Diagonal extendible multilinear operators between l(p)-spaces

[EN] We study extendibility of diagonal multilinear operators from l(p) to l(p) spaces. We determine the values of and for which every diagonal -linear operator is extendible, and those for which the only extendible ones are integral. We address the same question for multilinear forms on l(p).D. Carando, V. Dimant and R. Villafane were partially supported by CONICET PIP 0624 and UBACyT 20020100100746. P. Sevilla-Peris was supported by MICINN Project MTM2011-22417. R. Villafane has a doctoral fellowship from CONICET.Carando, D.; Dimant, V.; Sevilla Peris, P.; Villafañe, R. (2014). Diagonal extendible multilinear operators between l(p)-spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas (RACSAM). 108(2):541-555. https://doi.org/10.1007/s13398-013-0125-7S5415551082Alencar, R.: Multilinear mappings of nuclear and integral type. Proc. Am. Math. Soc. 94(1), 33–38 (1985)Alencar, R., Matos, M.: Some classes of multilinear mappings between Banach spaces. Pub. Dep. An. Mat. Univ. Complut. Madrid 12 (1989)Blei, R.C.: Multilinear measure theory and the Grothendieck factorization theorem. Proc. Lond. Math. Soc. 56(3), 529–546 (1988)Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. (2) 32(3), 600–622 (1931)Botelho, G., Braunss, H., Junek, H., Pellegrino, D.: Holomorphy types and ideals of multilinear mappings. Studia Math. 177, 43–65 (2006)Botelho, G., Michels, C., Pellegrino, D.: Complex interpolation and summability properties of multilinear operators. Rev. Mat. Complut. 23(1), 139–161 (2010)Botelho, G., Pellegrino, D.: Two new properties of ideals of polynomials and applications. Indag. Math. (N.S.) 16157–16169 (2005)Botelho, G., Pellegrino, D.: When every multilinear mapping is multiple summing. Math. Nachr. 282(10), 1414–1422 (2009)Çalışkan, E., Pellegrino, D.: On the multilinear generalizations of the concept of absolutely summing operators. Rocky Mt. J. Math. 37, 1137–1154 (2007)Carando, D.: Extendibility of polynomials and analytic functions on $$\ell _{p}$$ . Studia Math. 145(1), 63–73 (2001)Carando, D., Dimant, V., Muro, S.: Coherent sequences of polynomial ideals on Banach spaces. Math. Nachr. 282(8), 1111–1133 (2009)Carando, D., Dimant, V., Sevilla-Peris, P.: Limit orders and multilinear forms on $$\ell _p$$ spaces. Publ. Res. Inst. Math. Sci. 42(2), 507–522 (2006)Carando, D., Dimant, V., Sevilla-Peris, P.: Multilinear Hölder $$-$$ type inequalities on Lorentz sequence spaces. Studia Math. 195(2), 127–146 (2009)Carando, D., Lassalle, S.: Extension of vector-valued integral polynomials. J. Math. Anal. Appl. 307(1), 77–85 (2005)Carando, D., Sevilla-Peris, P.: Extendibility of bilinear forms on Banach sequence spaces. Israel J. Math., arXiv:1212.0777 (to appear)Castillo, J., García, R., Jaramillo, J.A.: Extension of bilinear forms on Banach spaces. Proc. Am. Math. Soc. 129(12), 3647–3656 (2001)Carl, B.: A remark on $$p$$ -integral and $$p$$ -absolutely summing operators from $$\ell _u$$ into $$\ell _v$$ . Studia Math. 57(3), 257–262 (1976)Defant, A., Floret, K.: Tensor norms and operator ideals. North Holland, Amsterdam (1993)Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995)Dineen, S.: Complex analysis on infinite dimensional spaces. Springer-Verlag, London (1999)Jarchow, H., Palazuelos, C., Pérez-García, D., Villanueva, I.: Hahn–Banach extension of multilinear forms and summability. J. Math. Anal. Appl. 336(2), 1161–1177 (2007)Kirwan, P., Ryan, R.: Extendibility of homogeneous polynomials on Banach spaces. Proc. Am. Math. Soc. 126(4), 1023–1029 (1998)König, H.: Diagonal and convolution operators as elements of operator ideals. Math. Ann. 218(2), 97–106 (1975)Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces II. Springer, Berlin (1977)Matos, M.: Absolutely summing holomorphic mappings. An. Acad. Bras. Ci. 68, 1–13 (1996)Maurizi, B., Queffélec, H.: Some remarks on the algebra of bounded Dirichlet series. J. Fourier Anal. Appl. 16(5), 676–692 (2010)Mujica, J.: Complex analysis in Banach spaces. North-Holland mathematics studies, vol. 120, North-Holland (1986)Pérez-García, D.: Comparing different classes of absolutely summing multilinear operators. Arch. Math. (Basel) 85(3), 258–267 (2005)García Pérez, D.: The trace class is a $$Q$$ -algebra. Ann. Acad. Sci. Fenn. Math. 31(2), 287–295 (2006)Pietsch, A.: Operator ideals. Mathematische monographien (Mathematical monographs), 16. VEB Deutscher Verlag der Wissenschaften, Berlin (1978)Pisier, G.: Counterexamples to a conjecture of Grothendieck. Acta Math. 151(3–4), 181–208 (1983)Ryan, R: Introduction to tensor products of Banach spaces. Springer monographs in mathematics, Springer-Verlag, London (2002)Villanueva, I.: Integral mappings between Banach spaces. J. Math. Anal. Appl. 279(1), 56–70 (2003

