On Paragrassmann Differential Calculus

Explicit general constructions of paragrassmann calculus with one and many variables are given. Relations of the paragrassmann calculus to quantum groups are outlined and possible physics applications are briefly discussed. This paper is the same as the original 9210075 except added Appendix and minor changes in Acknowledgements and References. IMPORTANT NOTE: This paper bears the same title as the Dubna preprint E5-92-392 but is NOT identical to it, containing new results, extended discussions, and references.Comment: 19p

Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators Cϕ : A (R) → A (R) for a real analytic self map ϕ : R → R as well as an isomorphic description of corresponding eigenspaces. We completely characterize those ϕ for which Abel’s equation f ◦ ϕ = f + 1 has a real analytic solution on the real line. We find cases when the operator Cϕ has roots using a constructed embedding of ϕ into the so-called real analytic iteration semigroups.(1) The research of the authors was partially supported by MEC and FEDER Project MTM2010-15200 and MTM2013-43540-P and the work of Bonet also by GV Project Prometeo II/2013/013. The research of Domanski was supported by National Center of Science, Poland, Grant No. NN201 605340. (2) The authors are very indebted to K. Pawalowski (Poznan) for providing us with references [26,27,47] and also explaining some topological arguments of [10]. The authors are also thankful to M. Langenbruch (Oldenburg) for providing a copy of [29].Bonet Solves, JA.; Domanski, P. (2015). Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory. 81(4):455-482. https://doi.org/10.1007/s00020-014-2175-4S455482814Abel, N.H.: Determination d’une function au moyen d’une equation qui ne contient qu’une seule variable. In: Oeuvres Complètes, vol. II, pp. 246-248. Christiania (1881)Baker I.N.: Zusammensetzung ganzer Funktionen. Math. Z. 69, 121–163 (1958)Baker I.N.: Permutable power series and regular iteration. J. Aust. Math. Soc. 2, 265–294 (1961)Baker I.N.: Permutable entire functions. Math. Z. 79, 243–249 (1962)Baker I.N.: Fractional iteration near a fixpoint of multiplier 1. J. Aust. Math. Soc. 4, 143–148 (1964)Baker I.N.: Non-embeddable functions with a fixpoint of multiplier 1. Math. Z. 99, 337–384 (1967)Baker I.N.: On a class of nonembeddable entire functions. J. Ramanujan Math. Soc. 3, 131–159 (1988)Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequ. Math. 61, 1–48 (2001)Belitskii G., Lyubich Y.: The Abel equation and total solvability of linear functional equations. Studia Math. 127, 81–97 (1998)Belitskii G., Lyubich Yu.: The real analytic solutions of the Abel functional equation. Studia Math. 134, 135–141 (1999)Belitskii G., Tkachenko V.: One-Dimensional Functional Equations. Springer, Basel (2003)Belitskii G., Tkachenko V.: Functional equations in real analytic functions. Studia Math. 143, 153–174 (2000)Bonet J., Domański P.: Power bounded composition operators on spaces of analytic functions. Collect. Math. 62, 69–83 (2011)Bonet J., Domański P.: Hypercyclic composition operators on spaces of real analytic functions. Math. Proc. Camb. Philos. Soc. 153, 489–503 (2012)Bracci, F., Poggi-Corradini, P.: On Valiron’s theorem. In: Proceedings of Future Trends in Geometric Function Theory. RNC Workshop Jyväskylä 2003, Rep. Univ. Jyväskylä Dept. Math. Stat., vol. 92, pp. 39–55 (2003)Contreras, M.D.: Iteración de funciones analíticas en el disco unidad. Universidad de Sevilla (2009). (Preprint)Contreras M.D., Díaz-Madrigal S., Pommerenke Ch.: Some remarks on the Abel equation in the unit disk. J. Lond. Math. Soc. 75(2), 623–634 (2007)Cowen C.: Iteration and the solution of functional equations for functions analytic in the unit disc. Trans. Am. Math. Soc. 265, 69–95 (1981)Cowen C.C., MacCluer B.D.: Composition operators on spaces of analytic functions. