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# Java Fill Triangle

Pascal's Triangle Using JAVA In mathematics , Pascal's triangle is a triangular array of the binomial coefficients. Balls are dropped onto the first peg and then bounce down to the bottom of the triangle where they collect in little bins. Sum of upper triangle = 1+2+3+4+6+7+8+6+5+1= 43. Note: If the path is not closed, the fill() method will add a line from the last point to the startpoint of the path to close the path (like closePath()), and then fill the path. That shape reference refers to the object of Triangle class at run time first and then object of Rectangle class and appropriate area method is called. In each of the examples below, the base is a side of the triangle. (N is the value inputted by the user). MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. The Graphics class is the abstract base class for all graphics contexts that allow an application to draw onto components that are realized on various devices, as well as onto off-screen images. Java is among the most popular programming languages out there, mainly because of how versatile and compatible it is. The dimensions of * the rectangle are chosen so that its sides are in proportion to * the "golden ratio" thought by the Greeks to represent the most * aesthetically pleasing geometry. The Adobe Acrobat User Community is a global resource for users of Acrobat and PDF, with free eseminars, tips, tutorials, videos and discussion forums. I hope you will find my work useful and beneficial. applet package examples. One of the famous one is its use with binomial equations. View a scaled diagram of the resulting triangle, or explore many other math calculators, as well as hundreds of other calculators addressing finance, health, fitness, and more. Join the DZone community and get the full member experience. While many programming environments will allow you to compile and run a program within the environment, you can also compile and run using Command Prompt. javac Cannot find my Example. Simply put, recursion is when a function calls itself. Java program to display Floyd's triangle, in the triangle, there are n integers in the nth row and a total of (n(n+1))/2 integers in n rows. If you were new to java or at the beginning stage then, Check – 500+ simple Java programs for beginners. Find file. Fill command, this is useful if you have created a geometric image (ie. java from §2. A student record contains the following information: student id, audit status, assignment score, and exam score. It is, as some wise people say, complete. If you were at newbie level to learn the Java programming then check out the following tutorial so that you will. The default color is black. Implement C++/Java/Python program to create a base class called shape. The other two values will be filled in. Doing this is very simple if you put a very little effort. Triangle Explorer is one of the Interactivate assessment explorers. Algorithm: find area of triangle using hero's formula in java (example) Read the sides of triangle from console. java, TestShape. Codility lessons solved in Java. The position is computed from adding offset to a reference point; the reference point is selected by the from_what argument. I also guide them in doing their final year projects. After this line of code has executed, you will end up with the triangle in its final form: You now have a triangle that has an outline and a fill. Figure shows the structure of pascal triangle having result up to six th rows. Syntax: triangle(x1, y1, x2, y2, x3, y3) Parameters. You may adjust the accuracy of your results. If he's using OpenGL or Direct3D, you don't fill a triangle with colour by colouring the pixels in the region like you generally do in a 2D library, you define a triangle with verticies and set the vertex colour. java, respectively. The Quincunx. java which has the main method to create a new Triangle object and to test what kind of Triangle it is. fillStyle will set the background property but will not fill. Hello Friends, I am Free Lance Tutor, who helped student in completing their homework. The numOfPoints gives the number of points (or to say, number of elements in the array) to join. Doing this is very simple if you put a very little effort. It consists of a compiler, interpreter, and disassembler, each of which is written in Java. x1, Triangle. By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy. The Equilateral Triangle is a triangle of equal sides and all angles are equal to 60 degrees. It actually consists of 2 triangle patterns, a triangle of empty spaces and a triangle of asterisks (*). The winding order of a hole does not have to be the same as the winding order of the boundary. C++ Example - C++ Program to Print Triangle of Stars. History Find file. We use cookies for various purposes including analytics. When you pass the x and y coords to rect and oval that point is the top left hand corner of the bounding rectangle of your shape and it can draw a rect or oval inside because there is only one possible way to do it. JOGL is the OpenGL binding for Java , mapping functions as closely as possible from C. This results in the fact that the flat side of the bottomflat triangle and also the flat side of the topflat triangle is drawn - so this falt edge it's plotted twice! Keep this is in mind - e. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. The intent of this project is to help you "Learn Android by Example" TM. The example shows how to print the triangle made of asterisks in Java. Fill color, specified as an RGB triplet, a hexadecimal color code, a color name, or a short name. In much of the Western world it is named after French mathematician Blaise Pascal , although other mathematicians studied it centuries before him in India , Iran , China , Germany , and Italy. In this tutorial we will see how to copy the content of one file to another file in java. The Quincunx. Note that you don't need two glBegin() functions to draw two triangles. As I mentioned in an earlier lesson, in his book, Java Foundation Classes in a Nutshell, David Flanagan tells us that the Java 2D definition of a Shape does not require the shape. How can I fill color on it? So far I can only success color the line but not fill the color. Website Home We would like to know how to draw a triangle. Answer import java. 17 LAB: Triangle comparison (classes) area Given class Triangle (in file Triangle. If you are using Windows, change the filename triangle. To access the libraries, you must add stdlib. Now I will show you two different ways to print Pascal's triangle in Java using a 2D array, up to N steps. IF: You need - create new solutions & upgrade your technology. Let us print a right angled triangle on the console in Java. Composite pattern should be applied only when the group of objects should behave as the single object. It is a simple pattern to print but helpful in learning how to create other patterns. It can never catch it. y1, Triangle. Area of a Triangle, Given 3 Sides, Heron's Formula - Duration: 13:39. One of the famous one is its use with binomial equations. Java Program Method 1 Simple Pascal’s triangle with no spacings. If there is no fillstyle then it will fill with black. Note: If the path is not closed, the fill() method will add a line from the last point to the startpoint of the path to close the path (like closePath()), and then fill the path. For example, if the distance between the height axis and a pyramid's side is 3 cm and the side's length is 5 cm, then the height will be: 5^2=3^2 + h^2 or h^2=25-9=16, hence h=4 cm. A triangle is a plane created by connecting three points. A from_what value of 0 measures from the beginning of the file, 1 uses the current file position, and 2 uses the end of the file as the reference point. jar bundles together all of our standard libraries into one file. If you need to calculate area of a triangle depending upon the input from the user, input() function can be used. closePath(). TriangleMesh. I have implemented this by using constructors, along with other utility methods. Note: All the shapes are drawn relative to the Java coordinate system. Hence, the problem of computing the closest distance to the edge of the triangle boils down to computing the distance of the point from each edge and then taking the minumum value. Java Code Examples: Ready to use Java examples which you can use directly into your Java programs. The triangle is composed of three edges and three nodes. Describe abstract class called Shape which has three subclasses say Triangle,Rectangle,Circle. A Graphics object encapsulates state information needed for the basic rendering operations that Java supports. y : ) For a triangle with a flat horizontal base and the 3rd point being above the middle of the base line, your 3 points are going to be:. C Programs : String Operations Without using Library Function. 17 LAB: Triangle comparison (classes) area Given class Triangle (in file Triangle. Triangle Explorer is one of the Interactivate assessment explorers. Path methods Methods to set different paths for objects. Functions are special values that encapsulate a piece of program. A simple program that I wrote using Java that will display Pascal Triangle using the value that is being given by the user. But because the type of ppoly1 and ppoly2 is pointer to Polygon (and not pointer to Rectangle nor pointer to Triangle), only the members inherited from Polygon can be accessed, and not those of the derived classes Rectangle and Triangle. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia (Iran), China, Germany, and Italy. Define one method area() in the abstract class and override this area() in these three subclasses to calculate for specific object i. With Highland Park to the east and Westwood-Roxhill to the west, the public right of way is central feature to the South Delridge corridor. Let us print a right angled triangle on the console in Java. JAVA: Write a program which creates a binary search tree of different shapes from a file. Java program to display Floyd's triangle, in the triangle, there are n integers in the nth row and a total of (n(n+1))/2 integers in n rows. A pascal triangle is started by first placing the number one in the first row and two ones in the second row. * @author Alina Lebron - Section 1 */ /** * Create a class that analyzes the type of triangle & finds the area */ public class Triangle {// create instance variables: private int sideA ; // the length of side A. 5 * base * height. Find the area of triangle and rectangle in java. I should put the declaration of class point in the header file and the definition of the methods in the definition file. Yellow warning triangle: Represents an application that cannot be identified because the certificate is untrusted or expired. Program/Source Code Here is the source code of the Java Program to Display Upper/Lower Triangle of a Matrix. length of the hypotenuse of a right triangle File I/O and Streams;. The winding order of a hole does not have to be the same as the winding order of the boundary. Pascal's triangle using recursive methods. Now I will show you two different ways to print Pascal's triangle in Java using a 2D array, up to N steps. The two middle terms are formed by adding one and two on either side and we get two three’s in between the two ones. JustSomeGuy wrote: Hi I'm writing a python application on my windows box and I want to draw a simple triangle on the screen How can I do this. I need to make a java program that inputs a word and then outputs a triangle of the word. The concept of the Sierpinski triangle is very simple: Take a triangle; Create four triangles out of that one by connecting the centers of each side. Similarly, to display the lower triangle check whether the sum of i and j indexes is greater than or equal to the order of the matrix if it is then print the element matrix[i][j]. com Free Programming Tutorials and Lessons By ProgrammingKnowledge. Website Home We would like to know how to draw a triangle. Use the following links to download an MS excel file that calculates the triangle area once the sides are input and also try the Java File that shows the relationship that Heron's formula has with a computer calculated area (i. ASCII stands for American Standard Code for Information Interchange. How can I fill color on it? So far I can only success color the line but not fill the color. Pascal's triangle Solution: Output: Download in file pdf Download in file pdf. java * Execution: java Triangle n * * Prints out an n-by-n triangular. Each element should have default values of 1. The Euler Line of a Triangle It's amazing how much geometry there is in the lowly triangle! Consider the triangle ABC below (colored magenta). Easy Tutor author of Program to draw a Triangle is from United States. pyw, suppressing the display of the console window. 2 Libraries. Java Tutorials lesson shows how to use the Graphics2D class to draw graphic primitives, arbitrary shapes, and to display graphics with outline and fill styles A browser with JavaScript enabled is required for this page to operate properly. The following example triangulates a polygon with a hole, and creates a triangle mesh primitive for the interior fill and two polyline primitives that outline the mesh. EDGE: If you want a distinct "edge" around the figure, also draw the figure in a contrasting color after the fill command. 2 million lines of source code examples to build from. In addition to bug fixes, this stable release supports Java 7 in addition to Java 5 and 6. Please try again later. Today I am going to show you how to print a Reverse Triangle in java. This entry was posted in LeetCode and tagged easy, Java, LeetCode, pascal's triangle on September 30, 2015 by cheonhyang. This results in the fact that the flat side of the bottomflat triangle and also the flat side of the topflat triangle is drawn - so this falt edge it's plotted twice! Keep this is in mind - e. The perimeter of a triangle is the sum of all three of the sides 1. Syntax: triangle(x1, y1, x2, y2, x3, y3); Parameters. The Floyd's triangle starts from 1, it is a right angled triangle, consisting of consecutive numbers. Except if he's using GDI (presumably 2D), then there are, as you pointed out, functions that create 2D filled triangles. Was it difficult to add to the file that used a function? Answer in a comment on the line below where you added the new function call. The way the entries are constructed in the table give rise to Pascal's Formula: Theorem 6. A good quickstart for the language itself is the Nutshell book mentioned in the bibliography. Java-When the class name and the filename of a given Java program. Once the path has been created, you can stroke or fill the path to render it. The area of the triangle is 14. Note: All the shapes are drawn relative to the Java coordinate system. However I dont know how to fill the triangle with one particular coloursay blue! I tried filling each line but then the lines dissappear. Produce a graphical representation of a Sierpinski triangle of order N in any orientation. This feature is not available right now. Test your Triangle class in a Drive program (in the same file) that prompts the user to enter the three sides of the triangle, the color, and whether or not the triangle is filled. Pseudocode (derived from pseudo and code) is a description of a computer programming algorithm that uses the structural conventions of programming languages, but omits detailed subroutines or language-specific syntax. *; class Triangle { public static String[]Triangles; public void drawTriangle() { //This method writes out the menu and asks for the input value. The first two arguments specify the first point, the middle two arguments specify the second point, and the last two arguments specify the third point. Doing this is very simple if you put a very little effort. Java program to calculate or to print area of a circle in a simple method. You need to just make reverse the inner two for loops. Compile the. What it does: The program prompts the user to enter three lengths for the sides. The file contains any number commands, * one per line. Java World Java knowledge sharing site Main menu. java, TestShape. The first is Rectangle and second is Triangle. Java Programming Code to Print Floyd Triangle. translate(50, 100); g2d. An amazing little machine created by Sir Francis Galton is a Pascal's Triangle made out of pegs. So we need three vertices on the left side representing one triangle and three vertices on the right side of the window representing the second triangle. fill() will fill the shape according to fillStyle. Leave the center triangle intact. Notes: All sides are measured in the same unit. The Sierpinski triangle fractal was first introduced in 1915 by Wacław Sierpiński. 