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  • Graph Demo
    use operators and where represents raising to a power it can use the functions sqrt cubert sin cos tan arctan arcsin ln log2 log10 abs round trunc floor and ceiling A function s argument must be enclosed in parentheses The C style ternary operator is also supported using boolean operators not or and The grapher attempts to draw vertical asymptotes and jump discontinuities correctly but it doesn t always work

    Original URL path: http://math.hws.edu/eck/js/graphs/grapher.html (2016-02-07)
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  • Family of Functions
    that can include parameters a b c and d as well as the variable x Drag the slider to change the parameter values For informatino about what is legal in a function definition see grapher html xmin xmax Apply Limits

    Original URL path: http://math.hws.edu/eck/js/graphs/function-family.html (2016-02-07)
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  • Derivatives Demo
    Derivatives Sorry but this page requires JavaScript For information about functions see the instructions on grapher html Try dragging your mouse on any graph to see tangent lines xmin xmax

    Original URL path: http://math.hws.edu/eck/js/graphs/derivatives.html (2016-02-07)
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  • Genetic Algorithms Demo Docs
    to the right It can also change its state by changing the number in its memory If it tries to move into a wall or onto a square that already contains another Eater it will not be allowed to move however it can still change its state If it moves onto a square containing a plant it eats the plant and scores a point Depending on menu settings the plant might immediately grow back somewhere else At the end of a year the fitness of the Eater depends the number of plants it has eaten plus one The 1 is added to avoid having a fitness of zero The more plants eaten the higher the fitness Now how does an Eater decide what to do on each turn It bases its decision on two things its current state and the item that it sees in front of it Its behavior is completely determined by a set of rules that tell it what to do for each possible combination of current state and item in view All the Eater does is follow its rules The rules differ from one Eater to another In fact the Eater s rules which completely determine the behavior of the Eater make up the Eater s genetic endowment what we will call its chromosome The chromosome consists of 64 rules one for each combination of one of 16 states and one of 4 items in view Each rule specifies two things an action and a new state The chromosome can be considered to be just a list of 128 numbers Here is what happens at the end of a year The average fitness of the current population is computed A new population is created by making copies of the Eaters in the current population An Eater s chance of being copied depends on its fitness Eaters that have fitness much below average are likely not to be reproduced at all although they always have some chance of reproduction Eaters with high fitness are likely to be copied several times Then a mutation operation is applied Each of the 128 numbers in each chromosome has a chance of being randomly changed This chance is 0 1 by default and is controlled by the Mutation Probability menu Finally a crossover operation is performed Pairs of Eaters in the new population are chosen at random and have a certain probability of being crossed over This probability is 80 by default and is controlled by the Crossover Probability menu Crossover here means that the chromosomes exchange some genetic material a random position between 1 and 128 is chosen and all data in the chromosomes after that position are swapped between the two chromosomes Note that the default mutation rate 0 1 seems very low but since it is appied to each of the 128 numbers in each chromosome it means that there is about a 1 in 8 chance that a given Eater will be mutated in some way This

    Original URL path: http://math.hws.edu/eck/js/genetic-algorithm/ga-info.html (2016-02-07)
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  • Pentominos Info
    can t be placed in black squares By using a large board and making a lot of black squares you can make oddly shaped and even disconnected areas for the program to try to fill For example The Run button will let the program work on the problem until it finds a solution at a speed determined by the speed menu The top entry in that menu Solutions Only NO PAUSE is special the program will show the solutions that it finds but it won t stop until it has checked every possibility and has found all the solutions Use this top speed to count solutions The second entry Run to next solution does just that it won t draw anything until it finds a solution or runs out of possibilities when it does it draws the solution and stops For the other five speeds the program shows every step in the solution process and it stops when it finds a solution or runs out of possibilities to try You can change the speed while the program is running Note that the number of moves that is reported by the program is the number of times that a piece was actually placed on the board It doesn t include checking whether a piece will fit or removing a piece during backtracking About the Check for Obvious Blocking Option Without this option you will sometimes see that the program seems to be doing something stupid You ll see an isolated group of one to four white squares where the program will never be able to place a pentomino but the program will be chugging away trying to place pentominoes in other parts of the board This is particularly obvious if you make a board with 3 rows and 20 columns where the problem is so bad that I have never had the patience to wait for a solution to be found This is why there is a 20 by 3 board in the Size menu rather than a 3 by 20 the 20 by 3 board is solved very quickly If you turn on the Check for Obvious Blocking option the program will check for blocking of this type every time it makes a move In fact it checks for any white area whose size is not a multiple of five or an even more complicated check when you haven t selected the maximum number of black squares This option can greatly decrease the number of moves needed to find solutions However the test itself is fairly complicated and so the net computer time spent searching for solutions is in many cases not very different About the Symmetry Check Option When this option is on the idea is to ignore solutions that are simply reflections or rotations of other solutions For some boards before any pieces are added when you rotate or reflect the board it looks the same after the operation as it did before This is called a symmetry

