For n = 2 we can show that
Formula 2) If v_{1}, v_{2}, ..., v_{k}is an orthonormal set and
then c_{1} = u • v_{1}, c_{2} = u • v_{2}, ..., c_{k} = u • v_{k}.
We shall give the proof of this statement only for the case k = 2. Suppose v_{1} and v_{2} are orthonormal and
We take the dot product of both sides with v_{1} and perform some elementary manipulations:
By assumption, v_{1} and v_{2} are orthonormal, and we can therefore substitue the relations v_{1} • v_{2} = 0 and v_{1} • v_{1} = 1 to obtain c_{1} = u • v_{1}. A similar argument, using v_{2}, shows that c_{2} = u • v_{2}.
If |v| = 1 then, by an application of formula (1), the projection of u onto v is (u • v)v; therefore, by formula (2):
[From textbook on Linear Algebra]
Homomorphisms of Inner Product Spaces account for, or correspond to, all possible bases eigenstates [superposition].
Inner automorphisms (self-isomorphic maps) represent all possible configurations of any given wavefunction of a system.
What does this have to with Galois Theory? A Galois group is one of automorphic maps. Is the hierarchy of extended fields equivalent to a composition series of factor groups, that is, homomorphisms?
Luc Steels writes: "A component has a particular functionality but this is not recognizable as a subfunction of the global functionality. Instead a component implements a behaviour whose side effect contributes to the global functionality. Each behaviour has a side effect and the sum of the side effects gives the desired functionality." (Steels: Towards a Theory of Emergent Functionality, 1990)
"In other words, the global or macroscopic functionality of a system with "emergent functionality" is the sum of all "side effects" of all emergent properties and functionalities."
Jeffrey Goldstein, in the journal Emergence, defined emergence as: "the arising of novel and coherent structures, patterns and properties during the process of self-organization in complex systems." (Wikipedia on Emergence)
Per his definition of emergence, Peter Corning also addresses emergence and evolution:
"[In] evolutionary processes, causation is iterative; effects are also causes. And this is equally true of the synergistic effects produced by emergent systems. In other words, emergence itself... has been the underlying cause of the evolution of emergent phenomena in biological evolution; it is the synergies produced by organized systems that are the key." ("The Re-Emergence of Emergence" [PDF File] by Peter Corning Ph.D.)
A vector space is spanned by its independent basis vectors. These components, through their interaction and interconnectivity, generate the whole of the space synergistically. Any single point contains the potentiality of the emergent property; that is, of being more than just a separate part. It's the mixture of these parts that produces the functionality of the entire vector space. Without the synergistic effect underlying the complexity of the whole, generated by the dimensional interdependence and "co-operation" of the entire set of basis vectors, the projections -- or "side effects" -- understood as individual emergent properties, would not collectively contribute to the macro identity of the space/system itself.
Emergent Properties and Processes
[The quoted material below is from the above Wikipedia article.]
"On the other hand, merely having a large number of interactions is not enough by itself to guarantee emergent behaviour; many of the interactions may be negligible or irrelevant, or may cancel each other out. In some cases, a large number of interactions can in fact work against the emergence of interesting behaviour, by creating a lot of "noise" to drown out any emerging "signal"; the emergent behaviour may need to be temporarily isolated from other interactions before it reaches enough critical mass to be self-supporting. Thus it is not just the sheer number of connections between components which encourages emergence; it is also how these connections are organised."
There must be meaning on the global level. Say you have a 1,000 piece jigsaw puzzle. You dump it on a table, a disorganized randomly associated pile of pieces. The end picture that appears on the box is that of the Taj Mahal. That is what represents the global emergent property, and it can only be realized when all the pieces are put in precise proximity to one another. Only then. So, as with a strange attractor, just having a significant threshold of energy passing through a material system is not, in itself, sufficient to generate an attractor -- the individual vortices must be in a proper interdependent relation with one another -- "a large number of interactions is not enough to guarantee emergent behaviour."
"The complex behaviour or properties are not a property of any single such entity, nor can they easily be predicted or deduced from behaviour in the lower-level entities: they are irreducible."
This is an interesting observation, especially when put in the context of the evolution of any particular, sufficiently complex organism. One of the arguments for Intelligent Design stipulates that a subtle sub-system, say, for instance, the blood-clotting system, of a human being could not have had its many parts evolve independently and then magically coalesce to produce such an intricate and elaborate mechanism for clotting blood. It's considerd an irreducible, a dynamic entirety in and of itself, whole and complete, coming into existence as one, at once. Hence: God did it.
A counter to this presumption is the existence of what Doctor Franklin M. Harold calls the Morphogenetic Field. It orchestrates and organizes the general shape of a cell, localizes position for all its sub-units -- of which there are thousands in the case of a human cell -- and regulates their integrated functionality and coordinates relationships on all scales and betweeen scales simultaneously. So, assuming the truth and reality of this field as a natural causal agency of emergent properties, we have to conclude that the finished cell comes into being from the top-down. That is, the cell and its intended function exists as an abstract template -- in meta-information space -- prior to the parts which constitute it. In other words, it exists unto itself, and because of this, creates life synergistically with all the inteconnecting and interdependent components acting as potentials.
Thinking of the morphogentic field as an emergent property itself, doesn't really do it justice. It's more like a law dictating the global properties of a complex system that would not emerge were it not for its purposeful oversight. But perhaps even law is too strong or misleading, too independently existing and fundamental sounding. According to Doctor Harold, it acts as an intermediary, guiding genetic expression through biochemical processes to the final form and function of a cell. So, in that sense, in the sense of coming into being as an effect of expression, we could say it is an emergent property, something that exists thanks to the combined working of its respective, pre-existing, basic components. But, it also takes on a life of its own insofar as it continues to maintain governance over structural integrity and metabolic efficacy for the lifetime of the cell.
On a purely physical, non-organic plane, the morphogenetic field behaves similarly to a magnetic field dictating the arrangement of a handful of iron particles thrown into its influence. The configuration is the emergent property. However, in this simple example, the final geometric shape is predictable, which cannot be said of emergent properties in general where unlike components work together to effect a pattern indeterminable from examining the parts.
"We can now see that the whole becomes not merely more, but very different from the sum of its parts." (Philip W. Anderson -- Nobel Laureate in Physics)
In some circles, consciousness is considered an emergent phenomenon. Is it not also top-down? Or rather, to use the current jargon -- supervenient? Suppose it were true that the nature of the cosmos is intrinsically one of consciousness the Mind -- seeking or creating organic units through which and by which to express itself. Negative feedback in constant tandem with positive amplification. The principle of evolution orchestrated by the conscious mind that is, in a very real sense, the universe at large.
New unpredictable emergent properties of a whole interacting with same of another, mutually disjoint, whole, synergistically can produce yet another set of emergent properties equally unpredictable and unprecedented. In fact, the unpredictability factor increases exponentially. How then could we reconstruct the universe from knowing a handful of fundamental laws? Arthur Koestler said as much.
In mathematics: all complex groups are constructed from the products of simple groups. And by simple is meant nondivisible, irreducible, prime -- a whole unto itself.
For a genuinely interesting and clearly written article on emergent phenomena, among other related concepts, check this out, especially if you have an interest in molecular biology. The primary author, of which there are three, is Professor Emeritus Bernard Testa