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wiki_27_chunk_20 | Automatic number announcement circuit | Long-distance carrier-specific Area code 700 is reserved for carrier-specific numbers operated by interstate long-distance providers, such as AT&T. With the exception of 1-700-555-4141 (which identifies the default interexchange carrier on a line), all of these are LD carrier-specific. Area code 700 is therefore rarely... | wikipedia |
wiki_27_chunk_21 | Automatic number announcement circuit | Area code 1-200 There is no non-geographic area code 200, although exchange 1-NPA-200-XXXX now exists in many local area codes (if it has not been explicitly reserved). The 1-200 area has occasionally been used as an unused space in which to place test numbers, but is rare as in most communities a 1- indicates a long-d... | wikipedia |
wiki_27_chunk_22 | Automatic number announcement circuit | Local numbers These are regular numbers within valid local exchanges in the communities listed. Many belong to competitive local exchange carriers or independent telephone company exchanges. Supposedly, a test call gives an automatic announcement. Some may announce caller ID instead of ANI; these will incur a toll (if ... | wikipedia |
wiki_27_chunk_23 | Automatic number announcement circuit | N-1-1 numbers These are mostly dead, except in rare locations where some of the standard information numbers (2-1-1 through 8-1-1) have not yet been assigned to their usual function. The corresponding test number will stop working when 2-1-1 becomes community info, 3-1-1 becomes city or county hall, 4-1-1 becomes direc... | wikipedia |
wiki_27_chunk_24 | Automatic number announcement circuit | US toll-free
Please note that it is always preferable to call the local ANAC; only if the local ANAC number can not be called is it advisable to call a toll-free ANAC number. It is also preferable to call an open ANAC rather than the password-protected one given below.
1-800-444-4444 MCI ANAC (no input needed) (Verifi... | wikipedia |
wiki_27_chunk_25 | Automatic number announcement circuit | The below numbers are not true ANAC numbers; however, they do read back one's phone number. These numbers provide valuable services to the customers they serve; it is, therefore, inadvisable to misuse them.
1-800-225-5313 BANK OF SOUTH SIDE VIRGINIA, FRAUD DEPT (press 1) (Verified March 9, 2021 from US and Canada)
1-... | wikipedia |
wiki_27_chunk_26 | Automatic number announcement circuit | Canada
The current use of exchange prefixes for each area code is listed by CNAC; if an exchange changes from "plant test" to reclaimed or active, any former test numbers with the associated prefix are invalidated. Commonly-used test numbers for major carriers (dialled with any of the local area codes, as 10 digits) in... | wikipedia |
wiki_27_chunk_27 | Automatic number announcement circuit | These numbers are carrier specific and may be blocked from some individual payphones. Additional plant test codes may be in use locally in some areas:
403: 555-0311 Alberta (GroupTel - may work in other parts of Canada - untested)
604: 1116 British Columbia (Telus)
604: 1211 British Columbia (Telus)
819: 959-1135 Most ... | wikipedia |
wiki_27_chunk_28 | Automatic number announcement circuit | Occasionally, a number in an existing, standard local exchange in the area is used. These will incur a toll (and might not work) outside their home area. Some may be announcing caller ID, which is not the same as ANI. As standard local calls, they are not accessible from ADSL "dry loop", inbound-only or unsubscribed li... | wikipedia |
wiki_27_chunk_29 | Automatic number announcement circuit | In Bell Canada territory, +1-areacode-320 was formerly reserved for 320-xxxx test numbers; these were moved to the 958-xxxx range and 320-xxxx reclaimed for use as a standard exchange. 958-ANAC was in use by Bell Canada (416 Toronto) but looks to have been replaced by 416-958-2580. The use of N11 prefixes (such as 3-1-... | wikipedia |
wiki_27_chunk_30 | Automatic number announcement circuit | Some lists erroneously mention 1-555-1313 as ANAC (506 New Brunswick). The purpose of +1-areacode-555-1313, a pay-per-use "name that number" reverse lookup information service introduced in the mid-1990s, differs from ANAC. ANAC announces the caller's own number; the reverse lookup gives the directory name for a listed... | wikipedia |
wiki_27_chunk_31 | Automatic number announcement circuit | United Kingdom
17070, Openreach Linetest Facilities
020 8759 9036, same recording as 17070 but useful on LLU and cable lines where 17070's functionality is limited. Not usable on mobiles.
