background preloader

Entropy (information theory)

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 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. Named after Boltzmann's H-theorem, Shannon defined the entropy H (Greek letter Eta) of a discrete random variable X with possible values {x1, ..., xn} and probability mass function P(X) as: When taken from a finite sample, the entropy can explicitly be written as . , with Related:  Wikipedia.orgMatematics

Nature (essay) Emerson by Eastman Johnson, 1846 "Nature" is an essay written by Ralph Waldo Emerson, and published by James Munroe and Company in 1836. It is in this essay that the foundation of transcendentalism is put forth, a belief system that espouses a non-traditional appreciation of nature.[1] Transcendentalism suggests that divinity suffuses all nature, and speaks to the notion that we can only understand reality through studying nature.[2] A visit to the Muséum National d'Histoire Naturelle in Paris inspired a set of lectures delivered in Boston and subsequently the ideas leading to the publication of Nature. Within this essay, Emerson divides nature into four usages; Commodity, Beauty, Language and Discipline. Henry David Thoreau had read "Nature" as a senior at Harvard College and took it to heart. Emerson followed the success of this essay with a famous speech entitled "The American Scholar". Emerson uses spirituality as a major theme in his essay, “Nature”. Jump up ^ Liebman, Sheldon W.

Go and mathematics As a result of its elegant and simple rules, the game of Go has long been an inspiration for mathematical research. Chinese scholars of the 11th century already published work on permutations based on the go board. In more recent years, research of the game by John H. Conway led to the invention of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals[1] being a specific example of its use in Go). Legal positions[edit] Since each location on the board can be either empty, black, or white, there are a total of 3N possible board positions on a board with N intersections. Game tree complexity[edit] The total number of possible games is a function both of the size of the board and the number of moves played. The total number of possible games can be estimated from the board size in a number of ways, some more rigorous than others. Positional complexity[edit] A game of Go See also[edit] Notes[edit] References[edit] External links[edit]

Teotihuacan Coordinates: Teotihuacan /teɪˌoʊtiːwəˈkɑːn/,[1] also written Teotihuacán (Spanish teotiwa'kan ), was a pre-Columbian Mesoamerican city located in the Basin of Mexico, 30 miles (48 km) northeast of modern-day Mexico City, which is today known as the site of many of the most architecturally significant Mesoamerican pyramids built in the pre-Columbian Americas. Apart from the pyramids, Teotihuacan is also anthropologically significant for its complex, multi-family residential compounds, the Avenue of the Dead, and the small portion of its vibrant murals that have been exceptionally well-preserved. Additionally, Teotihuacan exported a so-called "Thin Orange" pottery style and fine obsidian tools that garnered high prestige and widespread utilization throughout Mesoamerica.[2] Name[edit] This naming convention led to much confusion in the early 20th century, as scholars debated whether Teotihuacan or Tula-Hidalgo was the Tollan described by 16th–century chronicles. History[edit] Zenith[edit]

Science Mysteries, Fibonacci Numbers and Golden section in Nature Golden Ratio & Golden Section : : Golden Rectangle : : Golden Spiral Golden Ratio & Golden Section In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. Expressed algebraically: The golden ratio is often denoted by the Greek letter phi (Φ or φ). The figure of a golden section illustrates the geometric relationship that defines this constant. Golden Rectangle A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: j (one-to-phi), that is, 1 : or approximately 1:1.618. Golden Spiral In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to j, the golden ratio. Successive points dividing a golden rectangle into squares lie on a logarithmic spiral which is sometimes known as the golden spiral. Golden Ratio in Architecture and Art Here are few examples: Parthenon, Acropolis, Athens. Examples: Dr.

Emma Goldman Emma Goldman (June 27 [O.S. June 15], 1869 – May 14, 1940) was an anarchist political activist and writer. She played a pivotal role in the development of anarchist political philosophy in North America and Europe in the first half of the 20th century. In 1917, Goldman and Berkman were sentenced to two years in jail for conspiring to "induce persons not to register" for the newly instated draft. After their release from prison, they were arrested—along with hundreds of others—and deported to Russia. Initially supportive of that country's October Revolution which brought the Bolsheviks to power, Goldman reversed her opinion in the wake of the Kronstadt rebellion and denounced the Soviet Union for its violent repression of independent voices. Biography Family Taube's second marriage was arranged by her family and, as Goldman puts it, "mismated from the first".[5] Her second husband, Abraham Goldman, invested Taube's inheritance in a business that quickly failed. Adolescence Most and Berkman

Bethe Hagens - Planetary Grid - A New Synthesis by William Becker and Bethe Hagens Bethe Hagens The Planetary Grid: A New Synthesis Bill Becker (Professor of Industrial Design at the University of Illinois , Chicago ) and Bethe Hagens (Professor of Anthropology at Governors State University ) are a husband-wife team. "The experience of life in a finite, limited body is specifically for the purpose of discovering and manifesting supernatural existence within the finite." Introduction We've entitled our current exercise in planetary grid research "A New Synthesis" - and indeed we hope it is. Yet the events continue to be catalogued, with many reports suppressed or labelled "fraud" by orthodox scientists. In 1200 A D., a new energy began to move within the cultures of the West. Now it is 500 years later, and Leonardo's manifest symbol of individual view point and detachment has brought us to viewing video discs of the earth as seen from the moon. Planetary Grid Researchers: Prehistoric to Present

