The Experts below are selected from a list of 61485 Experts worldwide ranked by ideXlab platform
Yves Bertot - One of the best experts on this subject based on the ideXlab platform.
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Fixed Precision Patterns for the Formal Verification of Mathematical Constant Approximations
2015Co-Authors: Yves BertotAbstract:We describe two approaches for the computation of Mathematical Constant approximations inside interactive theorem provers. These two approaches share the same basis of fixed point computation and differ only in the way the proofs of correctness of the approximations are described. The first approach performs interval computations, while the second approach relies on bounding errors, for example with the help of derivatives. As an illustration, we show how to describe good approximations of the logarithm function and we compute π to a precision of a million decimals inside the proof system, with a guarantee that all digits up to the millionth decimal are correct. All these experiments are performed with the Coq system, but most of the steps should apply to any interactive theorem prover.
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CPP - Fixed Precision Patterns for the Formal Verification of Mathematical Constant Approximations
Proceedings of the 2015 Conference on Certified Programs and Proofs - CPP '15, 2015Co-Authors: Yves BertotAbstract:We describe two approaches for the computation of Mathematical Constant approximations inside interactive theorem provers. These two approaches share the same basis of fixed point computation and differ only in the way the proofs of correctness of the approximations are described. The first approach performs interval computations, while the second approach relies on bounding errors, for example with the help of derivatives. As an illustration, we show how to describe good approximations of the logarithm function and we compute -- to a precision of a million decimals inside the proof system, with a guarantee that all digits up to the millionth decimal are correct. All these experiments are performed with the Coq system, but most of the steps should apply to any interactive theorem prover.
Dirk Huylebrouck - One of the best experts on this subject based on the ideXlab platform.
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The Meta-golden Ratio Chi
2014Co-Authors: Dirk HuylebrouckAbstract:Based on artistic interpretations, art professor Christopher Bartlett (Towson University, USA) independently rediscovered a Mathematical Constant called the ‘meta-golden section’, which had been very succinctly described 2 years earlier by Clark Kimberling. Bartlett called it ‘the chi ratio’ and denoted it by (the letter following , the golden section, in the Greek alphabet). In contrast to mathematician Kimberling, Bartlett motivated his finding on artistic considerations. They may be subject to criticism similar to the ‘golden ratio debunking’, but here we focus on showing that his chi ratio is interesting as a number as such, with pleasant geometric properties, just as the golden ratio. Moreover, Bartlett’s construction of proportional rectangles using perpendicular diagonals, which is at the basis of his chi ratio, has interesting references in architecture and in art.
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Some Mathematical properties of the Bartlett chi ratio
arXiv: History and Overview, 2013Co-Authors: Dirk HuylebrouckAbstract:Based on artistic interpretations, art professor Christopher Bartlett (Towson University, USA) suggested introducing a new Mathematical Constant related to the golden ratio. He called it the chi ratio, as chi follows phi in the Greek alphabet. Without going too much into the artistic considerations, which may be subject to criticism similar to the golden ratio debunking, we focus on showing that the chi ratio is interesting as a number as such, with remarkable geometric properties, just as the golden ratio is.
Milan Perkovac - One of the best experts on this subject based on the ideXlab platform.
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Measurement of Mathematical Constant π and Physical Quantity Pi
Journal of Applied Mathematics and Physics, 2016Co-Authors: Milan PerkovacAbstract:Instead of calculating the number π in this article special attention is paid to the method of measuring it. It has been found that there is a direct and indirect measurement of that number. To perform such a measurement, a selection was made of some Mathematical and physical quantities which within themselves contain a number π. One such quantity is a straight angle Pi, which may be expressed as Pi = π rad. By measuring the angle, using the direct method, we determine the number π as π = arccos(-1). To implement an indirect measurement of the number π, a system consisting of a container with liquid and equating it with the measuring pot has been conceived. The accuracy of measurement by this method depends on the precision performance of these elements of the system.
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measurement of Mathematical Constant 960 and physical quantity pi
Journal of Applied Mathematics and Physics, 2016Co-Authors: Milan PerkovacAbstract:Instead of calculating the number π in this article special attention is paid to the method of measuring it. It has been found that there is a direct and indirect measurement of that number. To perform such a measurement, a selection was made of some Mathematical and physical quantities which within themselves contain a number π. One such quantity is a straight angle Pi, which may be expressed as Pi = π rad. By measuring the angle, using the direct method, we determine the number π as π = arccos(-1). To implement an indirect measurement of the number π, a system consisting of a container with liquid and equating it with the measuring pot has been conceived. The accuracy of measurement by this method depends on the precision performance of these elements of the system.
David H. Bailey - One of the best experts on this subject based on the ideXlab platform.
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A short history of formulas
2016Co-Authors: David H. BaileyAbstract:This note presents a short history Mathematical formulas involving the Mathematical Constant , and how they have been used in Mathematical research through the ages.
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A Compendium of BBP-Type Formulas for Mathematical Constants
2004Co-Authors: David H. BaileyAbstract:A 1996 paper by the author, Peter Borwein and Simon Ploue showed that any Mathematical Constant given by an innite series of a certain type has the property that its n-th digit in a particular number base could be calculated directly, without needing to compute any of the rst n 1 digits, by means of a simple algorithm that does not require multiple-precision arithmetic. Several such formulas were presented in that paper, including formulas for the Constants and log 2. Since then, numerous other formulas of this type have been found. This paper presents a compendium of currently known results of this sort, both proven and conjectured. Experimentally obtained results which are not yet proven have been checked to high precision and are marked with a ? =. Fully established results are as indicated in the citations and references below.
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The quest for PI
The Mathematical Intelligencer, 1997Co-Authors: David H. Bailey, Jonathan M. Borwein, Peter Borwein, Simon PlouffeAbstract:This article gives a brief history of the analysis and computation of the Mathematical Constant $\pi = 3.14159 \ldots$, including a number of the formulas that have been used to compute $\pi$ through the ages. Recent developments in this area are then discussed in some detail, including the recent computation of $\pi$ to over six billion decimal digits using high-order convergent algorithms, and a newly discovered scheme that permits arbitrary individual hexadecimal digits of $\pi$ to be computed.
Sanjay Kumar Khattri - One of the best experts on this subject based on the ideXlab platform.
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From Lobatto Quadrature to the Euler Constant e
PRIMUS, 2010Co-Authors: Sanjay Kumar KhattriAbstract:Abstract Based on the Lobatto quadrature, we develop several new closed form approximations to the Mathematical Constant e. For validating effectiveness of our approximations, a comparison of our results to the existing approximations is also presented. Another objective of our work is to inspire students to formulate other better approximations by using the presented approach. Because of the level of mathematics, the presented work is easily embraceable in an undergraduate class.
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New Approximations to the Mathematical Constant e
Journal of Mathematics Research, 2009Co-Authors: Sanjay Kumar KhattriAbstract:Based on the Newton-Cotes and Gaussian quadrature rules, we develop several new closed form approximations to the Mathematical Constant e. For validating effectiveness of our approximations, a comparison of our results to the existing approximations is also presented. Because of the level of mathematics, the presented work is easily embraceable in an undergraduate class. Another aim of this work is to encourage students for formulating other better approximations by using the suggested strategy.
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Sharp bounds on the Mathematical Constant e
2009Co-Authors: Sanjay Kumar KhattriAbstract:In this work, we construct sharp upper and lower bounds for the Euler Constant e. For obtaining these bounds, we use Lobotto and Gauss-Legendre quadrature rules.
