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.
Grzegorz Bancerek (2014)
Formalized Mathematics
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Two construction functors: simple term with a variable and compound term with an operation and argument terms and schemes of term induction are introduced. The degree of construction as a number of used operation symbols is defined. Next, the term context is investigated. An x-context is a term which includes a variable x once only. The compound term is x-context iff the argument terms include an x-context once only. The context induction is shown and used many times. As a key concept,...
Artur Korniłowicz (2010)
Formalized Mathematics
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We prove that basic arithmetic operations preserve continuity of functions.
Michał Trybulec (2009)
Formalized Mathematics
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This article introduces labelled state transition systems, where transitions may be labelled by words from a given alphabet. Reduction relations from [4] are used to define transitions between states, acceptance of words, and reachable states. Deterministic transition systems are also defined.
Przeworska-Rolewicz, Danuta (2004)
Fractional Calculus and Applied Analysis
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There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations...
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].
Saharon Shelah (2000)
Fundamenta Mathematicae
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Using a theorem from pcf theory, we show that for any singular cardinal ν, the product of the Cohen forcing notions on κ, κ < ν, adds a generic for the Cohen forcing notion on .
Combarro, E.F. (2004)
Sibirskij Matematicheskij Zhurnal
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Keiko Narita, Noboru Endou, Yasunari Shidama (2013)
Formalized Mathematics
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In this article, we describe the differential equations on functions from R into real Banach space. The descriptions are based on the article [20]. As preliminary to the proof of these theorems, we proved some properties of differentiable functions on real normed space. For the proof we referred to descriptions and theorems in the article [21] and the article [32]. And applying the theorems of Riemann integral introduced in the article [22], we proved the ordinary differential equations...
Takao Inoué, Adam Naumowicz, Noboru Endou, Yasunari Shidama (2011)
Formalized Mathematics
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In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).
Saharon Shelah (2000)
Fundamenta Mathematicae
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This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdotes and opinions. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history...