Corrected Evolutive Kendall's tau Coefficients for Incomplete Rankings with Ties: Application to Case of Spotify Lists

[EN] Mathematical analysis of rankings is essential for a wide range of scientific, public, and industrial applications (e.g., group decision-making, organizational methods, R&D sponsorship, recommender systems, voter systems, sports competitions, grant proposals rankings, web searchers, Internet streaming-on-demand media providers, etc.). Recently, some methods for incomplete aggregate rankings (rankings in which not all the elements are ranked) with ties, based on the classic Kendall's tau coefficient, have been presented. We are interested in ordinal rankings (that is, we can order the elements to be the first, the second, etc.) allowing ties between the elements (e.g., two elements may be in the first position). We extend a previous coefficient for comparing a series of complete rankings with ties to two new coefficients for comparing a series of incomplete rankings with ties. We make use of the newest definitions of Kendall's tau extensions. We also offer a theoretical result to interpret these coefficients in terms of the type of interactions that the elements of two consecutive rankings may show (e.g., they preserve their positions, cross their positions, and they are tied in one ranking but untied in the other ranking, etc.). We give some small examples to illustrate all the newly presented parameters and coefficients. We also apply our coefficients to compare some series of Spotify charts, both Top 200 and Viral 50, showing the applicability and utility of the proposed measures.This research was funded by the Spanish Government, Ministerio de Economia y Competividad, grant number MTM2016-75963-P.Pedroche Sánchez, F.; Conejero, JA. (2020). Corrected Evolutive Kendall's tau Coefficients for Incomplete Rankings with Ties: Application to Case of Spotify Lists. Mathematics. 8(10):1-30. https://doi.org/10.3390/math8101828S130810Diaconis, P., & Graham, R. L. (1977). Spearman’s Footrule as a Measure of Disarray. Journal of the Royal Statistical Society: Series B (Methodological), 39(2), 262-268. doi:10.1111/j.2517-6161.1977.tb01624.xMoreno-Centeno, E., & Escobedo, A. R. (2015). Axiomatic aggregation of incomplete rankings. IIE Transactions, 48(6), 475-488. doi:10.1080/0740817x.2015.1109737Criado, R., García, E., Pedroche, F., & Romance, M. (2013). A new method for comparing rankings through complex networks: Model and analysis of competitiveness of major European soccer leagues. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(4), 043114. doi:10.1063/1.4826446Fortune 500https://fortune.com/fortune500/Academic Ranking of World Universities ARWU 2020http://www.shanghairanking.com/ARWU2020.htmlCWTS Leiden Ranking 2020https://www.leidenranking.com/ranking/2020/listThe Hot 100https://www.billboard.com/charts/hot-100Fagin, R., Kumar, R., Mahdian, M., Sivakumar, D., & Vee, E. (2006). Comparing Partial Rankings. SIAM Journal on Discrete Mathematics, 20(3), 628-648. doi:10.1137/05063088xCook, W. D., Kress, M., & Seiford, L. M. (1986). An axiomatic approach to distance on partial orderings. RAIRO - Operations Research, 20(2), 115-122. doi:10.1051/ro/1986200201151Yoo, Y., Escobedo, A. R., & Skolfield, J. K. (2020). A new correlation coefficient for comparing and aggregating non-strict and incomplete rankings. European Journal of Operational Research, 285(3), 1025-1041. doi:10.1016/j.ejor.2020.02.027Pedroche, F., Criado, R., García, E., Romance, M., & Sánchez, V. E. (2015). Comparing series of rankings with ties by using complex networks: An analysis of the Spanish stock market (IBEX-35 index). Networks and Heterogeneous Media, 10(1), 101-125. doi:10.3934/nhm.2015.10.101Criado, R., García, E., Pedroche, F., & Romance, M. (2016). On graphs associated to sets of rankings. Journal of Computational and Applied Mathematics, 291, 497-508. doi:10.1016/j.cam.2015.03.009KENDALL, M. G. (1938). A NEW MEASURE OF RANK CORRELATION. Biometrika, 30(1-2), 81-93. doi:10.1093/biomet/30.1-2.81Kendall, M. G., & Smith, B. B. (1939). The Problem of $m$ Rankings. The Annals of Mathematical Statistics, 10(3), 