Constrained extrema of two variables functions

En este artículo se presenta el cálculo de máximos y mínimos condicionados para funciones de dos variable, es decir, cómo calcular los valores máximo y mínimo de una función de dos variables que satisfacen una igualdad determinada. Para ello se utilizan dos métodos: la reducción a un problema de obtención de máximos y mínimos de una función de una variable y el uso de multiplicadores de Lagrange.Herrero Debón, A. (2017). Constrained extrema of two variables functions. http://hdl.handle.net/10251/82485DE

Free extrema of two variables functions

En este artículo se presenta el concepto de máximo y mínimo relativo para una función de dos variables. Estos se obtienen a partir de los puntos críticos de la función, los cuales cumplen que las derivadas parciales primeras en esos puntos no exixten o son nulas. Finalmente se utiliza el hessiano asociado a la función para determinan si el punto crítico encontrado es un máximo, un mínimo o un punto de silla. En algunas ocasiones el hessiano no nos permotorá discriminar el caracter del punto crítico y deberemos utilizar otro tipo de criterios como por ejemplo la representación gráfica de la función.Herrero Debón, A. (2017). Free extrema of two variables functions. http://hdl.handle.net/10251/82481DE

Absolute extrema of two variables function

Se introduce el concepto de extremos absolutos para funciones de dos variables y cómo calcularlos cuando la región es cerrada y acotada. Para ello se ponen diversos ejemplos abarcando regiones cuya frontera son poligonales o curvas definidas por una función o combinación de ambas.Herrero Debón, A. (2016). Absolute extrema of two variables function. http://hdl.handle.net/10251/68276DE

Models cosmològics inhomogenis conformement plans

RESUM En aquest treball parlem de models cosmològics conformement plans en dos sentits, aquells que tenen una mètrica associada que és conformement plana en el sentit de mètrica d'espai-temps 4-dimensional, i aquells que tenen associades unes seccions espacials conformement planes en el sentit de mètriques 3-dimensionals. Ambdós sentits de mètriques conformement planes simplifiquen considerablement el tractament de les equacions d'Einstein al restringir-nos a espai-temps els quals tenen un únic grau de llibertat representat per una funció arbitrària de les coordenades. Als espais-temps conformement plans aquesta funció està representada pel factor de conformitat corresponent; mentre que als espais-temps amb seccions espacials conformement planes será el potencial gravitatori. En primer lloc estudiem els camps conformes Killing radials i les seues propietats (corbes integrals, caràcter causal, etc.) a lespai-temps de Minkowski, especialment aquestes que no són invariants sota conformitats (acceleració, expansió, ...). A continuació establim un mètode cinemàtic per a l'obtenció de factors de conformitat a espai-temps conformement plans. Aquest mètode consisteix en imposar diferents condicions sobre les propietats cinemàtiques associades a aquests camps. Imposem principalment tres condicions: acceleració nulla, la qual ens fa caure en el factor de conformitat de les mètriques de Robertson-Walker; expansió homogènia i seccions espacials amb curvatura seccional constant, les quals donen lloc a factors de conformitat despai-temps que generalitzen els universos de Robertson-Walker. El cas dexpansió homogènia és interpretable també mitjançant el transport Fermi de lacceleració al llarg de les corbes integrals del camp. Les dues últimes condicions considerades conjuntament porten a una subclasse dels universos de Stephani. En segon lloc tractem el cas corresponent a models cosmològics inhomogenis conformement plans en el sentit que admeten seccions espacials conformes a la mètrica