User manual for Iode - Department of Mathematics
Contents
1. Warning 1 1 e Your expression for f should involve the independent or dependent variables you have chosen or both but no other variables e Variable names should consist of only one letter to avoid the risk of conflicting with Iode s internal variables Enter display parameters Allows you to change the domain and range of the direction field and the number of line segments used to plot it Note that this will clear all plots in the graphics window The prompts are self explanatory Plot numerical solution To get a specific solution of the differential equation Sy f x y you also need to specify the initial data y z 9 Yo When you select Plot numerical solution lode asks you to input this information Then you are asked to input the color of the plot Remark 1 2 The algorithm and the step size used for actually computing the numerical solution are displayed above the menu and can be changed using the next option 11 Change numerical solver When you select 4 Change numerical solver you will be prompted for the numerical method and step size The step size should be a small positive number Some acceptable choices for the numerical method will be listed on the screen or see Appendix C for more information on solvers Plot arbitrary function If you know a formula for the exact solution of a differential equation then you might want to compare it with the numerical solution which is only an approxi2. adjusts to the size of the error so that successive plots may be displayed with different vertical scales Superimposing successive plots does not make sense if the scale changes and so this item will not superimpose plots regardless of the plotting options chosen in 7 Options 21 Show An This option plots the first Niaz Fourier cosine coefficients where N mar is the value of max harmonic which can be changed using menu option 7 Options It does not plot the contant term Ao Recall that the nth Fourier cosine coefficient is defined by 1 f n z A f x cos de where L x Xo 2 is the half period Show B_n Similar to the previous option but for the Fourier sine coefficients Ly fe Bn AN f x sin de Show C_n sqrt A_n 2 B_n 2 Similar to the previous two options The reason it is interesting to plot the values of C y A B2 is that A cos B sin SC cos Qn 4 2 where the angle an is defined by requiring cosa A C and sina B Cn Formula 4 2 is proved just by substituting the identity cos G a cos 3cosa sin sina on the right hand side The meaning of 4 2 is that C equals the amplitude of the combined oscillations at the nth frequency level in the Fourier series of f Options These options all deal with aspects of the display Play around and have fun Save plot See Section 1 22 CHAPTER 4 FOURIER SERIES Chapter 5 Partial
3. differential equations This module computes and plots solutions of the wave equation and heat diffusion equation Solutions are found on an interval 0 lt x lt L for times 0 lt t lt T The method is separation of variables Enter equation and boundary conditions When you select 1 Enter equation and boundary conditions you will be prompted first of all to choose either the heat equation U Kuzz or the wave equation 2 Utt C Urg Heat diffusion equation Then Iode asks for the diffusivity k followed by the boundary conditions Iode can deal with Dirichlet boundary conditions u 0 t 0 and u L t 0 with Neumann boundary conditions u 0 t 0 and u L t 0 and also with periodic boundary conditions u 0 t u L t and u 0 t uz L t Users are encouraged to create modules for other boundary conditions as well see dirichlet m as a guide to doing this Wave equation Then Iode asks for the wavespeed c followed by the bound ary conditions as above 23 24 CHAPTER 5 PARTIAL DIFFERENTIAL EQUATIONS Enter duration length and initial conditions After selecting 2 Enter duration length and initial conditions you will be asked to enter the duration T over which the numerical solution is to be plotted and then the length L of the interval Heat equation Then lode asks for the initial temperature or concentration function u x 0 f x Wave equation Then Iode asks for the initial displacement
4. t cx t ke t f t 3 1 Equations of this sort arise in models of simple mechanical vibrations and electrical circuits Incidentally the application to mechanical vibrations explains why the module has filename mvmenu m The module plots the forcing function f and it computes and plots nu merical solutions of the equation 3 1 The user can input the equation and the initial conditions and change the method used for computing numerical solutions After selecting 3 from the Iode main menu one sees the Second order linear ODEs menu the menu options are explained below The current ODE is always displayed above the menu along with the current values of the options for the numerical method The graphics window shows a plot of the forcing function f with the equation itself written across the top of the window The options in this module are either self explanatory or else are very similar to the corresponding parts of the direction fields module Enter differential equation When you select Enter differential equation you will be prompted for the six pieces of information needed to make sense of 3 1 an independent and a dependent variable the coefficients m c and k which are allowed to 17 18 CHAPTER 3 SECOND ORDER LINEAR ODES involve the independent variable but are usually just constants and the function f Otherwise this option is very similar to the corresponding part of the direction fields m
