Multiplicative numerical

When it is required to indicate the number of entire groups of atoms, the multiplicative numerals bis-, tris-, tetrakis-, pentakis-, and so on, are used (i.e., -kis is added starting from tetra-). The entity to which they refer is placed in parentheses. 

The combination of core and terminal unit names resembles conjunctive nomenclature. The multiplicity of branching (cascading) from the core unit is indicated by a bracketed numeral immediately following the name of the core unit if locants are necessary, they are also enclosed within the brackets, following and separated by a hyphen from the multiplicity numeral. 

In the numerical integration, instead of the boundary condition f(r)+ —> oo) = 1 a further initial condition f"(r]+ = 0) = const = c0 could be introduced. However this would require multiple estimations of c0, until the condition f (ri+ —> oo) = 1 is satisfied. The multiple numerical solutions can be avoided if the boundary value problem is traced back to an initial value problem. This is possible in a simple manner, as (3.183) remains invariant through the transformation f i)+) = cf(rj+) with fj+ = r)+ /c. This means it is transformed into an equation... 

The multiple numerical integration of system (2.55) on a computer for various initial conditions gives the dependence of on the initial parameters, and the cross section and microscopic rate constants are found fh>m formulae (2.8) and (2.21). 

Proportions of the various parts are expressed by Greek numerical prefixes but there are extremely important qualifications that mono (for unity) is usually omitted and that other numerical prefixes may also be omitted if no ambiguity results. Multiplicative numerical prefixes (bis, tris, tetrakis, etc,) are used when followed directly by another numerical prefix and may be used whenever ambiguity might otherwise be caused and prefixes may be delimited by parentheses to aid clarity further (examples are on pp. 28, 29). The terminal a of tetra, penta, etc., has normally been elided in English before another vowel in inorganic chemistry, but this is now expressly forbidden by the lUPAC 1965 revision. 

Finally, by volumetric imaging Three-dimensional information was obtained by stacking reflection tomograms from multiple planes. Using this stacking technique, cubic voxels were obtained and could be numerically dissected in any plane. Although there are several attractive features related to this technique, there are also several questions which need to be addressed before it can be used for industrial applications. For example, the applied sound field must be further characterized. 

The inner multiple integral is the transition state s density of states at energy , and also the numerator in... 

Venkatesh P K, Dean A M, Cohen M H and Carr R W 1999 Master equation analysis of intermolecular energy transfer in multiple-well, multiple-channel unimolecular reactions. II. Numerical methods and application to the mechanism of the C. + O2 reaction J. Chem. Phys. Ill 8313... 

The heightened appreciation of resonance problems, in particular, has been quite recent [63, 62], and contrasts the more systematic error associated with numerical stability that grows systematically with the discretization size. Ironically, resonance artifacts are worse in the modern impulse multiple-timestep methods, formulated to be symplectic and reversible the earlier extrapolative variants were abandoned due to energy drifts. 

T. R. Littell, R. D. Skeel, and M. Zhang. Error analysis of symplectic multiple time stepping. SIAM J. Numer. Anal., 34 1792-1807, 1997. 

Computational issues that are pertinent in MD simulations are time complexity of the force calculations and the accuracy of the particle trajectories including other necessary quantitative measures. These two issues overwhelm computational scientists in several ways. MD simulations are done for long time periods and since numerical integration techniques involve discretization errors and stability restrictions which when not put in check, may corrupt the numerical solutions in such a way that they do not have any meaning and therefore, no useful inferences can be drawn from them. Different strategies such as globally stable numerical integrators and multiple time steps implementations have been used in this respect (see [27, 31]). 

In order to obtain the fragmental constants, sets of molecules for which experimental log P values are available are cut into the predefined fragments and the numerical contribution of each fragment is determined by multiple hnear regression analysis. 

Table 7.1 presents us with something of a dilemma. We would obviously desire to explore i much of the phase space as possible but this may be compromised by the need for a sma time step. One possible approach is to use a multiple time step method. The underlyir rationale is that certain interactions evolve more rapidly with rime than other interaction The twin-range method (Section 6.7.1) is a crude type of multiple time step approach, i that interactions involving atoms between the lower and upper cutoff distance remai constant and change only when the neighbour list is updated. However, this approac can lead to an accumulation of numerical errors in calculated properties. A more soph sticated approach is to approximate the forces due to these atoms using a Taylor seri< expansion [Streett et al. 1978] ... 

One substitutes integral numbers (guesses) for z (oxygen) and y (nitrogen) until the divisor becomes an integral multiple of the numerator within 0.0002 mass unit. 

The multiple use of logarithms in the analysis presented by Fig. 4.9 obliterates much of the deviation between theory and experiment. More stringent tests can be performed by other numerical methods. 

Herbicides are also sometimes classified according to mode of action, selectivity, registered uses, and toxicity. The ever-increasing importance of herbicides and other pesticides and agrochemicals to a wide range of users, regulators, and researchers has led to the development of multiple and extensive computer databases. The primary database resources contain collected information relevant to herbicides, and numerous resource pubHcations are available to those needing information on the various aspects of herbicides (2). 

Triple-distiUed mercury is of highest purity, commanding premium prices. It is produced from primary and secondary mercury by numerous methods, including mechanical filtering, chemical and air oxidation of impurities, drying (qv), electrolysis, and most commonly multiple distillation. 

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