plana 3-dimensional. Ací s'hem plantejat un sistema més realista, estudiem la teoria relativista de les pertorbacions dun univers dEinstein-de Sitter. És a dir, tractem les equacions d'Einstein en forma evolutiva (formalisme 3+1) per a la mètrica dEinstein-de Sitter i on el contingut material estarà generat per una funció de distribució de matèria freda i sense collisions que verifica l'equació de Vlasov. Tractem per tant el problema dEinstein-Vlasov. Primerament tenim que les equacions dEinstein es redueixen a una única equació diferencial de segon ordre per al potencial gravitatori on els coeficients depenen del contingut energètic considerat. Aquests coeficients són determinats al resoldre lequació de Vlasov sobre una funció de distribució de matèria fosca sense collisions considerada com una pertorbació analítica en els moments del cas de quasi-equilibri. La resolució daquesta equació fins a tercer ordre en els moments ens porta a la construcció dun model sense anisotropies. A continuació, l'acoblament final de les equacions d'Einstein-Vlasov, una vegada obtinguda la forma de la densitat d'energia, la pressió mitjana i el fluxe de calor, ens porta a un problema de Cauchy de valors inicials per a levolució del potencial gravitatori. Finalment resolem lesmentat problema de Cauchy, en el cas en què fem una teoria linealitzada de levolució, utilitzant la tècnica de les transformades de Fourier. Com l'equació és de segon ordre, cal considerar dues condicions inicials, una sobre el potencial i una altra sobre la seua primera derivada temporal. Trobem la solució general daquest problema linealitzat a lespai real i en el cas de simetria esfèrica expressem aquesta solució en termes dintegrals unidimensionals de les condicions inicials. Laplicació a Cosmologia via el contrast de densitat ens permet estudiar la formació de cúmuls de galaxies i analitzar els resultats corresponents. Això porta a que la teoria relativista linealitzada de levolució de les pertorbacions de Friedmann pot donar creiximents considerables del contrast de densitat, la qual cosa pot ser relevant per a la formació destructures. __________________________________________________________________________________________________ SUMMARY Firstly, we study inhomogeneous cosmological models whose 4-dimensional metric is conformally flat. In these space-times we analyse the properties of radial conformal Killing vector fields (RCKF) as a way to obtain conformal factors. For example, the condition of nullity for the acceleration leads to the conformal factor of the Robertson-Walker metrics. Other conditions like homogeneous expansion or the existence of orthogonal (to a RCKF) surfaces with constant curvature allow obtaining conformal factors of space-times that generalise the Robertson-Walker universes. Secondly, we study cosmological models that admit conformally flat space-like sections. Concretely, we examine the relativistic theory of perturbations of an Einstein-de Sitter Universe in evolutive form. We consider that the energetic content is generated by a distribution function of collision-less cold matter (Einstein-Vlasov problem). The Einstein equations lead to a second order differential equation for the gravitational potential. We solve the Cauchy problem in the linealized case and for a model without anisotropies. It is necessary to consider two initial conditions, one of them over the potential and the other one over its first time derivative. In the spherical symmetric case the solution is expressed in terms of one-dimensional integrals of the initial conditions. This analysis leads to that the linealized relativistic theory of evolution of Friedmann perturbations can give big enhancements of the density contrast. This can be relevant to structure formation

