A contribution to the study of fuzzy metric spaces

[EN] We give some examples nad properties of fuzzy metric spaces, in the sense of George and Veramani, and characterize the To topological spaces which admit a compatible uniformity that has a countable transitive base, in terms of the fuzzy theory.Sapena Piera, A. (2001). A contribution to the study of fuzzy metric spaces. Applied General Topology. 2(1):63-75. doi:10.4995/agt.2001.3016SWORD63752

Espacios métricos fuzzy definidos por t-normas

Se prosigue con el estudio de los espacios métricos fuzzy introducidos por George y Veeramani. Se aportan nuevas propiedades y se tratan cuestiones como la completación, la continuidad uniforme y teoremas de punto fijo. Se introducen nuevos ejemplos (alguno de ellos de especial relevancia) y se dan resultados acerca de la precompacidad en espacios métricos fuzzy. Además, se desarrolla el estudio de las métricas fuzzy no arquimedianas y se aborda la cuestión de la completación de los espacios métricos fuzzy y se comprueba que, en este aspecto, existe una diferencia significativa con la teoría de los espacios métricos, pues no todo espacio métrico fuzzy admite completación. Se estudia la noción de continuidad uniforme y se definen los conceptos de equinormalidad y propiedad de Lebesgue para una métrica fuzzy ("análogos" a los clásicos) que permiten demostrar un teorema en el que se caracterizan los espacios métricos fuzzy en los que toda función real continua es uniformemente continua por el hecho de que la métrica fuzzy sea equinormal o cumpla la propiedad de Lebesgue. Además, se introduce el concepto de continuidad t-uniforme (que no tiene "homólogo" en la teoría clásica pero está estrechamente relacionado con la noción de contractividad que se aporta en el último capítulo) que permite caracterizar los espacios métricos fuzzy en los que toda función real continua es t-uniformemente continua mediante una adecuada definición de métrica fuzzy t-equinormal. Por último se introduce el concepto de aplicación contractiva fuzzy y se obtienen teoremas de punto fijo para este tipo de aplicaciones en espacios métricos fuzzy. Se establece que toda aplicación contractiva fuzzy en un espacio métrico fuzzy completo en el que toda sucesión contractiva fuzzy es una sucesión de Cauchy posee un único punto fijo.Sapena Piera, A. (2002). Espacios métricos fuzzy definidos por t-normas [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/5425Palanci

Fuzzy contractive sequences

[EN] In this paper we survey some results on contractive sequences in fuzzy metric spaces in the sense of George and Veeramani.This research is supported by Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P.Gregori Gregori, V.; Sapena Piera, A. (2017). Fuzzy contractive sequences. En Proceedings of the Workshop on Applied Topological Structures. Editorial Universitat Politècnica de València. 77-84. http://hdl.handle.net/10251/128042OCS778

How to Perform AMP? Cubic Adjustments for Improving the QoE

[EN] Adaptive Media Playout (AMP) consists of smoothly and dynamically adjusting the media playout rate to recover from undesired (e.g., buffer overflow/underflow or out-of-sync) situations. The existing AMP solutions are mainly characterized by two main aspects. The first one is their goal (e.g., keeping the buffers¿ occupancy into safe ranges or enabling media synchronization). The second one is the criteria that determine the need for triggering the playout adjustments (e.g., buffer fullness or asynchrony levels). This paper instead focuses on a third key aspect, which has not been sufficiently investigated yet: the specific adjustment strategy to be performed. In particular, we propose a novel AMP strategy, called Cubic AMP, which is based on employing a cubic interpolation method to adjust a deviated playout point to a given reference. On the one hand, mathematical analysis and graphical examples show that our proposal provides superior performance than other existing linear and quadratic AMP strategies in terms of the smoothness of the playout curve, while significantly outperforming the quadratic AMP strategy regarding the duration of the adjustment period and without increasing the computational complexity. It has also been proved and discussed that higher-order polynomial interpolation methods are less convenient than cubic ones. On the other hand, the results of subjective tests confirm that our proposal provides better Quality of Experience (QoE) than the other existing AMP strategies.This work has been funded, partially, by the “Fondo Europeo de Desarrollo Regional (FEDER)” and the Spanish Ministry of Economy and Competitiveness, under its R&D&I Support Program, in project with Ref. TEC2013-45492-R.Montagud, M.; Boronat, F.; Roig, B.; Sapena Piera, A. (2017). How to Perform AMP? Cubic Adjustments for Improving the QoE. Computer Communications. 103:61-73. https://doi.org/10.1016/j.comcom.2017.01.017S617310