5. u x 0 f x and the initial velocity u x 0 g x Remark 5 1 lode computes its approximate solutions by separation of vari ables For example for the heat equation with Dirichlet boundary conditions lode will use Nmaz Wet y be CALPE sin n 1 where Nmax is the value of Max harmonic which can be changed using menu item 6 Options and where the b are the Fourier sine coefficients of the initial value f x For the wave equation with Dirichlet boundary conditions the approximate solution is Nmax nret L nret nar UN on cost T 4 Bu sin T sin 7 n 1 where the b are the Fourier sine coefficients of the initial displacement f x and the B are the Fourier sine coefficients of the initial velocity g x Plot 3D This plots the graph z u x t of the approximate solution in 3 dimensions Plot snapshots t slices This plots the graph of u x t as a function of x for a discrete succession of t values The default plotting option is to step through the t values but animation can be chosen instead using menu item 6 Options 25 Plot sections x slices This plots the graph of u x t as a function of t for a discrete succession of x values The default plotting option is to step through the x values but animation can be chosen instead using menu item 6 Options Options The first option is to choose Max harmonic Nmax which specifies how many terms to take in the Four
6. which the phase portrait is shown and it is also the initial value of the independent variable when computing numerical solutions Most users will only work with autonomous systems in which case ty doesn t matter Plot numerical solution To get a specific solution of the system 2 1 you also need to specify the initial data xy x to and yo y to which say where the plot should start in the phase portrait When you select 3 Plot numerical solution Iode asks you to input the initial data and the t duration of the plot Think of the plot as tracing the path of a moving particle in which case the t duration denotes the time elapsed The t duration may be negative in which case the solution is computed backwards from to You are also asked to input the color of the plot Remark 2 2 The algorithm and the step size used for actually computing the numerical solution are displayed above the menu and can be changed using the next option 15 Change numerical solver See Section 1 Plot arbitrary trajectory As in Section 1 this feature will be most useful in situations where you know an exact formula for the solution and want to compare it to a numerical solution in order to estimate the accuracy of the numerical method Clear plot See Section 1 Save plot See Section 1 16 CHAPTER 2 PHASE PLANES Chapter 3 Second order linear ODEs This module deals with second order linear nonhomogeneous ODEs ma
7. User manual for lode text based interface Peter Brinkmann Robert Jerrard and Richard Laugesen Department of Mathematics University of Illinois Urbana Champaign U S A March 5 2010 Copyright c 2003 The Triode iode math uiuc edu Permission is granted to copy distribute and or modify this document under the terms of the GNU Free Documentation License Version 1 2 or any later version published by the Free Software Foundation with no Invariant Sec tions no Front Cover Texts and no Back Cover Texts A copy of the license is available at http www fsf org copyleft fdl html Contents e 5 QO gt Direction fields Phase planes Second order linear ODEs Fourier series Partial differential equations Purge temporary files Quit Installing Iode Running Iode Reference materials Structure of Iode 13 17 19 23 27 29 31 33 35 39 CONTENTS Overview lode rhymes with diode is a software package that enable students to ex plore direction fields phase planes second order linear ODEs Fourier series and heat and wave equations The name lode is supposed to be reminiscent of Illinois and ODE lode runs under either Matlab or Octave their programming languages are mostly compatible For instructions on downloading installing and running all the needed software see Appendix A If your computer already has Matlab or Octave installed then all you need to get is lo
8. ays displayed above the menu along with the current values of the options for the numerical method The graphics window shows a plot of the direction field for the ODE with the equation itself written across the top of the window When you first enter the module you will find Iode has already chosen a default ODE as well as reasonable option settings Next we explain how to change these settings Enter equation When you select 1 Enter equation you are prompted for three pieces of information an independent variable a dependent variable and a function When you input the names for the independent horizontal and depen dent vertical variables you are specifying the form in which you want the equation written for example 10 CHAPTER 1 DIRECTION FIELDS e dy dx f x y in which case the solution will be a function y of a variable x or e dx dt f t x in which case the solution will be a function x of a variable t In the first case you would input x for the independent variable and y for the dependent variable In the second case input t for the independent variable and x for the dependent variable Then Iode will prompt you to input the function f using valid Matlab or Octave syntax consult Appendix C for examples For example to study the equation dx dt xt you should input x t 2 Note that you do not input the letter f here You just input the expression that you want on the right hand side of the ODE
