The Mandelbrot Set Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah The Mandelbrot Set. Note: All of the Mandelbrot pictures on this page were generated with the applet on this page! You can click on any of them to see a large version, and you can use the applet to generate those very same pictures, or similar pictures all your own! The first picture ( No1 ) shows a small part of the Mandelbrot set (which is rendered in red). List of Contents What's so special about the Mandelbrot set? What is the Mandelbrot set? z(0) = z, z(n+1) = z(n)*z(n) + z, n=0,1,2, remains bounded. You may ask, what's so special about the particular iteration (1), and why do we use complex numbers instead of real ones. Much of the fascination of the Mandelbrot set stems from the fact that an extremely simple formula like (1) gives rise to an object of such great complexity. Consider this picture ( Title ). Now, I know you already clicked on that applet! This is what you should see. Max.
Mu-Ency -- The Encyclopedia of the Mandelbrot Set at MROB A second-order embedded Julia set This is a picture from the Mandelbrot Set, one of the most well-known fractal images in the world. (Click it for a larger version). The Mandelbrot Set is one of my hobbies, and I have collected a large amount of information about it. To organize that information I have created Mu-Ency, a large collection of text files linked to each other. Here are some entries from Mu-Ency: Mandelbrot Set: The mathematical definition. More Pictures: Some entries with pictures of parts of the Mandelbrot Set are: R2, Cusp, Embedded Julia set, 2-fold Embedded Julia set, 4-fold Embedded Julia set, Paramecia, R2.C(0), R2.C(1/3), R2.1/2.C(1/2), R2t series, Seahorse Valley, Delta Hausdorff Dimension, Exponential Map, Reverse Bifurcation. You can also look up specific terms in the index. Coordinates of the image above: Center: -1.769 110 375 463 767 385 + 0.009 020 388 228 023 440 i Width (and height): 0.000 000 000 000 000 160 Algorithm: distance estimator Iterations: 10000
Box-and-Whisker Plots: Interquartile Ranges and Outliers Box-and-Whisker Plots: Interquartile Ranges and Outliers (page 3 of 3) Sections: Quartiles, boxes, and whiskers, Five-number summary, Interquartile ranges and outliers The "interquartile range", abbreviated "IQR", is just the width of the box in the box-and-whisker plot. That is, IQR = Q3 – Q1. The IQR is the length of the box in your box-and-whisker plot. (Why one and a half times the width of the box? Find the outliers, if any, for the following data set: To find out if there are any outliers, I first have to find the IQR. Outliers will be any points below Q1 – 1.5×IQR = 14.4 – 0.75 = 13.65 or above Q3 + 1.5×IQR = 14.9 + 0.75 = 15.65. Then the outliers are at 10.2, 15.9, and 16.4. The values for Q1 – 1.5×IQR and Q3 + 1.5×IQR are the "fences" that mark off the "reasonable" values from the outlier values. By the way, your book may refer to the value of "1.5×IQR" as being a "step". Find the outliers and extreme values, if any, for the following data set, and draw the box-and-whisker plot.
Rare earth element As defined by IUPAC, a rare earth element (REE) or rare earth metal is one of a set of seventeen chemical elements in the periodic table, specifically the fifteen lanthanides, as well as scandium and yttrium. Scandium and yttrium are considered rare earth elements because they tend to occur in the same ore deposits as the lanthanides and exhibit similar chemical properties. List A table listing the seventeen rare earth elements, their atomic number and symbol, the etymology of their names, and their main usages (see also Applications of lanthanides) is provided here. Some of the rare earth elements are named after the scientists who discovered or elucidated their elemental properties, and some after their geographical discovery. A mnemonic for the names of the sixth-row elements in order is "Lately college parties never produce sexy European girls that drink heavily even though you look". Abbreviations The following abbreviations are often used: Spectroscopy
List of mathematical symbols When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Further, in mathematical logic, numerical equality is sometimes represented by "≡" instead of "=", with the latter representing equality of well-formed formulas. Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image. Guide This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. Basic symbols Symbols based on equality sign
Alcumus Alcumus offers students a customized learning experience, adjusting to student performance to deliver appropriate problems and lessons. Alcumus is specifically designed to provide high-performing students with a challenging curriculum appropriate to their abilities. Current features of Alcumus include: An intelligent learning system. As the student gets stronger, Alcumus automatically provides more challenging material, and conversely, if the student is having difficulty with a particular topic, Alcumus provides additional practice. Detailed progress reports. Students can track their performance in various subjects, and revisit problems and lessons at any time. Tools for teachers. Our Schola system allows teachers to monitor their students' progress and access Alcumus videos. Alcumus is currently free! You must have an account on the site in order to use Alcumus.
