Equations, mathematical interaction

The Hansch equation mathematically describes two events controlling the biological activity of a compound. The first involves cell or membrane penetration (influenced by the partition coefficient), and the second involves interaction of the active molecule at a receptor site (modified by variations in the Hammett sigma constant). 

Our analysis of the data for the dependence of Tg on Mn failed to reveal any quantitative structure-property relationships of sufficient accuracy for Kg and Kg" to allow the predictive use of Equation 6.7. There is considerable mathematical interaction between the effects of Kg and Kg". Small variations in the locations of the data points on the (Mn,Tg)-plane can cause large changes in the magnitudes of Kg and Kg", especially if Tg was only measured at a small number of values of Mn. Consequently, Equation 6.7 cannot be used to predict Tg as a function of Mn if experimental data are unavailable. However, if data are available, Equation 6.7 is... 

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. 

Electrostatics is the study of interactions between charged objects. Electrostatics alone will not described molecular systems, but it is very important to the understanding of interactions of electrons, which is described by a wave function or electron density. The central pillar of electrostatics is Coulombs law, which is the mathematical description of how like charges repel and unlike charges attract. The Coulombs law equations for energy and the force of interaction between two particles with charges q and q2 at a distance rn are... 

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

A potential energy function is a mathematical equation that allows for the potential energy, V, of a chemical system to be calculated as a function of its tliree-dimensional (3D) structure, R. The equation includes terms describing the various physical interactions that dictate the structure and properties of a chemical system. The total potential energy of a chemical system with a defined 3D strucmre, V(R)iai, can be separated into terms for the internal, V(/ )i,iBmai, and external, V(/ )extemai, potential energy as described in the following equations. 

Meteorology plays an important role in determining the height to which pollutants rise and disperse. Wind speed, wind shear and turbulent eddy currents influence the interaction between the plume and surroimding atmosphere. Ambient temperatures affect the buoyancy of a plume. However, in order to make equations of a mathematical model solvable, the plume rise is assumed to be only a function of the emission conditions of release, and many other effects are considered insignificant. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The major mechanism of a vapor cloud explosion, the feedback in the interaction of combustion, flow, and turbulence, can be readily found in this mathematical model. The combustion rate, which is primarily determined by the turbulence properties, is a source term in the conservation equation for the fuel-mass fraction. The attendant energy release results in a distribution of internal energy which is described by the equation for conservation of energy. This internal energy distribution is translated into a pressure field which drives the flow field through momentum equations. The flow field acts as source term in the turbulence model, which results in a turbulent-flow structure. Finally, the turbulence properties, together with the composition, determine the rate of combustion. This completes the circle, the feedback in the process of turbulent, premixed combustion in gas explosions. The set of equations has been solved with various numerical methods e.g., SIMPLE (Patankar 1980) SOLA-ICE (Cloutman et al. 1976). 

In 1914, F. W. Lanchester introduced a set of coupled ordinary differential equations-now commonly called the Lanchester Equationsl (LEs)-as models of attrition in modern warfare. Similar ideas were proposed around that time by [chaseSS] and [osip95]. These equations are formally equivalent to the Lotka-Volterra equations used for modeling the dynamics of interacting predator-prey populations [hof98]. The LEs have since served as the fundamental mathematical models upon which most modern theories of combat attrition are based, and are to this day embedded in many state-of-the-art military models of combat. [Taylor] provides a thorough mathematical discussion. 

Each of the processes shown in Figure 2.8 can be described by a Michaelis-Menten type of biochemical reaction, a standard generalized mathematical equation describing the interaction of a substrate with an enzyme. Michaelis and Men ten realized in 1913 that the kinetics of enzyme reactions differed from the kinetics of conventional... 

While mathematically attractive, this force law is of limited interest physically it represents only the interaction between permanent quadrupoles, and even this with neglect of angles of orientation. However, although the details of the dependence of viscosity upon temperature are affected by the force law used, the general form of the hydrodynamic equation in the Navier-Stokes approximation is not affected. 

