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wiki_41_chunk_17 | AIM (software) | Chat robots
AOL and various other companies supplied robots (bots) on AIM that could receive messages and send a response based on the bot's purpose. For example, bots could help with studying, like StudyBuddy. Some were made to relate to children and teenagers, like Spleak. | wikipedia |
wiki_41_chunk_18 | AIM (software) | Others gave advice. The more useful chat bots had features like the ability to play games, get sport scores, weather forecasts or financial stock information. Users were able to talk to automated chat bots that could respond to natural human language. They were primarily put into place as a marketing strategy and for u... | wikipedia |
wiki_41_chunk_19 | AIM (software) | Before the inclusions of such bots, the other bots DoorManBot and AIMOffline provided features that were provided by AOL for those who needed it. ZolaOnAOL and ZoeOnAOL were short-lived bots that ultimately retired their features in favor of SmarterChild. | wikipedia |
wiki_41_chunk_20 | AIM (software) | URI scheme
AOL Instant Messenger's installation process automatically installed an extra URI scheme ("protocol") handler into some Web browsers, so URIs beginning "aim:" could open a new AIM window with specified parameters. This was similar in function to the mailto: URI scheme, which created a new e-mail message usin... | wikipedia |
wiki_41_chunk_21 | AIM (software) | To specify a message body, the message parameter was used, so the link location would have looked like this:
aim:goim?screenname=notarealuser&message=This+is+my+message To specify an away message, the message parameter was used, so the link location would have looked like this:
aim:goaway?message=Hello,+my+name+is+Bi... | wikipedia |
wiki_41_chunk_22 | AIM (software) | To add a buddy, the addbuddy message was used, with the "screenname" parameter
aim:addbuddy?screenname=notarealuser
This type of link was commonly found on forum profiles to easily add contacts. | wikipedia |
wiki_41_chunk_23 | AIM (software) | Vulnerabilities
AIM had security weaknesses that have enabled exploits to be created that used third-party software to perform malicious acts on users' computers. Although most were relatively harmless, such as being kicked off the AIM service, others performed potentially dangerous actions, such as sending viruses. So... | wikipedia |
wiki_41_chunk_24 | AIM (software) | Users also have reported sudden additions of toolbars and advertisements from third parties in the newer version of AIM. Multiple complaints about the lack of control of third party involvement have caused many users to stop using the service. Extra features | wikipedia |
wiki_41_chunk_25 | AIM (software) | iPhone application
On March 6, 2008, during Apple Inc.'s iPhone SDK event, AOL announced that they would be releasing an AIM application for iPhone and iPod Touch users. The application was available for free from the App Store, but the company also provides a paid version, which displays no advertisements. Both were a... | wikipedia |
wiki_41_chunk_26 | AIM (software) | In 2011, AOL launched an overhaul of their Instant Messaging service. Included in the update was a brand new iOS application for iPhone and iPod Touch that incorporated all the latest features. A brand new icon was used for the application, featuring the new cursive logo for AIM. The user-interface was entirely redone ... | wikipedia |
wiki_41_chunk_27 | AIM (software) | Version 5.0.5, updated in March 2012, it supported more social stream features, much like Facebook and Twitter, as well as the ability to send voice messages up to 60 seconds long. iPad application
On April 3, 2010, Apple released the first generation iPad. Along with this newly released device AOL released the AIM app... | wikipedia |
wiki_41_chunk_28 | AIM (software) | AIM Express
AIM Express ran in a pop-up browser window. It was intended for use by people who are unwilling or unable to install a standalone application or those at computers that lack the AIM application. AIM Express supported many of the standard features included in the stand-alone client, but did not provide advan... | wikipedia |
wiki_41_chunk_29 | AIM (software) | AIM Pages
AIM Pages was a free website released in May 2006 by AOL in replacement of AIMSpace. Anyone who had an AIM user name and was at least 16 years of age could create their own web page (to display an online, dynamic profile) and share it with buddies from their AIM Buddy list. | wikipedia |
wiki_41_chunk_30 | AIM (software) | Layout
AIM Pages included links to the email and Instant Message of the owner, along with a section listing the owners "buddies", which included AIM user names. It was possible to create modules in a Module T microformat. Video hosting sites like Netflix and YouTube could be added to ones AIM Page, as well as other si... | wikipedia |
wiki_41_chunk_31 | AIM (software) | Discontinuation
By late 2007, AIM Pages had been discontinued. After AIM Pages shutdown, links to AIM Pages were redirected to AOL Lifestream, AOL's new site aimed at collecting external modules in one place, independent of AIM buddies. AOL Lifestream was shut down February 24, 2017. | wikipedia |
wiki_41_chunk_32 | AIM (software) | AIM for Mac
AOL released an all-new AIM for the Macintosh on September 29, 2008 and the final build on December 15, 2008. The redesigned AIM for Mac is a full universal binary Cocoa API application that supports both Tiger and Leopard — Mac OS X 10.4.8 (and above) or Mac OS X 10.5.3 (and above). On October 1, 2009, AOL... | wikipedia |
