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  • Discrete mathematics problem - Probability theory and counting

    - by Mohammad
    Hello All, I'm taking a discrete mathematics course, and I encountered a question and I need your help. I don't know if this is the right place for that though :) It says: Each user on a computer system has a password, which is six to eight characters long, where each character is an uppercase letter or a digit. Each password must contain at least one digit. How many possible passwords are there? The book solves this by adding the probabilities of having six,seven and eight characters long password. However, when he solves for probability of six characters he does this P6 = 36^6 - 26^6 and does P7 = 36^7 - 26^7 and P8 = 36^8 - 26^8 and then add them all I understand the solution, but my question is why doesn't calculating, P6 = 10*36^5 and the same for P7 and P8, work? 10 for the digit and 36 for the alphanumeric? Also, if anyone could give me another solution, other than the one in the book. Thank you very much :)

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  • Deriving arrays in mathematics

    - by Gio Borje
    So I found some similarities between arrays and set notation while learning about sets and sequences in precalc e.g. set notation: {a | cond } = { a1, a2, a3, a4, ..., an} given that n is the domain (or index) of the array, a subset of Natural numbers (or unsigned integer). Most programming languages would provide similar methods to arrays that are applied to sets e.g. upperbounds & lowerbounds; possibly suprema and infima too. Where did arrays come from?

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  • What does mathematics have to do with programming?

    - by Rory
    I just started a diploma in software development. Right now we're starting out with basic Java and such (so right from the bottom you might say) - which is fine, I have no programming experience apart from knowing how to do "Hello World" in Java. I keep hearing that mathematics is pertinent to coding, but how is it so? What general examples would show how mathematics and programming go together, or are reliant on one another? I apologize of my question is vague, I'm barely starting to get a rough idea of the kind of world I'm stepping into as a code monkey student...

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  • Mathematics for Computer Science

    - by jiewmeng
    I am going into university next year. I think maths would be one of the more important aspects of computer science? I recently saw the MIT Intro to Algorithms video on YouTube and the maths required is quite hardcore. I wonder what parts of maths do i need, probability, calculus, trigo etc. Will the book Concrete Mathematics - it claims to be foundation for computer science - on Amazon cover most of whats required?

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  • JavaOne 2012: Lessons from Mathematics

    - by darcy
    I was pleased to get notification recently that my bof proposal for Lessons from Mathematics was accepted for JavaOne 2012. This is a bit of a departure from the project-centric JavaOne talks I usually give, but whisps of this kind of material have appeared before. I'm looking forward to presenting material from linear algebra, stochastics, and numerical optimization that have influence my thinking about technical problems in the JDK and elsewhere.

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  • F# in ASP.NET, mathematics and testing

