Complexity order

A remarkable, but (at first sight, at least) naively unimpressive, feature of this rule is its class c4-like ability to give rise to complex ordered patterns out of an initially disordered state, or primordial soup. In figure 3.65, for example, which provides a few snapshot views of the evolution of four different random initial states taken during the first 50 iterations, we see evidence of the same typically class c4-like behavior that we have already seen so much of in one-dimensional systems. What distinguishes this system from all of the previous ones that we have studied, however, and makes this rule truly remarkable, is that Life has been proven to be capable of universal computation. 

Kinetic measurements, particularly the study of the rate-dependence on duster concentration can be very informative cluster-catalyzed reactions often display a first-order rate dependence on duster concentration, whereas fractional or complex orders of reaction are associated with fragmentation processes. 

The F -ion conductor first discovered by Faraday represents a more complex order-disorder transition to fast ionic conduction. At all temperatures, PbF2 is reported to have the fluorite structure in which the F ions occupy all the tetrahedral sites of a face-centred-cubic Pb -ion array however, the site potential of the Pb ions is asymmetric, and a measurement of the charge density with increasing temperature indicates that the F ions spend an increasing percentage of the time at the... 

Living systems are complex, ordered systems. This complexity and order is reflected in the molecules characteristic of life, in their interactions with each other, in the regulatory mechanisms that result from these interactions, and in the complex supramolecular structures characteristic of cells. Organization is also reflected in ordered metabolic and signaling pathways. Such complex, ordered structures and pathways are not characteristic of inanimate objects. 

It will be interesting to see how such treatments based on a sub-lattice model can be made more general. So far it has only been used for quite simple systems with ordered structures such as LI2 and LIq. It may be necessary to include more than four sub-lattices for complex ordered phases which are superstructures of these types, and to consider more than 1st or 2nd neighbour energies. Furthermore, the choice of clusters for the sro part must relate back to the sub-lattice model itself and it is difficult to see how the more complex clusters routinely handled by CVM models can be reconciled with the sub-lattice models used so far. 

Brad P. comments, Both the compnter chip and snow crystal are beantifnl because of their complexity, order and diversity—all within a small package. Both say volumes about the beauty and intelligence of their creators. ... 

The conversion from a connection table to other unambiguous representations is substantially more difficult. The connection table is the least structured representation and incorporates no concepts of chemical significance beyond the list of atoms, bonds, and connections. A complex set of rules must be applied in order to derive nomenclature and linear notation representations. To translate from these more structured representations to a connection table requires primarily the interpretation of symbols and syntax. The opposite conversion, from the connection table to linear notation, nomenclature, or coordinate representation first requires the detailed analysis of the connection table to identify appropriate substructural units. The complex ordering rules of the nomenclature or notation system or the esthetic rules for graphic display are then applied to derive the desired representation. 

In 1962, B. Josephson recognized the implications of the complex order parameter for the dynamics of the superconductor, and in particular when one considers a system consisting of two bulk conductors connected by a weak link." This research led to tile development of a series of weak link devices commonly called Josephson junctions. See also Josephson Tunnel-Junction. These devices hold much promise for achieving ultra high-speed computers where switching time is of the order of 1CT11 second. 

A number of strongly complexing inorganic anions are present in natural waters, such as HCOj/CO2-, Cl-, SO4-, PO4-, and so on. The complexation order of these... 

In this section we derive the effective Hamiltonian which will be the starting point for our further treatment. The strategy of the calculation is therefore separated into two steps. In the first step the system is treated in a mean-field-(MF) type approximation applied to a microscopic Hamiltonian. This leaves us with a slowly varying complex order parameter field for which we derive an effective Hamiltonian. The second step involves the consideration of the fluctuations of this order parameter. 

An alternative method is to start directly from the Ginzburg-Landau approach [12]. In this method, one introduces (in general complex) order parameters as in Table 8, and expands the free energy in powers of the order parameters. For real order parameters and up to quartic terms one obtains... 

Rule 4. Name the side chains by approved radical names and precede these with the correct position numbers. For the acyclics, the complexity order and the alphabetical order of prefixes will be identical, since only methyl and methylene side chains are involved — for example, 2,6-dimethyl in geraniolene (Nos. 1A and IB, Chart 3) and 7-methyl-3-methylene in myrcene (No. 2, Chart 3). 

Rule 6, To form the complete name of a compound, arrange the names of the substituents in the alphabetical or complexity order (in Chart 6 the Chemical Abstracts alphabetical order has been followed) then prefix this part of the name to the correct cyclo part of the name. Such a systematic cyclohexane-type name forms a single word. For example /S-cyclogeraniolene (Formula 31, Chart 6) is named... 

J. Thompson, A.R. Barnett, COULCC A continued-fraction algorithm for Coulomb functions of complex order with complex arguments, Comput. Phys. Comm. 36 (1985) 363. 

In the spirit of our model, two order parameters play a role the nematic tensorial order parameter Q,j and the smectic A complex order parameter practical reason we use the director fl and the modulus sW in the uniaxial nematic case... 

The rate of reaction can be of simple order, either first, second or non-integral, but it can also be complex. A complex order always means a complex mechanism, but a simple order need not mean a simple mechanism. 

In the macroscopic and phenomenological Ginzburg109-Landau description of superconductivity, a complex "order parameter" P(r) = j/(r) exp(i) is proposed, which equals zero above Tc and whose magnitude determines... 

Theoretically, the discussion must turn on the relative energies of complex ordered structures and of defective, random structures. Bertaut (5) (1953) attempted to compute the electrostatic lattice energy of the pyrrhotite phase FeSj.,4 (Fe+2o. 25, Fe+30.25 n + ) (S 2 n ) or (Fe+25, Fe+3, ) (S-28),for several alternative cases a completely disordered structure with higher-valent cations and vacancies randomly distributed over all cation sites and structures with the vacancies ordered into alternate cation sheets, with various hypotheses... 

The random solution of defects caused by lattice dissociation and nonstoichiometry has been discussed for CeCd 4 5. In addition to this random solution in an otherwise ordered lattice, these defects can themselves order to create very many new and complex ordered lattices which we have named microphases. Structures with very similar properties appear at cadmium concentrations both greater and less than that of CeCd4 5 if the CeCd4 5 structure were the common base of all the microphases, discontinuities would be expected at CeCd4>5 where the composition shifted from excess cadmium to deficient cadmium. CeCd4, however, is a reasonable common base. The microphases are the subject of another paper (6). 

The RPA can be improved on by the Landau-Ginzburg (LG) formalism [47] appropriate in a quasistatic regime. One introduces a complex order parameter i[f( ) (dimensions of energy) associated with Apld(jc), which can also be related to the amplitude of the lattice distortion [Eqs. (4 and 5)] qt oc e,2fc, vjf(jtj) + e 2kF i (xi). It is complex because the phase of the CDW or BOW at +2kF is independent of the one at -2kF. TTie partition function is expressed as a functional integral weighing all fluctuations in the order parameter Z = J3)i ie-p/w, where the free-energy functional is... 

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