Transformation of Functions

3. The transformation of the American class structure, 1960±1990 Two opposed images have dominated discussions of the transformation of class structures in developed capitalist societies. The ®rst of these is associated with the idea that contemporary technological changes are producing a massive transformation of social and economic structures that are moving us towards what is variously called a ``post-industrial society'' (Bell 1973), a ``programmed society'' (Touraine 1971), a ``service society'' (Singelmann 1978; Fuchs 1968) or some similar designation. The second image, rooted in classical Marxist visions of social change, argues that in spite of these transformations of the ``forces of production,'' we remain a capitalist society and the changes in that class structure thus continue to be driven by the fundamental ``laws of motion'' of capit- alism. The post-industrial scenario of social change generally envisions the class structure becoming increasingly less proletarianized, requiring higher and higher proportions of workers with technical expertise and demanding less mindless routine and more responsibility and knowl- edge. For some of these theorists, the central process underwriting this tendency is the shift from an economy centered on industrial production to one based on services. Other theorists have placed greater stress on the emancipatory effects of the technical±scienti®c revolution within material production itself. In either case the result is a trajectory of changes that undermines the material basis of alienation within produc- tion by giving employees progressively greater control over their condi- tions of work and freedom within work. In class terms, this augurs a decline in the working class and an expansion of various kinds of expert and managerial class locations. The classical Marxist image of transformation of class relations in capitalism is almost the negative of post-industrial theory: work is 56 becoming more proletarianized; technical expertise is being con®ned to a smaller and smaller proportion of the labor force; routinization of activity is becoming more and more pervasive, spreading to technical and even professional occupations; and responsibilities within work are becoming less meaningful. This argument was most clearly laid out in Braverman's (1974) in¯uential book, Labor and Monopoly Capital. The basic argument runs something like this: because the capitalist labor process is a process of exploitation and domination and not simply a technical process of production, capital is always faced with the problem of extracting labor effort from workers. In the arsenal of strategies of social control available to the capitalist class, one of the key weapons is the degradation of work, that is, the removal of skills and discretion from direct producers. The result is a general tendency for the proletar- ianzed character of the labor process to intensify over time. In terms of class structure, this implies that the working class will tend to expand, skilled employees and experts decline, and supervisory labor to increase as the demands of social control intensify. This chapter attempts to use quantitative data on the changes in distributions of people in the American class structure from 1960 to 1990 as a Transformation of Functions Transformation of Functions By: OpenStaxCollege (credit: "Misko"/Flickr) We all know that a flat mirror enables us to see an accurate image of ourselves and whatever is behind us When we tilt the mirror, the images we see may shift horizontally or vertically But what happens when we bend a flexible mirror? Like a carnival funhouse mirror, it presents us with a distorted image of ourselves, stretched or compressed horizontally or vertically In a similar way, we can distort or transform mathematical functions to better adapt them to describing objects or processes in the real world In this section, we will take a look at several kinds of transformations Graphing Functions Using Vertical and Horizontal Shifts Often when given a problem, we try to model the scenario using mathematics in the form of words, tables, graphs, and equations One method we can employ is to adapt the basic graphs of the toolkit functions to build new models for a given scenario There are systematic ways to alter functions to construct appropriate models for the problems we are trying to solve 1/66 Transformation of Functions Identifying Vertical Shifts One simple kind of transformation involves shifting the entire graph of a function up, down, right, or left The simplest shift is a vertical shift, moving the graph up or down, because this transformation involves adding a positive or negative constant to the function In other words, we add the same constant to the output value of the function regardless of the input For a function g(x) = f(x) + k, the function f(x) is shifted vertically k units See [link] for an example Vertical shift by k = of the cube root function f(x) = 3√x To help you visualize the concept of a vertical shift, consider that y = f(x) Therefore, f(x) + k is equivalent to y + k Every unit of y is replaced by y + k, so the y-value increases or decreases depending on the value of k The result is a