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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 used.
802: 1-700-222-2222 Vermont
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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-distance trunk call.
312: 1-200-555-1212 Chicago, Illinois
312: 1-200-8825 Chicago, Illinois (Last Four Change Rapidly)
708: 1-200-555-1212 Chicago/Elgin, Illinois
708: 1-200-8825 Chicago/Elgin, Illinois (Last Four Change Rapidly)
906: 1-200-222-2222 Marquette/Sault Ste. Marie, Michigan
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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 they work at all) for calls outside their home area. These are unverified; there is a risk these will be reassigned to individual subscribers:
209: 888-6945 Stockton, California (Reads ANAC and CNAM) (out of service, returns false answer supervision 2014)
334: 557-2311 Montgomery, Alabama (CLEC) (no answer, 2014)
334: 557-2411 Montgomery, Alabama (CLEC) (busy/no answer, 2013)
419: 353-1206 Bowling Green, Ohio (Frontier) (Verified April 2018)
503: 266-1021 Canby-Needy, Oregon (Canby Telephone Association, independent, returns ANI) (Verified November 2021, but does not supervise; will not work via Google Voice)
503: 697-0053 Clackamas/Lake Oswego, Oregon (Qwest, returns Caller ID) (Verified May 2018, but will only work when calling from the Centurylink Lake Oswego exchange)
505: 243-0049 Albuquerque, New Mexico (Quest, returns Caller ID) (Verified November 2021)
508: 200-5555 Worcester, Massachusetts (Dial 7 digits—City VZ landlines only?) (Verified September 2019 via Worcester 5ESS)
515: 280-1241 Des Moines, Iowa (Qwest, returns Caller ID, additional test menu) (Verified November 2021)
541: 330-0024 Bend, Oregon (Qwest) (Verified November 2021)
561: 364-1781 Boynton Beach, Florida (Bellsouth, West Palm Beach/Jupiter/Juno Beach, returns Caller ID) (Verified November 2021)
570: 674-0086 Dallas, Pennsylvania (Frontier/Commonwealth Telephone) (Verified November 2021)
602: 253-0227 Phoenix, Arizona (Qwest) (No answer, February 2018) (Reassigned to customer December 2019)
608: 884-1206 Edgerton, Wisconsin (Frontier North, returns Caller ID) (Verified November 2021)
702: 889-4579 Las Vegas, Nevada (CenturyLink) (no answer, 2014) (No answer, February 2018)
712: 563-1206 Audubon, Iowa (Windstream) (Verified November 2021)
747: 268-1966 La Cañada Flintridge, California (FPPTN/California Bell) (Verified November 2021)
806: 863-9999 Woodrow, Texas (South Plains Telephone Co-Op) (Verified May 2018, but does not supervise; will not work via Google Voice)
812: 462-1218 Terre Haute, Indiana (Frontier North) (no answer, 2014)
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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 directory info or 5-1-1 provides highway conditions, for instance. With rare exception, one should not expect these numbers to be valid.
402: 311 Lincoln, Nebraska (Verified 2016)
410: 811 Annapolis/Baltimore, Maryland
419: 311 Toledo, Ohio
434: 311 Danville, Virginia (Verizon)
501: 511 Arkansas
503: 611 Portland, Oregon
515: 811 Des Moines, Iowa
540: 311 Roanoke, Virginia (GTE) (Verified 2016)
703: 811 Alexandria/Arlington/Roanoke, Virginia
713: 811 Humble, Texas
810: 311 Pontiac/Southfield/Troy, Michigan
907: 811 Alaska
908: 311-MMYY Northern New Jersey (Embarq, now CenturyTel) (MMYY is current Month/Year)
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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) (Verified January 25, 2022 from Raleigh NC) (not reachable from Canada)
1-800-437-7950 MCI ANAC (no input needed) (Verified January 25, 2022 from Raleigh NC) (Verified March 9, 2021 from Canada)
1-800-223-1104 PASSWORD-PROTECTED ANAC 195632 (Verified June 20, 2020)
1-855-343-2255 TracFone ANAC (press 1 for English) (January 25, 2022 from Raleigh NC)
1-855-227-3250 Consolidated Communications ANAC (January 25, 2022 from Raleigh NC)
1-877-521-2311 CenturyLink (Verified March 9, 2021 from US & Canada)
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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)
1-800-225-5214 NATIONAL CAPITAL BANK OF WASHINGTON, FRAUD DEPT (press 1) (Verified April 2018)
1-800-444-2222 MCI customer service (business) (Verified March 9, 2021 from US and Canada)
1-800-444-3333 MCI customer service (residential) (Verified March 9, 2021 from US and Canada)
1-800-314-4258, 1-800-444-0800, 1-800-444-4444, 1-800-950-5555 and 1-888-624-9266 (press 2 at prompt) are also often listed as MCI customer service.
