Numerical breakdown

Equation 1.3 represents a system of usually several thousand coupled differential equations of second order. It can be solved only numerically in small time steps At via finite-difference methods [16]. There always the situation at t + At is calculated from the situation at t. Considering the very fast oscillations of covalent bonds, At must not be longer than about 1 fs to avoid numerical breakdown connected with problems with energy conservation. This condition imposes a limit of the typical maximum simulation time that for the above-mentioned system sizes is of the order of several ns. The limited possible size of atomistic polymer packing models (cf. above) together with this simulation time limitation also set certain limits for the structures and processes that can be reasonably simulated. Furthermore, the limited model size demands the application of periodic boundary conditions to avoid extreme surface effects. 

The percentage of word-oriented databases continues to increase at a faster pace than numeric databases. The number of image databases was 358 in 1992, greater than twentyfold increase from 1989. Audio databases rose from 1 in 1988 to 109 in 1992. A breakdown of the subclasses within word-oriented databases is given in Table 3, showing that full-text databases have surpassed bibhographic ones. Directory databases are the third most numerous. 

Bnich-kupfer, n. scrap copper, -last, /. breaking load, -metall, n. broken metal, scrap metal, -modul, m. modulus of rupture, -probe, /. breaking test, breakdown test, -punkt, m. breaking point, -riss, m. (Meial.) failure crack, -silber, n. broken silver, scrap silver, -spaonung,/. breaking stress tensile strength, -stein, m. quarry stone broken stone, -stelle,/. broken place, place of fracture. -strich, m. (Math.) fraction stroke (between numerator and denominator), -stiick, n. fragment shred, -stiicke, pi. debris scrap, -teil, m. fraction, -zahl, /. fractional number. 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

Examples of reversible breakdown of structure have been reported for procaine penicillin dispersions (7), for model systems of calcium carbonate in polybutene ( ), and for numerous other systems. During shear the particles are forced into contact with each other with sufficient kinetic energy to overcome any natural barrier against their displacement of a lyosphere around each individual particle. A dispersion which is inherently stable can thus be forced by shear into a condition of instability. 

Proteinase Any one of numerous enzymes that catalyze the breakdown of proteins also called protease. 

The plant is known to contain chelerythrine chloride, which inhibits the aggregation of rabbit platelet in vitro via inhibition on thromboxane formation and phosphoinosi-tides breakdown (30). Chelerythrine, which occurs in members of the family Papaver-aceae, has been reported to inhibit the enzymatic activity of protein kinase C and to exert cell-growth inhibitory effect via the induction of apoptosis in numerous cancer cell lines (31,32). What is the topoisomerase activity of chelerythrine ... 

The conclusions from this rather elementary survey of the symmetry constraint problem all point in the same general direction. The imposition of symmetry constraints (other than the Pauli principle) on a variationally-based model is either unnecessary or harmful. Far from being necessary to ensure the physical reality of the wave function, these constraints often lead to absurd results or numerical instabilities in the implementation. The spin eigenfunction constraint is only realistic when the electrons are in close proximity and in such cases comes out of the UHF calculation automatically. The imposition of molecular spatial symmetry on the AO basis is not necessary if that basis has been chosen carefully — i.e. is near optimum. Further, any breakdowns in the spatial symmetry of the AO basis are a useful indication that the basis has been chosen badly or is redundant. 

In each class the problem may be resolved into two essential parts (i) the breakdown of the organic compound under appropriate conditions to give a quantitative yield of fluoride ions in aqueous solution, and (ii) the determination of the concentration of these fluoride ions. Methods of breaking down the organic compounds were examined and the procedure adopted for the phosphorofluoridate was different from that used for the fluoroacetate series. From both, however, sodium fluoride was obtained as the breakdown product containing all the fluorine present. After numerous preliminary experiments we came to the conclusion that on the macro-scale a very convenient method of determining the quantity of fluoride ions in the products was by precipitation as lead chlorofluoride,2 PbCIF, which was then dissolved in dilute nitric acid and the chloride was determined by the Volhard method and calculated to the equivalent amount of fluorine. We determined carefully the conditions for the quantitative precipitation of lead chlorofluoride. 

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