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)Domański, P.: Notes on real analytic functions and classical operators. In: Topics in Complex Analysis and Operator Theory (Winter School in Complex Analysis and Operator Theory, Valencia, February 2010). Contemporary Math., vol. 561, pp. 3–47. Am. Math. Soc., Providence (2012)Domański P., Goliński M., Langenbruch M.: A note on composition operators on spaces of real analytic functions. Ann. Polon. Math. 103, 209–216 (2012)P. Domański M. Langenbruch 2003 Language="En"Composition operators on spaces of real analytic functions Math. Nachr. 254–255, 68–86 (2003)Domański P., Langenbruch M.: Coherent analytic sets and composition of real analytic functions. J. Reine Angew. Math. 582, 41–59 (2005)Domański P., Langenbruch M.: Composition operators with closed image on spaces of real analytic functions. Bull. Lond. Math. Soc. 38, 636–646 (2006)Domański P., Vogt D.: The space of real analytic functions has no basis. Studia Math. 142, 187–200 (2000)Fuks D.B., Rokhlin V.A.: Beginner’s Course in Topology. Springer, Berlin (1984)Greenberg M.J.: Lectures on Algebraic Topology. W. A. Benjamin Inc., Reading (1967)Hammond, C.: On the norm of a composition operator, PhD. dissertation, Graduate Faculty of the University of Virginia (2003). http://oak.conncoll.edu/cnham/Thesis.pdfHandt T., Kneser H.: Beispiele zur Iteration analytischer Funktionen. Mitt. Naturwiss. Ver. für Neuvorpommernund Rügen, Greifswald 57, 18–25 (1930)Heinrich T., Meise R.: A support theorem for quasianalytic functionals. Math. Nachr. 280(4), 364–387 (2007)Karlin S., McGregor J.: Embedding iterates of analytic functions with two fixed points into continuous group. Trans.Am. Math. Soc. 132, 137–145 (1968)Kneser H.: Reelle analytische Lösungen der Gleichung $${\varphi(\varphi(x))=e^x}$$ φ ( φ ( x ) ) = e x und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187, 56–67 (1949)Königs, G.: Recherches sur les intégrales de certaines équations fonctionnelles. Ann. Sci. Ecole Norm. Sup. (3) 1, Supplément, 3–41 (1884)Kuczma M.: Functional Equations in a Single Variable. PWN-Polish Scientific Publishers, Warszawa (1968)Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Cambridge University Press, Cambridge (1990)Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Milnor, J.: Dynamics in One Complex Variable. Vieweg, Braunschweig (2006)Schröder E.: über iterierte Funktionen. Math. Ann. 3, 296–322 (1871)Shapiro J.H.: Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics. Springer, New York (1993)Shapiro, J.H.: Notes on the dynamics of linear operators. Lecture Notes. http://www.mth.msu.edu/~hapiro/Pubvit/Downloads/LinDynamics/LynDynamics.htmlShapiro, J.H.: Composition operators and Schröder functional equation. In: Studies on Composition Operators (Laramie, WY, 1996), Contemp. Math., vol. 213, pp. 213–228. Am. Math. Soc., Providence (1998)Szekeres G.: Regular iteration of real and complex functions. Acta Math. 100, 203–258 (1958)Szekeres G.: Fractional iteration of exponentially growing functions. J. Aust. Math. Soc. 2, 301–320 (1961)Szekeres G.: Fractional iteration of entire and rational functions. J. Aust. Math. Soc. 4, 129–142 (1964)Szekeres G.: Abel’s equations and regular growth: variations on a theme by Abel. Exp. Math. 7, 85–100 (1998)Trappmann H., Kouznetsov D.: Uniqueness of holomorphic Abel function at a complex fixed point pair. Aequ. Math. 81, 65–76 (2011)Viro, O.: 1-manifolds. Bull. Manifold Atlas. http://www.boma.mpim-bonn.mpg.de/articles/48 (a prolonged version also http://www.map.mpim-bonn.mpg.de/1-manifolds#Differential_structures )Walker P.L.: A class of functional equations which have entire solutions. Bull. Aust. Math. Soc. 39, 351–356 (1988)Walker P.L.: The exponential of iteration of e x −1. Proc. Am. Math. Soc. 110, 611–620 (1990)Walker P.L.: On the solution of an Abelian functional equation. J. Math. Anal. Appl. 155, 93–110 (1991)Walker P.L.: Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57, 723–733 (1991

Global surveillance of cancer survival 1995-2009: analysis of individual data for 25,676,887 patients from 279 population-based registries in 67 countries (CONCORD-2)

