Maximum or minimum value

Specifications. Activated carbon producers furnish product bulletins that Hst specifications, usually expressed as a maximum or minimum value, and typical properties for each grade produced. Standards helpful in setting purchasing specifications for granular and powdered activated carbon products have been pubHshed (33,34). 

Further, since values of x for the critical point are intermediate between those on the two curves, it follows that the critical temperature and the critical pressure are maximum or minimum values, and we shall assume they are the former. The inequalities r2 > ri, S2 > Si now lead to the following inequalities, referring to the immediate vicinity of the critical point ... 

The problem is to find values for the variables v to vn that optimise the objective function that give the maximum or minimum value, within the constraints. 

The nature of the relationships and constraints in most design problems is such that the use of analytical methods is not feasible. In these circumstances search methods, that require only that the objective function can be computed from arbitrary values of the independent variables, are used. For single variable problems, where the objective function is unimodal, the simplest approach is to calculate the value of the objective function at uniformly spaced values of the variable until a maximum (or minimum) value is obtained. Though this method is not the most efficient, it will not require excessive computing time for simple problems. Several more efficient search techniques have been developed, such as the method of the golden section see Boas (1963b) and Edgar and Himmelblau (2001). 

The best cure for a rubber product is always a compromise, but optimum cure may be defined as that time of cure necessary to bring a preselected property of the vulcanisate to near maximum (or minimum) value, at the same time ensuring that the other properties are satisfactory. All the physical properties of a rubber do not reach their optimum values at the same time of cure, and therefore the time must be selected so that the properties are near their optima, the most weight being given to the property considered most important. 

In the simplest case, where (+)-AH and (-)AD are isotopically pure, a = [a]H[AH]0 and a2 = [a]D[AD]0 where a is the specific rotation of the AH and AD isotopomers, respectively, and [AH]0 and [AD]0 are the concentrations of the substrates in g ml-1 at time t = 0. When the substrate is neither isotopically nor enantiomerically pure, corrections must be made in calculating fli and a2 (Bergson et al., 1977). It is important to note that the pre-exponential factors, a and a2, which contain the information about the starting conditions, can be determined with high accuracy. The extreme, ae (the maximum or minimum value of the optical rotation in the optical rotation versus time plot) and the corresponding reaction time, te, are functions of the rate constant ratio (5 = kHlkD) (65) and the difference between the rate constants (66), respectively. 

The Monte Carlo analyses are used to observe how device tolerances can affect a design. There are two analyses that can be performed. The Worst Case analysis is used to find the maximum or minimum value of a parameter given device tolerances. Device tolerances are varied to their maximum or minimum limits such that the maximum or minimum of the specified parameter is found. The Monte Carlo analysis is used to find production yield. If the Worst Case analysis shows that not all designs will pass a specific criterion, the Monte Carlo analysis can be used to estimate what percentage will pass. The Monte Carlo analysis varies device parameters within the specified tolerance. The analysis randomly picks a value for each device that has tolerance and simulates the circuit using the random values. A specified output can be observed. 

The Worst Case analysis determines the absolute maximum or minimum value of a parameter for given component tolerances. 

Let us consider now, however, a more realistic model. Products quality specifications are not generally given as absolute quantities, but rather in terms of maximum or minimum values. Thus the linear equations in Eq. (1) become the linear inequalities 5... 

We often encounter situations in the physical sciences where we need to establish at which value(s) of an independent variable a maximum or minimum value in the function occurs. For example ... 

When we explore the nature and form of these and other multi-variable functions, we need to know how to locate specific features, such as maximum or minimum values. Clearly, functions of two variables, such as in the ideal gas equation above, require plots in three dimensions to display all their features (such plots appear as surfaces). Derivatives of such functions with respect to one of these (independent) variables are easily found by treating all the other variables as constants and finding the partial derivative with respect to the single variable of interest. 

At any point x = a, the function /(x) has maximum or minimum value if the following two conditions are satisfied. 

On-off control is implemented by a switch that keeps the manipulated variable at its maximum (or minimum) value when the controlled variable is below (or above) its set point. In most practical applications, there is a narrow band that the error must exceed before a change will occur. This band is known as the differential gap. 

If the factor being optimized (Cr) does not attain a usable maximum or minimum value, the solution for the dependent variable will indicate this... 

For the n-dimensional case, the region that is defined by the set of hyperplanes resulting from the linear constraints represents a convex set of all points which satisfy the constraints of the problem. If this is a bounded set, the enclosed space is a convex polyhedron, and, for the case of monotonically increasing or decreasing values of the objective function, the maximum or minimum value of the objective function will always be associated with a vertex... 

At a true optimum, the response will have a maximum (or minimum) value. This means that there will be an extremum point on the response surface. To... 

A wide variety of problems can be solved by finding maximum or minimum values of functions. For example, suppose it is desired to maximize the area of a rectangle inscribed in a semicircle. The area of the rectangle is given by A = 2xy. The semicircle is given by x + y = r, for y > 0, where r is the radius. To simplify the mathematics, note that A and A are both maximum for the same values of x and y, which occurs when the comer of the rectangle intersects the semicircle, that is, when y = r - x. Thus, we must find a maximum value of the function A = 4x (r -x ) = 4r x - 4x". The required condition... 

There are numerous practical applications in which it is desired to find the maximum or minimum value of a particular quantity. Such applications exist in economics, business, and engineering. Many can be solved using the methods of differential calculus described above. For example, in any manufacturing business it is usually possible to express profit as a function of the number of units sold. Finding a maximum for this function represents a straightforward way of maximizing profits. In... 

DMAX or DMIN returns, respectively, the maximum or minimum value in the specified field of all records that match the criteria. 

Finding a localization transformation is usually done by satisfying a given optimum criterion. Some well-known optimum criteria can be summarized as searching for the maximum or minimum value of the functional... 

The curvature coefficient, although not readily interpretable numerically, can be used to estimate the factor level that gives the maximum or minimum value of the response, and this occurs at the factor level value X = -b (The maxi-... 

How do we define peaks, ridges, passes, and downhill directions in a rigorous mathematical sense We need to locate points at which the electron density is at a maximum or minimum value. This is accomplished by finding the critical points of the density, that is, the point r where Vp(r) = 0, where the gradient is defined in Eq. [30]. 

Condition (a) specifies a constant concentration of solute along the sediment-water interface and within the burrow core. Condition (b) requires pore-water solute concentrations to go through a maximum or minimum value half-way between any two burrows. The last condition, (c), matches the bioturbated zone to the underlying unburrowed zone by requiring continuity in flux across the lower boundary at depth L. 

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