Input variable definition

Since it is undecidable whether a particular statement can ever be reached in any computation under any interpretation for any input, this condition will be undecidable and thus in our view undesirable. We prefer definitions to be effective, particularly definitions of the basic object under discussion. So this problem is avoided either by using the condition we have given which obviously implies the alternative one, or by always assuming all variables to be input variables, which shortcuts the whole problem. 

DEFINITION Suppose a program scheme P has n input variables and D is the domain of an interpretation I. We say that (P,I) halts everywhere (or converges everywhere) if for all inputs a in Dn the computation (P,I,a) halts. 

In the next definitions, let us assume that P is a scheme with n input variables, denoted by X, and m output variables, denoted by Z with X and Z... 

DEFINITION A Ianov scheme is a scheme with one input variable, x, one program... 

We made a slight departure from our previous assumptions. Notice that we have cheated a little in the definition of a scheme. We have not explained how the division into input variables, program variables, and output variables is to be carried out. In particular, we have not explained what the requirement that on every path... 

DEFINITION A program scheme augmented by one pushdown store has a special variable it which is not an input variable and whose contents is always a list and two additional instruction foiros ... 

Differential algebraic equation systems whose algebraic equations explicitly involve manipulated input variables are referred to as nonregular (Kumar and Daoutidis 1999a). See also Definition A.6. 

Suppose that we are interested in the effect of a subset of input variables, held in a vector xe, where e denotes the set of subscripts of the variables of interest. The vector of remaining variables is denoted by jc e. For example, when interest is in the effects of xi andX2 among d > 2 variables, we have e = 1,2 andxe = (jti, X2), whereupon x-e = (X3,..., Xd). Without loss of generality we rearrange the order of the input variables so that we can write x = (xe, x-e). To obtain a unique and workable definition of the effect of is essentially the problem of how to deal with the variables in x e. We next discuss several ways of approaching this problem. 

Definition 2.1 (A Channel with Side Information - SIC) A channel with side information is a memoryless channel with input variable X and output random variable Y, whose transition probability matrix pY xs y, s), depends on a side information variable S, which is distributed according to a p.d.f. ps s). 

A program s runtime may be analysed in terms of the size of its inputs. For the example program above, the preamble section (prior to the main simulation loop) contains a number of variable definitions, simple calculations, the initialisation of a stream for output, and a loop that calculates the values of the modified Yi coefficients. The time taken to perform the variable definitions, calculations and stream initialisation is effectively constant for a given machine it is dependent only on the specifics of the computer hardware and the compiler, and is independent of the program s inputs. The loop will take a time proportional to the size of the spatial grid, n (which is the number of loop iterations), plus some small constant time to initialise the loop variable. We may write that the runtime of the preamble section. 

Definition A.9 (Diagnostic model) A set of static or dynamic relations which link specific input variables— the symptoms— to specific output variables—the faults. 

Meaningfulness and validity of results. Note that in the current approach a continuous range of disturbances for one input variable is considered for the computation of the responsiveness time constant. In comparison with the generalized definition of... 

The definition of best, as stated above, is a multiobjective problem and the Pareto front provides one method to encapsulate the various optimal solutions. However, the sampling of this front to find a range of solutions is not always assured. Only localized areas of the front might be explored (e.g., by the use of small population sizes in each iteration, which reduces the breadth of solutions that can be followed in subsequent steps [49]) or solutions may be produced that only satisfy the easiest objective in the desirability criteria. The conceptually easiest solution is to include a (dis)similarity value as an input variable against which to optimize in the MPO but again, as noted earlier, there are difficulties in identifying which descriptors to use to define the correct notion of similarity. Other solutions cover ideas such as elitism or niching of compoimds from intermediate rounds of optimization to enhance the diversity of the final set of solutions. 

Simulation environment RAVEN is perceived by the user as a pool of tools and data. Any action in which the tools are applied to the data is considered a step in the RAVEN environment. For the scope of this paper, multiRun type of step will be described, since all others are either closely related (single run and adaptive run) or just used to perform data management and visualization. Firstly, the RAVEN input file associates the variable definition syntax to a set of PDFs and to a sampling strategy. The multiRun step is used to perform several runs (sampling) in a block of a model (e.g. in a MC sampling). 

For Step 5 the Np degrees of freedom are assigned by specifying a total of Np input variables to be either disturbance variables or manipulated variables. In general, disturbance variables are determined by other process units or by the environment. Ambient temperature and feed conditions determined by the operation of upstream processes are typical examples of disturbance variables. By definition, a disturbance variable d varies with time and is independent of the other Ny - 1 process variables. Thus, we can express the transient behavior of the disturbance variable as... 

By definition, disturbance variables are input variables that cannot be manipulated. Common DVs include ambient conditions and feed streams from upstream process units. 

The input variables are subdivided into controllable variables and parameters. The controllable variables are those quantities whose values can be varied by the experimenter as, for example, initial and boundary conditions of experiments. Empirical and physical constants that are assumed to remain unchanged in a given set or a subset of experiments constitute the parameters of the model. To clarify an ambiguity that may arise from the above definitions let us consider a rate constant that is expressed in the Arrhenius form k = A Qxp — EJRT). In isothermal modeling the rate constant at the assumed temperature, /c, is a parameter. However, if temperature is a controllable variable of the model, then the preexponential factor A and the activation energy are the parameters. 

The initial equation for Fq(X) will represent output given for the actual input starting at instruction 1. Each program variable is by definition assumed to be specified before it is computed on thus any initial value assigned to it is irrelevant. So the initial equation is ... 

With proportional control, the final control element has a definite position for each value of the measured variable. In other words, the output has a linear relationship with the input. Proportional band is the change in input required to produce a full range of change in the output due to the proportional control action. Or simply, it is the percent change of the input signal required to change the output signal from 0% to 100%. 

Following the approach worked out in Chapter 3, Table 4.1 shows that in the Kalecki-type formulation profits in each sector are defined in gross terms, consisting of expenditure on the replacement of existing constant capital and its expansion (C, + e/C.) whereas in Table 4.2 profits (/() are defined in net terms (dC, + dV,). The latter definition of profits is consistent with Marx s interpretation, with the total increment of capital identical to the volume of surplus value, after accounting for the replacement of current inputs of constant and variable capital. 

Process validation entails firstly the definition of both the critical and non-critical parameters. Qnce they are defined, emphasis can be directed to designing a program to validate these parameters. Some established steps involve the evaluation of process consistency over at least three batches, via the consideration of the processing steps and yield and comparing these with predetermined specifications. Some input parameters that may be considered as critical are temperature, flow rate, and stirring speed, and they are varied and checked against output variables such as yield, purity, and crystallization rate. 

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