BACKGROUND: Worldwide data for cancer survival are scarce. We aimed to initiate worldwide surveillance of cancer survival by central analysis of population-based registry data, as a metric of the effectiveness of health systems, and to inform global policy on cancer control. METHODS: Individual tumour records were submitted by 279 population-based cancer registries in 67 countries for 25·7 million adults (age 15-99 years) and 75,000 children (age 0-14 years) diagnosed with cancer during 1995-2009 and followed up to Dec 31, 2009, or later. We looked at cancers of the stomach, colon, rectum, liver, lung, breast (women), cervix, ovary, and prostate in adults, and adult and childhood leukaemia. Standardised quality control procedures were applied; errors were corrected by the registry concerned. We estimated 5-year net survival, adjusted for background mortality in every country or region by age (single year), sex, and calendar year, and by race or ethnic origin in some countries. Estimates were age-standardised with the International Cancer Survival Standard weights. FINDINGS: 5-year survival from colon, rectal, and breast cancers has increased steadily in most developed countries. For patients diagnosed during 2005-09, survival for colon and rectal cancer reached 60% or more in 22 countries around the world; for breast cancer, 5-year survival rose to 85% or higher in 17 countries worldwide. Liver and lung cancer remain lethal in all nations: for both cancers, 5-year survival is below 20% everywhere in Europe, in the range 15-19% in North America, and as low as 7-9% in Mongolia and Thailand. Striking rises in 5-year survival from prostate cancer have occurred in many countries: survival rose by 10-20% between 1995-99 and 2005-09 in 22 countries in South America, Asia, and Europe, but survival still varies widely around the world, from less than 60% in Bulgaria and Thailand to 95% or more in Brazil, Puerto Rico, and the USA. For cervical cancer, national estimates of 5-year survival range from less than 50% to more than 70%; regional variations are much wider, and improvements between 1995-99 and 2005-09 have generally been slight. For women diagnosed with ovarian cancer in 2005-09, 5-year survival was 40% or higher only in Ecuador, the USA, and 17 countries in Asia and Europe. 5-year survival for stomach cancer in 2005-09 was high (54-58%) in Japan and South Korea, compared with less than 40% in other countries. By contrast, 5-year survival from adult leukaemia in Japan and South Korea (18-23%) is lower than in most other countries. 5-year survival from childhood acute lymphoblastic leukaemia is less than 60% in several countries, but as high as 90% in Canada and four European countries, which suggests major deficiencies in the management of a largely curable disease. INTERPRETATION: International comparison of survival trends reveals very wide differences that are likely to be attributable to differences in access to early diagnosis and optimum treatment. Continuous worldwide surveillance of cancer survival should become an indispensable source of information for cancer patients and researchers and a stimulus for politicians to improve health policy and health-care systems

<em>COL4A2</em> mutations impair COL4A1 and COL4A2 secretion and cause hemorrhagic stroke.

Collagen, type IV, alpha 1 (COL4A1) and alpha 2 (COL4A2) form heterotrimers and are abundant components of basement membranes, including those of the cerebral vasculature. COL4A1 mutations are an increasingly recognized cause of multisystem disorders, including highly penetrant cerebrovascular disease and intracerebral hemorrhage (ICH). Because COL4A1 and COL4A2 are structurally and functionally associated, we hypothesized that variants in COL4A2 would also cause ICH. We sequence COL4A2 in 96 patients with ICH and identify three rare, nonsynonymous coding variants in four patients that are not present in a cohort of 144 ICH-free individuals. All three variants change evolutionarily conserved amino acids. Using a cellular assay, we show that these putative mutations cause intracellular accumulation of COL4A1 and COL4A2 at the expense of their secretion, which supports their pathogenecity. Furthermore, we show that Col4a2 mutant mice also have completely penetrant ICH and that mutations in mouse and human lead to retention of COL4A1 and COL4A2 within the endoplasmic reticulum (ER). Importantly, two of the three putative mutations found in patients trigger ER stress and activate the unfolded protein response. The identification of putative COL4A2 mutations that might contribute to ICH in human patients provides insight into the pathogenic mechanisms of this disease. Our data suggest that COL4A2 mutations impair COL4A1 and COL4A2 secretion and can also result in cytotoxicity. Finally, our findings suggest that, collectively, mutations in COL4A1 and COL4A2 contribute to sporadic cases of ICH