0808 170 7788, it does have a long introductory message, but it is useful on COCOTs which have 17070 barred.
18866, Same recording a... | wikipedia |
wiki_27_chunk_32 | Automatic number announcement circuit | Ireland
19 9000
This service announces the line number on all Eir lines, including lines where calls are carried by another provider using carrier preselect.
The same number also works for lines provided by local-loop unbundling. The number is called out without the leading 0. For example, 021 XXX XXXX is read back as ... | wikipedia |
wiki_27_chunk_33 | Automatic number announcement circuit | There is also an extended ANAC service for identifying which carrier handles calls. Dialling these numbers will cause the local switch to announce which carrier the calls are being routed through for a specific category of calls.
19 800 - International calls
19 822 - Local calls
19 801 - Calls to other parts of the Rep... | wikipedia |
wiki_27_chunk_34 | Automatic number announcement circuit | New Zealand
1956 or 0 (8) 320-1231 area code and number
1957 or 0 (8) 320-1234 local number Subscribers may also dial +64 (8) 320-1231 from overseas to test if the (CPN) Caller ID number is being passed on to New Zealand; this should announce the area code and local number as it appears on call display. | wikipedia |
wiki_27_chunk_35 | Automatic number announcement circuit | South Africa
+27 21 405 9111 Cape Town ANAC
+27 21 405 9116 Cape Town ANAC with callback
+27 10 130 0999 Johannesburg ANAC
+27 31 120 0999 Durban ANAC
+27 87 180 0999 VoIP ANAC
+27 84 190 0048 Mobile ANAC See also
Plant test number
Ringback number References External links
Automatic number announcement circuit numbe... | wikipedia |
wiki_28_chunk_0 | Associative algebra | In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field. The addition and multiplication operations together give A the structure of a ring; the addition and scalar multip... | wikipedia |
wiki_28_chunk_1 | Associative algebra | A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarif... | wikipedia |
wiki_28_chunk_2 | Associative algebra | Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative identity. For examples of this concept, if S is any ring with center C, then S is an a... | wikipedia |
wiki_28_chunk_3 | Associative algebra | Let R be a commutative ring (so R could be a field). An associative R-algebra (or more simply, an R-algebra) is a ring
that is also an R-module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies for all r in R and x, y in the alg... | wikipedia |
wiki_28_chunk_4 | Associative algebra | Equivalently, an associative algebra A is a ring together with a ring homomorphism from R to the center of A. If f is such a homomorphism, the scalar multiplication is (here the multiplication is the ring multiplication); if the scalar multiplication is given, the ring homomorphism is given by (See also below). Ever... | wikipedia |
wiki_28_chunk_5 | Associative algebra | As a monoid object in the category of modules
The definition is equivalent to saying that a unital associative R-algebra is a monoid object in R-Mod (the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative algebra is obtained b... | wikipedia |
wiki_28_chunk_6 | Associative algebra | Pushing this idea further, some authors have introduced a "generalized ring" as a monoid object in some other category that behaves like the category of modules. Indeed, this reinterpretation allows one to avoid making an explicit reference to elements of an algebra A. For example, the associativity can be expressed as... | wikipedia |
wiki_28_chunk_7 | Associative algebra | From ring homomorphisms
An associative algebra amounts to a ring homomorphism whose image lies in the center. Indeed, starting with a ring A and a ring homomorphism whose image lies in the center of A, we can make A an R-algebra by defining for all r ∈ R and x ∈ A. If A is an R-algebra, taking x = 1, the same formula ... | wikipedia |
wiki_28_chunk_8 | Associative algebra | If a ring is commutative then it equals its center, so that a commutative R-algebra can be defined simply as a commutative ring A together with a commutative ring homomorphism . | wikipedia |
wiki_28_chunk_9 | Associative algebra | The ring homomorphism η appearing in the above is often called a structure map. In the commutative case, one can consider the category whose objects are ring homomorphisms R → A; i.e., commutative R-algebras and whose morphisms are ring homomorphisms A → A that are under R; i.e., R → A → A is R → A (i.e., the coslice c... | wikipedia |
wiki_28_chunk_10 | Associative algebra | How to weaken the commutativity assumption is a subject matter of noncommutative algebraic geometry and, more recently, of derived algebraic geometry. See also: generic matrix ring. Algebra homomorphisms A homomorphism between two R-algebras is an R-linear ring homomorphism. Explicitly, is an associative algebra homom... | wikipedia |
wiki_28_chunk_11 | Associative algebra | The subcategory of commutative R-algebras can be characterized as the coslice category R/CRing where CRing is the category of commutative rings. Examples The most basic example is a ring itself; it is an algebra over its center or any subring lying in the center. In particular, any commutative ring is an algebra over a... | wikipedia |
wiki_28_chunk_12 | Associative algebra | Any ring A can be considered as a Z-algebra. The unique ring homomorphism from Z to A is determined by the fact that it must send 1 to the identity in A. Therefore, rings and Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent.
