Dynamics of the solutions of the water hammer equations

NOTICE: this is the author’s version of a work that was accepted for publication in Topology and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Topology and its Applications, [Volume 203, 15 April 2016, Pages 67-83] DOI10.1016/j.topol.2015.12.076¨[EN] In this note we provide a representation of the solution using an operator theoretical approach based on the theory of C-0-semigroups and cosine operator functions, when considering this phenomenon on a compressible fluid along an infinite pipe. We provide an integro-differential equation that represents this phenomenon and it only involves the discharge. In addition, the representation of the solution in terms of a specific C-0-semigroup lets us show that hypercyclicity and the topologically mixing property can occur when considering this phenomenon on certain weighted spaces of integrable and continuous functions on the real line. (C) 2016 Elsevier B.V. All rights reserved.The first and third authors are supported by MEC Projects MTM2010-14909 and MTM2013-47093-P. The first author is also supported by Programa de Investigación y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by CONICYT, under Fondecyt Grant number 1140258.Conejero, JA.; Lizama, C.; Ródenas Escribá, FDA. (2016). Dynamics of the solutions of the water hammer equations. Topology and its Applications. 203:67-83. https://doi.org/10.1016/j.topol.2015.12.076S678320

Chaos on Fuzzy Dynamical Systems

[EN] Given a continuous map f : X -> X on a metric space, it induces the maps f over bar :K(X) -> K(X), on the hyperspace of nonempty compact subspaces of X, and (f) over cap :F(X) -> F(X), on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u:X -> [0,1] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems (X,f), (K(X),f over bar ), and (F(X),(f) over cap). In particular, we considered several dynamical properties related to chaos: Devaney chaos, A-transitivity, Li-Yorke chaos, and distributional chaos, extending some results in work by Jardon, Sanchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451-463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).This work was supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00, and by Generalitat Valenciana, Projects PROMETEO/2017/102 and PROMETEU/2021/070.Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2021). Chaos on Fuzzy Dynamical Systems. Mathematics. 9(20):1-11. https://doi.org/10.3390/math9202629S11192

