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TheeXtensible Access Control Markup Language(XACML) is anXML-based standardmarkup languagefor specifyingaccess controlpolicies. The standard, published byOASIS, defines a declarative fine-grained, attribute-basedaccess controlpolicy language, an architecture, and a processing model describing how to evaluate access req... | https://en.wikipedia.org/wiki/XACML |
OpenAthensis anidentity and access managementservice, supplied byJisc, a Britishnot-for-profitinformation technology services company.Identity provider(IdP) organisations can keepusernamesin the cloud, locally or both. Integration withADFS,LDAPorSAMLis supported.[1]
OpenAthens for Publishers[2]software forservice prov... | https://en.wikipedia.org/wiki/OpenAthens |
MozillaPersonawas a decentralizedauthenticationsystem for the web, based on the open BrowserID protocol[1]prototyped byMozilla[2]and standardized byIETF.[3]It was launched in July 2011, but after failing to achieve traction, Mozilla announced in January 2016 plans to decommission the service by the end of the year.[4]
... | https://en.wikipedia.org/wiki/Mozilla_Persona |
TheCentral Authentication Service(CAS) is asingle sign-onprotocol for theweb.[1]Its purpose is to permit a user to access multiple applications while providing their credentials (such as user ID and password) only once. It also allows web applications to authenticate users without gaining access to a user's security cr... | https://en.wikipedia.org/wiki/Central_Authentication_Service |
IndieAuthis an open standarddecentralizedauthenticationprotocolthat usesOAuth2.0 and enables services to verify the identity of a user represented by aURL, as well as to obtain anaccess token, that can be used to access resources under the control of the user.[1][2][3]
IndieAuth is developed in theIndieWebcommunity an... | https://en.wikipedia.org/wiki/IndieAuth |
Aninformation card(ori-card) is a personal digital identity that people can use online, and the key component of an identity metasystem. Visually, each i-card has a card-shaped picture and a card name associated with it that enable people to organize their digital identities and to easily select one they want to use fo... | https://en.wikipedia.org/wiki/Information_card |
TheLiberty Alliance Projectwas an organization formed in September 2001 to establish standards, guidelines and best practices foridentity managementin computer systems.
It grew to more than 150 organizations, including technology vendors, consumer-facing companies, educational organizations and governments.
It releas... | https://en.wikipedia.org/wiki/Liberty_Alliance |
Light-weight Identity(LID), orLight Identity Management(LIdM), is anidentity managementsystem foronline digital identitiesdeveloped in part byNetMesh. It was first published in early 2005, and is the original URL-based identity system, later followed byOpenID. LID usesURLsas a verification of the user's identity, and m... | https://en.wikipedia.org/wiki/Light-weight_Identity |
Security Assertion Markup Language(SAML, pronouncedSAM-el,/ˈsæməl/)[1]is anopen standardfor exchangingauthenticationandauthorizationdata between parties, in particular, between anidentity providerand aservice provider. SAML is anXML-basedmarkup languagefor security assertions (statements that service providers use to m... | https://en.wikipedia.org/wiki/Security_Assertion_Markup_Language |
Shibbolethis asingle sign-onlog-in system for computer networks and theInternet. It allows people to sign in using just one identity to various systems run by federations of different organizations or institutions. The federations are often universities or public service organizations.
The ShibbolethInternet2middlewar... | https://en.wikipedia.org/wiki/Shibboleth_(software) |
SQRL(pronounced "squirrel")[2]orSecure, Quick, Reliable Login(formerlySecureQRLogin) is adraftopen standardfor securewebsiteloginandauthentication. Thesoftwaretypically uses alink of the schemesqrl://or optionally aQR code, where a user identifies via apseudonymouszero-knowledge proofrather than providing auser IDandpa... | https://en.wikipedia.org/wiki/SQRL |
WS-Federation(Web Services Federation) is anIdentity Federationspecification, developed by a group of companies:BEA Systems,BMC Software,CA Inc.(along with Layer 7 Technologies now a part of CA Inc.),IBM,Microsoft,Novell,Hewlett Packard Enterprise, andVeriSign. Part of the largerWeb Services Securityframework, WS-Feder... | https://en.wikipedia.org/wiki/WS-Federation |
Inchaos theory, thebutterfly effectis the sensitive dependence oninitial conditionsin which a small change in one state of adeterministicnonlinear systemcan result in large differences in a later state.