Graphene Aerogel – Lightest Material in The World A research team headed by Professor Gao Chao have developed ultra-light aerogel – it breaks the record of the world’s lightest material with surprising flexibility and oil-absorption. This progress is published in the “Research Highlights” column in Nature. Aerogel is the lightest substance recorded by Guinness Book of World Records. It couldn’t even cause deformation on dandelion flower fluffs. Gao Chao’s team has long been developing macroscopic graphene materials, such as one-dimensional graphene fibers and two-dimensional graphene films. In reported papers the carbon sponge developed by Gao’s team is the record holder of lightest material, with 0.16 mg/cubic centimeter, lower than the density of helium. The basic principle of developing aerogel is to remove solvent in the gel and retain the integrity. The title of review in Nature is “Solid carbon, springy and light”. The new material is just like a new-born baby. Source: Zhejiang University Related:
Outline of combinatorics The following outline is presented as an overview of and topical guide to combinatorics: Combinatorics – branch of mathematics concerning the study of finite or countable discrete structures. Essence of combinatorics Main article: Combinatorics Branches of combinatorics Multi-disciplinary fields that include combinatorics History of combinatorics Main article: History of combinatorics General combinatorial principles and methods Data structure concepts Problem solving as an art Living with large numbers Persons influential in the field of combinatorics Combinatorics scholars Journals Prizes See also References External links Combinatorics, a MathWorld article with many references.Combinatorics, from a MathPages.com portal.The Hyperbook of Combinatorics, a collection of math articles links.The Two Cultures of Mathematics by W.
TOP 10 IMPOSSIBLE INVENTIONS THAT WORK « Revolutionizing Awareness Searl Effects Generator by Jeane Manning When Leonardo da Vinci sketched out an impossible invention, fifteenth-century scholars probably put him down. Throughout history, experts tell innovators that their inventions are impossible. Perhaps in the 21st century the following inventions will be standard science, and a history student may wonder why 20th-century pundits disregarded them. This class of inventions could wipe out oil crises and help solve environmental problems. Forget the Rube Goldberg mechanical perpetual motion contraptions; they had to stop eventually. Inventors give various names to their space-energy converters. A spiritual commune in Switzerland had a tabletop free energy device running in greenhouses for years, but members feared that outsiders would turn the technology into weaponry. It may have been done before Tesla’s time. The garage inventors come from many backgrounds. One example is U.S. Look, Mom Earth, no power lines! 8. Like this:
Engineering ToolBox Space-Time Loops May Explain Black Holes Physics cannot describe what happens inside a black hole. There, current theories break down, and general relativity collides with quantum mechanics, creating what's called a singularity, or a point at whichthe equations spit out infinities. But some advanced physics theories are trying to bridge the gap between general relativity and quantum mechanics, tounderstand what's truly going on inside the densest objects in the universe. Recently, scientists applied a theory called loop quantum gravity to the case of black holes, and found that inside these objects, space and time may be extremely curved, but that gravity there is not infinite, as general relativity predicts. This was the first time scientists have applied the full loop quantum gravity theory to black holes, and the results were encouraging, researchers said. A black hole is created when a huge star runs out of fuel for nuclear fusion and collapses under its own gravity. "This model we've done is extremely simple," Pullin said.