9. bs x absolute value sqrt x square root yx 1 ifz gt 0 sign x signum function which equals Viel 1 ifz lt 0 sin x sinh x 35 36 APPENDIX C REFERENCE MATERIALS cos x trigonometric cosh x hyperbolic tan x functions tanh x trigonometric cot x x in radians coth x functions sec x sech x csc x csch x asin x asinh x acos x inverse acosh x inverse atan x trigonometric atanh x hyperbolic acot x functions acoth x trigonometric asec x asech x functions acsc x acsch x besselj nu z Bessel function of the first kind bessely nu z Bessel function of the second kind besseli nu z Modified Bessel function of the first kind besselk nu z Modified Bessel function of the second kind Example C 1 sin exp y 74 means sin e acos exp 1 7 1 means arccos e No matter whether you re using Octave or Matlab you can always find more information on a function by typing help function Additional lode functions The installation of lode includes the following functions useful for studying Fourier series and for creating initial values for partial differential equations hat x a b equals 0 for x lt a and x gt b and equals 1 for a lt x lt b To make sense hat requires a lt b triangle x a b m a triangular shaped function equalling 0 for x lt a then rising linearly to height 1 at x m and falling back linearly to zero at x b then equalling zero for x gt b To make se
10. de Experienced users can proceed directly to www math uiuc edu iode to download it The graphical user interface for lode is supported only under Matlab version 6 0 or later This manual describes the text based user interface which runs under earlier versions of Matlab and also under Octave The graphical user interface offers certain features and plotting options that are unavailable in the text based interface A good way to learn Iode is simply to explore it Choose various options and see what happens Also the built in help features in Matlab and Octave are your friends For example typing help euler at the prompt will give information on the module euler m Please let us know if you encounter problems with lode or if you have suggestions for improvement To contact us see www math uiuc edu iode From a programming perspective the main point of Iode is that it is mod ular and has well defined interfaces This has several useful consequences For example it is easy for the user to customize Iode or to create and plug in new modules Simple modules of this sort can be created even by users with out much programming experience In fact lode was born out of frustration with other educational packages that conceal their inner workings from the user The mathematical modules such as df m or euler m are usable in Mat 5 6 CONTENTS lab and Octave without the menus and much of the code is easily extensible Users are encourag
11. ed and in some of our classes expected to look at the code and modify it take it apart put it back together and so on For example Appendix D discusses how to create your own solver module for numerically solving ODEs On the other hand lode comes with a user interface that exposes most of the capabilities of the mathematical modules and so you can use lode even if you don t have any programming skills This manual will mostly be concerned with explaining the user interface lode main menu After launching the text based lode user interface see instructions in Ap pendix B we get to the main menu 1 Direction fields 2 Phase planes 3 Second order linear ODEs 4 Fourier series 5 Partial differential equations 6 Purge temporary files 7 Quit oh tat DA er PA vu Before looking at these options in detail we explain some features that are common to all parts of lode For example you will often be prompted to input some piece of information such as a number or a variable name or a function Whenever this happens lode will display the current value of the information inside square brackets For example the prompt Independent variable x asks you to input the name of the independent variable The current name is x which is displayed inside square brackets If in response you enter t and then hit return t will be the name for the independent variable from then on If in response you do not enter anything but ju
12. either Matlab or Octave If you wish to use the graphical user interface of Iode you will need Matlab Version 6 0 or later If your computer already has Matlab or Octave installed then you only need to install Iode Obtaining and installing Iode On your machine create a directory called my_iode or similar Then e Unix systems From www math uiuc edu iode download the file iode zip and unpack it into the directory my_iode with unzip or sim ilar e Windows systems From www math uiuc edu iode download the file iode_dos zip and unpack it into the directory my_iode with WinZip or similar lode is free software available under the GNU General Public License Obtaining and installing Matlab Matlab is commercial software available at www mathworks com for Unix Windows and Mac It is already installed on many Unix systems in universi ties Matlab is available to students at a discounted price see the downloads page at www math uiuc edu iode 31 32 APPENDIX A INSTALLING IODE After installing Matlab on a Windows system it is helpful to set it up to launch in the correct directory See Appendix B Obtaining and installing Octave Octave is free software and you can obtain it from www octave org for Unix Windows and Mac OS X Installing Octave on Unix systems If your home machine runs Unix then chances are that you are already expert enough to download and install Octave yourself following the directions at