Set-Valued Chaos in Linear Dynamics

[EN] We study several notions of chaos for hyperspace dynamics associated to continuous linear operators. More precisely, we consider a continuous linear operator on a topological vector space X, and the natural hyperspace extensions and of T to the spaces of compact subsets of X and of convex compact subsets of X, respectively, endowed with the Vietoris topology. We show that, when X is a complete locally convex space (respectively, a locally convex space), then Devaney chaos (respectively, topological ergodicity) is equivalent for the maps T, and . Also, under very general conditions, we obtain analogous equivalences for Li-Yorke chaos. Finally, some remarks concerning the topological transitivity and weak mixing properties are included, extending results in Banks (Chaos Solitons Fractals 25(3):681-685, 2005) and Peris (Chaos Solitons Fractals 26(1):19-23, 2005).The first author was partially supported by CNPq (Brazil) and by the EBW+ Project (Erasmus Mundus Programme). The second and third authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second author was partially supported by GVA, Project PROMETEOII/2013/013.Bernardes, NCJ.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2017). Set-Valued Chaos in Linear Dynamics. Integral Equations and Operator Theory. 88(4):451-463. https://doi.org/10.1007/s00020-017-2394-6S451463884Banks, J.: Chaos for induced hyperspace maps. Chaos Solitons Fractals 25(3), 681–685 (2005)Bauer, W., Sigmund, K.: Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79, 81–92 (1975)Bayart, F., Matheron, É.: Hypercyclic operators failing the hypercyclicity criterion on classical Banach spaces. J. Funct. Anal. 250(2), 426–441 (2007)Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li-Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35(6), 1723–1745 (2015)Bernardes Jr., N.C., Vermersch, R.M.: Hyperspace dynamics of generic maps of the Cantor space. Can. J. Math. 67(2), 330–349 (2015)Bès, J., Menet, Q., Peris, A., Puig, Y.: Strong transitivity properties for operators. Preprint ( arXiv:1703.03724 )Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167(1), 94–112 (1999)Bonet, J., Frerick, L., Peris, A., Wengenroth, J.: Transitive and hypercyclic operators on locally convex spaces. Bull. Lond. Math. Soc. 37, 254–264 (2005)de la Rosa, M., Read, C.: A hypercyclic operator whose direct sum $$T \oplus T$$ T ⊕ T is not hypercyclic. J. Oper. Theory 61(2), 369–380 (2009)Fedeli, A.: On chaotic set-valued discrete dynamical systems. Chaos Solitons Fractals 23(4), 1381–1384 (2005)Fu, H., Xing, Z.: Mixing properties of set-valued maps on hyperspaces via Furstenberg families. Chaos Solitons Fractals 45(4), 439–443 (2012)Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)Grivaux, S.: Hypercyclic operators, mixing operators, and the bounded steps problem. J. Oper. Theory 54, 147–168 (2005)Grosse-Erdmann, K.-G., Peris, A.: Weakly mixing operators on topological vector spaces. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 104(2), 413–426 (2010)Grosse-Erdmann, K.-G., Peris, A.: Linear Chaos. Universitext, Springer-Verlag, Berlin (2011)Guirao, J.L.G., Kwietniak, D., Lampart, M., Oprocha, P., Peris, A.: Chaos on hyperspaces. Nonlinear Anal. 71(1–2), 1–8 (2009)Hernández, P., King, J., Méndez, H.: Compact sets with dense orbit in $$2^X$$ 2 X . Topol. Proc. 40, 319–330 (2012)Herzog, G., Lemmert, R.: On universal subsets of Banach spaces. Math. Z. 229(4), 615–619 (1998)Hou, B., Tian, G., Shi, L.: Some dynamical properties for linear operators. Ill. J. Math. 53(3), 857–864 (2009)Hou, B., Tian, G., Zhu, S.: Approximation of chaotic operators. J. Oper. Theory 67(2), 469–493 (2012)Illanes, A., Nadler Jr., S.B.: Hyperspaces: Fundamentals and Recent Advances. Marcel Dekker Inc, New York (1999)Kupka, J.: On Devaney chaotic induced fuzzy and set-valued dynamical systems. Fuzzy Sets Syst. 177(1), 34–44 (2011)Kwietniak, D., Oprocha, P.: Topological entropy and chaos for maps induced on hyperspaces. Chaos Solitons Fractals 33(1), 76–86 (2007)Li, J., Yan, K., Ye, X.: Recurrence properties and disjointness on the induced spaces. Discrete Contin. Dyn. Syst. 35(3), 1059–1073 (2015)Liao, G., Wang, L., Zhang, Y.: Transitivity, mixing and chaos for a class of set-valued mappings. Sci. China Ser. A Math. 49(1), 1–8 (2006)Liu, H., Shi, E., Liao, G.: Sensitivity of set-valued discrete systems. Nonlinear Anal. 71(12), 6122–6125 (2009)Liu, H., Lei, F., Wang, L.: Li-Yorke sensitivity of set-valued discrete systems, J. Appl. Math. (2013). Article number: 260856Peris, A.: Set-valued discrete chaos. Chaos Solitons Fractals 26(1), 19–23 (2005)Peris, A., Saldivia, L.: Syndetically hypercyclic operators. Integral Equ. Oper. Theory 51, 275–281 (2005)Román-Flores, H.: A note on transitivity in set-valued discrete systems. Chaos Solitons Fractals 17(1), 99–104 (2003)Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc, New York (1991)Salas, H.: A hypercyclic operator whose adjoint is also hypercyclic. Proc. Am. Math. Soc. 112(3), 765–770 (1991)Shkarin, S.: Hypercyclic operators on topological vector spaces. J. Lond. Math. Soc. 86, 195–213 (2012)Wang, Y., Wei, G.: Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. Topol. Appl. 155(1), 56–68 (2007)Wang, Y., Wei, G., Campbell, W.H.: Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems. Topol. Appl. 156(4), 803–811 (2009)Wu, Y., Xue, X., Ji, D.: Linear transitivity on compact connected hyperspace dynamics. Dyn. Syst. Appl. 21(4), 523–534 (2012)Wu, X., Wang, J., Chen, G.: F-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429(1), 16–26 (2015

Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation

[EN] We study a third-order partial differential equation in the form $\tau u_{ttt} +\alpha u_{tt} -c^2 u_{xx} -b u_{xxt} =0, (1)$$that corresponds to the one-dimensional version of the Moore-Gibson-Thompson equation arising in high-intensity ultrasound and linear vibrations of elastic structures. In contrast with the current literature on the subject, we show that when the critical parameter$\gamma:=\alpha-\frac{\tauc^2}{b}$ is negative, the equation (1) admits an uniformly continuous, chaotic and topologically mixing semigroup on Banach spaces of Herzog s type.The first and third authors are supported in part by MEC Project MTM2013-47093-P. The second author is partially supported by Project Anillo ACT1112. The authors are grateful to an anonymous referee for helpful comments that improved the presentation of the paper.Conejero, JA.; Lizama, C.; Ródenas Escribá, FDA. (2015). Chaotic behaviour of the solutions of the Moore-Gibson-Thompson equation. Applied Mathematics & Information Sciences. 9(5):2233-2238. https://doi.org/10.12785/amis/090503S223322389

Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems

[EN] Let X be a compact metric space and a continuous map f:X-->X which defines a discrete dynamical system (X,f). The map f induces two natural maps, namely \bar{f}:K(X)-->K(X) on the hyperspace K(X) of non-empty compact subspaces of X and the Zadeh¿s extension \hat{f}:F(X)-->F(X) on the space F(X) of normal fuzzy set. In this work, we analyze the interaction of some orbit tracing dynamical properties, namely the specification and shadowing properties of the discrete dynamical system (X,f) and its induced discrete dynamical systems (K(X),\bar{f}) and (F(X),\hat{f}). Adding an algebraic structure yields stronger conclusions, and we obtain a full characterization of the specification property in the hyperspace, in the fuzzy space, and in the phase space X if we assume that the later is a convex compact subset of a (metrizable and complete) locally convex space and f is a linear operator.This work was supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GB-I00, and by Generalitat Valenciana, Project PROMETEU/2021/070.Bartoll Arnau, S.; Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2022). Orbit Tracing Properties on Hyperspaces and Fuzzy Dynamical Systems. Axioms. 11(12):1-11. https://doi.org/10.3390/axioms11120733111111

Sets of periods for chaotic linear operators

[EN] We provide a complete characterization of the possible sets of periods for Devaney chaotic linear operators on Hilbert spaces. As a consequence, we also derive this characterization for linearizable maps on Banach spaces.This work was supported by MICINN and FEDER, Projects MTM2016-75963-P and PID2019-105011GB-I00. The second and third authors were also supported by Generalitat Valenciana, Project PROMETEO/2017/102. We would to that the referees whose careful reading and observations produced an improvement in the presentation of the article.Conejero, JA.; Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2021). Sets of periods for chaotic linear operators. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 115(2):1-7. https://doi.org/10.1007/s13398-020-00996-zS17115

