Mathematical relationships

Ohm s law says that applying a potential V across an electrical resistor R induces a proportional current I. We can state this relationship mathematically as... 

As briefly mentioned in Section 6.4.3, the P-1 Method is sometimes based on empirical relationships. Mathematical expressions of P-1 damage curves are derived from test results. Refer to Baker 1983 and FACEDAP 1994 for further details. 

The potential energy surface (PES) is a central concept in computational chemistry. A PES is the relationship - mathematical or graphical - between the energy of a molecule (or a collection of molecules) and its geometry. 

Jacques Charles observed that for a fixed quantity of gas, the volume at constant pressure changes when temperature changes the volume increases (V f ) when the temperature increases (T f ) the volume decreases (V ) when the temperature decreases (T j). Although first described by Charles in 1787, it was not until 1802 that Joseph Gay-Lussac expressed the relationship mathematically. 

Express this relationship mathematically. To help you to do this, look at the mathematical form of Boyle s law, located after this investigation. 

In other words, as the pressure of a closed system increases, its volume decreases. If the pressure is decreased by half, the volume doubles. We can write this relationship mathematically by using the proportionality symbol, a. V a 1 IP means that volume is inversely proportional to the pressure. 

Schneider, F., Burrus, J., Wolf, S., Doligez, B. and P. Forbes, 1991. Modelling overpressures by efFective-stress/porosity relationship mathematical artefice or physical reality. Paper presented at the Norwegian Petroleum Society International Conference on Basin modelling Advances and applications, Stavanger, Norway, March 13-15, 1991 Schowalter, T.T., 1979. Mechanics of secondary hydrocarbon migration and entrapment. 

Quantitative Structure-Activity Relationships Mathematical (Empirical) Model - Employing the method of Free and Wilson, Beasley and Purcell have reported the successful prediction of the butyriicholinesterase inhibitory potency of l-decyl-3-(N-ethyl-N-methylcarbamoyl)piperidine hydrobromide. Three years after the predicted biochemical response was published, this compound was synthesized and evaluated biochemically. The observed response was found to be quantitatively near the predicted value. ... 

This chapter starts with some definitions, an important one being the thermodynamic system, and the macroscopic variables that characterize it. If we are considering a gas in our system, we will find that various mathematical relationships are used to relate the physical variables that characterize this gas. Some of these relationships— gas laws —are simple but inaccurate. Other gas laws are more complicated but more accurate. Some of these more complicated gas laws have experimentally determined parameters that are tabulated to be looked up later, and they may or may not have physical justification. We develop some relationships (mathematical ones) using some simple calculus. These mathematical manipulations will be useful in later chapters as we get deeper into thermodynamics. Finally, we introduce thermodynamics from a molecular point of view, because an acceptable model of thermodynamics must connect to the atomic theory of matter. 

Propositions have also formal logic as objects of a formal language. The form of a proposition depends on the type of logic. The elements of such language are either variables (cause-reason-input or effect-result-output), predictive relationships, mathematical symbols and operators, functions, quantifiers and constants. One can propose his/her view of point about an event and then s/he must defend it against the critical debates of others. 

In order to obtain structure-biological activity relationships mathematic methods have been used [191-193]. Quantitative correlations between molecular structure and pharmacokinetic and pharmacodynamic characteristics of fluoroquinolones in combination with informative hanometric approach have been used to forecast anti-pneumococcus activity [194]. Elucidation of the structure - activity relationships in the series of fluoroquinolones is the subject of numerous publications [195-197]. Dependence of antibacterial activity on the nature of substituents has been established for several series of bicyclic fluoroquinolones [11,198-200]. 

For each mechanism of inhibited oxidation, [RO ] can be related to InH and ROOM, expressing this relationship mathematically and solving the system of two differential equations, which describe oxygen absorption and inhibitor consumption, and to express the amount of absorbed oxygen through the amount of the consumed inhibitor. For example, in the case of mechanism V when reactions (8), (31), (-31), and (33) are key (see Table 11.1), we obtain... 

Power refers to the less dependent firm s abihty to influence the more dependent party s decision making. Though there are multiple conceptualizations of power in the extant hterature, we focus on the power derived fi om the relative dependence in a dyadic relationship. Mathematically, the source of power can be calculated as the difference between the dependence of firm A on B and the dependence of firm B on A. If the later is higher then firm A has power over firm B (Pfeffer, 1992). When the two firms are symmetrically dependent on each other, neither party has power over the other. 

