Input function

A biological neuron can be active (excited) or inactive (not excited). Similarly, the artificial neurons can also have different activation status. Some neurons can be programmed to have only two states (active/inactive) as the biological ones, but others can take any value within a certain range. The final output or response of a neuron (let us call it a) is determined by its transfer function, f, which operates on the net signal (Netj) received by the neuron. Hence the overall output of a neuron can be summarised as  

The numerical value of aj determines whether the neuron is active or not. The bias, 0j, should also be optimised during training [8]. The activation function, ranges currently from 0 to 1 or from — 1 to +1 (depending on the mathematical transfer function, /). When a, is 0 or — 1 the neuron is totally inactive, 

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The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

Common time domain input functions 3.4.1 The impulse function... 

Relationship between input function, system type and steady-state error 170... 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

When the administration is continuous, for example by oral infusion at a constant rate of k, the input function is given by ... 

The time-response characteristics of a model can be inferred from the poles, i.e., the roots of the characteristic polynomial. This observation is independent of the input function and singularly the most important point that we must master before moving... 

Since q and c0 are input functions, the linearized equations in deviation initial conditions are (with all apostrophes omitted in the notations) ... 

Example 4.7 We ll illustrate the results in this section with a numerical version of Example 4.5. Consider again two CSTR-in-series, with V] = 1 m3, V2 = 2 m3, k] =1 min-1, k2 =2 min-1, and initially at steady state, x, = 0.25 min, x2 = 0.5 min, and inlet concentration cos = 1 kmol/m3. Derive the transfer functions and state transition matrix where both c0 and q are input functions. 

The closed-loop system is stable if all the roots of the characteristic polynomial have negative real parts. Or we can say that all the poles of the closed-loop transfer function he in the left-hand plane (LHP). When we make this statement, the stability of the system is defined entirely on the inherent dynamics of the system, and not on the input functions, fn other words, the results apply to both servo and regulating problems. 

Using a forward-convolution program131 with instrumental and experimental parameter inputs (aperture sizes, flight distances, beam velocities, etc.), along with two center-of-mass (CM) input functions (the translational energy release distribution, P(E), and the CM angular distribution, T(0)), TOF spectra and lab angular distributions were calculated and compared... 

US) = input function into compartment 1 n = number of compartments... 

Disposition rate constants are functions of the intercompartmental transfer rate constants and exit rate constants and can be expressed as such by equating the denominators of Eqs. (5) and (6). Common input functions, in, are as follows. 

Input functions [i.e., I(t)], describing the rate at which the administered dose enters a compartment, may have various forms depending on the administration schedule. The input function /(f) is added to the appropriate mass balance equation and can describe any drug administration pattern. First-order absorption... 

System analysis techniques have been used to generate input functions for PB-PK models. Oral administration of carbon tetrachloride in different vehicles was successfully described by absorption input functions obtained by deconvolution and disposition decomposition methods [25,26],... 

THEOREM 6.21 Every partial recursive function from non-negative integers to ncn-negative integers can be expressed as f(n) = val(P,I,n) for P a Ianov scheme and I an interpretation permitting only functions px and x/p and predicate " P divides x " for every prime p, constant 1, and special input function 2X and output function logjX. ... 

FACT III If Ianov schemes are restricted to the interpretation I above, only a small subclass of the recursive functions are computed in the sense that g(n) = 1 + val(P,I,n) for (P,I,n) convergent and g(n) = 0 for (P,I,n) divergent is a total recursive function and "most" total recursive functions cannot be so expressed. However, if one selects as interpretation I the interpretation with functions px and x/p and predicates "p divides x" for every prime p and constant 1 as well as special input function 2X and output function log2x then every partial recursive function f(n) can be expressed as val(P,I,n) for a Ianov (single register) scheme P and this particular interpretation I. ... 

The value of cAo depends upon the input function (whether step or pulse), and the initial condition (cAj(0)) for each reactor must be specified. For a pulse input or step increase from zero concentration, cA,-(0) is zero for each reactor. For a washout study, Ai(O) is nonzero (Figure 19.5b), and cAo must equal zero. For integer values of N, a general recursion formula may be used to develop an analytical expression which describes the concentration transient following a step change. The following expressions are developed based upon a step increase from a zero inlet concentration, but the resulting equations are applicable to all types of step inputs. 

The convolution integral and the Exponential Piston Flow Model (EPM) were used to relate measured tracer concentrations to historical tracer input. The tritium input function is based on tritium concentrations measured monthly since the 1960s near Wellington, New Zealand. CFC and SF6 input functions are based on measured and reconstructed data from southern hemisphere sites. The EPM was applied consistently in this study because statistical justification for selection of some other response function requires a substantial record of time-series tracer data which is not yet available for the majority of NGMP sites, and for those NGMP sites with the required time-series data, the EPM and other response functions yield similar results for groundwater age. 

If the calorimeter could respond instantaneously to the heat effects associated with the addition of titrant, then the measured curve would coincide with the dashed lines in figure 11.5. The deviation of the data from this ideal behavior corresponds to periods in which the isothermal condition is not observed. When necessary, however, it is possible to use deconvolution techniques to generate the input function represented by the dashed line from the observed experimental curve. 

The input function is the product of amount in the central compartment Aj and the entry rate constant the output function is given by the amount in the effect compartment Ae and... 

A real sampler, as shown in Fig. 18.1, is closed for a finite period of time. This time of closure is usually small compared with the sampling period. Therefore the real sampler can be closely approximated by an impulse sampler. An impulse sampler is a device that converts a continuous input signal into a sequence of impulses or delta functions. Remember, these are impulses, not pulses. The height of each of these impulses is infinite. The width of each is zero. The area of the impulse or the strength of the impulse is equal to the magnitude of the input function at the sampling instant. 

A special kind of random noise, pseudo random noise, has the special property of not being really random. After a certain time interval, a sequence, the same pattern is repeated. The most suitable random input function used in CC is the Pseudo Random Binary Sequence (PRBS). The PRBS is a logical function, that has the combined properties of a true binary random signal and those of a reproducible deterministic signal. The PRBS generator is controlled by an internal clock a PRBS is considered with a sequence length N and a clock period t. It is very important to note that the estimation of the ACF, if computed over an integral number of sequences, is exactly equal to the ACF determined over an infinite time. 

Potentiostatic Transient Technique, In the potentiostatic technique the potential of the test electrode is controlled, while the current, the dependent variable, is measured as a function of time. The potential difference between the test electrode and the reference electrode is controlled by a potentiostat (Fig. 6.21). The input function, a constant potential, and the response function, i = f(t), are shown in Figure 6.22. 

Figure 6.23. Linear potential sweep voltammetry (a) input function (b) response function.

When a is large (> 100) then the peak is symmetrical with a mean value of 1.20. In the extreme as a - then the simulated peak approaches the input function which in our model simulation is a Dirac delta function at 0 = 1.2. In the real situation it should approach the true MWD of the polymer being analysed. As a decreases the simulated peak is first broadened and then skewed. It is apparent that the peak maximum shifts in a manner expected. 

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