Weak Sequential Convergence in Bounded Finitely Additive Measures

[EN] It is well known that a ¿-algebra ¿ of subsets of a set ¿ verifies both Nikodým property and property (G) for the Banach space ba(¿) of bounded finitely additive measures defined in ¿. A classic result of Valdivia stating that if a ¿-algebra ¿ is covered by an increasing sequence (¿n:n¿N) of subsets, there is p¿N such that ¿p is a Nikodým set for ba(¿) was complemented in Ferrando et al. (2020) proving that there exists p¿N such that ¿p is both a Nikodým and a Grothendieck set for ba(¿). Valdivia result was the first step to get that if (¿¿:¿¿N<¿) is a web in ¿ there exists a chain (¿n:n¿N) in N<¿ such that each ¿¿n, n¿N, is a Nikodým set for ba(¿). In this paper, we develop several properties in Banach spaces that enables us to complement the preceding web result extending the main result in Ferrando et al. (2020) proving that for each web (¿¿:¿¿N<¿) in a ¿-algebra ¿ there exists a chain (¿n:n¿N) in N<¿ such that each ¿¿n, n¿N, is both a Nikodým and a Grothendieck set for ba(¿). As an application we extend some results of classic Banach space theoryThe second author is supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.López Alfonso, S.; López Pellicer, M. (2020). Weak Sequential Convergence in Bounded Finitely Additive Measures. Vietnam Journal of Mathematics. 48(2):379-389. https://doi.org/10.1007/s10013-020-00387-2S379389482Arens, R.F., Kelley, J.L.: Characterizations of the space of continuous functions over a compact Hausdorff space. Trans. Am. Math. Soc. 62, 499–508 (1947)Diestel, J., Faires, B., Huff, R.: Convergence and boundedness of measures in non σ-complete algebras. Preprint (1976)Diestel, J., Uhl, J.J.: Vector Measures. Mathematical Surveys and Monographs, vol. 15. American Mathematical Society, Providence (1977)Dunford, N., Schwartz, J.T.: Linear Operators. Part I: General Theory. Wiley, New Jersey (1988)Fernández, J., Hui, S., Shapiro, H.: Unimodular functions and uniform boundedness. Publ. Mat. 33, 139–146 (1989)Ferrando, J.C., Ka̧kol, J., López-Pellicer, M.: On spaces Cb(X) weakly K-analytic. Math. Nachr. 290, 2612–2618 (2017)Ferrando, J.C., López-Alfonso, S., López-Pellicer, M.: On Nikodým and Rainwater sets for $ba(\mathcal {R})$ and a problem of M. Valdivia. Filomat 33, 2409–2416 (2019)Ferrando, J.C., López-Alfonso, S., López-Pellicer, M.: On the Grothendieck property (submited) (2020)Ferrando, J.C., López-Pellicer, M., Sánchez Ruiz, L.M.: Metrizable Barrelled Spaces. Pitman Research Notes in Mathematics Series, vol. 332. Longman, Harlow (1995)Ferrando, J.C., Sánchez Ruiz, L.M.: A survey on recent advances on the Nikodým boundedness theorem and spaces of simple functions. Rocky Mount. J. Math. 34, 139–172 (2004)Fonf, V.P.: On exposed and smooth points of convex bodies in Banach spaces. Bull. Lond. Math. Soc. 28, 51–58 (1996)Ka̧kol, J., López-Pellicer, M.: On Valdivia strong version of Nikodým boundedness property. J. Math. Anal. Appl. 446, 1–17 (2017)López-Alfonso, S., Mas, J., Moll, S.: Nikodým boundedness property and webs in σ-algebras. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110, 711–722 (2016)López-Alfonso, S.: On Schachermayer and Valdivia results in algebras of Jordan measurable sets. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110, 799–808 (2016)López-Pellicer, M.: Webs and bounded finitely additive measures. J. Math. Anal. Appl. 210, 257–267 (1997)Nygaard, O.: A strong uniform boundedness principle in Banach spaces. Proc. Am. Math. Soc. 129, 861–863 (2001)Nygaard, O.: Thick sets in Banach spaces and their properties. Quaest. Math. 29, 50–72 (2006)Plebanek, G., Sobota, D.: Countable tightness in the spaces of regular probability measures. Fund. Math. 229, 159–170 (2015)Rainwater, J.: Short notes: Weak convergence of bounded sequences. Proc. Am. Math. Soc. 14, 999–999 (1963)Schachermayer, W.: On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Diss. Math. (Rozprawy Mat.) 214, 1–33 (1982)Simons, S.: A convergence theorem with boundary. Pac. J. Math. 40, 703–708 (1972)Sobota, D., Zdomskyy, L.: The Nikodým property in the Sacks model. Topol. Appl. 230, 24–34 (2017)Talagrand, M.: Propriété de Nikodým and propriété de Grothendieck. Stud. Math. 78, 165–171 (1984)Valdivia, M.: On certain barrelled normed spaces. Ann. Inst. Fourier 29, 39–56 (1979)Valdivia, M.: On Nikodým boundedness property. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 107, 355–372 (2013

