Higher - Order Partial Differentiation 1 Noboru Endou

Several Differentiation Formulas of Special Functions

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

In this paper i, n, m are elements of N. The following propositions are true: (1) Let f be a set. Then f is a partial function from Rm to Rn if and only if f is a partial function from 〈Em, ‖ · ‖〉 to 〈En, ‖ · ‖〉. (2) Let n, m be non empty elements of N, f be a partial function from Rm to Rn, g be a partial function from 〈Em, ‖·‖〉 to 〈En, ‖·‖〉, x be an element of Rm, and y be a point of 〈Em, ‖ ·...

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

The notation and terminology used in this paper have been introduced in the following papers: [7], [15], [2], [3], [24], [4], [5], [1], [11], [16], [6], [9], [12], [17], [18], [10], [8], [23], [14], [21], [13], and [22]. For simplicity, we use the following convention: n, m denote non empty elements of N, i, j denote elements of N, f denotes a partial function from 〈Em, ‖ · ‖〉 to 〈En, ‖ · ‖〉, g...

Partial Differentiation on Normed Linear Spaces Rn

Let i, n be elements of N. The functor proj(i, n) yielding a function from Rn into R is defined by: (Def. 1) For every element x of Rn holds (proj(i, n))(x) = x(i). Next we state two propositions: (1) dom proj(1, 1) = R1 and rng proj(1, 1) = R and for every element x of R holds (proj(1, 1))(〈x〉) = x and (proj(1, 1))−1(x) = 〈x〉. (2)(i) (proj(1, 1))−1 is a function from R into R1, (ii) (proj(1, 1...

Definition of Flat Poset and Existence Theorems for Recursive

Reduction Systems and Idea of Knuth-Bendix Completion Algorithm By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Double Series and Sums By Noboru Endou . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Dual Spaces and Hahn-Banach Theorem By Keiko Narita, Noboru Endou, and Yasunari Shidama 69 Semiring of Sets By Roland Coghetto . ...