Extendibility of bilinear forms on banach sequence spaces

[EN] We study Hahn-Banach extensions of multilinear forms defined on Banach sequence spaces. We characterize c(0) in terms of extension of bilinear forms, and describe the Banach sequence spaces in which every bilinear form admits extensions to any superspace.The second author was supported by MICINN Project MTM2011-22417.DANIEL CARANDO; Sevilla Peris, P. (2014). Extendibility of bilinear forms on banach sequence spaces. Israel Journal of Mathematics. 199(2):941-954. https://doi.org/10.1007/s11856-014-0003-9S9419541992F. Albiac and N. J. Kalton, Topics in Banach Space Theory, Graduate Texts in Mathematics, Vol. 233, Springer, New York, 2006.R. Arens, The adjoint of a bilinear operation, Proceedings of the American Mathematical Society 2 (1951), 839–848.R. Arens, Operations induced in function classes, Monatshefte für Mathematik 55 (1951), 1–19.R. M. Aron and P. D. Berner, A Hahn-Banach extension theorem for analytic mappings, Bulletin de la Société Mathématique de France 106 (1978), 3–24.S. Banach, Sur les fonctionelles linéaires, Studia Mathematica 1 (1929), 211–216.S. Banach, Théorie des opérations linéaires, (Monogr. Mat. 1) Warszawa: Subwncji Funduszu Narodowej. VII, 254 S., Warsaw, 1932.D. Carando, Extendible polynomials on Banach spaces, Journal of Mathematical Analysis and Applications 233 (1999), 359–372.D. Carando, Extendibility of polynomials and analytic functions on l p, Studia Mathematica 145 (2001), 63–73.D. Carando, V. Dimant and P. Sevilla-Peris, Limit orders and multilinear forms on lp spaces, Publications of the Research Institute for Mathematical Sciences 42 (2006), 507–522.J. M. F. Castillo, R. García, A. Defant, D. Pérez-García and J. Suárez, Local complementation and the extension of bilinear mappings, Mathematical Proceedings of the Cambridge Philosophical Society 152 (2012), 153–166.J. M. F. Castillo, R. García and J. A. Jaramillo, Extension of bilinear forms on Banach spaces, Proceedings of the American Mathematical Society 129 (2001), 3647–3656.P. Cembranos and J. Mendoza, The Banach spaces ℓ ∞(c 0) and c 0(ℓ ∞) are not isomorphic, Journal of Mathematical Analysis and Applications 367 (2010), 461–463.A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Mathematics Studies, Vol. 176, North-Holland Publishing Co., Amsterdam, 1993.A. Defant and C. Michels, Norms of tensor product identities, Note di Matematica 25 (2005/06), 129–166.J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1995.D. J. H. Garling, On symmetric sequence spaces, Proceedings of the London Mathematical Society (3) 16 (1966), 85–106.A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79.H. Hahn, Über lineare Gleichungssysteme in linearen Räumen, Journal für die Reine und Angewandte Mathematik 157 (1927), 214–229.R. C. James, Bases and reflexivity of Banach spaces, Annals of Mathematics (2) 52 (1950), 518–527.H. Jarchow, C. Palazuelos, D. Pérez-García and I. Villanueva, Hahn-Banach extension of multilinear forms and summability, Journal of Mathematical Analysis and Applications 336 (2007), 1161–1177.W. B. Johnson and L. Tzafriri, On the local structure of subspaces of Banach lattices, Israel Journal of Mathematics 20 (1975), 292–299.P. Kirwan and R. A. Ryan, Extendibility of homogeneous polynomials on Banach spaces, Proceedings of the American Mathematical Society 126 (1998), 1023–1029.J. Lindenstrauss and A. Pełczyński, Absolutely summing operators in Lp-spaces and their applications, Studia Mathematica 29 (1968), 275–326.J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Vol. 97, Springer-Verlag, Berlin, 1979. Function spaces.G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, CBMS Regional Conference Series in Mathematics, Vol. 60, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986.M. Fernndez-Unzueta and A. Prieto, Extension of polynomials defined on subspaces, Mathematical Proceedings of the Cambridge Philosophical Society 148 (2010), 505–518.W. L. C. Sargent, Some sequence spaces related to the lp spaces, Journal of the London Mathematical Society 35 (1960), 161–171.N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 38, Longman Scientific & Technical, Harlow, 1989