1-800-660-2626, 1- 800-288-2020 AT&T Customer Service (Verified April 2018)
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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) include:
555-0311 Rogers (403 Alberta, 519 613 Ontario)
958-2580 Bell Canada (519 613 416 705 905 Ontario, 450 418 438 514 579 581 819 873 Quebec)
958-6111 Telus landline (403 780 Alberta, 250 BC)
959-4444 Manitoba Telecom Services (204 MB) (959 is used since 958 is a regular Winnipeg exchange, not a test prefix)
958-9999 Bell Aliant (506 NB, 709 NL)
958-2222 Eastlink
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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 of Outaouais region (Bell Canada)
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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 lines:
403: 705-0311 Calgary, Alberta (Allstream - gives "call cannot be completed as dialled" in other parts of Canada, identifying as Allstream, active November 2019)
416: 477-0034 Toronto, Ontario (Fibernetics - Verified January 2021)
416: 477-0035 Toronto, Ontario (Fibernetics - verified January 2021; this number allows you to leave a message for reasons not yet determined)
416: 981-0001 Toronto, Ontario (verified July 2010) (busy, September 2015)
418: 380-0099 Quebec City (Vidéotron - verified 2017)
905: 310-3789 Mississauga, ON (Now no longer includes loop line or ringback. In NPAs where Bell Canada is incumbent, 310-xxxx is assigned as a pseudo-tollfree exchange which may be called at local call rates from an entire area code.)
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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 3-1-1) for test numbers is also deprecated as 3-1-1 now often reaches city hall or municipal services while 2-1-1 is local community information.
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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 telephone number input by the user. 555-1313 is one of the rare uses of the 555 exchange for other than the standard 555-1212 directory information line.
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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 as 0808 170 7788 but a shorter number to remember.
020 8180 3803, Same recording as 0808 170 7788. These numbers are set up by a company offering low-charge calls in the UK, these numbers are meant to be used as a sort of operator routed through in order to qualify for these cheap calls. However, if the phone these numbers are dialled from is already registered with this company it will not announce the number.
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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 "21 XXX XXXX".
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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 Republic of Ireland, Irish mobile numbers and to landlines in Northern Ireland. Israel
*110 (Not working in all networks) Australia
1800 801 920
127 22 123(Telstra landlines only)
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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.
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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 numbers and recordings Telephone numbers
Telephony signals
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Associative algebra
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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 multiplication operations together give A the structure of a vector space over K. In this article we will also use the term [[algebra over a field|K-algebra]] to mean an associative algebra over the field K. A standard first example of a K-algebra is a ring of square matrices over a field K, with the usual matrix multiplication.
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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 clarification. In some areas of mathematics this assumption is not made, and we will call such structures non-unital associative algebras. We will also assume that all rings are unital, and all ring homomorphisms are unital.
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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 associative C-algebra. Definition
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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 algebra. (This definition implies that the algebra is unital, since rings are supposed to have a multiplicative identity.)
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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). Every ring is an associative -algebra, where denotes the ring of the integers. A is an associative algebra that is also a commutative ring.
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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 by replacing the category of abelian groups with the category of modules.
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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 follows. By the universal property of a tensor product of modules, the multiplication (the R-bilinear map) corresponds to a unique R-linear map
.
The associativity then refers to the identity:
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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 in turn defines a ring homomorphism whose image lies in the center.