Any ring of characteristic n is a (Z/nZ)-al... | wikipedia |
wiki_28_chunk_13 | Associative algebra | Representation theory | wikipedia |
wiki_28_chunk_14 | Associative algebra | The universal enveloping algebra of a Lie algebra is an associative algebra that can be used to study the given Lie algebra.
If G is a group and R is a commutative ring, the set of all functions from G to R with finite support form an R-algebra with the convolution as multiplication. It is called the group algebra of ... | wikipedia |
wiki_28_chunk_15 | Associative algebra | Analysis Given any Banach space X, the continuous linear operators A : X → X form an associative algebra (using composition of operators as multiplication); this is a Banach algebra.
Given any topological space X, the continuous real- or complex-valued functions on X form a real or complex associative algebra; here th... | wikipedia |
wiki_28_chunk_16 | Associative algebra | Geometry and combinatorics
The Clifford algebras, which are useful in geometry and physics.
Incidence algebras of locally finite partially ordered sets are associative algebras considered in combinatorics. | wikipedia |
wiki_28_chunk_17 | Associative algebra | Constructions
Subalgebras A subalgebra of an R-algebra A is a subset of A which is both a subring and a submodule of A. That is, it must be closed under addition, ring multiplication, scalar multiplication, and it must contain the identity element of A.
Quotient algebras Let A be an R-algebra. Any ring-theoretic ideal ... | wikipedia |
wiki_28_chunk_18 | Associative algebra | Separable algebra Let A be an algebra over a commutative ring R. Then the algebra A is a right module over with the action . Then, by definition, A is said to separable if the multiplication map splits as an -linear map, where is an -module by . Equivalently,
is separable if it is a projective module over ; thus, t... | wikipedia |
wiki_28_chunk_19 | Associative algebra | Commutative case
As A is Artinian, if it is commutative, then it is a finite product of Artinian local rings whose residue fields are algebras over the base field k. Now, a reduced Artinian local ring is a field and thus the following are equivalent
is separable.
is reduced, where is some algebraic closure of k.
