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Entering the World of Web Components - janmr blog
http://blog.janmr.com/2014/07/entering-the-world-of-web-components.html
Entering the World of Web Components. I am very excited about Web Components. It is going to fundamentally change the way we do web development. This post is going to contain miscellaneous information and links related to Web Components. The specification is still being developed. But the overall parts have been decided upon. To quote Introduction to Web Components. Web Components consists of five main parts:. Which define chunks of markup that are inert but can be activated for use later. On how to make...
blog.janmr.com
Bresenham's Line Algorithm - janmr blog
http://blog.janmr.com/2014/04/bresenhams-line-algorithm.html
In 1965 Jack Elton Bresenham published the paper. Algorithm for computer control of a digital plotter. In the IBM Systems Journal, volume 4, number 1. It explained how a line could be approximated on an integer grid. The algorithm is still used today as a rasterization. Technique for rendering lines on video displays or printers. As Bresenham’s paper suggests, however, it was originally devised for a plotter. Capable of moving from one grid point to one of the adjacent eight grid points. Delta x, Delta y).
blog.janmr.com
Time Budgets for the Web - janmr blog
http://blog.janmr.com/2015/04/time-budgets-for-the-web.html
Time Budgets for the Web. I was recently introduced to the so-called RAIL principle in the Udacity MOOC Browser Rendering Optimization. It is an acronym for R. Oad and introduces some time budgets to follow when creating a smooth and responsive web experience. It seems to have been a concept. Coined by the Google Chrome team and is also mentioned in a recent talk. By Google’s Paul Irish. As Paul Irish mentions in his talk, the RAIL principles are built upon research from 1993. When a page has been loaded...
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Basic Multiple-Precision Short Division - janmr blog
http://blog.janmr.com/2012/11/basic-multiple-precision-short-division.html
Basic Multiple-Precision Short Division. Let us consider short division, by which we mean a multiple-digit number. U = (u {m-1} ldots u 1 u 0) b. Divided by a single digit. See, e.g., post on number representation. U {m-1} neq 0. We are interested in a quotient. Q = lfloor u/v rfloor. 0 leq r v. B {m-1} leq u b m. We can deduce that. B {m-2} q b m. Can be represented using. Q = (q {m-1} ldots q 1 q 0) b. Q {m-1} = 0. Q {m-2} neq 0. We now have the following straightforward algorithm:. U {m-1} neq 0.
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MathJax, KaTeX and a lot of Math - janmr blog
http://blog.janmr.com/2015/01/mathjax-katex-and-a-lot-of-math.html
MathJax, KaTeX and a lot of Math. Prior to the current post, this blog contained 45 posts with a total of 2307 math items, where a math item is anything from single-letter variable identifiers to large, multi-line equations. That’s an avarage of 51 math items per post, ranging from a few posts containing no math at all to one. From the beginning I have used MathJax. To display the math (actually, the first few years I used jsMath. The HTML-CSS output is highly browser and client dependent. From basic...
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Deriving the Closed Form for Square Pyramidal Numbers - janmr blog
http://blog.janmr.com/2014/06/deriving-closed-form-for-square-pyramidal-numbers.html
Deriving the Closed Form for Square Pyramidal Numbers. The sum of the first. S n = sum {k=1} n k 2 = textstyle frac{1}{6} n (n 1) (2n 1) quad . Are called the square pyramidal numbers. Many different proofs exist. Seven different proofs can be found in Concrete Mathematics. And even a visual proof. Has been published (via @MathUpdate. One of the simplest proofs uses induction on. This approach assumes that you know (or guess) the correct formula beforehand, though. K 2 = sum {j=1} k (2j-1). A blog about ...
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Multiple-Precision Addition - janmr blog
http://blog.janmr.com/2011/10/multiple-precision-addition.html
This post will cover a basic addition algorithm for multiple-precision non-negative integers. The algorithm is based upon that presented in Section 4.3.1,. Of The Art of Computer Programming. Volume 2, by Donald E. Knuth. The notation and bounds used in this post were presented in a previous post. We consider adding two. U=(u {n-1} ldots u 1 u 0) b. V=(v {n-1} ldots v 1 v 0) b. B {n-1} leq u, v leq b n - 1. 2 b {n-1} leq u v leq 2 b n - 2. Which, when using the fact that. B {n-1} leq u v leq b {n 1} - 1.
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Thales' Theorem Illustrated - janmr blog
http://blog.janmr.com/2014/04/thales-theorem-illustrated.html
Thales’ Theorem: If A, B and C are points on a circle with the segment AC as a diameter, then the angle at B is a right angle (try dragging the point at B). Laquo; An Infinite Series Involving a Sideways Sum. Archimedes' Twin Circles Illustrated ». Comments powered by Disqus. A blog about mathematics and computer programming by Jan Marthedal Rasmussen. Subscribe in a reader.
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Basic Multiple-Precision Multiplication - janmr blog
http://blog.janmr.com/2011/11/basic-multiple-precision-multiplication.html
After addressing multiple-precision addition. We now turn to multiplication of two multiple-precision numbers. Once again, we use the number representation and notation introduced earlier. Several algorithms exist for doing multiple-precision multiplication. This post will present the basic, pencil-and-paper-like method. Basically, it consists of two parts: Multiplying a number by a single digit and adding together the sub-results, aligned appropriately. Multiple digits times a single digit. K {i 1} = lf...
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