Results 31 to 40 of about 602 (89)
International Scholarly Research Notices, Volume 2012, Issue 1, 2012., 2012 We study the properties of locally uniformly differentiable functions on 𝒩, a non‐Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In particular, we show that locally uniformly differentiable functions are C1, they include all polynomial functions, and they are closed under ...
Khodr Shamseddine +3 more
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Solutions of Smooth Nonlinear Partial Differential Equations
Abstract and Applied Analysis, Volume 2011, Issue 1, 2011., 2011 The method of order completion provides a general and type‐independent theory for the existence and basic regularity of the solutions of large classes of systems of nonlinear partial differential equations (PDEs). Recently, the application of convergence spaces to this theory resulted in a significant improvement upon the regularity of the solutions ...
Jan Harm van der Walt, Stephen Clark
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Topology in nonlinear extensions of hypernumbers
Discrete Dynamics in Nature and Society, Volume 2005, Issue 2, Page 145-170, 2005., 2005 Modern theory of dynamical systems is mostly based on nonlinear differential equations and operations. At the same time, the theory of hypernumbers and extrafunctions, a novel approach in functional analysis, has been limited to linear systems. In this paper, nonlinear structures are introduced in spaces of real and complex hypernumbers by extending ...
M. S. Burgin
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Recent progress in special Colombeau algebras: geometry, topology, and algebra [PDF]
Banach Center Publications, 2010 Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of modules over the ring of generalized numbers, and algebraic aspects of Colombeau theory.
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Travelling wave solutions to some PDEs of mathematical physics
International Journal of Mathematics and Mathematical Sciences, Volume 2004, Issue 21, Page 1105-1120, 2004., 2004 Nonlinear operations such as multiplication of distributions are not allowed in the classical theory of distributions. As a result, some ambiguities arise when we want to solve nonlinear partial differential equations such as differential equations of elasticity and multifluid flows, or some new cosmological models such as signature changing space ...
Kourosh Nozari, Ghasem Alizadeh Afrouzi
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Theory of hypernumbers and extrafunctions: Functional spaces and differentiation
Discrete Dynamics in Nature and Society, Volume 7, Issue 3, Page 201-212, 2002., 2002 The theory of hypernumbers and extrafunctions is a novel approach in functional analysis aimed at problems of mathematical and computational physics. The new technique allows operations with divergent integrals and series and makes it possible to distinct different kinds of convergence and divergence.
Mark Burgin
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Generalized solutions describing singularity interaction
International Journal of Mathematics and Mathematical Sciences, Volume 29, Issue 8, Page 481-494, 2002., 2002 We present a new method for constructing solutions to nonlinear evolutionary equations describing the propagation and interaction of nonlinear waves.
V. G. Danilov
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Stieltjes transforms on new generalized functions
International Journal of Mathematics and Mathematical Sciences, Volume 25, Issue 6, Page 421-427, 2001., 2001 We introduce a Stieltjes transform on the equivalence classes of a new generalized function which has been successfully developed by Colombeau. Subsets of rapid descent test functions, 𝒮(ℝn), as well as tempered distributions, 𝒮′(ℝn), are used to preserve Fourier analysis techniques.
John Schmeelk
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On Association in Colombeau Algebras Without Asymptotics [PDF]
, 2019 A recent variant of Colombeau algebras does not employ asymptotic estimates for its definition. We discuss how the concept of association with distributions transfers to this setting and why it still needs to be based on asymptotics.
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An embedding of Schwartz distributions in the algebra of asymptotic functions
International Journal of Mathematics and Mathematical Sciences, Volume 21, Issue 3, Page 417-428, 1998., 1997 We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper “asymptotic function,” similar to but different from J. F Colombeau′s algebras of new generalized functions.
Michael Oberguggenberger, Todor Todorov
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