... | wikipedia |
wiki_28_chunk_20 | Associative algebra | Noncommutative case
Since a simple Artinian ring is a (full) matrix ring over a division ring, if A is a simple algebra, then A is a (full) matrix algebra over a division algebra D over k; i.e., . More generally, if A is a semisimple algebra, then it is a finite product of matrix algebras (over various division k-alge... | wikipedia |
wiki_28_chunk_21 | Associative algebra | The fact that A is Artinian simplifies the notion of a Jacobson radical; for an Artinian ring, the Jacobson radical of A is the intersection of all (two-sided) maximal ideals (in contrast, in general, a Jacobson radical is the intersection of all left maximal ideals or the intersection of all right maximal ideals.) | wikipedia |
wiki_28_chunk_22 | Associative algebra | The Wedderburn principal theorem states: for a finite-dimensional algebra A with a nilpotent ideal I, if the projective dimension of as an -module is at most one, then the natural surjection splits; i.e., contains a subalgebra such that is an isomorphism. Taking I to be the Jacobson radical, the theorem says in pa... | wikipedia |
wiki_28_chunk_23 | Associative algebra | Let R be a Noetherian integral domain with field of fractions K (for example, they can be ). A lattice L in a finite-dimensional K-vector space V is a finitely generated R-submodule of V that spans V; in other words, . Let be a finite-dimensional K-algebra. An order in is an R-subalgebra that is a lattice. In general... | wikipedia |
wiki_28_chunk_24 | Associative algebra | A maximal order is an order that is maximal among all the orders. Related concepts
Coalgebras | wikipedia |
wiki_28_chunk_25 | Associative algebra | An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a... | wikipedia |
wiki_28_chunk_26 | Associative algebra | There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed above. Representations | wikipedia |
wiki_28_chunk_27 | Associative algebra | A representation of an algebra A is an algebra homomorphism ρ : A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being an algebra homomorphism means that ρ preserves the multiplicative operation (that is, ρ(xy) = ρ(x)ρ(y) for all x and y in A), and that ρ sends the uni... | wikipedia |
wiki_28_chunk_28 | Associative algebra | If A and B are two algebras, and ρ : A → End(V) and τ : B → End(W) are two representations, then there is a (canonical) representation A B → End(V W) of the tensor product algebra A B on the vector space V W. However, there is no natural way of defining a tensor product of two representations of a single associativ... | wikipedia |
wiki_28_chunk_29 | Associative algebra | Motivation for a Hopf algebra
Consider, for example, two representations and . One might try to form a tensor product representation according to how it acts on the product vector space, so that However, such a map would not be linear, since one would have for k ∈ K. One can rescue this attempt and restore linearity... | wikipedia |
wiki_28_chunk_30 | Associative algebra | Such a homomorphism Δ is called a comultiplication if it satisfies certain axioms. The resulting structure is called a bialgebra. To be consistent with the definitions of the associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A Hopf... | wikipedia |
wiki_28_chunk_31 | Associative algebra | Motivation for a Lie algebra One can try to be more clever in defining a tensor product. Consider, for example, so that the action on the tensor product space is given by . This map is clearly linear in x, and so it does not have the problem of the earlier definition. However, it fails to preserve multiplication: . Bu... | wikipedia |
wiki_28_chunk_32 | Associative algebra | Non-unital algebras Some authors use the term "associative algebra" to refer to structures which do not necessarily have a multiplicative identity, and hence consider homomorphisms which are not necessarily unital. One example of a non-unital associative algebra is given by the set of all functions f: R → R' whose limi... | wikipedia |
wiki_28_chunk_33 | Associative algebra | See also
Abstract algebra
Algebraic structure
Algebra over a field
Sheaf of algebras, a sort of an algebra over a ringed space Notes References Nathan Jacobson, Structure of Rings
James Byrnie Shaw (1907) A Synopsis of Linear Associative Algebra, link from Cornell University Historical Math Monographs.
Ross Stre... | wikipedia |
wiki_29_chunk_0 | Algebraic extension | In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental. For example, ... | wikipedia |
wiki_29_chunk_1 | Algebraic extension | All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraic numbers is an infinite algebraic extension of the rational numbers. | wikipedia |
wiki_29_chunk_2 | Algebraic extension | Let E be an extension field of K, and a ∈ E. If a is algebraic over K, then K(a), the set of all polynomials in a with coefficients in K, is not only a ring but a field: K(a) is an algebraic extension of K which has finite degree over K. The converse is not true. Q[π] and Q[e] are fields but π and e are transcendental ... | wikipedia |
wiki_29_chunk_3 | Algebraic extension | An algebraically closed field F has no proper algebraic extensions, that is, no algebraic extensions E with F < E. An example is the field of complex numbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but proving this in general requires some form of the axiom ... | wikipedia |
wiki_29_chunk_4 | Algebraic extension | Properties
The class of algebraic extensions forms a distinguished class of field extensions, that is, the following three properties hold:
If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.