The term is closely associated with the work of the mathematician and meteorologistEdward Norton Lorenz. He noted th... | https://en.wikipedia.org/wiki/Butterfly_effect |
Incryptography,confusionanddiffusionare two properties of a securecipheridentified byClaude Shannonin his 1945 classified reportA Mathematical Theory of Cryptography.[1]These properties, when present, work together to thwart the application ofstatistics, and other methods ofcryptanalysis.
Confusion in asymmetric ciphe... | https://en.wikipedia.org/wiki/Confusion_and_diffusion |
Incomputing, awordis anyprocessordesign's natural unit of data. A word is a fixed-sizeddatumhandled as a unit by theinstruction setor the hardware of the processor. The number ofbitsor digits[a]in a word (theword size,word width, orword length) is an important characteristic of any specific processor design orcomputer ... | https://en.wikipedia.org/wiki/Word_(computer_architecture) |
Incomputer security,challenge-response authenticationis a family of protocols in which one party presents a question ("challenge") and another party must provide a valid answer ("response") to beauthenticated.[1]
The simplest example of a challenge-response protocol ispasswordauthentication, where the challenge is ask... | https://en.wikipedia.org/wiki/Challenge%E2%80%93response_authentication |
Incomputer science, ahash listis typically alistofhashesof the data blocks in a file or set of files. Lists of hashes are used for many different purposes, such as fast table lookup (hash tables) and distributed databases (distributed hash tables).
A hash list is an extension of the concept of hashing an item (for ins... | https://en.wikipedia.org/wiki/Hash_list |
Linked timestampingis a type oftrusted timestampingwhere issued time-stamps are related to each other. Each time-stamp would contain data that authenticates the time-stamp before it, the authentication would be authenticating the entire message, including the previous time-stamps authentication, making a chain. This ma... | https://en.wikipedia.org/wiki/Linked_timestamping |
This is a comprehensive list ofvolunteer computingprojects, which are a type ofdistributed computingwhere volunteers donate computing time to specific causes. The donated computing power comes from idleCPUsandGPUsinpersonal computers,video game consoles,[1]andAndroid devices.
Each project seeks to utilize the computin... | https://en.wikipedia.org/wiki/List_of_volunteer_computing_projects |
Incryptography, abrute-force attackconsists of an attacker submitting manypasswordsorpassphraseswith the hope of eventually guessing correctly. The attacker systematically checks all possible passwords and passphrases until the correct one is found. Alternatively, the attacker can attempt to guess thekeywhich is typic... | https://en.wikipedia.org/wiki/Brute_force_attack |
ECRYPT(European Network of Excellence in Cryptology) was a 4-yearEuropeanresearch initiative launched on 1 February 2004 with the stated objective of promoting the collaboration of European researchers ininformation security, and especially incryptologyanddigital watermarking.
ECRYPT listed five core research areas, t... | https://en.wikipedia.org/wiki/ECRYPT |
Incryptography, asponge functionorsponge constructionis any of a class ofalgorithmswith finiteinternal statethat take an inputbit streamof any length and produce an output bit stream of any desired length. Sponge functions have both theoretical and practical uses. They can be used to model or implement manycryptographi... | https://en.wikipedia.org/wiki/Sponge_function |
Multi-factor authentication(MFA;two-factor authentication, or2FA) is anelectronic authenticationmethod in which a user is granted access to awebsiteorapplicationonly after successfully presenting two or more distinct types of evidence (orfactors) to anauthenticationmechanism. MFA protectspersonal data—which may include... | https://en.wikipedia.org/wiki/Multi-factor_authentication |
LinOTPis Linux-based software to manage authentication devices fortwo-factor authenticationwithone time passwords.
It is implemented as a web service based on the python frameworkPylons. Thus it requires a web server to
run in.
