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Bill the Lizard: Six Visual Proofs

Bill the Lizard: Six Visual Proofs
1 + 2 + 3 + ... + n = n * (n+1) / 2 1 + 3 + 5 + ... + (2n − 1) = n2 Related posts:Math visualization: (x + 1)2 Further reading:Proof without Words: Exercises in Visual ThinkingQ.E.D.: Beauty in Mathematical Proof Also, my thanks go out to fellow redditor cnk for improvements made to the 1/3 + 1/9 + 1/27 + 1/81 + ... = 1/2 graphic.

The Geomblog Learn Japanese Online for Free - it's fun with easy flash quizes copyright - J.P.Gourret - 2000 - Maillage multirésolution de sur (Chapitre I-Part 1)....Chapitre I - Introduction (Chapitre II-Part 1)...Chapitre II - Programmation (Chapitre II-Part 2)..........II-1- La bibliothèque graphique et son interface de programmation (Chapitre II-Part 3)....................II-1-1- Ossature classique d'un programme (Chapitre II-Part 4)....................II-1-2- Opérations successives réalisées sur les données (Chapitre II-Part 5)....................II-1-3- Commandes ..........................................................II-1-3-a- Mode immédiat ..........................................................II-1-3-b- Liste d'affichage (Chapitre II-Part 6)....................II-1-4- Exemple de programme (Chapitre II-Part 7)..........II-2- Fonctions pour modéliser l.éclairement et les sources de lumière (Chapitre II-Part 8)....................II-2-1- Réflexion diffuse due à la lumière ambiante (Chapitre II-Part 9)....................II-2-2- Réflexion diffuse due à une source localisée (Chapitre II-Part 14).........II-4- Calcul matriciel

Auslan - Signbank Blind Deconvolution To this point, we have studied restoration techniques assuming that we knew the blurring function h . Actually, we have also assumed that we knew the image spectral density Suu and Spectral noise Snn as well. This section will focus on some techniques for estimating h based on our degraded image. Two restoration filters will be the basis for our procedures. We use as our degradation model the standard idea that our input image is blurred through convolution with a low pass LSI filter (h) and then Gaussian Noise is added to the result. where H is the Fourier Transform of h, and Suu and Snn are defined as above. The above images were generated using wien.m. Next we will examine the effectiveness of Power Spectrum Equalization. Note the similarity to Wiener Filtering, but we only use the magnitude of H. The above images were generated using pse.m. Next we will perform the same restoration using estimated spectral noise. Our second approach came from the Lim text.

Game Theory First published Sat Jan 25, 1997; substantive revision Wed May 5, 2010 Game theory is the study of the ways in which strategic interactions among economic agents produce outcomes with respect to the preferences (or utilities) of those agents, where the outcomes in question might have been intended by none of the agents. The meaning of this statement will not be clear to the non-expert until each of the italicized words and phrases has been explained and featured in some examples. Doing this will be the main business of this article. First, however, we provide some historical and philosophical context in order to motivate the reader for the technical work ahead. 1. The mathematical theory of games was invented by John von Neumann and Oskar Morgenstern (1944). Despite the fact that game theory has been rendered mathematically and logically systematic only since 1944, game-theoretic insights can be found among commentators going back to ancient times. 2. 2.1 Utility

Inverse Filtering If we know of or can create a good model of the blurring function that corrupted an image, the quickest and easiest way to restore that is by inverse filtering. Unfortunately, since the inverse filter is a form of high pass filer, inverse filtering responds very badly to any noise that is present in the image because noise tends to be high frequency. In this section, we explore two methods of inverse filtering - a thresholding method and an iterative method. Method 1: Thresholding Theory We can model a blurred image by where f is the original image, b is some kind of a low pass filter and g is our blurred image. But how do we find h? In the ideal case, we would just invert all the elements of B to get a high pass filter. So the higher we set , the closer H is to the full inverse filter. Implementation and Results Since Matlab does not deal well with infinity, we had to threshold B before we took the inverse. where n is essentially and is set arbitrarily close to zero for noiseless cases. where

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