Distinguished Property in Tensor Products and Weak* Dual Spaces

[EN] A local convex space $E$ is said to be distinguished if its strong dual $E_{\beta }^{\prime }$ has the topology $\beta (E^{\prime },(E_{\beta}^{\prime })^{\prime })$, i.e., if $E_{\beta }^{\prime }$ is barrelled. The distinguished property of the local convex space $C_{p}\left( X\right)$ of real-valued functions on a Tychonoff space $X$, equipped with the pointwise topology on $X$, has recently aroused great interest among analysts and $C_{p}$-theorists, obtaining very interesting properties and nice characterizations. For instance, it has recently been obtained that a space $C_{p}\left( X\right)$ is distinguished if and only if any function $f\in \mathbb{R}^{X}$ belongs to the pointwise closure of a pointwise bounded set in $C\left( X\right)$. The extensively studied distinguished properties in the injective tensor products $C_{p}\left( X\right) \otimes _{\varepsilon }E$ and in $C_{p}(X,E)$ contrasts with the few distinguished properties of injective tensor products related to the dual space $L_{p}\left( X\right)$ of $C_{p}\left( X\right)$ endowed with the weak* topology, as well as to the weak* dual of $C_{p}(X,E)$. To partially fill this gap, some distinguished properties in the injective tensor product space $L_{p}\left(X\right) \otimes _{\varepsilon }E$ are presented and a characterization of the distinguished property of the weak* dual of $C_{p}(X,E)$ for wide classes of spaces $X$ and $E$ is provided.This research was funded for the second named author by grant PGC2018-094431-B-I00 of Ministry of Science, Innovation and Universities of Spain.López Alfonso, S.; López Pellicer, M.; Moll López, SE. (2021). Distinguished Property in Tensor Products and Weak* Dual Spaces. Axioms. 10(3):1-7. https://doi.org/10.3390/axioms100301511710