    - by DigiMortal
    Starting from Visual Studio 2010 F# is full member of .NET Framework languages family. It is functional language with syntax specific to functional languages but I think it is time for us also notice and study functional languages. In this posting I will show you some examples about cool things other people have done using F#. F# and ASP.NET As I am ASP/ASP.NET MVP I am – of course – interested in how people use different languages and technologies with ASP.NET. C# MVP Tomáš Petrícek writes about developing ASP.NET MVC applications using F#. He also shows how to use LINQ To SQL in F# (using F# PowerPack) and provides sample solution and Visual Studio 2010 template for F# MVC web applications. You may also find interesting how you can create controllers in F#. Excellent work, Tomáš! Vladimir Matveev has interesting example about how to use F# and ApplicationHost class to process ASP.NET requests ouside of IIS. This is simple and very straight-forward example and I strongly suggest you to take a look at it. Very cool example is project Strom in Codeplex. Storm is web services testing tool that is fully written on F#. Take a look at this site because Codeplex offers also source code besides binaries. Math Functional languages are strong in fields like mathematics and physics. When I wrote my C# example about BigInteger class I found out that recursive version of Fibonacci algorithm in C# is not performing well. In same time I made same experiment on F# and in F# there were no performance problems with recursive version. You can find F# version of Fibonacci algorithm from Bob Palmer’s blog posting Fibonacci numbers in F#. Although golden spiral is useful for solving many problems I looked for some practical code example and found one. Kean Walmsley published in his Through the Interface blog very interesting posting Creating Fibonacci spirals in AutoCAD using F#. There are also other cool examples you may be interested in. Using numerical components by Extreme Optimization  it is possible to make some numerical integration (quadrature method) using F# (also C# example is available). fsharp.it introduces factorials calculation on F#. Robert Pickering has made very good work on programming The Game of Life in Silverlight and F# – I definitely suggest you to try out this example as it is very illustrative too. Who wants something more complex may take a look at Newton basin fractal example in F# by Jonathan Birge. Testing After some searching and surfing I found out that there is almost everything available for F# to write tests and test your F# code. FsCheck - FsCheck is a port of Haskell's QuickCheck. Important parts of the manual for using FsCheck is almost literally "adapted" from the QuickCheck manual and paper. Any errors and omissions are entirely my responsibility. FsTest - This project is designed to Language Oriented Programming constructs around unit testing and behavior testing in F#. The goal of this project is to create a Domain Specific Language for testing F# code in a way that makes sense for functional programming. FsUnit - FsUnit makes unit-testing with F# more enjoyable. It adds a special syntax to your favorite .NET testing framework. xUnit.NET - xUnit.net is a developer testing framework, built to support Test Driven Development, with a design goal of extreme simplicity and alignment with framework features. It is compatible with .NET Framework 2.0 and later, and offers several runners: console, GUI, MSBuild, and Visual Studio integration via TestDriven.net, CodeRush Test Runner and Resharper. It also offers test project integration for ASP.NET MVC. Getting started Well, as a first thing you need Visual Studio 2010. Then take a look at these resources: F# samples @ MSDN Microsoft F# Developer Center @ MSDN F# Language Reference @ MSDN F# blog F# forums Real World Functional Programming: With Examples in F# and C# (Amazon) Happy F#-ing! :)

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  • count of distinct acyclic paths from A[a,b] to A[c,d]?

    - by Sorush Rabiee
    I'm writing a sokoban solver for fun and practice, it uses a simple algorithm (something like BFS with a bit of difference). now i want to estimate its running time ( O and omega). but need to know how to calculate count of acyclic paths from a vertex to another in a network. actually I want an expression that calculates count of valid paths, between two vertices of a m*n matrix of vertices. a valid path: visits each vertex 0 or one times. have no circuits for example this is a valid path: but this is not: What is needed is a method to find count of all acyclic paths between the two vertices a and b. comments on solving methods and tricks are welcomed.

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  • count of paths from A[a,b] to A[c,d] without duplicating?

    - by Sorush Rabiee
    I write a sokoban solver for fun and practice, it uses a simple algorithm (something like BFS). now i want to estimate its running time ( O and omega). but i need to know how to calculate count of paths from a vertex to another in a network. each path from a to b is a sequence of edges with no circuit. for example this is a correct path: http://www.imgplace.com/viewimg143/4789/501k.png but this is not: http://www.imgplace.com/viewimg143/6140/202.png

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  • Counting problem: possible sudoko tables?

    - by Sorush Rabiee
    Hi, I'm working on a sudoko solver. my method is using a game tree and explore possible permutations for each set of digits by DFS Algorithm. in order to analyzing problem, i want to know what is the count of possible valid and invalid sudoko tables? - a 9*9 table that have 9 one, 9 two, ... , 9 nine. (this isn't exact duplicate by this question) my solution is: 1- First select 9 cells for 1s: (*) 2- and like (1) for other digits (each time, 9 cells will be deleted from remaining available cells): C(81-9,9) , C(81-9*2,9) .... = 3- finally multiply the result by 9! (permutation of 123456789 in (*)) this is not equal to accepted answer of this question but problems are equivalent. what did i do wrong?

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  • Extreme Optimization Numerical Libraries for .NET – Part 1 of n

