Mathematical techniques

Spectral features and their corresponding molecular descriptors are then applied to mathematical techniques of multivariate data analysis, such as principal component analysis (PCA) for exploratory data analysis or multivariate classification for the development of spectral classifiers [84-87]. Principal component analysis results in a scatter plot that exhibits spectra-structure relationships by clustering similarities in spectral and/or structural features [88, 89]. 

Dence, J. B., 1975. Mathematical Techniques in Chemistry, Wiley, New York. 

The reaction coordinate is one specific path along the complete potential energy surface associated with the nuclear positions. It is possible to do a series of calculations representing a grid of points on the potential energy surface. The saddle point can then be found by inspection or more accurately by using mathematical techniques to interpolate between the grid points. 

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

So what is the total uncertainty when using this pipet to deliver two successive volumes of solution from the previous discussion we know that the total uncertainty is greater than 0.000 mL and less than 0.012 mL. To estimate the cumulative effect of multiple uncertainties, we use a mathematical technique known as the propagation of uncertainty. Our treatment of the propagation of uncertainty is based on a few simple rules that we will not derive. A more thorough treatment can be found elsewhere. ... 

A mathematical technique for fitting an equation, such as that for a straight line, to experimental data. 

Both equation 11 and the two-dimensional counterpart of equation 9 can be solved by several standard mathematical techniques, one of the more useful being that of conformal mapping. A numerical solution is often more practical for compHcated configurations. 

In some cases, however, it is possible, by analysing the equations of motion, to determine the criteria by which one flow pattern becomes unstable in favor of another. The mathematical technique used most often is linearised stabiHty analysis, which starts from a known solution to the equations and then determines whether a small perturbation superimposed on this solution grows or decays as time passes. 

Heat Exchangers Using Non-Newtonian Fluids. Most fluids used in the chemical, pharmaceutical, food, and biomedical industries can be classified as non-Newtonian, ie, the viscosity varies with shear rate at a given temperature. In contrast, Newtonian fluids such as water, air, and glycerin have constant viscosities at a given temperature. Examples of non-Newtonian fluids include molten polymer, aqueous polymer solutions, slurries, coal—water mixture, tomato ketchup, soup, mayonnaise, purees, suspension of small particles, blood, etc. Because non-Newtonian fluids ate nonlinear in nature, these ate seldom amenable to analysis by classical mathematical techniques. 

Mathematical and Computational Implementation. Solution of the complex systems of partial differential equations governing both the evolution of pollutant concentrations and meteorological variables, eg, winds, requires specialized mathematical techniques. Comparing the two sets of equations governing pollutant dynamics (eq. 5) and meteorology (eqs. 12—14) shows that in both cases they can be put in the form ... 

H. C. Andrews, Introduction to Mathematical Techniques in Pattern Recognition, Wiley-Interscience, New York, 1972. 

Most flow sheets have one or mote recycles, and trial-and-ettot becomes necessary for the calculation of material and energy balances. The calculations in a block sequential simulator ate repeated in this trial-and-ettot process. In the language of numerical analysis, this is known as convergence of the calculations. There ate mathematical techniques for speeding up this trial-and-ettot process, and special hypothetical calculation units called convergence, or recycle, units ate used in calculation flow diagrams that invoke special calculation routines. 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

With many variables and constraints, linear and nonlinear programming may be applicable, as well as various numerical gradient search methods. Maximum principle and dynamic programming are laborious and have had only limited applications in this area. The various mathematical techniques are explained and illustrated, for instance, by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-Hill, 1988). 

Westerterp et al. (1984 see Case Study 4, preceding) conclude, Thanks to mathematical techniques and computing aids now available, any optimization problem can be solved, provided it is reahstic and properly stated. The difficulties of optimization lie mainly in providing the pertinent data and in an adequate construc tion of the objective function. ... 

The development of a quantitative estimate of risk based on engineering evaluation and mathematical techniques for combining estimates of incident consequences and frequencies... 

Developments in experimental and mathematical techniques in the 1970s have initiated an interest in the development of better laboratory reactors for catal5d ic studies. Besides the many publications on new reactors for general or special tasks, quite a few review articles have been published on the general subject of laboratory reactors for catalytic studies. 

For different regions in the flow field in front of an expanding piston, separate solutions in the form of asymptotic expansions may be developed. An overall solution can be constructed by matching these separate solutions. This mathematical technique was employed by several authors including Guirao et al. (1976), Gorev and Bystrov (1985), Deshaies and Clavin (1979), Cambray and Deshaies (1978), and Cambray et al. (1979). 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

If V is not a function of time, the Schrodinger equation can be simplified using the mathematical technique known as separation of variables. If we write the wavefunction as the product of a spatial function and a time function ... 

Six isotopes of element 106 are now known (see Table 31.8) of which the most recent has a half-life in the range 10-30 s, encouraging the hope that some chemistry of this fugitive species might someday be revealed. This heaviest isotope was synthsised by the reaction Cm( Ne,4n) 106 and the present uncertainty in the half-life is due to the very few atoms which have so far been observed. Indeed, one of the fascinating aspects of work in this area is the development of philosophical and mathematical techniques to define and deal with the statistics of a small number of random events or even of a single event. 

Frequency-domain data are obtained by converting time-domain data using a mathematical technique referred to as Fast Fourier Transform (FFT). FFT allows each vibration component of a complex machine-train spectrum to be shown as a discrete frequency peak. The frequency-domain amplitude can be the displacement per unit time related to a particular frequency, which is plotted as the Y-axis against frequency as the X-axis. This is opposed to time-domain spectrums that sum the velocities of all frequencies and plot the sum as the Y-axis against time... 

Mathematical techniques allow us to quantify total displacement caused by all vibrations, to convert the displacement measurements to velocity or acceleration, to separate this data into its components using FFT analysis, and to determine the amplitudes and phases of these functions. Such quantification is necessary if we are to isolate and correct abnormal vibrations in machinery. 

A mathematical technique used to convert a time-domain plot into its unique frequency components. 

Using the mathematical technique of dimensionless group analysis, the rate of mass transport (/ m) in terms of moles per unit area per unit time can be shown to be a function of these variables, which when grouped together can be related to the rate by a power term. For many systems under laminar flow conditions it has been shown that the following relationship holds ... 

The chromatographic system should, wherever possible, be optimized to obtain complete resolution of the mixture and not place reliance on mathematical techniques to aid in the analysis. 

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