The Specification Property for C0-Semigroups

[EN] We study one of the strongest versions of chaos for continuous dynamical systems, namely the specification property. We extend the definition of specification property for operators on a Banach space to strongly continuous one-parameter semigroups of operators, that is, C0-semigroups. In addition, we study the relationships of the specification property for C0-semigroups (SgSP) with other dynamical properties: mixing, Devaney's chaos, distributional chaos, and frequent hypercyclicity. Concerning the applications, we provide several examples of semigroups which exhibit the SgSP with particular interest on solution semigroups to certain linear PDEs, which range from the hyperbolic heat equation to the Black-Scholes equation.The authors were supported by MINECO, Projects MTM2013-47093-P and MTM2016-75963-P. The second and third authors were also supported by Generalitat Valenciana, Projects PROMETEOII/2013/013 and PROMETEO/2017/102. We are indebted to the referee whose valuable comments produced an improvement in the presentation of the paper.Bartoll Arnau, S.; Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2019). The Specification Property for C0-Semigroups. Mediterranean Journal of Mathematics. 16(3):1-12. https://doi.org/10.1007/s00009-019-1353-7S112163Albanese, A.A., Barrachina, X., Mangino, E.M., Peris, A.: Distributional chaos for strongly continuous semigroups of operators. Commun. Pure Appl. Anal. 12, 2069–2082 (2013)Aroza, J., Kalmes, T., Mangino, E.: Chaotic $$C_0$$-semigroups induced by semiflows in Lebesgue and Sobolev spaces. J. Math. Anal. Appl. 412, 77–98 (2014)Badea, C., Grivaux, S.: Unimodular eigenvalues, uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation-stability and chaos. Discrete Contin. Dyn. Syst. 29, 67–79 (2011)Bartoll, S., Martínez-Giménez, F., Peris, A.: The specification property for backward shifts. J. Differ. Equ. Appl. 18, 599–605 (2012)Bartoll, S., Martínez-Giménez, F., Peris, A.: Operators with the specification property. J. Math. Anal. Appl. 436, 478–488 (2016)Bayart, F., Bermúdez, T.: Semigroups of chaotic operators. Bull. Lond. Math. Soc. 41, 823–830 (2009)Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bayart, F., Ruzsa, I.Z.: Difference sets and frequently hypercyclic weighted shifts. Ergod. Theory Dyn. Syst. 35, 691–709 (2015)Bermúdez, T., Bonilla, A., Conejero, J.A., Peris, A.: Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Stud. Math. 170, 57–75 (2005)Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265, 2143–2163 (2013)Bernardes Jr., N.C., Bonilla, A., Müller, V., Peris, A.: Li–Yorke chaos in linear dynamics. Ergod. Theory Dyn. Syst. 35, 1723–1745 (2015)Bernardes Jr., N.C., Bonilla, A., Peris, A., Wu, X.: Distributional chaos for operators on Banach spaces. J. Math. Anal. Appl. 459, 797–821 (2018)Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergod. Theory Dyn. Syst. 27, 383–404 (2007)Bowen, R.: Topological entropy and axiom $${\rm A}$$. In: Global Analysis (Proc. Sympos. Pure Math., vol. XIV, Berkeley, Calif., 1968), pp. 23–41. Amer. Math. Soc., Providence (1970)Bowen, R.: Periodic orbits for hyperbolic flows. Am. J. Math. 94, 1–30 (1972)Chakir, M., EL Mourchid, S.: Strong mixing Gaussian measures for chaotic semigroups. J. Math. Anal. Appl. 459, 778–788 (2018)Conejero, J.A., Lizama, C., Murillo-Arcila, M., Peris, A.: Linear dynamics of semigroups generated by differential operators. Open Math. 15, 745–767 (2017)Conejero, J.A., Müller, V., Peris, A.: Hypercyclic behaviour of operators in a hypercyclic $$C_0$$-semigroup. J. Funct. Anal. 244, 342–348 (2007)Conejero, J.A., Peris, A.: Hypercyclic translation $$C_0$$-semigroups on complex sectors. Discrete Contin. Dyn. Syst. 25, 1195–1208 (2009)Conejero, J.A., Peris, A., Trujillo, M.: Chaotic asymptotic behaviour of the hyperbolic heat transfer equation solutions. Int. J. Bifur. Chaos Appl. Sci. Eng. 20, 2943–2947 (2010)Costakis, G., Peris, A.: Hypercyclic semigroups and somewhere dense orbits. C. R. Math. Acad. Sci. Paris 335, 895–898 (2002)Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17, 793–819 (1997)Emamirad, H., Goldstein, G., Goldstein, J.A.: Chaotic solution for the Black–Scholes equation. Proc. Am. Math. Soc. 140, 2043–2052 (2012)Goldstein, J.A., Mininni, R.M., Romanelli, S.: A new explicit formula for the solution of the Black–Merton–Scholes equation. In: Infinite Dimensional Stochastic Analysis, World Series Publ., pp. 226–235 (2008)Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Universitext, Springer-Verlag London Ltd., London (2011)Herzog, G.: On a universality of the heat equation. Math. Nachr. 188, 169–171 (1997)Mangino, E.M., Peris, A.: Frequently hypercyclic semigroups. Stud. Math. 202, 227–242 (2011)Mangino, E.M., Murillo-Arcila, M.: Frequently hypercyclic translation semigroups. Stud. Math. 227, 219–238 (2015)Murillo-Arcila, M., Peris, A.: Strong mixing measures for $$C_0$$-semigroups. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109, 101–115 (2015)Oprocha, P.: Specification properties and dense distributional chaos. Discrete Contin. Dyn. Syst. 17, 821–833 (2007)Rudnicki, R.: Chaoticity and invariant measures for a cell population model. J. Math. Anal. Appl. 339, 151–165 (2012)Yin, Z., Wei, Y.: Recurrence and topological entropy of translation operators. J. Math. Anal. Appl. 460, 203–215 (2018

