What is nonlinear?
In geometry, linearity refers to Euclidean objects: lines, planes, (flat) th
ree-dimensional space, etc.--these objects appear the same no matter how we
examine them. A nonlinear object, a sphere for example, looks different on d
ifferent scales--when looked at closely enough it looks like a plane, and fr
om a far enough distance it looks like a point.
In algebra, we define linearity in terms of functions that have the property
f(x+y) = f(x)+f(y) and f(ax) = af(x). Nonlinear is defined as the negation
of linear. This means that the result f may be out of proportion to the inpu
t x or y. The result may be more than linear, as when a diode begins to pass
current; or less than linear, as when finite resources limit Malthusian pop
ulation growth. Thus the fundamental simplifying tools of linear analysis ar
e no longer available: for example, for a linear system, if we have two zero
s, f(x) = 0 and f(y) = 0, then we automatically have a third zero f(x+y) = 0
(in fact there are infinitely many zeros as well, since linearity implies t
hat f(ax+by) = 0 for any a and b). This is called the principle of superposi
tion--it gives many solutions from a few. For nonlinear systems, each soluti
on must be fought for (generally) with unvarying ardor!
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在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? "
然而我知道 當我正在日耳曼深處的黑森林
繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離.........
遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來
於是僅以 一顆量子渾沌
一本符號動力學 祝那發生在週一下午的新生
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※ 發信站: 批踢踢實業坊(ptt.csie.ntu.edu.tw)
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