    - by JoshReuben
    While many of my colleagues are fascinated in constructing the ultimate ViewModel or ServiceBus, I feel that this kind of plumbing code is re-invented far too many times – at some point in the near future, it will be out of the box standard infra. How many times have you been to a customer site and built a different variation of the same kind of code frameworks? How many times can you abstract Prism or reliable and discoverable WCF communication? As the bar is raised for whats bundled with the framework and more tasks become declarative, automated and configurable, Information Systems will expose a higher level of abstraction, forcing software engineers to focus on more advanced computer science and algorithmic tasks. I've spent the better half of the past decade building skills in .NET and expanding my mathematical horizons by working through the Schaums guides. In this series I am going to examine how these skillsets come together in the implementation provided by ExtremeOptimization. Download the trial version here: http://www.extremeoptimization.com/downloads.aspx Overview The library implements a set of algorithms for: linear algebra, complex numbers, numerical integration and differentiation, solving equations, optimization, random numbers, regression, ANOVA, statistical distributions, hypothesis tests. EONumLib combines three libraries in one - organized in a consistent namespace hierarchy. Mathematics Library - Extreme.Mathematics namespace Vector and Matrix Library - Extreme.Mathematics.LinearAlgebra namespace Statistics Library - Extreme.Statistics namespace System Requirements -.NET framework 4.0  Mathematics Library The classes are organized into the following namespace hierarchy: Extreme.Mathematics – common data types, exception types, and delegates. Extreme.Mathematics.Calculus - numerical integration and differentiation of functions. Extreme.Mathematics.Curves - points, lines and curves, including polynomials and Chebyshev approximations. curve fitting and interpolation. Extreme.Mathematics.Generic - generic arithmetic & linear algebra. Extreme.Mathematics.EquationSolvers - root finding algorithms. Extreme.Mathematics.LinearAlgebra - vectors , matrices , matrix decompositions, solvers for simultaneous linear equations and least squares. Extreme.Mathematics.Optimization – multi-d function optimization + linear programming. Extreme.Mathematics.SignalProcessing - one and two-dimensional discrete Fourier transforms. Extreme.Mathematics.SpecialFunctions

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  • Street-Fighting Mathematics

    Sanjoy Mahajan's new book lays out practical tools for educated guessing and down-and-dirty problem-solving Problem solving - Math - Recreations - Competitions - Methods and Theories

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  • Applications: The mathematics of movement, Part 1

