Results 1 to 10 of about 195 (57)
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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A correspondence principle for the Gowers norms [PDF]
Journal of Logic and Analysis, 2009 The Furstenberg Correspondence shows that certain "local behavior" of dynamical system is equivalent to the behavior of sufficiently large finite systems. The Gowers uniformity norms, however, are not local in the relevant sense.
H. Towsner
semanticscholar +1 more source
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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Algebraic properties of external numbers
Journal of Logic and Analysis, 2011 Neutrices and external numbers were proposed as models of orders of magnitude within nonstandard analysis. We show that the external numbers form a commutative regular semigroup for addition and that the external numbers which are not neutrices form a ...
Bruno Miguel Antunes Dinis, I. Berg
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Mixed characteristic homological theorems in low degrees [PDF]
, 2002 Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps ...
Schoutens, Hans
core +3 more sources
Axiom of Choice in nonstandard set theory
Journal of Logic and Analysis, 2012 We verify that the best-known nonstandard set theories: IST, BST, and HST, with the Axiom of Choice deleted, are conservative extensions of ZF + Boolean Prime Ideal Theorem.
K. Hrbacek
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