wiki_41_chunk_33 | AIM (software) | AIM real-time IM
This feature is available for AIM 7 and allows for a user to see what the other is typing as it is being done. It was developed and built with assistance from Trace Research and Development Centre at University of Wisconsin–Madison and Gallaudet University. The application provides visually impaired us... | wikipedia |
wiki_41_chunk_34 | AIM (software) | AIM to mobile (messaging to phone numbers)
This feature allows text messaging to a phone number (text messaging is less functional than instant messaging). Discontinued features | wikipedia |
wiki_41_chunk_35 | AIM (software) | AIM Phoneline
AIM Phoneline was a Voice over IP PC-PC, PC-Phone and Phone-to-PC service provided via the AIM application. It was also known to work with Apple's iChat Client. The service was officially closed to its customers on January 13, 2009. The closing of the free service caused the number associated with the ser... | wikipedia |
wiki_41_chunk_36 | AIM (software) | Launched on May 16, 2006, AIM Phoneline provided users the ability to have several local numbers, allowing AIM users to receive free incoming calls. The service allowed users to make calls to landlines and mobile devices through the use of a computer. The service, however, was only free for receiving and AOL charged us... | wikipedia |
wiki_41_chunk_37 | AIM (software) | AIM Call Out
AIM Call Out is a discontinued Voice over IP PC-PC, PC-Phone and Phone-to-PC service provided by AOL via its AIM application that replaced the defunct AIM Phoneline service in November 2007. It did not depend on the AIM client and could be used with only an AIM screenname via the WebConnect feature or a de... | wikipedia |
wiki_41_chunk_38 | AIM (software) | Security
On November 4, 2014, AIM scored one out of seven points on the Electronic Frontier Foundation's secure messaging scorecard. AIM received a point for encryption during transit, but lost points because communications are not encrypted with a key to which the provider has no access, i.e., the communications are n... | wikipedia |
wiki_41_chunk_39 | AIM (software) | See also
Comparison of cross-platform instant messaging clients
List of defunct instant messaging platforms References External links 1997 software Android (operating system) software
Instant Messenger
BlackBerry software
Classic Mac OS instant messaging clients
Computer-related introductions in 1997
Cross-platform s... | wikipedia |
wiki_42_chunk_0 | Ackermann function | In computability theory, the Ackermann function, named after Wilhelm Ackermann, is one of the simplest and earliest-discovered examples of a total computable function that is not primitive recursive. All primitive recursive functions are total and computable, but the Ackermann function illustrates that not all total co... | wikipedia |
wiki_42_chunk_1 | Ackermann function | Its value grows rapidly, even for small inputs. For example, is an integer of 19,729 decimal digits (equivalent to 265536−3, or 22222−3). | wikipedia |
wiki_42_chunk_2 | Ackermann function | History
In the late 1920s, the mathematicians Gabriel Sudan and Wilhelm Ackermann, students of David Hilbert, were studying the foundations of computation. Both Sudan and Ackermann are credited with discovering total computable functions (termed simply "recursive" in some references) that are not primitive recursive. S... | wikipedia |
wiki_42_chunk_3 | Ackermann function | and for p > 2 it extends these basic operations in a way that can be compared to the hyperoperations: (Aside from its historic role as a total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although not as seamlessl... | wikipedia |
wiki_42_chunk_4 | Ackermann function | In On the Infinite, David Hilbert hypothesized that the Ackermann function was not primitive recursive, but it was Ackermann, Hilbert's personal secretary and former student, who actually proved the hypothesis in his paper On Hilbert's Construction of the Real Numbers. Rózsa Péter and Raphael Robinson later developed a... | wikipedia |
wiki_42_chunk_5 | Ackermann function | In 1963 R.C. Buck based an intuitive two-variable variant on the hyperoperation sequence: Compared to most other versions Buck's function has no unessential offsets: Many other versions of Ackermann function have been investigated. Definition Definition: as m-ary function
Ackermann's original three-argument function... | wikipedia |
wiki_42_chunk_6 | Ackermann function | The Ackermann function has also been expressed in relation to the hyperoperation sequence: or, written in Knuth's up-arrow notation (extended to integer indices ): or, equivalently, in terms of Buck's function F: Definition: as iterated 1-ary function
Define as the n-th iterate of : Iteration is the process of compos... | wikipedia |
wiki_42_chunk_7 | Ackermann function | The function then becomes a sequence of unary functions, defined from iteration: As function composition is associative, the last line can as well be Computation
The recursive definition of the Ackermann function can naturally be transposed to a term rewriting system (TRS). TRS, based on 2-ary function
The definition ... | wikipedia |
wiki_42_chunk_8 | Ackermann function | WHILE stackLength <> 1
{
POP 2 elements;
PUSH 1 or 2 or 3 elements, applying the rules r1, r2, r3
} The pseudocode is published in . For example, on input , Remarks
The leftmost-innermost strategy is implemented in 225 computer languages on Rosetta Code.