On Four Classical Measure Theorems

[EN] A subset B of an algebra A of subsets of a set Omega has property (N) if each B-pointwise bounded sequence of the Banach space ba(A) is bounded in ba(A), where ba(A) is the Banach space of real or complex bounded finitely additive measures defined on A endowed with the variation norm. B has property (G) [(VHS)] if for each bounded sequence [if for each sequence] in ba(A) the B-pointwise convergence implies its weak convergence. B has property (sN) [(sG) or (sVHS)] if every increasing covering {B-n:n is an element of N} of B contains a set B-p with property (N) [(G) or (VHS)], and B has property (wN) [(wG) or (wVHS)] if every increasing web {Bn(1)n(2)...n(m) : n(i) is an element of N,1 <= i <= m,m is an element of N} of B contains a strand {B-p1p2...pm: m is an element of N} formed by elements B-p1p2...pm with property (N) [(G) or (VHS)] for every m is an element of N. The classical theorems of Nikodym-Grothendieck, Valdivia, Grothendieck and Vitali-Hahn-Saks say, respectively, that every sigma-algebra has properties (N), (sN), (G) and (VHS). Valdivia's theorem was obtained through theorems of barrelled spaces. Recently, it has been proved that every sigma-algebra has property (wN) and several applications of this strong Nikodym type property have been provided. In this survey paper we obtain a proof of the property (wN) of a sigma-algebra independent of the theory of locally convex barrelled spaces which depends on elementary basic results of Measure theory and Banach space theory. Moreover we prove that a subset B of an algebra A has property (wWHS) if and only if B has property (wN) and A has property (G).This research was funded for the second named author by grant PGC2018-094431-B-I00 of Ministry of Science, Innovation and Universities of Spain.López Alfonso, S.; López Pellicer, M.; Moll López, SE. (2021). On Four Classical Measure Theorems. Mathematics. 9(5):1-17. https://doi.org/10.3390/math90505261179

On Grothendieck Sets

[EN] We call a subset M of an algebra of sets A a Grothendieck set for the Banach space ba (A) of bounded finitely additive scalar-valued measures on A equipped with the variation norm if each sequence fmn g n =1 in ba(A) which is pointwise convergent on M is weakly convergent in ba(A), i. e., if there is m 2 ba (A) such that mn ( A) ! m ( A) for every A 2M then mn ! m weakly in ba(A). A subset M of an algebra of sets A is called a Nikodym set for ba(A) if each sequence fm n g n =1 in ba(A) which is pointwise bounded on M is bounded in ba (A). We prove that if S is a s-algebra of subsets of a set W which is covered by an increasing sequence fS n : n 2 Ng of subsets of S there exists p 2 N such that S p is a Grothendieck set for ba(A). This statement is the exact counterpart for Grothendieck sets of a classic result of Valdivia asserting that if a s-algebra S is covered by an increasing sequence fS n : n 2 Ng of subsets, there is p 2 N such that S p is a Nikodym set for ba (S). This also refines the Grothendieck result stating that for each s -algebra S the Banach space ` (S) is a Grothendieck space. Some applications to classic Banach space theory are given.This research was funded by grant PGC2018-094431-B-I00 of Ministry of Scence, Innovation and universities of Spain.Ferrando, JC.; López Alfonso, S.; López Pellicer, M. (2020). On Grothendieck Sets. Axioms. 9(1):1-7. https://doi.org/10.3390/axioms9010034S1791Valdivia, M. (1979). On certain barrelled normed spaces. Annales de l’institut Fourier, 29(3), 39-56. doi:10.5802/aif.752Ferrando, J. C., López-Alfonso, S., & López-Pellicer, M. (2019). On Nikodým and Rainwater sets for ba (R) and a problem of M. Valdivia. Filomat, 33(8), 2409-2416. doi:10.2298/fil1908409fLópez-Alfonso, S. (2015). On Schachermayer and Valdivia results in algebras of Jordan measurable sets. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 110(2), 799-808. doi:10.1007/s13398-015-0267-xFerrando, J. C., & Ruiz, L. M. S. (2004). A Survey on Recent Advances on the Nikodým Boundedness Theorem and Spaces of Simple Functions. Rocky Mountain Journal of Mathematics, 34(1). doi:10.1216/rmjm/1181069896Valdivia, M. (2012). On Nikodym boundedness property. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 107(2), 355-372. doi:10.1007/s13398-012-0081-7Ferrando, J. C., Ka̧kol, J., & López-Pellicer, M. (2017). On spaces Cb(X) weakly K-analytic. Mathematische Nachrichten, 290(16), 2612-2618. doi:10.1002/mana.201600406Rainwater, J. (1963). Weak convergence of bounded sequences. Proceedings of the American Mathematical Society, 14(6), 999. doi:10.1090/s0002-9939-1963-0155171-9Simons, S. (1972). A convergence theorem with boundary. Pacific Journal of Mathematics, 40(3), 703-708. doi:10.2140/pjm.1972.40.703Drewnowski, L., Florencio, M., & Paúl, P. J. (1994). Barrelled subspaces of spaces with subseries decompositions or Boolean rings of projections. Glasgow Mathematical Journal, 36(1), 57-69. doi:10.1017/s0017089500030548Saxon, S. A. (1972). Nuclear and product spaces, Baire-like spaces, and the strongest locally convex topology. Mathematische Annalen, 197(2), 87-106. doi:10.1007/bf0141958