If E and F are algebraic extensions of K in a common overfield C, ... | wikipedia |
wiki_29_chunk_5 | Algebraic extension | This fact, together with Zorn's lemma (applied to an appropriately chosen poset), establishes the existence of algebraic closures. Generalizations Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of M into N is called an algebraic extension if for every x in N there is a fo... | wikipedia |
wiki_29_chunk_6 | Algebraic extension | is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The Galois group of N over M can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case. See also
In... | wikipedia |
wiki_30_chunk_0 | List of artificial intelligence projects | The following is a list of current and past, non-classified notable artificial intelligence projects. Specialized projects Brain-inspired
Blue Brain Project, an attempt to create a synthetic brain by reverse-engineering the mammalian brain down to the molecular level.
Google Brain A deep learning project part of Goog... | wikipedia |
wiki_30_chunk_1 | List of artificial intelligence projects | 4CAPS, developed at Carnegie Mellon University under Marcel A. Just
ACT-R, developed at Carnegie Mellon University under John R. Anderson.
AIXI, Universal Artificial Intelligence developed by Marcus Hutter at IDSIA and ANU.
CALO, a DARPA-funded, 25-institution effort to integrate many artificial intelligence approac... | wikipedia |
wiki_30_chunk_2 | List of artificial intelligence projects | Games
AlphaGo, software developed by Google that plays the Chinese board game Go.
Chinook, a computer program that plays English draughts; the first to win the world champion title in the competition against humans.
Deep Blue, a chess-playing computer developed by IBM which beat Garry Kasparov in 1997.
FreeHAL, a s... | wikipedia |
wiki_30_chunk_3 | List of artificial intelligence projects | Internet activism
Serenata de Amor, project for the analysis of public expenditures and detect discrepancies. | wikipedia |
wiki_30_chunk_4 | List of artificial intelligence projects | Knowledge and reasoning
Braina, an intelligent personal assistant application with a voice interface for Windows OS.
Cyc, an attempt to assemble an ontology and database of everyday knowledge, enabling human-like reasoning.
Eurisko, a language by Douglas Lenat for solving problems which consists of heuristics, inclu... | wikipedia |
wiki_30_chunk_5 | List of artificial intelligence projects | Motion and manipulation
AIBO, the robot pet for the home, grew out of Sony's Computer Science Laboratory (CSL).
Cog, a robot developed by MIT to study theories of cognitive science and artificial intelligence, now discontinued. Music
Melomics, a bioinspired technology for music composition and synthesization of mus... | wikipedia |
wiki_30_chunk_6 | List of artificial intelligence projects | AIML, an XML dialect for creating natural language software agents.
Apache Lucene, a high-performance, full-featured text search engine library written entirely in Java.
Apache OpenNLP, a machine learning based toolkit for the processing of natural language text. It supports the most common NLP tasks, such as tokeniz... | wikipedia |
wiki_30_chunk_7 | List of artificial intelligence projects | Other
1 the Road, the first novel marketed by an AI.
Synthetic Environment for Analysis and Simulations (SEAS), a model of the real world used by Homeland security and the United States Department of Defense that uses simulation and AI to predict and evaluate future events and courses of action. Multipurpose projects | wikipedia |
wiki_30_chunk_8 | List of artificial intelligence projects | Software libraries
Apache Mahout, a library of scalable machine learning algorithms.
Deeplearning4j, an open-source, distributed deep learning framework written for the JVM.
Keras, a high level open-source software library for machine learning (works on top of other libraries).
Microsoft Cognitive Toolkit (previou... | wikipedia |
wiki_30_chunk_9 | List of artificial intelligence projects | GUI frameworks
Neural Designer, a commercial deep learning tool for predictive analytics.
Neuroph, a Java neural network framework.
OpenCog, a GPL-licensed framework for artificial intelligence written in C++, Python and Scheme.
PolyAnalyst: A commercial tool for data mining, text mining, and knowledge management.
... | wikipedia |
wiki_30_chunk_10 | List of artificial intelligence projects | Cloud services
Data Applied, a web based data mining environment.
Grok, a service that ingests data streams and creates actionable predictions in real time.