LinOTP is mainly developed by the German company KeyIdentity GmbH. Its core components ar... | https://en.wikipedia.org/wiki/LinOTP |
The following is a generalcomparison of OTP applicationsthat are used to generateone-time passwordsfortwo-factor authentication(2FA) systems using thetime-based one-time password(TOTP) or theHMAC-based one-time password(HOTP) algorithms.
by 2Stable[45] | https://en.wikipedia.org/wiki/Comparison_of_OTP_applications |
HMAC-based one-time password(HOTP) is aone-time password(OTP) algorithm based onHMAC. It is a cornerstone of theInitiative for Open Authentication(OATH).
HOTP was published as an informationalIETFRFC4226in December 2005, documenting the algorithm along with a Java implementation. Since then, the algorithm has been ado... | https://en.wikipedia.org/wiki/HMAC-based_one-time_password_algorithm |
TheFIDO("Fast IDentity Online")Allianceis an open industry association launched in February 2013 whose stated mission is to develop and promoteauthenticationstandards that "help reduce the world’s over-reliance onpasswords".[1]FIDO addresses the lack of interoperability among devices that use strong authentication and... | https://en.wikipedia.org/wiki/FIDO_Alliance |
The list below includes the names of notable ofpassword managerswith their Wikipedia articles. | https://en.wikipedia.org/wiki/List_of_password_managers |
Apassword manageris a software program to preventpassword fatiguebyautomatically generating,autofillingand storingpasswords.[1][2]It can do this forlocal applicationsorweb applicationssuch asonline shopsorsocial media.[3]Web browserstend to have a built-in password manager. Password managers typically require a user to... | https://en.wikipedia.org/wiki/Password_manager |
Incommunicationsandinformation processing,codeis a system of rules to convertinformation—such as aletter,word, sound, image, orgesture—into another form, sometimesshortenedorsecret, for communication through acommunication channelor storage in astorage medium. An early example is an invention oflanguage, which enabled ... | https://en.wikipedia.org/wiki/Code |
Communicationis commonly defined as the transmission ofinformation. Its precise definition is disputed and there are disagreements about whetherunintentionalor failed transmissions are included and whether communication not only transmitsmeaningbut also creates it.Models of communicationare simplified overviews of its ... | https://en.wikipedia.org/wiki/Communication |
Trench codes(a form ofcryptography) werecodesused for secrecy by field armies inWorld War I.[1][2]Messages by field telephone, radio and carrier pigeons could be intercepted, hence the need for tacticalWorld War I cryptography. Originally, the most commonly used codes were simple substitution codes, but due to the rela... | https://en.wikipedia.org/wiki/Trench_code |
The vulnerability ofJapanese naval codesand ciphers was crucial to the conduct ofWorld War II, and had an important influence on foreign relations between Japan and the west in the years leading up to the war as well. Every Japanese code was eventually broken, and the intelligence gathered made possible such operations... | https://en.wikipedia.org/wiki/JN-25 |
see also
TheZimmermann telegram(orZimmermann noteorZimmermann cable) was a secret diplomatic communication issued from theGerman Foreign Officeon January 17, 1917, that proposed a military contract between theGerman EmpireandMexicoif theUnited StatesenteredWorld War Iagainst Germany. With Germany's aid, Mexicowould re... | https://en.wikipedia.org/wiki/Zimmermann_telegram |
Acode talkerwas a person employed by the military during wartime to use a little-known language as a means of secret communication. The term is most often used for United States service members during theWorld Warswho used their knowledge ofNative American languagesas a basis to transmit coded messages. In particular, ... | https://en.wikipedia.org/wiki/Code_talkers |
ATM SafetyPIN softwareis asoftware applicationthat allows users ofautomated teller machines(ATMs) to alert law enforcement of a forced cash withdrawal (such as in arobbery) by entering theirpersonal identification number(PIN) in reverse order.[1]The system was patented byIllinoislawyer Joseph Zingher (U.S. patent 5,73... | https://en.wikipedia.org/wiki/ATM_SafetyPIN_software |
Atransaction authentication number(TAN) is used by someonline bankingservices as a form ofsingle useone-time passwords(OTPs) to authorizefinancial transactions. TANs are a second layer of security above and beyond the traditional single-passwordauthentication.