1 Pascal's Formula top Let n and r be positive integers and suppose r £ n. If you managed to draw a triangle or a rectangle just like we did then congratulations, you managed to make it past one of the hardest parts of modern OpenGL: drawing your first triangle. 70 In this program, area of the triangle is calculated when three sides are given using Heron's formula. Just as the 100-th triangle number is half of 100 times 101, the 5-th triangle is half of 5 times 6, or 15, and the 9-th triangle number is half of 9 times 10, or 45. Be sure to get the latest edition. As an applet, it draws the triangle. Write a Java application that prints the first 10 lines of Pascals Triangle. You can only upload videos smaller than 600MB. h" header file in c. To create a new class, under the Geometry2 folder, right-click the Geometry2 sub-folder -> New -> Java Class Set the Name to Cylinder and click Finish; Change the file as follows:. Computers can only understand numbers, so an ASCII code is the numerical representation of a character such as 'a' or '@' or an action of some sort. Step 3: Planet Sphere. Well organized and easy to understand Web building tutorials with lots of examples of how to use HTML, CSS, JavaScript, SQL, PHP, Python, Bootstrap, Java and XML. Line //3 is the main() method, which is always the entry point into a Java program. With the code above the origin point 5,5 is included in the filled blue rectangle, however the argument rectangle's bottom right corner of 95,50 (5+90 , 45+5) is outside the filled area. Create a Dimension data type to store the length of the triangle legs. This Circle class is a "building block" and is meant to be used in another program. Binomial Coefficient (Pascal Triangle) objects in Java An exercise on our Java course from which our delegates gain a lot of appreciation is the one in which I ask them to set up an array of arrays containing Pascal's Triangle of Binomial Coefficients. Simple Inheritance using Java; Simple form Validation Using Jquery; Pascal Triangle Using 2 dimensional Array in JAVA; Access Database using PHP; Using same record set multiple times in PHP. For this you just need to understand the working of for-loop. 17) Create a class Shape which consists one final method area and volume (). Following example demonstrates how to create an applet which will have fill color in a rectangle using setColor(), fillRect() methods of Graphics class to fill color in a Rectangle. Divide it into 4 smaller congruent triangle and remove the central triangle. Scroll down more for the other style. paintComponents(g);. Loading Unsubscribe from Neeraj Bahl? JAVA - How To Design Login And Register Form In Java Netbeans - Duration: 44:14. But before learning this topic first read carefully if, if. You must use an array; do not use individual instance variables. javac Cannot find my Example. Explanation: This program will create a pattern which consists of the Pascal triangle. • public Triangle (int s1, int s2, int s3) - Sets up a triangle with the specified side lengths. else, for loop and while loop concept in detail. Each row of a Pascals Triangle can be calculated from the previous row so the core of the solution is a method that calculates a row based on the previous row which is passed as input. Write a java program to split a given text file into n parts. Compile the. C Programming language tutorial, Sample C programs, C++ Programs, Java Program, Interview Questions, C graphics programming, Data Structures, Binary Tree, Linked List, Stack, Queue, Header files, Design Patterns in Java, Triangle and Star pyramid pattern, Palindrome anagram Fibonacci programs, C puzzles. Write a C and Java program to print Right-angled Triangle pattern formed by star(*) character. Answer import java. What is Applet in Java with Example? What is Applet in Java? Applets in Java Applet Tag Parameters,Applet Tag in HTML Java - Opening a URL from an Applet Java - Opening a url in new window from an applet Tag Parameters: The Tag Applet versus Application Java - Read file Applet What is an Applet - Java Applet Tutorial java. This guide will walk you through the steps need to compute the area of a triangle using the Heron's Formula in NetBeans. Once the path has been created, you can stroke or fill the path to render it. Create a Dimension data type to store the length of the triangle legs. 2 thousand tutorials and articles to learn from. Question: Try "java Arithmetic 2 4 *" (in CMD shell and Eclipse/NetBeans) and explain the result obtained. In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. The first two arguments specify the first point, the middle two arguments specify the second point, and the last two arguments specify the third point. glut or freeGlut are a much easier learning curve). It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1 in the top left corner, there are n integers in the nth row and a sum of (n(n+1))/2 integers in n rows. Java - Pyramid Triangle pattern programs in Java with explanation - QA With Experts. Write a Java Program to find Area of Equilateral Triangle, Perimeter, Semi Perimeter, and Altitude of Equilateral Triangle with example. Easy Tutor author of Program to draw a Triangle is from United States. The following code sample illustrates how this can be achieved. The two middle terms are formed by adding one and two on either side and we get two three’s in between the two ones. Java can be used for a large number of things, including software development, mobile applications, and large systems development. After this line of code has executed, you will end up with the triangle in its final form: You now have a triangle that has an outline and a fill. Since the area of a parallelogram is A = B * H, the area of a triangle must be one-half the area of a parallelogram. The fill() method fills the current drawing (path). Graphics Class: Graphics Context and Custom Painting. Servis de sayelerinde kusursuz işledi. How to Create a Java Class in Netbeans to Compute the Area of a Triangle Using Heron's Formula. How can I fill color on it? So far I can only success color the line but not fill the color. java_triangle_classifier. Draw Triangle. junit " first & then the below class inside it :. To the right, you see this canvas with the default grid overlayed. Handwriting readiness - fill a triangle with straight lines - Log in or Become a Member to download. closePath(). The first two arguments specify the first point, the middle two arguments specify the second point, and the last two arguments specify the third point. İçeri ilk girdiğimiz andan itibaren tüm ekip bizi son derece kibar ve sıcak karşıladı. I am trying to fill a triangle. Note that you don't need two glBegin() functions to draw two triangles. Write a static method called trianglePerimeter that takes in the lengths of the three sides of the triangle and returns the perimeter of the triangle which is cal- culated by adding up the three sides. Notes: All sides are measured in the same unit. Experiment with this file. OpenGL is a standardized API used to draw 3D graphics to the screen (similar to DirectX / Direct3D). *; import java. If we know the side, then calculate the area of an Equilateral Triangle using the below formula. java /* * This Class is to take the Triangle parameter and compute * Design a class named Triangle that extends * GeometricObject. java * * Play chaos game on triangle to produce Sierpinski triangle. Produce a graphical representation of a Sierpinski triangle of order N in any orientation. Select Archive Format. It consists of a compiler, interpreter, and disassembler, each of which is written in Java. java source file. Java Graphics How to - Draw a triangle. It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1 in the top left corner, there are n integers in the nth row and a sum of (n(n+1))/2 integers in n rows. 0 to represent the three sides of the triangle. Syntax: triangle(x1, y1, x2, y2, x3, y3); Parameters. It is, as some wise people say, complete. Enter values three of the six sides and angles of the triangle and the other three values will be computed. In Java 2D, you generally create a Shape object, then call either the draw or fill method of the Graphics2D object, supplying the Shape object as an argument. The Sierpinski triangle fractal was first introduced in 1915 by Wacław Sierpiński. The theorem states that in any triangle, the square of the side opposite to a right angle, equals to the sum of the squares of the two remaining sides. The winding order of a hole does not have to be the same as the winding order of the boundary. To change the file object’s position, use f. The perimeter of a triangle is the sum of all three of the sides 1. Pascal's triangle is a set of numbers arranged in the form of a triangle. 5 * base * height. DrJava is a lightweight programming environment for Java designed to foster test-driven software development. It actually consists of 2 triangle patterns, a triangle of empty spaces and a triangle of asterisks (*). (N is the value inputted by the user). OK, I Understand. Compile "Circle. (For eg: Point. Easy Tutor says. We need to do a little bit edit to the fill triangle's code. Simple and Easy Core Java Tutorial to help beginners learn java. fillStyle will set the background property but will not fill. These computations can be found in Edge. , image and sound files), and a manifest file describing the special features of the archive. Steps on how to draw a Rectangle, Circle or Basic Shape on PDF Page: Choose Insert Menu > Drawing > Select Rectangle , Circle or Basic Shape. Java Graphics How to - Draw a triangle. Getting the Java examples to work. I tried to do it myself using. /* * File: FeltBoard. Test your Triangle class in a Drive program (in the same file) that prompts the user to enter the three sides of the triangle, the color, and whether or not the triangle is filled. Programming Pascals Triangle using eclipse java - posted in Programming: This is for my computer science class, but I am having a hard time understanding how to even start this program. A Computer Science portal for geeks. So, altogether, the above pyramid pattern is a combination of 3 triangle patterns (symbol ‘-‘ is used to represent an empty space), as illustrated below: So now, can you guess how many for loops do you need to produce the pyramid pattern. If you need to calculate area of a triangle depending upon the input from the user, input() function can be used. To print pascal triangle in Java Programming, you have to use three for loops and start printing pascal triangle as shown in the following example. Learn how to donate to SCORE International. It also defines a method called area that computes the area of an object. java from §2. Using those values, we will calculate the Perimeter of a triangle, Semi Perimeter of a triangle, and then Area of a Triangle. Drawing our triangle. So first of all, you have to include the stdio header file using the "include" preceding by # which tells that the header file needs to be process before compilation, hence named preprocessor directive. Taminations Loading. We do this by creating a buffer:. *; class Triangle { public static String[]Triangles; public void drawTriangle() { //This method writes out the menu and asks for the input value. Pascal's triangle is a set of numbers arranged in the form of a triangle. Notes: All sides are measured in the same unit. A simple program that I wrote using Java that will display Pascal Triangle using the value that is being given by the user. The author has concocted an implement using AWT components for drawing basic geometric shapes on its panel using a mouse. How to Create a Java Class in Netbeans to Compute the Area of a Triangle Using Heron's Formula. This is a relationship is what Java Inheritance is all about. Pascals Triangle | Learn Java by Example November 23, 2015 at 1:16 pm If you found this problem a little challenging then we would suggest having a look at some simpler examples of printing triangles using Java. Create a Dimension data type to store the length of the triangle legs. JOGL (Java OpenGL) Tutorial. Sierpinski triangle is created as follows: Begin with an equilateral triangle, which is Sierpinski triangle of order 0. In each of the examples below, the base is a side of the triangle. There is no difficulty in applying loops for them, what we need to do is to make some little changes in the for loop conditions and it will work fine. Loading Unsubscribe from Neeraj Bahl? JAVA - How To Design Login And Register Form In Java Netbeans - Duration: 44:14. The Graphics class is the abstract base class for all graphics contexts that allow an application to draw onto components that are realized on various devices, as well as onto off-screen images. A Graphics object encapsulates state information needed for the basic rendering operations that Java supports. I was not able to find a solution for this problem. The next step is to give this triangle to OpenGL. Explanation: This program will create a pattern which consists of the Pascal triangle. Thus, the formula for the area of a triangle is: or where b is the base, h is the height and · means multiply. A graphics context provides the capabilities of drawing on the screen. This tutorial will guide you from simple functions to more advanced features. Related Links. Sierpinski triangle/Graphical You are encouraged to solve this task according to the task description, using any language you may know. The order of the vertices used in the calculation will affect the direction of the normal (in or out of the face w. In this tutorial, we will discuss Java code to Floyd’s triangle star pattern using nested for loop. Now we will draw a new shape tringle. Give exactly 3 pieces of information, including at least one side. A good quickstart for the language itself is the Nutshell book mentioned in the bibliography. A JAR file is simply a ZIP file that contains classes, possibly other files that a program may need (e. As I mentioned in an earlier lesson, in his book, Java Foundation Classes in a Nutshell, David Flanagan tells us that the Java 2D definition of a Shape does not require the shape. The Equilateral Triangle is a triangle of equal sides and all angles are equal to 60 degrees. On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Doing this is very simple if you put a very little effort. Pascal's triangle Solution: Output: Download in file pdf Download in file pdf. We will learn how to create hollow triangle pattern in Java language using nested for loop-Java code to hollow triangle pattern. Create Point data types to store the location of the three vertices of the triangle. Write a Java Program to Print Pascal Triangle using Recursion Following Java Program ask to the user to enter the number of line/row upto which the Pascal’s triangle will be printed to print the Pascal’s triangle on the screen. To print pascal triangle in Java Programming, you have to use three for loops and start printing pascal triangle as shown in the following example. Floyd’s triangle can be defined as a series of numbers which are sequentially spread across a series of rows. import java. Graphics class comes with two methods and one constructor to draw polygons. copy and paste in notepad, How to print out a triangle in java java source code: How to print a triangle in java. It has many interpretations. Explanation. To create a new class, under the Geometry2 folder, right-click the Geometry2 sub-folder -> New -> Java Class Set the Name to Cylinder and click Finish; Change the file as follows:. If you were at newbie level to learn the Java programming then check out the following tutorial so that you will. The sum of the internal angles that exist at the vertices always total the same number for every triangle — 180 degrees, or $\\pi$ radians.