For all the computation of takes no more than steps.... | wikipedia |
wiki_42_chunk_9 | Ackermann function | TRS, based on iterated 1-ary function
The definition of the iterated 1-ary Ackermann functions leads to different reduction rules As function composition is associative, instead of rule r6 one can define Like in the previous section the computation of can be implemented with a stack. Initially the stack contains the t... | wikipedia |
wiki_42_chunk_10 | Ackermann function | On input the successive stack configurations are The corresponding equalities are When reduction rule r7 is used instead of rule r6, the replacements in the stack will follow The successive stack configurations will then be The corresponding equalities are | wikipedia |
wiki_42_chunk_11 | Ackermann function | Remarks
On any given input the TRSs presented so far converge in the same number of steps. They also use the same reduction rules (in this comparison the rules r1, r2, r3 are considered "the same as" the rules r4, r5, r6/r7 respectively). For example, the reduction of converges in 14 steps: 6 × r1, 3 × r2, 5 × r3. The... | wikipedia |
wiki_42_chunk_12 | Ackermann function | TRS, based on hyperoperators
As — or — showed explicitly, the Ackermann function can be expressed in terms of the hyperoperation sequence: or, after removal of the constant 2 from the parameter list, in terms of Buck's function Buck's function , a variant of Ackermann function by itself, can be computed with the foll... | wikipedia |
wiki_42_chunk_13 | Ackermann function | These rules take care of the base case A(0,n), the alignment (n+3) and the fudge (-3). Example Compute The matching equalities are
when the TRS with the reduction rule is applied: when the TRS with the reduction rule is applied: | wikipedia |
wiki_42_chunk_14 | Ackermann function | Remarks
The computation of according to the rules {b1 - b5, b6, r8 - r10} is deeply recursive. The maximum depth of nested s is . The culprit is the order in which iteration is executed: . The first disappears only after the whole sequence is unfolded.
