Spaces of modular forms. Modular curves and dimensions

The Modularity Theorem states that all rational elliptic curve arise from modular forms. In 1995, Andrew Wiles proved a special case of this theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves, completing the proof of Fermat’s Last Theorem after some 350 years. Later, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extended Wiles’s techniques to prove completely the Modularity Theorem. In this work we explain a complex analytic version of this notable theorem.El Teorema de Modularidad afirma que todas las curvas elípticas racionales surgen de formas modulares. En 1995, Andrew Wiles probó un caso especial de este teorema (entonces conocido como la conjetura de Taniyama–Shimura) para curvas elípticas semiestables, completando así la prueba del Ultimo Teorema de Fermat después de unos 350 años. Más tarde, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor extendieron las técnicas de Wiles para probar completamente el Teorema de Modularidad. En este trabajo nosotros explicamos una versión analítica compleja de este notable teorema.Universidad de Sevilla. Máster Universitario en Matemática

MIMO PID Controller Tuning Method for Quadrotor Based on LQR/LQG Theory

In this work, a new pre-tuning multivariable PID (Proportional Integral Derivative) controllers method for quadrotors is put forward. A procedure based on LQR/LQG (Linear Quadratic Regulator/Gaussian) theory is proposed for attitude and altitude control, which suposes a considerable simplification of the design problem due to only one pretuning parameter being used. With the aim to analyze the performance and robustness of the proposed method, a non-linear mathematical model of the DJI-F450 quadrotor is employed, where rotors dynamics, together with sensors drift/bias properties and noise characteristics of low-cost commercial sensors typically used in this type of applications are considered. In order to estimate the state vector and compensate bias/drift effects in the measures, a combination of filtering and data fusion algorithms (Kalman filter and Madgwick algorithm for attitude estimation) are proposed and implemented. Performance and robustness analysis of the control system is carried out by employing numerical simulations, which take into account the presence of uncertainty in the plant model and external disturbances. The obtained results show the proposed controller design method for multivariable PID controller is robust with respect to: (a) parametric uncertainty in the plant model, (b) disturbances acting at the plant input, (c) sensors measurement and estimation errors

Dual Stochastic Natural Gradient Descent

Although theoretically appealing, Stochastic Natural Gradient Descent (SNGD) is computationally expensive, it has been shown to be highly sensitive to the learning rate, and it is not guaranteed to be convergent. Convergent Stochastic Natural Gradient Descent (CSNGD) aims at solving the last two problems. However, the computational expense of CSNGD is still unacceptable when the number of parameters is large. In this paper we introduce the Dual Stochastic Natural Gradient Descent (DSNGD) where we take benefit of dually flat manifolds to obtain a robust alternative to SNGD which is also computationally feasible.Comment: 16 page

AutoTuning Environment for Static Obstacle Avoidance Methods Applied to USVs

This work is focused on reactive Static Obstacle Avoidance (SOA) methods used to increase the autonomy of Unmanned Surface Vehicles (USVs). Currently, there are multiple approaches to avoid obstacles, which can be applied to different types of USV. In order to assist in the choice of the SOA method for a particular vessel and to accelerate the pretuning process necessary for its implementation, this paper proposes a new AutoTuning Environment for Static Obstacle Avoidance (ATESOA) methods applied to USVs. In this environment, a new simplified modelling of a LIDAR (Laser Imaging Detection and Ranging) sensor is proposed based on numerical simulations. This sensor model provides a realistic environment for the tuning of SOA methods that, due to its low load computation, is used by evolutionary algorithms for the autotuning. In order to analyze the proposed ATESOA, three SOA methods were adapted and implemented to consider the measurements given by the LIDAR model. Furthermore, a mathematical model is proposed and evaluated for using as USV in the simulation enviroment. The results obtained in numerical simulations show how the new ATESOA is able to adjust the SOA methods in scenarios with different obstacle distributions

Expansions in series of varying Laguerre polynomials and some applications to molecular potentials

11 pages, no figures.-- MSC2000 codes: 81V55, 33C45, 81Q10.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001).MR#: MR1985711 (2004f:33026)Zbl#: Zbl 1017.81048The expansion of a large class of functions in series of linearly varying Laguerre polynomials, i.e., Laguerre polynomials whose parameters are linear functions of the degree, is found by means of the hypergeometric functions approach. This expansion formula is then used to obtain the Brown–Carlitz generating function (which gives a characterization of the exponential function) and the connection formula for these polynomials. Finally, these results are employed to connect the bound states of the quantum–mechanical potentials of Morse and Pöschl–Teller, which are frequently used to describe molecular systems.This work has been partially supported by the Junta de Andalucía, under the research Grants FQM0207 (J.S.D. and J.S.R.) and FQM0229 (P.L.A.); the Spanish MCyT projects BFM2001-3878-C02 (J.S.R., J.S.D. and P.L.A.) and BFM2000-0206-C04-01 (J.S.R.); and the European Union project INTAS 2000-272 (J.S.D. and J.S.R.).Publicad

A Robust Reactive Static Obstacle Avoidance System for Surface Marine Vehicles

This paper is centered on the guidance systems used to increase the autonomy of unmanned surface vehicles (USVs). The new Robust Reactive Static Obstacle Avoidance System (RRSOAS) has been specifically designed for USVs. This algorithm is easily applicable, since previous knowledge of the USV mathematical model and its controllers is not needed. Instead, a new estimated closed-loop model (ECLM) is proposed and used to estimate possible future trajectories. Furthermore, the prediction errors (due to the uncertainty present in the ECLM) are taken into account by modeling the USV's shape as a time-varying ellipse. Additionally, in order to decrease the computation time, we propose to use a variable prediction horizon and an exponential resolution to discretize the decision space. As environmental model an occupancy probability grid is used, which is updated with the measurements generated by a LIDAR sensor model. Finally, the new RRSOAS is compared with other SOA (static obstacle avoidance) methods. In addition, a robustness study was carried out over a set of random scenarios. The results obtained through numerical simulations indicate that RRSOAS is robust to unknown and congested scenarios in the presence of disturbances, while offering competitive performance with respect to other SOA methods