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# JEE Novice - (19)

Algebra Level 4

$\Large -m\leq x - 20\sqrt{m}$

Find the smallest value of $x$ which satisfies the above inequality for all non-negative '$m$'.

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# The generic fibre is a genus 1 curve

Let $S/\mathbb{C}$ be a smooth projective surface and let $C/\mathbb{C}$ be a smooth projective curve such that there exists a surjective morphism $\pi: S \to C$. Are the following conditions equivalent and if so, why?

1. The generic fibre $S_\eta \to \text{Spec}(\mathbb{C}(C))$ is a (smooth) genus 1 curve.
2. For all but finitely many closed points $P$ of $C$, the fibre $S_P$ is an elliptic curve.
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Yes if you suppose $O_C\to \pi_*O_S$ is an isomorphism. By the generic smoothness, we know that all fibers are smooth except for a proper closed subset of (so finitely many points in) $C$. Moreover, as $\pi$ is flat, the fibers have the same arithmetic genus. The hypothesis on $\pi_*O_S$ implies that all fibers are connected (Zariski's connectedness theorem). So for smooth fibers, arithmetic genus equal to geometric genus.
(1) implies (2): if the generic fiber has genus 1, then all smooth fibers are curves of genus $1$ over $\mathbb C$, so they are elliptic curves.
(2) implies (1): if one fiber is elliptic, then the generic fiber has genus $1$.
Without the hypothesis on $\pi_*O_S$, the implication from (1) to (2) is not true: consider a non trivial cover $C' \to C$, a surface $S$ fibered over $C'$ and compose $S\to C'\to C$. Then most of the fibers of $S\to C$ are not connected and therefore are not elliptic curves.
Edit the counterexample may not be one, it depends on what you call a genus $1$ curve. If it is geometrically connected, then (1) implies the condition on $\pi_*O_S$ and hence (2). On the other hand, the method of the "counterexample" shows that (2) implies (1) (and $O_C\simeq\pi_*O_S$).


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# Monetary Policy and Redistribution in Open Economies

Available as: PDF

Globalization has not integrated households equally into the international goods and financial markets. Because of this uneven international integration, external shocks to the economy can have different impacts across households. How does monetary policy influence these effects?

We build a model of a small open economy featuring uneven international integration across households. Then we analyze the difference between a floating exchange rate regime and a fixed exchange rate regime to study the distributional effects of external shocks on household consumption.