The computation according to the rules {b1 - b5, b7, r8 - r10} is... | wikipedia |
wiki_42_chunk_15 | Ackermann function | Huge numbers
To demonstrate how the computation of results in many steps and in a large number: Table of values
Computing the Ackermann function can be restated in terms of an infinite table. First, place the natural numbers along the top row. To determine a number in the table, take the number immediately to the lef... | wikipedia |
wiki_42_chunk_16 | Ackermann function | The numbers here which are only expressed with recursive exponentiation or Knuth arrows are very large and would take up too much space to notate in plain decimal digits. Despite the large values occurring in this early section of the table, some even larger numbers have been defined, such as Graham's number, which can... | wikipedia |
wiki_42_chunk_17 | Ackermann function | General remarks
It may not be immediately obvious that the evaluation of always terminates. However, the recursion is bounded because in each recursive application either decreases, or remains the same and decreases. Each time that reaches zero, decreases, so eventually reaches zero as well. (Expressed more tec... | wikipedia |
wiki_42_chunk_18 | Ackermann function | Not primitive recursive The Ackermann function grows faster than any primitive recursive function and therefore is not itself primitive recursive. Specifically, one shows that to every primitive recursive function there exists a non-negative integer such that for all non-negative integers , Once this is established, ... | wikipedia |
wiki_42_chunk_19 | Ackermann function | and show that contains all primitive recursive functions. The latter is achieved by showing that contains the constant functions, the successor function, the projection functions and that it is closed under the operations of function composition and primitive recursion. Inverse
Since the function considered above g... | wikipedia |
wiki_42_chunk_20 | Ackermann function | This inverse appears in the time complexity of some algorithms, such as the disjoint-set data structure and Chazelle's algorithm for minimum spanning trees. Sometimes Ackermann's original function or other variations are used in these settings, but they all grow at similarly high rates. In particular, some modified fun... | wikipedia |
wiki_42_chunk_21 | Ackermann function | This function arises in more precise analyses of the algorithms mentioned above, and gives a more refined time bound. In the disjoint-set data structure, m represents the number of operations while n represents the number of elements; in the minimum spanning tree algorithm, m represents the number of edges while n repr... | wikipedia |
wiki_42_chunk_22 | Ackermann function | The inverse of the Ackermann function is primitive recursive. Use as benchmark
The Ackermann function, due to its definition in terms of extremely deep recursion, can be used as a benchmark of a compiler's ability to optimize recursion. The first published use of Ackermann's function in this way was in 1970 by Dragoș V... | wikipedia |
wiki_42_chunk_23 | Ackermann function | Sundblad's seminal paper was taken up by Brian Wichmann (co-author of the Whetstone benchmark) in a trilogy of papers written between 1975 and 1982. See also Computability theory
Double recursion
Fast-growing hierarchy
Goodstein function
Primitive recursive function
Recursion (computer science) Notes References Bi... | wikipedia |
wiki_42_chunk_24 | Ackermann function | External links
An animated Ackermann function calculator
Ackerman function implemented using a for loop
Scott Aaronson, Who can name the biggest number? (1999)
Ackermann functions. Includes a table of some values.
Hyper-operations: Ackermann's Function and New Arithmetical Operation
Robert Munafo's Large Nu... | wikipedia |
wiki_42_chunk_25 | Ackermann function | Arithmetic
Large integers
Special functions
Theory of computation
Computability theory | wikipedia |
wiki_43_chunk_0 | AMOS (programming language) | AMOS BASIC is a dialect of the BASIC programming language implemented on the Amiga computer. AMOS BASIC was published by Europress Software and originally written by François Lionet with Constantin Sotiropoulos in the year 1990. AMOS was considered to be a fast language. It also had 3D capabilities. History AMOS is a d... | wikipedia |
wiki_43_chunk_1 | AMOS (programming language) | AMOS competed on the Amiga platform with Acid Software's Blitz BASIC. Both BASICs differed from other dialects on different platforms, in that they allowed the easy creation of fairly demanding multimedia software, with full structured code and many high-level functions to load images, animations, sounds and display th... | wikipedia |
wiki_43_chunk_2 | AMOS (programming language) | The original AMOS was a BASIC interpreter which, whilst working fine, suffered the same disadvantages of any language being run interpretively. By all accounts, AMOS was extremely fast among interpreted languages, being speedy enough that an extension called AMOS 3D could produce playable 3D games even on plain 7 MHz 6... | wikipedia |