    - by TechTwaddle
    Before you continue reading this post, a suggestion; if you haven’t read “Programming Windows Phone 7 Series” by Charles Petzold, go read it. Now. If you find 150+ pages a little too long, at least go through Chapter 5, Principles of Movement, especially the section “A Brief Review of Vectors”. This post is largely inspired from this chapter. At this point I assume you know what vectors are, how they are represented using the pair (x, y), what a unit vector is, and given a vector how you would normalize the vector to get a unit vector. Our task in this post is simple, a marble is drawn at a point on the screen, the user clicks at a random point on the device, say (destX, destY), and our program makes the marble move towards that point and stop when it is reached. The tricky part of this task is the word “towards”, it adds a direction to our problem. Making a marble bounce around the screen is simple, all you have to do is keep incrementing the X and Y co-ordinates by a certain amount and handle the boundary conditions. Here, however, we need to find out exactly how to increment the X and Y values, so that the marble appears to move towards the point where the user clicked. And this is where vectors can be so helpful. The code I’ll show you here is not ideal, we’ll be working with C# on Windows Mobile 6.x, so there is no built-in vector class that I can use, though I could have written one and done all the math inside the class. I think it is trivial to the actual problem that we are trying to solve and can be done pretty easily once you know what’s going on behind the scenes. In other words, this is an excuse for me being lazy. The first approach, uses the function Atan2() to solve the “towards” part of the problem. Atan2() takes a point (x, y) as input, Atan2(y, x), note that y goes first, and then it returns an angle in radians. What angle you ask. Imagine a line from the origin (0, 0), to the point (x, y). The angle which Atan2 returns is the angle the positive X-axis makes with that line, measured clockwise. The figure below makes it clear, wiki has good details about Atan2(), give it a read. The pair (x, y) also denotes a vector. A vector whose magnitude is the length of that line, which is Sqrt(x*x + y*y), and a direction ?, as measured from positive X axis clockwise. If you’ve read that chapter from Charles Petzold’s book, this much should be clear. Now Sine and Cosine of the angle ? are special. Cosine(?) divides x by the vectors length (adjacent by hypotenuse), thus giving us a unit vector along the X direction. And Sine(?) divides y by the vectors length (opposite by hypotenuse), thus giving us a unit vector along the Y direction. Therefore the vector represented by the pair (cos(?), sin(?)), is the unit vector (or normalization) of the vector (x, y). This unit vector has a length of 1 (remember sin2(?) + cos2(?) = 1 ?), and a direction which is the same as vector (x, y). Now if I multiply this unit vector by some amount, then I will always get a point which is a certain distance away from the origin, but, more importantly, the point will always be on that line. For example, if I multiply the unit vector with the length of the line, I get the point (x, y). Thus, all we have to do to move the marble towards our destination point, is to multiply the unit vector by a certain amount each time and draw the marble, and the marble will magically move towards the click point. Now time for some code. The application, uses a timer based frame draw method to draw the marble on the screen. The timer is disabled initially and whenever the user clicks on the screen, the timer is enabled. The callback function for the timer follows the standard Update and Draw cycle. private double totLenToTravelSqrd = 0; private double startPosX = 0, startPosY = 0; private double destX = 0, destY = 0; private void Form1_MouseUp(object sender, MouseEventArgs e) {     destX = e.X;     destY = e.Y;     double x = marble1.x - destX;     double y = marble1.y - destY;     //calculate the total length to be travelled     totLenToTravelSqrd = x * x + y * y;     //store the start position of the marble     startPosX = marble1.x;     startPosY = marble1.y;     timer1.Enabled = true; } private void timer1_Tick(object sender, EventArgs e) {     UpdatePosition();     DrawMarble(); } Form1_MouseUp() method is called when ever the user touches and releases the screen. In this function we save the click point in destX and destY, this is the destination point for the marble and we also enable the timer. We store a few more values which we will use in the UpdatePosition() method to detect when the marble has reached the destination and stop the timer. So we store the start position of the marble and the square of the total length to be travelled. I’ll leave out the term ‘sqrd’ when speaking of lengths from now on. The time out interval of the timer is set to 40ms, thus giving us a frame rate