Almost sure-sign convergence of Hardy-type Dirichlet series

[EN] Hartman proved in 1939 that the width of the largest possible strip in the complex plane on which a Dirichlet series is uniformly a.s.- sign convergent (i.e., converges uniformly for almost all sequences of signs epsilon (n) = +/- 1) but does not convergent absolutely, equals 1/2. We study this result from a more modern point of view within the framework of so-called Hardytype Dirichlet series with values in a Banach space.Supported by CONICET-PIP 11220130100329CO, PICT 2015-2299 and UBACyT 20020130100474BA. Supported by MICINN MTM2017-83262-C2-1-P. Supported by MICINN MTM2017-83262-C2-1-P and UPV-SP20120700.Carando, D.; Defant, A.; Sevilla Peris, P. (2018). Almost sure-sign convergence of Hardy-type Dirichlet series. Journal d Analyse Mathématique. 135(1):225-247. https://doi.org/10.1007/s11854-018-0034-yS2252471351A. Aleman, J.-F. Olsen, and E. Saksman, Fourier multipliers for Hardy spaces of Dirichlet series, Int. Math. Res. Not. IMRN 16 (2014), 4368–4378.R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series Studia Math. 175 (2006), 285–304.F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136 (2002), 203–236.F. Bayart, A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, Monomial series expansion of Hp-functions and multipliers ofHp-Dirichlet series, Math. Ann. 368 (2017), 837–876.F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda, The Bohr radius of the n-dimensional polydisk is equivalent to $$\sqrt {\left( {\log n} \right)/n}$$ ( log n ) / n , Adv. Math. 264 (2014), 726–746.F. Bayart, H. Queffélec, and K. Seip, Approximation numbers of composition operators on Hp spaces of Dirichlet series, Ann. Inst. Fourier (Grenoble) 66 (2016), 551–588.H. F. Bohnenblust and E. Hille. On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (1931), 600–622.H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variablen in der Theorie der Dirichlet–schen Reihen $$\sum {\frac{{{a_n}}}{{{n^s}}}}$$ ∑ a n n s , Nachr. Ges.Wiss. Göttingen, Math. Phys. Kl., 1913, pp. 441–488.D. Carando, A. Defant, and P. Sevilla-Peris, Bohr’s absolute convergence problem for Hp- Dirichlet series in Banach spaces, Anal. PDE 7 (2014), 513–527.D. Carando, A. Defant, and P. Sevilla-Peris, Some polynomial versions of cotype and applications, J. Funct. Anal. 270 (2016), 68–87.B. J. Cole and T. W. Gamelin, Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc. (3) 53 (1986), 112–142.R. de la Bretèche. Sur l’ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134 (2008), 141–148.A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounäies, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (2011), 485–497.A. Defant, D. García, M. Maestre, and D. Pérez-García, Bohr’s strip for vector valued Dirichlet series, Math. Ann. 342 (2008), 533–555.A. Defant, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231 (2012), 2837–2857.A. Defant and A. Pérez, Hardy spaces of vector-valued Dirichlet series, StudiaMath. (to appear), 2018 DOI: 10.4064/sm170303-26-7.A. Defant, U. Schwarting, and P. Sevilla-Peris, Estimates for vector valued Dirichlet polynomials, Monatsh. Math. 175 (2014), 89–116.J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators, Cambridge University Press, Cambridge, 1995.P. Hartman, On Dirichlet series involving random coefficients, Amer. J. Math. 61 (1939), 955–964.H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in L2(0, 1), Duke Math. J. 86 (1997), 1–37.A. Hildebrand, and G. Tenenbaum, Integers without large prime factors, J. Thor. Nombres Bordeaux 5 (1993), 411–484.S. V. Konyagin and H. Queffélec, The translation 1/2 in the theory of Dirichlet series, Real Anal. Exchange 27 (2001/02) 155–175.J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces. II, Springer-Verlag, Berlin, 1979.B. Maurizi and H. Queffélec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal. Appl. 16 (2010), 676–692.H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.H. Queffélec and M. Queffélec, Diophantine Approximation and Dirichlet Series, Hindustan Book Agency, New Delhi, 2013.G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge University Press, Cambridge, 1995

