Mathematical derivative

Use of random flight statistics to derive rg for the coil assumes the individual segments exclude no volume from one another. While physically unrealistic, this assumption makes the derivation mathematically manageable. Neglecting this volume exclusion means that coil dimensions are underestimated by the random fight model, but this effect can be offset by applying the result to a solvent in which polymer-polymer contacts are somewhat favored over polymer-solvent contacts. 

The amount by which a plume rises above the top of a chimney can be derived mathematically. 

In his theory, A.F. Huxley suggested that a myosin crossbridge attaches to actin and produces an amount of force which is proportional to the amount of strain on the crossbridge, that is to the amount by which it is distorted (Figure 5). He also suggested that crossbridges were only able to attach over a range of preferred distortions. From these basic ideas he derived mathematical expressions for force... 

Atmospheric Reaeration. Interfacial properties and phenomena that govern oxygen concentrations in river systems include 1) oxygen solubility (temperature, partial pressure and surface dependency), 2) rate of dissolution of oxygen (saturation level, temperature and surface thin film dependency, i.e., ice, wind), and 3) transport of oxygen via mixing and molecular diffusion. A number of field and empirically derived mathematical relationships have been developed to describe these processes and phenomena, the most common of which is (32) ... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

In lack of analytical or numerical methods to obtain the spectra of complicated Hamiltonians, Wigner and Dyson analyzed ensembles of random matrices and were able to derive mathematical expressions. A Gaussian random matrix ensemble consists of square matrices with their matrix elements drawn from a Gaussian distribution... 

In order to derive mathematical equations able to describe the movement of a species towards or from an electrode surface it would be necessary to know the physical laws which govern the three modes of the mass transport. 

The conditions for synergism in surface tension reduction efficiency, mixed micelle formation, and Surface tension reduction effectiveness in aqueous solution have been derived mathematically together with the properties of the surfactant mixture at the point of maximum synergism. This treatment has been extended to liquid-liquid (aqueous solution/hydrocarbon) systems at low surfactant concentrations.) The effect of chemical structure and molecular environment on the value of B is demonstrated and discussed. 

One approach to developing mathematical models is to begin with one that contains a relatively detailed description of the physical system and then to derive simpler models by identifying those elements that can be approximated while still retaining the essential behavior of the system (see, for example, Aris, 1978). This is the approach that we will follow here. Our particular interest will be in deriving mathematical models of packed bed reactors that are appropriate for use in designing control systems. Thus, we will be interested in models capable of simulating dynamic behavior. 

Mathematical models do not, in a literal sense, provide new information. They are rather a means for making more effective use of the information from which they were derived. Mathematical models, just as physical models, are rarely perfect in the accuracy and completeness of their correspondence to a real situation. When based largely on empirical evidence, the model must be restricted in use to the range of the original data. Often, with more effort and cost, a model can be improved, but there is always a point beyond which further refinement is unwarranted for the purpose at hand. 

If the variation were completely unpredictable, there would be no hope of rational planning to take it into account. Usually, however, although it is not possible to predict that a given occurrence will certainly happen, it is possible to assign a probability for any particular occurrence. If this is done for all possible occurrences, then, in effect, a probability distribution function has been defined. Certain types of such distributions can be derived mathematically to fit special situations. The normal, Poisson, and binomial distributions are frequently encountered in practice. 

In contrast to the mechanical and rheological properties of materials, which have defined physical meanings, no such definitions exist for the psychophysical assessment of equivalent textural properties of foods. To identify material properties, or combinations of these, which are able to model sensory assessments requires a mixture of theory and experimentation. Scientific studies of food texture began during the twentieth century by the analysis of the rheological properties of liquid or semi-solid foods. In particular Kokini14 combined theoretical and experimental approaches in order to identify appropriate rheological parameters from which to derive mathematical models for textural attributes of liquid and semi-solid foods, namely, thickness, smoothness and creaminess. 

Although the values of K will depend on the nature of Q, Z, T, and v, there is, for each v, a special situation, the ideal random case, where the sorting of the substituents about the central atom follows the laws of random statistics. For this case, the Z and T substituents become arranged about the Q in a completely random fashion irrespective of other substituents which are attached to Q. The K values for the ideal random case may be derived mathematically. If it is assumed that the Z/T atom ratio is p/q (with p + q = 1), the probability P of having a central atom Q with i Z substituents and (v — i) T substituents in the v sites subject to redistribution is... 

Structural analysis, initially developed on an intuitive basis, later became identified with variational calculus, in which the Ritz procedure is used to minimize a functional derived mathematically or arrived at directly from physical principles. By substituting the final solutions into the variational statement of the problem and minimizing the latter, the FEM equations are obtained. Example 15.2 gives a very simple demonstration of this procedure. 

In the late 1700s, Charles Augustin Coulomb derived mathematical equations for attraction and repulsion between electrically charged objects. 

With the help of the Maxwell equation Eq. 3.8, it is possible to derive mathematical expressions for the three types of velocities, viz., the most probable velocity, cp the average velocity, and the root-mean-square velocity, /2. These expressions are as follows ... 

Molecular orbitals (MOs) are derived mathematically by a linear combination of the wavefunctions for the atomic orbitals (AOs) of the individual atoms in a molecule. Usually, only the atomic orbitals of the valence electrons are considered, because these are the electrons involved in bonding. We can visualize the formation of MOs as proceeding from overlap of the AOs of the valence electrons. 

Patel S, Kaushal A. Bansal A. Effect of particle size and compression force on compaction behavior and derived mathematical parameters of compressibility. Pharm. Res. 2007 24(1), 111-24. 

There are three classes of SI units base, derived, and supplemental units. A base unit was decreed by the CGPM for each of eight fundamental but dimensionally independent physical quantities. These units are listed in Table 1-2. A derived unit is derived mathematically from two or more base units (Table 1 -3). A supplemental unit is a unit that conforms to the SI but that has not been classified as either base or derived. At present only the radian (for plane angles) and the steradian (for solid angles) are classified this way. 

The closed-loop gains can be derived mathematically from the open-loop gains. For a two by two process, the variations AC, and ACj resulting from changes in both manipulated variables are given by the equations... 

The fundamental reaction mechanism for the free-radical oxidation of hydrocarbons has been used to relate the consumption of oxygen to the formation of oxidation products in polypropylene. A kinetic interpretation is based on the steady-state approximation equating the rates of the initiation and termination reactions. With this approach it is possible to derive mathematical equations describing the consumption of oxygen or the formation of specific oxidation products. To solve the equations it is necessary to determine the most likely route for initiation of oxidation. The initiation mechanism chosen is the bimolecular reaction of hydroperoxides, reaction (1 ) of Scheme 1.55, with a rate coefficient k. ... 

Distribution of Total Number of Premalignant Cells Conditional on No Prior Malignancies. To model screening for premalignant lesions in asymptomatic individuals with no prior history of cancer. Jeon et al. (2008) derived mathematical expressions for the size distribution of premalignant lesions, conditional on no prior malignancy in the tissue of interest. 

Ransohoff and Radke [1988] derived mathematical expressions for average liquid velocity in comers bounded by liquid-vapor interfaces. They used a detailed numerical scheme to solve the Navier-Stokes equations for the assumed geometry and boundary conditions. Their results were reduced to the general form ... 

---