【鼎革‧革鼎】︰ Raspbian Stretch 《六之 J.3‧MIR-13.5始 》

梅花

踏雪尋梅

320px-LT-SEM_snow_crystal_magnification_series-3

傳說『梅花詩』有十首是宋代『卜算』以及『易理』 的大師『邵康節』先生之所著。這十首詩詞每首四句詩文,每一句詩中文字,都是用來預言之『意境』與『意象』。也許他想『推測』的是假使『時空變換』後,誰將會是如何的『興起』又誰將會如何『衰亡 』之事!!

宋‧邵康節‧梅花詩


蕩蕩天門萬古開,幾人歸去幾人來;
山河雖好非完璧,不信黃金是禍胎。

湖山一夢事全非,再見雲龍向北飛;
三百年來終一日,長天碧水歎瀰瀰。

天地相乘數一原,忽逢甲子又興元;
年華二八乾坤改,看盡殘花總不言。

畢竟英雄起布衣,朱門不是舊皇畿;
飛來燕子尋常事,開到李花春已非。

胡兒騎馬走長安,開闢中原海境寬;
洪水乍平洪水起,清光宜向漢中看。

漫天一白漢江秋,憔悴黃花總帶愁;
吉曜半升箕斗隱,金烏起滅海山頭。

雲霧蒼茫各一天,可憐西北起烽煙;
東來暴客西來盜,還有胡兒在眼前。

如棋世事局初殘,共濟和衷卻大難;
豹死猶留皮一襲,最佳秋色在長安。

火龍蟄起燕門秋,原璧應難趙氏收;
一院奇花春有主,連宵風雨不須愁。

數點梅花天地春,欲將剝復問前因;
宸中自有承平日,四海為家孰主賓。

─── 《踏雪尋梅!!

 

雪泥有鴻爪,哪個留蹤跡?山中無歲月,什人知興替!

聽調辨曲,聞聲合拍,談何容易??!!

踏雪尋梅

 

思始 □ ○ 概念緣起!!??

Functional decomposition

In mathematics, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition.

This process of decomposition may be undertaken to gain insight into the identity of the constituent components which may reflect individual physical processes of interest. Also functional decomposition may result in a compressed representation of the global function, a task which is feasible only when the constituent processes possess a certain level of modularity (i.e., independence or non-interaction).

Interactions between the components are critical to the function of the collection. All interactions may not be observable, but possibly deduced through repetitive perception, synthesis, validation and verification of composite behavior.

Basic mathematical definition

For a multivariate function y=f(x_{1},x_{2},\dots ,x_{n}), functional decomposition generally refers to a process of identifying a set of functions  \{g_{1},g_{2},\dots g_{m}\} such that

f(x_{1},x_{2},\dots ,x_{n})=\phi (g_{1}(x_{1},x_{2},\dots ,x_{n}),g_{2}(x_{1},x_{2},\dots ,x_{n}),\dots g_{m}(x_{1},x_{2},\dots ,x_{n}))

where  \phi is some other function. Thus, we would say that the function  f is decomposed into functions  \{g_{1},g_{2},\dots g_{m}\}. This process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions  g_{i} into a collection of constituent functions  \{h_{1},h_{2},\dots h_{p}\}such that

g_{i}(x_{1},x_{2},\dots ,x_{n})=\gamma (h_{1}(x_{1},x_{2},\dots ,x_{n}),h_{2}(x_{1},x_{2},\dots ,x_{n}),\dots h_{p}(x_{1},x_{2},\dots ,x_{n}))

where  \gamma is some other function. Decompositions of this kind are interesting and important for a wide variety of reasons. In general, functional decompositions are worthwhile when there is a certain “sparseness” in the dependency structure; that is, when constituent functions are found to depend on approximately disjoint sets of variables. Thus, for example, if we can obtain a decomposition of x_{1}=f(x_{2},x_{3},\dots ,x_{6}) into a hierarchical composition of functions  \{g_{1},g_{2},g_{3}\} such that  x_{1}=g_{1}(x_{2}) x_{2}=g_{2}(x_{3},x_{4},x_{5}) x_{5}=g_{3}(x_{6}), as shown in the figure at right, this would probably be considered a highly valuable decomposition.

An example of a sparsely connected dependency structure. Direction of causal flow is upward.

Philosophical considerations

The philosophical antecedents and ramifications of functional decomposition are quite broad, as functional decomposition in one guise or another underlies all of modern science. Here we review just a few of these philosophical considerations.

Reductionist tradition

One of the major distinctions that is often drawn between Eastern philosophy and Western Philosophy is that the Eastern philosophers tended to espouse ideas favoring holism while the Western thinkers tended to espouse ideas favoring reductionism. This distinction between East and West is akin to other philosophical distinctions such as realism vs. anti-realism). Some examples of the Eastern holistic spirit:

  • “Open your mouth, increase your activities, start making distinctions between things, and you’ll toil forever without hope.” — The Tao Te Ching of Lao Tzu (Brian Browne Walker, translator)
  • “It’s a hard job for [people] to see the meaning of the fact that everything, including ourselves, depends on everything else and has no permanent self-existence.” Majjhima Nikaya (Anne Bankroft, translator)
  • “A name is imposed on what is thought to be a thing or a state and this divides it from other things and other states. But when you pursue what lies behind the name, you find a greater and greater subtlety that has no divisions…” Visuddhi Magga (Anne Bankroft, translator)

The Western tradition, from its origins among the Greek philosophers, preferred a position in which drawing correct distinctions, divisions, and contrasts was considered the very pinnacle of insight. In the Aristotelian/Porphyrian worldview, to be able to distinguish (via strict proof) which qualities of a thing represent its essence vs. property vs. accident vs. definition, and by virtue of this formal description to segregate that entity into its proper place in the taxonomy of nature — this was to achieve the very height of wisdom.

……

Applications

………

Signal processing

Functional decomposition is used in the analysis of many signal processing systems, such as LTI systems. The input signal to an LTI system can be expressed as a function,  f(t). Then  f(t) can be decomposed into a linear combination of other functions, called component signals:

f(t)=a_{1}\cdot g_{1}(t)+a_{2}\cdot g_{2}(t)+a_{3}\cdot g_{3}(t)+\dots +a_{n}\cdot g_{n}(t)

Here,  \{g_{1}(t),g_{2}(t),g_{3}(t),\dots ,g_{n}(t)\} are the component signals. Note that \{a_{1},a_{2},a_{3},\dots ,a_{n}\} are constants. This decomposition aids in analysis, because now the output of the system can be expressed in terms of the components of the input. If we let  T\{\} represent the effect of the system, then the output signal is  T\{f(t)\}, which can be expressed as:

T\{f(t)\}=T\{a_{1}\cdot g_{1}(t)+a_{2}\cdot g_{2}(t)+a_{3}\cdot g_{3}(t)+\dots +a_{n}\cdot g_{n}(t)\}
=a_{1}\cdot T\{g_{1}(t)\}+a_{2}\cdot T\{g_{2}(t)\}+a_{3}\cdot T\{g_{3}(t)\}+\dots +a_{n}\cdot T\{g_{n}(t)\}

In other words, the system can be seen as acting separately on each of the components of the input signal. Commonly used examples of this type of decomposition are the Fourier series and the Fourier transform.

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