of about ~25fps. In the timer callback, we update the marble position and draw the marble. We know what DrawMarble() does, so here, we’ll only look at how UpdatePosition() is implemented; private void UpdatePosition() {     //the vector (x, y)     double x = destX - marble1.x;     double y = destY - marble1.y;     double incrX=0, incrY=0;     double distanceSqrd=0;     double speed = 6;     //distance between destination and current position, before updating marble position     distanceSqrd = x * x + y * y;     double angle = Math.Atan2(y, x);     //Cos and Sin give us the unit vector, 6 is the value we use to magnify the unit vector along the same direction     incrX = speed * Math.Cos(angle);     incrY = speed * Math.Sin(angle);     marble1.x += incrX;     marble1.y += incrY;     //check for bounds     if ((int)marble1.x < MinX + marbleWidth / 2)     {         marble1.x = MinX + marbleWidth / 2;     }     else if ((int)marble1.x > (MaxX - marbleWidth / 2))     {         marble1.x = MaxX - marbleWidth / 2;     }     if ((int)marble1.y < MinY + marbleHeight / 2)     {         marble1.y = MinY + marbleHeight / 2;     }     else if ((int)marble1.y > (MaxY - marbleHeight / 2))     {         marble1.y = MaxY - marbleHeight / 2;     }     //distance between destination and current point, after updating marble position     x = destX - marble1.x;     y = destY - marble1.y;     double newDistanceSqrd = x * x + y * y;     //length from start point to current marble position     x = startPosX - (marble1.x);     y = startPosY - (marble1.y);     double lenTraveledSqrd = x * x + y * y;     //check for end conditions     if ((int)lenTraveledSqrd >= (int)totLenToTravelSqrd)     {         System.Console.WriteLine("Stopping because destination reached");         timer1.Enabled = false;     }     else if (Math.Abs((int)distanceSqrd - (int)newDistanceSqrd) < 4)     {         System.Console.WriteLine("Stopping because no change in Old and New position");         timer1.Enabled = false;     } } Ok, so in this function, first we subtract the current marble position from the destination point to give us a vector. The first three lines of the function construct this vector (x, y). The vector (x, y) has the same length as the line from (marble1.x, marble1.y) to (destX, destY) and is in the direction pointing from (marble1.x, marble1.y) to (destX, destY). Note that marble1.x and marble1.y denote the center point of the marble. Then we use Atan2() to get the angle which this vector makes with the positive X axis and use Cosine() and Sine() of that angle to get the unit vector along that same direction. We multiply this unit vector with 6, to get the values which the position of the marble should be incremented by. This variable, speed, can be experimented with and determines how fast the marble moves towards the destination. After this, we check for bounds to make sure that the marble stays within the screen limits and finally we check for the end condition and stop the timer. The end condition has two parts to it. The first case is the normal case, where the user clicks well inside the screen. Here, we stop when the total length travelled by the marble is greater than or equal to the total length to be travelled. Simple enough. The second case is when the user clicks on the very corners of the screen. Like I said before, the values marble1.x and marble1.y denote the center point of the marble. When the user clicks on the corner, the marble moves towards the point, and after some time tries to go outside of the screen, this is when the bounds checking comes into play and corrects the marble position so that the marble stays inside the screen. In this case the marble will never travel a distance of totLenToTravelSqrd, because of the correction is its position. So here we detect the end condition when there is not much change in marbles position. I use the value 4 in the second condition above. After experimenting with a few values, 4 seemed to work okay. There is a small thing missing in the code above. In the normal case, case 1, when the update method runs for the last time, marble position over shoots the destination point. This happens because the position is incremented in steps (which are not small enough), so in this case too, we should have corrected the marble position, so that the center point of the marble sits exactly on top of the destination point. I’ll add this later and update the post. This has been a pretty long post already, so I’ll leave you with a video of how this program looks while running. Notice in the video that the marble moves like a bot, without any grace what so ever. And that is because the speed of the marble is fixed at 6. In the next post we will see how to make the marble move a little more elegantly. And also, if Atan2(), Sine() and Cosine() are a little too much to digest, we’ll see how to achieve the same effect without using them, in the next to next post maybe. Ciao!