    Original URL path: http://math.hws.edu/eck/js/pentominos/pentominos-info.html (2016-02-07)
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  • Planar Symmetry Groups
    of a pattern that has 137 rotation symmetries and no reflection symmetries The groups that we get when we add reflection symmetry are called the dihedral groups For example the dihedral group D 137 is the symmetry group of a pattern that has 137 rotation symmetries and 137 reflection symmetries The rotation and dihedral groups are sometimes referred to collectively as rosette groups The web page rosette html lets you draw patterns with symmetry group R n or D n for n up to 20 There is a set of radio buttons that allows you to select the number of rotation symmetries and there is a checkbox that adds reflection symmetries When you draw something in the white area all the symmetric transformations of the thing that you draw are also shown If we drop the requirement that a pattern should have a center point that is left in place by all symmetry operations then we get new kinds of symmetry In particular we get translation symmetries A translation operation moves everything by a certain distance in a certain direction If a pattern has one translation symmetry it automatically has an infinite number of them For example if moving everything one inch to the right is a symmetry of a pattern then moving two inches to the right or three inches or four inches are also symmetries A pattern that has a translation symmetry is necessarily infinite Here is an example of a pattern that has a horizontal translation symmetry You have to imagine the pattern extending infinitely to the left and right This image was made with the program frieze html which lets you draw patterns that have horizontal translation symmetries The symmetry groups for the patterns in that program are called frieze groups A frieze group includes translations symmetries in one direction but not in a second independent direction Furthermore the group is discrete in the sense that there is a minimum translation distance that is a symmetry For example this picture has arbitrarily small horizontal translation symmetries so its symmetry group is not a frieze group With this restriction there are only seven different frieze groups They have names like p1m1 and pma2 though sometimes they are simply called F 1 through F 7 In the program there is a set of radio buttons for selecting the frieze group that you want to use The simplest frieze group p111 has translation symmetry and no other symmetry The other groups add additional symmetries For example p112 adds a 180 degree rotation symmetry 180 degrees is the only possible rotation symmetry in a frieze group p1m1 adds a reflection through a vertical line while pm11 adds reflection through a horizontal line and pmm2 adds both An m stands for mirror and indicates a reflection symmetry The groups p1a1 and pma2 add another kind of symmetry called a glide reflection A glide reflection consists of a translation followed by a reflection over a line that is parallel to the direction

    Original URL path: http://math.hws.edu/eck/js/symmetry/symmetry-info.html (2016-02-07)
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  • Preface from TMCM
    first time through you should pay close attention to the main ideas The next two chapters explain how computers can be built step by step out of very simple components By the end of chapter 3 you will understand how a physical object can be built to execute an arbitrarily complex set of program instructions This is the most technical part of the book If you decide to skip over it you will not be at a great disadvantage in the rest of the text However you will miss some really neat ideas and I encourage you to browse through Section 2 1 and the beginnings of Sections 3 1 and 3 3 at least And of course you can read the chapter summaries Chapter 4 on Theoretical Computers shows that all computers from the simple model computer constructed in Chapter 3 to the most advanced supercomputer are really equally powerful except for their speed and the amount of memory they have Furthermore they are all subject to certain surprising limitations on the problems they can solve The idea of computational universality covered in Section 4 1 is quite important the rest of the chapter is interesting but not vital to later chapters The next chapter turns for the first time to real computers It surveys their history examines their social impact and discusses how practical machines differ from the simplified model computers considered in the previous chapters Chapters 6 7 and 8 cover computer programming Chapter 6 introduces the basic concepts such as variables loops and decisions Chapter 7 concentrates on methods for writing very large or complex programs And Chapter 8 finishes by looking at some of the many different languages available for writing programs The last section of the book Chapters 9 through 12 deals with applications of computing After a general survey of applications in Chapter 9 the next three chapters cover three of the most important and exciting areas of computer science computer graphics parallel and distributed processing and artificial intelligence These four chapters can be read in any order The book is supplemented with a set of computer programs and with lab worksheets based on those programs The programs are currently available only for Macintosh computers but I am working to make them available to run under Windows as well The programs are closely tied to the ideas covered in the text They are not essential to understanding the material in the book but the hands on experience they give could certainly help to make some of the ideas presented here more accessible They are also as far as I can judge them myself rather fun The programs include xLogicCircuits which lets you build and run simulated circuits made from AND OR and NOT gates like those discussed in Chapter 2 xComputer which implements the model computer xComputer constructed in Chapter 3 xTuringMachine in which you can enter rule tables for Turing Machines as discussed in Chapter 4 and watch them as they move

    Original URL path: http://math.hws.edu/TMCM/preface.html (2016-02-07)
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  • Table of Contents of TMCM
    1 3 Instructions Subroutines etc 1 4 Handling Complexity Chapter Summary Questions Chapter 2 Teaching Silicon to Compute 2 1 Logical Circuitry 2 2 Arithmetic 2 3 Circuits that Remember Chapter Summary Questions Chapter 3 Building a Computer 3 1 Basic Design 3 2 Fetching and Executing 3 3 Self control 3 4 Postscript Assembly Language Chapter Summary Questions Chapter 4 Theoretical Computers 4 1 Simulation and Universality 4 2 Turing Machines 4 3 Unsolvable Problems Chapter Summary Questions Chapter 5 Real Computers 5 1 A Brief History 5 2 Usable Computers 5 3 Computers and Society Chapter Summary Questions Chapter 6 Programming 6 1 The Power of Names 6 2 Taking Control 6 3 Building Programs Chapter Summary Questions Chapter 7 Subroutines and Recursion 7 1 Writing and Using Subroutines 7 2 Real Programs 7 3 Recursion 7 4 Postscript Implementation Issues Chapter Summary Questions Chapter 8 Real Programming Languages 8 1 Virtual Machines 8 2 The Other Half of Programming 8 3 Escape from the von Neumann Machine Chapter Summary Questions Chapter 9 Applications 9 1 The Works 9 2 Off the Desktop 9 3 Postscript Analysis of Algorithms Chapter Summary Questions Chapter 10 Cooperating Computers 10 1

    Original URL path: http://math.hws.edu/TMCM/TOC.html (2016-02-07)
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