Microscopic cutoff

The crossover equation contains the parameters Ct Cp, iZ, and A, but with two constrains (aq = 6 and uq = 9) imposed by the classical van der Waals equation. Hence, there still remain two microscopic parameters that cannot be specified on the basis of the classical equation the parameter A, which represents a microscopic cutoff wave number made dimensionless by dividing with and the parameter q = which is inversely... 

The derivation of the correlation for the excess viscosity A (p, T) from experimental data in general is much easier, since for the viscosity the critical enhancement is just restricted to a very narrow region around the critical point. After the correlations for the excess thermal conductivity and the excess viscosity have been determined, the coincidence of the theoretical results calculated for the thermal conductivity in the critical region with those obtained experimentally could be optimized by adjusting the value of the parameter a microscopic cutoff distance in the crossover function... 

In practice, however, all these numerical techniques use a discretization of space, real-space and pseudospectral methods additionally discretize the chain contour. Thereby, a microscopic cutoff is introduced via the numerical methodology. 

To relate the instantaneous densities of the ensemble of molecules to the external fields, one has to introduce a microscopic cutoff. Either one utilizes interactions with a finite range in Eqs. (5.21) and (5.24) or one regularizes the 8-function that appear in Eq. (5.5) by a smoothing function of finite support. We wUl show that both techniques are equivalent. 

This shows that in general the one dimensional profile is unstable to perturbations of the contact line. A more detailed stability analysis has been performed in reference (37) which shows that the wavelength of the fastest growing mode is of the order of the size of the bump 1 and also varies slightly with the microscopic cutoff k of Tanner law (the thickness of the precursor film). 

Although somewhat arbitrary, the 0.45/rm "cutoff" between dissolved and particulate organic is for the most part convenient. For example, particles above about 1.0/im are observable with a microscope and tend to settle in seawater. Particles less than 1.0/rm are submicroscopic and generally sink very slowly and disperse as a result of Brownian motion. In addition, particles less than 0.45 fall below the range of most living... 

Here, Amici is the microscopic area which is defined within the fractal limits y, the dimensionless geometrical parameter X0, the spatial outer (upper) cutoff /fmacr, the macroscopic area which is defined beyond the fractal limits (= Aex in Eq. 15) and Xi represents the spatial inner (lower) cutoff. 

Optical effects due to refraction and total internal reflection have been observed in dye-loaded zeolite L crystals of 2.5 p,m length and 1.4 p-m diameter by means of an optical microscope equipped with polarizers, a narrow band, and cutoff filters [23]. An astonishing effect taking place in an Ox +-loaded crystal is seen in Fig. 15. Looking at the polarized red emission, a homogeneous intensity distri-... 

In the context of the model presented above, the microscopic relaxation time of a water molecule is equal to the cutoff time of the scaling in time domain To-For the most hydrophilic polymer, PVA, the strong interaction between the polymer and the water molecule results in the greatest value of microscopic relaxation time To, only 10% less than the macroscopic relaxation time of the bulk water. The most hydrophobic polymer, PVP, has the smallest value of a single water molecule microscopic relaxation time, which is almost equal to the microscopic relaxation time of bulk water (see Table III). Therefore, weakening the hydrophilic properties (or intensifying the hydrophobic properties) results in a decreasing of interaction between the water and the polymer and consequently in the decrease of To-... 

Fig. 4.10. First confocal epi-iUumination microscope with dichroic mirror and beam splitter with two RCA avalange photodiodes and identical cutoff filters. Building year 1988 and still in use...

The term NAcct> is multiplied by M to account for the difference between the NA on the object side of the microscope objective and the NA on the image side of the microscope objective. The sixth factor which influences the effective numerical aperture is the numerical aperture of microscope objective (NA cro)- These multiple NA terms can be interpreted as multiple low-pass spatial frequency filtering operations of the transverse light amplitude field. The effective spatial frequency is roughly limited by the lowest spatial cutoff frequency or the smallest NA term. Hence, the effective NA (AMeff) of the whole... 

In eq 10.82, u 0.472 is a universal coupling constant, u a scaled system-dependent coupling constant, and A a dimensionless cutoff wave number related to the microscopic (molecular) length scale = vq /tcA for the lattice gas Vo is the volume of the unit cell and for fluids vq is the molecular volume. As a measure of a distance from the critical point we consider a parameter k, related to the inverse correlation length, which in the classical limit is proportional to the square root of the inverse susceptibility x and which has the form ... 

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