Variable instance

A variable that appears in both sides of a statement is called a feedback variable. Any element of the set A (F(i)) i G / is called a variable instance. Variables with common name but diiferent index functions are considered as... 

Let a variable instance computed at a point ii use the value of a variable instance generated at another point i2> The vector d = i2-ii is called dependence vector. The dependence graph (DG) is a directed graph, where each node corresponds to a point i G and each edge corresponds to a DV joining two dependent points of the index space. 

In order to transform a nested loop with linear dependencies into an equivalent URE form, the propagation space of each variable should be determined and the localization of the data movements should be performed. Generally, propagation exists when the propagation space of a variable instance contains more than one node of the index space. The propagation in the index space can take two different forms, namely broadcast operations and Jan-in . More specifically, broadcasting occurs when a value of an instance is distributed to many nodes of the space, while fan-in occurs when different values of an instance are concentrated from many nodes to one node. The latter is the main feature of WSACs [20]. 

In this case, the DVs can be seen as integer vectors that express the movement of the value of a variable instance within its propagation space. Consider any two nodes ii — (in,. ., Hiv) and i2 = ( 21, , i2iv) included in the propagation space of a variable. The coordinates of ii and 12 are not independent since equation (1) holds for both nodes. Therefore, the following equations should be satisfied ... 

The space-time mapping of the parametric URE derived in the previous step is accomplished by decomposing the index space to independent subsets of variable instances. The number of these subsets depends on the DVs and can be modified by alternative selection of the URE parameters. This results in a variety of array architectures in terms of size, PE utilization, and interconnection patterns. This method is complementary to the approaches introduced in chapters 3, 4, and 6, as indicated there. However, many of the proposed techniques can be combined. 

The overall design method developed at University of Patras (see chapter 5) is also not directly oriented to our particular application domain. This alternative method exploits the existence of independent subsets of variable instances, but these are not present in many video processing kernels, so no gains are possible here. Also, since the approach is partly latency-driven, an architecture with less than 100% PE load would be the result. However, the extension to bit-level pipelining can be integrated with the approach presented here to arrive at architectures with extremely high clock frequencies. This was not required for the ME demonstrator. 

In the same section, we also see that the source of the appropriate analytic behavior of the wave function is outside its defining equation (the Schibdinger equation), and is in general the consequence of either some very basic consideration or of the way that experiments are conducted. The analytic behavior in question can be in the frequency or in the time domain and leads in either case to a Kramers-Kronig type of reciprocal relations. We propose that behind these relations there may be an equation of restriction, but while in the former case (where the variable is the frequency) the equation of resh iction expresses causality (no effect before cause), for the latter case (when the variable is the time), the restriction is in several instances the basic requirement of lower boundedness of energies in (no-relativistic) spectra [39,40]. In a previous work, it has been shown that analyticity plays further roles in these reciprocal relations, in that it ensures that time causality is not violated in the conjugate relations and that (ordinary) gauge invariance is observed [40]. 

Relational databases can store unlimited numbers of results for every sample and unlimited samples for every request. The advantage of a relational DBMS over a more traditional hierarchical system, in which data sets may contain other data sets, is that the design of the database only has to consider relationships between data elements, not the number of instances for any given variable. 

In steam stimulation, heat and drive energy are suppHed in the form of steam injected through weUs into the tar sand formation. In most instances, the injection pressure must exceed the formation fracture pressure in order to force the steam into the tar sands and into contact with the oil. When sufficient heating has been achieved, the injection weUs are closed for a soak period of variable length and then allowed to produce, first applying the pressure created by the injection and then using pumps as the weUs cool and production declines. 

The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimension equation resolves into an equation containing only one or two variables. 

With many variables and constraints, linear and nonlinear programming may be applicable, as well as various numerical gradient search methods. Maximum principle and dynamic programming are laborious and have had only limited applications in this area. The various mathematical techniques are explained and illustrated, for instance, by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-Hill, 1988). 

Variables It is possible to identify a large number of variables that influence the design and performance of a chemical reactor with heat transfer, from the vessel size and type catalyst distribution among the beds catalyst type, size, and porosity to the geometry of the heat-transfer surface, such as tube diameter, length, pitch, and so on. Experience has shown, however, that the reactor temperature, and often also the pressure, are the primary variables feed compositions and velocities are of secondary importance and the geometric characteristics of the catalyst and heat-exchange provisions are tertiary factors. Tertiary factors are usually set by standard plant practice. Many of the major optimization studies cited by Westerterp et al. (1984), for instance, are devoted to reactor temperature as a means of optimization. 

