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Analytical properties of power series on LeviCivita fields
 Ann. Math. Blaise Pascal
, 2005
"... A detailed study of power series on the LeviCivita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are i ..."
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A detailed study of power series on the LeviCivita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and reexpandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable. 1
Generalized power series on a nonArchimedean field
, 2006
"... Power series with rational exponents on the real numbers field and the LeviCivita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we ..."
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Cited by 7 (7 self)
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Power series with rational exponents on the real numbers field and the LeviCivita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the LeviCivita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the LeviCivita field; and we study the differentiability properties of such functions within their domain of convergence.
Analysis on the LeviCivita field, a brief overview
 CONTEMPORARY MATH., VOLUME 508,
, 2010
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On the topological structure of the LeviCivita field
 J. Math. Anal. Appl
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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Cited by 5 (5 self)
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevierâ€™s archiving and manuscript policies are
Absolute and relative extrema, the mean value theorem and the inverse function theorem for analytic functions on a LeviCivita field
, 2011
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The implicit function theorem in a nonArchimedean setting
, 2009
"... In this paper, the inverse function theorem and the implicit function theorem in a nonArchimedean setting will be discussed. We denote by N any nonArchimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properti ..."
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Cited by 1 (1 self)
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In this paper, the inverse function theorem and the implicit function theorem in a nonArchimedean setting will be discussed. We denote by N any nonArchimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from N n to N m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from N n to N n and the implicit function theorem for functions from N n to N m with m<n.
OneVariable and MultiMariable Calculus on a NonArchimedean Field Extension of the Real Numbers
, 2013
"... New elements of calculus on a complete real closed nonArchimedean field extension F of the real numbers R will be presented. It is known that the total disconnectedness of F in the topology induced by the order makes the usual (topological) notions of continuity and differentiability too weak to ex ..."
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New elements of calculus on a complete real closed nonArchimedean field extension F of the real numbers R will be presented. It is known that the total disconnectedness of F in the topology induced by the order makes the usual (topological) notions of continuity and differentiability too weak to extend real calculus results to F. In this paper, we introduce new stronger concepts of continuity and differentiability which we call derivate continuity and derivate differentiability [2, 12]; and we show that derivate continuous and differentiable functions satisfy the usual addition, product and composition rules and that ntimes derivate differentiable functions satisfy a Taylor formula with remainder similar to that of the real case. Then we generalize the definitions of derivate continuity and derivate differentiability to multivariable Fvalued functions and we prove related results that are useful for doing analysis on F and F n in general.