Tracy Harms-3 wrote:
>
> The fact that operators (adverbs and conjunctions) can produce
> entities of any type (noun, verb, adverb, conjunction) is among the
> most powerful aspects of J. Here is is most apparent that J
> facilitates higher orders of abstraction.
>
> K.E. Iverson, writing on the history of APL, noted that the first four
> operators emerged through independent efforts, and were only later
> recognized as instances of a common class. It's a huge leap from the
> initial hand-crafting of those first few operators (embedded in the
> language) to the freedom, with J, to define custom operators.
>
> K.E.I. credited Oliver Heaviside with the innovation of operators.
> Heaviside was also among the early developers of vectors and vector
> calculus. His vector solution to Maxwell's equations was a significant
> advancement. There was a strong emphasis ine Heaviside's work to
> simplify, to approach mathematics with an exploratory attitude, and to
> also see it as a tool for practical application. All these strike me
> as concordant with the spirit of APL.
>
>
>
A few quotes from the net about (the) Heaviside('s method) and notation:
> This reproduction of Heaviside's article is an unedited copy of the
> original, except that I have converted some formulas and all vector
> equations appearing in the article to modern mathematical notation.
> [http://www.as.wvu.edu/coll03/phys/www/OJ/Heavisid.htm]
>
> Between 1880 and 1887, Heaviside developed the operational calculus
> (involving the D notation for the differential operator, which he is
> credited with creating)[...]
> [http://en.wikipedia.org/wiki/Oliver_Heaviside]
>
> These operators occurring in interpolation theory are fundamentally
> different from those of Heaviside's methods; here the fundamental operator
> is D, whereas in Heaviside's methods it is p^-1, which is not simply the
> inverse of D because the two do not commute.
> ["Methods of Mathematical Physics" by H. Jeffreys, B. Jeffreys, B.
> Swirles; p.266]
>
> Heaviside introduced the unit step function, U(t), [...]. The derivative
> of U(t) is the Dirac delta function, δ(t), [...]. These strange functions
> are very widely used in physics and engineering today, and avoid much
> cumbersome notation.
>
> Heaviside used the operator p to represent differentiation: p = d/dt.
> Then, 1/p must represent integration from 0 to t, since (1/p)p f(t) =
> f(t). Operators are treated like algebraic symbols, and manipulated by the
> usual rules--we have just had an example. Then, pU(t) = δ(t), and
> (1/p)U(t) = ∫(0,t)U(t)dt = t. To make it easy to write in HTML, the limits
> of the integral in parentheses follow the integral sign.
> [http://mysite.du.edu/~jcalvert/math/laplace.htm , emphasize added.]
>
Never-ending reinvention of notations, including my own writing of "p to the
power minus one" as p^-1 in the third quote above.
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