Thus, (MSF) should in practice be regarded as a given or predetermined variable, and Eq. (9-117) accordingly becomes more useful if it is rearranged. For instance, the values of contribution efficiency for a given value of (MSF) are related to the number of elapsed payback periods by... 

The type of evaporator to be used and the materials of construc tion are generally selected on the basis of past experience with the material to be concentrated. The method of feeding can usually be decided on the basis of known feed temperature and the properties of feed and produc t. However, few of the listed variables are completely independent. For instance, if a large number of effects is to be used, with a consequent low temperature drop per effect, it is impractical to use a natural-circiilation evaporator. It expensive materials of construction are desirable, it may be found that the forced-circulation evaporator is the cheapest and that only a few effec ts are justifiable. 

Normally when a small change is made in the condition of a reactor, only a comparatively small change in the response occurs. Such a system is uniquely stable. In some cases, a small positive perturbation can result in an abrupt change to one steady state, and a small negative perturbation to a different steady condition. Such multiplicities occur most commonly in variable temperature CSTRs. Also, there are cases where a process occurring in a porous catalyst may have more than one effectiveness at the same Thiele number and thermal balance. Some isothermal systems likewise can have multiplicities, for instance, CSTRs with rate equations that have a maximum, as in Example (d) following. 

Assume a continuous release of pressurized, hquefied cyclohexane with a vapor emission rate of 130 g moLs, 3.18 mVs at 25°C (86,644 Ib/h). (See Discharge Rates from Punctured Lines and Vessels in this sec tion for release rates of vapor.) The LFL of cyclohexane is 1.3 percent by vol., and so the maximum distance to the LFL for a wind speed of 1 iti/s (2.2 mi/h) is 260 m (853 ft), from Fig. 26-31. Thus, from Eq. (26-48), Vj 529 m 1817 kg. The volume of fuel from the LFL up to 100 percent at the moment of ignition for a continuous emission is not equal to the total quantity of vapor released that Vr volume stays the same even if the emission lasts for an extended period with the same values of meteorological variables, e.g., wind speed. For instance, in this case 9825 kg (21,661 lb) will havebeen emitted during a 15-min period, which is considerablv more than the 1817 kg (4005 lb) of cyclohexane in the vapor cloud above LFL. (A different approach is required for an instantaneous release, i.e., when a vapor cloud is explosively dispersed.) The equivalent weight of TNT may be estimated by... 

Turboexpander sensitivity to process gas inlet pressure. As previously mentioned, variable speed turboexpanders are more sensitive to changes in normal operating conditions. Tlie pattern of TTE degradation, however, is the same as for constant speed turboexpanders (Figure 7-13a). In other words, TTE is more sensitive to pressure drop than to pressure rise. For instance, a 20% drop in gas inlet pressure will reduce TTE to 90% of the design value, whereas a 20% increase in gas inlet pressure reduces TTE to 99% of the design value. 

Individuals differ in their sensitivity to odor. Figure 14-7 shows a typical distribution of sensitivities to ethylsulfide vapor (17). There are currently no guidelines on inclusion or exclusion of individuals with abnormally high or low sensitivity. This variability of response complicates the data treatment procedure. In many instances, the goal is to determine some mean value for the threshold representative of the panel as a whole. The small size of panels (generally fewer than 10 people) and the distribution of individual sensitivities require sophisticated statistical procedures to find the threshold from the responses. 

Differential temperature as well as differential pressure can be used as a primary control variable. In one instance, it was hard to meet purity on a product in a column having close boiling components. The differential temperature across several bottom section trays was found to be the key to maintaining purity control. So a column side draw flow higher in the column was put on control by the critical temperature differential. This controlled the liquid reflux running down to the critical zone by varying the liquid drawn off at the side draw. This novel scheme solved the control problem. 

The final variable to be mentioned here is the presence of impurities. These may be metallic fragments residual from Ziegler-type processes or they can be trace materials incorporated into the polymer chain. Such impurities as catalyst fragments and carbonyl groups incorporated into the chain can have a serious adverse influence on the power factor of the polymer, whilst in other instances impurities can have an effect on aging behaviour. 

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