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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 .
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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 category of the category of commutative rings under R.) The prime spectrum functor Spec then determines an anti-equivalence of this category to the category of affine schemes over Spec R.
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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 homomorphism if The class of all R-algebras together with algebra homomorphisms between them form a category, sometimes denoted R-Alg.
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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 any of its subrings. Other examples abound both from algebra and other fields of mathematics. Algebra
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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)-algebra in the same way.
Given an R-module M, the endomorphism ring of M, denoted EndR(M) is an R-algebra by defining (r·φ)(x) = r·φ(x).
Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication. This coincides with the previous example when M is a finitely-generated, free R-module.
In particular, the square n-by-n matrices with entries from the field K form an associative algebra over K.
The complex numbers form a 2-dimensional commutative algebra over the real numbers.
The quaternions form a 4-dimensional associative algebra over the reals (but not an algebra over the complex numbers, since the complex numbers are not in the center of the quaternions).
The polynomials with real coefficients form a commutative algebra over the reals.
Every polynomial ring R[x1, ..., xn] is a commutative R-algebra. In fact, this is the free commutative R-algebra on the set {x1, ..., xn}.
The free R-algebra on a set E is an algebra of "polynomials" with coefficients in R and noncommuting indeterminates taken from the set E.
The tensor algebra of an R-module is naturally an associative R-algebra. The same is true for quotients such as the exterior and symmetric algebras. Categorically speaking, the functor that maps an R-module to its tensor algebra is left adjoint to the functor that sends an R-algebra to its underlying R-module (forgetting the multiplicative structure).
The following ring is used in the theory of λ-rings. Given a commutative ring A, let the set of formal power series with constant term 1. It is an abelian group with the group operation that is the multiplication of power series. It is then a ring with the multiplication, denoted by , such that determined by this condition and the ring axioms. The additive identity is 1 and the multiplicative identity is . Then has a canonical structure of a -algebra given by the ring homomorphism On the other hand, if A is a λ-ring, then there is a ring homomorphism giving a structure of an A-algebra.
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Representation theory
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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 G. The construction is the starting point for the application to the study of (discrete) groups.
If G is an algebraic group (e.g., semisimple complex Lie group), then the coordinate ring of G is the Hopf algebra A corresponding to G. Many structures of G translate to those of A.
A quiver algebra (or a path algebra) of a directed graph is the free associative algebra over a field generated by the paths in the graph.
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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 the functions are added and multiplied pointwise.
The set of semimartingales defined on the filtered probability space (Ω, F, (Ft)t ≥ 0, P) forms a ring under stochastic integration.
The Weyl algebra
An Azumaya algebra
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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.
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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 I in A is automatically an R-module since r · x = (r1A)x. This gives the quotient ring A / I the structure of an R-module and, in fact, an R-algebra. It follows that any ring homomorphic image of A is also an R-algebra.
Direct products The direct product of a family of R-algebras is the ring-theoretic direct product. This becomes an R-algebra with the obvious scalar multiplication.
Free products One can form a free product of R-algebras in a manner similar to the free product of groups. The free product is the coproduct in the category of R-algebras.
Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras for more details. Given a commutative ring R and any ring A the tensor product R ⊗Z A can be given the structure of an R-algebra by defining r · (s ⊗ a) = (rs ⊗ a). The functor which sends A to R ⊗Z A is left adjoint to the functor which sends an R-algebra to its underlying ring (forgetting the module structure). See also: Change of rings.
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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, the -projective dimension of A, sometimes called the bidimension of A, measures the failure of separability. Finite-dimensional algebra Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring.
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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.
for some n.
is the number of -algebra homomorphisms .
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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-algebras), the fact known as the Artin–Wedderburn theorem.
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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.)
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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 particular that the Jacobson radical is complemented by a semisimple algebra. The theorem is an analog of Levi's theorem for Lie algebras. Lattices and orders
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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, there are a lot fewer orders than lattices; e.g., is a lattice in but not an order (since it is not an algebra).
