Bound, defined

The difference between the bounds defined by the simple models can be large, so that more advanced theories are needed to predict the transverse modulus of unidirectional composites from the constituent properties and fiber volume fractions (1). The Halpia-Tsai equations (50) provide one example of these advanced theories ia which the rule of mixtures expressions for the extensional modulus and Poisson s ratio are complemented by the equation... 

Abgratschrott, m. trimmings, abgreifen, v.t. read off wear out. tbgrenzbar, a. definable, limitable. tbgrenzen, z.t. mark off bound define. Abgrenzung, /. demarcation, boundary. 

No doubt you are ready to complain that this generalization isn t really true Quite the opposite The generalization states a regularity discovered among all the observations available, and as long as observations are restricted to objects in the list, the generalization is applicable. A generalization is reliable within the bounds defined by the experiments that led to the rule. 

Bound. Define best upper bound fv = min, fVj and delete (fathom) all subregions with lower bounds/ >fu. If the remaining sub-regions satisfy/ >/[Pg.66]

There are some special but quite widespread cases where only lower or upper bounds are imposed. For the case where only lower bounds define the subregion, Equation 8.22 becomes... 

Upper and lower bounds defined in advance. In order to reach this condition, the sum of tlte upper bound for each factor plus the lower bounds for the remaining factors must not be greater than one, i.e. for three factors... 

Distance measures give 0 for identical structures and have an upper bound defined by the property space. The Euclidean and Hamming distances are the most common ... 

Feed composition. Demonstrate control of feed composition to maintain corrosion and transport of salts within the bounds defined by the preceding criteria. Real-time indicators of satisfactory feed composition include reactor temperature and pressure profiles and effluent pH and turbidity. 

The combination of the equality constraints, inequality constraints, and lower and upper bounds defines a feasible region. A feasible solution is one that satisfies the equality constraints, the inequality constraints, and the upper and lower bounds for a feasible set of decision variables. If the solution also minimizes (or maximizes) the objective function, it is a local optimal solution. Sometimes, other local optimal solutions exist in the feasible region, with one or more being a global optimal solution. 

Lower bound Defined wifh respect to a problem. A lower bound on the resources (time or space) needed to solve a specified problem has the property that the problem cannot be solved by any algorithm that uses less resources than the lower bound. 

Support for rule inheritance Creation of marketed intent will be defined within the bounds defined by the engineered intent. 

In all cases, data points lying outside the whisker limits are conventionally denoted by crosses or dots, often in another colour. Such points can be labelled as outliers. Of the available definitions, the most commonly encountered box-and-whisker plots use whisker bounds defined by the first two mles. Typical box-and-whisker plots are shown in Fig. 1.6. These box-and-whisker plots were created using the interquartile range for the data points. 

The use of isotopic substitution to detennine stmctures relies on the assumption that different isotopomers have the same stmcture. This is not nearly as reliable for Van der Waals complexes as for chemically bound molecules. In particular, substituting D for H in a hydride complex can often change the amplitudes of bending vibrations substantially under such circumstances, the idea that the complex has a single stmcture is no longer appropriate and it is necessary to think instead of motion on the complete potential energy surface a well defined equilibrium stmcture may still exist, but knowledge of it does not constitute an adequate description of the complex. 

Apparently, the most natural choice for the electronic basis functions consist of the adiabatic functions / and tli defined in the molecule-bound frame. By making use of the assumption that A" is a good quantum number, we can write the complete vibronic basis in the form... 

In contrast to the point charge model, which needs atom-centered charges from an external source (because of the geometry dependence of the charge distribution they cannot be parameterized and are often pre-calculated by quantum mechanics), the relatively few different bond dipoles are parameterized. An elegant way to calculate charges is by the use of so-called bond increments (Eq. (26)), which are defined as the charge contribution of each atom j bound to atom i. 

For an irreversible reaction Pj - 0 at the center of the pellet when the size of the pellet becomes very large. Thus p — YT p at the center of large pellets. Clearly, from (11.45), the pressure rises towards this value on moving into the pellet when n > 1 and falls to it when n < 1. Thus we can define the following bounds for the pressure... 

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