Plasma Growth Factors in Cerebral Palsy

The use of plasma growth factors is opening a new field of clinical application in medicine, developing a new discipline called regenerative medicine. In many fields such as traumatology, dental implantology or anesthesia, the use of this biotechnology is improving the quality of life of patients, through techniques that are not invasive but with extraordinary functional results. A discipline where this type of procedure opens an interesting field of application is undoubtedly neurology, especially those processes of ischemic or hypo-anoxic origin such as cerebral palsy, where recent studies point to an improvement of cognitive abilities in patients, together with specific neurorehabilitation therapies

Clinical Applications of Plasma Growth Factors

The use of plasma rich in growth factors has become a technique increasingly used in various fields of medicine. Since its inception in use in sports medicine and dental implants in the mid-80s, gradually it has expanded its field of use in clinical specialties. The power cell tropism for certain tissues, attributed to growth factors, has currently talked of a new medical discipline as Regenerative Medicine. Not only has experienced an ever increasing boom in various medical specialties, but simultaneously has increased exponentially types and methodology for obtaining application forms even for the same pathology. So much so that now its use has exceeded the capacity to produce scientific evidence for successful clinical application

Power-aggregation of pseudometrics and the McShane-Whitney extension theorem for Lipschitz p-convace maps

[EN] Given a countable set of families {Dk:k¿N} of pseudometrics over the same set D, we study the power-aggregations of this class, that are defined as convex combinations of integral averages of powers of the elements of ¿kDk. We prove that a Lipschitz function f is dominated by such a power-aggregation if and only if a certain property of super-additivity involving the powers of the elements of ¿kDk is fulfilled by f. In particular, we show that a pseudo-metric is equivalent to a power-aggregation of other pseudometrics if this kind of domination holds. When the super-additivity property involves a p-power domination, we say that the elements of Dk are p-concave. As an application of our results, we prove under this requirement a new extension result of McShane-Whitney type for Lipschitz p-concave real valued maps.Both authors gratefully acknowledge the support of the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigaciones and FEDER under each grants MTM2015-64373-P (MINECO/FEDER, UE) and MTM2016-77054-C2-1-P.Rodríguez López, J.; Sánchez Pérez, EA. (2020). Power-aggregation of pseudometrics and the McShane-Whitney extension theorem for Lipschitz p-convace maps. Acta Applicandae Mathematicae. 170:611-629. https://doi.org/10.1007/s10440-020-00349-3S611629170Dahia, E., Achour, D., Rueda, P., Sánchez Pérez, E.A., Yahi, R.: Factorization of Lipschitz operators on Banach function spaces. Math. Inequal. Appl. 21(4), 1091–1104 (2018)Beliakov, G.: Optimization and aggregation functions. In: Lodwick, W.A., Kacprzyk, J. (eds.) Fuzzy Optimization: Recent Advances and Applications, pp. 77–108. Springer, Berlin (2010)Beliakov, G., Bustince Sola, H., Calvo Sánchez, T.: A Practical Guide to Averaging Functions, vol. 329. Springer, Heidelberg (2016)Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch domination theorem. J. Math. Anal. Appl. 365(1), 269–276 (2010)Botelho, G., Pellegrino, D., Rueda, P.: On Pietsch measures for summing operators and dominated polynomials. Linear Multilinear Algebra 62(7), 860–874 (2014)Chávez-Domínguez, J.A.: Duality for Lipschitz p-summing operators. J. Funct. Anal. 261, 387–407 (2011)Chávez-Domínguez, J.A.: Lipschitz $p$-convex and $q$-concave maps (2014). arXiv:1406.6357Defant, A.: Variants of the Maurey-Rosenthal theorem for quasi Köthe function spaces. Positivity 5, 153–175 (2001)Defant, A., Floret, K.: Tensor Norms and Operator Ideals. Elsevier, Amsterdam (1992)Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge University Press, Cambridge (1995)Dugundji, J.: Topology. Allyn and Bacon, Needham Heights (1966)Farmer, J., Johnson, W.: Lipschitz $p$-summing operators. Proc. Am. Math. Soc. 137(9), 2989–2995 (2009)Juutinen, P.: Absolutely minimizing Lipschitz extensions on a metric space. Ann. Acad. Sci. Fenn., Math. 27, 57–67 (2002)Marler, T.R., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2010)Mustata, C.: Extensions of semi-Lipschitz functions on quasi-metric spaces. Rev. Anal. Numér. Théor. Approx. 30(1), 61–67 (2001)Mustata, C.: On the extremal semi-Lipschitz functions. Rev. Anal. Numér. Théor. Approx. 31(1), 103–108 (2002)Pellegrino, D., Santos, J.: Absolutely summing multilinear operators: a panorama. Quaest. Math. 34(4), 447–478 (2011)Romaguera, S., Sanchis, M.: Semi-Lipschitz functions and best approximation in quasi-metric spaces. J. Approx. Theory 103, 292–301 (2000)Willard, S.: General Topology. Addison-Wesley, Reading (1970)Yahi, R., Achour, D., Rueda, P.: Absolutely summing Lipschitz conjugates. Mediterr. J. Math. 13(4), 1949–1961 (2016