Geometric measure theory

Federer, H. (1969). Geometric measure theory. Springer, New York. 

In our earlier work, we have shown that the surface area in Eq. 12.1 can be evaluated via a two-dimensional (2D) integral for arbitrarily shaped molecules [123,124]. For variation purposes, the total free functional must be set up as a 3D integral in M. To this end, we take advantage of geometric measure theory by considering the mean surface area [74] and the coarea formula [151] ... 

More recently new mathematical techniques of differential geometry have been developed which required the use of currents, varifolds, and geometric measure theory. They are being applied with some success to the Plateau problem, the determination of the minimum area contained by a boundary. 

Noncrossing rule, geometric phase theory, 2 Nondemolition measurements, phase interference, 207... 

Interfadal tension between two fluid phases is a definite and accurately measurable property depending on the properties of both phases. Also, the contact angle, depending now on the properties of the three phases, is an accurately measurable property. Experimental approaches are described, e.g., in Refs. 8,60, and 63 and in Ref. 62, where especially detailed discussion of the Wilhehny technique is presented. Theories such as harmonic mean theory, geometric mean theory, and acid base theory (reviewed, e.g., in Refs. 8, 20, and 64) allow calculation of the soHd surface energy (because it is difficult to directly measure) from the contact angle measurements with selected test liquids with known surface tension values. These theories require introduction of polar and dispersive components of the surface free energy. 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

The molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

Fractal theory is a relatively new field of geometry, formulated by Mandelbrot [196] for irregular rough-surfaced objects. The major properties of such objects are the dependence of the measured length (perimeter), surface, or volume on the scale of measurement and geometrical self-similarity... 

It has been mentioned above that with nonuniform catalyst surfaces, as they mostly occur in practice, the above equations give merely an upper limit of the velocity although in such catalysts the true surface would be expected to be greater than the geometrical one. In nearly all the cases where a reaction has been measured on different catalysts, it has been found that fco is by no means a universal constant as would have to be expected from the simple theory. There is a regular connection between fco and the activation energy, of the general form ... 

Mott transition, 25 170-172 paramagnetic states, 25 148-161, 165-169 continuum model, 25 159-161 ESR. studies, 25 152-157 multistate model, 25 159 optical spectra, 25 157-159 and solvated electrons, 25 138-142 quantitative theory, 25 138-142 spin-equilibria complexes, 32 2-3, see also specific complex four-coordinated d type, 32 2 implications, 32 43-44 excited states, 32 47-48 porphyrins and heme proteins, 32 48-49 electron transfer, 32 45-46 race-mization and isomerization, 32 44—45 substitution, 32 46 in solid state, 32 36-39 lifetime limits, 32 37-38 measured rates, 32 38-39 in solution, 32 22-36 static properties electronic spectra, 32 12-13 geometric structure, 32 6-11 magnetic susceptibility, 32 4-6 vibrational spectra, 32 13 summary and interpretation... 

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