We find that the fixed exchange rate regime amplifies aggregate fluctuations following the external shocks but causes more even consumption responses across households. When considering distributional effects, our quantitative results show the following:

• Under the floating exchange rate regime, taking the increase in foreign demand for domestically produced goods as an example, wages in the tradable goods sector increase significantly. Households working in this sector increase their consumption by much more than those working in the non-tradable goods sector do. Yet this shock also leads to an appreciation of the domestic currency.
• Under the fixed exchange rate regime, the monetary authority lowers the interest rate to avoid such a domestic currency appreciation. This stimulates overall demand, indirectly increases wages in the non-tradable goods sector and reduces the difference in household consumption responses.

These results indicate a trade-off between maintaining overall stabilization and controlling consumption inequality.

JEL Code(s): E, E3, E32, E5, E52, F, F4, F41, F44

DOI: https://doi.org/10.34989/swp-2022-6


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## Calculating activation energy from Arrhenius equation

Arrhenius Equation: $\ln k = - \frac{E_{a}}{RT} + \ln A$

Nancy Le - 1F
Posts: 52
Joined: Fri Sep 29, 2017 7:07 am

### Calculating activation energy from Arrhenius equation

If given 2 rate constants and the corresponding temperatures, like in exercise 61, does it matter which one we assign as k1,k2,T1,T2?

Angela G 2K
Posts: 30
Joined: Fri Sep 29, 2017 7:06 am

### Re: Calculating activation energy from Arrhenius equation

Just tried switching k1, k2, T1, T2 and the answer was the same, so I guess the order doesn't matter. Just make sure the you use the correct temperature for their respective rate constants.

Mathematically it works out, since a higher temperature gives a larger rate constant. If you use the bigger temp/rate constant as k2/T2, then you'll get a positive multiplying a positive, and if you switch it and make it k1/T1 instead, you'll get a negative multiplying a negative, resulting in the same positive answer.


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## anonymous 4 years ago A man 6 feet tall walks at a rate of 13 feet per second away from a light that is 15 feet above the ground. When he is 8 feet from the base of the light, at what rate is the tip of his shadow moving?

1. anonymous

HINT HINT: Use similar Triangles!

2. anonymous

yup..gigles is right

3. anonymous

another hint. the rate of change will be a constant, the "when he is 8 feet" will have nothing to do with it.

4. anonymous

|dw:1320959290284:dw| use $\frac{x}{6}=\frac{x+y}{15}$ to get a relation between x and y. you know $y'=13$ and you want $x'$


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Now showing items 3167-3186 of 3388

• #### Thermal, rheological, and mechanical properties of a polymer composite cured at staged cure cycles

(Wichita State University, 2010-05)
Thermal, rheological, and mechanical properties of a polymer composite cured at different one-stage and two-stage cure cycles were studied in this dissertation. A commercial carbon-fiber prepreg, Cycom 977-2 UD, was used. ...
• #### Thermal/photo/electrochemical catalytic process for clean energy

(Elsevier Ltd, 2022-11-05)
[No abstract available]
• #### Thermally induced loads of fastened hybrid composite/aluminum structures

(American Institute of Aeronautics and Astronautics, 2008-03)
Large composite structures have been increasingly used in the aviation industry. New applications of composite materials include primary structures such as aircraft fuselages. These large composite parts are sometimes ...
• #### Thermo-mechanical analysis of metallized stereolithography EDM electrodes

(2005-12)
EDM is widely accepted in tool and die industries for its versatility in machining hardened steels. Its ability to machine heat treated hardened steels eliminates the conventional method of heat treatment after machining ...
• #### Thermo-mechanical simulation of underwater friction stir welding of low carbon steel

(MDPI, 2021-08-30)
This article investigates the flow of materials and weld formation during underwater friction stir welding (UFSW) of low carbon steel. A thermo-mechanical model is used to understand the relation between frictional heat ...
• #### Thermodynamic modeling of in-situ rocket propellant fabrication on Mars

(Elsevier, 2022-05-20)
Summary In-situ resource utilization (ISRU) to refuel rockets on Mars will become critical in the future. The current effort presents a thorough feasibility analysis of a scalable, Matlab-based, integrated ISRU framework ...
• #### Thermoset composite recycling - Driving forces, development, and evolution of new opportunities

(SAGE, 2018-04-01)
Thermoset composites represent a substantial challenge for recycling, even as composite products increase in market interest. The concept of putting all future thermoset composite products into landfills over the next ...
• #### Thickness measurement of thin films by x-ray absorption

(IEEE, 1991-01)
An x-ray diffraction method for determining thicknesses of thin films grown on single-crystal substrates is presented. The equations, based on the kinematical theory of x-ray diffraction and the mosaic crystal model, were ...
• #### Thin is green: Leveraging the thin-client paradigm for sustainable mobile computing

(Elsevier B.V., 2015-07)
Progress in the area of environmental sustainability for the mobile computing industry could be achieved by making advancement on two fronts: reducing the energy consumed by individual devices throughout their life cycle ...
• #### Threat modeling of para-virtualized environments

(Wichita State University, 2011-12)
Organizations are looking at various cost effective methods to reduce the overall cost of data storage systems. This measure is taken essentially to reduce the hardware that is currently being used for hosting servers. In ...
• #### A three-dimensional approach towards measuring sender anonymity

(Wichita State University, 2010-12)
Privacy plays an indispensable role over internet transactions. There are existing methods in place which enhance the level of privacy over WorldWide Web (WWW). Anonymity intends to hide the identity of the user during ...
• #### Three-dimensional fluid-structure-interaction simulation of tilting disk mechanical heart valve

(American Society of Mechanical Engineers, 2014)
The current computational effort will focus on the numerical analysis of current tiling disk MHVs. In this work an implicit fluid-structure-interaction (FSI) simulation of the Bjork-Shiley design was carried out using ...
• #### Three-dimensional quadcopter modeling and simulation using real-time brain machine interface

(Wichita State University, 2019-12)
The recent progresses in the area of neuro-robotics science and technology has provided the possibility to explore its practicability in aviation, especially in the area of flight controls through brain machine interfaces. ...
• #### Three-dimensional structured superhydrophobic electrospun nanofiber mats for environmental remediations

(Wichita State University, 2019-05)
This research is focused on the production of nanofibers from the electrospinning process with and without a Teflon™ emulsion. Research on superhydrophobic surfaces and its applications in different fields has garnered ...
• #### Three-stage algorithms for the large-scale dynamic vehicle routing problem with industry 4.0 approach

(Taylor & Francis, 2022-08-29)
Companies are eager to have a smart supply chain especially when they have a dynamic system. Industry 4.0 is a concept which concentrates on mobility and real-time integration. Thus, it can be considered as a necessary ...
• #### Throughput comparison of Shuffle-exchange networks with additional stages due to resource scheduling

(IEEE, 2017)
Several approaches are introduced to increase the throughput and reliability of Multistage Interconnection Networks (MINs), mostly by changing the network architecture. When multiple sources in such a network try to send ...
• #### Throughput-otimal LIFO policy for bounded delay in the presence of heavy-tailed traffic

(IEEE, 2016)
Scheduling is one of the most important resource allocation for networked systems. Conventional scheduling policies are primarily developed under light-tailed (LT) traffic assumptions. However, recent empirical studies ...
• #### Ti$_3$C$_2$T$_x$ MXene framework materials: Preparation, properties and applications in energy and environment

(Elsevier Inc., 2022-11-05)
Solar energy as a clean energy plays an important role in producing sufficient energy for both home and industry usage. It’s an alternative for fossil fuels which is damaging the environment by emission of lots of greenhouse ...
• #### Tight bounds on one-pass map-reduce algorithms for matrix multiplication

(Wichita State University, 2015-12)
MapReduce is an effi t parallel computation model introduced by Google, for performing many large-scale computations, including matrix multiplication. Matrix multiplication can be done using either an one-pass or a two-pass ...
• #### TiiCH : Interactive Whiteboard System

(2011-12-15)
This TIICH system is designed to eliminate the use of chalkboards and whiteboards. Letting teachers fully enter the electronic world but at a minimal cost. We wanted to let teachers teach their class, write their notes, ...


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## Fatherland and HHhH

March 6, 2014

by Robert Harris; Laurent Binet & Sam Taylor

These two books are tied together by the name of Reinhard Heydrich. I can’t think of a polite way of describing Heydrich. He was one of the architects of the Holocaust, a fervid Nazi, and an all-round total bastard. Hitler called him “the man with the iron heart”, which gives you some kind of idea of what kind of a git he was.

In the real world, Heydrich dies in 1942 from injuries sustained during an assassination attempt by Czech and Slovak commandos while he was “Acting Reich Protector of Bohemia and Moravia”. In the world of Fatherland, he survives, the Germans win the Second World War and Europe languishes under Nazi rule, with a sympathetic administration in the USA led by Kennedy (père rather than fils) providing no effective check on their activities. Heydrich continues in his role as second-in-command of the SS and goes on being as much as a bastard as ever.

Fatherland was published in 1992, so I’m a little late to the party (for a change). Alternative history novels set around WW2 are a popular genre, but Harris does something interesting and different. He manages to avoid any of the obvious missteps in representing a Nazified Europe by writing what starts out as a straight police procedural. The Thousand Year Reich still needs plods, apparently. Well, it’s a police state, so you do need some police. The slight twist is that the Kripo (Kriminalpolizei) was subsumed into the SS in 1939, thus becoming a sister agency to the Gestapo and so under the overall control of Himmler and his sidekick Heydrich. Our hero, Xavier March, wears the feared uniform of the SS, rationalising that he’d rather do some good wearing the uniform than no good at all.