Parallel CT image reconstruction based on GPUs

[EN] In X-ray computed tomography (CT) iterative methods are more suitable for the reconstruction of images with high contrast and precision in noisy conditions from a small number of projections. However, in practice, these methods are not widely used due to the high computational cost of their implementation. Nowadays technology provides the possibility to reduce effectively this drawback. It is the goal of this work to develop a fast GPU-based algorithm to reconstruct high quality images from under sampled and noisy projection data.Research supported by ANITRAN Project PROMETEO/2010/039.Flores, LA.; Vidal Gimeno, VE.; Mayo Nogueira, P.; Ródenas Escribá, FDA.; Verdú Martín, GJ. (2014). Parallel CT image reconstruction based on GPUs. Radiation Physics and Chemistry. 95(1):247-250. https://doi.org/10.1016 / j.radphyschem.2013.03.011S24725095

Integración múltiple y vectorial

Este texto presenta una exposición detallada de la integración múltiple, los fundamentos de la teoría de campos y las integrales de línea y de superficie. Se ha añadido también un apéndice acerca de superficies cuádricas en el espacio tridimensional. La teoría de campos escalares y vectoriales tiene aplicaciones en estática, dinámica, acústica, elasticidad, fluidos, transmisión del calor, electromagnetismo, entre otras muchas áreas. Cada capítulo comienza exponiendo las definiciones, enunciados y demostraciones, ilustrándolos con gráficas y ejemplos clarificadores; a continuación se presenta una colección de problemas resueltos y se termina con un listado de problemas propuestos, de dificultad variada. Se presentan las definiciones y los enunciados correctamente, pero no se incluyen las demostraciones rigurosas de todos los resultados necesarios en la exposición. Damos la mayor información posible, con la máxima claridad, incluyendo sólo algunas demostraciones sencillas y muchos ejemplos, problemas y gráficasBonet Solves, JA.; Calvo Roselló, V.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2023). Integración múltiple y vectorial. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/19807

Restauración de Imágenes Médicas con Diferentes Tipos de Ruido

Las imágenes obtenidas por rayos X o computarizada tomografía computarizada (CT) en condiciones adversas, pueden estar contaminadas con ruido que puede afectar a la detección de enfermedades. Un gran número de técnicas de procesamiento de imágenes han sido propuestas para eliminar el ruido. Estas técnicas dependen del tipo de ruido presente en la imagen. En este trabajo, se propone un método para reducir el ruido gaussiano, impulsivo y speckle. Este filtro, llamado PGMFDNL combina un filtro de difusión no lineal con peer group y fuzzy. El filtro propuesto es capaz de reducir eficazmente el ruido de la imagen sin ningún tipo de información acerca del ruido presente en la imagen. Como resultado, el método propuesto obtiene un buen rendimiento en los diferentes tipos de ruido.Este trabajo fue financiado por el ministerio español de ciencia e innovación: Proyecto TIN2011-26254), ANITRAN PROMETEO/2010/039, Proyecto TIN2008-06570-C04-04, y DGEST ITCG a través del programa PROMEP (México).Sanchez, G.; Vidal Gimeno, VE.; Verdú Martín, GJ.; Mayo Nogueira, P.; Ródenas Escribá, FDA. (2013). Restauración de Imágenes Médicas con Diferentes Tipos de Ruido. http://hdl.handle.net/10251/48285