Watson, a pilot service by IBM to uncover and share data-driven insights, and to spur cognitive applications. See also
Comparison of cognitive architectures
C... | wikipedia |
wiki_31_chunk_0 | Analytic geometry | In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is th... | wikipedia |
wiki_31_chunk_1 | Analytic geometry | Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply:... | wikipedia |
wiki_31_chunk_2 | Analytic geometry | Ancient Greece
The Greek mathematician Menaechmus solved problems and proved theorems by using a method that had a strong resemblance to the use of coordinates and it has sometimes been maintained that he had introduced analytic geometry. | wikipedia |
wiki_31_chunk_3 | Analytic geometry | Apollonius of Perga, in On Determinate Section, dealt with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. Apollonius in the Conics further developed a method that is so similar to analytic geometry that his... | wikipedia |
wiki_31_chunk_4 | Analytic geometry | Persia
The 11th-century Persian mathematician Omar Khayyam saw a strong relationship between geometry and algebra and was moving in the right direction when he helped close the gap between numerical and geometric algebra with his geometric solution of the general cubic equations, but the decisive step came later with D... | wikipedia |
wiki_31_chunk_5 | Analytic geometry | Western Europe Analytic geometry was independently invented by René Descartes and Pierre de Fermat, although Descartes is sometimes given sole credit. Cartesian geometry, the alternative term used for analytic geometry, is named after Descartes. Descartes made significant progress with the methods in an essay titled La... | wikipedia |
wiki_31_chunk_6 | Analytic geometry | La Geometrie, written in his native French tongue, and its philosophical principles, provided a foundation for calculus in Europe. Initially the work was not well received, due, in part, to the many gaps in arguments and complicated equations. Only after the translation into Latin and the addition of commentary by van ... | wikipedia |
wiki_31_chunk_7 | Analytic geometry | Pierre de Fermat also pioneered the development of analytic geometry. Although not published in his lifetime, a manuscript form of Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci) was circulating in Paris in 1637, just prior to the publication of Descartes' Discourse. Clearly written and well r... | wikipedia |
wiki_31_chunk_8 | Analytic geometry | Coordinates In analytic geometry, the plane is given a coordinate system, by which every point has a pair of real number coordinates. Similarly, Euclidean space is given coordinates where every point has three coordinates. The value of the coordinates depends on the choice of the initial point of origin. There are a va... | wikipedia |
wiki_31_chunk_9 | Analytic geometry | The most common coordinate system to use is the Cartesian coordinate system, where each point has an x-coordinate representing its horizontal position, and a y-coordinate representing its vertical position. These are typically written as an ordered pair (x, y). This system can also be used for three-dimensional geome... | wikipedia |
wiki_31_chunk_10 | Analytic geometry | In polar coordinates, every point of the plane is represented by its distance r from the origin and its angle θ, with θ normally measured counterclockwise from the positive x-axis. Using this notation, points are typically written as an ordered pair (r, θ). One may transform back and forth between two-dimensional Cart... | wikipedia |
wiki_31_chunk_11 | Analytic geometry | In cylindrical coordinates, every point of space is represented by its height z, its radius r from the z-axis and the angle θ its projection on the xy-plane makes with respect to the horizontal axis. Spherical coordinates (in a space) In spherical coordinates, every point in space is represented by its distance ρ from ... | wikipedia |
wiki_31_chunk_12 | Analytic geometry | In analytic geometry, any equation involving the coordinates specifies a subset of the plane, namely the solution set for the equation, or locus. For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal. These points form a line, and y = x is ... | wikipedia |
wiki_31_chunk_13 | Analytic geometry | Usually, a single equation corresponds to a curve on the plane. This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x2 + y2 = 0 specifies only the single point (0, 0). In three dimensions, a single equation usually gives a surface, and a curve must be specified as the ... | wikipedia |
wiki_31_chunk_14 | Analytic geometry | Lines in a Cartesian plane, or more generally, in affine coordinates, can be described algebraically by linear equations. In two dimensions, the equation for non-vertical lines is often given in the slope-intercept form: where:
m is the slope or gradient of the line.
b is the y-intercept of the line.