It should be stressed, however, that the introduction of the operator 2(k) in the present context is purely for mathematical convenience. All the subsequent development could also be carried out without its introduction. It is only when we consider the interaction of the quantized electromagnetic field with charged particles that the potentials assume new importance—at least in the usual formulation with its particular way of fixing the phase factors in the operators of the charged fields—since the potentials themselves then appear in the equations of motion of the interacting electromagnetic and matter fields. 

Fugacity of a Component in a Gaseous Mixture One could guess that the determination of fugacities, /, for the individual components in a gaseous mixture can become complicated as one takes into account the different types of interactions that are present. The mathematical relationship that applies is obtained by starting with the defining equations... 

The parameter a in Equation (11.6) is positive for electrophobic reactions (5r/5O>0, A>1) and negative for electrophilic ones (3r/0Oelectrochemical promotion behaviour is frequently encountered, leading to volcano-type or inverted volcano-type behaviour. However, even then equation (11.6) is satisfied over relatively wide (0.2-0.3 eV) AO regions, so we limit the present analysis to this type of promotional kinetics. It should be remembered thatEq. (11.6), originally found as an experimental observation, can be rationalized by rigorous mathematical models which account explicitly for the electrostatic dipole interactions between the adsorbates and the backspillover-formed effective double layer, as discussed in Chapter 6. 

The time taken for an analyte to elute from a chromatographic column with a particular mobile phase is termed its retention time, fan- Since this will vary with column length and mobile phase flow rate, it is more useful to use the capacity factor, k. This relates the retention time of an analyte to the time taken by an unretained compound, i.e. one which passes through the column without interacting with the stationary phase, to elute from the column under identical conditions (to). This is represented mathematically by the following equation ... 

In chemical kinetics, one finds linked sets of differential equations expressing the rates of change of the interacting species. Overall, mathematical models have been exceedingly successfiil in depicting the broad outlines of an enormously diverse variety of phenomena in nature. Some scientists have even commented in surprise at how well mathematics works in describing nature. So successful have these mathematical models been that their use has spread from the hard sciences to areas as diverse as economics and the analysis of athletic performance [3]. 

Both of the above approaches rely in most cases on classical ideas that picture the atoms and molecules in the system interacting via ordinary electrical and steric forces. These interactions between the species are expressed in terms of force fields, i.e., sets of mathematical equations that describe the attractions and repulsions between the atomic charges, the forces needed to stretch or compress the chemical bonds, repulsions between the atoms due to then-excluded volumes, etc. A variety of different force fields have been developed by different workers to represent the forces present in chemical systems, and although these differ in their details, they generally tend to include the same aspects of the molecular interactions. Some are directed more specifically at the forces important for, say, protein structure, while others focus more on features important in liquids. With time more and more sophisticated force fields are continually being introduced to include additional aspects of the interatomic interactions, e.g., polarizations of the atomic charge clouds and more subtle effects associated with quantum chemical effects. Naturally, inclusion of these additional features requires greater computational effort, so that a compromise between sophistication and practicality is required. 

For freely suspended bioparticles the most likely flow stresses are perceived to be either shear or normal (elongation) stresses caused by the local turbulent flow. In each case, there are a number of ways of describing mathematically the interactions between turbulent eddies and the suspended particles. Most methods however predict the same functional relationship between the prevailing turbulent flow stresses, material properties and equipment parameters, the only difference between them being the constant of proportionality in the equations. Typically, in the viscous dissipation subrange, theory suggests the following relationship for the mean stress [85] ... 

The beauty of finite-element modelling is that it is very flexible. The system of interest may be continuous, as in a fluid, or it may comprise separate, discrete components, such as the pieces of metal in this example. The basic principle of finite-element modelling, to simulate the operation of a system by deriving equations only on a local scale, mimics the physical reality by which interactions within most systems are the result of a large number of localised interactions between adjacent elements. These interactions are often bi-directional, in that the behaviour of each element is also affected by the system of which it forms a part. The finite-element method is particularly powerful because with the appropriate choice of elements it is easy to accurately model complex interactions in very large systems because the physical behaviour of each element has a simple mathematical description. 

He describes molecular populations mathematically in the way physicists calculate classical dynamic systems. Very exact dynamic equations are devised, while the laws of interaction are left very general. This leads to a general theory of molecular systems, which makes it possible to define what is understood by the origin of metabolism (Dyson, 1999). 

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