Sobre soluciones reflexivas de la ecuación matricial AXB = C

El problema de resolver la ecuación matricial AXB = C ha sido estudiado en la literatura. Algunos autores han buscado la solución general de este problema mientras que otros han considerado algún tipo de restricción sobre la solución buscada (simétrica, definida positiva, etc.) o bien sobre las matrices conocidas (por ejemplo, siendo B la matriz identidad y A una reflexión generalizada). En este trabajo se estudia dicho problema buscando soluciones X que sean reflexivas con respecto a una reflexión generalizada tripotente P (es decir, la matriz X ∈ C n×n debe cumplir la condición X = P XP) siendo P ∈ C n×n una matriz Hermítica y tripotente conocida

New matrix partial order based spectrally orthogonal matrix decomposition

[EN] We investigate partial orders on the set of complex square matrices and introduce a new order relation based on spectrally orthogonal matrix decompositions. We also establish the relation of this concept with the known orders.The research of the first author was supported by the Grants [grant number RFBR-15-01-01132], [grant number MD-962.2014.1]. The second and third authors have been partially supported by Ministerio de Economia y Competitividad from Spain, DGI [grant number MTM2013-43678-P].Guterman, A.; Herrero Debón, A.; Thome, N. (2016). New matrix partial order based spectrally orthogonal matrix decomposition. Linear and Multilinear Algebra. 64(3):362-374. https://doi.org/10.1080/03081087.2015.1041365S362374643Meyer, C. (2000). Matrix Analysis and Applied Linear Algebra. doi:10.1137/1.9780898719512Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Baksalary, O. M., & Trenkler, G. (2010). Core inverse of matrices. Linear and Multilinear Algebra, 58(6), 681-697. doi:10.1080/03081080902778222Baksalary, O. M., & Trenkler, G. (2014). On a generalized core inverse. Applied Mathematics and Computation, 236, 450-457. doi:10.1016/j.amc.2014.03.048Hernández, A., Lattanzi, M., Thome, N., & Urquiza, F. (2012). The star partial order and the eigenprojection at 0 on EP matrices. Applied Mathematics and Computation, 218(21), 10669-10678. doi:10.1016/j.amc.2012.04.034Hernández, A., Lattanzi, M., & Thome, N. (2013). On a partial order defined by the weighted Moore–Penrose inverse. Applied Mathematics and Computation, 219(14), 7310-7318. doi:10.1016/j.amc.2013.02.010Hernández, A., Lattanzi, M., & Thome, N. (2015). Weighted binary relations involving the Drazin inverse. Applied Mathematics and Computation, 253, 215-223. doi:10.1016/j.amc.2014.12.102Lebtahi, L., Patrício, P., & Thome, N. (2013). The diamond partial order in rings. Linear and Multilinear Algebra, 62(3), 386-395. doi:10.1080/03081087.2013.779272Malik, S. B., Rueda, L., & Thome, N. (2013). Further properties on the core partial order and other matrix partial orders. Linear and Multilinear Algebra, 62(12), 1629-1648. doi:10.1080/03081087.2013.839676Rakić, D. S., & Djordjević, D. S. (2012). Space pre-order and minus partial order for operators on Banach spaces. Aequationes mathematicae, 85(3), 429-448. doi:10.1007/s00010-012-0133-2Nambooripad, K. S. S. (1980). The natural partial order on a regular semigroup. Proceedings of the Edinburgh Mathematical Society, 23(3), 249-260. doi:10.1017/s0013091500003801Mitra, S. K. (1987). On group inverses and the sharp order. Linear Algebra and its Applications, 92, 17-37. doi:10.1016/0024-3795(87)90248-

Nonnegative singular control systems using the Drazin projector

In this work we study conditions for guaranteeing the nonnegativity of a discrete-time singular control system. A first approach can be found in the literature for general systems, using the whole coefficient matrices. Also, the particular case of matrices of index 1 has been treated by using a block decomposition and the group-projector of the matrix that gives the singularity to the system. In order to complete this study, an analysis of the nonnegativity of a singular control system for matrices having arbitrary index is done by means of the core nilpotent decomposition. This technique allows us to reduce the size of the original matrices, improving the results where the whole coefficients are involved.This paper was partially supported by the Ministry of Education of Spain (grant DGI MTM2010-18228).Herrero Debón, A.; Thome Coppo, NJ. (2013). Nonnegative singular control systems using the Drazin projector. Applied Mathematics Letters. 26:799-803. doi:10.1016/j.aml.2013.03.0047998032