TANs provide additional security because they act as a fo... | https://en.wikipedia.org/wiki/Transaction_authentication_number |
TheAztec Codeis amatrix codeinvented by Andrew Longacre, Jr. and Robert Hussey in 1995.[1]The code was published byAIM, Inc.in 1997. Although the Aztec Code waspatented, that patent was officially madepublic domain.[2]The Aztec Code is also published as ISO/IEC 24778:2024 standard. Named after the resemblance of the ce... | https://en.wikipedia.org/wiki/Aztec_Code |
AData Matrixis atwo-dimensional codeconsisting of black and white "cells" or dots arranged in either asquareorrectangularpattern, also known as amatrix. The information to be encoded can be text or numeric data. Usual data size is from a few bytes up to 1556bytes. The length of the encoded data depends on the number of... | https://en.wikipedia.org/wiki/Data_Matrix |
High Capacity Color Barcode(HCCB) is a technology developed byMicrosoftfor encoding data in a2D "barcode"using clusters of colored triangles instead of the square pixels conventionally associated with 2D barcodes orQR codes.[1]Data density is increased by using a palette of 4 or 8 colors for the triangles, although HCC... | https://en.wikipedia.org/wiki/High_Capacity_Color_Barcode |
JAB Code (Just Another Barcode)is a colour 2D matrix symbology made of colour squares arranged in either square or rectangle grids. It was developed byFraunhofer Institute for Secure Information Technology SIT[de].[1]
The code contains one primary symbol and optionally multiple secondary symbols. The primary symbol co... | https://en.wikipedia.org/wiki/JAB_Code |
PDF417is a stacked linearbarcodeformat used in a variety of applications such as transport, identification cards, and inventory management. "PDF" stands forPortable Data File, while "417" signifies that each pattern in the code consists of 4 bars and spaces in a pattern that is 17 units (modules) long.
The PDF417 symb... | https://en.wikipedia.org/wiki/PDF417 |
QRpediais amobile Web-based system which usesQR codesto deliverWikipediaarticles to users, in their preferred language.[1][2][3]A typical use is onmuseum labels, linking to Wikipedia articles about the exhibited object. QR codes can easily be generated to link directly to anyUniform Resource Identifier(URI), but the QR... | https://en.wikipedia.org/wiki/QRpedia |
SnapTag, invented by SpyderLynk, is a2D mobile barcodealternative similar to aQR code, but that uses an icon or company logo and code ring rather than a square pattern of black dots.[1][2]
Similar to a QR code, SnapTags can be used to take consumers to a brand’s website, but can also facilitatemobile purchases,[3]coup... | https://en.wikipedia.org/wiki/SnapTag |
ASPARQCodeis amatrix code(or two-dimensionalbar code)encodingstandard that is based on the physicalQR Codedefinition created by Japanese corporationDenso-Wave.
The QR Code standard as defined by Denso-Wave in ISO/IEC 18004 covers the physical encoding method of a binary data stream.[1]However, the Denso-Wave standard ... | https://en.wikipedia.org/wiki/SPARQCode |
Touchatag(previouslyTikiTag[1]) was anRFIDservice for consumers, application developers and operators/enterprises created byAlcatel-Lucent. Consumers could use RFID tags to trigger what touchatag calledApplications, which could include opening a webpage, sending a text message, shutting down the computer, or running a ... | https://en.wikipedia.org/wiki/Touchatag |
MaxiCodeis apublic domain, machine-readable symbol system originally created by theUnited Parcel Service(UPS) in 1992.[1]Suitable for tracking and managing the shipment of packages, it resembles anAztec CodeorQR code, but uses dots arranged in ahexagonal gridinstead ofsquare grid. MaxiCode has been standardised underIS... | https://en.wikipedia.org/wiki/MaxiCode |
Anauthenticatoris a means used to confirm a user's identity,[1][2]that is, to perform digital authentication. A person authenticates to a computer system or application by demonstrating that he or she has possession and control of an authenticator.[3][4]In the simplest case, the authenticator is a commonpassword.
