes this result is also established, and some partial results are given towards its validity. Among these results, the simplest, but non-trivial, new case is also proven to be true. In a different vein, this thesis considers certain generalizations theta(q) of the classical theta q-series. We show under conditions on the coefficients of the series that theta(q) is neither rational nor quadratic irrational for each integer q>1.
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Landstedt, Erik. "Parametric Geometry of Numbers and Exponents of Diophantine Approximation." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-388506.

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Holmin, Samuel. "Geometry of numbers, class group statistics and free path lengths." Doctoral thesis, KTH, Matematik (Avd.), 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-177888.

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This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies. In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants. In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices. In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures. In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.

QC 20151204

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Thiel, Carsten [Verfasser], and Martin [Akademischer Betreuer] Henk. "Adelic convex geometry of numbers / Carsten Thiel. Betreuer: Martin Henk." Magdeburg : Universitätsbibliothek, 2014. http://d-nb.info/1054638128/34.

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Pham, Van Anh. "Loop Numbers of Knots and Links." TopSCHOLAR®, 2017. http://digitalcommons.wku.edu/theses/1952.

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This thesis introduces a new quantity called loop number, and shows the conditions in which loop numbers become knot invariants. For a given knot diagram D, one can traverse the knot diagram and count the number of loops created by the traversal. The number of loops recorded depends on the starting point in the diagram D and on the traversal direction. Looking at the minimum or maximum number of loops over all starting points and directions, one can define two positive integers as loop numbers of the diagram D. In this thesis, the conditions under which these loop numbers become knot invariants are identified. In particular, the thesis answers the question when these numbers are invariant under flypes in the diagram D.
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Hahn, Marvin Anas Verfasser], and Hannah [Akademischer Betreuer] [Markwig. "Combinatorics and degenerations in algebraic geometry : Hurwitz numbers, Mustafin varieties and tropical geometry / Marvin Anas Hahn ; Betreuer: Hannah Markwig." Tübingen : Universitätsbibliothek Tübingen, 2018. http://d-nb.info/1199355968/34.

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Conley, Randolph M. "A survey of the Minkowski?(x) function." Morgantown, W. Va. : [West Virginia University Libraries], 2003. http://etd.wvu.edu/templates/showETD.cfm?recnum=3055.

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Mohammed, Dilbak. "Generalised Frobenius numbers : geometry of upper bounds, Frobenius graphs and exact formulas for arithmetic sequences." Thesis, Cardiff University, 2015. http://orca.cf.ac.uk/98161/.

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Given a positive integer vector ${\ve a}=(a_{1},a_{2}\dots,a_k)^t$ with \bea 1< a_{1}<\cdots
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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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Johnson, Jamie. "Continued Radicals." TopSCHOLAR®, 2005. http://digitalcommons.wku.edu/theses/240.

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If a1, a2, . . . , an are nonnegative real numbers and fj(x) = paj + x, then f1o f2o· · · fn(0) is a nested radical with terms a1, . . . , an. If it exists, the limit as n ! 1 of such an expression is a continued radical. We consider the set of real numbers S(M) representable as an infinite nested radical whose terms a1, a2, . . . are all from a finite set M. We give conditions on the set M for S(M) to be (a) an interval, and (b) homeomorphic to the Cantor set.
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Books on the topic "Numbers and Geometry"

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Stillwell, John. Numbers and geometry. New York: Springer, 1998.

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1922-, Lekkerkerker C. G., and Lekkerkerker C. G. 1922-, eds. Geometry of numbers. 2nd ed. Amsterdam: North-Holland, 1987.

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Stillwell, John. Numbers and Geometry. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0687-3.

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Stillwell, John. Numbers and Geometry. New York, NY: Springer New York, 1998.

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Olds, C. D. The geometry of numbers. Washington, DC: Mathematical Association of America, 2000.

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Complex numbers and geometry. Washington, D.C: Mathematical Association of America, 1994.

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Florian, Luca, and Somer Lawrence, eds. 17 lectures on Fermat numbers: From number theory to geometry. New York: Springer, 2001.

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Siegel, Carl Ludwig. Lectures on the geometry of numbers. Berlin: Springer-Verlag, 1989.

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Hlawka, Edmund. Geometric and analytic numbertheory. Berlin: Springer-Verlag, 1991.

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Stillwell, John. Elements of algebra: Geometry, numbers, equations. New York: Springer-Verlag, 1994.

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Book chapters on the topic "Numbers and Geometry"

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Stillwell, John. "Geometry." In Numbers and Geometry, 37–67. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0687-3_2.

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Vorobiew, Nicolai N. "Fibonacci Numbers and Geometry." In Fibonacci Numbers, 125–47. Basel: Birkhäuser Basel, 2002. http://dx.doi.org/10.1007/978-3-0348-8107-4_5.

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Stillwell, John. "Complex Numbers." In Numbers and Geometry, 215–45. New York, NY: Springer New York, 1998. http://dx.doi.org/10.1007/978-1-4612-0687-3_7.

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Lawson, Mark V. "Complex Numbers." In Algebra & Geometry, 171–86. 2nd ed. Second edition. | Boca Raton : Chapman & Hall/CRC Press, 2021.: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003098072-8.

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Alpay, Daniel. "Complex Numbers: Geometry." In A Complex Analysis Problem Book, 61–86. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0078-5_2.

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Alpay, Daniel. "Complex Numbers: Geometry." In A Complex Analysis Problem Book, 65–91. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-42181-0_2.

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Hlawka, Edmund, Rudolf Taschner, and Johannes Schoißengeier. "Geometry of Numbers." In Geometric and Analytic Number Theory, 38–70. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/978-3-642-75306-0_3.

