<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">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].</abstract>

Faculty of Engineering, Ibaraki University, Hitachi, Japan

Faculty of Engineering, Graduate School of Ibaraki University, Hitachi, Japan

Riemann Integral of Functions R into C

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">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]).</abstract>

Sperner's Lemma

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">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.</abstract>

Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">In this article, we define and develop differentiation of vector-valued functions on <italic>n</italic>-dimensional real normed linear spaces (refer to [16] and [17]).</abstract>

Inaba 2205, Wing-Minamikan Nagano, Nagano, Japan

Gifu National College of Technology, Japan

Differentiation of Vector-Valued Functions on <italic>n</italic>-Dimensional Real Normed Linear Spaces

<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">The article provides counting derangements of finite sets and counting non bijective functions. We provide a recursive formula for the number of derangements of a finite set, together with an explicit formula involving the number e. We count the number of non-one-to-one functions between to finite sets and perform a computation to give explicitely a formalization of the birthday problem. The article is an extension of [10].</abstract>

Counting Derangements, Non Bijective Functions and the Birthday Problem

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Volume 18 (2010): Issue 4 (January 2010)

Riemann Integral of Functions R into C

Sperner's Lemma

Probability Measure on Discrete Spaces and Algebra of Real-Valued Random Variables

Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

Counting Derangements, Non Bijective Functions and the Birthday Problem

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