On Nikodym and Rainwater sets for ba (R) and a Problem of M. Valdivia

[EN] If R is a ring of subsets of a set Omega and ba (R) is the Banach space of bounded finitely additive measures defined on R equipped with the supremum norm, a subfamily Delta of R is called a Nikodym set for ba (R) if each set {mu(a) : alpha is an element of A} in ba (R) which is pointwise bounded on Delta is norm-bounded in ba (R). If the whole ring R is a Nikodym set, R is said to have property (N), which means that R satisfies the Nikodym-Grothendieck boundedness theorem. In this paper we find a class of rings with property (N) that fail Grothendieck's property (G) and prove that a ring R has property (G) if and only if the set of the evaluations on the sets of R is a so-called Rainwater set for ba(R). Recalling that R is called a (wN)-ring if each increasing web in R contains a strand consisting of Nikodym sets, we also give a partial solution to a question raised by Valdivia by providing a class of rings without property (G) for which the relation (N) double left right arrow (wN) holds.The first and the third authors are supported by Grant PGC2018-094431-B-I00 of the Ministry of Science, Innovation and Universities of Spain.Ferrando, JC.; López Alfonso, S.; López Pellicer, M. (2019). On Nikodym and Rainwater sets for ba (R) and a Problem of M. Valdivia. Filomat. 33(8):2409-2416. https://doi.org/10.2298/FIL1908409FS2409241633

A Survey on Nikodým and Vitali-Hahn-Saks Properties

[EN] Let ba(A) be the Banach space of the real (or complex) finitely additive measures of bounded variation defined on an algebra A of subsets of Omega and endowed with the variation norm. . A subset B of A is a Nikod ́ym set for ba(A) if each B-pointwise bounded subset M of ba(A) is uniformly bounded on A and B is a strong Nikod´ym set for ba(A) if each increasing covering (Bm)1 m=1 of B contains a Bn which is a Nikod´ym set for ba(A). If, additionally, the Nikod´ym subset B verifies that the sequential B-pointwise convergence in ba(A) implies weak convergence then B has the Vitali-Hahn-Saks property, (VHS ) in brief, and B has the strong (VHS ) property if for each increasing covering (Bm)1m=1 of B there exists Bq that has (VHS ) property Motivated by Valdivia result that every -algebra has strong Nikod´ym property and by his 2013 open question concerning that if Nikod´ym property in an algebra of subsets implies strong Nikod´ym property we survey this Valdivia theorem and we get that in a strong Nikod´ym set the (VHS ) property implies the strong (VHS ) property.Research supported for the second named author by Grant PGC2018-094431-B-I00 of Ministry of Science, Innovation & Universities of Spain.López Alfonso, S.; López-Pellicer, M.; Mas Marí, J. (2021). A Survey on Nikodým and Vitali-Hahn-Saks Properties. Montes Taurus Journal of Pure and Applied Mathematics. 3(3):112-121. http://hdl.handle.net/10251/181186S1121213

