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! -- 在细雨的午後 书页里悉哩哩地传来 " 周期3 = ? " 然而我知道 当我正在日耳曼深处的黑森林 继续发掘海森堡未曾做过的梦时 康德的诺言早已远离......... 远来的传教士静静地看着山涧不断反覆叠代自己的 过去 现在 和 未来 於是仅以 一颗量子浑沌 一本符号动力学 祝那发生在周一下午的新生 --

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