13. gorithm m in your lode directory The structure of my_algorithm m should follow that of euler m You will simply need to change the update steps which in euler m are k1 feval fs x tc i x x hxk1 Then after creating the file my_algorithm m you can input my_algorithm when lode prompts you for a numerical method Appendix D Structure of lode For users who like to look under the hood we now provide some information on the structure of lode We encourage users to modify lode and create new modules Figure D 1 breaks lode down into its constituent modules Each module occurs as an m file with the same name For example dfmenu in the figure means the file dfmenu m that comes as part of the lode package There are two kinds of module The first kind are the menus and aux iliaries providing the user interface The second and main kind are the mathematical modules which actually perform the calculations needed to solve the differential equations numerically and calculate the Fourier coeffi cients numerically and so on The mathematical modules euler m and rk m are particularly instructive they implement the Euler and Runge Kutta methods for solving a first order ODE You can implement different solver methods yourself as outlined in Appendix C In order to develop a better understanding of what the various modules do you can look at the file doc txt Together with Figure D 1 it should give you a rather complete pict
14. ier series that are used to construct approximate solutions The remaining options all deal with aspects of the display and are self explanatory Save plot See Section 1 26 CHAPTER 5 PARTIAL DIFFERENTIAL EQUATIONS Chapter 6 Purge temporary files Unfortunately Octave and Gnuplot generate a large number of temporary files and they don t always remove these files properly A prolonged lode session or a session involving extremely complex plots can then fill up your file system with temporary files This is a known problem of Octave and will be corrected in a future version If Octave has accumulated too many temporary files your plots will be left incomplete or might not get updated correctly You can fix the problem by exiting to the main menu and choosing 6 Purge temporary files After deleting the temporary files you can continue your work with lode If you are using Octave and Gnuplot without the user interface of Iode it may be a good idea to remove temporary files every once in a while by issuing the command purge_tmp_files 27 28 CHAPTER 6 PURGE TEMPORARY FILES Chapter 7 Quit This quits out of lode returning the user to an Octave prompt Incidentally when running lode under Octave Ctrl D is a shortcut for quitting out of any module or out of Octave itself More drastically Ctrl C will immediately terminate Iode 29 30 CHAPTER 7 QUIT Appendix A Installing Iode lode runs under
15. mation to see how accurate the numerical solution is This helps indicate how reliable the numerical solution will be on other problems where you do not know the exact solution The item 5 Plot arbitrary function allows you to plot the exact solution Actually it will plot any function for which you input the formula For example to plot y sin x you just input sin x Consult Appendix C for more examples of functions you can use Clear plots Clears all plots of functions from the graphics window while leaving the direction field intact Save plot This feature is only implemented for Octave because Matlab s way of saving plots is quite different This shouldn t be a problem because Matlab s user interface has a menu item for saving plots If you select 7 Save plot you will be prompted for two pieces of information first a file name without any extension and then an extension For example suppose that at the first prompt you enter my_plot and at the second prompt you hit return to keep the default extension eps Then a file called my_plot eps will be created containing your plot in a format known as Encapsulated PostScript To view or print your file use the program 12 CHAPTER 1 DIRECTION FIELDS Ghostview For example on a Unix system you would get to a Unix prompt outside of Octave and type ghostview my_plot eps Remark 1 3 Only the graphics display is saved in the file not the lode settings that gene
16. nse triangle requires a lt m lt b The parameter m is optional and defaults to the midpoint a 37 bump x a b m is a continuously differentiable function that equals 0 for x lt a and x gt b is positive for a lt x lt b has a maximum of 1 at x m and is strictly increasing between a and m and strictly decreasing between m and b To make sense bump requires a lt m lt b The parameter m is optional and defaults to ott Logical expressions in Matlab Octave and Iode Expressions like x gt 2 are treated as logical functions and return a value of either 1 true or 0 false So x gt 2 is the step function that equals E22 f 0 otherwise Example C 2 Logical functions help us create functions defined in pieces ift lt 4 0 otherwise t72 t lt 4 means l because t lt 4 equals 1 if t lt 4 and equals 0 otherwise Color codes recognized by Matlab Octave and Iode blue green red cyan magenta yellow black wK BORO o Note that yellow plots are usually hard to see due to lack of contrast 38 APPENDIX C REFERENCE MATERIALS Some solver codes recognized by lode euler Euler method for systems of first order equations rk Runge Kutta for systems of first order equations In addition to the pre installed solvers listed above you can also create your own For example if you devise a new method for solving differential equations then you could program it in a new file called my_al