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A maximal order is an order that is maximal among all the orders. Related concepts
Coalgebras
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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 linear map (i. e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
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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
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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 unit of A to the unit of End(V) (that is, to the identity endomorphism of V).
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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 associative algebra in such a way that the result is still a representation of that same algebra (not of its tensor product with itself), without somehow imposing additional conditions. Here, by tensor product of representations, the usual meaning is intended: the result should be a linear representation of the same algebra on the product vector space. Imposing such additional structure typically leads to the idea of a Hopf algebra or a Lie algebra, as demonstrated below.
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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 by imposing additional structure, by defining an algebra homomorphism Δ: A → A ⊗ A, and defining the tensor product representation as
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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 algebra is a bialgebra with an additional piece of structure (the so-called antipode), which allows not only to define the tensor product of two representations, but also the Hom module of two representations (again, similarly to how it is done in the representation theory of groups).
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Associative algebra
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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: . But, in general, this does not equal . This shows that this definition of a tensor product is too naive; the obvious fix is to define it such that it is antisymmetric, so that the middle two terms cancel. This leads to the concept of a Lie algebra.
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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 limit as x nears infinity is zero. Another example is the vector space of continuous periodic functions, together with the convolution product.
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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 Street (1998) Quantum Groups: an entrée to modern algebra'', an overview of index-free notation. Algebras
Algebraic geometry
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Algebraic extension
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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, the field extension R/Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C/R and Q()/Q are algebraic, where C is the field of complex numbers.
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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.
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Algebraic extension
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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 over Q.
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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 of choice. An extension L/K is algebraic if and only if every sub K-algebra of L is a field.
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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, then the compositum EF is an algebraic extension of K.
If E is an algebraic extension of F and E > K > F then E is an algebraic extension of K. These finitary results can be generalized using transfinite induction:
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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 formula p with parameters in M, such that p(x) is true and the set
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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
Integral element
Lüroth's theorem
Galois extension
Separable extension
Normal extension Notes References Field extensions
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List of artificial intelligence projects
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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 Google X attempting to have intelligence similar or equal to human-level.
Human Brain Project
NuPIC, an open source implementation by Numenta of its cortical learning algorithm. Cognitive architectures
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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 approaches (natural language processing, speech recognition, machine vision, probabilistic logic, planning, reasoning, many forms of machine learning) into an AI assistant that learns to help manage your office environment.
CHREST, developed under Fernand Gobet at Brunel University and Peter C. Lane at the University of Hertfordshire.
CLARION, developed under Ron Sun at Rensselaer Polytechnic Institute and University of Missouri.
CoJACK, an ACT-R inspired extension to the JACK multi-agent system that adds a cognitive architecture to the agents for eliciting more realistic (human-like) behaviors in virtual environments.
Copycat, by Douglas Hofstadter and Melanie Mitchell at the Indiana University.
DUAL, developed at the New Bulgarian University under Boicho Kokinov.
FORR developed by Susan L. Epstein at The City University of New York.
IDA and LIDA, implementing Global Workspace Theory, developed under Stan Franklin at the University of Memphis.
OpenCog Prime, developed using the OpenCog Framework.
Procedural Reasoning System (PRS), developed by Michael Georgeff and Amy L. Lansky at SRI International.
Psi-Theory developed under Dietrich Dörner at the Otto-Friedrich University in Bamberg, Germany.
R-CAST, developed at the Pennsylvania State University.
Soar, developed under Allen Newell and John Laird at Carnegie Mellon University and the University of Michigan.
Society of mind and its successor the Emotion machine proposed by Marvin Minsky.
Subsumption architectures, developed e.g. by Rodney Brooks (though it could be argued whether they are cognitive).
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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 self-learning conversation simulator (chatterbot) which uses semantic nets to organize its knowledge to imitate a very close human behavior within conversations.
Halite, an artificial intelligence programming competition created by Two Sigma.
Libratus, a poker AI that beat world-class poker players in 2017, intended to be generalisable to other applications.
Quick, Draw!, an online game developed by Google that challenges players to draw a picture of an object or idea and then uses a neural network to guess what the drawing is.
