Functions of two variables

We shall now proceed to investigate potential functions K(x c) dependent on many state variables and on parameters, where x = (xj. x ), c = (cl5. ck). 

Similarly to the case of functions of on variable, three fundamental situations may take place  

From the standpoint of elementary catastrophe theory, the functions having degenerate critical points are most interesting. As follows from Section 2.2, in gradient systems catastrophes may happen only in a case when the system is described by a potential function having a degenerate critical point. 

Such a function may be included into a structurally stable parameter--dependent family of functions which will be considered to be a potential function. The state of a physical system will be determined from the condition of the minimum of a potential function having a degenerate critical point, defining the catastrophe surface M. 

It will appear that on evolution of a system generated by continuous variations of control parameters and proceeding on the catastrophe surface a continuous evolution of the system on this surface is not always possible. In other words, in some circumstances a catastrophe must occur — the system has to leave for some time the surface of the potential minimum. 

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According to this concept, a reduced property is expressed as a function of two variables, and I/, and of the acentric factor, cd ... 

Thus, the harmonic function >P(2 ,y) is a function of two variables which can be determined from the boundary conditions. This follows also from the fact that If the distribution of is described only by harmonic functions, the other stress components do not develop in cylinders [2]. 

A TOFD or B-Scan image is a discrete image defined as a function/of two variables on a finite and discrete domain D of dimensions MxN. 

The Co-Occurrence Matrix is a function of two variables i and j, the intensities of two pixels, each in E it takes its elements in N (set of natural integers). 

The dimensionality of a potential energy surface depends on the number of degrees of freedom in a molecule. If Vp s is a function of two variables, then a plot of the potential energy surface represents a 3D space. 

State Functions State functions depend only on the state of the system, not on past history or how one got there. If r is a function of two variables, x and y, then z x,y) is a state function, since z is known once X and y are specified. The differential of z is... 

Taylor Series The Taylor series for a function of two variables, expanded about the point (xq, yo), is... 

There is no single formula for estimating resource requirements, which are a function of two variables whaf s planned (PSM goals) and whaTs already... 

Potential energy surfaces show many fascinating features, of which the most important for chemists is a saddle point. At any stationary point, both df/dx and df /Sy are zero. For functions of two variables f(x, y) such as that above, elementary calculus texts rarely go beyond the simple observation that if the quantity... 

The concept of homogeneity naturally extends to functions of more than one variable. For example, a generalized homogeneous function of two variables, f(x,y), can be written in the form... 

It follows that 1/T is the integrating factor of SQ. Now since SQ is a function of two variables (in the simple case of a homogeneous fluid), and since the integrating factor of such a magni-" tude is usually also a function of the same two variables, we must regard the proposition that the integrating factor of SQ is a function of one variable only as expressing a physical, not a mathematical, truth. 

If equation (2.51) is the total differential for as a function of two variables, 1 and 2, we can expect that its partial derivatives (d E/d Zi) and (<9 /c> 2)5 can be expressed as functions of only those two variables. That is, — ( , 2). Thus, derivatives of (<9 /<9 ) and (d E/d Zi)- with respect to variables other than 1 and 2 should be zero. As we consider the implications of this statement, it is important to note that a change can be made independently in the r variable of one subsystem without affecting that of the other, but a change in 0 will affect both subsystems (since 0 is the same in both subsystems). Therefore, we must consider the implications for c and 0 separately in the analysis that follows. 

The complexation of Pu(IV) with carbonate ions is investigated by solubility measurements of 238Pu02 in neutral to alkaline solutions containing sodium carbonate and bicarbonate. The total concentration of carbonate ions and pH are varied at the constant ionic strength (I = 1.0), in which the initial pH values are adjusted by altering the ratio of carbonate to bicarbonate ions. The oxidation state of dissolved species in equilibrium solutions are determined by absorption spectrophotometry and differential pulse polarography. The most stable oxidation state of Pu in carbonate solutions is found to be Pu(IV), which is present as hydroxocarbonate or carbonate species. The formation constants of these complexes are calculated on the basis of solubility data which are determined to be a function of two variable parameters the carbonate concentration and pH. The hydrolysis reactions of Pu(IV) in the present experimental system assessed by using the literature data are taken into account for calculation of the carbonate complexation. 