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Shafarevich, Igor R. "Intersection Numbers." In Basic Algebraic Geometry 1, 233–82. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-37956-7_4.

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Shafarevich, Igor R. "Intersection Numbers." In Basic Algebraic Geometry 1, 223–73. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-642-57908-0_4.

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Pach, János. "Crossing Numbers." In Discrete and Computational Geometry, 267–73. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-540-46515-7_23.

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Conference papers on the topic "Numbers and Geometry"

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Eichhorn, Jürgen. "Characteristic numbers, bordism theory and the Novikov conjecture for open manifolds." In Geometry and Topology of Manifolds. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2007. http://dx.doi.org/10.4064/bc76-0-14.

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Aliev, Iskander, and Martin Henk. "s-Frobenius Numbers: Optimal Lower Bound." In Annual International Conference on Computational Mathematics, Computational Geometry & Statistics. Global Science and Technology Forum (GSTF), 2012. http://dx.doi.org/10.5176/2251-1911_cmcgs25.

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Dutour, Mathieu, and Konstantin Rybnikov. "A New Algorithm in Geometry of Numbers." In 4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007). IEEE, 2007. http://dx.doi.org/10.1109/isvd.2007.6.

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Wu, Wen-tsun. "On generalized Chern classes and Chern numbers of irreducible complex algebraic varieties with arbitrary singularities." In Geometry and Topology of Manifolds. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2007. http://dx.doi.org/10.4064/bc76-0-12.

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Lando, Sergei K. "Hurwitz Numbers: On the Edge Between Combinatorics and Geometry." In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010). Published by Hindustan Book Agency (HBA), India. WSPC Distribute for All Markets Except in India, 2011. http://dx.doi.org/10.1142/9789814324359_0153.

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Johansen, Stein E. "Negative approach: Implications for the relation between number theory and geometry, including connection to Santilli mathematics, from revolving geometric generation of natural numbers with recurrent outshooting leaving prime numbers." In 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014. AIP Publishing LLC, 2014. http://dx.doi.org/10.1063/1.4904610.

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Kim, J. H., A. J. H. Frijns, S. V. Nedea, and A. A. van Steenhoven. "The Effects of Geometry and Knudsen Numbers on Micro- and Nanochannel Flows." In ASME 2011 9th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2011. http://dx.doi.org/10.1115/icnmm2011-58073.

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In this work we use a three dimensional Molecular Dynamics simulation method to study the effect of different geometries and Knudsen number regimes on the gas flow in micro-nanochannels. Argon molecules have been used for the simulations. Thermal wall and diffusive-specular wall types were used for the boundaries of the channels. The velocity profiles in the channel were obtained and analyzed with three different channel geometries that are commonly used in the industry: circular, rectangular (square), and slit channel. We found that when using the same driving force, the maximum velocity of the flow increases when the geometry changes in the order from circular geometry to rectangular geometry to slit geometry, where the latter becomes 1.2∼1.5 times as large compared with either the rectangular or circular channel. While the absolute values of the velocity profiles show a distinct difference according to the different geometries, geometry effect on the shape of the velocity profile also shows interesting features. Rectangular tube shows much flatter profile compared with the other two channels. Also the effect of the size of the channels and different Knudsen numbers on the velocity profiles is investigated. Two different sizes were used here: 100nm and 10nm corresponding to typical sizes of a nano channel and carbon nanotubes. We found that the Knudsen number has an effect on the slip and maximum flow velocity for the slit geometry even for higher Knudsen number. For the Kn higher than approximately 3, it was found that the Knudsen number has a small influence on the slip flow velocity for the circular channel and rectangular channel than for lower Knudsen number.
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Hemmerling, Marco. "Architecture by numbers. An interdisciplinary approach towards computational design and architectural geometry." In XXI Congreso Internacional de la Sociedad Iberoamericana de Gráfica Digital. São Paulo: Editora Blucher, 2017. http://dx.doi.org/10.5151/sigradi2017-024.

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Johansen, Stein E. "Positive approach: Implications for the relation between number theory and geometry, including connection to Santilli mathematics, from Fibonacci reconstitution of natural numbers and of prime numbers." In 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014. AIP Publishing LLC, 2014. http://dx.doi.org/10.1063/1.4904611.

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Safitri, Lidia, Wahyu Sukartiningsih, and Sri Setyowati. "The Effect of Geometry Box Media on the Ability to Know the Concept of Numbers and Geometric Shapes." In Proceedings of the 2nd International Conference on Education Innovation (ICEI 2018). Paris, France: Atlantis Press, 2018. http://dx.doi.org/10.2991/icei-18.2018.40.

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Reports on the topic "Numbers and Geometry"

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Li, Howell, Jijo K. Mathew, Woosung Kim, and Darcy M. Bullock. Using Crowdsourced Vehicle Braking Data to Identify Roadway Hazards. Purdue University, 2020. http://dx.doi.org/10.5703/1288284317272.

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Modern vehicles know more about the road conditions than transportation agencies. Enhanced vehicle data that provides information on “close calls” such as hard braking events or road conditions during winter such as wheel slips and traction control will be critical for improving safety and traffic operations. This research applied conflict analyses techniques to process approximately 1.5 million hard braking events that occurred in the state of Indiana over a period of one week in August 2019. The study looked at work zones, signalized intersections, interchanges and entry/exit ramps. Qualitative spatial frequency analysis of hard-braking events on the interstate demonstrated the ability to quickly identify temporary and long-term construction zones that warrant further investigation to improve geometry and advance warning signs. The study concludes by recommending the frequency of hard-braking events across different interstate routes to identify roadway locations that have abnormally high numbers of “close calls” for further engineering assessment.
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