275-287. doi:10.1214/aoms/1177732186Bogart, K. P. (1973). Preference structures I: Distances between transitive preference relations†. The Journal of Mathematical Sociology, 3(1), 49-67. doi:10.1080/0022250x.1973.9989823Bogart, K. P. (1975). Preference Structures. II: Distances Between Asymmetric Relations. SIAM Journal on Applied Mathematics, 29(2), 254-262. doi:10.1137/0129023Cicirello, V. (2020). Kendall tau sequence distance: Extending Kendall tau from ranks to sequences. EAI Endorsed Transactions on Industrial Networks and Intelligent Systems, 7(23), 163925. doi:10.4108/eai.13-7-2018.163925Armstrong, R. A. (2019). Should Pearson’s correlation coefficient be avoided? Ophthalmic and Physiological Optics, 39(5), 316-327. doi:10.1111/opo.12636Redman, W. (2019). An O(n) method of calculating Kendall correlations of spike trains. PLOS ONE, 14(2), e0212190. doi:10.1371/journal.pone.0212190Pihur, V., Datta, S., & Datta, S. (2009). RankAggreg, an R package for weighted rank aggregation. BMC Bioinformatics, 10(1). doi:10.1186/1471-2105-10-62Pnueli, A., Lempel, A., & Even, S. (1971). Transitive Orientation of Graphs and Identification of Permutation Graphs. Canadian Journal of Mathematics, 23(1), 160-175. doi:10.4153/cjm-1971-016-5Gervacio, S. V., Rapanut, T. A., & Ramos, P. C. F. (2013). Characterization and Construction of Permutation Graphs. Open Journal of Discrete Mathematics, 03(01), 33-38. doi:10.4236/ojdm.2013.31007Golumbic, M. C., Rotem, D., & Urrutia, J. (1983). Comparability graphs and intersection graphs. Discrete Mathematics, 43(1), 37-46. doi:10.1016/0012-365x(83)90019-5Emond, E. J., & Mason, D. W. (2002). A new rank correlation coefficient with application to the consensus ranking problem. Journal of Multi-Criteria Decision Analysis, 11(1), 17-28. doi:10.1002/mcda.313Spotify Reports Second Quarter 2020 Earningshttps://newsroom.spotify.com/2020-07-29/spotify-reports-second-quarter-2020-earningsCompany infohttps://newsroom.spotify.com/company-info/Bussines Wirehttps://www.businesswire.com/news/home/20200429005216/en/Swanson, K. (2013). A Case Study on Spotify: Exploring Perceptions of the Music Streaming Service. Journal of the Music and Entertainment Industry Educators Association, 13(1), 207-230. doi:10.25101/13.10Microsoft Retires Groove Music Service, Partners with Spotifyhttps://www.theverge.com/2017/10/2/16401898/microsoft-groove-music-pass-discontinued-spotify-partnerSpotify Launches on PlayStation Music Todayhttps://blog.playstation.com/2015/03/30/spotify-launches-on-playstation-music-today/You Can Now Share Music from Spotify to Facebook Storieshttps://techcrunch.com/2019/08/30/you-can-now-share-music-from-spotify-to-facebook-storiesMähler, R., & Vonderau, P. (2017). Studying Ad Targeting with Digital Methods: The Case of Spotify. Culture Unbound, 9(2), 212-221. doi:10.3384/cu.2000.1525.1792212Analyzing Spotify Data. Exploring the Possibilities of User Data from a Scientific and Business Perspective. (Supervised by Sandjai Bhulai). Report from Vrije Universiteit Amsterdamhttps://www.math.vu.nl/~sbhulai/papers/paper-vandenhoven.pdfGreenberg, D. M., Kosinski, M., Stillwell, D. J., Monteiro, B. L., Levitin, D. J., & Rentfrow, P. J. (2016). The Song Is You. Social Psychological and Personality Science, 7(6), 597-605. doi:10.1177/1948550616641473Spotify Charts Regionalhttps://spotifycharts.com/regionalSpotify Charts Launch Globally, Showcase 50 Most Listened to and Most Viral Tracks Weeklyhttps://www.engadget.com/2013-05-21-spotify-charts-launch.htmlSpotify says its Viral-50 chart reaches the parts other charts don’thttps://musically.com/2014/07/15/spotify-says-its-viral-50-chart-reaches-the-parts-other-charts-dont/Spotify Reveals New Viral 50 Charthttps://www.musicweek.com/news/read/spotify-launches-the-viral-50-chart/059027Reports Results for Fiscal Second Quarter Ended 31 March 2020https://www.wmg.com/news/warner-music-group-corp-reports-results-fiscal-second-quarter-ended-march-31-2020-34751COVID-19’s Effect on the Global Music Business, Part 1: Genrehttps://blog.chartmetric.com/covid-19-effect-on-the-global-music-business-part-1-genre/Top 200https://spotifycharts.com/regional/global/weeklySpotify Chartshttps://spotifycharts.com/viral