Stockfish AI, an open source chess engine currently ranked the highest in many computer chess rankings.
TD-Gammon, a program that learned to play world-class backgammon partly by playing against itself (temporal difference learning with neural networks).
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Internet activism
Serenata de Amor, project for the analysis of public expenditures and detect discrepancies.
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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, including some for how to use and change its heuristics.
Google Now, an intelligent personal assistant with a voice interface in Google's Android and Apple Inc.'s iOS, as well as Google Chrome web browser on personal computers.
Holmes a new AI created by Wipro.
Microsoft Cortana, an intelligent personal assistant with a voice interface in Microsoft's various Windows 10 editions.
Mycin, an early medical expert system.
Open Mind Common Sense, a project based at the MIT Media Lab to build a large common sense knowledge base from online contributions.
P.A.N., a publicly available text analyzer.
Siri, an intelligent personal assistant and knowledge navigator with a voice-interface in Apple Inc.'s iOS and macOS.
SNePS, simultaneously a logic-based, frame-based, and network-based knowledge representation, reasoning, and acting system.
Viv (software), a new AI by the creators of Siri.
Wolfram Alpha, an online service that answers queries by computing the answer from structured data.
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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 music, where computers develop their own style, rather than mimic musicians. Natural language processing
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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 tokenization, sentence segmentation, part-of-speech tagging, named entity extraction, chunking and parsing.
Artificial Linguistic Internet Computer Entity (A.L.I.C.E.), an award-winning natural language processing chatterbot.
Cleverbot, successor to Jabberwacky, now with 170m lines of conversation, Deep Context, fuzziness and parallel processing. Cleverbot learns from around 2 million user interactions per month.
ELIZA, a famous 1966 computer program by Joseph Weizenbaum, which parodied person-centered therapy.
GPT-3, a 2020 language model developed by OpenAI that can produce text difficult to distinguish from that written by a human.
Jabberwacky, a chatbot by Rollo Carpenter, aiming to simulate natural human chat.
Mycroft, a free and open-source intelligent personal assistant that uses a natural language user interface.
PARRY, another early chatterbot, written in 1972 by Kenneth Colby, attempting to simulate a paranoid schizophrenic.
SHRDLU, an early natural language processing computer program developed by Terry Winograd at MIT from 1968 to 1970.
SYSTRAN, a machine translation technology by the company of the same name, used by Yahoo!, AltaVista and Google, among others.
ASR-automated speech recognization System.
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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
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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 (previously known as CNTK), an open source toolkit for building artificial neural networks.
OpenNN, a comprehensive C++ library implementing neural networks.
PyTorch, an open-source Tensor and Dynamic neural network in Python.
TensorFlow, an open-source software library for machine learning.
Theano, a Python library and optimizing compiler for manipulating and evaluating mathematical expressions, especially matrix-valued ones.
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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.
RapidMiner, an environment for machine learning and data mining, now developed commercially.
Weka, a free implementation of many machine learning algorithms in Java.
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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
Comparison of deep-learning software References External links
AI projects on GitHub
AI projects on SourceForge Artificial intelligence projects
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Analytic geometry
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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 the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
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Analytic geometry
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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: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom. History
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Analytic geometry
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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.
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Analytic geometry
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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 work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.
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Analytic geometry
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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 Descartes. Omar Khayyam is credited with identifying the foundations of algebraic geometry, and his book Treatise on Demonstrations of Problems of Algebra (1070), which laid down the principles of analytic geometry, is part of the body of Persian mathematics that was eventually transmitted to Europe. Because of his thoroughgoing geometrical approach to algebraic equations, Khayyam can be considered a precursor to Descartes in the invention of analytic geometry.
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Analytic geometry
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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 Geometrie (Geometry), one of the three accompanying essays (appendices) published in 1637 together with his Discourse on the Method for Rightly Directing One's Reason and Searching for Truth in the Sciences, commonly referred to as Discourse on Method.
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Analytic geometry
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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 Schooten in 1649 (and further work thereafter) did Descartes's masterpiece receive due recognition.
