Moment first

Another important characteristic of the late stages of phase separation kinetics, for asynnnetric mixtures, is the cluster size distribution fimction of the minority phase clusters n(R,z)dR is the number of clusters of minority phase per unit volume with radii between R and + cW. Its zeroth moment gives the mean number of clusters at time r and the first moment is proportional to die mean cluster size. 

Experimental access to the probabilities P(E ,E) for energy transfer in large molecules usually involves teclmiques providing just the first moment of this distribution, i.e. the average energy (AE) transferred in a collision. Such methods include UV absorption, infrared fluorescence and related spectroscopic teclmiques [11. 28. 71. 72, 73 and 74]. More advanced teclmiques, such as kinetically controlled selective ionization (KCSI [74]) have also provided infonnation on higher moments of P(E ,E), such as ((AE) ). 

The numerical value of the exponent k determines which moment we are defining, and we speak of these as moments about the value chosen for M. Thus the mean is the first moment of the distribution about the origin (M = 0) and is the second moment about the mean (M = M). The statistical definition of moment is analogous to the definition of this quantity in physics. When Mj = 0, Eq. (1.11) defines the average value of M this result was already used in writing Eq. (1.6) with k = 2. 

The dipole moment (A) of a molecule is the first moment of the elec tric charge density of a molecule. Paraffins have dipole moments of zero, while dipole moments of almost all hydrocarbons are small. McClellan lists many dipole moments. The computer method of Dixon and Jurs" is the most useful method for predicting dipole moments. Lyman et al. give other methods of calculation. 

It should be noted also that the intercept is difficult to determine accurately because of large potential experimental error in observing the time of the start of filtration and the time-volume correspondence during the first moments when the filtration rate is high. The value of / calculated from the intercept may vaiy appreciably from test to test, and will almost always be different from the value measured with clean medium in a permeability test. 

Residence time, mean The average time spent by the molecules in a vessel. Mathematically, it is the first moment of the effluent concentration from a vessel with impulse input, or ... 

A measure of bias d is the first moment of the distribution of these differences, or the average difference ... 

The average nonuniform permeability is spatially dependent. For a homogeneous but nonuniform medium, the average permeability is the correct mean (first moment) of the permeability distribution function. Permeability for a nonuniform medium is usually skewed. Most data for nonuniform permeability show permeability to be distributed log-normally. The correct average for a homogeneous, nonuniform permeability, assuming it is distributed log-normally, is the geometric mean, defined as ... 

The mean residenee time [ is obtained from Equation 8-39, whieh is the first moment. 

The first moment and synonyms is the location at which the curve, if cut out, would balance on a knife-edge. 

A generating function is defined by equation 2.5-47. To illustrate it use. Table 2.> 2 gives the generating function for an exponential distribution as -A/(0-X). Each moment i.s obtained by successive differentiations. Equation 2.5-48 shows how to obtain the first moment. By taking the limit of higher derivatives higher moments are found. 

The moments describe the characteristics of a sample or distribution function. The mean, which locates the average value on the measurement axis, is the first moment of values measured about the origin. The mean is denoted by p for the population and X for the sample and is given for a continuous random variable by... 

One solution to the volume problem was proposed using moment analysis. The steady-state volume of distribution (Vss) can be derived from the area under the curve (AUC) and the area under the first moment curve (AUMC). 

The interpretation of the higher-order moments an is simplified if they are first centered about the first moment. To this end, we define the wth central moment pn of the distribution function or, equivalently,... 

Moment Analysis. The zeroth moment is the molar concentration of polymer and is expressed by Equation 5. The first moment is proportioned to the mass of polymer formed and is related to monomer concentration Equation 3, The second moment WA(t) is expressed by... 

To describe single-point measurements of a random process, we use the first-order probability density function p/(/). Then p/(/) df is the probability that a measurement will return a result between / and / -I- df. We can characterize a random process by its moments. The nth moment is the ensemble average of /", denoted (/"). For example, the mean is given by the first moment of the probability density function. 

The first moment is the mean of the distribution or the mean residence time. 

Roughly speaking, the first moment, t, measures the size of a residence time distribution, while higher moments measure its shape. The ability to characterize shape is enhanced by using moments about the mean ... 

The entire residence time distribution can be made dimensionless. A normalized distribution has the residence time replaced by the dimensionless residence time, X = t/t. The first moment of a normalized distribution is 1, and all the moments are dimensionless. Normalized distributions allow flow systems to be compared in a manner that is independent of their volume and throughput. For example, all CSTRs have the same normalized residence time distribution, W(x) = exp(—t). Similarly, all PFRs have f(r) = S(x — 1). 

Analysis of most (perhaps 65%) pharmacokinetic data from clinical trials starts and stops with noncompartmental analysis (NCA). NCA usually includes calculating the area under the curve (AUC) of concentration versus time, or under the first-moment curve (AUMC, from a graph of concentration multiplied by time versus time). Calculation of AUC and AUMC facilitates simple calculations for some standard pharmacokinetic parameters and collapses measurements made at several sampling times into a single number representing exposure. The approach makes few assumptions, has few parameters, and allows fairly rigorous statistical description of exposure and how it is affected by dose. An exposure response model may be created. With respect to descriptive dimensions these dose-exposure and exposure-response models... 

Yu-Lu plan is a variant of the Julian one and its remarkable material efficiency is primarily due to a combined high atom economy and high reaction yields. The Trost plan has the best performing molecular weight first moment parameter. The Mukai plan has the highest degree of convergence. 

---