Möbius strip

The MTR network adopted a system of recirculated magnetic plastic cards when it started operations in 1979. These cards were either used as single journey tickets or as stored value tickets. The KCRC adopted the same magnetic cards in 1984, and the stored value version was renamed Common Stored Value Tickets. In 1989, the Common Stored Value Tickets system was extended to KMB buses providing a feeder service to MTR/KCR stations and to Citybus, and was also extended to a limited number of non-transport applications, such as payments at ...

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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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The Octopus system was launched in 1997, and 3 million cards were issued within the first three months. The main reason for the quick success of the system was that the MTR and KCR required that all holders of common stored value tickets replace their tickets with Octopus cards in three months or have their tickets made obsolete, thus forcing their combined base of 3.3 million commuters (2.2 million for the MTR, 1.1 million for the KCR) to switch quickly. Another reason is the coin shortage in Hong Kong after 1997, there is a belief that old ...

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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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Mass transit agencies have been using stored value, pre-paid cards for electronic ticketing since the 1970s. This market started to move from magnetic stripe technology to smart cards since the early 1990s; Hong Kong was actually the first major system to change over. The Sony FeliCa technology used by Octopus is also used by Singapore's EZ-link card for its MRT and bus systems, Japan's Suica on the JR East, as well as the Nagasaki Smart Card system in Nagasaki, Japan. All these however use more up-to-date versions of the technology, ...

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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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Mondex specifically cited the widespread popularity of Octopus as the reason for withdrawing from the Hong Kong market in 2002. This is despite the fact that they launched their cards one year before the Octopus (in 1996), and had the backing of two of Hong Kong's biggest banks, HSBC and its subsidiary Hang Seng Bank. Academic studies suggest that the biggest cause was the lack of a compelling reason on the commuters' part to adopt the Mondex system, unlike Octopus, which had the solid backing of public tra ...

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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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The Chinese name for the Octopus card literally means "eight places pass." Eight is a significant number in Chinese in that it is often used to indicate "many." For instance, the Chinese phrase 四面八方 ("four sides eight directions") is a common expression meaning "in all directions." Eight is also considered a lucky number in Chinese culture. The English name "Octopus card" is derived from the use of the number eight since an octopus has eight tentacles. The name is also particularly appropriate since "octopus" has the connotation of b ...

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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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Let be a differential 2-form defined on the surface S, and let be an orientation preserving parametrization of S with (s,t) in D. Then, the surface integral of f on S is given by where is the surface normal to S. Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as componen ...

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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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A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space which we inhabit; or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional "string" to three dimensional "ball" makes an ob ...

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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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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, some 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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The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1 with its centre at the origin. If x and y are the coordinates of a point on the circle, then we have x² + y² = 1. Locally, the circle resembles a line, which is one-dimensional. In other words, only one coordinate is needed to describe the circle locally. Consider, for instance, the top part of the circle, for which the y-coordinate is positive (the yellow part i ...

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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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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 homeomor ...

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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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Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to itself acquire a sense of ordering, which we can think of as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, we can choose charts so that overlapping regions agree on their "handedness"; these are orientable manifo ...

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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

Read more here: » Manifold: Encyclopedia II - Manifold - Orientability

The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rn. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rn. These homeomorphisms are the charts of the manifold. Usually additional t ...

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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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In the special case when the bundle E in question is the tangent bundle of a compact, oriented, r-dimensional manifold, the Euler class is an element of the top cohomology of the manifold, which is naturally identified with the integers by evaluating cohomology classes on the fundamental homology class. Under this identification, the Euler class of the ta ...

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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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A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space in which we live. Or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional string to three dimensional ball makes an ob ...

See also:

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

Read more here: » Manifold: Encyclopedia II - Manifold - Introduction

Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group which acts continuously on the fiber space F on the left. We lose nothing if we require G to act effectively on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle (E, B, π, F) is a local trivialization such that for any two overlapping charts (Ui, φi) and (USee also:

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

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A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do not in general have globally-defined sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology. Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle ...

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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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The "figure 8" immersion of the Klein bottle has a particularly simple parametrization: In this immersion, the self-intersection circle is a geometric circle in the XY plane. The positive constant r is the radius of this circle. The parameter u gives the angle in the XY plane, and v specifies ...

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

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Maurits Cornelis, or Mauk as he was to be nicknamed, was born in Leeuwarden (Friesland), the Netherlands. He was the youngest son of civil engineer George Arnold Escher and his second wife, Sarah Gleichman. In 1903, the family moved to Arnhem where he took carpentry and piano lessons until the age of thirteen. From 1912 until 1918 he attended secondary school. Though he excelled at drawing, his grades were generally poor, and he was required to repeat the second form. In 1919 Escher attended the Haarlem School of Architectur ...

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M.C. Escher, M.C. Escher - Youth, M.C. Escher - Marriage and later life, M.C. Escher - Works, M.C. Escher - Selected list of works, M.C. Escher - References in popular culture, M.C. Escher - Bibliography

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Consider a vector field v on S, that is, for each x in S, v(x) is a vector. Then the integral of v over S is called the flux. Imagine that we have a fluid flowing through S, such that v(x) determines the direction and velocity of the fluid at x. Then the flux is the quantity of fluid flowing through S in unit amount of time. This illustration implies that if the vector field is tangent to S at each point, then the integral of the v ...

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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

Read more here: » Surface integral: Encyclopedia II - Surface integral - Surface integrals of vector fields

Escher travelled to Italy regularly in the following years. It was in Italy that he first met Jetta Umiker, the woman whom he married in 1924. The young couple settled down in Rome and stayed there until 1935, when the political climate under Mussolini became unbearable. The family next moved to Château-d'Œx, Switzerland where they remained for two years. Escher, who had been very fond of and inspired by the landscape in Italy, was decidedly unhappy in Switzerland, so in 1937, the family moved again, to Ukkel, a small town near Brus ...

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M.C. Escher, M.C. Escher - Youth, M.C. Escher - Marriage and later life, M.C. Escher - Works, M.C. Escher - Selected list of works, M.C. Escher - References in popular culture, M.C. Escher - Bibliography

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Let E = B × F and let π : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle. Perhaps the simplest example of a nontrivial bundle E is the Möbius strip. The Möbius strip has a circle for a base B and a line segment for the fiber F. A neighborhood U of a point is an arc; in the picture, this is th ...

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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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Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components. We use the following notation. sphere: S torus: T Klein bottle: K Projective plane: P Facts: S # S = S S # M = M P # P = K P # K = P # T We use a shorthand ...

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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

Read more here: » Surface: Encyclopedia II - Surface - Connected sum of surfaces