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http://hdl.handle.net/11320/3543
2019-09-18T07:51:18ZProbability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables
http://hdl.handle.net/11320/3578
Tytuł: Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables
Autorzy: Okazaki, Hiroyuki; Shidama, Yasunari
Abstrakt: In this article we continue formalizing probability and randomness started in [13], where we formalized some theorems concerning the probability and real-valued random variables. In this paper we formalize the variance of a random variable and prove Chebyshev's inequality. Next we formalize the product probability measure on the Cartesian product of discrete spaces. In the final part of this article we define the algebra of real-valued random variables.2010-01-01T00:00:00ZSperner's Lemma
http://hdl.handle.net/11320/3574
Tytuł: Sperner's Lemma
Autorzy: Pąk, Karol
Abstrakt: In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function ƒ, which for an arbitrary vertex υ of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains υ, we can find a simplex S of B which satisfies ƒ(S) = K (see [10]).2010-01-01T00:00:00ZDifferentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces
http://hdl.handle.net/11320/3577
Tytuł: Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces
Autorzy: Inoué, Takao; Endou, Noboru; Shidama, Yasunari
Abstrakt: In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).2010-01-01T00:00:00ZRiemann Integral of Functions R into C
http://hdl.handle.net/11320/3576
Tytuł: Riemann Integral of Functions R into C
Autorzy: Miyajima, Keiichi; Kato, Takahiro; Shidama, Yasunari
Abstrakt: In this article, we define the Riemann Integral on functions R into C and proof the linearity of this operator. Especially, the Riemann integral of complex functions is constituted by the redefinition about the Riemann sum of complex numbers. Our method refers to the [19].2010-01-01T00:00:00Z