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  • Applications: The Mathematics of Movement, Part 2

    - by TechTwaddle
    In part 1 of this series we saw how we can make the marble move towards the click point, with a fixed speed. In this post we’ll see, first, how to get rid of Atan2(), sine() and cosine() in our calculations, and, second, reducing the speed of the marble as it approaches the destination, so it looks like the marble is easing into it’s final position. As I mentioned in one of the previous posts, this is achieved by making the speed of the marble a function of the distance between the marble and the destination point. Getting rid of Atan2(), sine() and cosine() Ok, to be fair we are not exactly getting rid of these trigonometric functions, rather, replacing one form with another. So instead of writing sin(?), we write y/length. You see the point. So instead of using the trig functions as below, double x = destX - marble1.x; double y = destY - marble1.y; //distance between destination and current position, before updating marble position distanceSqrd = x * x + y * y; double angle = Math.Atan2(y, x); //Cos and Sin give us the unit vector, 6 is the value we use to magnify the unit vector along the same direction incrX = speed * Math.Cos(angle); incrY = speed * Math.Sin(angle); marble1.x += incrX; marble1.y += incrY; we use the following, double x = destX - marble1.x; double y = destY - marble1.y; //distance between destination and marble (before updating marble position) lengthSqrd = x * x + y * y; length = Math.Sqrt(lengthSqrd); //unit vector along the same direction as vector(x, y) unitX = x / length; unitY = y / length; //update marble position incrX = speed * unitX; incrY = speed * unitY; marble1.x += incrX; marble1.y += incrY; so we replaced cos(?) with x/length and sin(?) with y/length. The result is the same.   Adding oomph to the way it moves In the last post we had the speed of the marble fixed at 6, double speed = 6; to make the marble decelerate as it moves, we have to keep updating the speed of the marble in every frame such that the speed is calculated as a function of the length. So we may have, speed = length/12; ‘length’ keeps decreasing as the marble moves and so does speed. The Form1_MouseUp() function remains the same as before, here is the UpdatePosition() method, private void UpdatePosition() {     double incrX = 0, incrY = 0;     double lengthSqrd = 0, length = 0, lengthSqrdNew = 0;     double unitX = 0, unitY = 0;     double speed = 0;     double x = destX - marble1.x;     double y = destY - marble1.y;     //distance between destination and marble (before updating marble position)     lengthSqrd = x * x + y * y;     length = Math.Sqrt(lengthSqrd);     //unit vector along the same direction as vector(x, y)     unitX = x / length;     unitY = y / length;     //speed as a function of length     speed = length / 12;     //update marble position     incrX = speed * unitX;     incrY = speed * unitY;     marble1.x += incrX;     marble1.y += incrY;     //check for bounds     if ((int)marble1.x < MinX + marbleWidth / 2)     {         marble1.x = MinX + marbleWidth / 2;     }     else if ((int)marble1.x > (MaxX - marbleWidth / 2))     {         marble1.x = MaxX - marbleWidth / 2;     }     if ((int)marble1.y < MinY + marbleHeight / 2)     {         marble1.y = MinY + marbleHeight / 2;     }     else if ((int)marble1.y > (MaxY - marbleHeight / 2))     {         marble1.y = MaxY - marbleHeight / 2;     }     //distance between destination and marble (after updating marble position)     x = destX - (marble1.x);     y = destY - (marble1.y);     lengthSqrdNew = x * x + y * y;     /*      * End Condition:      * 1. If there is not much difference between lengthSqrd and lengthSqrdNew      * 2. If the marble has moved more than or equal to a distance of totLenToTravel (see Form1_MouseUp)      */     x = startPosX - marble1.x;     y = startPosY - marble1.y;     double totLenTraveledSqrd = x * x + y * y;     if ((int)totLenTraveledSqrd >= (int)totLenToTravelSqrd)     {         System.Console.WriteLine("Stopping because Total Len has been traveled");         timer1.Enabled = false;     }     else if (Math.Abs((int)lengthSqrd - (int)lengthSqrdNew) < 4)     {         System.Console.WriteLine("Stopping because no change in Old and New");         timer1.Enabled = false;     } } A point to note here is that, in this implementation, the marble never stops because it travelled a distance of totLenToTravelSqrd (first if condition). This happens because speed is a function of the length. During the final few frames length becomes very small and so does speed; and so the amount by which the marble shifts is quite small, and the second if condition always hits true first. I’ll end this series with a third post. In part 3 we will cover two things, one, when the user clicks, the marble keeps moving in that direction, rebounding off the screen edges and keeps moving forever. Two, when the user clicks on the screen, the marble moves towards it, with it’s speed reducing by every frame. It doesn’t come to a halt when the destination point is reached, instead, it continues to move, rebounds off the screen edges and slowly comes to halt. The amount of time that the marble keeps moving depends on how far the user clicks from the marble. I had mentioned this second situation here. Finally, here’s a video of this program running,

