Value function

This geometry can be described by vector-valued function ... 

Figure B3.6.2. Local mterface position in a binary polymer blend. After averaging the interfacial profile over small lateral patches, the interface can be described by a single-valued function u r. (Monge representation). Thennal fluctuations of the local interface position are clearly visible. From Wemer et al [49].

Here a symmetric projection step is used to enforce conservation of energy. Let a(g,p) and b q,p) be two vector-valued functions such that (p a q,p) + U q) b q,p)) is bounded away from zero. Then we propose the following modified midpoint method,... 

We recall some definitions which are useful in the work to follow. The smallest a-algebra containing all compact sets in r 9r is called the Borel a-algebra (Landkof, 1966). Any a-additive real-valued function defined on the Borel a-algebra which is finite for all compact sets B c r 9r is called a measure on 9r. Thus, for a measure p and a set A, the a-additivity means... 

Whereas a linear relation between flow stress and lattice-parameter change is obeyed for any single solute element in nickel, the change in yield stress for various solutes in nickel is not a single-valued function of the lattice parameter, but depends directly on the position of the solute in the Periodic Table... 

The value functions appearing in equation 3 may be expanded in Taylor series about x and, because the concentration changes effected by a single stage are relatively small, only the first nonvanishing term is retained. When the value of is replaced by its material balance equivalent, ie, equation 4 ... 

The Value Function. The value function itself is defined, as has been indicated above, by the second-order differential ... 

In the design of cascades, a tabulation of p x) and of p (x) is useful. The solution of the above differential equation contains two arbitrary constants. A simple form of this solution results when the constants are evaluated from the boundary conditions u(0.5) = u (0.5) = 0. The expression for the value function is then ... 

However, recalling the definition of the value function, equation 11, and assuming that the value of a is the same for all stages, the integral maybe written in the form ... 

A norm on iT is a real-valued function/defined on iT with the following properties ... 

A scalar-valued function/(/4) of one symmetric second-order tensor A is said to be symmetric if... 

A scalar-valued function f(A, B) of two symmetric second-order tensors A and B is said to be isotropic if... 

Analogous results are available for scalar-valued functions of more than two tensor variables, see, e.g., [20]. 

See Benderskii et al. [1993] for exceptions to this statement the period may be not a good parameter to use for parametrizing this family because the action may be not a single-valued function of the period however this difficulty is easily circumvented by parametrizing the solutions, for example, by their energy. 

A relatively simple approach suggests itself if the interfaces are known to be almost flat. In that case, the interface position can be described by a single-valued function z(x,y), where (A,y) are cartesian coordinates on a flat parallel reference plane. The functional (21) can be approximated by... 

A random variable is a real-valued function defined over tlie sample space S of a random experiment (Note tliat tliis application of probability tlieorem to plant and equipment failures, i.e., accidents, requires tliat tlie failure occurs randomly. 

An important tool in the study of matrices is provided by vector norms and matrix norms. A vector norm j... is any real valued function of the elements satisfying the following three conditions ... 

We begin our discussion of random processes with a study of the simplest kind of distribution function. The first-order distribution function Fx of the time function X(t) is the real-valued function of a real-variable defined by6... 

The most important characteristic of self information is that it is a discrete random variable that is, it is a real valued function of a symbol in a discrete ensemble. As a result, it has a distribution function, an average, a variance, and in fact moments of all orders. The average value of self information has such a fundamental importance in information theory that it is given a special symbol, H, and the name entropy. Thus... 

Mutual information is thus a random variable since it is a real valued function defined on the points of an ensemble. Consequently, it has an average, variance, distribution function, and moment generating function. It is important to note that mutual information has been defined only on product ensembles, and only as a function of two events, x and y, which are sample points in the two ensembles of which the product ensemble is formed. Mutual information is sometimes defined as a function of any two events in an ensemble, but in this case it is not a random variable. It should also be noted that the mutual... 

Since the energies are single-valued functions of the volumes and entropies of the substance in the two states, and since the equations (9), (10), (11) give three relations between the four quantities vi, Si, v2, S2, it is evident that if one is fixed the others are determined, so that the two points lie on two definite curves on the model ... 

This notation admits of generalisation (Gibbs, 1876). The total energy of a homogeneous fluid is a continuous and single-valued function of the masses mi, m2, m3,. . mh of its constituents, of the total volume Y, and the total entropy S ... 

Emil Bose (1910) maintains that Zawidski s calculations, with Margules solution with only a few coefficients, are not satisfactory, and proposes to find the partial pressures by a graphical method which consists in drawing the two partial pressure curves so that the sum of their ordinates is everywhere equal to the ordinate of the (known) total pressure curves. The Duhem equation shows that pi, p2 are positive, continuous, and smgle-valued functions of a , so that only one decomposition of the total pressure curve has any physical significance, and for every value of x ... 

In general, for polymerization reactions, the heat generation rate is not a single-valued function of temperature, g(t), but also a function of monomer and catalyst concentrations, f(c). This is particularly important in high conversion reactions where a certain amount of peaking can be tolerated. 

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