In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to X
- f : I → X.
The initial point of the path is f(0) and the terminal point is f(1). One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path. Note that a path is not just a subset of X which "looks like" a curve, it also includes a parameterization. For example, the maps f(x) = x and g(x) = x2 represent two different paths from 0 to 1 on the real line.
A loop in a space X based at x ∈ X is a path from x to x. A loop may be equally well regarded as a map f : I → X with f(0) = f(1) or as a continuous map from the unit circle S1 to X
- f : S1 → X.
This is because S1 may be regarded as a quotient of I under the identification 0 ∼ 1. The set of all loops in X forms a space called the loop space of X.
A topological space for which there exists a path connecting any two points is said to be path-connected. Any space may be broken up into a set of path-connected components. The set of path-connected components of a space X is often denoted π0(X);.
One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X is a topological space with basepoint x0, then a path in X is one whose initial point is x0. Likewise, a loop in X is one that is based at x0.
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Homotopy of paths [link]
Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.
Specifically, a homotopy of paths, or path-homotopy, in X is a family of paths ft : I → X indexed by I such that
- ft(0) = x0 and ft(1) = x1 are fixed.
- the map F : I × I → X given by F(s, t) = ft(s) is continuous.
The paths f0 and f1 connected by a homotopy are said to homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.
The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path f under this relation is called the homotopy class of f, often denoted [f].
Path composition [link]
One can compose paths in a topological space in an obvious manner. Suppose f is a path from x to y and g is a path from y to z. The path fg is defined as the path obtained by first traversing f and then traversing g:
- Failed to parse (Missing texvc executable; please see math/README to configure.): fg(s) = \begin{cases}f(2s) & 0\leq s \leq \frac{1}{2} \\ g(2s-1) & \frac{1}{2} \leq s \leq 1.\end{cases}
Clearly path composition is only defined when the terminal point of f coincides with the initial point of g. If one considers all loops based at a point x0, then path composition is a binary operation.
Path composition, whenever defined, is not associative due to the difference in parametrization. However it is associative up to path-homotopy. That is, [(fg)h] = [f(gh)]. Path composition defines a group structure on the set of homotopy classes of loops based at a point x0 in X. The resultant group is called the fundamental group of X based at x0, usually denoted π1(X,x0).
In situations calling for associativity of path composition "on the nose," a path in X may instead be defined as a continuous map from an interval [0,a] to X for any real a ≥ 0. A path f of this kind has a length |f| defined as a. Path composition is then defined as before with the following modification:
- Failed to parse (Missing texvc executable; please see math/README to configure.): fg(s) = \begin{cases}f(s) & 0\leq s \leq |f| \\ g(s-|f|) & |f| \leq s \leq |f|+|g|\end{cases}
Whereas with the previous definition, f, g, and fg all have length 1 (the length of the domain of the map), this definition makes |fg| = |f| + |g|. What made associativity fail for the previous definition is that although (fg)h and f(gh) have the same length, namely 1, the midpoint of (fg)h occurred between g and h, whereas the midpoint of f(gh) occurred between f and g. With this modified definition (fg)h and f(gh) have the same length, namely |f|+|g|+|h|, and the same midpoint, found at (|f|+|g|+|h|)/2 in both (fg)h and f(gh); more generally they have the same parametrization throughout.
Fundamental groupoid [link]
There is a categorical picture of paths which is sometimes useful. Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of X. Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x0 in X is just the fundamental group based at X. More generally, one can define the fundamental groupoid on any subset A of X, using homotopy classes of paths joining points of A. This is convenient for the Van Kampen's Theorem.
References [link]
- Ronald Brown, Topology and groupoids, Booksurge PLC, (2006).
- Peter May, A concise course in algebraic topology, University of Chicago Press, (1999).
- James Raymond Munkres, Topology 2ed, Prentice Hall, (2000).
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Pathé
Pathé or Pathé Frères (French pronunciation: [pate fʁɛʁ], styled as PATHÉ!) is the name of various French businesses that were founded and originally run by the Pathé Brothers of France starting in 1896. In the early 1900s, Pathé became the world's largest film equipment and production company, as well as a major producer of phonograph records. In 1908, Pathé invented the newsreel that was shown in cinemas prior to a feature film.
