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Discrete Math Lecture Notes

Discrete Math Lecture Notes
Related:  Mathematicsmath and physics

K-MODDL > Tutorials > Reuleaux Triangle If an enormously heavy object has to be moved from one spot to another, it may not be practical to move it on wheels. Instead the object is placed on a flat platform that in turn rests on cylindrical rollers (Figure 1). As the platform is pushed forward, the rollers left behind are picked up and put down in front. An object moved this way over a flat horizontal surface does not bob up and down as it rolls along. Is a circle the only curve with constant width? How to construct a Reuleaux triangle To construct a Reuleaux triangle begin with an equilateral triangle of side s, and then replace each side by a circular arc with the other two original sides as radii (Figure 4). The corners of a Reuleaux triangle are the sharpest possible on a curve with constant width. Other symmetrical curves with constant width result if you start with a regular pentagon (or any regular polygon with an odd number of sides) and follow similar procedures. Figure 1: Platform resting on cylindrical roller

Calculating the Distance to the Horizon For Any Game Home Up Site Map Assumptions | Method 1 | Method 2 Method 1 | Method 2 This is all based on the assumption that the horizon is the point on the world's surface at which the line of sight of the viewer, whatever their height, becomes parallel (tangential) to the surface of the world, and meets the surface of the world (so that the viewer cannot see any further than it). Note that I do not mention units in any of the equations on this page. Assumptions | Method 2 For a right-angled triangle: Where: R is the longest side (the hypotenuse), x and y are the other two sides. Using this equation on the triangle in the figure above, the longest side is the radius of the planet plus the height of the observer (r + h) , and the other two sides are d and r . Or, re-arranged: Or: So the total distance to the horizon is given by: This equation will work for any size world, and any height of observer. Assumptions | Method 1 Back to My Roleplaying Page .

Pauls Online Math Notes Coaching with the Johari Window We live in a society where we all present a persona, a mask to the world. This protects us from breaking some of the social and legal rules that structure our society. I can also present us with some limitations in behaviours that can restrict both our performance and our ability to succeed. The Johari window is a useful tool to apply in the coaching domain. Imaginatively named after it’s inventors, Joseph Luft and Harry Ingram it can help people understand elements of their behaviour, what they see of themselves, how it could be perceived by others and what isn’t being seen by anyone. Consider the framework in figure 1 below. Figure 1: The Johari Window (Adapted) We all behave in a way that is visible to both ourselves and others. We do, however, hide some of the things we believe (our personality and our desires) from other people and this is the ‘Private Area’. People’s behaviour can often be interpreted in a manner different to what was intended. The final area is the ‘Hidden Area’.

Engineering ToolBox Elementary Concepts in Statistics In this introduction, we will briefly discuss those elementary statistical concepts that provide the necessary foundations for more specialized expertise in any area of statistical data analysis. The selected topics illustrate the basic assumptions of most statistical methods and/or have been demonstrated in research to be necessary components of our general understanding of the "quantitative nature" of reality (Nisbett, et al., 1987). We will focus mostly on the functional aspects of the concepts discussed and the presentation will be very short. Further information on each of the concepts can be found in statistical textbooks. What are Variables? Variables are things that we measure, control, or manipulate in research. Correlational vs. Most empirical research belongs clearly to one of these two general categories. Dependent vs. Independent variables are those that are manipulated whereas dependent variables are only measured or registered. Measurement Scales Magnitude (or "size").

Free Online Course Materials | Courses | MIT OpenCourseWare - Aurora factoids > googol / googolplex Words of wisdom are spoken by children at least as often as by scientists. The name 'googol' was invented by a child (Dr Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested 'googol' he gave a name for a still larger number: 'Googolplex'. A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out. Mathematics and the Imagination