Integration of multi-sensorial effects in synchronised immersive hybrid TV scenarios

[EN] Traditionally, TV media content has exclusively involved 2D or 3D audiovisual streams consumed by using a simple TV device. However, in order to generate more immersive media consumption experiences, other new types of content (e.g., omnidirectional video), consumption devices (e.g., Head Mounted Displays or HMD) and solutions to stimulate other senses beyond the traditional ones of sight and hearing, can be used. Multi-sensorial media content (a.k.a. mulsemedia) facilitates additional sensory effects that stimulate other senses during the media consumption, with the aim of providing the consumers with a more immersive and realistic experience. They provide the users with a greater degree of realism and immersion, but can also provide greater social integration (e.g., people with AV deficiencies or attention span problems) and even contribute to creating better educational programs (e.g., for learning through the senses in educational content or scientific divulgation). Examples of sensory effects that can be used are olfactory effects (scents), tactile effects (e.g., vibration, wind or pressure effects), and ambient effects (e.g., temperature or lighting). In this paper, a solution for providing multi-sensorial and immersive hybrid (broadcast/broadband) TV content consumption experiences, including omnidirectional video and sensory effects, is presented. It has been designed, implemented, and subjectively evaluated (by 32 participants) in an end-to-end platform for hybrid content generation, delivery and synchronised consumption. The satisfactory results which were obtained regarding the perception of fine synchronisation between sensory effects and multimedia content, and regarding the users' perceived QoE, are summarised and discussed.This work was supported in part by the "Vicerrectorado de Investigacion de la Universitat Politecnica de Valencia'' under Project PAID-11-21 and Project PAID-12-21.Marfil, D.; Boronat, F.; González-Salinas, J.; Sapena Piera, A. (2022). Integration of multi-sensorial effects in synchronised immersive hybrid TV scenarios. IEEE Access. 10:79071-79089. https://doi.org/10.1109/ACCESS.2022.319417079071790891

IDMS solution for hybrid broadcast broadband delivery within the context of HbbTV standard

"© 2019 IEEE. Personal use of this material is permitted. Permissíon from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertisíng or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works."[EN] Inter-destination media synchronization (IDMS) is a key requirement to enable successful networked shared media experiences between remote users. This paper presents an adaptive, accurate and standard-compliant IDMS solution for hybrid broadcast and broadband delivery. Apart from providing multi- and cross-technology support, the presented IDMS solution is able to accomplish synchronization when different formats/versions of the same, or even related, contents are being played out in a shared session. It is also able to independently manage the playout processes of different groups of users. The IDMS solution has been integrated within an end-to-end platform, which is compatible with the hybrid broadcast broadband TV standard. It has been applied to digital video broadcasting-terrestrial technology and tested for a Social TV scenario, by also including an ad-hoc chat tool as an interaction channel. The results of the conducted (objective and subjective) evaluations prove the statisfactory behavior and performance of the IDMS solution and platform as well as in terms of the perceived quality of experience.This work was supported by Generalitat Valenciana, Investigacion competitiva proyectos, through the Research and Development Program "Grants for research groups to be consolidated, AICO/2017," under Grant AICO/2017/059.Marfil-Reguero, D.; Boronat, F.; Montagud, M.; Sapena Piera, A. (2019). IDMS solution for hybrid broadcast broadband delivery within the context of HbbTV standard. IEEE Transactions on Broadcasting. 65(4):645-663. https://doi.org/10.1109/TBC.2018.2878285S64566365

