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: .
TV Wild :: Home Reaction Diffusion-Java This applet displays a reaction-diffusion system. Begin exploring by using the "Preset" choice at the bottom - and the "Restart" button below it - to see some of the possible configurations. Configurable parameters include a number of manual colour-map controls (on the left) - and six main parameters of the reaction-diffusion system (on the right). Take care if using the other controls on the right. They are usually very sensitive. The model is a cellular automaton, based on the von-Neumann neighbourhood. It is based on the Gray-Scott model, and was taken from John E. More details about the type of system used can be found at the [Xmorphia web site]. Toroidial boundary constraints are appled, so the images will tesselate seamlessly. Optionally, a bumpmap technique is employed to give the images a sense of depth. Geometric note Also, the "saturated" state in the automaton is often a chequer-board pattern.
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:
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
Rod calculus Rod calculus or rod calculation is the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty and Yuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie. Japanese counting board with grids Hardware[edit] The basic equipment for carrying out rod calculus is a bundle of counting rods and a counting board. In 1971 Chinese archaeologists unearthed a bundle of well preserved animal bone counting rods stored in a silk pouch from a tomb in Qian Yang county in Shanxi province, dated back to the first half of Han dynasty(206 BC – 8AD). Software[edit] Rod Numerals[edit] Displaying Numbers[edit] representation of the number 231 For numbers larger than 9, a decimal system is used. Displaying Zeroes[edit] 日
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
Earth’s Calendar Year: 4.5 billion years compressed into 12 months - Biomimicry 3.8 Life has an incredible amount to teach us about living well on planet Earth, in no small part due to the fact that it’s been thriving here for 3.85 billion years. But, how long is that really? If we take the age of Earth (4.5 billion years) and compress it into one year, we can better grasp the time-tested wisdom of our fellow planet-mates. Earth was “born” on January 1, 4.54 billion years ago. If we compress 4.54 billion years into one year, that means 144 years is 1 second. It’s not until July that multi-cellular organisms appear. Earth’s Calendar Year plainly shows our age as a species relative to the much, much older forms of life on Earth. There are 30 million other species on the planet today. There’s no guarantee that we, or any of the other species around us, will remain. Is the Earth here for us? Here’s a look at milestones worth celebrating:
De l’esthétique des fractions continues Lundi 19 décembre 2011 1 19 /12 /Déc /2011 11:00 La récente livraison de BibNum (à laquelle j’ai contribué) me fait fantasmer sur les fractions continues. Au XVIIIe et au XIXe siècles, les mathématiciens manipulaient couramment ces objets, quasiment abandonnés depuis ! Galois himself n’est pas en reste, puisque son premier article, à 18 ans, porte sur les fractions continues. On peut développer un nombre rationnel en fraction continue – ainsi du nombre 3,14159 approximation de π (voir dans l’analyse de l’article BibNum la façon dont on utilise la division euclidienne pour trouver cette fraction, unique) On peut aussi développer en fractions continues des nombres solutions d’équations polynomiales (nombres algébriques). Quelle élégance, une fraction continue (infinie, dans ce cas) rien qu’avec des 1 ! nettement plus élégant (à mon goût !) Marcher sur les traces de Galois pour découvrir ces notions peut motiver certains élèves ! L'oeil malicieux de Galois, né en 1811, réactualisé en 2011
Reference Pieces on Space One interesting question about the assumptions for Euclid's system of geometry is the difference between the "axioms" and the "postulates." "Axiom" is from Greek axíôma, "worthy." An axiom is in some sense thought to be strongly self-evident. A "postulate," on the other hand, is simply postulated, e.g. Given below is the axiomatization of geometry by David Hilbert (1862-1943) in Foundations of Geometry (Grundlagen der Geometrie), 1902 (Open Court edition, 1971). I. Hilbert's Comments: If Archimedes' Axiom is dropped [non-Archimedean geometry], then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less than two right angles. Hilbert's comments should serve to remind us that not only the Parallel Postulate can be denied without contradiction. Philosophy of Science, Space and Time Philosophy of Science Home Page Copyright (c) 1996, 1999, 2001 Kelley L. Three Points in Kant's Theory of Space and Time Metaphysics