17. odule described in Section 1 Enter domain and range See Section 1 Plot numerical solution See Section 1 Change numerical solver See Section 1 Plot arbitrary function See Section 1 Clear plots See Section 1 Save plot See Section 1 Chapter 4 Fourier series The Fourier series module can compute and graph the Fourier coefficients of a periodic function f and can plot partial sums of the Fourier series as well as the difference between the function f and partial sums of its Fourier series The user inputs the function f decides how many coefficients are to be computed and can then step through the graphs of the partial sums For concreteness here we will write f x for the function being considered even though the user can change the name of the independent variable from x to something else if desired After selecting 4 from the lode main menu you will see the Fourier series menu and a graphics window At the top of the graphics window and also above the Fourier series menu you will see displayed a function f x and the value of the top harmonic The function is written in the form F x a valid Matlab or Octave expression for o lt x lt x extended periodically where x and x are numbers Remark 4 1 The interval 7 lt x lt 2 is part of the definition of the function For example it tells us that the period of the function is P 11 Xo See Section 4 for fur
18. ports an error when processing the command iode you can launch the text based interface right away by typing iodetxt Running lode under Windows Log in to the machine and launch Matlab or Octave through the Start Programs menu Remark B 1 Important note Before launching Matlab or Octave for the very first time it is helpful to right click on the program name in the Start Programs menu then click on Properties and put the cor rect filepath into the Start in box Your filepath should look something 33 34 APPENDIX B RUNNING IODE like C My_files MathClass my_iode The point is that from now on when you launch Matlab or Octave it will be able to find the iode files Once you have gotten Matlab or Octave running just type iode at the Matlab or Octave prompt If this does not work then you probably need to change into the directory my_iode where you have stored the Iode files You can do this from within Matlab or Octave using Unix commands such as cd my_iode at the Matlab or Octave prompt Appendix C Reference materials Mathematical expressions in Matlab Octave and lode For simple expressions we use the usual keyboard characters 2x x means 2x x73 1 6 means x 1 6 Or instead of the usual division we can use left division pi 3 means 3 3 pi also means 3 Built in functions exp x exponential e log x natural logarithm ln x log10 x base 10 logarithm log a
19. rait for the system with the system itself written across the top of the window When you first enter the module you will find Iode has already chosen a system and chosen the option settings Next we explain how to change these settings Enter functions When you select 1 Enter functions you will be prompted to input five pieces of information an independent variable two dependent variables the horizontal and vertical variables and the two functions F and G Most 13 14 CHAPTER 2 PHASE PLANES users will only work with autonomous systems in which case the expressions that you enter for F and G will involve only the dependent variables not the independent one Remark 2 1 Note for advanced users lode can also handle nonautonomous systems in which the expressions for f and g depend also on the independent variable t lode will correctly compute numerical solutions for such systems but will not draw the full phase portrait which is three dimensional The phase portrait that Iode displays in this case is the one at t t0 see Sec tion 2 In all other respects this option is similar to the corresponding part of the direction fields module described in Section 1 Enter display parameters Allows you to specify the region shown in the phase portrait and to change the number of line segments used to plot the phase portrait The prompts are self explanatory The parameter tg is the value of the independent variable for
20. rated it In the text based user interface of lode there is no way to save your current settings and come back to them later The graphical user interface does offer this feature For most purposes you should use the default extension eps Other valid extensions include fig for Xfig pictures choose this if you would like to edit your plots with Xfig gif for gif files not recommended due to patent issues and png for Portable Network Graphics choose this if you would like to include your plots in web pages and such Warning 1 4 When you save a plot if you enter the same name as an existing file then the existing file will be overwritten with no warning Chapter 2 Phase planes This module deals with autonomous systems of two equations that is sys tems of equations of the form dx y TE G x y 2 1 F x y Fr The module creates a graphics window showing the phase portrait associated with such a system and it computes and plots numerical solutions of the system in this phase plane The user can input the system and the initial conditions and change the method used for computing numerical solutions After selecting 2 from the Iode main menu one sees the phase planes menu the menu options are explained below The current system of equa tions is always displayed above the menu along with the current values of the options for the numerical method The graphics window shows a plot of the phase port
