Chaos theory A double rod pendulum animation showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a completely different trajectory. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions. Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior can be observed in many natural systems, such as weather and climate.[6][7] This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Introduction[edit] Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic dynamics[edit] The map defined by x → 4 x (1 – x) and y → x + y mod 1 displays sensitivity to initial conditions. In common usage, "chaos" means "a state of disorder".[9] However, in chaos theory, the term is defined more precisely. where , and , is: .
Central limit theorem The central limit theorem has a number of variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions, given that they comply with certain conditions. In more general probability theory, a central limit theorem is any of a set of weak-convergence theorems. Central limit theorems for independent sequences[edit] Classical CLT[edit] Let {X1, ..., Xn} be a random sample of size n — that is, a sequence of independent and identically distributed random variables drawn from distributions of expected values given by µ and finite variances given by σ2. of these random variables. Lindeberg–Lévy CLT. In the case σ > 0, convergence in distribution means that the cumulative distribution functions of √n(Sn − µ) converge pointwise to the cdf of the N(0, σ2) distribution: for every real number z, where Φ(x) is the standard normal cdf evaluated at x. Lyapunov CLT[edit] Let
Simon Newcomb _ wikipedia Canadian-American mathematician Simon Newcomb (March 12, 1835 – July 11, 1909) was a Canadian–American astronomer, applied mathematician, and autodidactic polymath. He served as Professor of Mathematics in the United States Navy and at Johns Hopkins University. Though Newcomb had little conventional schooling, he completed a BSc at Harvard in 1858. Biography[edit] Early life[edit] Simon Newcomb was born in the town of Wallace, Nova Scotia. Newcomb seems to have had little conventional schooling and was taught by his father. Newcomb taught for two years in Maryland, from 1854 to 1856; for the first year in a country school in Massey's Cross Roads, Kent County, then for a year nearby in Sudlersville in Queen Anne's County. In particular he read Isaac Newton's Principia (1687) at this time. Newcomb independently studied mathematics and physics. Peirce family[edit] Career in astronomy[edit] In the prelude to the American Civil War, many US Navy staff with Southern backgrounds left the service.
TV Wild :: Home The dynamics of correlated novelties : Scientific Reports Human activities data We begin by analyzing four data sets, each consisting of a sequence of elements ordered in time: (1) Texts: Here the elements are words. A novelty in this setting is defined to occur whenever a word appears for the first time in the text; (2) Online music catalogues: The elements are songs. The rate at which novelties occur can be quantified by focusing on the growth of the number D(N) of distinct elements (words, songs, wikipages, tags) in a temporally ordered sequence of data of length N. Gutenberg42 (a), (f), Last.fm43 (b), (g), Wikipedia44 (c), (h), del.icio.us45 (d), (i) datasets, and the urn model with triggering (e), (j). Full size image (281 KB) A second statistical signature is given by the frequency of occurrence of the different elements inside each sequence of data. Next we examine the four data sets for a more direct form of evidence of correlations between novelties. All the data sets display the predicted correlations among novelties. is drawn again.
Hasse diagram Hasse diagrams are named after Helmut Hasse (1898–1979); according to Birkhoff (1948), they are so-called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams; they appear, e.g., in Vogt (1895). Although Hasse diagrams were originally devised as a technique for making drawings of partially ordered sets by hand, they have more recently been created automatically using graph drawing techniques.[1] The phrase "Hasse diagram" may also refer to the transitive reduction as an abstract directed acyclic graph, independently of any drawing of that graph, but this usage is eschewed here. A "good" Hasse diagram[edit] Although Hasse diagrams are simple as well as intuitive tools for dealing with finite posets, it turns out to be rather difficult to draw "good" diagrams. The following example demonstrates the issue. . Upward planarity[edit] Notes[edit] References[edit] External links[edit] Related media at Wikimedia Commons:
Statistical Data Mining Tutorials Advertisment: In 2006 I joined Google. We are growing a Google Pittsburgh office on CMU's campus. We are hiring creative computer scientists who love programming, and Machine Learning is one the focus areas of the office. We're also currently accepting resumes for Fall 2008 intenships. If you might be interested, feel welcome to send me email: awm@google.com . The following links point to a set of tutorials on many aspects of statistical data mining, including the foundations of probability, the foundations of statistical data analysis, and most of the classic machine learning and data mining algorithms. These include classification algorithms such as decision trees, neural nets, Bayesian classifiers, Support Vector Machines and cased-based (aka non-parametric) learning. I hope they're useful (and please let me know if they are, or if you have suggestions or error-corrections).
George Shakan | Math and Machine Learning Blog Nature's fireworks: the best meteor showers coming in 2015 Watching meteors in the night sky can be fun, although typically you only see a few flashes an hour. But there are certain times of the year when you can see many more – events known as meteor showers. These are caused by Earth moving through streams of debris left behind by passing comets and asteroids. Such showers are typically active for several days or even weeks, but usually feature short, sharp climaxes, or “maxima,” with their best rates being visible on just a single night. The coming year promises to be another good one for meteor buffs, so here’s a guide on what to expect and how best to enjoy nature’s own firework displays. Alpha Centaurids: maximum February 8 Click to enlarge Extending from January 28 to February 21, and peaking on February 8, this shower is known for producing brightly coloured fireballs with long-lasting trains varying from a few seconds to several minutes. This year, unfortunately, the moon will greatly hamper observations. Lyrids: maximum April 22
Boulanger: la saga continue (2/2) PART 2 Les mystères du monde continu Je vous ai montré dans le billet précédent pourquoi en mélangeant de façon très méthodique une image faite de pixels (j’étale dans un sens, je replie et je recommence) on revient tôt ou tard à la l’image initiale. Mais aujourd’hui nous allons voir que cet éternel recommencement est un privilège réservé aux images numériques. Même si on savait répéter une telle transformation avec une précision parfaite sur une image réelle, faite d’un continuum de points, on ne retrouverait jamais l’image de départ. Pétrissage décimalPour vous le montrer simplement, je vais modifier légèrement la méthode de pétrissage: le boulanger étire son carré de 10 fois sa longueur initiale (au lieu de deux) et il coupe les neuf morceaux qui dépassent afin de retrouver la forme initiale du carré une fois empilés. C’est un peu plus compliqué à première vue mais vous allez vite comprendre pourquoi ça simplifie les calculs… L’irréversibilité et la flèche du temps Tags: chaos
Magician-turned-mathematician uncovers bias in coin flipping By ESTHER LANDHUIS Persi Diaconis has spent much of his life turning scams inside out. In 1962, the then 17-year-old sought to stymie a Caribbean casino that was allegedly using shaved dice to boost house odds in games of chance. In the mid-1970s, the upstart statistician exposed some key problems in ESP research and debunked a handful of famed psychics. Diaconis set out to test what he thought was obvious -- that coin tosses, the currency of fair choices, couldn't be biased. Wrong. A decade later, in 2002, a large manufacturer of card-shuffling machines for casinos summoned Diaconis to determine whether their new automated shufflers truly randomized the deck. Diaconis had no idea this mission would prompt a career shift. At 24, Diaconis began taking evening math classes at the City College of New York. But D'Aristotile, now a math professor at the State University of New York-Plattsburgh, saw that the kid had chutzpah. But Diaconis wasn't satisfied. Of flipping coins and falling cats