Loss of GATA-1 full length as a cause of Diamond-Blackfan anemia phenotype

Mutations in the hematopoietic transcription factor GATA-1 alter the proliferation/differentiation of hemopoietic progenitors. Mutations in exon 2 interfere with the synthesis of the full-length isoform of GATA-1 and lead to the production of a shortened isoform, GATA-1s. These mutations have been found in patients with Diamond-Blackfan anemia (DBA), a congenital erythroid aplasia typically caused by mutations in genes encoding ribosomal proteins. We sequenced GATA-1 in 23 patients that were negative for mutations in the most frequently mutated DBA genes. One patient showed a c.2T > C mutation in the initiation codon leading to the loss of the full-length GATA-1 isoform

Weighted Banach spaces of harmonic functions

“The final publication is available at Springer via http://dx.doi.org/10.1007/s13398-012-0109-z."We study Banach spaces of harmonic functions on open sets of or endowed with weighted supremum norms. We investigate the harmonic associated weight defined naturally as the analogue of the holomorphic associated weight introduced by Bierstedt, Bonet, and Taskinen and we compare them. We study composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions characterizing the continuity, the compactness and the essential norm of composition operators among these spaces in terms of associated weights.The research of the first author was partially supported by MEC and FEDER Project MTM2010-15200 and by GV project ACOMP/2012/090.Jorda Mora, E.; Zarco García, AM. (2014). Weighted Banach spaces of harmonic functions. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 108(2):405-418. https://doi.org/10.1007/s13398-012-0109-zS4054181082Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Berlin (2001)Bierstedt, K.D., Bonet, J., Galbis, A.: Weighted spaces of holomorphic functions on balanced domains. Mich. Math. J. 40(2), 271–297 (1993)Bierstedt, K.D., Bonet, J., Taskinen, J.: Associated weights and spaces of holomorphic functions. Stud. Math. 127(2), 137–168 (1998)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 54(1), 70–79 (1993)Bonet, J., Domański, P., Lindström, M.: Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions. Can. Math. Bull. 42(2), 139–148 (1999)Bonet, J., Domański, P., Lindström, M.: Weakly compact composition operators on weighted vector-valued Banach spaces of analytic mappings. Ann. Acad. Sci. Fenn. Math. Ser. A I 26, 233–248 (2001)Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 64, 101–118 (1998)Bonet, J., Friz, M., Jordá, E.: Composition operators between weighted inductive limits of spaces of holomorphic functions. Publ. Math. Debr. Ser. A 67, 333–348 (2005)Boyd, C., Rueda, P.: The v-boundary of weighted spaces of holomorphic functions. Ann. Acad. Sci. Fenn. Math. 30, 337–352 (2005)Boyd, C., Rueda, P.: Complete weights and v-peak points of spaces of weighted holomorphic functions. Isr. J. Math. 155, 57–80 (2006)Boyd, C., Rueda, P.: Isometries of weighted spaces of harmonic functions. Potential Anal. 29(1), 37–48 (2008)Carando, D., Sevilla-Peris, P.: Spectra of weighted algebras of holomorphic functions. Math. Z. 263, 887–902 (2009)Contreras, M.D., Hernández-Díaz, G.: Weighted composition operators in weighted Banach spaces of analytic functions. J. Aust. Math. Soc. Ser. A 69(1), 41–60 (2000)García, D., Maestre, M., Rueda, P.: Weighted spaces of holomorphic functions on Banach spaces. Stud. Math. 138(1), 1–24 (2000)García, D., Maestre, M., Sevilla-Peris, P.: Composition operators between weighted spaces of holomorphic functions on Banach spaces. Ann. Acad. Sci. Fenn. Math. 29, 81–98 (2004)Gunning, R., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence (2009)Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs (1962)Krantz, S.G.: Function Theory of Several Complex Variables. AMS, Providence (2001)Lusky, W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Stud. Math. 175(1), 19–45 (2006)Meise, R., Vogt, D.: Introduction to Functional Analysis. Oxford University Press, Oxford (1997)Montes-Rodríguez, A.: Weight composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(2), 872–884 (2000)Ng, K.F.: On a theorem of Diximier. Math. Scand. 29, 279–280 (1972)Rudin, W.: Real and Complex Analysis. MacGraw-Hill, NY (1970)Rudin, W.: Functional analysis. In: International series in pure and applied mathematics, 2nd edn. McGraw-Hill, Inc., New York (1991)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299(300), 256–279 (1978)Shields, A.L., Williams, D.L.: Bounded projections and the growth of harmonic conjugates in the unit disc. Mich. Math. J. 29, 3–25 (1982)Zheng, L.: The essential norms and spectra of composition operators on $$H^\infty$$ . Pac. J. Math. 203(2), 503–510 (2002