Bernard of Clairvaux Bernard of Clairvaux, O.Cist (1090 – August 20, 1153) was a French abbot and the primary builder of the reforming Cistercian order. After the death of his mother, Bernard sought admission into the Cistercian order. "Three years later, he was sent to found a new abbey at an isolated clearing in a glen known as the Val d'Absinthe, about 15 km southeast of Bar-sur-Aube. On the death of Pope Honorius II on 13 February 1130, a schism broke out in the Church. Following the Christian defeat at the Siege of Edessa, the pope commissioned Bernard to preach the Second Crusade. Early life (1090–1113)[edit] Bernard would expand upon Anselm of Canterbury's role in transmuting the sacramentally ritual Christianity of the Early Middle Ages into a new, more personally held faith, with the life of Christ as a model and a new emphasis on the Virgin Mary. Bernard was only nineteen years of age when his mother died. Abbot of Clairvaux (1115–28)[edit] The beginnings of Clairvaux Abbey were trying and painful.

Kepler’s Laws – One Minute Astronomer Once Kepler got his hands on Tycho’s measurements, he worked diligently to make sense of the data and to develop a solid framework for the workings of the solar system. He succeeded. Working for more than a decade, crunching numbers with pen and paper, he laid out three simple mathematical laws that account for the motion of the planets. *** See the Deep Sky with Your Telescope! A concise guide to observing the universe beyond our solar system. Kepler’s First Law As mentioned in the short biography of Kepler, his first law was a result of a great insight: that planets did not necessarily move in perfect circular orbits, which had been long assumed. Kepler’s First Law states that planets move around the Sun in an elliptical orbit with the Sun at one focus of the ellipse. Kepler’s Second Law As you may have surmised, if a planet traverses an elliptical orbit, its distance from the Sun changes during an orbit. “Who cares”, you might ask? Kepler’s Third Law

List of best-selling singles The artists with the most singles (including featured credits) on this list is Rihanna with nine entries, followed by Katy Perry with eight entries, Beyoncé and Elvis Presley next on the list with six entries each. Lady Gaga, Flo Rida, Britney Spears and Michael Jackson (appearing twice as lead member of The Jackson 5) each appear five times. Given the recent trend that commercial music download sites increasingly sell mostly single tracks rather than whole albums, the list is split between digital singles and physical (CD and vinyl) singles.[1] Portable audio players, which make it extremely easy to load and play songs from many different artists, are claimed to be a major factor behind this trend. In italics, singles whose listed sales are the compilation of sales in available markets instead of a worldwide sales figure. Best-selling physical singles[edit] 15 million copies or more[edit] 10–14.9 million copies[edit] 7–9.9 million copies[edit] 5–6.9 million copies[edit] Notes[edit]

Kepler–Bouwkamp constant A sequence of inscribed polygons and circles. Numerical value of the Kepler–Bouwkamp constant[edit] The decimal expansion of the Kepler–Bouwkamp constant is (sequence A085365 in OEIS) If the product is taken over the odd primes, the constant is obtained (sequence A131671 in OEIS). See also[edit] References[edit] Finch, S. Valentin Turchin Valentin Fyodorovich Turchin (Russian: Валенти́н Фёдорович Турчи́н, 1931 – 7 April 2010) was a Soviet and American cybernetician and computer scientist. He developed the Refal programming language, the theory of metasystem transitions and the notion of supercompilation. As such he can be seen as a pioneer in Artificial Intelligence and one of the visionaries at the basis of the Global brain idea. Biography[edit] Turchin was born in 1931 in Podolsk, Soviet Union. In the 1960s, Turchin became politically active. He came to New York where he joined the faculty of the City University of New York in 1979. His son, Peter Turchin, is a world renowned specialist in population dynamics and mathematical modeling of historical dynamics. Work[edit] The philosophical core of Turchin's scientific work is the concept of the metasystem transition, which denotes the evolutionary process through which higher levels of control emerge in system structure and function. Major publications[edit] Valentin F.

Journal of Humanistic Mathematics - an online-only, open access, peer reviewed journal | Journals at Claremont | Claremont Colleges Editors Follow Welcome to the Journal of Humanistic Mathematics, an online-only, open-access, peer-reviewed journal. Please feel free to browse our website and add your email to our database if you would like to receive a Table of Contents for each issue in your email. We would love to hear from you! Please email the editors (Mark Huber at and Gizem Karaali at, or submit a response directly to any of the published articles or essays (you can use the Submit A Response link on the relevant page). Journal of Humanistic Mathematics Claremont Center for the Mathematical Sciences 640 North College Avenue Claremont, CA 91711 United States of America JHM is an online-only, open-access, peer-reviewed journal. Front Matter Editorial Article Report Exposition The World of Mathematics Book Review Poetry Fiction Announcement