x is the indepe... | wikipedia |
wiki_31_chunk_15 | Analytic geometry | In a manner analogous to the way lines in a two-dimensional space are described using a point-slope form for their equations, planes in a three dimensional space have a natural description using a point in the plane and a vector orthogonal to it (the normal vector) to indicate its "inclination". | wikipedia |
wiki_31_chunk_16 | Analytic geometry | Specifically, let be the position vector of some point , and let be a nonzero vector. The plane determined by this point and vector consists of those points , with position vector , such that the vector drawn from to is perpendicular to . Recalling that two vectors are perpendicular if and only if their dot product... | wikipedia |
wiki_31_chunk_17 | Analytic geometry | Conversely, it is easily shown that if a, b, c and d are constants and a, b, and c are not all zero, then the graph of the equation is a plane having the vector as a normal. This familiar equation for a plane is called the general form of the equation of the plane. In three dimensions, lines can not be described by a ... | wikipedia |
wiki_31_chunk_18 | Analytic geometry | where:
x, y, and z are all functions of the independent variable t which ranges over the real numbers.
(x0, y0, z0) is any point on the line.
a, b, and c are related to the slope of the line, such that the vector (a, b, c) is parallel to the line. Conic sections In the Cartesian coordinate system, the graph of a qua... | wikipedia |
wiki_31_chunk_19 | Analytic geometry | As scaling all six constants yields the same locus of zeros, one can consider conics as points in the five-dimensional projective space The conic sections described by this equation can be classified using the discriminant If the conic is non-degenerate, then:
if , the equation represents an ellipse;
if and , the eq... | wikipedia |
wiki_31_chunk_20 | Analytic geometry | A quadric, or quadric surface, is a 2-dimensional surface in 3-dimensional space defined as the locus of zeros of a quadratic polynomial. In coordinates , the general quadric is defined by the algebraic equation Quadric surfaces include ellipsoids (including the sphere), paraboloids, hyperboloids, cylinders, cones, and... | wikipedia |
wiki_31_chunk_21 | Analytic geometry | In analytic geometry, geometric notions such as distance and angle measure are defined using formulas. These definitions are designed to be consistent with the underlying Euclidean geometry. For example, using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the ... | wikipedia |
wiki_31_chunk_22 | Analytic geometry | while the angle between two vectors is given by the dot product. The dot product of two Euclidean vectors A and B is defined by where θ is the angle between A and B. Transformations Transformations are applied to a parent function to turn it into a new function with similar characteristics. The graph of is changed by ... | wikipedia |
wiki_31_chunk_23 | Analytic geometry | There are other standard transformation not typically studied in elementary analytic geometry because the transformations change the shape of objects in ways not usually considered. Skewing is an example of a transformation not usually considered.
For more information, consult the Wikipedia article on affine transform... | wikipedia |
wiki_31_chunk_24 | Analytic geometry | For example, the parent function has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant. In general, if , then it can be transformed into . In the new ... | wikipedia |
wiki_31_chunk_25 | Analytic geometry | Transformations can be applied to any geometric equation whether or not the equation represents a function.
Transformations can be considered as individual transactions or in combinations. Suppose that is a relation in the plane. For example, is the relation that describes the unit circle. Finding intersections of g... | wikipedia |
wiki_31_chunk_26 | Analytic geometry | For example, might be the circle with radius 1 and center : and might be the circle with radius 1 and center . The intersection of these two circles is the collection of points which make both equations true. Does the point make both equations true? Using for , the equation for becomes or which is true, so... | wikipedia |
wiki_31_chunk_27 | Analytic geometry | Substitution: Solve the first equation for in terms of and then substitute the expression for into the second equation: We then substitute this value for into the other equation and proceed to solve for : Next, we place this value of in either of the original equations and solve for : So our intersection has two ... | wikipedia |
wiki_31_chunk_28 | Analytic geometry | Elimination: Add (or subtract) a multiple of one equation to the other equation so that one of the variables is eliminated. For our current example, if we subtract the first equation from the second we get . The in the first equation is subtracted from the in the second equation leaving no term. The variable has b... | wikipedia |
wiki_31_chunk_29 | Analytic geometry | Finding intercepts One type of intersection which is widely studied is the intersection of a geometric object with the and coordinate axes. The intersection of a geometric object and the -axis is called the -intercept of the object.
The intersection of a geometric object and the -axis is called the -intercept of the ... | wikipedia |
wiki_31_chunk_30 | Analytic geometry | In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve at a point on the cu... | wikipedia |
wiki_31_chunk_31 | Analytic geometry | As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane tha... | wikipedia |
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