Time functions

From the time function F t) and the calculation of [IT], the values of G may be found. One way to calculate the G matrix is by a fast Fourier technique called the Cooley-Tukey method. It is based on an expression of the matrix as a product of q square matrices, where q is again related to N by = 2 . For large N, the number of matrix operations is greatly reduced by this procedure. In recent years, more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform. 

Ignition time function (30 seconds allowed for establishing flame on both detectors or the unit will be shut down after several tries). 

Ignition time function (to de-energize ignition at the proper time). 

A prominent part of many of the techniques is separation of variables. In that method, the deflection variables, or the variation In deflection variables, are arbitrarily separated into functions of plate coordinate x alone times functions of y alone. Wang [5-8] determined that separation of variables leads to exact solutions for some classes of plate problems, but does not for others, I.e., the deflections are not always separable. A specific example of an approximate use of separation of variables due to Ashton [5-9] will be discussed in Section 5.3.2. Other exact uses of the method abound throughout Section 5.3 through 5.5. 

The quality of the packed bed may also be determined by frontal analysis where the sample is applied until it reaches a plateau to give the residence time function and then the solution is momentarily switched to wash to give the washout function. The latter is used to calculate the plate height of the column... 

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

If V is not a function of time, the Schrodinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function ... 

This can be a real problem for baghouses that rely on automatic timers to control cleaning frequency. The use of a timing function to control cleaning frequency is not recommended unless the dust load is known to be consistent. A better approach is to use differential-pressure gages to physically measure the pressure drop across the filter media to trigger the cleaning process based on preset limits. 

If the flow is accompanied with CBA decomposition, the G value in Eq. (5) should be substituted with its time function, G(t). In the general case, thermal decomposition of a solid substance with gas emission is a heterogeneous topochemical reaction [22]. Kinetic curves of such reactions are S -shaped the curves representing reaction rate changes in time pass a maximum. At unchanging temperature, the G(t) function for any CBA is easily described with the Kolrauch exponential function [20, 23, 24] ... 

Deformation or fracture. The failure of a plastic product in the performance of its normal long-time function is usually caused by... 

As a simple illustration of this philosophy, we analyze one aspect of the following data-smoothing device. The input to this device is a time function X(t) and the output Y(t) is given by the formula... 

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 significance of this definition can be understood most easily by reference to Tig. 3-1, which depicts a typical section of a time function X(t) and the corresponding function C X(t)] drawn for a particular... 

The important point we wish to re-emphasize here is that a random process is specified or defined by giving the values of certain averages such as a distribution function. This is completely different from the way in which a time function is specified i.e., by giving the value the time function assumes at various instants or by giving a differential equation and boundary conditions the time function must satisfy, etc. The theory of random processes enables us to calculate certain averages in terms of other averages (known from measurements or by some indirect means), just as, for example, network theory enables us to calculate the output of a network as a function of time from a knowledge of its input as a function of time. In either case some information external to the theory must be known or at least assumed to exist before the theory can be put to use. 

One further point is worth mentioning at this time. The definition of a distribution function (3-5) involves the taking of a limit and, consequently, brings up the question of the existence of this limit. The limit will not, in general, exist for all possible time functions X(t), and the investigation of conditions for its existence is a legitimate mathematical problem. However, questions of this sort are quite beside the point in the present context. We are not really interested in knowing how to specify time functions in such a way that their distribution functions exist. Instead, we want to know how to specify a function Fx in such a way that it is the distribution function of... 

Viewed in a somewhat different light, the point here is that we are not trying to fix, once and for all, the class of functions X(t) for which all of our results will be valid but, rather, we limit our attention in any specific situation to those time functions for which certain time averages (such as a distribution function) assume specified values. This being the case, the only existence question of importance reduces to making sure that the underlying class of functions is not vacuous. 

Random Variables.—An interesting and useful interpretation of the theorem of averages is to regard it as a means for calculating the distribution functions of certain time functions Y(t) that are related to a time function X(t) whose distribution function is known. More precisely, if Y(t) is of the form Y(t) = [X(t)], then the theorem of averages enables us to calculate the distribution function of Y(t)... 

Note carefully that the same random variable (function) may have many different distribution functions depending on the distribution function of the underlying function X(t). We will avoid confusion on this point by adopting the convention that, in any one problem, and unless an explicit statement to the contrary is made, all random variables are to be used in conjunction with a time function X(t) whose distribution function is to be the same in all expressions in which it appears. With this convention, the notation F is just as unambiguous as the more cumbersome notation so that we are free to make use of whichever seems more appropriate in a given situation. 

The notion of the distribution function of a random variable is also useful in connection with problems where it is not possible or convenient to subject the underlying function X(t) to direct measurements, but where certain derived time functions of the form Y(t) = [X(t)] are available for observation. The theorem of averages then tdls us what averages of X(t) it is possible to calculate when all that is known is the distribution function of . The answer is quite simple if / denotes (almost) apy real-valuqd function of a real variable, then all X averages of the form... 

There is one further point that is worth mentioning in connection with the random variable concept. We have repeatedly stressed the fact that the theory of random processes is primarily concerned with averages of time functions and not with their detailed structure. The same comment applies to random variables. The distribution function of a random variable (or perhaps some other less complete information about averages) is the quantity of interest not its functional form. The functional form of the random variable is only of interest insofar as it enables us to derive its distribution function from the known distribution function of the underlying time function X(t). It is the relationship between averages of various time functions that is of interest and not the detailed relationship between the time functions themselves. 

The problem just considered can be generalized in a useful way by assuming that we want to predict the value of a time function Y at time t from our knowledge of the value of a different time function X at time t. For example, X(t) could be a noise voltage measured at some point in an electrical network and F(f) the noise voltage measured at a... 

We turn now to a study of time averages that involve the values of several different time functions at several instants of time. A quite general average of this type is of the form 20... 

We shall limit ourselves to averages involving two different time functions. The extension to more than two functions is obvious. 

The notions of random variable and mathematical expectation also cany over to the multidimensional case. A function of n + m-real variables is called a random variable when it is used to generate a new time function Z(t) from the time functions X(t) and Y(t) by means of the equation... 

The multidimensional theorem of averages can be used to calculate the higher-order joint distribution functions of derived sets of time functions, each of which is of the form... 

Another instructive example concerns the joint distribution function of the pair of time functions Zx(t) and Z2(t) defined by... 

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