Mathematical moments

Suppose that ip and / are expressed as functions of (r, c,t), then a mathematical moment can be defined by the following ensemble average ... 

It is evident that below count no. 22, there is no calibration. Assuming the extrapolated part of the calibration curve, is correct, the mathematical moments of the MW distribution were calculated according to Equations (5.1)-(5.5), where Q/s are zeroth to fourth moments. A, is the relative MW based on the polystyrene standard and Wj is the mass fraction of polymer having MW, A,. The corresponding curves for the area-integration are shown in Figures 5.3a-e [5]. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

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 ... 

Reduce >f minimal cui failure data Mathematical combination of uncertainties output includes two moments of minimal cutsets and the lop event Johnson, empirical C le multiple. sy.siem fiiiJLuizjn with multiple data input descriptions can fit Johnsem-type distribution to the top event 1 t brnia... 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The sufficient and necessary condition is therefore Cb iCa. As a consequence of imposing the more restrictive condition, which is obviously not correct throughout most of the reaction, it is possible for mathematical inconsistencies to arise in kinetic treatments based on the steady-state approximation. (The condition Cb = 0 is exact only at the moment when Cb passes through an extremum and at equilibrium.)... 

Kgp values can be used to make predictions as to whether or not a precipitate will form when two solutions are mixed. To do this, we follow an approach very similar to that used in Chapter 12, to determine the direction in which a system will move to reach equilibrium. We work with a quantity Q, which has the same mathematical form as K. The difference is that the concentrations that appear in Q are those that apply at a particular moment. Those that appear in are equilibrium concentrations. Putting it another way, the value of Q is expected to change as a precipitation reaction proceeds, approaching Ksp and eventually becoming equal to it. 

In a typical chapter, you will find ten or more Examples, each designed to illustrate a particular principle. These have answers, screened in colon Most of them contain a Strategy statement, which describes the reasoning behind the Solution. You should find it helpful to get into the habit of working all problems this way. First, spend a few moments deciding how the problem should be solved. Then, and only then, set up the mathematics to solve it... 

El theory In all materials (plastics, metals, wood, etc.) elementary mechanical theory demonstrates that some shapes resist deformation from external loads. This phenomenon stems from the basic physical fact that deformation in beam or sheet sections depends upon the mathematical product of the modulus of elasticity (E) and the moment of inertia (I), commonly expressed as EL This theory has been applied to many different constructions including sandwich panels. 

One aspect of the mathematical treatment of the quantum mechanical theory is of particular interest. The wavefunction of the perturbed molecule (i.e. the molecule after the radiation is switched on ) involves a summation over all the stationary states of the unperturbed molecule (i.e. the molecule before the radiation is switched on ). The expression for intensity of the line arising from the transition k —> n involves a product of transition moments, MkrMrn, where r is any one of the stationary states and is often referred to as the third common level in the scattering act. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

The present section analyzes the above concepts in detail. There are many different mathematical methods for analyzing molecular weight distributions. The method of moments is particularly easy when applied to a living pol5mer polymerization. Equation (13.30) shows the propagation reaction, each step of which consumes one monomer molecule. Assume equal reactivity. Then for a batch polymerization. 

C. H. Bamford and H. Tompa, J. Polymer Sci.j 10, 345 (1953), first derive the moments of the distribution for the case of chain transfer to polymer. They then obtain the molecular weight distribution from these moments by appropriate mathematical methods. Their procedure should be applicable to a wide variety of polymerization mechanisms. 

See also the theoretical description of a micro reactor for optical photocatalytic dissociation of non-linear molecules in [140]. Here, a mathematical model for a novel type of micro reactor is given. Rotating non-linear molecules at excitation of valent vibrations are considered, having a magnetic moment. Resonance decay of molecules can be utilized with comparatively weak external energy sources only. 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

In classical control theory, we make extensive use of Laplace transform to analyze the dynamics of a system. The key point (and at this moment the trick) is that we will try to predict the time response without doing the inverse transformation. Later, we will see that the answer lies in the roots of the characteristic equation. This is the basis of classical control analyses. Hence, in going through Laplace transform again, it is not so much that we need a remedial course. Your old differential equation textbook would do fine. The key task here is to pitch this mathematical technique in light that may help us to apply it to control problems. 

The development of the methods described in Section 9.2 was an important step in modeling polarization because it led to accurate calculations of molecular polarizability tensors. The most serious issue with those methods is known as the polarization catastrophe since they are unable to reproduce the substantial decrease of the total dipole moment at distances close to contact as obtained from ab initio calculations. As noted by Applequist et al. [49], and Thole [50], a property of the unmodified point dipole is that it may originate infinite polarization by the cooperative interaction of the two induced dipoles in the direction of the line connecting the two. The mathematical origins of such singularities are made more evident by considering a simple system consisting of two atoms (A and B) with isotropic polarizabilities, aA and c b. The molecular polarizability, has two components, one parallel and one perpendicular to the bond axis between A and B,... 

The advantage of Raman spectromicroscopy is that very small specimens can be studied while still allowing the determination of the second and fourth moments of the ODF. However, the expressions for the Raman intensities are more complex since the optical effects induced by the microscope objective have to be considered. Although the corrections may be small, they are not necessarily negligible [59]. This problem was first treated by Turrell [59-61] and later by Sourisseau and coworkers [5]. Turrell has mathematically quantified the depolarization of the incident electric field in the focal plane of the objective and the collection efficiency of the scattered light by high numerical aperture objectives. For brevity, only the main results of the calculations will be presented. Readers interested in more details are referred to book chapters and reviews of Turrell or Sourisseau [5,59,61]. The intensity in Raman spectromicroscopy is given by [59-61]... 

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