The plot isn’t terribly unpredictable, although it’s uncovered in stages, so it’s not immediately obvious what’s going on. An upcoming summit between Hitler and Kennedy precipitates mild discomfort within the Nazi hierarchy over the Final Solution: everyone in Germany talks about the Jews “going to the East”, if they talk about the Jews at all, but that’s not quite euphemistic enough a cover-up for the kind of high-level negotiations that are coming up. Heydrich conceives a spectacularly brilliant and quintessentially Nazi solution: let’s destroy all the documents (well, that bit isn’t quite so typical Nazi, but this is a serious enough problem that we can tolerate a little disruption in the paper trail), and murder all the high-level bureaucrats involved (basically everyone who was at the Wannsee Conference, apart from Heydrich and the higher higher-ups, of course). Cue slapstick confusion between Gestapo on the one hand (killing inconvenient bureaucrats) and Kripo on the other (trying to figure out why all these fat old Nazis are getting bumped off).

As one might imagine, things don’t end so well for March. At the end of the book, after a good going-over from Heydrich and his Gestapo buddies, it looks like he won’t be drawing a whole lot of that SS pension. However, he has succeeded in getting information about the Holocaust out (not called that in Alternative Earth Germany, of course), which will soon lead to widespread condemnation of the Nazi regime, a breakdown in negotiations between the US and Germany and a new world order. You think? Well, perhaps not. If experience on Real Earth is any guide, clear and detailed documentation of atrocities usually leads to, well, not much. A stiff editorial in the Guardian. Questions in the Lords. That sort of thing. Real change, not so much.

Fortunately, of course, Heydrich didn’t survive. The evil shit died in 1942. HHhH (Himmlers Hirn heißt Heydrich) is real history, rather than alternative history, and tells the story of how that happened.

Now, a straight “history of the killing of Reinhard Heydrich” might be of interest to historians, but HHhH goes well beyond that.

Historical fiction, alternative history and “real” history have something of a funny relationship. Writing good historical fiction is exceedingly difficult. If you write a conventional novel, you get to choose the plot, the characters, the events you portray, how you render the dialogue, basically everything that goes into your book. To a great extent, you can do the same in alternative history. You choose a jumping off point where your alternative world diverges from ours and off you go. You can use historical characters without worrying too much about historical authenticity – who can say exactly how Heydrich would have responded in any given situation, if it’s a situation that never existed in the real world and thus to which there can be no witnesses? You can just make it up.

For a historical novel, you certainly don’t get to choose the plot, and you’re constrained as far as characters and events go too. You can do what C. J. Sansom does in his Matthew Shardlake novels or Patrick O’Brien in the Aubrey-Maturin novels and invent incidental characters to focus on, using the history more as a backdrop than an integral part of the novel. That gives you a great deal of freedom to write what you will while still exploiting the atmosphere and mores of the period you’re setting the novel in to add a bit of glamour. Or, you can do what Hilary Mantel did in her Wolfe Hall novels. This involves thousands of index cards recording the most trivial of recorded events in the lives of the most minor of characters of the period and an effort to weave all of those strands into a sort of “maximally historical” narrative. The fact that the Wolfe Hall novels are successful as novels within these constraints is a testament to Mantel’s genius.

And then there’s real history, where you may have to forego some of the requirements of good writing in order to present the known facts in sufficient detail to support whatever thesis you want to present. At this level, you care about details. Counting the buttons on uniforms in archive photographs may tell you which factory produced those uniforms and when, giving insight into logistics and supply. Trawling through thousands of pages of agricultural production records may help you to identify a hidden famine. Of course, you then take away the scaffolding, hiding the details. The process of tracking down all of those facts, the worries that you may have missed something, the controversies, the lacunae, the not-quite-justified leaps of logic, all of those are swept under the carpet.

HHhH doesn’t do this sweeping away. The scaffolding is there in plain sight in the form of short “writing of” chapters interspersed between the conventional narrative chapters describing the assassination attempt. It works really well, mostly because of the worried, slightly paranoid tone that Binet has. He worries about the number of buttons. He worries about what person X said to person Y on occasion Z. He doesn’t want to say that a certain German officer on the Eastern Front was driving an Opel if he doesn’t have incontrovertible evidence that said officer really was driving an Opel.

Now, handled badly, this kind of thing could descend into a sort of annoying historical pettifoggery that would serve only to distract from the real story. Here though, it’s handled well, and serves more to emphasise the malleability of history and the importance of rigour. On the face of it, whether it was an Opel or a Volkswagen doesn’t matter. But where do you stop? If you write, definitively, that it was an Opel, then some future historian researching German industrial production during WW2 may take that as evidence that Opel were producing a given vehicle at a given time from a given factory. Which may be untrue. And which may lead this hypothetical historian to draw incorrect conclusions. So that would be bad history.

What makes all this the exact opposite of annoying, indeed incredibly engaging, is that Biney truly appears to care about the people whose stories he has taken it upon himself to tell. Getting things wrong due to inattention or lack of diligence would be an insult to the people of Lidice, and to the memories of the Czech and Slovak soldiers (and many others) who gave their lives in the effort to remove Heydrich’s boot from their peoples’ shoulders.

Binet was writing HHhH around the time when Jonathan Littell was being lauded to the skies for Les Bienveillantes (in English, The Kindly Ones). Dealing as it does with the Holocaust, the war in Russia and other matters WW2-ish, this was of great interest to Binet. There are some very funny, although eventually unpublished (you can find them on the web with a little Googling) sections of HHhH where Binet frets about the likely reception of his book after the success of Littell’s book, and he has a good little bitching session about Littell’s slapdash approach to historical accurary. Slapdash in comparison to Binet, that is, so probably well within the bounds of factual accuracy any sane person would expect from a historical novel. The Opel example comes from this stuff – Littell talks about some officer’s car and Binet starts to worry that he’s missed a source that describes the car that this (real) officer was driving at the time. And if he’s missed that, what ever else might he have missed?

Now, I thought that The Kindly Ones was a brilliant novel, and the voice of the narrator was really strong and interesting (the phrase “unreliable narrator” doesn’t really do justice to secretly homosexual ex-SS officers with direct responsibility for the killing of Jews during the Holocaust living in hiding in France…), but HHhH really is something sui generis. Binet pulls off a great trick, in making us care about counting buttons (metaphorically) by tying that quest for historical fidelity to a respect for and love of the people whose stories he tells.

## Using Images in Haddock Documentation on Hackage

February 21, 2014

A useful but little used feature of Haddock is the ability to include inline images in Haddock pages. Here are a few examples. You can use images for diagrams or for inserting mathematics into your documentation in a readable way. In order to use an image, you just put <<path-to-image>> in a Haddock comment. The image can be of any format that browsers support: PNG, JPEG, SVG, whatever.

Until the most recent (1.18) version of Cabal, using this feature was kind of a pain because there was no way to ensure that your image files ended up in a reasonable place in the documentation tree. That meant either hosting the images somewhere other than Hackage, or inserting the image data directly into your Haddock comments, which is painful in the extreme.

Now though, Cabal has a new extra-doc-files option you can specify in the Cabal file for your package, which lists files that should be copied from your source tree into the documentation tarball. That means you can get your image files into just the right place with no pain at all.

As an example, in the arb-fft package I use a couple of SVG images to render some equations in the documentation. In the source tree there’s a directory called doc-formulae that contains a couple of bits of LaTeX and a Makefile that processes them and uses dvisvgm to turn the resulting DVI output into SVG files. The Cabal file contains a line saying extra-doc-files: doc-formulae/*.svg which ensures that the SVG images end up in the Haddock documentation tarball. I can then refer to them in Haddock comments as something like <<doc-formulae/fft-formula.svg>>, which is really convenient.

This should now work for any new uploads to Hackage after a fix last night by Duncan Coutts.

## Functional Differential Geometry

February 17, 2014

by Gerald Jay Sussman & Jack Wisdom (with Will Farr)

This book follows very much in the mould of Sussman & Wisdom’s Structure and Interpretation of Classical Mechanics in that it’s an attempt to take a body of mathematics (differential geometry here, classical mechanics in SICM) and to “computationalise” it, i.e. to use computer programs to make the mathematical structures involved manifest in a way that a more traditional approach to these subjects doesn’t.

This computational viewpoint is a very powerful one to adopt, for a number of reasons.

## The Storage Capacity of Neural Systems

February 8, 2014

I recently read The Quest for Artificial Intelligence by Nils Nilsson. Interrupting a fairly linear history of the AI field is an interlude on some more philosophical questions about artificial intelligence, minds and thought. Searle’s Chinese Room, things like that.

I got to thinking about some of these questions on my daily peregrinations with Winnie, and I started wondering about the scales that are involved in most discussions of minds and AI. Searle talks about an individual person in his Chinese Room, which makes the idea of some sort of disembodied intelligence actually understanding the Chinese sentences it’s responding to seem pretty absurd. But is that in any sense a realistic representation of a brain or a human mind?

In keeping with my upbringing as a baby physicist, I’m going to take a very reductionist approach to this question. I want to get some idea of exactly what the information storage capacity of a human brain is. I’m deliberately not going to think about information processing speeds (because that would involve too much thinking about switching rates, communication between different parts of brains, and other biological things about which I know very little). I’ll treat this in the spirit of a Fermi problem, which means I’ll be horrifyingly slapdash with anything other than powers of ten. There will be big numbers.

