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## Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]

The equation (4) on the page 178 of the paper previously published has to be corrected. We had only handled the case of the Farey vertices for which $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)\in\mathbb{N}^{*}$. In fact we had to distinguish two cases: $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)\in\mathbb{N}^{*}$ and $\min\left(\left\lfloor\dfrac{2m}{sr'}\right\rfloor,\left\lfloor\dfrac{n}{s'r}\right\rfloor \right)=0$. However, we highlight the correct results of the original paper and its applications. We underline that in this work, we still brought several contributions. These contributions are: applying the fundamental formulas of Graph Theory to the Farey diagram of order $(m,n)$, finding a good upper bound for the degree of a Farey vertex and the relations between the Farey diagrams and the linear diophantine equations.
Combinatorial number theory, Farey diagrams, Theoretical computer sciences, Discrete planes, Diophantine equations, Arithmetical geometry, Combinatorial geometry, Discrete geometry, Graph theory in computer sciences
• [1] D. M. Acketa, J. D. Žunic, On the number of linear partitions of the (m; n)-grid, Inform. Process. Lett. 38(3) (1991) 163–168.
• [2] T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.
• [3] T. Asano, N. Katoh, Variants for the Hough transform for line detection, Comput. Geom. 6(4) (1996) 231–252.
• [4] C. A. Berenstein, D. Lavine, On the number of digital straight line segments, IEEE Trans. Pattern Anal. Mach. Intell. 10(6) (1988) 880–887.
• [5] J. M. Chassery, D. Coeurjolly, I. Sivignon, Duality and geometry straightness, characterization and envelope, Discrete geometry for computer imagery, 1–16, Lecture Notes in Comput. Sci., 4245, Springer, Berlin, 2006.
• [6] J. M. Chassery, A. Montanvert, Geometrical representation of shapes and objects for visual perception, Geometrical representation of shapes and objects for visual perception. In: Geometric Reasoning for Perception and Action, vol. 708 of LNCS, pp. 163–182. Springer, Berlin, 1993.
• [7] D. Coeurjolly, Algorithmique et géométrie discrete pour la caractérisation des courbes et des surfaces, Phd-Thesis, Université Lumière-Lyon II, 2002.
• [8] D. Coeurjolly, I. Sivignon, F. Dupont, F. Feschet, J. -M. Chassery, On digital plane preimage structure, Discrete Appl. Math. 151(1-3) (2005) 78–92.
• [9] A. Daurat, M. Tajine, M. Zouaoui, About the frequencies of some patterns in digital planes application to area estimators, Comput. Graph. 33(1) (2009) 11–20.
• [10] I. Debled-Rennesson, Etude et reconnaissance des droites et plans discrets, PhD thesis, Université Louis-Pasteur, 1995.
• [11] E. Domenjoud, D. Jamet, D. Vergnaud, L. Vuillon, Enumeration formula for (2; n)-cubes in discrete planes, Discrete Appl. Math. 160(15) (2012) 2158–2171.
• [12] G. H. Hardy, E. M. Wright, Introduction À La Théorie Des Nombres, Traduction de François Sauvageot, Springer, 2007.
• [13] P. Haukkanen, J. K. Merikoski, Asymptotics of the number of threshold functions on a twodimensional rectangular grid, Discrete Appl. Math. 161(1-2) (2013) 13–18.
• [14] A. Hatcher, Topology of Numbers, Unpublished book, in preparation; see http://www.math.cornell.edu/~hatcher/TN/TNpage.html.
• [15] W. Hou, C. Zhang, Parallel-beam ct reconstruction based on mojette transform and compressed sensing, Int. J. Comput. Electr. Eng. 5(1) (2013) 83–87.
• [16] Y. Kenmochi, L. Buzer, A. Sugimoto, I. Shimizu, Digital planar surface segmentation using local geometric patterns, Discrete geometry for computer imagery, 322–333, Lecture Notes in Comput. Sci., 4992, Springer, Berlin, 2008.
• [17] D. Khoshnoudirad, Farey lines defining Farey diagrams and application to some discrete structures, Appl. Anal. Discrete Math. 9(1) (2015) 73–84.