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  • Applications: The Mathematics of Movement, Part 3

    - by TechTwaddle
    Previously: Part 1, Part 2 As promised in the previous post, this post will cover two variations of the marble move program. The first one, Infinite Move, keeps the marble moving towards the click point, rebounding it off the screen edges and changing its direction when the user clicks again. The second version, Finite Move, is the same as first except that the marble does not move forever. It moves towards the click point, rebounds off the screen edges and slowly comes to rest. The amount of time that it moves depends on the distance between the click point and marble. Infinite Move This case is simple (actually both cases are simple). In this case all we need is the direction information which is exactly what the unit vector stores. So when the user clicks, you calculate the unit vector towards the click point and then keep updating the marbles position like crazy. And, of course, there is no stop condition. There’s a little more additional code in the bounds checking conditions. Whenever the marble goes off the screen boundaries, we need to reverse its direction.  Here is the code for mouse up event and UpdatePosition() method, //stores the unit vector double unitX = 0, unitY = 0; double speed = 6; //speed times the unit vector double incrX = 0, incrY = 0; private void Form1_MouseUp(object sender, MouseEventArgs e) {     double x = e.X - marble1.x;     double y = e.Y - marble1.y;     //calculate distance between click point and current marble position     double lenSqrd = x * x + y * y;     double len = Math.Sqrt(lenSqrd);     //unit vector along the same direction (from marble towards click point)     unitX = x / len;     unitY = y / len;     timer1.Enabled = true; } private void UpdatePosition() {     //amount by which to increment marble position     incrX = speed * unitX;     incrY = speed * unitY;     marble1.x += incrX;     marble1.y += incrY;     //check for bounds     if ((int)marble1.x < MinX + marbleWidth / 2)     {         marble1.x = MinX + marbleWidth / 2;         unitX *= -1;     }     else if ((int)marble1.x > (MaxX - marbleWidth / 2))     {         marble1.x = MaxX - marbleWidth / 2;         unitX *= -1;     }     if ((int)marble1.y < MinY + marbleHeight / 2)     {         marble1.y = MinY + marbleHeight / 2;         unitY *= -1;     }     else if ((int)marble1.y > (MaxY - marbleHeight / 2))     {         marble1.y = MaxY - marbleHeight / 2;         unitY *= -1;     } } So whenever the user clicks we calculate the unit vector along that direction and also the amount by which the marble position needs to be incremented. The speed in this case is fixed at 6. You can experiment with different values. And under bounds checking, whenever the marble position goes out of bounds along the x or y direction we reverse the direction of the unit vector along that direction. Here’s a video of it running;   Finite Move The code for finite move is almost exactly same as that of Infinite Move, except for the difference that the speed is not fixed and there is an end condition, so the marble comes to rest after a while. Code follows, //unit vector along the direction of click point double unitX = 0, unitY = 0; //speed of the marble double speed = 0; private void Form1_MouseUp(object sender, MouseEventArgs e) {     double x = 0, y = 0;     double lengthSqrd = 0, length = 0;     x = e.X - marble1.x;     y = e.Y - marble1.y;     lengthSqrd = x * x + y * y;     //length in pixels (between click point and current marble pos)     length = Math.Sqrt(lengthSqrd);     //unit vector along the same direction as vector(x, y)     unitX = x / length;     unitY = y / length;     speed = length / 12;     timer1.Enabled = true; } private void UpdatePosition() {     marble1.x += speed * unitX;     marble1.y += speed * unitY;     //check for bounds     if ((int)marble1.x < MinX + marbleWidth / 2)     {         marble1.x = MinX + marbleWidth / 2;         unitX *= -1;     }     else if ((int)marble1.x > (MaxX - marbleWidth / 2))     {         marble1.x = MaxX - marbleWidth / 2;         unitX *= -1;     }     if ((int)marble1.y < MinY + marbleHeight / 2)     {         marble1.y = MinY + marbleHeight / 2;         unitY *= -1;     }     else if ((int)marble1.y > (MaxY - marbleHeight / 2))     {         marble1.y = MaxY - marbleHeight / 2;         unitY *= -1;     }     //reduce speed by 3% in every loop     speed = speed * 0.97f;     if ((int)speed <= 0)     {         timer1.Enabled = false;     } } So the only difference is that the speed is calculated as a function of length when the mouse up event occurs. Again, this can be experimented with. Bounds checking is same as before. In the update and draw cycle, we reduce the speed by 3% in every cycle. Since speed is calculated as a function of length, speed = length/12, the amount of time it takes speed to reach zero is directly proportional to length. Note that the speed is in ‘pixels per 40ms’ because the timeout value of the timer is 40ms.  The readability can be improved by representing speed in ‘pixels per second’. This would require you to add some more calculations to the code, which I leave out as an exercise. Here’s a video of this second version,

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  • Stairway to T-SQL DML Level 5: The Mathematics of SQL: Part 2

    Joining tables is a crucial concept to understanding data relationships in a relational database. When you are working with your SQL Server data, you will often need to join tables to produce the results your application requires. Having a good understanding of set theory, and the mathematical operators available and how they are used to join tables will make it easier for you to retrieve the data you need from SQL Server.

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  • Stairway to T-SQL DML Level 4: The Mathematics of SQL: Part 1

    A relational database contains tables that relate to each other by key values. When querying data from these related tables you may choose to select data from a single table or many tables. If you select data from many tables, you normally join those tables together using specified join criteria. The concepts of selecting data from tables and joining tables together is all about managing and manipulating sets of data. In Level 4 of this Stairway we will explore the concepts of set theory and mathematical operators to join, merge, and return data from multiple SQL Server tables. Get Smart with SQL Backup Pro Powerful centralised management, encryption and more.SQL Backup Pro was the smartest kid at school Discover why.

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  • What is the the relation between programming and mathematics?