shift upward or downward A General Note Vertical Shift Given a function f(x), a new function g(x) = f(x) + k, where k is a constant, is a vertical shift of the function f(x) All the output values change by k units If k is positive, the graph will shift up If k is negative, the graph will shift down Adding a Constant to a Function To regulate temperature in a green building, airflow vents near the roof open and close throughout the day [link] shows the area of open vents V (in square feet) throughout the day in hours after midnight, t During the summer, the facilities manager decides to try 2/66 Transformation of Functions to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night Sketch a graph of this new function We can sketch a graph of this new function by adding 20 to each of the output values of the original function This will have the effect of shifting the graph vertically up, as shown in [link] Notice that in [link], for each input value, the output value has increased by 20, so if we call the new function S(t), we could write 3/66 Transformation of Functions S(t) = V(t) + 20 This notation tells us that, for any value of t, S(t) can be found by evaluating the function V at the same input and then adding 20 to the result This defines S as a transformation of the function V, in this case a vertical shift up 20 units Notice that, with a vertical shift, the input values stay the same and only the output values change See [link] t 10 17 19 24 V(t) 0 220 220 0 S(t) 20 20 240 240 20 20 How To Given a tabular function, create a new row to represent a vertical shift Identify the output row or column Determine the magnitude of the shift Add the shift to the value in each output cell Add a positive value for up or a negative value for down Shifting a Tabular Function Vertically A function f(x) is given in [link] Create a table for the function g(x) = f(x) − x f(x) 11 The formula g(x) = f(x) − tells us that we can find the output values of g by subtracting from the output values of f For example: f(2) = Given g(x) = f(x) − Given transformation g(2) = f(2) − =1−3 = −2 Subtracting from each f(x) value, we can complete a table of values for g(x) as shown in [link] x 4/66 Transformation of Functions f(x) 11 g(x) −2 Analysis As with the earlier vertical shift, notice the input values stay the same and only the output values change Try It The function h(t) = − 4.9t2 + 30t gives the height h of a ball (in meters) thrown upward from the ground after t seconds Suppose the ball was instead thrown from the top of a 10-m building Relate this new height function b(t) to h(t), and then find a formula for b(t) b(t) = h(t) + 10 = − 4.9t2 + 30t + 10 Identifying Horizontal Shifts We just saw that the vertical shift is a change to the output, or outside, of the function We will now look at how changes to input, on the inside of the function, change its graph and meaning A shift to the input results in a movement of the graph of the function left or right in what is ... Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). Chapter 4. Integration of Functions 4.0 Introduction Numerical integration,which is also called quadrature, has a history extending back to the invention of calculus and before. The fact that integrals of elementary functions could not, in general, be computed analytically, while derivatives could be, served to give the field a certain panache, and to set it a cut above the arithmetic drudgery of numerical analysis during the whole of the 18th and 19th centuries. With the invention of automatic computing, quadrature became just one numer- ical task among many, and not a very interesting one at that. Automatic computing, even the most primitivesort involvingdesk calculators and roomsfull of “computers” (that were, until the 1950s, people rather than machines), opened to feasibility the much richer field of numerical integration of differential equations. Quadrature is merely the simplest special case: The evaluation of the integral I =  b a f(x)dx (4.0.1) is precisely equivalent to solving for the value I ≡ y(b) the differential equation dy dx = f(x)(4.0.2) with the boundary condition y(a)=0 (4.0.3) Chapter 16 of this book deals with the numerical integration of differential equations. In that chapter, much emphasis is given to the concept of “variable” or “adaptive” choices of stepsize. We will not, therefore, develop that material here. If the function that you propose to integrate is sharply concentrated in one or more peaks, or if its shape is not readily characterized by a single length-scale, then it is likely that you should cast the problem in the form of (4.0.2)–(4.0.3) and use the methods of Chapter 16. The quadrature methods in this chapter are based, in one way or another, on the obvious device of adding up the value of the integrand at a sequence of abscissas within the range of integration. The game is to obtain the integral as accurately as possible with the smallest number of function evaluations of the integrand. Just as in the case of interpolation (Chapter 3), one has the freedom to choose methods 129 130 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). of various orders, with higher order sometimes, but not always, giving higher accuracy. “Romberg