Multiplicative structures of hypercyclic functions for convolution operators

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains.Comment: 12 page

A freshmen mentoring program at the Universitat Politècnica de València over the period 2000-2010

Purpose: We quantify the dedication of mentors and mentees in a double-mentor program for freshmen. We also analyze the fulfillment of the expectations of all the participants and the utility of each mentor during the meetings. Design/methodology/approach: A questionnaire addressed to the participants was designed and took to all of them. Findings: We see that the mentees are highly satisfied with the mentoring program. Despite of mentors want mentees to participate more intensively in the program, mentees consider that their participation in the program is sufficient. Student and teacher mentors agree that freshmen could take more profit of the program. We also observe that the student mentor is slightly more implicated in the program than the teacher mentor. Research limitations/implications: Our results depend on the opinion of the mentors and the mentees. It would be fruitful to know the opinion of the freshmen who are not enrolled in the program. Practical implications: Our research confirms the validity of the system. We hope that our conclusions will be fruitful for other institutions that would like to implement a double-mentor program for freshmen. Originality/value: Many mentoring programs have been designed basing on a big brother/sister system. We analyze the development of a double-mentor program. The last law concerning university students, approved in Spain in 2011, indicates that mentoring programs should be designed in the frame of every degree, and they should be conducted by teachers and by technical staff. With our formula, both approaches are combined.Peer Reviewe

Dynamics of multidimensional Césaro operators

[EN] We study the dynamics of the multi-dimensional Cesar degrees integral operator on L-P (I-n), for I the unit interval, 1 = 2, that is defined as C(f)(x(1),...,x(n)) = 1/x(1)x(2)...x(n) integral(x1)(0) ... integral(x1)(0) f(u(1),...,u(n))du(1)...du(n) for f is an element of L-p(I-n). This operator is already known to be bounded. As a consequence of the Eigenvalue Criterion, we show that it is hypercyclic as well. Moreover, we also prove that it is Devaney chaotic and frequently hypercyclic.The first author was supported by MEC, grant MTM201675963-P. The third author was supported by grant MTM2015-65825-P.Conejero, JA.; Mundayadan, A.; Seoane-Sepúlveda, JB. (2019). Dynamics of multidimensional Césaro operators. Bulletin of the Belgian Mathematical Society Simon Stevin. 26(1):11-20. http://hdl.handle.net/10251/159145S112026

Shock wave formation in compliant arteries

[EN] We focus on the problem of shock wave formation in a model of blood flow along an elastic artery. We analyze the conditions under which this phenomenon can appear and we provide an estimation of the instant of shock formation. Numerical simulations of the model have been conducted using the Discontinuous Galerkin Finite Element Method. The results are consistent with certain phenomena observed by practitioners in patients with arteriopathies, and they could predict the possible formation of a shock wave in the aorta.C. Rodero and I. Garcia-Fernandez are supported by Projects TIN2014-59932-JIN (MINECO/FEDER, EU) and CIB16-BM019 (IISCII). J. A. Conejero is supported by MEC, Project MTM2016-75963-P.Rodero, C.; Conejero, JA.; García-Fernández, I. (2019). Shock wave formation in compliant arteries. Evolution Equations and Control Theory (Online). 8(1):221-230. https://doi.org/10.3934/eect.2019012S2212308

A pre-processing procedure for the implementation of the greedy rank-one algorithm to solve high-dimensional linear systems

