Functions mathematical

Many basic mathematical functions already exists(FFT, loops, booleans,. ), which simplifies the work of the programmer. 

W.Gautschi, in Handbook of Mathematical Functions, edited by Abramowitz M. and Stegun I.A. (Dover, New York,1964), p.295. 

Figure 7-9 shows two mathematical functions used to describe the potential energy cuiwe for the variation of the distance between two bound atoms within a molecule. 

In the style of the Darwinian Theory, the quality of a chromosome is called its fitness. The quality or fitness of a ehromosome is usually caleulated with the help of an objeetive function, which is a mathematical function indicating how good the solution, and thus the chromosome, is for the optimization problem. This computation of the fitness is done for each chromosome in each population,... 

Th is discussion focuses on th e individual compon en ts of a typical molecular mechanics force field. It illustrates the mathematical functions used, wdi y those functions are chosen, and the circiim -Stan ces u n der wh ich the fun ction s become poor approxirn atiori s. Part 2 of th is book, Theory and Melhadx, includes details on the implementation of the MM+,. AM BHR, RlO-g and OPl.S force fields in HyperChem. 

We do not know the orbitals of the electrons either. (An orbital, by the way, is not a ball of fuzz, it is a mathematical function.) We can reasonably assume that the ground-state orbitals of electrons I and 2 are similar but not identical to the Is orbital of hydrogen. The Slater-type orbitals... 

The second approximation in HF calculations is due to the fact that the wave function must be described by some mathematical function, which is known exactly for only a few one-electron systems. The functions used most often are linear combinations of Gaussian-type orbitals exp(—nr ), abbreviated GTO. The wave function is formed from linear combinations of atomic orbitals or, stated more correctly, from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree-Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO—3G or 6—311++g. Basis sets are discussed further in Chapters 10 and 28. 

Gaussian theory (Gl, G2, G3) a method for extrapolating from ah initio results to an estimation of the exact energy Gaussian-type orbital (GTO) mathematical function for describing the wave function of an electron in an atom... 

Attempts have been made to devise mathematical functions to represent the distributions that are found experimentally. The mathematical treatment is necessarily based on the assumption that the number of particles in the sample is large enough for statistical considerations to be applicable. With the SOO-member sample of the previous section one could not expect any more than approximate agreement between mathematical prediction and experiment. 

Analog signal. A signal that can be expressed as a continuously variable mathematical function of time. 

A continuous distribution function is a mathematical function which gives N as a function of M. This is the most general way of describing the... 

Abramowitz, M., and I. A. Stegun. Handbook of Mathematical Functions, National Bureau of Standards, Washington, D.C. (1964). 

Luke, Y. L. Mathematical Functions and Their Applications, Academic, New York (1975). 

Normal Distribution of Observations Many types of data follow what is called the gaussian, or bell-shaped, curve this is especially true of averages. Basically, the gaussian curve is a purely mathematical function which has very specif properties. However, owing to some mathematically intractable aspects primary use of the function is restricted to tabulated values. 

Distillation columns have four or more closed loops—increasing with the number of product streams and their specifications—all of which interact with each other to some extent. Because of this interaction, there are many possible ways to pair manipulated and controlled variables through controllers and other mathematical functions with widely differing degrees of effectiveness. Columns also differ from each other, so that no single rule of configuring control loops can be apphed successfully to all. The following rules apply to the most common separations. 

The unknown parameters of the model, such as film thicknesses, optical constants, or constituent material fractions, are varied until a best fit between the measured P and A and the calculated P/ and A/ is found, where m signifies a quantity that is measured. A mathematical function called the mean squared error (MSE) is used as a measure of the goodness of the fit ... 

The following are some of the mathematical functions used by LINGO ... 

While orbitals may be useful for qualitative understanding of some molecules, it is important to remember that they are merely mathematical functions that represent solutions to the Hartree-Fock equations for a given molecule. Other orbitals exist which will produce the same energy and properties and which may look quite different. There is ultimately no physical reality which can be associated with these images. In short, individual orbitals are mathematical not physical constructs. 

The purpose of our study was to model the steady-state (capillary) flow behavior of TP-TLCP blends by a generalized mathematical function based on some of the shear-induced morphological features. Our attention was primarily confined to incompatible systems. 

Abramowitz, M., andSegun, I.A., 96A,Handbookof Mathematical Functions, Dover, New York, fifth edition. 

Choosing models that have parameters that can be related to biology is preferable to generic mathematical functions that may give better fits. 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

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