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！！
思始 □ ○ 概念緣起！！？？
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 , functional decomposition generally refers to a process of identifying a set of functions such that
where is some other function. Thus, we would say that the function is decomposed into functions . This process is intrinsically hierarchical in the sense that we can (and often do) seek to further decompose the functions into a collection of constituent functions such that
where 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 into a hierarchical composition of functions such that , , , 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.
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.
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.
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, . Then can be decomposed into a linear combination of other functions, called component signals:
Here, are the component signals. Note that 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 represent the effect of the system, then the output signal is , which can be expressed as:
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.