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Integrability Formulas. Part III

References [1] Czesław Byliński. Partial functions. Formalized Mathematics , 1( 2 ):357-367, 1990. [2] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics , 8( 1 ):93-102, 1999. [3] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics , 9( 2 ):281-284, 2001

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Integrability Formulas. Part I

References [1] Czesław Byliński. Partial functions. Formalized Mathematics , 1( 2 ):357-367, 1990. [2] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics , 8( 1 ):93-102, 1999. [3] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics , 9( 2 ):281-284, 2001

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Integrability Formulas. Part II

References [1] Czesław Byliński. Partial functions. Formalized Mathematics , 1( 2 ):357-367, 1990. [2] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics , 8( 1 ):93-102, 1999. [3] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics , 9( 2 ):281-284, 2001

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Several Integrability Formulas of Special Functions

Byliński. Partial functions. Formalized Mathematics , 1(2):357-367, 1990. [6] Czesław Byliński. and Piotr Rudnicki. Bounding boxes for compact sets in ε 2 . Formalized Mathematics , 6(3):427-440, 1997. [7] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics , 8(1):93-102, 1999. [8] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from $R to $R and integrability

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Several Integrability Formulas of Special Functions. Part II

(1):93-102, 1999. [5] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics , 9(2):281-284, 2001. [6] Artur Korniłowicz and Yasunari Shidama. Inverse trigonometric functions arcsin and arccos. Formalized Mathematics , 13(1):73-79, 2005. [7] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics , 1(2):273-275, 1990

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High-Order Variational Time Integrators for Particle Dynamics

References 1. J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., vol. 10, pp. 357{514, 2001. 2. C. Kane, J. E. Marsden, M. Ortiz, and M. West, Variational integrators and the Newmark algorithmfor conservative and dissipative mechanical systems, Internat. J. Numer. Methods Engrg., vol. 49, no. 10, pp. 1295{1325, 2000. 3. E. Hairer, C. Lubich, and G. Wanner, Geometric numerical integration, vol. 31 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2010

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Several Integrability Formulas of Some Functions, Orthogonal Polynomials and Norm Functions

(1):93-102, 1999. [5] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics , 9(2):281-284, 2001. [6] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics , 1(1):35-40, 1990. [7] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics , 1(2):273-275, 1990. [8] Jarosław Kotowicz. Partial

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Integrability and the Integral of Partial Functions from R into R1

boxes for compact sets in ε 2 . Formalized Mathematics , 6(3):427-440, 1997. [10] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics , 8(1):93-102, 1999. [11] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability for partial functions from R to R and integrability for continuous functions. Formalized Mathematics , 9(2):281-284, 2001. [12] Noboru Endou, Katsumi Wasaki, and Yasunari

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Tractability of Multivariate Integration Using Low-Discrepancy Sequences

REFERENCES [1] ATANASSOV, E. I.: On the discrepancy of the Halton sequences , Mathematica Balkanica New Series, 18 (1-2) (2004), 15–32. [2] DICK, J.—PILLICHSHAMMER, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration , Cambridge University Press, 2010. [3] DICK, J.—NIEDERREITER, H.—PILLICHSHAMMER, F.: Weighted star discrepancy of digital nets in prime bases . In: Monte Carlo and quasi-Monte Carlo methods 2004, Springer, (2006), 77–96. [4] DUSART, P.: The k th prime is greater than k (ln k + ln ln k

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Riemann Integral of Functions from ℝ into Real Banach Space
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. [15] Noboru Endou and Artur Korniłowicz. The definition of the Riemann definite integral and some related lemmas. Formalized Mathematics , 8( 1 ):93-102, 1999. [16] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Scalar multiple of Riemann definite integral. Formalized Mathematics , 9( 1 ):191-196, 2001. [17] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Darboux’s theorem. Formalized Mathematics , 9( 1 ):197-200, 2001. [18] Noboru Endou, Katsumi Wasaki, and Yasunari Shidama. Definition of integrability

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