Conjuntos que determinan la acotación uniforme

The classical Nykodym theorem (1933) asserts that a set H of countably additive complex measures defined on a sigma-algebra S which is bounded for each element of S, then H is uniformly bounded on S. It is well known that this theorem fails if we replace the sigma-algebra S simply by an algebra. Let A be the algebra of subsets of a nonempty set, and consider the Banach space ba(A) of all real (or complex) finitely additive measures of bounded variation defined on A. A subset B of A is said to have the N-property (Nikodym property) if every B-pointwise bounded subset M of ba(A) is uniformly bounded on A. Recall the classical Nikodym-Dieudonné-Grothendieck's theorem which says that each sigma-algebra has the N-property. Moreover B is said to have the strong N-property if for each increasing countable covering (B_{m})_{m} of B there exists B_{n} which has the N-property. Valdivia proved in 1979 that each sigma-algebra has the strong N-property. The aforementioned Valdivia's theorem motivated to prove that each sigma-algebra S of subsets of a set has web-N-property, that is, if (B_{m_1})_{m_1} is an increasing countable covering of S and if (B_{m_1},_{m_2},....,_{m_p},_{m_{p+1}})_{m_{p+1}} is an increasing countably covering of B_{m_1},_{m_2},....,_{m_p}, for each natural numbers p, mi, with i = 1, 2,..., p, then there exists a sequence (n_{r})_{r} such thatB_{n_1},_{n_2},....,_{n_r} has the N-property for every r = 1, 2, 3, ...... . In this thesis it is proved that nearly all infinite chains in the increasing web (B_{m_1},_{m_2},....,_{m_p}: m_i=1,2,... , i=1,2,...,p, and p=1,2,.....} are composed of sets that have web-N-property. In the main result in this thesis it is proved that the algebra J(K) of Jordan measurables subsets of a compact k-dimensional interval K contained in R^k has the web-N-property. This result imporves the 2013 Valdivia's theorem stating that J(K) has the strong Nikodym property, which in turns was a grest improvement of Schachermayer's result of N property of J([0; 1]). The analysis of the demonstration of this result has allowed us give a sufficient condition in an algebra of subsets of a set that implies the property **wN. This sufficient condition is verified by each sigma-algebra as well as by the algebra J(K), whence the properties wN of any sigma-algebra and of J(*K) could have been presented like corollaries of this sufficient condition. It has seemed more natural to follow the chronological order, such as it has done in the Thesis. In the chapter 5 the problem proposed by Valdivia in 2013 is considered, i.e., to prove if it is true or not that when an algebra A of subsets have the property N then it has the property sN. In the section 5.2 this problem is considered in the most general context of normed spaces, because a subset B of an algebra A has the property N if each subset M of finitely additive bounded measures pointwise bounded in the sets of B verifies that M is a bounded subset of the Banach space of bounded finitely additive measures in the algebra A endowed with the supreme norm, what carries to the study of the sets DAU that determine the uniform boundedness of a normed space. Several applications of the obtained results to problems of location of vectorial averages and of convergence of sequences as well as several open problems are presented. The limitation of number of characters prevents to comment other results. We finish this summary indicating that we have proved that the properties wN, w(sN) and w(wN) are equivalents.El clásico teorema de Nykodym (1933) afirma que si un subconjunto H de medias complejas numerablemente aditivas definidas en una sigma-algebra S está acotado en cada elemento de S, entonces