• [18] A. O. Matveev, Relative blocking in posets, J. Comb. Optim. 13(4) (2007) 379–403. Corrigendum: arXiv:math/0411026.
• [19] A. O. Matveev, A note on Boolean lattices and Farey sequences, Integers. 7(A20) (2007) 1–7.
• [20] A. O. Matveev, A note on Boolean lattices and Farey sequences II, Integers. 8(A24) (2008) 1–8.
• [21] A. O. Matveev, Neighboring fractions in Farey subsequences, arXiv:0801.1981, 2008.
• [22] M. D. Mcllroy, A note on discrete representation of lines, AT&T Tech. J. 64(2) (1985) 481–490.
• [23] E. Remy, E. Thiel, Structures dans les sphéres de chanfrein, RFIA. (2000) 483–492.
• [24] E. Remy and E. Thiel, Exact medial axis with euclidean distance, Image Vision Comput. 23(2) (2005) 167–175.
• [25] I. Svalbe, A. Kingston, On correcting the unevenness of angle distributions arising from integer ratios lying in restricted portions of the Farey plane, Combinatorial image analysis, 110–121, Lecture Notes in Comput. Sci., 3322, Springer, Berlin, 2005.
• [26] E. Szemerédi, W. T. Trotter Jr., Extremal problems in discrete geometry, Combinatorica. 3(3-4) (1983) 381–392.
• [27] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, 1995.
• [28] E. Thiel, Les distances de chanfrein en analyse d’images:fondements et applications,PhD thesis, Université Joseph-Fourier-Grenoble I, 1994.
• [29] E. Thiel, A. Montanvert, Chamfer masks: discrete distance functions, geometrical properties and optimization, Pattern Recognition, 1992. Vol. III. Conference C: Image, Speech and Signal Analysis, Proceedings., 11th IAPR International Conference on. IEEE, 1992.
• [30] R. Tomás, From Farey sequences to resonance diagrams, Phys. Rev. ST Accel. Beams. 17(1) (2014) 014001-1–014001-3.
• [31] R. Tomás, Asymptotic behavior of a series of Euler’s totient function $\varphi(k)$ times the index of ${1}/{k}$ in a Farey sequence, arXiv:1406.6991, 2014.
• [32] V. Trifonov, L. Pasqualucci, R. Dalla-Favera, R. Rabadan, Fractal-like distributions over the rational numbers in high-throughput biological and clinical data, Sci. Rep. 1(591) (2011).
Subjects Engineering Articles Author: Daniel Khoshnoudirad Publication Date : May 15, 2016
 Bibtex @correction { jacodesmath285455, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2016}, volume = {3}, pages = {105 - 123}, doi = {10.13069/jacodesmath.00924}, title = {Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]}, key = {cite}, author = {Khoshnoudirad, Daniel} } APA Khoshnoudirad, D . (2016). Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190] . Journal of Algebra Combinatorics Discrete Structures and Applications , 3 (2) , 105-123 . DOI: 10.13069/jacodesmath.00924 MLA Khoshnoudirad, D . "Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]" . Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 105-123 Chicago Khoshnoudirad, D . "Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]". Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 105-123 RIS TY - JOUR T1 - Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190] AU - Daniel Khoshnoudirad Y1 - 2016 PY - 2016 N1 - doi: 10.13069/jacodesmath.00924 DO - 10.13069/jacodesmath.00924 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 105 EP - 123 VL - 3 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.00924 UR - https://doi.org/10.13069/jacodesmath.00924 Y2 - 2020 ER - EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190] %A Daniel Khoshnoudirad %T Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190] %D 2016 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 3 %N 2 %R doi: 10.13069/jacodesmath.00924 %U 10.13069/jacodesmath.00924 ISNAD Khoshnoudirad, Daniel . "Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]". Journal of Algebra Combinatorics Discrete Structures and Applications 3 / 2 (May 2016): 105-123 . https://doi.org/10.13069/jacodesmath.00924 AMA Khoshnoudirad D . Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(2): 105-123. Vancouver Khoshnoudirad D . Erratum to “A further study for the upper bound of the cardinality of Farey vertices and applications in discrete geometry” [J. Algebra Comb. Discrete Appl. 2(3) (2015) 169-190]. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(2): 105-123.

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