Acoustic Localization of Bragg Peak Proton Beams for Hadrontherapy Monitoring

[EN] Hadrontherapy makes it possible to deliver high doses of energy to cancerous tumors by using the large energy deposition in the Bragg-peak. However, uncertainties in the patient positioning and/or in the anatomical parameters can cause distortions in the calculation of the dose distribution. In order to maximize the effectiveness of heavy particle treatments, an accurate monitoring system of the deposited dose depending on the energy, beam time, and spot size is necessary. The localized deposition of this energy leads to the generation of a thermoacoustic pulse that can be detected using acoustic technologies. This article presents different experimental and simulation studies of the acoustic localization of thermoacoustic pulses captured with a set of sensors around the sample. In addition, numerical simulations have been done where thermo-acoustic pulses are emitted for the specific case of a proton beam of 100 MeV.This research was funded by the Spanish Agencia Estatal de Investigacion, grant number FPA2015-65150-C3-2-P (MINECO/FEDER).Otero-Vega, JE.; Felis-Enguix, I.; Ardid Ramírez, M.; Herrero Debón, A. (2019). Acoustic Localization of Bragg Peak Proton Beams for Hadrontherapy Monitoring. Sensors. 19(9):1-13. https://doi.org/10.3390/s19091971S113199Kundu, T. (2014). Acoustic source localization. Ultrasonics, 54(1), 25-38. doi:10.1016/j.ultras.2013.06.009Bortfeld, T. (1997). An analytical approximation of the Bragg curve for therapeutic proton beams. Medical Physics, 24(12), 2024-2033. doi:10.1118/1.598116Ahmad, M., Xiang, L., Yousefi, S., & Xing, L. (2015). Theoretical detection threshold of the proton-acoustic range verification technique. Medical Physics, 42(10), 5735-5744. doi:10.1118/1.4929939Knapp, C., & Carter, G. (1976). The generalized correlation method for estimation of time delay. IEEE Transactions on Acoustics, Speech, and Signal Processing, 24(4), 320-327. doi:10.1109/tassp.1976.1162830Adrián-Martínez, S., Bou-Cabo, M., Felis, I., Llorens, C. D., Martínez-Mora, J. A., Saldaña, M., & Ardid, M. (2015). Acoustic Signal Detection Through the Cross-Correlation Method in Experiments with Different Signal to Noise Ratio and Reverberation Conditions. Lecture Notes in Computer Science, 66-79. doi:10.1007/978-3-662-46338-3_7Felis, I., Martínez-Mora, J., & Ardid, M. (2016). Acoustic Sensor Design for Dark Matter Bubble Chamber Detectors. Sensors, 16(6), 860. doi:10.3390/s16060860Bragg, W. H., & Kleeman, R. (1905). XXXIX. On the α particles of radium, and their loss of range in passing through various atoms and molecules. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 10(57), 318-340. doi:10.1080/14786440509463378Janni, J. F. (1982). Energy loss, range, path length, time-of-flight, straggling, multiple scattering, and nuclear interaction probability. Atomic Data and Nuclear Data Tables, 27(2-3), 147-339. doi:10.1016/0092-640x(82)90004-3Jones, K. C., Seghal, C. M., & Avery, S. (2016). How proton pulse characteristics influence protoacoustic determination of proton-beam range: simulation studies. Physics in Medicine and Biology, 61(6), 2213-2242. doi:10.1088/0031-9155/61/6/2213Lai, H. M., & Young, K. (1982). Theory of the pulsed optoacoustic technique. The Journal of the Acoustical Society of America, 72(6), 2000-2007. doi:10.1121/1.388631Sigrist, M. W. (1986). Laser generation of acoustic waves in liquids and gases. Journal of Applied Physics, 60(7), R83-R122. doi:10.1063/1.337089Tam, A. C. (1986). Applications of photoacoustic sensing techniques. Reviews of Modern Physics, 58(2), 381-431. doi:10.1103/revmodphys.58.381Xiang, L., Han, B., Carpenter, C., Pratx, G., Kuang, Y., & Xing, L. (2012). X-ray acoustic computed tomography with pulsed x-ray beam from a medical linear accelerator. Medical Physics, 40(1), 010701. doi:10.1118/1.4771935Assmann, W., Kellnberger, S., Reinhardt, S., Lehrack, S., Edlich, A., Thirolf, P. G., … Parodi, K. (2015). Ionoacoustic characterization of the proton Bragg peak with submillimeter accuracy. Medical Physics, 42(2), 567-574. doi:10.1118/1.4905047De Bonis, G. (2009). Acoustic signals from proton beam interaction in water—Comparing experimental data and Monte Carlo simulation. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 604(1-2), S199-S202. doi:10.1016/j.nima.2009.03.072Kraan, A. C., Battistoni, G., Belcari, N., Camarlinghi, N., Cirrone, G. A. P., Cuttone, G., … Rosso, V. (2014). Proton range monitoring with in-beam PET: Monte Carlo activity predictions and comparison with cyclotron data. Physica Medica, 30(5), 559-569. doi:10.1016/j.ejmp.2014.04.003Patch, S. K., Hoff, D. E. M., Webb, T. B., Sobotka, L. G., & Zhao, T. (2017). Two-stage ionoacoustic range verification leveraging Monte Carlo and acoustic simulations to stably account for tissue inhomogeneity and accelerator-specific time structure - A simulation study. Medical Physics, 45(2), 783-793. doi:10.1002/mp.12681Lehrack, S., Assmann, W., Bertrand, D., Henrotin, S., Herault, J., Heymans, V., … Parodi, K. (2017). Submillimeter ionoacoustic range determination for protons in water at a clinical synchrocyclotron. Physics in Medicine & Biology, 62(17), L20-L30. doi:10.1088/1361-6560/aa81f8Hickling, S., Lei, H., Hobson, M., Léger, P., Wang, X., & El Naqa, I. (2017). Experimental evaluation of x-ray acoustic computed tomography for radiotherapy dosimetry applications. Medical Physics, 44(2), 608-617. doi:10.1002/mp.12039Ardid, M., Felis, I., Martínez-Mora, J. A., & Otero, J. (2017). Optimization of Dimensions of Cylindrical Piezoceramics as Radio-Clean Low Frequency Acoustic Sensors. Journal of Sensors, 2017, 1-8. doi:10.1155/2017/817967