wiki_43_chunk_3 | AMOS (programming language) | To simplify animation of sprites, AMOS included the AMOS Animation Language (AMAL), a compiled sprite scripting language which runs independently of the main AMOS BASIC program. It was also possible to control screen and "rainbow" effects using AMAL scripts. AMAL scripts in effect created CopperLists, small routines ex... | wikipedia |
wiki_43_chunk_4 | AMOS (programming language) | After the original version of AMOS, Europress released a compiler (AMOS Compiler), and two other versions of the language: Easy AMOS, a simpler version for beginners, and AMOS Professional, a more advanced version with added features, such as a better IDE, ARexx support, a new UI API and new flow control constructs. Ne... | wikipedia |
wiki_43_chunk_5 | AMOS (programming language) | Perhaps AMOS BASIC's biggest disadvantage, stemming from its Atari ST lineage, was its incompatibility with the Amiga's operating system functions and interfaces. Instead, AMOS BASIC controlled the computer directly, which caused programs written in it to have a non-standard user interface, and also caused compatibilit... | wikipedia |
wiki_43_chunk_6 | AMOS (programming language) | On the 4 April 2019, François Lionet announced the release of AMOS2 on his website amos2.org. AMOS2 replaces STOS and AMOS together, using JavaScript as its code interpreter, making the new development system independent and generally deployed in internet browsers. Amos 2 is now called AOZ Studio. Its website is at htt... | wikipedia |
wiki_43_chunk_7 | AMOS (programming language) | Miggybyte
Scorched Tanks
Games by Vulcan Software, amongst which was the Valhalla trilogy
Amiga version of Ultimate Domain (called Genesia) by Microïds
Flight of the Amazon Queen, by Interactive Binary Illusions
Extreme Violence, included on an Amiga Power cover disk
Jetstrike, a commercial game by Rasputin Softw... | wikipedia |
wiki_43_chunk_8 | AMOS (programming language) | External links
Source code for AMOS Professional 68000 ASM from pianetaamiga.it (archived, ZIP)
Source code for AMOS and STOS 68000 ASM from clickteam.com (archived, ZIP)
The AMOS Factory (an AMOS support/community site)
Amigacoding website (contains in-depth info and references for AMOS)
History of STOS and AMOS... | wikipedia |
wiki_43_chunk_9 | AMOS (programming language) | BASIC programming language family
Video game development software
Amiga development software
Software using the BSD license
Programming languages created in 1990 | wikipedia |
wiki_44_chunk_0 | Adoptionism | Adoptionism, also called dynamic monarchianism, is an early Christian nontrinitarian theological doctrine, which holds that Jesus was adopted as the Son of God at his baptism, his resurrection, or his ascension. | wikipedia |
wiki_44_chunk_1 | Adoptionism | Definition
Adoptionism is one of two main forms of monarchianism (the other is modalism which considers God to be one while working through the different "modes" or "manifestations" of God the Father, God the Son, and God the Holy Spirit, without limiting his modes or manifestations). Adoptionism denies the eternal pre... | wikipedia |
wiki_44_chunk_2 | Adoptionism | History Early Christianity Adoptionism and High Christology | wikipedia |
wiki_44_chunk_3 | Adoptionism | Bart Ehrman holds that the New Testament writings contain two different Christologies, namely a "low" or adoptionist Christology, and a "high" or "incarnation Christology." The "low Christology" or "adoptionist Christology" is the belief "that God exalted Jesus to be his Son by raising him from the dead," thereby raisi... | wikipedia |
wiki_44_chunk_4 | Adoptionism | According to the "evolutionary model" c.q. "evolutionary theories," as proposed by Bousset, followed by Brown, the Christological understanding of Christ developed over time, from a low Christology to a high Christology, as witnessed in the Gospels. According to the evolutionary model, the earliest Christians believed ... | wikipedia |
wiki_44_chunk_5 | Adoptionism | Since the 1970s, the late datings for the development of a "high Christology" have been contested, and a majority of scholars argue that this "High Christology" existed already before the writings of Paul. This "incarnation Christology" or "high Christology" did not evolve over a longer time, but was a "big bang" of id... | wikipedia |
wiki_44_chunk_6 | Adoptionism | According to Ehrman, these two Christologies existed alongside each other, calling the "low Christology" an "adoptionist Christology, and "the "high Christology" an "incarnation Christology." | wikipedia |
wiki_44_chunk_7 | Adoptionism | New Testamental epistles
Adoptionist theology may also be reflected in canonical epistles, the earliest of which pre-date the writing of the gospels. The letters of Paul the Apostle, for example, do not mention a virgin birth of Christ. Paul describes Jesus as "born of a woman, born under the law" and "as to his human ... | wikipedia |
wiki_44_chunk_8 | Adoptionism | The Book of Hebrews, a contemporary sermon by an unknown author, describes God as saying "You are my son; today I have begotten you." Shepherd of Hermas