H está uniformente acotado en S. Es bien conocido que este teorema no es cierto en general si se sustituye la sigma-álgebra S por un álgebra. Sea A un álgebra de subconjuntos de un conjunto no vacío, y consideremos el espacio de Banach ba(A) de las medias reales (o complejas) finitamente aditivas de variación acotada definidas en A. Un subconjunto B of A se dice que tiene la propiedad N (propiedad de Nikodym) si para cada subconjunto M of ba(A) que sea B-puntualmente acotado se tiene que M es uniformemente acotado en A. Recordemos que el clásico teorema de Nikodym-Dieudonné-Grothendieck's dice que cada sigma-algebra tiene la propiedad N. Además se dice que B tiene la propiedad N-fuerte si cada para cada cubrimiento numerable creciente (B_{m})_{m} de B existe B_{n} que tiene la propiedad N. Valdivia demmostró en 1979 que cada sigma-algebra tiene la propiedad N-property. Este teorema de Valdivia motivó demostrar que cada sigma-algebra S de subconjuntos de un conjunto tiene la propiedad N para mallas crecientes, es decir, si (B_{m_1})_{m_1} es un cubrimiento numerable creciente de S y si (B_{m_1},_{m_2},....,_{m_p},_{m_{p+1}})_{m_{p+1}} es un cubrimiento numerable creciente de B_{m_1},_{m_2},....,_{m_p}, para cada números naturales p, mi, con i=1, 2,..., p, entonces existe una sucesión (n_{r})_{r} tal que B_{n_1},_{n_2},....,_{n_r} tiene la propiedad N para cada r = 1, 2, 3, ...... . En la tesis se prueba que casi todas las cadenas infinitas en una malla creciente (B_{m_1},_{m_2},....,_{m_p}: m_i=1,2,... , i=1,2,...,p, and p=1,2,.....} están compuestas de conjuntos que tienen la propiedad N para mallas crecientes. El resultado principal de la tesis prueba que el algebra J(K) de los subconjuntos Jordan medibles de un intervalo compacto k-dimensional K contenido en R^k tiene la propiedad N para mallas crecientes. Este resultado mejora el resultado de Valdivia de 2013 de que J (K) tiene la propiedad fuerte de Nikodym, que a su vez mejoraba un resultado anterior de Schachermayer, quien probó que J ([0; 1]) tiene la propiedad N. El análisis de la demostración de este resultado nos ha permitido dar una condición suficiente en un álgebra de subconjuntos de un conjunto que implica la propiedad wN. Esta condición suficiente la verifican tanto las sigma-álgebras como el álgebra J (K), por lo que las propiedades wN de cualquier sigma-álgebra y de J(K) se podían haber presentado como corolarios de dicha condición suficiente. Ha parecido más natural seguir el orden cronológico, tal como se ha hecho en la Tesis. En el capítulo 5 se considera el problema planteado por Valdivia en 2013. Consiste en averiguar si el que un álgebra A de conjuntos tenga la propiedad N implica o no el tener la propiedad sN. En la sección 5.2 se considera este problema en el contexto más general de los espacios normados, pues un subconjunto B de un álgebra A tiene la propiedad N si cada subconjunto M de medidas acotadas, finitamente aditivas y puntualmente acotadas en el conjunto de funciones características de los conjuntos de B verifica que M es un subconjunto acotado del espacio de Banach de dichas medias finitamente aditivas y acotadas definidas en A con la norma supremo, lo que lleva al estudio de los conjuntos DAU que determinan la acotación uniforme en un espacio normado. Se presentan varias aplicaciones de los resultados obtenidos a problemas de localización de medias vectoriales y de convergencia de sucesioens de medias y varios problemas abiertos. La limitación de número de caracteres impide comentar otros resultados. Terminamos este resumen indicando que hemos probado que las propiedades wN, w(sN) y w(wN) son equivalentes.El clàssic teorema de Nykodym (1933) afirma que si un subconjunt H de mesures complexes numerablement aditives