On completeness in metric spaces and fixed point theorems

[EN] Complete ultrametric spaces constitute a particular class of the so called, recently, G-complete metric spaces. In this paper we characterize a more general class called weak G-complete metric spaces, by means of nested sequences of closed sets. Then, we also state a general fixed point theorem for a self-mapping of a weak G-complete metric space. As a corollary, every asymptotically regular self-mapping of a weak G-Complete metric space has a fixed point.V. Gregori acknowledges the support of the Ministry of Economy and Competitiveness of Spain under Grant MTM2015-64373-P (MINECO/Feder, UE). J.J. Minana acknowledges financial support from the Spanish Ministry of Economy and Competitiveness under Grants TIN2016-81731-REDT (LODISCO II) and AEI/FEDER, UE funds, by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by Project Ref. PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and by project ROBINS. The latter has received research funding from the EU H2020 framework under GA 779776. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Roig, B.; Sapena Piera, A. (2018). On completeness in metric spaces and fixed point theorems. Results in Mathematics. 73(4):1-13. https://doi.org/10.1007/s00025-018-0896-4113734Bourbaki, N.: Topologie Générale II. Herman, Paris (1974)Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–469 (1969)Browder, F.E., Petryshyn, W.V.: The solution by iteration of nonlinear functional equations in Banach spaces. Bull. Am. Math. Soc. 72, 571–575 (1966)Edelstein, M.: On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, 74–79 (1962)Fang, J.X.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 46(1), 107–113 (1992)Grabiec, M.: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385–389 (1989)Gregori, V., Sapena, A.: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 125, 245–252 (2002)Gregori, V., Miñana, J.-J., Morillas, S., Sapena, A.: Cauchyness and convergence in fuzzy metric spaces. RACSAM 111(1), 25–37 (2017)Gregori, V., Miñana, J-J., Sapena, A.: On Banach contraction principles in fuzzy metric spaces. Fixed Point Theory (to appear)Kelley, J.: General Topology. Van Nostrand, Princeton (1955)Matkowski, J.: Integrable solutions of functional equations. Dissertationes Mathematicae (Rozprawy Matematyczne) 127, 1–63 (1975)Mihet, D.: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 144, 8431–439 (2004)Steen, L.A., Seebach, J.A.: Counterexamples in Topology, 2nd edn. Springer, Berlin (1978)Tirado, P.: On compactness and G-completeness in fuzzy metric spaces. Iran. J. Fuzzy Syst. 9(4), 151–158 (2012)Tirado, P.: Contraction mappings in fuzzy quasimetric spaces and $$[0,1]$$ [ 0 , 1 ] -fuzzy posets. Fixed Point Theory 13(1), 273–283 (2012)Vasuki, R., Veeramani, P.: Fixed points theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets Syst. 135(3), 415–417 (2003

Characterizing a class of completable fuzzy metric spaces

In this paper we give a characterization of the class of completable strong (non-Archimedean) fuzzy metric spaces, in the sense of George and Veeramani.Gregori Gregori, V.; Miñana, JJ.; Morillas, S.; Sapena Piera, A. (2016). Characterizing a class of completable fuzzy metric spaces. Topology and its Applications. 203:3-11. doi:10.1016/j.topol.2015.12.070S31120

