Möbius strip

Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and two directions, infinite ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

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Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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An on-loan Octopus card can be purchased at Mass Transit Railway (MTR) and Kowloon Canton Railway (KCR) stations. No identification is required. If an owner loses a card, only the stored value of the card is lost. This type of Octopus card is anonymous; no personal information, bank account or credit card details are stored on the card. Making or recording a payment using the card (eg. by passing through a MTR or ferry ticket gate, boarding a bus, alighting from a tram, or purchasing items from various outlets) is done by holding the ...

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Octopus card, Octopus card - Name and logo, Octopus card - Obtaining and using an Octopus card, Octopus card - Fare grades, Octopus card - Personalised cards, Octopus card - Automatic Add Value Service, Octopus card - Souvenir cards, Octopus card - Special purpose cards, Octopus card - Octopus gadgets, Octopus card - Refunding an Octopus card, Octopus card - Back-end technology and operations, Octopus card - Clearing and settlement, Octopus card - Privacy and encryption, Octopus card - Operator, Octopus card - History, Octopus card - Adoption of the Octopus card, Octopus card - Comparison with other electronic cash systems, Octopus card - Comparison with other transit card systems, Octopus card - Future developments, Octopus card - Further information

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Topologically, the Klein bottle can be defined as the square [0,1] × [0,1] with sides identified by the relations (0,y) ~ (1,y) for 0 ≤ y ≤ 1 and (x,0) ~ (1-x,1) for 0 ≤ x ≤ 1, as in the following diagram: Like the Möbius strip, the Klein bottle is a two-dimensional differentiable manifold which is not orientable. Unlike the Möbius strip, the Klein bottle is a closed manifold, meaning it is a compact manifold withou ...

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Klein bottle, Klein bottle - Properties, Klein bottle - Dissection, Klein bottle - Parametrization

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Let us notice that we defined the surface integral by using a parametrization of the surface S. We know that a given surface might have several parametrizations. For example, if we move the locations of the North Pole and South Pole on a sphere, the latitude and longitude change for all the points on the sphere. A natural question is then whether the definition of the surface integral depends on the chosen parametrization. For integrals of scalar fields, the answer to this question is simple, the value of the surface integral will ...

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Surface integral, Surface integral - Surface integrals of scalar fields, Surface integral - Surface integrals of vector fields, Surface integral - Surface integrals of differential 2-forms, Surface integral - Theorems involving surface integrals, Surface integral - Advanced issues

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In three dimensions, every figure which possesses a plane of symmetry or a center of symmetry is achiral. (A plane of symmetry of a figure F is a plane P, such that F is invariant under the mapping , when P is chosen to be the x-y-plane of the coordinate system. A center of symmetry of a figure See also:

Chirality mathematics, Chirality mathematics - Chirality and symmetry group, Chirality mathematics - Chirality in three dimensions, Chirality mathematics - Chirality in two dimensions, Chirality mathematics - Knot theory

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Jokes often depend for humour on the unexpected, the mildly taboo (which can include the distasteful or socially improper), or the playing on stereotypes and other cultural myths. Many jokes fit into more than one category. Joke - Yo' mama jokes. Main article: The dozens. Jokes of this kind originate in the dozens, an African-American custom with West African roots in which two competitors -- usually males -- go head to head in a competition of comedic, often ribald, trash-talk. The target of ...

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Joke, Joke - Psychology of jokes, Joke - Types of jokes, Joke - Yo' mama jokes, Joke - Political jokes, Joke - Question–answer, Joke - Dirty jokes, Joke - Sick jokes, Joke - Little Johnny jokes, Joke - Ethnic jokes, Joke - Sexist jokes, Joke - Less offensive versions, Joke - Blonde jokes, Joke - Jokes about animals, Joke - Shaggy dog stories, Joke - You have two cows, Joke - Duck jokes, Joke - Religion in jokes, Joke - Other classes of jokes

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In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open neighbourhood homeomorphic to either an open subset of E2 (Euclidean 2-space) or an open subset of the closed half of E2. The set of points which have an open neighbourhood homeomorphic to En is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (1 ...

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Surface, Surface - Examples, Surface - Definition, Surface - Classification of closed surfaces, Surface - Compact surfaces, Surface - Embeddings in R³, Surface - Differential geometry, Surface - Some models, Surface - Fundamental polygon, Surface - Connected sum of surfaces, Surface - Algebraic surface

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A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : E → B is a continuous surjection satisfying a local triviality condition outlined below. B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map. We shall assume in what ...

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Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

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Infinity - Ancient view of infinity. The earliest known documented knowledge of infinity is presented in the Hindu Yajur Veda (ca. 1800 BC - 800 BC) which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". The Indian Jaina mathematical text Surya Prajinapti (ca. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite. It recognises five different types of infinity: infinite in one and tw ...

See also:

Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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In common parlance, infinity is often used in a hyperbolic sense. For example, "The movie was infinitely boring, but we had to wait forever to get tickets." In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. In theory, as long as there is no external interaction, the loop will continue to run for all time. In practice however, most programming loops co ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting). It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. It is for example presumed impossible for any body to have infinite mass or infinite energy. There exists the concept of infinite en ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Infinity - Infinity in real analysis. In real analysis, the symbol , called "infinity", denotes an unbounded limit. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. Points labeled and can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers. Adding algebraic properties to this gives us the extended real numbers. We can also treat and as the same, leading to the one-point compact ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomo ...