February 2, 2014

After releasing the arb-fft package last week, a couple of issues were pointed out by Daniel Díaz, Pedro Magalhães and Carter Schonwald. I’ve made another couple of releases to fix these things. They were mostly little Haskell things that I didn’t know about, so I’m going to quickly describe them here. (I’m sure these are things that most people know about already…)

## Some Thoughts on the MOOCopalypse

January 31, 2014

I was obscurely disappointed to discover that I wasn’t the first person to come up with “MOOCopalypse”. I think I could claim first dibs on “MOOCaclysm” if I wanted, since there are no Google hits for it, but it’s not quite as good.

Unless you’ve been living under a rock for the last couple of years, you’ve probably heard of MOOCs (Massive Open Online Courses). These are courses offered online by universities or other suppliers, intended to bring high-quality higher education resources to a wide and varied audience. I’ve done a few of these courses: Peter Norvig and Sebastian Thrun’s Stanford AI course, one on fantasy and science fiction, and more recently Jeff Ullman’s automata theory course and another on general game playing (about which I’ll write something once I’ve had some time to work on the software I developed as part of the course).

There has been a bit of a hoo-ha about MOOCs recently, mostly because Udacity, the company founded by Sebastian Thrun (High Priest and foremost evangelist of MOOCdom, part-time Googler, ex-Stanford) has stepped back from its original lofty goals to concentrate on vocational and professional training. This is obviously a bit of a comedown: from Opening the Academy to the World to “Introduction to Salesforce App Development” (the first course listed on Udacity’s site as I write this).

There’s been a fair amount of Schadenfreude floating around about this retreat/re-positioning. When he left Stanford to found Udacity, there was a lot of high-minded talk from Thrun about the possibilities of MOOCs that frankly sounded a bit snake-oily. He appeared very well-intentioned, and the mission was undoubtedly worthy, but it was hard to believe that it was really going to work. Recent events, in particular a poorly executed attempt to replace some “normal” courses at San Jose State University with MOOCs, seem to show that this was all another unfortunate case of “all talk, no trousers” from the worthies of Silicon Valley.

So what really went wrong? Tressie McMillan Cottom has done some great work analysing both the SJSU fiasco and the wider question of MOOCs – start here and read everything she has to say. Her analysis is informed by a lot of sociological thinking that I don’t know anything about, and she’s spent a lot more time thinking about these issues than I have, but there are a few obvious things to say about the whole MOOC project that I’d like to cover. My teaching experience is all in physics and mathematics, so is probably of more or less no relevance to any other field.

First, let me say that personally, I really like MOOCs. I’ve learnt quite a bit of interesting material using them. But I’ve had the benefit of ten years of post-secondary education, plus a number of years working as a research scientist. I don’t think I’m really in the target audience for the kind of worldwide dissemination of academic knowledge that MOOCs have been sold as providing. After spending so long “learning to learn”, I can teach myself subjects I’m interested in from books or Wikipedia or even original research literature. But I can only do that because I’ve had the good fortune to be able to spend so much time developing these independent study skills. I also have enough background knowledge that I can usually deal with infelicities in presentation and can usually figure out what it’s safe to ignore, what I need to do further reading about, and so on. These are all skills that it takes time to develop.

The first and most obvious problem with MOOCs involves material barriers to access. These come in at least two forms. If you don’t have easy access to a computer and a fast internet connection, most online courses aren’t going to be usable. If you need to use a computer in a public library, it’s not going to be practical to watch 3+ hours of video lectures per week and to do homework exercises as well. The second form of material barrier is that of time. Even if you have access to a computer and internet connection, if you have a full-time job (or jobs) and/or caring responsibilities, simply making the time in your schedule to follow a MOOC may be impossible. I’m an independent software contractor, which means that I can convince myself that time spent on relevant (or semi-relevant!) MOOCs can be counted as continuing professional education so it’s (sort of) work time, but if I had a “real” full-time job, I would find it hard to do these things. If I had a couple of low-paid jobs plus child-care responsibilities, it would be absolutely impossible. Any approach to education that doesn’t address these accessibility issues is never going to be universal, and is never going to meet the originally stated goals of the MOOC movement.

I’ve almost certainly done a lot less teaching than Thrun, but I have done a bit, and I’ve taught in a range of different settings. I’ve done university lecturing in Canada (a second year differential equations course that was fun for everyone involved, and an instance of Calculus I that was as painful for my students as for me); I’ve done tutorial teaching in Oxford (teaching mathematical physics and quantum mechanics to undergraduates on a one-to-one or one-instructor-to-two-students basis); and I spent a few months teaching maths in a classroom setting in a further education college in Wales (this was a course primarily intended for mature students who wanted to return to university to study biology after a period in work, and was intended to equip the students with the mathematical knowledge they would need to start an undergraduate degree course).

What did I learn from those teaching experiences? The first thing is that lecturing is relatively ineffective as a means of transferring knowledge from instructure to students. As an instructor confronted with a room of 80-90 students, all you can do is present the material in a fairly conventional way. There’s a constant conflict between presenting more examples (students in beginning undergraduate mathematics courses love examples) and going too fast. There’s also the need to be entertaining enough to keep students engaged with the material. You can try as hard as you like to make the material “relevant” (whatever that means), but the sad truth is that, for Calculus I, for example, you do need to talk about ten different kinds of subsitutions for different kinds of integrals, and that is unavoidably tedious. And of what benefit to students are lectures really? A good lecturer can give a different viewpoint on the material compared to the textbook, but that presupposes that the students have the time and inclination to read the textbook to give themselves a baseline for that different viewpoint.

I would have liked to try a “flipped classroom” approach, but that wasn’t really feasible in the setting we had.

So lecturing isn’t particularly effective. This seems to correspond with what other people find (whatever you think of the “Learning Pyramid”, putting lecturing at the bottom in terms of effectiveness seems about right). What about the most effective teaching I’ve done? Without a doubt, this was the small group tutorial teaching I did in Oxford. In this setting, you have one or two students who’ve done some work (in this case, it was undergraduate physics work of various kinds), handed it in for you to mark, and you meet for an hour or so to talk about the topic at hand. I was on the receiving end of this instructional style as an undergraduate myself, and it can be quite intimidating. There’s nowhere to hide if you don’t know something. Given a well-intentioned instructor though, it can be amazingly effective. If a student doesn’t understand a concept the way that it’s been presented in lectures or in their textbooks, the instructor can explain it some other way, using some other example that might be more accessible.

It’s worth saying that this instructional setting can also be stressful for the instructor: there’s absolutely no possibility of bluffing – you need to know the material backwards and forwards and moreover, you need to know seven different ways to look at anything so that you choose another angle to explain things if the first few approaches don’t take. The peak of my understanding of quantum mechanics was definitely the two or three months when I was teaching it to undergraduates at St. Catherine’s in Oxford. It’s a really great feeling to see understanding dawn in someone as you explain really quite difficult concepts to them, adapting your explanations to your model of their understanding. It feels like real TEACHING! That’s an experience that’s sadly not available in most other settings.

When I was teaching in Canada, I really tried to push the benefits of office hours to students. There were a few who turned up religiously at every session, and I really think they got some benefit from it. Office hours aren’t quite as good as tutorial-style teaching, mostly because you have many more students to deal with, so it’s hard to remember what issues individual students have trouble with, but you can use time at the blackboard with students to try to develop some problem-solving skills and “learning to learn” and “learning to think” skills. When a student asks “How do I solve this problem?”, you can hand them a piece of chalk, sit yourself down and say, “Well, how would you start?”. From there, you can guide them as little or as much as they need, you can suggest things, you can ask them why they take a particular approach to a problem (and so diagnose misconceptions that they might have) and, if you do it right (and you’re lucky), you can give the student the feeling of accomplishment of solving a problem that they didn’t know how to solve.

So, what does this all have to do with MOOCs? Most MOOCs have video lectures, on-line homeworks composed of multiple choice or short answer questions (or, in some cases, peer marked written answers), and discussion forums (which may or may not be monitored by the instructors). There may be a recommended textbook (which may be another financial barrier to entry for some students) or lecture notes available as well. There’s no space in this setup for small group teaching or interactive feedback and explanation. You watch the lectures, try the exercises, and look for help on the discussion forums. If you’re lucky, your fellow students or the instructors will answer your questions. I’d contend that in that setting, there’s really relatively little teaching going on.

What’s more, the resources offered by MOOCs are wholely inadequate for the claimed target audience. Video lectures, self study with exercises and notes and some non-realtime online discussion might be enough for those of us fortunate enough to have been primed for it by a lifetime of “in person” education, but for students who haven’t benefited from that, these resources just aren’t enough. There are a whole range of “soft” skills that are needed: you need to know how to listen to lectures, to deal with errors by the lecturer, to know when obscure points can be disregarded until later, to know how to look for and how to understand further information if the lectures aren’t enough or if you have lacunae in your knowledge that you need to fill to understand the lectures. Those independent study skills aren’t things you can teach using the current MOOC model. Once you have those skills, MOOCs can be pretty useful.

Which all makes me wonder just what the hell was Thrun thinking? The promise of MOOCs providing high quality higher education to everyone seems like a pipe dream. The students most in need of access to MOOCs are the students least likely to have the material resources to access them. The students who could benefit most from MOOCs are the students least likely to have the pre-requisite skills needed to benefit from the limited range of teaching possibilities currently offered by MOOCs. It just seems kind of silly to make these great claims for an unproved teaching platform, one that on the face of it simply doesn’t have a hope of meeting the needs of the people who need it most.