The 2nd-century work Shepherd of Hermas may also have taught that Jesus was a virtuous man filled with the Holy Spirit and adopted as the Son. While the Shepherd of He... | wikipedia |
wiki_44_chunk_9 | Adoptionism | Theodotus of Byzantium
Theodotus of Byzantium (fl. late 2nd century), a Valentinian Gnostic, was the most prominent exponent of adoptionism. According to Hippolytus of Rome (Philosophumena, VII, xxiii) Theodotus taught that Jesus was a man born of a virgin, according to the Council of Jerusalem, that he lived like othe... | wikipedia |
wiki_44_chunk_10 | Adoptionism | Adoptionism was declared heresy at the end of the 3rd century and was rejected by the Synods of Antioch and the First Council of Nicaea, which defined the orthodox doctrine of the Trinity and identified the man Jesus with the eternally begotten Son or Word of God in the Nicene Creed. The belief was also declared hereti... | wikipedia |
wiki_44_chunk_11 | Adoptionism | The Ebionites were a Jewish Christian movement that existed during the early centuries of the Christian Era. They show strong similarities with the earliest form of Jewish Christianity, and their specific theology may have been a "reaction to the law-free Gentile mission." They regarded Jesus as the Messiah while rejec... | wikipedia |
wiki_44_chunk_12 | Adoptionism | Distinctive features of the Gospel of the Ebionites include the absence of the virgin birth and of the genealogy of Jesus; an Adoptionist Christology, in which Jesus is chosen to be God's Son at the time of his Baptism; the abolition of the Jewish sacrifices by Jesus; and an advocacy of vegetarianism. Spanish Adoptioni... | wikipedia |
wiki_44_chunk_13 | Adoptionism | Spanish Adoptionism was a theological position which was articulated in Umayyad and Christian-held regions of the Iberian peninsula in the 8th and 9th centuries. The issue seems to have begun with the claim of archbishop Elipandus of Toledo that – in respect to his human nature – Christ was adoptive Son of God. Another... | wikipedia |
wiki_44_chunk_14 | Adoptionism | Despite the shared name of "adoptionism" the Spanish Adoptionist Christology appears to have differed sharply from the adoptionism of early Christianity. Spanish advocates predicated the term adoptivus of Christ only in respect to his humanity; once the divine Son "emptied himself" of divinity and "took the form of a s... | wikipedia |
wiki_44_chunk_15 | Adoptionism | Historically, many scholars have followed the Adoptionists' Carolingian opponents in labeling Spanish Adoptionism as a minor revival of “Nestorian” Christology. John C. Cavadini has challenged this notion by attempting to take the Spanish Christology in its own Spanish/North African context in his study, The Last Chris... | wikipedia |
wiki_44_chunk_16 | Adoptionism | Scholastic Neo-adoptionism
A third wave was the revived form ("Neo-adoptionism") of Peter Abelard in the 12th century. Later, various modified and qualified Adoptionist tenets emerged from some theologians in the 14th century. Duns Scotus (1300) and Durandus of Saint-Pourçain (1320) admit the term Filius adoptivus in ... | wikipedia |
wiki_44_chunk_17 | Adoptionism | Modern adoptionist groups
A form of adoptionism surfaced in Unitarianism during the 18th century as denial of the virgin birth became increasingly common, led by the views of Joseph Priestley and others. | wikipedia |
wiki_44_chunk_18 | Adoptionism | A similar form of adoptionism was expressed in the writings of James Strang, a Latter Day Saint leader who founded the Church of Jesus Christ of Latter Day Saints (Strangite) after the death of Joseph Smith in 1844. In his Book of the Law of the Lord, a purported work of ancient scripture found and translated by Stran... | wikipedia |
wiki_44_chunk_19 | Adoptionism | See also
Adoptivi
Arianism
Binitarianism Notes References Sources
Printed sources Philip Schaff History of the Christian Church, Volume IV, 1882.
(6th German edition, translated by George Ogg) Web sources External links Adoptionism in Catholic Encyclopedia
Adoptionism in Christian Cyclopedia
Chapter XI. Doct... | wikipedia |
wiki_45_chunk_0 | Algebraic closure | In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemma or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic... | wikipedia |
wiki_45_chunk_1 | Algebraic closure | The algebraic closure of a field K can be thought of as the largest algebraic extension of K.
To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K.
The algebraic closure of K is also the sm... | wikipedia |
wiki_45_chunk_2 | Algebraic closure | The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite. | wikipedia |
wiki_45_chunk_3 | Algebraic closure | Examples
The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.
The algebraic closure of the field of rational numbers is the field of algebraic numbers.
There are many countable algebraically closed fields within the complex numbers, and stric... | wikipedia |
wiki_45_chunk_4 | Algebraic closure | Existence of an algebraic closure and splitting fields
Let be the set of all monic irreducible polynomials in K[x].
For each , introduce new variables where .