In three-dimensional experiments, two different 2D experiments are combined, so three frequency coordinates are involved. In general, the 3D experiment may be made up of the preparation, evolution (mixing periods of the first 2D experiment, combined with the evolution t ), mixing, and detection ( ) periods of the second 2D experiment. The 3D signals are therefore recorded as a function of two variable evolution times, t and <2, and the detection time %. This is illustrated in Fig. 6.1. 

For a function of N variables one needs a (N+l)-dimensional geometric figure or simplex to use and select points on the vertices to evaluate the function to be minimized. Thus, for a function of two variables an equilateral triangle is used whereas for a function of three variables a regular tetrahedron. 

Edgar and Himmelblau (1988) demonstrate the use of the method for a function of two variables. Nelder and Mead (1965) presented the method for a function of N variables as a flow diagram. They demonstrated its use by applying it to minimize Rosenbrock s function (Equation 5.22) as well as to the following functions ... 

In the case of a function of two variables the direction is from the rejected vertex through the middle of the line of the triangle that is opposite to this point. The new point together with the previous two points define a new equilateral triangle. 

An important class of the constrained optimization problems is one in which the objective function, equality constraints and inequality constraints are all linear. A linear function is one in which the dependent variables appear only to the first power. For example, a linear function of two variables x and x2 would be of the general form ... 

The half-life (fi/2) of a drug is a function of two variables clearance and volume of distribution. Half-life is directly related to volume of distribution (VD) and inversely related to clearance (CL) ... 

If a function of two variables is quadratic or approximated by a quadratic function /(x) = b0 + bxxx + 62 2 + 11 1 + 22 2 + 12 1 2 then the eigenvalues of H(x) can be calculated and used to interpret the nature of fix) at x. Table 4.2 lists some conclusions that can be reached by examining the eigenvalues of H(x) for a function of two variables, and Figures 4.12 through 4.15 illustrate the different types of surfaces corresponding to each case that arises for quadratic function. By... 

Figure 4.16 illustrates the character of ffx) if the objective function is a function of a single variable. Usually we are concerned with finding the minimum or maximum of a multivariable function fix)- The problem can be interpreted geometrically as finding the point in an -dimension space at which the function has an extremum. Examine Figure 4.17 in which the contours of a function of two variables are displayed. 

A function of two variables with a single stationary point (the extremum). 

Examine Figure 5.7 in which contours of a function of two variables are displayed ... 

Methods of experimental design discussed in most basic statistics books can be applied equally well to minimizing fix) (see Chapter 2). You evaluate a series of points about a reference point selected according to some type of design such as the ones shown in Figure 6.1 (for an objective function of two variables). Next you move to the point that improves the objective function the most, and repeat. 

Graphical representation of a function of two variables reduced to a function of one variable by direct substitution. The unconstrained minimum is at (0,0), the center of the contours. 

Now the concentration of a transient absorbing species varies with time, c(t), and so A is a function of two variables, namely time and... 

Equation (131) can be used for the evaluation of VF. The use of this equation has been simplified by having a graphical solution, which represents the drop diameter graphically as a function of two variables,... 

For a second-order transition, this problem is conveniently studied in terms of the order parameter distribution function, Pi,( ). Finite size scaling theory implies that near the critical point P/.( ) longer depends on the three variables L, 1 — T/T separately but rather is a scaled function of two variables (1 — only where v is the critical exponent of... 

As the state of a thermodynamic system generally is a function of more than one independent variable, it is necessary to consider the mathematical techniques for expressing these relationships. Many thermodynamic problems involve only two independent variables, and the extension to more variables is generally obvious, so we will limit our illustrations to functions of two variables. 

That is, the order of differentiation is immaterial for any function of two variables. Therefore, if dL is exact. Equation (2.23) is correct [8]. 

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