System Theory and Physical Systems  

In General
$ System: A relation SX × Y for some two sets X = ×iI Vi (input set) and Y = ×iI Wi (output set).
$ State set and response function: A pair C, R with R: (C × X) → Y and

(x, y) ∈ S  iff  there exists  cC  such that  R(c, x) = y .

* Conditions: Any system has a C and R; A state set can be defined, e.g., by the set of functions

C:= {fC | fc: XY, fcS} .

$ Composition of systems: Given two systems, S1 ⊂ (X1 × Z21) × (Y1 × Z12) and S2 ⊂ (X2 × Z12) × (Y2 × Z21), the system S = S1S2 ⊂ (X1 × X2) × (Y1 × Y2) is given by

S:= { ((x1, x2), (y1, y2)) | ∃ z1Z12, z2Z21, such that ((x1, z2), (y1, z1)) ∈ S1, ((x2, z1), (y2, z2)) ∈ S2} .

* Symmetry: Given a group G acting on X × Y, a system S is G-symmetric iff

for all gG, xX, yY,   (gx, gy) ∈ S iff (x, y) ∈ S .

@ References: Smullyan 61; Mesarović & Takahara 75.
> Online resources: see Wikipedia page.

Systems in Physics > s.a. classical systems; Emergent Systems; Isolated Systems; Open Systems; state of a system.
* Ingredients: A physical system has a state, including internal and external correlations, and internal and external interactions.
@ General references: Szabó IJTP(86) [and elementary objects]; Aerts & Pulmannová JMP(06)-a0811 [state property systems]; Lee & Hoban EPTCS(16)-a1606 [information content, communication complexity].
@ Dimensionality of a system: Wolf & Pérez-García PRL(09)-a0902 [quantum systems, from evolution]; Gallego et al PRL(10)-a1010, Hendrych et al a1111 [tests].
@ In quantum theory: Dugić & Jeknić IJTP(06)qp/05 [and decoherence theory]; Dugić & Jeknić-Dugić IJTP(08)qp/06 [information-theoretic arguments]; Seidewitz FP(11)-a1002 [systems, subsystems, and their interactions]; > s.a. quantum foundations [ontology]; quantum systems.


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