I don’t for a moment doubt Thrun’s good intentions, but good intentions don’t get you very far. The SJSU situation, where all of these difficulties of access came to a head because of an attempt to replace some conventional courses with MOOCs (with students paying for them), seems like the result of gross hubris on the part of Thrun’s company, Udacity. At least Thrun now seems to recognise this. He said, “We were on the front pages of newspapers and magazines, and at the same time, I was realizing, we don’t educate people as others wished, or as I wished. We have a lousy product.” It’s a shame he and Udacity didn’t recognise that before they exposed a whole bunch of SJSU students to their lousy product. These things have real-life consequences for real students and you shouldn’t use students as experimental subjects to help improve your product, which is what seems to be happening.

One final question that I’ve been mulling over a little: Can MOOCs really help with broadening access to education? Setting aside the material barriers to access, which are important but not something that MOOCs are any kind of solution for, what about the other barriers? I could parade my ignorance of educational psychology and pull some ideas out of my hat, but it turns out that there are people who do actual real research on these questions! If you’re at all interested in this question, you should head to another of Tressie McMillan Cottom’s articles here and read some of the links. There’s a lot of good stuff there. It seems that there’s some hope. It would be depressing if technology couldn’t help. But it’s going to require something a little more subtle than gung-ho evangelism and “disruption”.

January 26, 2014

After thirteen blog articles and about two-and-a-half weeks’ equivalent work time spread out over three months, I think it’s time to stop with the FFT stuff for a while. I told myself when I started all this that if I could get within a factor of 20 of the performance of FFTW, I would be “quite obscenely pleased with myself”. For most $N$, the performance of our FFT code is within a factor of 5 or so of the performance of the vector-fftw package. For a pure Haskell implementation with some but not lots of optimisation work, that’s not too shabby.

But now, my notes are up to nearly 90 pages and, while there are definitely things left to do, I’d like to move on for a bit and do some other data analysis tasks. This post is just to give a quick round-up of what we’ve done, and to lay out the things remaining to be done (some of which I plan to do at some point and some others that I’d be very happy for other people to do!). I’ve released the latest version of the code to Hackage. The package is called arb-fft. There is also an index of all the posts in this series.

## Lessons learned

The lessons I’ve taken from this exercise are all things that more experienced Haskellers have known for some time, but it’s interesting to see how things play out in a realistic example:

• Haskell is a great language for writing clear code. The earlier versions of the FFT code were, in many cases, more or less direct transcriptions of my mental models of the way these algorithms should work. This isn’t such a big deal for a relatively simple example like the FFT, but when you’re dealing with more complicated problems, that clarity of expression is really valuable.

• QuickCheck is a wonderful testing tool and Criterion is a wonderful benchmarking tool. But everyone knows that!

• Clear code isn’t necessarily fast code, but when it comes time to optimise, Haskell allows you to isolate any “nastiness” you introduce. By this, I mean for example, that where I’ve fallen back on using stateful calculations and mutable vectors, the ST monad allows this stateful computation to be “locked away” so that no state leaks out into the surrounding pure code. This isolation makes reasoning about code much easier than in a comparable piece of procedural code — you know, because the type system guarantees it, that no references can leak from your stateful code: compare the situation in C++, where you need to be very careful to manage aliasing yourself because the language definition provides only limited scope to the compiler to help you.

• Having a language that allows you to switch seamlessly between symbolic and numeric computation is nice. Quite a lot of the planning code is classic combinatorial code (multiset permutations and compositions of integers), and this can be expressed very clearly and easily in a functional language. The same goes for the “toy algebra system” code in the earlier articles where we used a limited set of algebraic structures to help understand the Cooley-Tukey decomposition of the Fourier matrix. And although these things are quick and easy to write, they can then be carried forward into more efficient representations.

Another lesson, not confined to Haskell or functional programming: good performance comes first from good algorithms, not from tweaking your code. There’s no point in optimising an $O(N^2)$ algorithm if it’s known that there’s an $O(N \log N)$ algorithm out there. If you do that, you’re just generating waste heat in your CPU.

A related point is that good performance quite often requires you to look beyond “textbook” algorithms. I’ve yet to find a textbook reference to Rader’s algorithm or any of the other prime-length FFT algorithms. I don’t know what the more recent editions of Numerical Recipes say, but my copy of the second edition, in reference to using the fast Fourier transform for anything other than power-of-two lengths, basically just says “Don’t do that.” If you want fast, you need to look around a bit, read some literature, and implement good algorithms. Code tweaking comes later!

These are rated from 1-5 for Difficulty, Effort, Value and Fun:

#### Haskell genfft for “bottomlets” (D4/E4/V5/F4)

Writing a Haskell version of FFTW’s genfft utility for producing the optimised straight-line “codelets” used for the base transforms would be interesting, and it would remove all the embarrassing things in Numeric.FFT.Special, which are copied more or less wholesale from FFTW1. And once there’s a Haskell genfft, it should be possible to generate truly optimal plans at compile-time by running the genfft code as Template Haskell.

#### Haskell genfft for “twiddlets” (D4/E3/V5/F3)

I’ve only written specialised straight-line code for what I’ve been calling “bottomlets”, i.e. the base transforms used at the bottom of the recursive decomposition of the input vector. The intermediate Danielson-Lanczos steps all use generic unspecialised code. It ought to be possible to produce specialised versions of the Danielson-Lanczos steps for specific sub-vector lengths, which might give good speed-ups for some input lengths. Once there’s a genfft for “bottomlets”, it ought to be relatively straightforward to extend that to “twiddlets” as well.

#### Truly optimal planning (D3/E3/V3/F3)

Having a Haskell genfft for both “bottomlets” and “twiddlets” opens up the possibility of doing what I’d call truly optimal planning, in the sense that, as well as testing different plans as is done now, it would also be possible to construct specialised straight-line codelets for custom input sizes. Need a custom FFT of length 437? That’s $19 \times 23$, so we could construct (at compile-time) bottomlets and twiddlets of sizes 19 and 23 and benchmark to see which ordering is quicker, avoiding the use of Rader’s algorithm for most prime factors. Similarly, for more composite input lengths, we could try various combinations of custom bottomlet and twiddlet sizes, instead of relying only on a pre-selected list of sizes for specialisation.

#### Low-level optimisation (D4/E2/V5/F2)

I’ve really done relatively little optimisation on the arb-fft code: using unboxed mutable vectors is about as far as it goes. The empirical plan generation also helps quite a bit, but there is definitely more that could be done in terms of lower-level optimisation. I’ve not even looked at strictness, and haven’t even bothered with more profiling after optimisation, so there are probably a few things that could be done to squeeze some more performance out. Low-level optimisation of Haskell isn’t really something that I know a whole lot about, so I’ve left it to one side for now.

(There’s another optimisation task that really ought to be done too: for large problem sizes, planning can take a long time, particularly for problem sizes with large prime factors. I’ve added some code to filter out ridiculous plans where possible, i.e. ones that are guaranteed to be slow because of huge Danielson-Lanczos steps, but it’s not always possible to avoid looking at all of these, and they’re unavoidably slow. Better heuristics for planning and switching to a simplified timing approach instead of using Criterion would speed this up a lot.)

#### Real-to-real, etc. FFTs (D3/E4/V4/F0)

So far, I’ve only implemented complex-to-complex FFTs. For a production library, I’d also need to implement real-to-complex transforms along with the various flavours of real-to-real transforms (cosine, sine and Hartley). None of these things are terribly difficult to do, but they are really pretty tedious exercises in data management. No fun at all!

#### Parallelisation (D3/E2/V3/F2)

There are a lot of cheap opportunities for parallelisation in the divide-and-conquer approach of the Cooley-Tukey algorithm, and this is a place where it might be quite easy to get performance that exceeds that of FFTW (by cheating, of course — getting single processor performance that’s competitive with FFTW would take some doing!).

Of these tasks, I’ll probably have a go at the Haskell genfft stuff at some point and I might have a bit of a poke at some more optimisation efforts. I don’t think I’ll do the real-to-real FFTs unless someone finds themselves in desperate need of a pure Haskell FFT library with real-to-real transforms and offers to pay me for it! Parallelisation probably isn’t worth the effort unless single-processor performance can be made more competitive with FFTW.

Now though, it’s time for a break from FFT endeavours. Although today, I just learnt about this very interesting looking paper…

1. Extra embarrassing because I made a mistake in translating the biggest and most complicated of these things to Haskell and it took ages to track down the (one-character) typo. This was in a single function with around 800 local variables, all called things like t1Y or t2t or t2T. A Haskell genfft would be nice.

## Haskell FFT 13: Optimisation Part 3

January 26, 2014

In this article, we’ll finish up with optimisation, tidying up a couple of little things and look at some “final” benchmarking results. The code described here is the pre-release-4 version in the GitHub repository. I’m not going to show any code snippets in this section, for two reasons: first, if you’ve been following along, you should be familiar enough with how things work to find the sections of interest; and second, some of the optimisation means that the code is getting a little more unwieldy, and it’s getting harder to display what’s going on in small snippets. Look in the GitHub repository if you want to see the details.

## Haskell FFT 12: Optimisation Part 2

January 20, 2014

In the last article, we did some basic optimisation of our FFT code. In this article, we’re going to look at ways of reordering the recursive Cooley-Tukey FFT decomposition to make things more efficient. In order to make this work well, we’re going to need more straight line transform “codelets”. We’ll start by looking at our $N=256$ example in detail, then we’ll develop a general approach.

January 17, 2014

I haven’t done one of these for a long time, but I have a big old pile of interesting things I’ve read recently. I might make this a weekly thing – Harry Connolly does a Friday Randomness roundup and there’s always something interesting or weird there. I don’t think I can quite attain his level of randomness, so I’ll have to settle for “worth reading” as a criterion.