Acoustic Bragg Peak Localization in Proton Therapy Treatment: Simulation Studies

[EN] A full chain simulation of the acoustic hadron therapy monitoring for brain tumors is presented in this work. For the study, a proton beam of 100 MeV was considered. In the first stage, Geant4 was used to simulate the energy deposition and to study the behavior of the Bragg peak. The energy deposition in the medium produced local heating that can be considered instantaneous with respect to the hydrodynamic time scale producing a sound pressure wave. The resulting thermoacoustic signal was subsequently obtained by solving the thermoacoustic equation. The acoustic propagation was simulated by the Finite Element Method (FEM) in the brain and the skull, where a set of piezoelectric sensors were placed. Lastly, the final received signals in the sensors were processed in order to reconstruct the position of the thermal source and, thus, to determine the feasibility and accuracy of acoustic beam monitoring in hadron therapy.Otero, J.; Felis, I.; Ardid Ramírez, M.; Herrero Debón, A.; Merchán, JA. (2019). Acoustic Bragg Peak Localization in Proton Therapy Treatment: Simulation Studies. MDPI. 1-7. https://doi.org/10.3390/ecsa-6-06533S1

Análisis del cambio repentino a docencia remota por la COVID-19 en los resultados de aprendizaje: caso de dos asignaturas anuales básicas en Grados de Ingeniería

[EN] In this article we analyse the experience in two basic annual subjects in Engineering Degrees with the aim of determining if there are differences in the degree of achievement of the learning objectives, due to the sudden change to virtual teaching obliged by the COVID-19 lockdown. With this aim, a statistical analysis of the grades obtained by the students is carried out, comparing the periods of in-class teaching with those of forced remote teaching in contrast with the experience of previous courses, where virtual teaching was absent. Results of this analysis do not hold significant differences in the grades of the students associated with the vast transformation in teaching and learning process. Even though the results cannot necessarily conclude that there have not been changes in learning outcomes, they are indicative of a reasonably good adaptability of the teaching and learning process to the sudden new context given by the pandemics.[ES] En este artículo analizamos la experiencia en dos asignaturas básicas anuales en Grados en Ingenierías con el objetivo de determinar si se observan diferencias en el grado de consecución de los objetivos de aprendizaje debido al repentino cambio a docencia virtual por el confinamiento de la COVID-19. Para ello se hace un análisis estadístico de las calificaciones obtenidas por los estudiantes, comparando los periodos de docencia presencial con los de docencia remota forzada y contrastándolos con la experiencia de los cursos precedentes, donde la docencia virtual no era presente. Los resultados de este análisis no sustentan diferencias significativas en las calificaciones de los estudiantes asociados a esta transformación en la docencia. Si bien los resultados no concluyen unívocamente que no haya habido cambios en los resultados de aprendizaje, sí que son indicativos de una buena adaptabilidad de la docencia ante el nuevo contexto de pandemia, en la que se ha sabido acompasar el nivel de exigencia a los condicionantes del entorno en situación de crisis sobrevenida.S. Ardid agradece el apoyo del Programa CIDEGENT de la Generalitat Valenciana, CIDEGENT/2019/043.Ardid Ramírez, M.; Ardid Ramírez, JS.; Herrero Debón, A. (2021). Análisis del cambio repentino a docencia remota por la COVID-19 en los resultados de aprendizaje: caso de dos asignaturas anuales básicas en Grados de Ingeniería. En IN-RED 2021: VII Congreso de Innovación Edicativa y Docencia en Red. Editorial Universitat Politècnica de València. 931-940. https://doi.org/10.4995/INRED2021.2021.13448OCS93194