Inhomogeneity, time

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

Analogous considerations apply to spatially distributed reacting media where diffusion is tire only mechanism for mixing chemical species. Under equilibrium conditions any inhomogeneity in tire system will be removed by diffusion and tire system will relax to a state where chemical concentrations are unifonn tliroughout tire medium. However, under non-equilibrium conditions chemical patterns can fonn. These patterns may be regular, stationary variations of high and low chemical concentrations in space or may take tire fonn of time-dependent stmctures where chemical concentrations vary in botli space and time witli complex or chaotic fonns. 

Because the time at high temperature is much less, austenite is produced, which is chemically inhomogeneous especially with undissolved carbides, and has a fine grain crystal size. The formation of the hard martensite requites more rapid cooling than for conventional hardening. Thus case hardening by heat treatment intrinsically requites that the surface region to be hardened be relatively thin and cooled rapidly. 

Selection of appropriate time intei vals for increment extractions relates to property variation (inhomogeneity) within material flow streams. Ten minute extraction intei vals are generally adequate to obtain suitably representative samples from material flows under practical circumstances. Precise determination of extraction intei vals consistent with individual apphcations can be calculatedthrough autocorrelation of historical sampling data, a statistical method described in references (Gy, Pitard). 

There are advantages to direct solid sampling. Sample preparation is less time consuming and less prone to contamination, and the analysis of microsamples is more straightforward. However, calibration may be more difficult than with solution samples, requiring standards that are matched more closely to the sample. Precision is typically 5% to 10% because of sample inhomogeneity and variations in the sample vaporization step. 

Laterally inhomogeneous films and patterned structures of microelectronic and optoelectronic applications require small measuring spots. Today s measurements in 50 pm X 50 pm areas are standard for p-spot spectroscopic ellipsometers used in fa-blines. Areas more than ten times smaller can be analyzed by use of discrete-wave-length ellipsometers equipped with laser-light sources. 

In contrast, because of the spatially variable (inhomogeneous) nature of material in a composite stiffener, the bending stiffness cannot be separated into a material factor times a geometric term as in Equation (7.6). Instead, the composite stiffener bending stiffness is... 

Note that the scalar product is formally the same as in the nonrela-tivistic case it is, however, now required to be invariant under all orthochronous inhomogeneous Lorentz transformations. The requirement of invariance under orthochronous inhomogeneous Lorentz transformations stems of course from the homogeneity and isotropy of space-time, send corresponds to the assertion that all origins and orientation of the four-dimensional space time manifold are fully equivalent for the description of physical phenomena. 

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