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Displaying similar documents to “ The Correspondence Between n -dimensional Euclidean Space and the Product of n Real Lines ”

The Sum and Product of Finite Sequences of Complex Numbers

Keiichi Miyajima, Takahiro Kato (2010)

Formalized Mathematics

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This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11].

On linear operators and functors extending pseudometrics

C. Bessaga (1993)

Fundamenta Mathematicae

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For some pairs (X,A), where X is a metrizable topological space and A its closed subset, continuous, linear (i.e., additive and positive-homogeneous) operators extending metrics for A to metrics for X are constructed. They are defined by explicit analytic formulas, and also regarded as functors between certain categories. An essential role is played by "squeezed cones" related to the classical cone construction. The main result: if A is a nondegenerate absolute neighborhood retract for...

Bertrand’s Ballot Theorem

Karol Pąk (2014)

Formalized Mathematics

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In this article we formalize the Bertrand’s Ballot Theorem based on [17]. Suppose that in an election we have two candidates: A that receives n votes and B that receives k votes, and additionally n ≥ k. Then this theorem states that the probability of the situation where A maintains more votes than B throughout the counting of the ballots is equal to (n − k)/(n + k). This theorem is item #30 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. ...

Some existence results for solutions of differential inclusions with retardations

L. H. Erbe, W. Krawcewicz, Shaozhu Chen (1991)

Annales Polonici Mathematici

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Using the topological transversality method of Granas we prove an existence result for a system of differential inclusions with retardations of the form y'' ∈ F(t,y,y',Φ(y)). The result is applied to the study of the existence of solutions to an equation of the trajectory of an r-stage rocket with retardations.

Arithmetical transfinite induction and hierarchies of functions

Z. Ratajczyk (1992)

Fundamenta Mathematicae

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We generalize to the case of arithmetical transfinite induction the following three theorems for PA: the Wainer Theorem, the Paris-Harrington Theorem, and a version of the Solovay-Ketonen Theorem. We give uniform proofs using combinatorial constructions.

Mizar Analysis of Algorithms: Algorithms over Integers

Grzegorz Bancerek (2008)

Formalized Mathematics

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This paper is a continuation of [5] and concerns if-while algebras over integers. In these algebras the only elementary instructions are assignment instructions. The instruction assigns to a (program) variable a value which is calculated for the current state according to some arithmetic expression. The expression may include variables, constants, and a limited number of arithmetic operations. States are functions from a given set of locations into integers. A variable is a function...