Dependent variable from mathematical function

Any governing model equations have to be supplemented by initial and boundary conditions, all together called side conditions. Their definition means imposing certain conditions on the dependent variable and/or functions of it (e.g. its derivative) on the boundary (in time and space) for uniqueness of solution. A proper choice of side conditions is crucial and usually represents a significant portion of the computational effort. Simply speaking, boundary conditions are the mathematical description of the different situations that occur at the boundary of the chosen domain that produce different results within the same physical system (same governing equations). A proper and accurate specification of the boundary conditions is necessary to produce relevant results from the calculation. Once the mathematical expressions of all boundary conditions are defined the so-called properly-posed problem is reached. Moreover, it must be noted that in fuel cell modeling there are various... 

A mathematical function of one independent variable is a rule that generates a unique value of a dependent variable from a given value of an independent variable. It is as though... 

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... 

The function ij/(r, 9, p) (clearly ij/ could also be expressed in Cartesians), depends functionally on r, 6, p and parametrically on n, l and inm for each particular set (n. I, mm ) of these numbers there is a particular function with the spatial coordinates variables r, 6, p (or x, y, z). A function like /rsiiir is a function of x and depends only parametrically on k. This ij/ function is an orbital ( quasi-orbit the term was invented by Mulliken, Section 4.3.4), and you are doubtless familiar with plots of its variation with the spatial coordinates. Plots of the variation of ij/2 with spatial coordinates indicate variation of the electron density (recall the Bom interpretation of the wavefunction) in space due to an electron with quantum numbers n, l and inm. We can think of an orbital as a region of space occupied by an electron with a particular set of quantum numbers, or as a mathematical function ij/ describing the energy and the shape of the spatial domain of an electron. For an atom or molecule with more than one electron, the assignment of electrons to orbitals is an (albeit very useful) approximation, since orbitals follow from solution of the Schrodinger equation for a hydrogen atom. 

Partial and total order ranking strategies, which from a mathematical point of view are based on elementary methods of Discrete Mathematics, appear as an attractive and simple tool to perform data analysis. Moreover order ranking strategies seem to be a very useful tool not only to perform data exploration but also to develop order-ranking models, being a possible alternative to conventional QSAR methods. In fact, when data material is characterised by uncertainties, order methods can be used as alternative to statistical methods such as multiple linear regression (MLR), since they do not require specific functional relationship between the independent variables and the dependent variables (responses). 

Continuous functions are ones in which the dependent variable changes smoothly and continuously for smooth and continuous changes of the independent variables. Figures 2.1 and 2.2 represent continuous functions, but Figure 2.3 represents a function which is continuous for x 7 a but shows a discontinuity from —00 to -hoo at X = a. The mathematical definition of continuity is that /(x) is continuous at x = a if /(a) is defined, and if lima a /(x) = /(a). 

Simple linear regression analysis provides bivariate statistical tools essential to the applied researcher in many instances. Regression is a methodology that is grounded in the relationship between two quantitative variables (y, x) such that the value of y (dependent variable) can be predicted based oti the value of X (independent variable). Determining the mathematical relationship between these two variables, such as exposure time and lethality or wash time and logic microbial reductions, is very common in applied research. From a mathematical perspective, two types of relationships must be discussed (1) a functional relationship and (2) a statistical relationship. Recall that, mathematically, a functional relationship has the form... 

When the dependence of the spectroscopic intensity from every chro-mophore on at least one experimental variable can be described by a highly specific mathematical function, then the approach known as global analysis is preferred. When this condition is not known to be met, but spectroscopic intensity is separately linear in functions of two or more experimental variables, then the multilinear models described in this chapter are valuable. 

Transfer function A mathematical function describing the numerical relationship between one/several independent variables and one (several) dependent variable(s). Commonly used to infer past values of an environmental variable (e.g., pH, salinity) from the composition of fossil assemblages (e.g., diatom abundances). 

An object from the BzzNo IL inearRegress ion class includes numerous constructors and functions that allow us to select the most appropriate criterion to estimate the model parameters. In the following example, the target is to minimize the deviation between the mathematical model and the experimental data. The function where the dependent variables are calculated must have the following argument ... 

From the values of dependent variables yj obtained from experimental results (Table 2), the values of the coefficients of the regressive function (1) were calculated (Table 3). As the values of the indexes of multiple correlation I y show, the mathematical model describes the effects of the system composition on its characteristics in an extremely fitting manner. An exception is the case of elongation at break (yg) where the value of I, y is... 

Usually, in these kinds of titrations, the quantity that is experimentally measured as a function of the volume v of titrant solution added is pAg = —log[Ag+] (Fig. 36.2). A good reason to choose pAg as the dependent variable lies in the fact that it is directly given by a silver electrode. [To be rigorous from a physical point of view, pAg (whose value is given by a silver electrode) is actually —log( Ag+) and not —log[Ag+] (furthermore, from the mathematical standpoint, the argument of a logarithm must be a dimensionless number).]... 

If a function is continuous, the dependent variable does not change abruptly for a small change in the independent variable. If you are drawing a graph of a continuous function, you will not have to draw a vertical step in your curve. We define continuity with a mathematical limit. In a limiting process, an independent variable is made to approach a given value, either from the positive side or the negative side. We say that a function f x) is continuous at x = a if... 

It is commonly observed that the temperature and frequency dependence of polymer relaxations are related. This is expressed qualitatively as the time-temperature superposition principle, or the frequency-temperature equivalence, or the method of reduced variables. A mathematical way to describe this behavior is to note that if the dispersion relation for the relaxation [eqs. 29,30, and 44] depends on frequency and temperature only through the product of frequency and a function of temperature, cor T), then the effect of a change in frequency is indistinguishable from a change in temperature. In other words, a measurement... 

In the sixteenth century, as a consequence of problems of motion, mathematics entered into the study of variable magnitudes and functions. In this instance, the abstract concepts of "variables" and "functions" have their real counterparts in the mutual dependencies of such parameters as time, distance, and velocity. From this concern with functions grew the branch of mathematics termed "analysis."... 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

The first difficulty derives from the fact that given any values of the macroscopic expected values (restricted only by broad moment inequality conditions), a probability density always exists (mathematically) giving rise to these expected values. This means that as far as the mathematical framework of dynamics and probability goes, the macroscopic variables could have values violating the laws of phenomenological physics (e.g., the equation of state, Newton s law of heat conduction, Stokes law of viscosity, etc.). In other words, there is a macroscopic dependence of macroscopic variables which reflects nothing in the microscopic model. Clearly, there must exist a principle whereby nature restricts the class of probability density functions, SF, so as to ensure the observed phenomenological dependences. 

To simplify mathematical manipulations, let us consider now the case of equal diffusion coefficients, Da = D, in which case the similar correlation functions just coincide, Xv r),T) = X(t),t). Taking into account the definition of correlation length Id = VDt, where D = Da + D = 2D a, as well as time-dependence of new variables r) and r, one gets from (5.1.2) to (5.1.4) a set of equations... 

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