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.
Living Earth Simulator will simulate the entire world Described as a “knowledge collider,” and now with a pledge of one billion euros from the European Union, the Living Earth Simulator is a new big data and supercomputing project that will attempt to uncover the underlying sociological and psychological laws that underpin human civilization. In the same way that CERN’s Large Hadron Collider smashes together protons to see what happens, the Living Earth Simulator (LES) will gather knowledge from a Planetary Nervous System (PNS — yes, really) to try to predict societal fluctuations such as political unrest, economic bubbles, disease epidemics, and so on. The scale of the LES, when it’s complete, will be huge. It is hoped that supercomputing centers all over the world will chip in with CPU time, and data will be corralled from existing projects and a new Global Participatory Platform, which is basically open data on a worldwide scale. The project also has commercial backing from Microsoft Research, IBM, Yahoo Research, and others.
Home - Math 106 Visualizing a function can give a mathematician enormous insight into the function's algebraic and geometrical properties. The easiest way to see what a function looks like is to use a computer as a graphing tool. At times, this technique is the most useful, but drawing the function yourself is always the best way to get a feeling for why the function looks the way it does when graphed. Interactive Mathematics Miscellany and Puzzles Bifurcation In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter is varied. More generally, a bifurcation is a separation of a structure into two branches or parts. , where denotes the real part, exhibits a bifurcation along the negative real axis and
9 Mental Math Tricks Math can be terrifying for many people. This list will hopefully improve your general knowledge of mathematical tricks and your speed when you need to do math in your head. 1. Multiplying by 9, or 99, or 999 Multiplying by 9 is really multiplying by 10-1. So, 9×9 is just 9x(10-1) which is 9×10-9 which is 90-9 or 81. Let’s try a harder example: 46×9 = 46×10-46 = 460-46 = 414. One more example: 68×9 = 680-68 = 612. To multiply by 99, you multiply by 100-1. So, 46×99 = 46x(100-1) = 4600-46 = 4554. Multiplying by 999 is similar to multiplying by 9 and by 99. 38×999 = 38x(1000-1) = 38000-38 = 37962. 2. To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges. Let me illustrate: To multiply 436 by 11 go from right to left. First write down the 6 then add 6 to its neighbor on the left, 3, to get 9. Write down 9 to the left of 6. Then add 4 to 3 to get 7. Then, write down the leftmost digit, 4. So, 436×11 = is 4796. Let’s do another example: 3254×11. 3. 4. 5.
Catastrophe theory In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry. Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s. It considers the special case where the long-run stable equilibrium can be identified with the minimum of a smooth, well-defined potential function (Lyapunov function). Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. Elementary catastrophes[edit] Potential functions of one active variable[edit] Fold catastrophe[edit] Stable and unstable pair of extrema disappear at a fold bifurcation Cusp catastrophe[edit] Swallowtail catastrophe[edit] Arnold's notation[edit]
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