defiides en una sigma-àlgebra S és acotat en cada element de S, aleshores H és uniforment acotat en S. És ben conegut que aquest teorema no és cert en general si es sustitueix la sigma-àlgebra S simplement per una àlgebra. Siga A una àlgebra de subconjunts d'un conjunt no buit, i considerem l'espai de Banach ba(A) de les mesures reals (o complexes) finitament aditives de variació acotada definides en A. Un subconjunt B de A es diu que té la propietat N (propietat de Nikodym) si cada subconjunt M de ba(A) que siga B-puntualment acotat es té que M és uniformement acotat en A. Recordem que el clàssic teorema de Nikodym-Dieudonné-Grothendieck's diu que cada sigma-àlgebra té la propietat N. A més a més es diu que B té la propietat forta N si per a cada cubrimient numerable creixent (B_{m})_{m} de B existeix B_{n} que té la propietat N. Valdivia va provar en 1979 que cada sigma-àlgebra té la propietat N. L'esmentat teorema de Valdivia va motivar demostrar que cada sigma-àlgebra S de subconjunts de un conjunt té la propietat N per a malles creixents, és dir, si (B_{m_1})_{m_1} és un cubrimient numerable creixent de S i si (B_{m_1},_{m_2},....,_{m_p},_{m_{p+1}})_{m_{p+1}} és un cubrimient numerable creixent de B_{m_1},_{m_2},....,_{m_p}, per a cada nombres naturals p, mi, amb i = 1, 2,..., p, aleshores existeix una successió (n_{r})_{r} tal que B_{n_1},_{n_2},....,_{n_r} té la propietat N per a cada r = 1, 2, 3, ...... . En la tesi es prova que gairebé totes les cadenes infinites en una malla creixent (B_{m_1},_{m_2},....,_{m_p}: m_i=1,2,... , i=1,2,...,p, and p=1,2,.....} estan composades de conjunts que tenen la propietat N per a malles creixents. El resultat principal de la tesi prova que l'àlgebra J(K) dels subconjunts Jordan mesurables d'un interval compacte k-dimensional K contingut en R^k té la propietat N per a malles creixents. Aquest resultat millora el resultat de Valdivia de 2013 de que J (K) té la propietat forta de Nikodym, que alhora millorava un resultat anterior de Schachermayer, qui va provar que J([0; 1]) té la propietat N. L'anàlisi de la demostració d'aquest resultat ens ha permès donar una condició suficient en un àlgebra de subconjunts d'un conjunt que implica la propietat wN. Aquesta condició suficient la verifiquen tant les sigma-àlgebres com l'àlgebra J(K), per la qual cosa les propietats wN de qualsevol sigma-àlgebra i de J(K) es podien haver presentat com a corol·laris d'aquesta condició suficient. Ha semblat més natural seguir l'ordre cronològic, tal com s'ha fet en la Tesi. En el capítol 5 es considera el problema plantejat per Valdivia en 2013. Consisteix a esbrinar si el que un àlgebra A de conjunts tinga la propietat N implica o no el tenir la propietat sN. En la secció 5.2 es considera aquest problema en el context més general dels espais normados, doncs un subconjunt B d'un àlgebra A té la propietat N si cada subconjunt M de mesures acotades, finitament additives i puntualment acotades en el conjunt de funcions característiques dels conjunts de B verifica que M és un subconjunt acotat de l'espai de Banach d'aquestes mesure finitament additives i acotades definides en A amb la norma suprem, la qual cosa porta a l'estudi dels conjunts DAU que determinen l'acotació uniforme en un espai normat. Es presenten diverses aplicacions dels resultats obtinguts a problemes de localització de mitjanes vectorials i de convergència de **sucesioens de mitjanes i diversos problemes oberts. La limitació de nombre de caràcters impedeix comentar altres resultats. Acabem aquest resum indicant que hem provat que les propietats wN, w(sN) i w(wN) són equivalents.López Alfonso, S. (2016). Conjuntos que determinan la acotación uniforme [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/75266TESI