Once the production potential of the producing wells is insufficient to maintain the plateau rate, the decline periodbegins. For an individual well in depletion drive, this commences as soon as production starts, and a plateau for the field can only be maintained by drilling more wells. Well performance during the decline period can be estimated by decline curve analysis which assumes that the decline can be described by a mathematical formula. Examples of this would be to assume an exponential decline with 10% decline per annum, or a straight line relationship between the cumulative oil production and the logarithm of the water cut. These assumptions become more robust when based on a fit to measured production data. 

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

The next and very important step is to make a decision about the descriptors we shall use to represent the molecular structures. In general, modeling means assignment of an abstract mathematical object to a real-world physical system and subsequent revelation of some relationship between the characteristics of the object on the one side, and the properties of the system on the other. 

Note that the mathematical symbol V stands for the second derivative of a function (in this case with respect to the Cartesian coordinates d fdx + d jdy + d jdz y, therefore the relationship stated in Eq. (41) is a second-order differential equation. Only for a constant dielectric Eq.(41) can be reduced to Coulomb s law. In the more interesting case where the dielectric is not constant within the volume considered, the Poisson equation is modified according to Eq. (42). 

Building a QSPR model consists of three steps descriptor calculation, descriptor analysis and optimization, and establishment of a mathematical relationship between descriptors and property. 

A series of monographs and correlation tables exist for the interpretation of vibrational spectra [52-55]. However, the relationship of frequency characteristics and structural features is rather complicated and the number of known correlations between IR spectra and structures is very large. In many cases, it is almost impossible to analyze a molecular structure without the aid of computational techniques. Existing approaches are mainly based on the interpretation of vibrational spectra by mathematical models, rule sets, and decision trees or fuzzy logic approaches. 

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]. 

When structure-property relationships are mentioned in the current literature, it usually implies a quantitative mathematical relationship. Such relationships are most often derived by using curve-fitting software to find the linear combination of molecular properties that best predicts the property for a set of known compounds. This prediction equation can be used for either the interpolation or extrapolation of test set results. Interpolation is usually more accurate than extrapolation. 

Another way of predicting liquid properties is using QSPR, as discussed in Chapter 30. QSPR can be used to And a mathematical relationship between the structure of the individual molecules and the behavior of the bulk liquid. This is an empirical technique, which limits the conceptual understanding obtainable. However, it is capable of predicting some properties that are very hard to model otherwise. For example, QSPR has been very successful at predicting the boiling points of liquids. 

Very often in practice a relationship is found (or known) to exist between two or more variables. It is frequently desirable to express this relationship in mathematical form by determining an equation connecting the variables. 

How do we find the best estimate for the relationship between the measured signal and the concentration of analyte in a multiple-point standardization Figure 5.8 shows the data in Table 5.1 plotted as a normal calibration curve. Although the data appear to fall along a straight line, the actual calibration curve is not intuitively obvious. The process of mathematically determining the best equation for the calibration curve is called regression. 

Theoretical Models of the Response Surface Mathematical models for response surfaces are divided into two categories those based on theory and those that are empirical. Theoretical models are derived from known chemical and physical relationships between the response and the factors. In spectrophotometry, for example, Beer s law is a theoretical model relating a substance s absorbance. A, to its concentration, Ca... 

This last m/z value is easy to measure accurately, and, if its relationship to the true mass is known (n = 10), then the true mass can be measured very accurately. The multicharged ions have typical m/z values of <3000 Da, which means that conventional quadrupole or magnetic-sector analyzers can be used for mass measurement. Actually, the spectrum consists of a series of multicharged protonated molecular ions [M + nWY for each component present in the sample. Each ion in the series differs by plus and minus one charge from adjacent ions ([M + uH] + n -an integer series for example, 1, 2, 3,. .., etc.). Mathematical transformation of the spectrum produces a true molecular mass profile of the sample (Figure 40.5). 

A formal mathematical analysis of the flow in the concentric cylinder viscometer yields the following relationship between the experimental variables and the viscosity ... 

There is probably no area of science that is as rich in mathematical relationships as thermodynamics. This makes thermodynamics very powerful, but such an abundance of riches can also be intimidating to the beginner. This chapter assumes that the reader is familiar with basic chemical and statistical thermodynamics at the level that these topics are treated in physical chemistry textbooks. In spite of this premise, a brief review of some pertinent relationships will be a useful way to get started. 

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