1. Dogs Are People, Too: Functional MRI for dogs! And the dogs get to choose whether or not to play, which is really great.

2. The Mother of All Squid Builds a Library: Lovely short story.

3. What the World Eats: Twelve families from very different parts of the world photographed with a week’s worth of groceries. I keep going back to look at them.

4. Meat Atlas: Friends of the Earth and the Böll Foundation produced this report about global meat production, the economics, the environmental impacts, and so on.

5. Hofstadter interview in The Atlantic: After reading this long interview with Douglas Hoftstadter, I was inspired to go back and re-read Gödel, Escher, Bach and I now have a pile of other Hofstadter things to read. He’s a very interesting guy.

6. The Guardian: Ask a Grown-Up: This is something The Guardian has been doing for a while, but I’ve only just seen it. It’s a great idea. Children can write in with questions about anything at all, and they get answers from experts in whatever field they’re asking about (or sometimes slightly random celebrities, but mostly the answerers know what they’re on about). Best one I’ve seen so far: Douglas Hofstadter (yes, him again!) answering a six-year-old’s question “Does my cat Oscar know he’s a cat?”!

7. The UK National Archives: I’ve not had much of a chance to rummage around in here yet, but it looks like an amazing resource where I could probably waste infinite amounts of time.

8. How Doctors Die: End-of-life care choices and the differences between what doctors recommend to their patients and what they do themselves. Pretty grim, but needs to be talked about.

9. Early Onset Dementia: More grimness, also needing to be talked about. Early onset dementia (not just Alzheimer’s – the article is about something different) is just terrifying. Imagine losing the ability to recognise all of your loved ones, to know who you are or where you are, and this not to be something that happens towards the end of your life, but at a time when you might conceivably have to live like that for 40 or 50 years. Now imagine that it’s not you that has this problem, but your partner…

10. Implementing a JIT Compiled Language with Haskell and LLVM: On a completely different topic, this is a Haskell rendering of a long LLVM tutorial that looks really useful. I’ve only read part of it so far, but it’s good.


text:
How To Solve An Initial Value Problem

There is one differential equation that everybody probably knows, that is Newton’s Second Law of Motion.If an object of mass $$m$$ is moving with acceleration $$a$$ and being acted on with force $$F$$ then Newton’s Second Law tells us.As you will see most of the solution techniques for second order differential equations can be easily (and naturally) extended to higher order differential equations and we’ll discuss that idea later on.

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A linear differential equation is any differential equation that can be written in the following form.

$\begin \left( t \right)\left( t \right) \left( t \right)\left( t \right) \cdots \left( t \right)y'\left( t \right) \left( t \right)y\left( t \right) = g\left( t \right) \label\end$ The important thing to note about linear differential equations is that there are no products of the function, $$y\left( t \right)$$, and its derivatives and neither the function or its derivatives occur to any power other than the first power.

The first definition that we should cover should be that of differential equation.

A differential equation is any equation which contains derivatives, either ordinary derivatives or partial derivatives.

$\begin F = ma \label \end$ To see that this is in fact a differential equation we need to rewrite it a little.

List Essay Linking Words - How To Solve An Initial Value Problem

First, remember that we can rewrite the acceleration, $$a$$, in one of two ways.

$\beginay'' by' cy = g\left( t \right)\label\end$ $\begin\sin \left( y \right)\frac = \left( \right)\frac \label\end$ $\begin 10y''' - 4y' 2y = \cos \left( t \right) \label\end$ $\begin\frac = \frac\label\end$ $\begin = \label\end$ $\begin\frac = 1 \frac \label\end$ The order of a differential equation is the largest derivative present in the differential equation.

In the differential equations listed above $$\eqref$$ is a first order differential equation, $$\eqref$$, $$\eqref$$, $$\eqref$$, $$\eqref$$, and $$\eqref$$ are second order differential equations, $$\eqref$$ is a third order differential equation and $$\eqref$$ is a fourth order differential equation.

$\begina = \frac\hspace\hspace\,\,\,\,\,\,a = \frac \label\end$ Where $$v$$ is the velocity of the object and $$u$$ is the position function of the object at any time $$t$$.

We should also remember at this point that the force, $$F$$ may also be a function of time, velocity, and/or position.

Comments How To Solve An Initial Value Problem

• Differential Equations - WolframAlpha Examples

Answers to differential equations problems. Solve ODEs, linear, nonlinear, ordinary and numerical differential equations, Bessel. Specify initial values.…

• Solving an initial value problem with ode45 - UMD MATH

You need to download an m-file. You have to download the m-file dirfield.m. format short g. Initial value problem. We consider the initial value problem. y'=y-t.…

• Initial value problems for system of differential-algebraic.

Sep 6, 2018. In this paper, we discuss a Maple package, deaSolve, of the symbolic algorithm for solving an initial value problem for the system of linear.…

• Example Solving a First-Order ODE Initial Value Problem

The output, fyk, is a function of a function, so you must specify the value of the parameter for which you would like the solution function. 4. Assign the result to an.…

• Initial value problems - Scholarpedia

Feb 25, 2007. that are to hold on a finite interval t_0, t_f\. An initial value problem specifies the solution of interest by an initial condition yt_0 = A\.…

• Solving initial value problem of ordinary differential equations.

In this paper, on the basis of Monte Carlo method, a new way to solve initial value problem of ordinary differential equations was presented. The authors have.…

• Initial Value Problem - Teaching Concepts with Maple.

The initial value problem for a driven damped oscillator is solved with the ODE Analyzer Assistant, which provides the analytic solution and its graph. A stepwise.…


text:
## Stream: new members

### Topic: Magic dot in place of proof?

#### Mark Dickinson (Nov 05 2018 at 21:50):

Looking through the standard library, I see this in basic.lean:

lemma not_succ_le_zero : ∀ (n : ℕ), succ n ≤ 0 → false
.

What's the magic dot on the second line? Is this syntactic sugar for a particular tactic or set of tactics?

#### Scott Morrison (Nov 05 2018 at 21:51):

no, it just says "this is the end of the definition", and causes lean to realise that there are no cases to prove!

magic. :-)

#### Mark Dickinson (Nov 05 2018 at 21:58):

Ah, thank you! I did try searching the archives, but it's not easy to figure out how to search for a single period.

#### Patrick Massot (Nov 05 2018 at 21:59):

I searched for the name of this lemma, because I was pretty sure it was already discussed

#### Mark Dickinson (Nov 05 2018 at 21:59):

Yep, I should have thought to use not_succ_le_zero as a search term ..

#### Patrick Massot (Nov 05 2018 at 21:59):

And there are very few opportunities to use this trick

No problem

#### Bryan Gin-ge Chen (Nov 05 2018 at 23:37):

I've sometimes found it useful when rewriting a proof, e.g. instead of temporarily commenting out the rest of a proof, I just insert a ..

#### Kevin Buzzard (Nov 05 2018 at 23:40):

I usually insert a #exit

Last updated: May 14 2021 at 07:19 UTC


text:
# Are there non-reflexive modules isomorphic to their bi-dual?

Let $M$ be an $R$-module. We say that $M$ is reflexive if the natural map $M\rightarrow M^{**}$ is an isomorphism.

I'd like to know if there exists a module isomorphic to its bi-dual but not reflexive, do you know an example?

-
Yes. In fact there is an example of a Banach space $J$ (the James space) that is isometrically isomorphic to $J^{\ast\ast}$ but for which the image of $J \to J^{\ast\ast}$ has codimension 1. See mathoverflow.net/questions/43986/… –  Faisal Sep 20 '11 at 21:51
For finitely generated modules over a Noetherian ring, no such examples exist. A student of Huneke proved this around 2004, but I don't think he ever published it. (It's possible it was already known at that time, but I never found a reference.) –  Graham Leuschke Sep 21 '11 at 0:23

As a specific example, take E to be a stationary and costationary subset of $\omega_1$, and let $X=\omega_1 + 1\backslash E$, given the subspace topology (of the space $\omega_1 +1$ with the interval topology). Then $C(X, Z)$ (continuous functions to the integers with the discrete topology) is such a group. In fact for this group $C(X,Z)^{**} = \sigma[C(X,Z)] \oplus Z$ where $\sigma$ is the natural map, so that $C(X,Z)$ is not reflexive, but $C(X,Z)\oplus Z \cong C(X,Z)$ because for example $C(X,Z)$ has $Z^\omega$ as a summand.


text:
# Laplace Transform help

GreenLRan
I have two subsystems that I'm trying to transform into the frequency domain, and trying to obtain the transfer function for each.

I have an equation (for system 1)

m$$\ddot{x}$$ -k(x2-x1) - b*$$\dot{x}$$1 = f

Since this is for system 1, when I take the laplace transform of it, do i have x2 as a function of s (x2(s)) or do i treat x2 as a constant?

Thanks

m$$\ddot{x}$$ -k(x2-x1) - b*$$\dot{x}$$1 = f


text:
# Is the the complement of the Halting problem NP-hard [duplicate]

Let us define the language (the complement of the Halting Problem):

$$LOOPING = \left\{ \langle M,x \rangle | M \ doesn't \ halt \ on \ x\right\}$$

The question is if $LOOPING$ is NP-Hard. I'm pretty clueless about the answer. I know the the Halting problem is NP-Hard, but I can't seem to find a reduction from an NP-Hard problem to $LOOPING$.

## marked as duplicate by quicksort, David Richerby, Evil, Community♦Dec 24 '17 at 9:21

• I think you mean "$M$ doesn't halt on $x$". – Yuval Filmus Dec 23 '17 at 13:49