Elementos finitos en fenómenos de transmisión

La forma de proceder de la teoría de Elementos Finitos recuerda la aplicación del Cálculo Diferencial en la resolución de problemas físicos, cuyo origen se sitúa en el siglo XVII, por lo que podemos suponer que, como sucedió con el Cálculo Diferencial, los distintos problemas estudiados con Elementos Finitos motivarán el desarrollo de mucho trabajo matemático para analizar la dependencia de la solución de los parámetros iniciales, para acotar el error de la solución aproximada obtenida y para mejorar el ajuste, sobre todo cuando la geometría del problema es compleja. Es indudable que se han logrado grandes avances en los últimos treinta años en la teoría de Elementos Finitos, pero el examen de las numerosas buenas revistas especializadas en esta teoría permite intuir que nos encontramos ante un nuevo desafío intelectual para las próximas décadas. En este artículo introductorio de divulgación se desarrollar la aplicación del método de los Elementos Finitos en la resolución aproximada de la ecuación del calor, se deduce obteniendo la ecuación matricial de transmisión del calor en elementos rectángulares y triangulares. Esta ecuación matricial se aplica en un sencillo problema bidimensional, refinando el mallado incial. Además, se muestra la aplicación programa Ansys en la resolución de problemas de transmisin de calor. Es bien conocido que las ecuaciones diferenciales que rigen los fenómenos de transmisión son muy similares, por lo que lo expuesto en este artículo para fenómenos de transmisión del calor es muy similar en problemas de fluidos no viscosos, en el estudio de la difusión de un fluido en un medio poroso, así como en el estudio de la torsión, entre otros problemas físicos.López Pellicer, M.; López Alfonso, S. (2011). Elementos finitos en fenómenos de transmisión. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid. 105(1):49-75. http://hdl.handle.net/10251/62374S4975105

Elementos finitos en problemas mecánicos

La descomposición de una estructura en elementos facilita la aplicación de los principios físicos en cada elemento. Luego se deben ensamblar las ecuaciones obtenidas y se generan grandes sistemas de ecuaciones lineales, para los que, afortudamente, se dispone de procedimientos computacionales que dan soluciiones muy aproximadas con relativa rapidez. En este artículo divulgativo e introductorio se estudia con detalle una sencilla estructura de barras enlazadas que trabajan a compresión o a tracción, se deduce la ecuación matricial de rigidez y se comparan los resultados obtenidos por cálculo manual con los resultados obtenidos con el programa Ansys. También se presentan ejemplos bidimensionales y tridimensionales.López Pellicer, M.; López Alfonso, S. (2012). Elementos finitos en problemas mecánicos. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales de Madrid. 105(2):227-240. http://hdl.handle.net/10251/62375S227240105

Strengthening mathematical skills through MOOCs: a case study

[EN] Recently there has been a huge development in Massive Open Online Courses (MOOC) with the aim to ease and complement the learning process, especially at university level. In this context we presented four MOOCs entitled Basic Mathematics: Numbers and Terminology, Differentiability, Integrals and Algebra on the platforms UPV[X] and EdX, aiming to match the freshmen s level in mathematics at engineering grades. We have used these courses to reinforce theoretical knowledge during the first year of university and to promote them as an educational complement among the students showing more difficulties in mathematics. The implementation of these MOOCs as an element of the learning process has brought new methodological opportunities. The resources and tools offered make learning a more social and collaborative process as connect students with each other, allowing new methodologies focused on problem-solving techniques. In addition, although the MOOCs impose a sequence of contents, this is usually quite adaptable and contributes to the individualization of learning, allowing students to work at their own pace and in an environment of their choice. The procedure has been based on tracking students with lower academic performance or those showing mathematical gaps and offering them the opportunity to reinforce such knowledge through the use of specific MOOCs. The process has been done mainly online but with periodic meetings with the teachers to evaluate student progress. This methodology (based in a blended learning methodology) is intended to enhance the motivation and improve the performance of the students, avoiding dropouts. The results of the students joining these courses are presented versus the results from the students that did not participated.The authors would like to thank the Department of Applied Mathematics for the Teaching Innovation Projects, PID-DMA-2014, which funds this research.López Alfonso, S.; Moll López, SE.; Sánchez, S.; Vega Fleitas, E. (2016). Strengthening mathematical skills through MOOCs: a case study. International Journal for e-Learning Security. 6(1):488-493. https://doi.org/10.20533/ijds.2046.4568.2016.0062S4884936