A Characterization of Strong Completeness in Fuzzy Metric Spaces

[EN] Here, we deal with the concept of fuzzy metric space(X,M,*), due to George and Veeramani. Based on the fuzzy diameter for a subset ofX, we introduce the notion of strong fuzzy diameter zero for a family of subsets. Then, we characterize nested sequences of subsets having strong fuzzy diameter zero using their fuzzy diameter. Examples of sequences of subsets which do or do not have strong fuzzy diameter zero are provided. Our main result is the following characterization: a fuzzy metric space is strongly complete if and only if every nested sequence of close subsets which has strong fuzzy diameter zero has a singleton intersection. Moreover, the standard fuzzy metric is studied as a particular case. Finally, this work points out a route of research in fuzzy fixed point theory.Juan-Jose Minana acknowledges financial support from FEDER/Ministerio de Ciencia, Innovacion y Universidades-Agencia Estatal de Investigacion/Proyecto PGC2018-095709-B-C21, and by Spanish Ministry of Economy and Competitiveness under contract DPI2017-86372-C3-3-R (AEI, FEDER, UE). This work was also partially supported by Programa Operatiu FEDER 2014-2020 de les Illes Balears, by project PROCOE/4/2017 (Direccio General d'Innovacio i Recerca, Govern de les Illes Balears), and by projects ROBINS and BUGWRIGHT2. These two latest projects have received funding from the European Union's Horizon 2020 research and innovation program under grant agreements Nos. 779776 and 871260, respectively. This publication reflects only the authors views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Roig, B.; Sapena Piera, A. (2020). A Characterization of Strong Completeness in Fuzzy Metric Spaces. Mathematics. 8(6):1-11. https://doi.org/10.3390/math8060861S11186Menger, K. (1942). Statistical Metrics. Proceedings of the National Academy of Sciences, 28(12), 535-537. doi:10.1073/pnas.28.12.535George, A., & Veeramani, P. (1994). On some results in fuzzy metric spaces. Fuzzy Sets and Systems, 64(3), 395-399. doi:10.1016/0165-0114(94)90162-7Gregori, V., & Romaguera, S. (2000). Some properties of fuzzy metric spaces. Fuzzy Sets and Systems, 115(3), 485-489. doi:10.1016/s0165-0114(98)00281-4Gregori, V. (2002). On completion of fuzzy metric spaces. Fuzzy Sets and Systems, 130(3), 399-404. doi:10.1016/s0165-0114(02)00115-xAtanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87-96. doi:10.1016/s0165-0114(86)80034-3Gregori, V., Romaguera, S., & Veeramani, P. (2006). A note on intuitionistic fuzzy metric spaces☆. Chaos, Solitons & Fractals, 28(4), 902-905. doi:10.1016/j.chaos.2005.08.113Gregori, V., & Sapena, A. (2018). Remarks to «on strong intuitionistic fuzzy metrics». Journal of Nonlinear Sciences and Applications, 11(02), 316-322. doi:10.22436/jnsa.011.02.12Abu-Donia, H. M., Atia, H. A., & Khater, O. M. A. (2020). Common fixed point theorems in intuitionistic fuzzy metric spaces and intuitionistic (ϕ,ψ)-contractive mappings. Journal of Nonlinear Sciences and Applications, 13(06), 323-329. doi:10.22436/jnsa.013.06.03Gregori, V., & Miñana, J.-J. (2016). On fuzzy ψ -contractive sequences and fixed point theorems. Fuzzy Sets and Systems, 300, 93-101. doi:10.1016/j.fss.2015.12.010Miheţ, D. (2007). On fuzzy contractive mappings in fuzzy metric spaces. Fuzzy Sets and Systems, 158(8), 915-921. doi:10.1016/j.fss.2006.11.012Wardowski, D. (2013). Fuzzy contractive mappings and fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 222, 108-114. doi:10.1016/j.fss.2013.01.012Gregori, V., Miñana, J.-J., Morillas, S., & Sapena, A. (2016). Cauchyness and convergence in fuzzy metric spaces. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 111(1), 25-37. doi:10.1007/s13398-015-0272-0Gregori, V., & Miñana, J.-J. (2017). Strong convergence in fuzzy metric spaces. Filomat, 31(6), 1619-1625. doi:10.2298/fil1706619gGrabiec, M. (1988). Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems, 27(3), 385-389. doi:10.1016/0165-0114(88)90064-4George, A., & Veeramani, P. (1997). On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems, 90(3), 365-368. doi:10.1016/s0165-0114(96)00207-2Miheţ, D. (2008). Fuzzy -contractive mappings in non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 159(6), 739-744. doi:10.1016/j.fss.2007.07.006Vasuki, R., & Veeramani, P. (2003). Fixed point theorems and Cauchy sequences in fuzzy metric spaces. Fuzzy Sets and Systems, 135(3), 415-417. doi:10.1016/s0165-0114(02)00132-xGregori, V., & Romaguera, S. (2004). Characterizing completable fuzzy metric spaces. Fuzzy Sets and Systems, 144(3), 411-420. doi:10.1016/s0165-0114(03)00161-1Gregori, V., Miñana, J.-J., & Morillas, S. (2012). Some questions in fuzzy metric spaces. Fuzzy Sets and Systems, 204, 71-85. doi:10.1016/j.fss.2011.12.008Ricarte, L. A., & Romaguera, S. (2014). A domain-theoretic approach to fuzzy metric spaces. Topology and its Applications, 163, 149-159. doi:10.1016/j.topol.2013.10.014Gregori, V., López-Crevillén, A., Morillas, S., & Sapena, A. (2009). On convergence in fuzzy metric spaces. Topology and its Applications, 156(18), 3002-3006. doi:10.1016/j.topol.2008.12.043Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64. doi:10.1007/bf00531809Shukla, S., Gopal, D., & Sintunavarat, W. (2018). A new class of fuzzy contractive mappings and fixed point theorems. Fuzzy Sets and Systems, 350, 85-94. doi:10.1016/j.fss.2018.02.010Beg, I., Gopal, D., Došenović, T., … Rakić, D. (2018). α-type fuzzy H-contractive mappings in fuzzy metric spaces. Fixed Point Theory, 19(2), 463-474. doi:10.24193/fpt-ro.2018.2.37Zheng, D., & Wang, P. (2019). Meir–Keeler theorems in fuzzy metric spaces. Fuzzy Sets and Systems, 370, 120-128. doi:10.1016/j.fss.2018.08.014Rakić, D., Došenović, T., Mitrović, Z. D., de la Sen, M., & Radenović, S. (2020). Some Fixed Point Theorems of Ćirić Type in Fuzzy Metric Spaces. Mathematics, 8(2), 297. doi:10.3390/math802029