Today, Pathé is a major film production and distribution company, owns a great number of cinema chains, across Europe but mainly in France, including 66% of the Les Cinémas Gaumont Pathé a joint venture between Pathé and the Gaumont Film Company, and several television networks across Europe. It is the second oldest still-operating film company in the world, predating Universal Studios and Paramount Pictures, second only to the French Gaumont Film Company studio.
History
The company was founded as Société Pathé Frères (Pathé Brothers Company) in Paris, France on 28 September 1896, by the four brothers Charles, Émile, Théophile and Jacques Pathé. During the first part of the 20th century, Pathé became the largest film equipment and production company in the world, as well as a major producer of phonograph records.
Path
Path may refer to:
In mathematics and computing:
Physical paths:
Other uses:
See also
Environment variable
Environment variables are a set of dynamic named values that can affect the way running processes will behave on a computer.
They are part of the environment in which a process runs. For example, a running process can query the value of the TEMP environment variable to discover a suitable location to store temporary files, or the HOME or USERPROFILE variable to find the directory structure owned by the user running the process.
They were introduced in their modern form in 1979 with Version 7 Unix, so are included in all Unix operating system flavors and variants from that point onward including Linux and OS X. From PC DOS 2.0 in 1982, all succeeding Microsoft operating systems including Microsoft Windows, and OS/2 also have included them as a feature, although with somewhat different syntax, usage and standard variable names.
Details
In all Unix and Unix-like systems, each process has its own separate set of environment variables. By default, when a process is created, it inherits a duplicate environment of its parent process, except for explicit changes made by the parent when it creates the child. At the API level, these changes must be done between running fork and exec. Alternatively, from command shells such as bash, a user can change environment variables for a particular command invocation by indirectly invoking it via env or using the ENVIRONMENT_VARIABLE=VALUE <command> notation. All Unix operating system flavors, DOS, and Windows have environment variables; however, they do not all use the same variable names. A running program can access the values of environment variables for configuration purposes.
Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.
Definition
The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.
Topology (musical ensemble)
Topology is an indie classical quintet from Australia, formed in 1997. A leading Australian new music ensemble, they perform throughout Australia and abroad and have to date released 12 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensemble at the University of Western Sydney. The group works with composers including Tim Brady in Canada, Andrew Poppy, Michael Nyman, and Jeremy Peyton Jones in the UK, and Terry Riley, Steve Reich, Philip Glass, Carl Stone, and Paul Dresher in the US, as well as many Australian composers.
In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.
Members
Bernard studied viola at the Queensland Conservatorium (B.Mus 1987) and at Michigan State University (Master of Music 1993) with John Graham and Robert Dan. He studied in summer schools with Kim Kashkashian (Aldeborough), the Alban Berg Quartet and the Kronos Quartet. While in the US, he played with the Arlington Quartet, touring the US and UK. He was a violist in the Queensland Philharmonic Orchestra from 1994-2000, and is now Associate Principal Violist of the Queensland Orchestra, playing solo parts in works such as the sixth Brandenburg Concerto. He has directed several concerts for the Queensland Philharmonic’s Off the Factory Floor chamber series.
Topology (electrical circuits)
The topology of an electronic circuit is the form taken by the network of interconnections of the circuit components. Different specific values or ratings of the components are regarded as being the same topology. Topology is not concerned with the physical layout of components in a circuit, nor with their positions on a circuit diagram. It is only concerned with what connections exist between the components. There may be numerous physical layouts and circuit diagrams that all amount to the same topology.
Strictly speaking, replacing a component with one of an entirely different type is still the same topology. In some contexts, however, these can loosely be described as different topologies. For instance, interchanging inductors and capacitors in a low-pass filter results in a high-pass filter. These might be described as high-pass and low-pass topologies even though the network topology is identical. A more correct term for these classes of object (that is, a network where the type of component is specified but not the absolute value) is prototype network.
