Quasi-continuous variables

Simultaneously, the probability distribution over the C quasi-continuous variables x = Xat ... 

Continuous distribution functions Some experiments, such as liquid chromatography or mass spectrometry, allow for the determination of continuous or quasi-continuous distribution functions, which are readily obtained by a transition from the discrete property variable X to the continuous variable X and the replacement of the discrete statistical weights g, by the continuous probability density g(X). For simplicity, we assume g(X) as being normalized J ° g(X)dX = 1. Averages and moments of a quantity Y(X) are defined by analogy to the discrete case as... 

The only pieces of hardware needed for photo-CIDNP are a light source and an unmodified NMR spectrometer. Pulsed lasers are most convenient for illumination, as they allow both time-resolved experiments (when the laser flash is followed by an acquisition pulse after a variable time delay) and steady-state ones (when the laser is triggered with a high repetition rate, thus providing quasi-continuous excitation). All the examples of this work draw on the second variant. Nevertheless, they yield kinetic information about much faster processes than would be observable by direct... 

Consider now a quasi-two dimensional crystal of finite thickness. The basic cell vector a, perpendicular to the surface is chosen equal to this thickness. This crystal is handled by setting = 1. The diffraction is then still sharply peaked in both directions parallel to the surface, but the Laue condition on Qj (= Q ) is relaxed, and the intensity is continuous in the out-of-plane direction the reciprocal space is made of rods perpendicular to the surface plane. If we still define / by Q.a, = 2M, I is now taken as a continuous variable since intensity is present for non-integer values of /. The intensity is now given by ... 

In the thermodynamic limit (i.e., as n — oo) it is convenient to replace the discrete variables n by their (quasi-) continuous counterparts p — n ln so that the double sums can be replaced by double integrals,... 

The presence of non-interacting components in a macromrriecular system of different molecular weight or density, or both. Quasi-continuous distribution of mucin molecular weights arising from variability in carbohydrate side-chain composition. 

Transformations between types of fluid phase behavior is closely related to the so-called Tamily concept, originally introduced by Schneider [8,9]. A transformation of type-II throu type-IV to type-III fluid phase behavior is known to occur, for example, for the series of binary systems CO2 + alkane with increasing carbon number of the alkane. The system CO2 + dodecane shows type-II [10], the system CO2 + tridecane belong to type-IV [11] and the system CO2 + tetradecane shows type-III fluid phase behavior [22]. A summary of all binary CEP data for the systems CO2 + alkane can be found in Miller and Luks [12]. A quasi-binary investigation of the system CO2 + alkane by de Loos et al. [13] indicated the occurrence of one TCP and one DCEP. Stamoulis [14] estimated the TCP at N p = 12.33 and Ttcp = 317.5 K, and the DCEP at Ndcep = 13.55 and Tdcep 296.0 K. Note, that in this theoretical (quasi-binary) approach the carbon number of the alkane does not necessarily have to take integer values, but can be considered as a continuous variable. 

In the following we shall call the function gj(eO) the density-of-sites function. As the energy spectrum is a quasi-continuum (for a disordered system) we can treat e as a continuous variable and drop the index j in eqs. 16 and 17. The thermodynamic properties of the disordered metal-hydrogen system can then be deduced from the following expression for the total fraction of occupied sites. 

Palakodaty and coworkers (15) used such cosolvent distfibution in defining the process variables for crystallizing lactose monohydrate from aqueous media in a quasi-continuous operation and reported the effect of methanol and ethanol as cosolvents on the crystallized product. The flow conditions of different... 

Under dynamic or quasi-steady-state conditions, a continuously monitored process will reveal changes in the operating conditions. When the process is sampled regularly, at discrete periods of time, then along with the spatial redundancy previously defined, we will have temporal redundancy. If the estimation methods presented in the previous chapters were used, the estimates of the desired process variables calculated for two different times, t and t2, are obtained independently, that is, no previous information is used in the generation of estimates for other times. In other words, temporal redundancy is ignored and past information is discarded. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

HCV shows pronounced genetic variability. The cause lies in the fact that HCV mutates swiftly and continuously under immune pressure, so that several variants result (known as quasi species). The mutation rate is about 10 times higher than that of HBV. Worldwide, different virus isolates have been found and characterized in great detail. The genes for the envelope proteins El and E2 seem to be particularly variable. Up to now, 6 genotypes and 13 subtypes have been distin-... 

A system of molecules with various conformational isomers is a mixture from the point of view of phenonwnological thermodynamics. The composition cannot, however, be specified because the components of the mixture, i.e. the physically distinguishable conformational isomers, are continuously interconverting in a kind of quasi-chemical reaction that depends on the thermal agitation. The conformational isomerism gives the system internal d ees of freedom that are described phenomenologically by so-called internal variables. The mean mass concentrations or mole fractions of the conformational isomers can, for example, be chosen as internal variables. If the conformational degrees of freedom are coupled, linear combinations of these variables are determinant. Their composition may depend on external ables such as temperature, pressure, mechanical stress, etc. 

We denote AG per unit volume of solution by AG. Thermodynamics tells us that AG of a quasi-binary solution is a function of T, p, and 4>i. For a given polymer species it also may depend on the distribution of relative chain length in the polymer mij re. Thus, when the pressure effect is not considered, the basic variables for AG of quasi-binaiy solutions are T, , and /(P) as the basic variables. For a paucidisperse polymer, q is finite and /(P) is represented by q delta functions with different strength, while for a truly poly-disperse polymer, q is infinitely large and /(P) becomes a continuous function of P. Thus, AG for quasi-binary solutions of a truly polydisperse polymer is a functional with respect to /(P), and requites sophisticated mathematics for its treatment. 

Thermodynamics is the study of equilibrium at a macroscopic level. When a system is in mechanical equilibrium, there is no net force imbalance that causes motion. Complete thermodynamic equilibrium is more extensive and requires not only mechanical equilibrium but also thermal, phase, and chemical equihbrium. We can use classical thermodynamics to analyze chemically reacting and nonequiUbrium flows, such as those in fuel cells, but are restricted to only the quasi-equilibrium beginning and end states of the process, with no details of the reaction itself. Thermodynamics can tell us the potential for reaction and direction of spontaneous reaction, but not how fast the reaction will occur. Classical thermodynamics also assumes a continuous fluid, meaning that there are enough molecules of a substance to yield accurate values of thermodynamic variables like pressure and temperature. As such, classical thermodynamics is generally inappropriate for use with microscopic-level molecular charge transfer processes and electrochemical reactions. 

---