Procyanidin B3 Prevents Articular Cartilage Degeneration and Heterotopic Cartilage Formation in a Mouse Surgical Osteoarthritis Model

Osteoarthritis (OA) is a common disease in the elderly due to an imbalance in cartilage degradation and synthesis. Heterotopic ossification (HO) occurs when ectopic masses of endochondral bone form within the soft tissues around the joints and is triggered by inflammation of the soft tissues. Procyanidin B3 (B3) is a procyanidin dimer that is widely studied due to its high abundance in the human diet and antioxidant activity. Here, we evaluated the role of B3 isolated from grape seeds in the maintenance of chondrocytes in vitro and in vivo. We observed that B3 inhibited H2O2-induced apoptosis in primary chondrocytes, suppressed H2O2- or IL-1ß−induced nitric oxide synthase (iNOS) production, and prevented IL-1ß−induced suppression of chondrocyte differentiation marker gene expression in primary chondrocytes. Moreover, B3 treatment enhanced the early differentiation of ATDC5 cells. To examine whether B3 prevents cartilage destruction in vivo, OA was surgically induced in C57BL/6J mice followed by oral administration of B3 or vehicle control. Daily oral B3 administration protected articular cartilage from OA and prevented chondrocyte apoptosis in surgically-induced OA joints. Furthermore, B3 administration prevented heterotopic cartilage formation near the surgical region. iNOS protein expression was enhanced in the synovial tissues and the pseudocapsule around the surgical region in OA mice fed a control diet, but was reduced in mice that received B3. Together, these data indicated that in the OA model, B3 prevented OA progression and heterotopic cartilage formation, at least in a part through the suppression of iNOS. These results support the potential therapeutic benefits of B3 for treatment of human OA and heterotopic ossification

Holomorphic functions with large cluster sets

We study linear and algebraic structures in sets of bounded holomorphic functions on the ball which have large cluster sets at every possible point (i.e., every point on the sphere in several complex variables and every point of the closed unit ball of the bidual in the infinite dimensional case). We show that this set is strongly (Formula presented.) -algebrable for all separable Banach spaces. For specific spaces including (Formula presented.) or duals of Lorentz sequence spaces, we have strongly (Formula presented.) -algebrability and spaceability even for the subalgebra of uniformly continous holomorphic functions on the ball.Fil: Alves, Thiago R.. Universidad Federal del Amazonas.; BrasilFil: Carando, Daniel Germán. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin