Monotone response function

The calibration curve of an analytical procedure is the relationship between the assay response (signal, e.g., area under the curve, peak height, and absorption) and the concentration (quantity) of the analyte in the sample. The calibration curve should preferably be a simple, monotonic response function that gives accurate measurements. 

For a reaction with positive gas mole change, Eq. (47) indicates that Kx decreases with pressure. Because ce is a monotonically increasing function of Kx, the equilibrium extent of a reaction with positive Avgas always decreases as pressure is increased. This is an example of Le Chatelier s principle, which states that a reaction at equilibrium shifts in response to a change in external conditions in a way that moderates the change. In this case, because the reaction increases the number of moles of gas and thus the pressure, the reaction shifts back to reactants. The isothermal compressibility of a reactive system can, therefore, be much greater than that of a nonreactive system. This effect can be dramatic in systems with condensed phases. For example, in the calcium carbonate dissociation discussed in Example 12, if the external pressure is raised above the dissociation pressure of C02, the system will compress down to the volume of the solid. Of course, a similar effect is observed in simple vaporization or sublimation equilibrium. As the pressure on water at 100°C is increased above 1.0 atm, all vapor is removed from the system. 

Since the displacement response function does not depend on the bath temperature [Eq. (115) or (116)], Eq. (126) displays the above quoted property that, at any fixed time t,D(t) is a monotonic increasing function of the temperature. In the infinitely short memory limit, taking into account the corresponding expression (79) of %xx, one gets from Eq. (126)... 

Ideally a single-particle counter should have a response function monotonically dependent on particle size exemplified by that portion of Figure I for n = 1.57 — 0.56i and a < 17 but with zero response for all other a. The real situation however is quite different as indicated. Two types of deviations from desired behavior can be identified. 

Differential scarming calorimetry (DSC) (22) is based on measuring changes in heat capacity because the temperature of a system is varied monotonically. As heat capacity is a thermodynamic response function, it is expected to exhibit critical behavior close to a phase transition boundary, which in turn can be detected by DSC. Consequently, DSC is a common way to detect phase behavior. 

This equation, formulated for liquids, also describes the behavior of solids if q = 00 is assumed in the latter case. Experimentally it is found that J(t) — t/r[ — Jg is a monotonous increasing function of time that reaches the value Jj as t 00. The function l (t) modulates the entropic response to the shear stress the time dependence of this function is indicated in Figure 5.15. Since W( — 0) is a monotonous decreasing function of time, d it — Q)/d(t — 0) is an increasing function of time whose slope increases considerably as 0 approaches t. Consequently, (0 behaves as a memory function that mod-... 

That is, as the independent variable increases, the general shape of the response function is either mono-tonically increasing to a value of 1 or monotonically... 

We assume in this separation that the spatial and temporal aspects of the fluctuations are independent. The term f Q) is the magnetic form factor and is related to the spatial extent of unpaired electrons. The form factor can be approximated as f Q) = jo) + Cx jj), where j ) are Bessel transforms of the single-electron density, U r), and Ci is a coefficient normally between 0 (for spin only systems) and 2. In most cases the form factor is a monotonically falling function with Q, although there are some important exceptions. For a fuller discussion and complete references, see Lander (1993). Due to the finite mass of the neutron, investigations at 0=0 are impossible. If we consider the electrons contributing to the dynamical susceptibility, which describes the temporal and spatial behavior of the magnetic response of a solid, x" Q, T)- then we frequently think... 

Many sensor-transmitters have overdamped dynamics and exhibit monotonic responses to a step change in the variable being measured. Thus, it is reasonable to model this type of measurement dynamics as a first-order transfer function between the actual value y and the measured value... 

This is a curious result, as it indicates that a nonlinear property can be calculated from linear data, but it has been found to describe accurately the response of polymeric liquids at sufficiently low shear rates. The low-shear-rate limiting behavior indicated by the above equations, which involves monotone increasing functions, is always shown in plots of nonlinear data the nonlinear responses involve overshoots in the material functions, but should always start out at short times, when the strain is still small, by following the low-shear-rate, LVE curve. Then, as the shear rate increases, the nonlinear data fall below the linear envelopes at shorter and shorter times [44,45]. These features can be seen in Fig. 10.9, which shows the data of Menezes and Graessley [46] for shear and first normal stress difference in start-up of simple shear. The dashed lines are calculated from the linear spectrum using Eqs.4.8 and 10.49. As expected, the... 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

In this section, by applying the heterodyne interferometry to a mixed gas of H2 and D2 molecules, we probe attosecond dynamics of nuclear wavepackets in the molecules. We find that not only the single molecule responses but also the propagation effects of harmonics differ between the two isotopes and that to discuss dynamics of molecules in the single molecule responses, the propagation effects need to be excluded from the raw harmonic signals. The measured relative phase as well as intensity ratio are found to be monotonic functions of the harmonic order and are successfully reproduced by applying... 

Temperature may significantly affect chemoreception. For instance, electrical responses to amylacetate delivered to olfactoiy receptors of a tortoise, Gopherus polyphemus, were little affected by air temperatures between 20 and 30 °C at the nares but changed considerably above and below that range. Up to - -35 °C and down to -1-10 °C, the olfactory response was a monotonic slowly decreasing function of temperature (Tucker, 1963 see also Grundvig etal., 1967). 

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