50 Kick-Ass Websites You Need to Know About It's time to update the entries in your browser's links toolbar. But with recent estimates putting the size of the internet at well more than 100 million distinct websites, it's getting harder and harder to get a handle on all the great stuff that's out there. That's why we've compiled this list. And unlike some lists you may have seen, which try to name the very "best" websites, but end up just telling you a lot of stuff you already know, we've chosen instead to highlight 50 of our favorite sites that fly under most people's radar. Think of it as the Maximum PC blog roll (remember those?). You might have heard of some of these sites, but we'll bet you haven't heard of all them. Demoscene.tv See What Can Be Done with 4 Kilobytes If you’re any kind of nerd at all, you probably know about the demoscene, where talented programmers create complex videos rendered in real-time, stored in incredibly small files. lite.Facebook.com Clutter-Free Social Networking You can admit it. Soyouwanna.com
Entropy (information theory) 2 bits of entropy. A single toss of a fair coin has an entropy of one bit. A series of two fair coin tosses has an entropy of two bits. The number of fair coin tosses is its entropy in bits. This random selection between two outcomes in a sequence over time, whether the outcomes are equally probable or not, is often referred to as a Bernoulli process. This definition of "entropy" was introduced by Claude E. Entropy is a measure of unpredictability of information content. Now consider the example of a coin toss. English text has fairly low entropy. If a compression scheme is lossless—that is, you can always recover the entire original message by decompressing—then a compressed message has the same quantity of information as the original, but communicated in fewer characters. Shannon's theorem also implies that no lossless compression scheme can compress all messages. Here E is the expected value operator, and I is the information content of X.[8][9] I(X) is itself a random variable. .
What does 0^0 (zero raised to the zeroth power) equal? Why do mathematicians and high school teachers disagree Clever student: I know! Now we just plug in x=0, and we see that zero to the zero is one! Cleverer student: No, you’re wrong! which is true since anything times 0 is 0. Cleverest student : That doesn’t work either, because if then is so your third step also involves dividing by zero which isn’t allowed! and see what happens as x>0 gets small. So, since = 1, that means that High School Teacher: Showing that approaches 1 as the positive value x gets arbitrarily close to zero does not prove that . is undefined. does not have a value. Calculus Teacher: For all , we have Hence, That is, as x gets arbitrarily close to (but remains positive), stays at On the other hand, for real numbers y such that , we have that That is, as y gets arbitrarily close to Therefore, we see that the function has a discontinuity at the point . but when we approach (0,0) along the line segment with y=0 and x>0 we get Therefore, the value of is going to depend on the direction that we take the limit. that will make the function ! . as is whatever
cooking conversions e^(i theta) Consider the function on the right hand side (RHS) f(x) = cos( x ) + i sin( x )Differentiate this function f ' (x) = -sin( x ) + i cos( x) = i f(x)So, this function has the property that its derivative is i times the original function. What other type of function has this property?A function g(x) will have this property if dg / dx = i g This is a differential equation that can be solved with seperation of variables (1/g) dg = i dx (1/g) dg = i dx ln| g | = i x + C | g | = ei x + C = eC ei x | g | = C2 ei x g = C3 ei xSo we need to determine what value (if any) of the constant C3 makes g(x) = f(x). If we set x=0 and evaluate f(x) and g(x), we get f(x) = cos( 0 ) + i sin( 0 ) = 1 g(x) = C3 ei 0 = C3 These functions are equal when C3 = 1.Therefore, cos( x ) + i sin( x ) = ei x (This is the usual justification given in textbooks.)By use of Taylors Theorem, we can show the following to be true for all real numbers: sin x = x - x3/3! e^(i) = 1 + (i) + (i)2/2! e^(i) = 1 + i - 2/2!
Vi Hart: Math Doodling Remember that video about doodling dragons and fractals and stuff? I finally finished part 2! Here is a magnet link so you can dowload it via torrent. Here it is on YouTube: You can tell I worked on it for a long time over many interruptions (travelling and other stuff), because in order to keep myself from hating what was supposed to be a quick easy part 2, I had to amuse myself with snakes. Here was part 1, via Torrent or YouTube. 10 Search Engines to Explore the Invisible Web Not everything on the web will show up in a list of search results on Google or Bing; there are lots of places that their web crawlers cannot access. To explore the invisible web, you need to use specialist search engines. Here are our top 12 services to perform a deep internet search. What Is the Invisible Web? Before we begin, let's establish what does the term "invisible web" refer to? Simply, it's a catch-all term for online content that will not appear in search results or web directories. There are no official data available, but most experts agree that the invisible web is several times larger than the visible web. The content on the invisible web can be roughly divided into the deep web and the dark web. The Deep Web The deep web made up of content that typically needs some form of accreditation to access. If you have the correct details, you can access the content through a regular web browser. The Dark Web The dark web is a sub-section of the deep web. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Hammack Home This book is an introduction to the standard methods of proving mathematical theorems. It has been approved by the American Institute of Mathematics' Open Textbook Initiative. Also see the Mathematical Association of America Math DL review (of the 1st edition), and the Amazon reviews. The second edition is identical to the first edition, except some mistakes have been corrected, new exercises have been added, and Chapter 13 has been extended. (The Cantor-Bernstein-Schröeder theorem has been added.) Order a copy from Amazon or Barnes & Noble for $13.75 or download a pdf for free here. Part I: Fundamentals Part II: How to Prove Conditional Statements Part III: More on Proof Part IV: Relations, Functions and Cardinality Thanks to readers around the world who wrote to report mistakes and typos! Instructors: Click here for my page for VCU's MATH 300, a course based on this book.
Weierstrass functions Weierstrass functions are famous for being continuous everywhere, but differentiable "nowhere". Here is an example of one: It is not hard to show that this series converges for all x. In fact, it is absolutely convergent. It is also an example of a fourier series, a very important and fun type of series. It can be shown that the function is continuous everywhere, yet is differentiable at no values of x. Here's a graph of the function. You can see it's pretty bumpy. Below is an animation, zooming into the graph at x=1. Wikipedia and MathWorld both have informative entries on Weierstrass functions. back to Dr.
List of emoticons A simple smiley This is a list of notable and commonly used emoticons or textual portrayals of a writer's mood or facial expression in the form of icons. The Western use of emoticons is quite different from Eastern usage, and Internet forums, such as 2channel, typically show expressions in their own ways. In recent times, graphic representations, both static and animated, have taken the place of traditional emoticons in the form of icons. Emoticons can generally be divided into two groups: Western or Horizontal (mainly from America and Europe), and Eastern or Vertical (mainly from east Asia). Western The emoticon in Western style is written most often from left to right as though the head is rotated counter-clockwise 90 degrees. Eastern Eastern emoticons generally are not rotated, and may include non-Latin characters to allow for additional complexity. Unicode characters References
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. The reason is that cylindrical rollers have a circular cross section, and a circle is closed curve "with constant width." What does that mean? 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. Here is another really surprising method of constructing curves with constant width:
Bicycle Maintenance Guide and Riding Tips This page was last updated 24 May 2007. I often get asked about bicycle maintenance and repairs, and tips for how to ride efficiently. This little manual is intended as a summary of what I have learned over the years. Gothic Architecture Pictures All text and pictures © QT Luong. See conditions for use of pictures. There are 28 pictures on this page out of 228 pictures of Gothic Architecture. Eco-friendly Gallery | Environment Team