integration,” which is discussed in §4.3, is a general formalism for making use of integration methods of a variety of different orders, and we recommend it highly. Apart from the methods of this chapter and of Chapter 16, there are yet other 130 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). of various orders, with higher order sometimes, but not always, giving higher accuracy. “Romberg integration,” which is discussed in §4.3, is a general formalism for making use of integration methods of a variety of different orders, and we recommend it highly. Apart from the methods of this chapter and of Chapter 16, there are yet other methods for obtaining integrals. One important class is based on function approximation. We discuss explicitly the integration of functions by Chebyshev approximation (“Clenshaw-Curtis” quadrature) in §5.9. Although not explicitly discussed here, you ought to be able to figure out how to do cubic spline quadrature using the output of the routine spline in §3.3. (Hint: Integrate equation 3.3.3 over x analytically. See [1] .) Some integrals related to Fourier transforms can be calculated using the fast Fourier transform (FFT) algorithm. This is discussed in §13.9. Multidimensional integrals are another whole multidimensional bag of worms. Section 4.6 is an introductory discussion in this chapter; the important technique of Monte-Carlo integration is treated in Chapter 7. CITED REFERENCES AND FURTHER READING: Carnahan, B., Luther, H.A., and Wilkes, J.O. 1969, Applied Numerical Methods (New York: Wiley), Chapter 2. Isaacson, E., and Keller, H.B. 1966, Analysis of Numerical Methods (New York: Wiley), Chapter 7. Acton, F.S. 1970, Numerical Methods That Work ; 1990, corrected edition (Washington: Mathe- matical Association of America), Chapter 4. Stoer, J., and Bulirsch, R. 1980, Introduction to Numerical Analysis (New York: Springer-Verlag), Chapter 3. Ralston, A., and Rabinowitz, P. 1978, A First Course in Numerical Analysis , 2nd ed. (New York: McGraw-Hill), Chapter 4. Dahlquist, G., and Bjorck, A. 1974, Numerical Methods (Englewood Cliffs, NJ: Prentice-Hall), § 7.4. Kahaner, D., Moler, C., and Nash, S. 1989, Numerical Methods and Software (Englewood Cliffs, NJ: Prentice Hall), Chapter 5. Forsythe, G.E., Malcolm, M.A., and Moler, C.B. 1977, Computer Methods for Mathematical Computations (Englewood Cliffs, NJ: Prentice-Hall), § 5.2, p. 89. [1] Davis, P., and Rabinowitz, P. 1984, Methods of Numerical Integration , 2nd ed. (Orlando, FL: Academic Press). 4.1 Classical Formulas for Equally Spaced Abscissas Where would any book on numerical analysis be without Mr. Simpson and his “rule”? The classical formulas for integrating a function whose value is known at equally spaced steps have a certain elegance about them, and they are redolent with historical association. Through them, the modern numerical analyst communes with the spirits of his or her predecessors back across the centuries, as far as the time of Newton, if not farther. Alas, times do change; with the exception of two of the most modest formulas (“extended trapezoidal rule,” equation 4.1.11, and “extended 4.1 Classical Formulas for Equally Spaced Abscissas 131 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is Description of Functions Description of Functions 11.2003 Edition sinumerik SINUMERIK 802D SINUMERIK 802D base line Preface, Table of Contents EMERGENCY STOP (N2) 1 Axis Monitoring (A3) 2 Velocities, Setpoint/Actual-Value Systems, Closed-Loop Control (G2) 3 Acceleration (B2) 4 Spindle (S1) 5 Rotary Axes (R2) 6 Transverse Axes (P1) 7 Reference Point Approach (R1) 8 Manual Traversing and Hand- wheel Traversing (H1) 9 Operating Modes, Program Mode (K1) 10 Feed (V1) 11 Continous Path Mode, Exact Stop and LookAhead (B1) 12 Output of Auxiliary Functions to the PLC (H2) 13 Tool Compensation (W1) 14 Measuring (M5) 15 Compensation (K3) 16 Traversing to Fixed Stop (F1) 17 Kinematic Transformations (M1) 18 Various Interface Signals (A2) 19 PLC User Interface 20 Various Machine Data 21 Glossary, Index 11.03 Edition SINUMERIK 802D Description of Functions SINUMERIK  Documentation Printing history Brief details of this edition and previous edition are listed below. The status of each edition is shown by the code in the “Remarks” column. Status code in the “Remarks” column: A New documentation . . . . B Unrevised reprint with new Order No . . . . C Revised edition with new status. . . . . . If actual changes have been made on the page since the last edition, this is indicated by a new edition coding in the header on that page. Edition Order No. Remarks 11.00 6FC5 697-2AA10-0BP0 A 10.02 6FC5 697-2AA10-0BP1 C 11.03 6FC5 697-2AA10-0BP2 C This Manual is included on the documentation on CD ROM (DOCONCD) E Trademarks SIMATICr, SIMATIC HMIr, SIMATIC NETr, SIROTECr, SINUMERIKr and SIMODRIVEr are registered trademarks of Siemens. Third parties using for their own purpose any other names in this document which refer to trademarks might infringe upon the rights of trademark owners. This publication was produced with Interleaf V 7. The reproduction, transmission or use of this document or its contents is not permitted without express written authority.Offenders will be made liable for damages. All rights, including rights created by patent grant or registration of utility model or design, are reserved.  