Iterated function systems

Barnsley, M.F., and S. Demko. 1985. Iterated function systems and the global construction of fractals. Proc. R. Soc. London Ser. A 399 243-275. 

Iterated Function Systems. Iterated function systems make it possible to re-create natural objects using mathematical descriptions collected into data sets with accompanying rules for computation. This process is also known as fractal image compression, and it is the technology that makes computer-generated landscapes and visual effects in film possible. Very simply, a table of numbers (or matrix formulation) is created describing the affine transformations desired. These follow the conventional order of scalings, reflections, rotations, and translations. 

The Feedback Loop. The iterated feedback loop is a fundamental concept of fractal geometry. A feedback loop is a transformational system where the input of a cycle (or mathematical function) gives a different output, which becomes the input of a new transformation. Feedback processes are fundamental tools for studying natural systems. Examples of iterated mathematical systems can be found in history... 

In the final version the arrows and the keys are changed according to Fig. 4.3. The essential input for the definition of the vehicle system (ITEM) is the functional concept. Since ISO 26262 does not cover the hazards, which result from a correct functioning system, the functional concept itself is required to completely describe the free from danger of intended function and its structure. This won t be possible at an early stage of the product development. Therefore, we are forced to see all activities as continuous iterations for which the obtained insights have to be proven... 

In the first iteration the system requirements, design limitations, architectural assumptions and other constraints should be derived to functional or logical elements of the electronic. Architecture assumptions for the electronic, which are in line with the system limits (see Fig. 4.72) are the basis for this break-down. Base on new insights of other impacts specification of the elements gets more and more mature in further iterations. During those iterations and their verifications, the functional requirements and also the requirements for mechanical devices such as plugs, housing, circuit board, and fuses etc. have to become sufficient consistent. 

The first central issue in functional analysis is then Starting out with a complex requirements definition document (RDD) (i.e. with numerous and interacting requirements on the functionality), how do we select a set of functional elements and their interactions such that the resulting functional system satisfies the RDD We could try to pick a subset from a set of previously used elements and let them interact in a particular fashion. But we are faced with exactly the same problem as we encountered with the bottom-up approach to design in the physical domain, as discussed in Sec. B5.2. It is unlikely that the emergent behaviour of the functional system will meet all the requirements in the RDD. It will require a number of iterations to achieve that, and even then we have no way of knowing and demonstrating that we have found the best solution (the second central issue in functional analysis, to be discussed in the next section). 

It is important to stress that unnecessary thermodynamic function evaluations must be avoided in equilibrium separation calculations. Thus, for example, in an adiabatic vapor-liquid flash, no attempt should be made iteratively to correct compositions (and K s) at current estimates of T and a before proceeding with the Newton-Raphson iteration. Similarly, in liquid-liquid separations, iterations on phase compositions at the current estimate of phase ratio (a)r or at some estimate of the conjugate phase composition, are almost always counterproductive. Each thermodynamic function evaluation (set of K ) should be used to improve estimates of all variables in the system. 

In his early survey of computer experiments in materials science , Beeler (1970), in the book chapter already cited, divides such experiments into four categories. One is the Monte Carlo approach. The second is the dynamic approach (today usually named molecular dynamics), in which a finite system of N particles (usually atoms) is treated by setting up 3A equations of motion which are coupled through an assumed two-body potential, and the set of 3A differential equations is then solved numerically on a computer to give the space trajectories and velocities of all particles as function of successive time steps. The third is what Beeler called the variational approach, used to establish equilibrium configurations of atoms in (for instance) a crystal dislocation and also to establish what happens to the atoms when the defect moves each atom is moved in turn, one at a time, in a self-consistent iterative process, until the total energy of the system is minimised. The fourth category of computer experiment is what Beeler called a pattern development... 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

Now that the values of Pp and n have been determined, we can model the system as described by Section 11.2. Wfith the proper cost functions, die total annualized cost TAC of the system can be evaluated for this iteration. Next, a new value of Cp is selected and steps 2-10 are repeated. The TAC of the system can be plotted vs. Cp (or any of the 10 other variables) to determine the minimum TAC of the system. 

Establishing a calibration function with one single broad distributed sample is an alternative to traditional peak postion calibration of SEC systems with a set of narrow distributed standards. An obvious advantage of this technique is time for peak position calibration elution profiles for the set of standards need to be determined for broad standard calibration the elution profile of one sample needs to be determined only. Establishing a linear calibration function with a broad distributed standard includes startup information [M (true), Mn(true)] and an iterative (repeat.. . until) algorithm ... 

It may happen that many steps are needed before this iteration process converges, and the repeated numerical solution of Eqs. III.21 and III.18 becomes then a very tedious affair. In such a case, it is usually better to try to plot the approximate eigenvalue E(rj) as a function of the scale factor rj, particularly since one can use the value of the derivative BE/Brj, too. The linear system (Eq. III. 19) may be written in matrix form HC = EC and from this and the normalization condition Ct C = 1 follows... 

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