Let R be the polynomial ring over K generated by for all and all . Write | wikipedia |
wiki_45_chunk_5 | Algebraic closure | with .
Let I be the ideal in R generated by the . Since I is strictly smaller than R,
Zorn's lemma implies that there exists a maximal ideal M in R that contains I.
The field K1=R/M has the property that every polynomial with coefficients in K splits as the product of and hence has all roots in K1. In the same way,... | wikipedia |
wiki_45_chunk_6 | Algebraic closure | It can be shown along the same lines that for any subset S of K[x], there exists a splitting field of S over K. | wikipedia |
wiki_45_chunk_7 | Algebraic closure | Separable closure
An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separable extensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable... | wikipedia |
wiki_45_chunk_8 | Algebraic closure | The separable closure is the full algebraic closure if and only if K is a perfect field. For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension. In general, the absolute Galois group of K is the Galois group of Ksep over K. See also
Algebraical... | wikipedia |
wiki_46_chunk_0 | Advanced Power Management | Advanced power management (APM) is an API developed by Intel and Microsoft and released in 1992 which enables an operating system running an IBM-compatible personal computer to work with the BIOS (part of the computer's firmware) to achieve power management. Revision 1.2 was the last version of the APM specification, r... | wikipedia |
wiki_46_chunk_1 | Advanced Power Management | APM uses a layered approach to manage devices. APM-aware applications (which include device drivers) talk to an OS-specific APM driver. This driver communicates to the APM-aware BIOS, which controls the hardware. There is the ability to opt out of APM control on a device-by-device basis, which can be used if a driver w... | wikipedia |
wiki_46_chunk_2 | Advanced Power Management | Communication occurs both ways; power management events are sent from the BIOS to the APM driver, and the APM driver sends information and requests to the BIOS via function calls. In this way the APM driver is an intermediary between the BIOS and the operating system. Power management happens in two ways; through the a... | wikipedia |
wiki_46_chunk_3 | Advanced Power Management | In APM 1.0 and APM 1.1, power management is almost fully controlled by the BIOS. In APM 1.2, the operating system can control PM time (e.g. suspend timeout). Power management events
There are 12 power events (such as standby, suspend and resume requests, and low battery notifications), plus OEM-defined events, that ca... | wikipedia |
wiki_46_chunk_4 | Advanced Power Management | APM functions
There are 21 APM function calls defined that the APM driver can use to query power management statuses, or request power state transitions. Example function calls include letting the BIOS know about current CPU usage (the BIOS may respond to such a call by placing the CPU in a low-power state, or returnin... | wikipedia |
wiki_46_chunk_5 | Advanced Power Management | System power states
APM defines five power states for the computer system:
Full On: The computer is powered on, and no devices are in a power saving mode.
APM Enabled: The computer is powered on, and APM is controlling device power management as needed.
APM Standby: Most devices are in their low-power state, the CPU... | wikipedia |
wiki_46_chunk_6 | Advanced Power Management | Device power states
APM also defines power states that APM-aware hardware can implement. There is no requirement that an APM-aware device implement all states. The four states are:
Device On: The device is in full power mode.
Device Power Managed: The device is still powered on, but some functions may not be availabl... | wikipedia |
wiki_46_chunk_7 | Advanced Power Management | CPU
The CPU core (defined in APM as the CPU clock, cache, system bus and system timers) is treated specially in APM, as it is the last device to be powered down, and the first device to be powered back up. The CPU core is always controlled through the APM BIOS (there is no option to control it through a driver). Driver... | wikipedia |
wiki_46_chunk_8 | Advanced Power Management | In ATA drives
The ATA specification defines APM provisions for hard drives via the subcommand , which specifies a trade-off between spin-down frequency and always-on performance. Unlike the BIOS-side APM, the ATA APM has never been deprecated. Aggressive spin-down frequencies may reduce drive lifespan by unnecessarily ... | wikipedia |
wiki_46_chunk_9 | Advanced Power Management | See also
Active State Power Management - hardware power management protocol for PCI Express
Advanced Configuration and Power Interface (ACPI) - successor to APM
Green computing
Power management
BatteryMAX (idle detection) References External links
APM V1.2 Specification (RTF file). BIOS | wikipedia |
wiki_47_chunk_0 | Alternative algebra | In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associa... | wikipedia |
wiki_47_chunk_1 | Alternative algebra | Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by
.
By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equiv... | wikipedia |
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