[EN] Algorithms that use tensors are increasingly important due to the goodness of this operation when performing calculations with large amounts of data. Among them, we find the algorithms that search for the solution of a linear system in separated form, where the Greedy Rank-One Update method stands out, the starting point of the famous PGD family (from its acronym, Proper Generalized Decomposition). When the matrices of these systems have the particular structure of a Laplacian-type matrix, the convergence of the previous methods is faster and more accurate. The Laplacian Decomposition Algorithm calculates the Laplacian matrix that best approximates a given square matrix. When the residue of this approximation is small, we will be able to solve the linear system associated with a Laplacian-type matrix and thus obtain an approximation of the solution of the original system, with a lower computational cost. In this paper we prove that the discretization of a general Partial Differential Equation of the second order can be written as a linear system with a Laplacian-type matrix.J. A. Conejero acknowledges funding from grant PID2021-124618NB-C21, funded by MCIN/AEI/ 10.13039/501100011033, and by ERDF: A way of making Europe , by the European Union ; M. Mora-Jimenez was supported by the Generalitat Valenciana and the European Social ¿ Fund under grant number ACIF/2020/269; A. Falco was supported by the MICIN grant number ¿ RTI2018-093521-B-C32 and Universidad CEU Cardenal Herrera under grant number INDI22/15Conejero, JA.; Falcó, A.; Mora-Jiménez, M. (2023). A pre-processing procedure for the implementation of the greedy rank-one algorithm to solve high-dimensional linear systems. AIMS Mathematics. 8(11):25633-25653. https://doi.org/10.3934/math.20231308256332565381

Fractional vs. ordinary control systems: What does the fractional derivative provide?

[EN] The concept of a fractional derivative is not at all intuitive, starting with not having a clear geometrical interpretation. Many different definitions have appeared, to the point that the need for order has arisen in the field. The diversity of potential applications is even more overwhelming. When modeling a problem, one must think carefully about what the introduction of fractional derivatives in the model can provide that was not already adequately covered by classical models with integer derivatives. In this work, we present some examples from control theory where we insist on the importance of the non-local character of fractional operators and their suitability for modeling non-local phenomena either in space (action at a distance) or time (memory effects). In contrast, when we encounter completely different nonlinear phenomena, the introduction of fractional derivatives does not provide better results or further insight. Of course, both phenomena can coexist and interact, as in the case of hysteresis, and then we would be dealing with fractional nonlinear models.Conejero, JA.; Franceschi, J.; Picó-Marco, E. (2022). Fractional vs. ordinary control systems: What does the fractional derivative provide?. Mathematics. 10(15):1-18. https://doi.org/10.3390/math10152719118101

Convolution operators supporting hypercyclic algebras

[EN] We show that any convolution operator induced by a non-constant polynomial that vanishes at zero supports a hypercyclic algebra. This partially solves a question raised by R. AronThis work is supported in part by MICINN and FEDER, Project MTM2013-47093-P, and by GVA, Project ACOMP/2015/005.Bès, JP.; Conejero, JA.; Papathanasiou, D. (2017). Convolution operators supporting hypercyclic algebras. Journal of Mathematical Analysis and Applications. 445(2):1232-1238. https://doi.org/10.1016/j.jmaa.2016.01.029S12321238445

An algorithm for self-organization of driverless vehicles of a car-rental service