21. st hit return then Iode will simply keep using the current name x for the independent variable Iode also retains the current value if it cannot make sense of the informa tion you provide For example a common mistake is to forget to type to indicate multiplication If you are asked to enter a function and type 2xy as 2xy then Iode will report an error and will retain the previous function The correct expression is 2 x y 8 CONTENTS The current value gives you a clue as to what kind of information lode wants from you For example if the current value is a number then you should probably enter a number If the information that lode seeks is not a number but instead say a color code then lode will display a list of acceptable responses Under Octave Ctrl D is a shortcut for quitting from any module in Iode or from Octave itself More drastically Ctrl C will immediately terminate lode Chapter 1 Direction fields This module deals with general first order ODEs that is equations of the form a T fen 1 1 The module plots the associated direction or slope field in a separate graph ics window and it computes and plots numerical solutions of the equation The user can input the equation and the initial condition and change the method used for computing numerical solutions After selecting 1 from the lode main menu one sees the direction fields menu the menu options are explained below The current ODE is alw
22. ther discussion The graphics window shows a plot of f in blue over two periods The red plot is a partial sum of the Fourier series of f that is a plot of N A 3 LilAncos Basin 4 1 19 20 CHAPTER 4 FOURIER SERIES where N top harmonic Here A and B are the Fourier coefficients and L is the half period L x x 2 The top harmonic number N tells you the highest frequency that is included in the partial sum Enter function Select 1 Enter function to enter a new function The function is the periodic extension of a formula given by the user on an interval given by the user So you are prompted not only for a formula for the function but also for the left and right endpoints of the interval You are also given the chance to change the name of the independent variable Plot partial sums This displays partial sums of the Fourier series in red in the graphics window Try hitting f for forward a few times and see how the display changes Notice that the value of the top harmonic as shown in the graphics window goes up when you enter f and goes down when you enter b Various options relating to these plots can be set using menu option 7 Options Plot errors This menu item is very similar to the previous one It displays the error A N 5 gt A cos B sin m 1 between f and partial sum of its Fourier series The vertical scale of the plot automatically
23. ure of the structure of lode Another useful file to look at is sample m which contains the transcript of a short Octave session that shows how to use the direction fields module and numerical solvers without resorting to any menus Any manual can only take you so far If you have followed this manual up to this point you are ready to go exploring on your own Good luck and have fun 39 40 APPENDIX D STRUCTURE OF IODE lode Text Version af solplot ES euler pp 7 N ae on wee p k ppmenu trajplot mvmenu iodetxt EN Pr mp men slideshow o Dd A dirichlet choose OS e neumann art er fs 7 See wave O most coeff a oe nn sinef periodic trapsum Figure D 1 Relationships between the most important modules of the text based version of lode
24. www octave org If yourun into trouble your local Unix geeks will be happy to help Installing Octave on Windows systems You can find an easy to install Windows executable at http www site uottawa ca adler octave After downloading the exe file chances are that your browser will offer to run the installation program If not then you should double click on the file to activate the installation If something goes wrong with the installation of Octave you may be able to get some help at http octave sourceforge net Octave Windows htm After installing Octave under Windows it is helpful to set it up to launch in the correct directory See Appendix B Appendix B Running lode Running lode under Unix Log in to the machine and get to a Unix prompt Change directory to get into my_iode by using a Unix command such as cd my_iode Then launch Matlab or Octave by typing matlab or octave at the prompt Note that the order of operations matters here you need to change into the directory containing lode before you launch Matlab or Octave Once you have gotten Matlab or Octave running just type iode at the Matlab or Octave prompt The graphical user interface for lode will launch if 1t can otherwise the text based user interface will launch Some develop ment versions of Octave sometimes fail to work with the part of lode that determines whether the graphical user interface can be launched If you are using a version of Octave that re
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