    - by Math Grad
    Programmers seem to think that their work is quite mathematical. I understand this when you try to optimize something in performance, find the most efficient alogithm, etc.. But it patently seems false when you look at a billing application for a shop, or a systems software riddled with I/O calls. So what is it exactly? Is computation and associated programming really mathematical? Here I have in mind particularly the words of the philosopher Schopenhauer in mind: That arithmetic is the basest of all mental activities is proved by the fact that it is the only one that can be accomplished by means of a machine. Take, for instance, the reckoning machines that are so commonly used in England at the present time, and solely for the sake of convenience. But all analysis finitorum et infinitorum is fundamentally based on calculation. Therefore we may gauge the “profound sense of the mathematician,” of whom Lichtenberg has made fun, in that he says: “These so-called professors of mathematics have taken advantage of the ingenuousness of other people, have attained the credit of possessing profound sense, which strongly resembles the theologians’ profound sense of their own holiness.” I lifted the above quote from here. It seems that programmers are doing precisely the sort of mechanized base mental activity the grand old man is contemptuous about. So what exactly is the deal? Is programming really the "good" kind of mathematics, or just the baser type, or altogether something else just meant for business not to be confused with a pure discipline?

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  • Mathematics - Why is Differential Calculus (MVP) in PHP a tabu?

    - by Email
    Hi I want to do a Mean-Variance-Optimization (Markowitz) but i never found anything written in php that does this. MVP needs differential calculus. Can it be done in php and why arent there any classes/works from universities? For a webapplication (regarding performance) would another language be the better choice to handle heavy calculations? Thanks so much for any help/answer on this

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  • Learning Basic Mathematics

    - by NeedsToKnow
    I'm going to just come out and say it. I'm 20 and can't do maths. Two years ago I passed the end-of-high-school mathematics exam (but not at school), and did pretty well. Since then, I haven't done a scrap of mathematics. I wondered just how bad I had gotten, so I was looking at some simple algebra problems. You know, the kind you learn halfway through highschool. 5(-3x - 2) - (x - 3) = -4(4x + 5) + 13 Couldn't do them. I've got half a year left until I start a Computer Science undergraduate degree. I love designing and creating programs, and I remember I loved mathematics back when I did it. Basically, I've had a pretty bad education, but I want to be knowledgable in these areas. I was thinking of buying some high school textbooks and reading them, but I'm not sure this is the right way to go. I need to start off at some basic level and work towards a greater understanding. My question is: What should I study, how should I study, and what books can you recommend? Thanks!

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  • Music and Mathematics. Finding the natural scale matemathically. Is this correct?

    - by Alfonso de la Osa
    Hi! I wrote this post Music and Mathematics, finding the Natural and the Pentathonic scales. Central A at 383,56661 Hz. Is a method to find the Natural scale. I want to discuss it and find if its true. This is the code of the reasoning in js. <script> var c = 1.714285714285714; var tot = 0; var scale = []; while(tot < (14 - c)){ tot += c; scale.push(Math.round(tot)); } if(scale.length == 8){ document.write(scale + " " + c + "<br />"); } </script>

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  • Music and Mathematics. Finding the natural scale generator. The best way?

    - by Alfonso de la Osa
    Hi! I wrote this post Music and Mathematics, finding the Natural and the Pentatonic scales. Is a method to find the Natural scale. I want to discuss it and find if its true. This is the code of the reasoning in js. <script> var c = 12/7; var tot = 0; var scale = []; while(tot < (14 - c)){ tot += c; scale.push(Math.round(tot)); } if(scale.length == 8){ document.write(scale + " " + c + "<br />"); } </script>

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  • Mathematics for Computer Science Students

    - by Ender
    To cut a long story short, I am a CS student that has received no formal Post-16 Maths education for years. Right now even my Algebra is extremely rusty and I have a couple of months to shape up my skills. I've got a couple of video lectures in my bookmarks, consisting of: Pre-Calculus Algebra Calculus Probability Introduction to Statistics Differential Equations Linear Algebra My aim as of today is to be able to read the CLRS book Introduction to Algorithms and be able to follow the Mathematical notation in that, as well as being able to confidently read and back-up any arguments written in Mathematical notation. Aside from these video lectures, can anyone recommend any good books to help teach someone wishing to go from a low-foundation level to a more advanced level of Mathematics? Just as a note, I've taken a first-year module in Analytical Modelling, so I understand some of the basic concepts of Discrete Mathematics. EDIT: Just a note to those that are looking to learn Linear Algebra using the Video Lectures I have posted up. Peteris Krumins' Blog contains a run-through of these lecture notes as well as his own commentary and lecture notes, an invaluable resource for those looking to follow the lectures too.

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