Lebesgue’s dominated convergence theorem in Bishop’s style

Coen, Claudio Sacerdoti; Zoli, Enrico

doi:10.1016/j.apal.2011.06.020

Cited by 3 publications

(1 citation statement)

References 4 publications

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“…[9], for more recent developments in constructive integration cf. [4], [6], [8], [11], [14], [15], [16], [17], [18], [22]). Within the framework of constructive mathematics, it is a little bit delicate to speak about integrability, especially if one adopts intuitionistic mathematics in the sense of L. E. J. Brouwer (cf.…”

Section: Introductionmentioning

confidence: 99%

The swap of integral and limit in constructive mathematics

Taschner

2010

Mathematical Logic Quarterly

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who, during his whole life, stood up for the Riemann integral.Integration within constructive, especially intuitionistic mathematics in the sense of L. E. J. Brouwer, slightly differs from formal integration theories: Some classical results, especially Lebesgue's dominated convergence theorem, have to be substituted by appropriate alternatives. Although there exist sophisticated, but rather laborious proposals, e.g. by E. Bishop and D. S. Bridges (cf.[2]), the reference to partitions and the Riemann-integral, also with regard to the results obtained by R. Henstock and J. Kurzweil (cf.[9], [12]), seems to give a better direction. Especially, convergence theorems can be proved by introducing the concept of "equi-integrability".The paper is strongly motivated by Brouwer's result that each function fully defined on a compact interval has necessarily to be uniformly continuous. Nevertheless, there are, with only one exception (a corollary of Theorem 4.2), no references to the fan-theorem or to bar-induction. Therefore, the whole paper can be read within the setting of Bishop's access to constructive mathematics. Nothing of genuine full-fledged Brouwerian intuitionism is used for the main results in this note.

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Section: Introductionmentioning

confidence: 99%

The swap of integral and limit in constructive mathematics

Taschner

2010

Mathematical Logic Quarterly

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One-Dimensional Modelling of the Pressure Loss in Concrete Pumping and Experimental Verification

Zhao,

Gao,

Wan

et al. 2024

Preprint

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An accurate formula with high precision for calculating the pressure loss in concrete pumping plays a significant guiding role in the design and service process of pump trucks. Based on the flow characteristics of concrete pumping, a one-dimensional model for the pressure loss was developed in which both the viscous force of the mortar in the lubrication layer and the blocking effect of coarse aggregate particles were considered. First, the complex geometrical shapes of the aggregate particles were reconstructed by using a HandySCAN noncontact 3D laser scanner and the inverse modelling software Geomagic Design X. Then, the equivalent spherical size of uneven aggregate particles was calculated according to the equal hydraulic radius principle. The blocking effect of the aggregate particles was converted into the wall roughness. Finally, an explicit expression for the pressure loss in concrete pumping was deduced by using Modi’s formula, Bernoulli’s equation and Darcy’s formula, and the calculated value was compared with the measured value at a corresponding experimental site. The results indicated that the pressure loss values calculated with the one-dimensional flow model, which are closer to the actual pumping pressure loss values, are more accurate than those obtained with a theoretical empirical formula.

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A constructive and formal proof of Lebesgue’s Dominated Convergence Theorem in the interactive theorem prover Matita

Coen

Tassi

2008

Journal of Formalized Reasoning; Vol 1

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Lebesgue’s dominated convergence theorem in Bishop’s style

Cited by 3 publications

References 4 publications

The swap of integral and limit in constructive mathematics

The swap of integral and limit in constructive mathematics

One-Dimensional Modelling of the Pressure Loss in Concrete Pumping and Experimental Verification

A constructive and formal proof of Lebesgue’s Dominated Convergence Theorem in the interactive theorem prover Matita

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