Results 1 to 10 of about 36 (36)
The explicit formula for Gauss-Jordan elimination applied to flexible systems
Special Matrices, 2022 Flexible systems are obtained from systems of linear equations by adding to the elements of the coefficient matrix and the right-hand side scalar neutrices, which are convex groups of (non-standard) real numbers.
Tran Nam Van +2 more
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Infinitesimals via Cauchy sequences: Refining the classical equivalence
Open Mathematics, 2021 A refinement of the classic equivalence relation among Cauchy sequences yields a useful infinitesimal-enriched number system. Such an approach can be seen as formalizing Cauchy’s sentiment that a null sequence “becomes” an infinitesimal.
Bottazzi Emanuele, Katz Mikhail G.
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A topological interpretation of three Leibnizian principles within the functional extensions [PDF]
Logical Methods in Computer Science, 2018 Three philosophical principles are often quoted in connection with Leibniz: "objects sharing the same properties are the same object" (Identity of indiscernibles), "everything can possibly exist, unless it yields contradiction" (Possibility as ...
Marco Forti
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An algebraic model for the propagation of errors in matrix calculus
Special Matrices, 2020 We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) additive group.
Van Tran Nam, van den Berg Imme
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Metric completions, the Heine-Borel property, and approachability
Open Mathematics, 2020 We show that the metric universal cover of a plane with a puncture yields an example of a nonstandard hull properly containing the metric completion of a metric space.
Kanovei Vladimir +2 more
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Calculus using proximities: a mathematical approach in which students can actually prove theorems
Open Mathematics, 2017 Teaching and learning calculus are notoriously difficult and the didactic solutions may involve resorting to intuitive but vague definitions or informal gestures offered as proofs.
O’Donovan Richard
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Time averaging for functional differential equations
Journal of Applied Mathematics, Volume 2003, Issue 1, Page 1-16, 2003., 2003 We present a result on the averaging for functional differential equations on finite time intervals. The result is formulated in both classical mathematics and nonstandard analysis; its proof uses some methods of nonstandard analysis.
Mustapha Lakrib
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The method of averaging and functional differential equations with delay
International Journal of Mathematics and Mathematical Sciences, Volume 26, Issue 8, Page 497-511, 2001., 2001 We present a natural extension of the method of averaging to fast oscillating functional differential equations with delay. Unlike the usual approach where the analysis is kept in an infinite‐dimensional Banach space, our analysis is achieved in ℝn. Our results are formulated in classical mathematics. They are proved within Internal Set Theory which is
Mustapha Lakrib
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Generalized solutions of variational problems and applications
Advances in Nonlinear Analysis, 2018 Ultrafunctions are a particular class of generalized functions defined on a hyperreal field ℝ*⊃ℝ{\mathbb{R}^{*}\supset\mathbb{R}} that allow to solve variational problems with no classical solutions.
Benci Vieri +2 more
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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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