Discontinuous function

Taking the corresponding derivatives, we again arrive at Equation (1.33). The system (1.31) is written for points where the density 6 is defined. However, there are exceptions for instance, an interface between media with different densities. Fig. 1.6c, since in such places the density of masses is a discontinuous function. Now, making use of Equation (1.30), it is easy to derive a surface analogy of Equation (1.31). Let us calculate the circulation along the path shown in Fig. 1.6c. From Equation (1.29) it follows ... 

Besides, the potential and its first derivatives are continuous at boundaries where a volume density is discontinuous function. It is obvious that in this case the solution of the forward problem is unique. Now consider a completely different situation, when a density 5 q) is given only inside some volume V surrounded by a surface S, Fig. 1.8a. Inasmuch as the distribution of masses outside V is unknown, it is natural to expect that Poisson s equation does not uniquely define the potential U, and in order to illustrate this fact let us represent its solution as a sum ... 

The gradient search methods require derivatives of the objective functions whereas the direct methods are derivative-free. The derivatives may be available analytically or otherwise they are approximated in some way. It is assumed that the objective function has continuous second derivatives, whether or not these are explicitly available. Gradient methods are still efficient if there are some discontinuities in the derivatives. On the other hand, direct search techniques, which use function values, are more efficient for highly discontinuous functions. 

In the examples described above the resulting probability distributions were discontinuous functions. However, it is not difficult to imagine cases in which the distributions become continuous in the limit of an infinite - or at least a very large - number of trials. Sucb is the case in the application of statistical arguments to problems in thermodynamics, as outlined in Section 10.5. 

Step and impulse inputs. These discontinuous functions are used particularly in Chapter 5. Their definitions and transforms are ... 

In carrying out analytical or numerical optimization you will find it preferable and more convenient to work with continuous functions of one or more variables than with functions containing discontinuities. Functions having continuous derivatives are also preferred. Case A in Figure 4.1 shows a discontinuous function. Is case B also discontinuous ... 

Notice that we have approximated a discontinuous function by a continuous one. It turns out that any function in L —1, 1] can be approximated by trigonometric polynomials — this is one of the important results of the theory of Fourier series. ... 

The error of approximation in the class of smooth coefficients. The main point of the theory is the accurate account of the accuracy of the uniform scheme (16)—(17) in the class of continuous and discontinuous functions k(x), q(x) and f(x). In preparation for this, let u(x) be an exact solution of the original problem... 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

Convergence of the actual solution to the self-similar one over time occurs in a way similar to convergence of a Fourier series to a discontinuous function as the number of terms in the sum increases the well-known Gibbs phenomenon leads to the fact that, near the discontinuity, for any number of terms, the maximum difference between the series and the function does not approach zero however, the width of the region in which the series differs noticeably from the function approaches zero as the number of terms increases. 

A.I. Vol pert and S.I. Khudyav, Analysis in Class of Discontinuous Functions and Equations of Mathematical Physics, Martinus Nijhoff, Dordrecht, 1985 Nauka, Moscow, 1975 (in Russian). 

Consider sensory hairs 2 pm in diameter that are 20 pm apart. Inserting these values into equation (21.22) and assuming that D is 2.5 x 10 6 m2/s (the diffusion coefficient for bombykol, the main component of the commercial silkmoth sex pheromone Adam and Delbriick, 1968) results in the prediction that these hairs are likely to interfere with each other s odorant interception when the air speed between the hairs is below 0.0125 m/s. This is not a discontinuous function - the sensory hairs will interfere with each other more at slower speeds and less at faster speeds. Another way of appreciating what this means quantitatively is to recognize that the root mean square displacement of a molecule (considering movement in one dimension) is... 

The cost equations thus written are discontinuous functions of the size of the units which compose the trains. A mathematical minimization of any of these equations may not lead to a practical minimum. It may indicate only the domain where less-expensive solutions may exist. The practical alternative is to draw flow schemes which are equivalent to the process under investigation. The economic analysis of these schemes terminates with the selection of one which requires the minimum capital investment and operating costs. 

The occasional mathematician will point out the hazards of taking the derivative of a discontinuous function with respect to a discontinuous variable. Easy-going types will be satisfied with the explanation that for astronomically large numbers of possible states, the function W and the variables are effectively continuous. Sticklers for mathematical rigor will have to find satisfaction elsewhere. 

Setpoint profiles almost always consist of straight-line segments (either horizontal or sloped), and they are programmed as discontinuous functions. 

Excel Tip. Discontinuous functions in your Solver model can cause problems. They can be either discontinuous mathematical functions such as TAN, which has a discontinuity at nil, or worksheet functions that are inherently "discontinuous", such as IF, ABS, INT, ROUND, CHOOSE, LOOKUP, HLOOKUPor VLOOKUP. 

From the examples given by Engels, it would appear that the transformation of quantity into quality covers two cases, one being a special instance of the other. First, the water-into-ice example suggests the notion of a discontinuous functional link between the independent variable... 

The method is applicable to nonconvex objective functions with multiple optima and to nondifferentiable (discontinuous) functions and it may be used for discrete (combinatorial) and continuous variables optimization. 

The idea of a lattice, which expresses the translational periodicity within a crystal as the systematic repetition of the molecular contents of a unit cell, is a salient concept in X-ray diffraction analysis. A lattice, mathematically, is a discrete, discontinuous function. A lattice is absolutely zero everywhere except at very specific, predictable, periodically distributed points where it takes on a value of one. We can begin to see, from the discussion... 

In order to complete the mathematical formulation of the problem, appropriate initial and boimdary conditions (Chapter 2, Section 2.1.4, and Chapter 6, Section 6.2) must be used. Finally, we must define the fimctional space in which solutions of a partial differential equation are sought. In the case of Eq. 7.1, solutions will be sought in the space of (discontinuous functions) with bounded variations, denoted BV(n) in mathematics [3]. Discontinuities should be allowed, for reasons made clear in the next section. Roughly speaking, the variation of a function in a domain Q is the integral of the norm of its gradient (in the sense of distributions) over Q. If the variations are boimded, neither oscillations nor discontinuities can develop too much, which is required in the present case. In fact, the variation should decrease in the course of time. 

Integrals of singular or discontinuous functions (Problems 2, 3, 7 and 19) appear to be rather difficult problems for DQXG2. The present AQN9D can effectively approximate these problems by the combination of the Ninomiya and the FLR methods and by virtue of the detection of extraordinary points in the Ninomiya scheme. 

The Xj. is considered a generalized function which is especially useful in connection with differentiation of discontinuous functions within an integral. The ensemble average of ip is then defined by ... 

It is noted that the original Reynolds axioms are not applicable to discontinuous functions as normally occur across the interfaces in multiphase flow. As a remedy. Drew [54] extended these functions making them continuous by use of the generalized function concept coimecting the functions of the continuous phases on each side of the interface across the interface. Hence the discontinuous functions are modified to be continuous but locally very steep functions across the interface. Formally the averaging axioms can then be extended to include the interfaces, giving rise to the modified formulations of the axioms. 

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