Siemens AG 2003. All rights reserved. Other functions not described in this documentation might be executable in the control. This does not, however, represent an obligation to supply such functions with a new control or when servicing. We have checked that the contents of this document correspond to the hardware and software described. Nonetheless, differences might exist and therefore we cannot guarantee that they are completely identical. The information contained in this document is, however, reviewed regularly and any necessary changes will be included in the next edition. We welcome suggestions for improvement. Subject to change without prior notice. Siemens -AktiengesellschaftOrder No.: 6FC5 697-2AA10 - 0BP2 Printed in the Federal Republic of Germany 3ls v SINUMERIK 802D, 802D base line 6FC5 697-2AA10-0BP2 (11.03) (DF) Preface Notes for the reader The descriptions of functions are only valid for or up to the specified software release. When new software releases are issued, the relevant descriptions of functions must be requested. Old descriptions of functions can only partially be used for new software releases. Note Other functions not described in this documentation might be executable in the control. This does not, however, represent an obligation to supply such functions with a new control or when servicing. Technical notes Notations The following notations and abbreviations are used in this Documentation: S PLC interface signals -> IS ”Signal name” (signal data) Example: IS ”Feed override“ (VB380x 0000) The variable byte is in the range “at axis“, “x” stands for the axis: 0 axis 1 1 aixs 2 n axis n+1. S Machine data -> Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMs visit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). Chapter 10. Minimization or Maximization of Functions 10.0 Introduction In a nutshell: You are given a single function f that depends on one or more independent variables. You want to find the value of those variables where f takes on a maximum or a minimum value. You can then calculate what value of f is achieved at the maximum or minimum. The tasks of maximization and minimization are trivially related to each other, since one person’s function f could just as well be another’s −f. The computational desiderata are the usual ones: Do it quickly, cheaply, and in small memory. Often the computational effort is dominated by the cost of evaluating f (and also perhaps its partial derivatives with respect to all variables, if the chosen algorithm requires them). In such cases the desiderata are sometimes replaced by the simple surrogate: Evaluate f as few times as possible. An extremum (maximum or minimum point) can be either global (truly the highest or lowest function value) or local (the highest or lowest in a finite neighborhood and not on the boundary of that neighborhood). (See Figure 10.0.1.) Finding a global extremum is, in general, a very difficult problem. Two standard heuristics are widely used: (i) find local extrema starting from widely varying starting values of the independent variables (perhaps chosen quasi-randomly, as in §7.7), and then pick the most extreme of these (if they are not all the same); or (ii) perturb a local extremum by taking a finite amplitude step away from it, and then see if your routine returns you to a better point, or “always” to the same one. Relatively recently, so-called “simulated annealing methods” (§10.9) have demonstrated important successes on a variety of global extremization problems. Our chapter title could just as well be optimization, which is the usual name for this very large field of numerical research. The importance ascribed to the various tasks in this field depends strongly on the particular interests of whom you talk to. Economists, and some engineers, are particularly concerned with constrained optimization, where there are apriorilimitations on the allowed values of independent variables. For example, the production of wheat in the U.S. must be a nonnegative number. One particularly well-developed area of constrained optimization is linear programming, where both the function to be optimized and the constraints happen to be linear functions of the independent variables. Section 10.8, which is otherwise somewhat disconnected from the restof the material that we have chosen to include in this chapter, implements the so-called “simplex algorithm” for linear programming problems. 394 10.0 Introduction 395 Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machine- readable files (including this one) to ... both of the functions We can see the horizontal shift in each point 8/66 Transformation of Functions Identifying a Horizontal Shift of a Toolkit Function [link] represents a transformation of the... units: (−1, 0) → (−1, −3) 12/66 Transformation of Functions [link] shows the graph of h Try It 13/66 Transformation of Functions Given f(x) = |x|, sketch a graph of h(x) = f(x − 2) + Identifying... percentage of mastery that can be achieved after t practice sessions This is a transformation of the function f(t) = 2t shown in [link] Sketch a graph of k(t) 20/66 Transformation of Functions