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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 received, the Introduction also laid the groundwork for analytical geometry. The key difference between Fermat's and Descartes' treatments is a matter of viewpoint: Fermat always started with an algebraic equation and then described the geometric curve that satisfied it, whereas Descartes started with geometric curves and produced their equations as one of several properties of the curves. As a consequence of this approach, Descartes had to deal with more complicated equations and he had to develop the methods to work with polynomial equations of higher degree. It was Leonhard Euler who first applied the coordinate method in a systematic study of space curves and surfaces.
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Analytic geometry
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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 variety of coordinate systems used, but the most common are the following: Cartesian coordinates (in a plane or space)
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Analytic geometry
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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 geometry, where every point in Euclidean space is represented by an ordered triple of coordinates (x, y, z). Polar coordinates (in a plane)
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Analytic geometry
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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 Cartesian and polar coordinates by using these formulae: This system may be generalized to three-dimensional space through the use of cylindrical or spherical coordinates. Cylindrical coordinates (in a space)
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Analytic geometry
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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 the origin, the angle θ its projection on the xy-plane makes with respect to the horizontal axis, and the angle φ that it makes with respect to the z-axis. The names of the angles are often reversed in physics. Equations and curves
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Analytic geometry
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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 said to be the equation for this line. In general, linear equations involving x and y specify lines, quadratic equations specify conic sections, and more complicated equations describe more complicated figures.
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Analytic geometry
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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 intersection of two surfaces (see below), or as a system of parametric equations. The equation x2 + y2 = r2 is the equation for any circle centered at the origin (0, 0) with a radius of r. Lines and planes
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Analytic geometry
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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 independent variable of the function y = f(x).
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Analytic geometry
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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".
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Analytic geometry
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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 is zero, it follows that the desired plane can be described as the set of all points such that (The dot here means a dot product, not scalar multiplication.)
Expanded this becomes which is the point-normal form of the equation of a plane. This is just a linear equation:
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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 single linear equation, so they are frequently described by parametric equations:
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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 quadratic equation in two variables is always a conic section – though it may be degenerate, and all conic sections arise in this way. The equation will be of the form
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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 equation represents a circle, which is a special case of an ellipse;
if , the equation represents a parabola;
if , the equation represents a hyperbola;
if we also have , the equation represents a rectangular hyperbola. Quadric surfaces
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Analytic geometry
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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 planes. Distance and angle
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Analytic geometry
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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 formula which can be viewed as a version of the Pythagorean theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula where m is the slope of the line. In three dimensions, distance is given by the generalization of the Pythagorean theorem:
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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 standard transformations as follows: Changing to moves the graph to the right units.
Changing to moves the graph up units.
Changing to stretches the graph horizontally by a factor of . (think of the as being dilated)
Changing to stretches the graph vertically.
Changing to and changing to rotates the graph by an angle .
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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 transformations.
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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 transformed function, is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative values, the function is reflected in the -axis. The value compresses the graph of the function horizontally if greater than 1 and stretches the function horizontally if less than 1, and like , reflects the function in the -axis when it is negative. The and values introduce translations, , vertical, and horizontal. Positive and values mean the function is translated to the positive end of its axis and negative meaning translation towards the negative end.
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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 geometric objects For two geometric objects P and Q represented by the relations and the intersection is the collection of all points which are in both relations.
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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 is in the relation . On the other hand, still using for the equation for becomes or which is false. is not in so it is not in the intersection. The intersection of and can be found by solving the simultaneous equations: Traditional methods for finding intersections include substitution and elimination.
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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 points:
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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 been eliminated. We then solve the remaining equation for , in the same way as in the substitution method: We then place this value of in either of the original equations and solve for : So our intersection has two points: For conic sections, as many as 4 points might be in the intersection.
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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 object. For the line , the parameter specifies the point where the line crosses the axis. Depending on the context, either or the point is called the -intercept. Tangents and normals Tangent lines and planes
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Analytic geometry
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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 curve if the line passes through the point on the curve and has slope where f is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
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wikipedia
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wiki_31_chunk_31
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Analytic geometry
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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 that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. Normal line and vector
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wikipedia
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