Fundamental Group of n-sphere for n ≥ 2
Marco Riccardi, Artur Korniłowicz (2012)
Formalized Mathematics
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Triviality of fundamental groups of spheres of dimension greater than 1 is proven, [17]
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Displaying similar documents to “Complex Integral”
Fundamental Group of n-sphere for n ≥ 2
More on the Continuity of Real Functions
Keiko Narita, Artur Kornilowicz, Yasunari Shidama (2011)
Formalized Mathematics
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In this article we demonstrate basic properties of the continuous functions from R to Rn which correspond to state space equations in control engineering.
Higher-Order Partial Differentiation
Noboru Endou, Hiroyuki Okazaki, Yasunari Shidama (2012)
Formalized Mathematics
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In this article, we shall extend the formalization of [10] to discuss higher-order partial differentiation of real valued functions. The linearity of this operator is also proved (refer to [10], [12] and [13] for partial differentiation).
The Cauchy-Riemann Differential Equations of Complex Functions
Hiroshi Yamazaki, Yasunari Shidama, Yatsuka Nakamura, Chanapat Pacharapokin (2009)
Formalized Mathematics
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In this article we prove Cauchy-Riemann differential equations of complex functions. These theorems give necessary and sufficient condition for differentiable function.
Riemann Integral of Functions from R into n -dimensional Real Normed Space
Keiichi Miyajima, Artur Korniłowicz, Yasunari Shidama (2012)
Formalized Mathematics
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In this article, we define the Riemann integral on functions R into n-dimensional real normed space and prove the linearity of this operator. As a result, the Riemann integration can be applied to the wider range. Our method refers to the [21].
Second-Order Partial Differentiation of Real Ternary Functions
Takao Inoué (2010)
Formalized Mathematics
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In this article, we shall extend the result of [17] to discuss second-order partial differentiation of real ternary functions (refer to [7] and [14] for partial differentiation).
Lebesgue's Convergence Theorem of Complex-Valued Function
Keiko Narita, Noboru Endou, Yasunari Shidama (2009)
Formalized Mathematics
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In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences...
Partial Differentiation of Real Binary Functions
Bing Xie, Xiquan Liang, Hongwei Li (2008)
Formalized Mathematics
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In this article, we define two single-variable functions SVF1 and SVF2, then discuss partial differentiation of real binary functions by dint of one variable function SVF1 and SVF2. The main properties of partial differentiation are shown [7].MML identifier: PDIFF 2, version: 7.9.03 4.104.1021
More on Continuous Functions on Normed Linear Spaces
Hiroyuki Okazaki, Noboru Endou, Yasunari Shidama (2011)
Formalized Mathematics
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In this article we formalize the definition and some facts about continuous functions from R into normed linear spaces [14].
Sorting by Exchanging
Grzegorz Bancerek (2011)
Formalized Mathematics
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We show that exchanging of pairs in an array which are in incorrect order leads to sorted array. It justifies correctness of Bubble Sort, Insertion Sort, and Quicksort.
On -maximal curves of genus .
Abdón, Miriam, Torres, Fernando (2005)
Beiträge zur Algebra und Geometrie
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