Monotone function

The heat capacity of an ideal vapor is a monotonic function of temperature in this work it is expressed by the empirical relation... 

A major contribution to the rational organization of contact angle data was made by Zisman and co-workers. They observed that cos 6 (advancing angle) is usually a monotonic function of 7l for a homologous series of liquids. The proposed function was... 

In the range of operating temperatures and compositions, the equilibrium relations are monotonic functions of temperature of the MSA. This is typically true. For instance, normally in gas absorption Henry s coefficient monotonically decreases as the temperature of the MSA is lowered while for stripping the gas-liquid distribution coefficient monotonically increases as the temperature of the stripping agent is increased. 

To start the review of the PIMC results [328], we note that the detailed study of the quantum APR model (Eq. (41)) was partly motivated by the strong changes in shape of the orientational order parameter as a function of temperature as the rotational constant was increasing from its classical value 0 = 0 (see Fig. 3 in Ref. 327). For small enough 0 it was found that the order parameter decays monotonically with increasing temperature, similarly to the classical case. This is qualitatively different for larger 0, where T ) becomes a non-monotonic function of temperature. 

In a continuous game both the choice of strategy and the payoff as a function of that choice are continuous. The latter is particularly important because a discontinuous payoff function may not yield a solution. Thus, instead of a matrix [ow], a function M(x,y) gives the payoff each time a strategy is chosen (i.e., the value of x and y are fixed). The strategy of each player in this case is defined as a member of the class D of probability distribution functions that are defined as continuous, real-valued, monotonic functions such that... 

Their impact and friction sensitivity data appear to be monotonic functions of corrected OB, but the relation between corrected OB and Trauzl Block data appears to be complex... 

There is nothing in Equations 1-8 which is an all-or-none situation. There are no positive feedback loops which might cause some kind of flip-flop of states of operation of the system. There are some possibilities for saturation phenomena but all relationships are graded. Overall, transient or steady-state, the changes of concentration of P-myosin are continuous, monotonic functions of the intracellular Ca ion concentration. On this basis it is more appropriate to say that smooth muscle contraction is modulated rather than triggered by Ca ion. 

The monotone function (21) is quite applicable when r[x) is a fastly varying function of the variable x and the condition R < I fails to hold at some points with no influence on the accuracy to a considerable extent. 

According to Eq. (1.3.31) logy should be a monotonous function of /7 which obviously is not the case. A possible remedy of this situation would be, for example, the introduction of an additional linear term on the right-hand side of that equation,... 

Physical hardness can be defined to be proportional, and sometimes equal, to the chemical hardness (Parr and Yang, 1989). The relationship between the two types of hardness depends on the type of chemical bonding. For simple metals, where the bonding is nonlocal, the bulk modulus is proportional to the chemical hardness density. The same is true for non-local ionic bonding. However, for covalent crystals, where the bonding is local, the bulk moduli may be less appropriate measures of stability than the octahedral shear moduli. In this case, it is also found that the indentation hardness—and therefore the Mohs scratch hardness—are monotonic functions of the chemical hardness density. 

Solar-type stars start destroying Li soon after their arrival on the ZAMS. The difference in the average Li between the Pleiades and the 2 Gyr clusters implies depletion timescales of the order of 1 Gyr. After 2 Gyr they become extremely long for part of the stars (Li depletion virtually stops) or very short for another fraction. The Sun belongs to the second category and, as such, it is not representative of all stars with the same mass, age, and metallicity. Li depletion is clearly not a monotonic function of age and at least one additional... 

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... 

As shown above, the size and distribution of minute particles are conveniently investigated by high-resolution STEM with a HAADF detector (60,63). The intensity in HAADF images is a monotonic function of the sample thickness and atomic number, a pre-requisite for the electron tomography experiments described below. 

Let

monotonous function and note

inverse function which is assumed to be single-valued (i.e., a value of the independent variable is associated with one value of the dependent variable). If the random variable X has the density function /v(x), then Y = density function fY(y) given by... 

The useful equations (4.3.6) and (4.3.7), which are valid only for smooth monotonous functions, can be translated into relationships between the corresponding sample statistics... 

The basic GC-model of the Constantinou and Gani method (Eq. 1) as presented above provides the basis for the formulation of the solvent replacement problem as a MILP-optimization problem. For purposes of simplicity, in this chapter, only the first-order approximation is taken into consideration (that is, W is equal to zero). In this way, the functions of the target properties of the generated molecules (solvent replacements) are written as monotonic functions of the property values, thereby, leading to a linear right hand side of the property constraints (property model equation), as follows,... 

In Eq. 2, index p indicates a specific target property from the NP number of properties considered, pk is the value of the property k for the given compound and fk(pt) is the corresponding monotonic functional transformation of this value. 

In principle, one can carry out a four-dimensional optimization in which the four parameters are varied subject to constraints (< 1 and P4 < 1 ), to minimize the deposition time with the non-uniformity bounded e.g., MN < 3. However, objective function evaluations involve solutions of the Navier-Stokes and species balance equations and are computationally expensive. Instead, Brass and Lee carry out successive unidirectional optimizations, which show the key trends and lead to excellent designs. A summary of the observed trends is shown in Table 10.4-1. Both the deposition rate and the non-uniformity are monotonic functions of the geometric parameters within the bounds considered, with the exception that the non-uniformity goes through a minimum at optimal values of P3 and P4. 

By hypothesis, the regular term Ehph(Z) is a (smooth) monotonic function without... 

The configurational entropy per occupied lattice site (i.e., per unit mass rather than per unit volume) is by definition a monotonic function of temperature, and, of course, the fluid entropy deduced from calorimetric measurements also has this monotonic property. Figure 5a compares the mass and site configurational entropies computed from the LCT as functions of temperature T for the F-F and... 

A cornerstone of condensed phase reaction theory is the Kramers-Grote-Hynes theory. In a seminal paper Kramers solved the Fokker-Plank equation in two limiting cases, for high and low friction, by assiuning Markovian dynamics y(t) 5(t). He foimd that the rate is a non-monotonic function of the friction ( Kramers turnover .) Further progress was made by Grote and Hynes - who... 

It was found that the rate of hydrate formation jn the photolysis of dimethyluracil in acetonitrile was zero, and the rate in acetonitrile-water mixtures was a monotonic function of water concentration... 

Nonetheless, as on the low current branch, the concentration variations within the charged layers are monotonic functions of lVj (this time, however, nonlinear). In particular, there exists a critical (N4T) such that... 

The following observation can be made about (4.4.1), (4.4.2). From (4.4.2d) j is a monotonic function of V, bounded from above by jum, termed the limiting current density, such that... 

The general tendency in aqueous organic solvents (11) is for the free energies of transfer to be monotonic functions of solvent composition. Often the enthalpy of transfer shows a maximum in the water-rich region, and for salts of small ions there may be a further reversal in the organic-rich region. 

---