On Principal Fuzzy Metric Spaces

[EN] In this paper, we deal with the notion of fuzzy metric space (X, M, *), or simply X, due to George and Veeramani. It is well known that such fuzzy metric spaces, in general, are not completable and also that there exist p-Cauchy sequences which are not Cauchy. We prove that if every p-Cauchy sequence in X is Cauchy, then X is principal, and we observe that the converse is false, in general. Hence, we introduce and study a stronger concept than principal, called strongly principal. Moreover, X is called weak p-complete if every p-Cauchy sequence is p-convergent. We prove that if X is strongly principal (or weak p-complete principal), then the family of p-Cauchy sequences agrees with the family of Cauchy sequences. Among other results related to completeness, we prove that every strongly principal fuzzy metric space where M is strong with respect to an integral (positive) t-norm * admits completion.Samuel Morillas acknowledges financial support from Ministerio de Ciencia e Innovacion of Spain under grant PID2019-107790RB-C22 funded by MCIN/AEI/10.13039/501100011033. JuanJose Minana acknowledges financial support from Proyecto PGC2018-095709-B-C21 financiado por MCIN/AEI/10.13039/501100011033 y FEDER "Una manera de hacer Europa" and from project BUGWRIGHT2. This last project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 871260. Also acknowledge support of Generalitat Valenciana under grant CIAICO/2021/137. This publication reflects only the authors' views and the European Union is not liable for any use that may be made of the information contained therein.Gregori Gregori, V.; Miñana, J.; Morillas, S.; Sapena Piera, A. (2022). On Principal Fuzzy Metric Spaces. Mathematics. 10(16):1-10. https://doi.org/10.3390/math10162860110101