[EN] The solution of fleet assignment problems is one of the core points in the management of a car-rental company. These problems arise from the restrictions imposed by the acceptance of reservations of the customers and from the necessity of an arrangement of the cars at the different depots for attending these bookings. However, with the coming of driverless cars, the rearrangement of vehicles among the depots can be simplified since no staff must be responsible of these tasks. In this note, we show a heuristic algorithm that permits to minimize the number of cars that have to be subcontracted from an external provider in order to attend a list of reservations from customers. Its efficacy is shown in comparison with the solution given by the integer linear programming method. These results can be of interest for other autonomous systems that need to be controlled.The first author is supported by MEC Project MTM2013-47093-P. The third author is supported by MEC Project MTM2012-33073. When adapting the paper to the template we missed out the acknowledgements to the research projects that support our research.Conejero, JA.; Jordan-Lluch, C.; Sanabria-Codesal, E. (2016). An algorithm for self-organization of driverless vehicles of a car-rental service. Nonlinear Dynamics. 84(1):107-114. https://doi.org/10.1007/s11071-015-2237-4S107114841Abara, J.: Applying integer linear programming to the fleet assignment problem. Interfaces 19(4), 20–28 (1989)Ahuja, R.K., Magnati, T.L., Orlin, J.B.: Networks Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)Bodin, L., Golden, B., Assad, A., Ball, M.: Routing and scheduling of vehicles and crews: the state of the art. Comput. Oper. Res. 10, 63–212 (1983)Cacchiani, V., Caprara, A., Toth, P.: A Lagrangian heuristic for a train-unit assignment problem. Discrete Appl. Math. 161, 1707–1718 (2013)Cacchiani, V., Caprara, A., Toth, P.: Solving a real-world train unit assignment problem. Math. Program. Ser. B 124, 207–231 (2010)CNN. Driverless car tech gets serious at CES http://edition.cnn.com/2014/01/09/tech/innovation/self-driving-cars-ces/ . Last visit on June 18th, 2015Conejero, J.A., Jordan, C., Sanabria-Codesal, E.: An iterative algorithm for the management of an electric-car-rental service. J. Appl. Math. 2014, Article ID 483734, 11 (2014)DARPA Grand Challenge. http://en.wikipedia.org/wiki/2005_DARPA_Grand_Challenge#2005_Grand_Challenge . Last visit on June 18th, 2015Fraile, A., Larrod, E., Magren, A., Sicilia, J.A.: Decision model for siting transport and logistic facilities in urban environments: A methodological approach. J. Comput. Appl. Math. doi: 10.1016/j.cam.2014.12.012Evans, J.R., Minieka, E.: Optimization Algorithms for Networks and Graphs. Dekker, USA (1992)Fink, A., Reiners, T.: Modeling and solving the short-term car rental logistics problem. Transp. Res. Part E 42, 272–292 (2006)Google driverless cars http://en.wikipedia.org/wiki/Google_driverless_car . Last visit on June 18th, (2015)Heikkilä, S.: Mobility as a service A proposal for action for the public administration. Case Helsinki. Master Thesis. School of Engineering. Aalto UniversityHertz, A., Schindl, D., Zufferey, N.: A solution method for a car fleet management problem with maintenance constraints. J. Heuristics 15, 425–450 (2009)Kennington, J.L., Helgason, R.V.: Algorithms for Network Programming. Wiley, New York (1980)Lohatepanont, M., Barnhart, C.: Airline schedule planning: integrated models and algorithms for schedule design and fleet assignment. Transp. Sci. 38, 19–32 (2004)Mobincity Project. http://www.mobincity.eu/ . Last visit on June 18th, (2015)RelayRides. https://relayrides.com/ Last visit on June 18th, (2015)ROADEF Challenge. Description of the problem. http://www.prism.uvsq.fr/~vdc/ROADEF/CHALLENGES/1999/ (1999) Last visit on June 18th, (2015)Superhub Project: SUstainable and PERsuasive Human Users moBility in future cities. http://superhub-project.eu/ . Last visit on June, 18th (2015)Yan, S., Chen, C.H.: Optimal flight scheduling models for cargo airlines under alliances. J. Sched. 11, 175–186 (2008)Yan, S., Tang, C.H., Lee, M.C.: A flight scheduling model for Taiwan airlines under market competitions. Omega 35, 175–186 (2007)Yan, S., Tseng, C.H.: A passenger demand based model for airline flight scheduling. Comput. Oper. Res. 29, 1559–1581 (2002)Yang, Y., Jin, W., Hao, X.: Car rental logistics problem: A review of literature. IEEE International Conference on Service Operations and Logistics, and Informatics, pp. 2815 - 2819. E-ISBN: 978-1-4244-2013-1 (2008

Recovering discrete delayed fractional equations from trajectories

[EN] We show how machine learning methods can unveil the fractional and delayed nature of discrete dynamical systems. In particular, we study the case of the fractional delayed logistic map. We show that given a trajectory, we can detect if it has some delay effect or not and also to characterize the fractional component of the underlying generation model.Ministerio de Ciencia e Innovacion-Agencia Estatal de Investigacion, Grant/Award Number: PID2021-124618NB-C21; European UnionConejero, JA.; Garibo-I-Orts, Ó.; Lizama, C. (2023). Recovering discrete delayed fractional equations from trajectories. Mathematical Methods in the Applied Sciences. https://doi.org/10.1002/mma.922