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NonStandard Analysis in ACL2
, 2001
"... ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complexrational numbers and excluding the real and complex irrationals. In conjunction ..."
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Cited by 18 (7 self)
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ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complexrational numbers and excluding the real and complex irrationals. In conjunction with the arithmetic completion axioms, this numeric type system makes it possible to prove the nonexistence of specific irrational numbers, such as √2. This paper describes ACL2(r), a version of ACL2 with support for the real and complex numbers. The modifications are based on nonstandard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.
Continuity and differentiability in ACL2
 ComputerAided Reasoning: ACL2 Case Studies, chapter 18
, 2000
"... This case study shows how ACL2 can be used to reason about the real and complex numbers, using nonstandard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems ..."
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Cited by 9 (5 self)
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This case study shows how ACL2 can be used to reason about the real and complex numbers, using nonstandard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems of analysis, such as the intermediatevalue and meanvalue theorems.
Mechanical Verification of a Square Root Algorithm Using Taylor’s Theorem
 In Formal Methods in Computer Aided Design (FMCAD'02
, 2002
"... Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theore ..."
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Cited by 5 (1 self)
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Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of nonstandard analysis, as implemented in ACL2(r). Since Taylor series of a given order have less accuracy than Chebyshev series in general, we used hundreds of Taylor series generated by ACL2(r) to evaluate the error of a Chebyshev series. 1
Taylor's Formula with Remainder
 In Proceedings of the Third International Workshop of the ACL2 Theorem Prover and its Applications
, 2002
"... In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder. ..."
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Cited by 4 (4 self)
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In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder.
Natural and Formal Infinities
 Educational Studies in Mathematics 48 (2001
"... Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fis ..."
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Cited by 4 (1 self)
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Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fischbein et al., 1979, p. 31). The formal approach to mathematics in the twentieth century attempted to rationalize these inconsistencies by selecting a finite list of specific properties (or axioms) from which the conception of a formal infinity is built by formal deduction. By beginning with different properties of finite numbers, such as counting, ordering or arithmetic, different formal systems may be developed. Counting and ordering lead to cardinal and ordinal number theory and the properties of arithmetic lead to ordered fields that may contain infinite and infinitesimal quantities. Cardinal and ordinal numbers can be added and multiplied but not divided or subtracted. The operations of cardinals are commutative, but the operations of ordinals are not. Meanwhile an ordered field has a full system of arithmetic in which the reciprocals of
Theory Extension in ACL2(r)
"... Abstract. ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper pres ..."
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Cited by 2 (1 self)
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Abstract. ACL2(r) is a modified version of the theorem prover ACL2 that adds support for the irrational numbers using nonstandard analysis. It has been used to prove basic theorems of analysis, as well as the correctness of the implementation of transcendental functions in hardware. This paper presents the logical foundations of ACL2(r). These foundations are also used to justify significant enhancements to ACL2(r). 1.
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
NONSTANDARD ANALYSIS, FRACTAL PROPERTIES AND BROWNIAN MOTION
, 2007
"... In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather tha ..."
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In this paper I explore a nonstandard formulation of Hausdorff dimension. By considering an adapted form of the counting measure formulation of Lebesgue measure, I prove a nonstandard version of Frostman’s lemma and show that Hausdorff dimension can be computed through a counting argument rather than by taking the infimum of a sum of certain covers. This formulation is then applied to obtain a simple proof of the doubling of the dimension of certain sets under a Brownian motion. Keywords: Frostman’s lemma, Nonstandard Hausdorff dimension, Brownian motion 1
RESEARCH ARTICLES On Hyperbolic Fixed Points in Ultrametric Dynamics ∗
, 2010
"... Abstract—Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding l ..."
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Abstract—Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective. DOI: 10.1134/S2070046610030052 Key words: dynamical system, linearization, conjugacy, ultrametric field.