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Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts, atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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The Hitchhiker's Guide to the Galaxy contains the following definition of infinity: "Bigger than the biggest thing ever and then some, much bigger than that, in fact really amazingly immense, a totally stunning size, real 'Wow, that's big!' time. Infinity is just so big that by comparison, bigness itself looks really titchy. Gigantic multiplied by colossal multiplied by staggeringly huge is the s ...

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Infinity, Infinity - History, Infinity - Ancient view of infinity, Infinity - Views from the Renaissance to modern times, Infinity - Modern philosophical views, Infinity - Infinity symbol, Infinity - Mathematical infinity, Infinity - Infinity in real analysis, Infinity - Infinity in complex analysis, Infinity - Arithmetic properties of infinity, Infinity - Infinity in set theory, Infinity - Mathematics without infinity, Infinity - Use of infinity in common speech, Infinity - Physical infinity, Infinity - Infinity in cosmology, Infinity - Three types of infinities, Infinity - Infinity in science fiction, Infinity - Note

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Why we laugh has been the subject of serious academic study, examples being: Sigmund Freud's "Jokes and Their Relationship to the Unconscious". Marvin Minsky in Society of Mind. Marvin suggests that laughter has a specific function related to the human brain. In his opinion jokes and laughter are mechanisms for the brain to learn Nonsense. For that reason, he argues, jokes are usually not as funny when you hear them repeatedly. Edward de Bono in "The mechanism of ...

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Joke, Joke - Psychology of jokes, Joke - Types of jokes, Joke - Mathematical jokes, Joke - Yo' mama jokes, Joke - Political jokes, Joke - Self-deprecating humor, Joke - Question–answer, Joke - Dirty jokes, Joke - Sick jokes, Joke - Little Johnny jokes, Joke - Ethnic jokes, Joke - Sexist jokes, Joke - Less offensive versions, Joke - Blonde jokes, Joke - Jokes about animals, Joke - Shaggy dog stories, Joke - You have two cows, Joke - Duck jokes, Joke - Religion in jokes, Joke - Other classes of jokes, Joke - External links

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Let X be a Hausdorff space. A homeomorphism from an open subset U⊂X to a subset of C is called a chart. Two charts f and g whose domains intersect are said to be compatible if the maps f o g−1 and g o f −1 are holomorphic over their domains. If A is a collection of compatible charts and if any x in X is in the domain of some f in A, then we say that A is an atlas. When we endow XSee also:

Riemann surface, Riemann surface - Formal definition, Riemann surface - Examples, Riemann surface - Properties and further definitions, Riemann surface - History, Riemann surface - In art and literature

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The Euler class e(E) is an element of the integral cohomology group , constructed as follows. An orientation of E amounts to a continuous choice of generator of the cohomology of each fiber F relative to its zero element F0. This induces an orientation class in the cohomology of E relative to the zero section E0< ...

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Euler class, Euler class - Formal definition, Euler class - Properties, Euler class - Relations to other invariants, Euler class - Example: Line bundles over the circle

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The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium. Bernhard Riemann was the first to do extensive work that required a generalization of manifolds to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translates as "manifoldness". In his Göttingen inaugural lecture, Riemann states the possible values a p ...

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Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

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As of 2005, Octopus Cards Limited (OCL), the operator of Octopus, is a joint-venture between six transit companies, namely MTR Corporation (57.4%), KCR (22.1%), Kowloon Motor Bus (12.4%), Citybus (5%), NWFB (3.05%) and First Ferry (0.05%). Since the Government of Hong Kong owns nearly three-quarters of MTR and 100% of KCR, it is the biggest effective shareholder in the company, although the business is operated on a commercial basis. OCL has been aggressively expanding the use of Octopus in Hong Kong, and has won a number of contr ...

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Octopus card, Octopus card - Name and logo, Octopus card - Obtaining and using an Octopus card, Octopus card - Fare grades, Octopus card - Personalised cards, Octopus card - Automatic Add Value Service, Octopus card - Souvenir cards, Octopus card - Special purpose cards, Octopus card - Octopus gadgets, Octopus card - Refunding an Octopus card, Octopus card - Back-end technology and operations, Octopus card - Clearing and settlement, Octopus card - Privacy and encryption, Octopus card - Operator, Octopus card - History, Octopus card - Adoption of the Octopus card, Octopus card - Comparison with other electronic cash systems, Octopus card - Comparison with other transit card systems, Octopus card - Future developments, Octopus card - Further information

Read more here: » Octopus card: Encyclopedia II - Octopus card - Operator

The Octopus system was designed by AES ProData (Hong Kong) Limited, now known as ERG Transit Systems, a member of the ERG Group based in Perth, Western Australia. ERG was contracted for the initial design and building of the back-end systems. Operations, maintenance and development is undertaken by Octopus, and in 2005, Octopus replaced the central transaction clearing house with its own system. The Octopus card uses the Sony 13.56 MHz FeliCa radio frequency identification (RFID) chip (and other related technology); and Hong Kong is t ...

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Octopus card, Octopus card - Name and logo, Octopus card - Obtaining and using an Octopus card, Octopus card - Fare grades, Octopus card - Personalised cards, Octopus card - Automatic Add Value Service, Octopus card - Souvenir cards, Octopus card - Special purpose cards, Octopus card - Octopus gadgets, Octopus card - Refunding an Octopus card, Octopus card - Back-end technology and operations, Octopus card - Clearing and settlement, Octopus card - Privacy and encryption, Octopus card - Operator, Octopus card - History, Octopus card - Adoption of the Octopus card, Octopus card - Comparison with other electronic cash systems, Octopus card - Comparison with other transit card systems, Octopus card - Future developments, Octopus card - Further information

Read more here: » Octopus card: Encyclopedia II - Octopus card - Back-end technology and operations