Keiko Narita, Artur Korniłowicz, Yasunari Shidama (2012)
Formalized Mathematics
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In control engineering, differentiable partial functions from R into Rn play a very important role. In this article, we formalized basic properties of such functions.
Katuhiko Kanazashi, Hiroyuki Okazaki, Yasunari Shidama (2013)
Formalized Mathematics
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In this article, we formalize continuous differentiability of realvalued functions on n-dimensional real normed linear spaces. Next, we give a definition of the Ck space according to [23].
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).
Xiquan Liang, Piqing Zhao, Ou Bai (2010)
Formalized Mathematics
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In this article, we first extend several basic theorems of the operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors: e1, e2, e3 and the definition of vector function in the same spaces are introduced. By dint of unit vector the main operation properties as well as the differentiation formulas of vector function are shown [12].
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]).
Takao Inoué, Noboru Endou, Yasunari Shidama (2010)
Formalized Mathematics
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In this article, we define and develop differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).
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).
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...
K. Spallek (1991)
Annales Polonici Mathematici
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A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however, is not yet completely determined. The following gives as a corollary a first essential answer. It is shown, more generally,that a locally compact topological transformation group, operating effectively on a differentiable space X (which...
Noboru Endou, Yasunari Shidama (2013)
Formalized Mathematics
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In this article we formalized the Fréchet differentiation. It is defined as a generalization of the differentiation of a real-valued function of a single real variable to more general functions whose domain and range are subsets of normed spaces [14].
Gerd Herzog (1998)
Colloquium Mathematicae
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We prove that the initial value problem x’(t) = f(t,x(t)), is uniquely solvable in certain ordered Banach spaces if f is quasimonotone increasing with respect to x and f satisfies a one-sided Lipschitz condition with respect to a certain convex functional.