Unimodality for the free sum and product of Reflexive and Birkhoff polytopes
DOI:
https://doi.org/10.29304/jqcm.2022.14.2.936Keywords:
reflexive polytopes, Birkhoff polytope, , free sum, product, h-vector, unimodalityAbstract
Scholars have recently become interested in the importance of the reflexive and Birkhoff polytopes in a variety of applications in our daily lives. Unanswered queries and educated guesses abound in reflexive polytopes. We use the free sum and product for reflexives polytopes, as well as the product for two Birkhoff polytopes, and the proven theorem to get specific unimodality results. The computations are acquired via algorithms.
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References
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[17] Shatha A. Salman, Iman S. Hadeed,”Fourier Transform of a Polytope and its Applications in Geometric Combinatorics”, Second Al-Sadiq International Conference on Multidisciplinary in IT and Communication Science and Applications, IEEE, 2017.
[18] Shatha A. Salman, "Lattice Point and its Application in RSA Cryptosystem", Energy Procedia, 157, pp.39–42, 2019.
[19] Shatha A. Salman, Iman S. Hadeed, "An algorithm for a modified computation of Dedekind sums", Journal of Al Qadisiyah for Computer Science and Mathematics, 12, no. 1, pp. 49–53, 2020.
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[26] A. Paffenholz, "Faces of Birkhoff polytopes",The electronic journal of combinatorics ,vol.22(1), 2015.
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[28] V. Klee & C. Witzgall, " Facets and vertices of transportation polytopes", Mathematics of the decision sciences1, pp.257-282,1968.
[2] M.Beck, et al, "Integer points in polyhedral", American Mathematical Society, 2006.
[3] C.Hasse and I.V. Melnikov, "The reflexive dimension of a lattice polytope", Annals of Combinatorics, vol.10, no.2, pp. 211-217, 2006.
[4] S.A.Salman and N.Rashid; "On the bounded of vertices for reflexive polytope", International Journal of Applied Mathematics & Statistical Sciences (IJAMSS), vol.4, no.2, pp. 43-48, 2015.
[5] S.A.Salman and N.Rashid," The Maximal Number of Vertices of the Reflexive Polytope", Eng&Tech. Journal, vol.33, part (B), no. 4, 2015.
[6] R.A. Brualdi and P.M Gibson "Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function" , Journal of Combinatorial Theory, Series A, 22(2), pp.194-230 ,1977.
[7] P.Diaconis & A.Gangolli " Rectangular arrays with fixed margins", In Discrete probability and algorithms, Springer, New York, NY, pp. 15-41,1995.
[8] B.J.Braun, "Ehrhart theory for lattice polytopes", Ph.D. thesis, Department of Mathematics, Washington University, 2007.
[9] S.A.Salman Computing "The Number of Integral Points in4-dimensional Ball Using Tutte Polynomial", Eng. &Tech. Journal, 33, pp.1420-1429, 2015.
[10] R.P. Stanley, "Decompositions of rational convex polytopes." Ann. Discrete Math 6.6,pp. 333-342,1980.
[11] R.P. Stanley, "Log-concave and unimodal sequences in algebra, combinatorics, and geometry", Ann. New York Acad. Sci 576.1, pp.500-535. 1989.
[12] T. Hibi," Dual polytopes of rational convex polytopes". Combinatorica, 12(2), pp. 237–240, 1992.
[13] S.Payne & M.Mustata ; " Ehrhart polynomials and stringy Betti numbers." Mathematische Annalen 333.4, pp. 787-795. 2005.
[14] Shatha A.S. and Fatema A.S., "Ehrhart Polynomials of a Cyclic Polytopes", Eng & Tec Journal, 27(14),pp. 2624-2631, 2009.
[15] Shatha A.S, "On the number of lattice points in n-dimensional space with an application", International Journal of Mathematics and Computer Science, 16, no. 2, pp. 705–712, 2021.
[16] Shatha A.S et al, "A new algorithm for water retention on magic square", Indonesian Journal of Electrical Engineering and Computer Science, Vol.2,No.2, pp.1062-1069, 2020
[17] Shatha A. Salman, Iman S. Hadeed,”Fourier Transform of a Polytope and its Applications in Geometric Combinatorics”, Second Al-Sadiq International Conference on Multidisciplinary in IT and Communication Science and Applications, IEEE, 2017.
[18] Shatha A. Salman, "Lattice Point and its Application in RSA Cryptosystem", Energy Procedia, 157, pp.39–42, 2019.
[19] Shatha A. Salman, Iman S. Hadeed, "An algorithm for a modified computation of Dedekind sums", Journal of Al Qadisiyah for Computer Science and Mathematics, 12, no. 1, pp. 49–53, 2020.
[20] C.A. Athanasiadis, "Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley" ,PP.163-174, 2005.
[21] M. Beck, and D. Pixton. "The Ehrhart polynomial of the Birkhoff polytope", Discrete & Computational Geometry 30, no.4, PP. 623-637,2003.
[22] M.Beck and S.Robins, "Computing the continuous discretely", Berlin: Springer science+ Business media, LLC, 2007.
[23] M.Øbro, "Classification of smooth Fanopolytopes", Ph.D. thesis, Department. Of Mathematical Sciences, university of Aarhus, 2007.
[24] A.Higashitani, "Classification of Ehrhart polynomials of integral simplices", Discrete Mathematics & Theoretical Computer Science, 2012.
[25] R.P.Stanly, "On Hillbert function of a graded cohen-Macauly of a graded Cohen-Macaulay domain", journal of pure and applied algebra, vol.73, no.3, pp.307-314, 1991.
[26] A. Paffenholz, "Faces of Birkhoff polytopes",The electronic journal of combinatorics ,vol.22(1), 2015.
[27] P.Diaconis & A.Gangolli, "Rectangular arrays with fixed margins", In Discrete probability and algorithms . Springer, New York, NY.pp.15-41,1995.
[28] V. Klee & C. Witzgall, " Facets and vertices of transportation polytopes", Mathematics of the decision sciences1, pp.257-282,1968.
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Published
2022-06-04
How to Cite
Sadiq, F. A., & Salman, S. A. (2022). Unimodality for the free sum and product of Reflexive and Birkhoff polytopes. Journal of Al-Qadisiyah for Computer Science and Mathematics, 14(2), Math Page 1–8. https://doi.org/10.29304/jqcm.2022.14.2.936
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