Usin... | https://en.wikipedia.org/wiki/Authenticator |
Ahardware security module(HSM) is a physical computing device that safeguards and manages secrets (most importantlydigital keys), and performsencryptionand decryption functions fordigital signatures, strong authentication and other cryptographic functions.[1]These modules traditionally come in the form of a plug-in car... | https://en.wikipedia.org/wiki/Hardware_security_module |
Initiative for Open Authentication(OATH) is an industry-wide collaboration to develop an openreference architectureusingopen standardsto promote the adoption of strong authentication. It has close to thirty coordinating and contributing members and is proposing standards for a variety of authentication technologies, wi... | https://en.wikipedia.org/wiki/Initiative_for_Open_Authentication |
Asoftware token(a.k.a.soft token) is a piece of atwo-factor authenticationsecurity device that may be used to authorize the use of computer services.[1]Software tokens are stored on a general-purpose electronic device such as adesktop computer,laptop,PDA, ormobile phoneand can be duplicated. (Contrasthardware tokens, w... | https://en.wikipedia.org/wiki/Software_token |
Botanis aBSD-licensedcryptographicand TLS library written inC++11. It provides a wide variety of cryptographic algorithms, formats, and protocols, e.g.SSL and TLS. It is used in theMonotonedistributed revision controlprogram, theOpenDNSSECsystem, andISC's Kea DHCP serveramong other projects.
The project was originally... | https://en.wikipedia.org/wiki/Botan_(programming_library) |
multiOTPis an open source PHP class, a command line tool, and a web interface that can be used to provide an operating-system-independent, strongauthenticationsystem. multiOTP isOATH-certified since version 4.1.0 and is developed under theLGPLlicense. Starting with version 4.3.2.5, multiOTP open source is also availabl... | https://en.wikipedia.org/wiki/MultiOTP |
The following is a generalcomparison of OTP applicationsthat are used to generateone-time passwordsfortwo-factor authentication(2FA) systems using thetime-based one-time password(TOTP) or theHMAC-based one-time password(HOTP) algorithms.
by 2Stable[45] | https://en.wikipedia.org/wiki/Comparison_of_TOTP_applications |
Authentication(fromGreek:αὐθεντικόςauthentikos, "real, genuine", from αὐθέντηςauthentes, "author") is the act of proving anassertion, such as theidentityof a computer system user. In contrast withidentification, the act of indicating a person or thing's identity, authentication is the process of verifying that identity... | https://en.wikipedia.org/wiki/Authentication#Authentication_factors |
Electronic authenticationis the process of establishing confidence in user identities electronically presented to aninformation system.[1]Digital authentication,ore-authentication,may be used synonymously when referring to theauthenticationprocess that confirms or certifies a person's identity and works. When used in c... | https://en.wikipedia.org/wiki/Electronic_authentication |
Intelecommunications,out-of-bandactivity is activity outside a definedfrequency band, or, metaphorically, outside of any primarycommunication channel. Protection fromfalsingis among its purposes. | https://en.wikipedia.org/wiki/Out-of-band |
Reliance authenticationis a part of the trust-based identity attribution process whereby a second entity relies upon theauthenticationprocesses put in place by a first entity. The second entity creates a further element that is unique and specific to its purpose, that can only be retrieved or accessed by the authentica... | https://en.wikipedia.org/wiki/Reliance_authentication |
Universal 2nd Factor(U2F) is an open standard that strengthens and simplifiestwo-factor authentication(2FA) using specializedUniversal Serial Bus(USB),near-field communication(NFC), orBluetooth Low Energy(BLE) devices based on similar security technology found insmart cards.[1][2][3][4][5]It is succeeded by theFIDO2 Pr... | https://en.wikipedia.org/wiki/Universal_2nd_Factor |
Identity threat detection and response (ITDR)is acybersecuritydiscipline that includes tools and best practices to protectidentity managementinfrastructure from attacks. ITDR can block and detectthreats, verify administrator credentials, respond to various attacks, and restore normal operations.[1]Common identity threa... | https://en.wikipedia.org/wiki/Identity_threat_detection_and_response |
Abstract algebrais the subject area ofmathematicsthat studiesalgebraic structures, such asgroups,rings,fields,modules,vector spaces, andalgebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for mani... | https://en.wikipedia.org/wiki/List_of_abstract_algebra_topics |
The followingoutlineis provided as an overview of and guide tocategory theory, the area of study inmathematicsthat examines in anabstractway the properties of particular mathematical concepts, by formalising them as collections ofobjectsandarrows(also calledmorphisms, although this term also has a specific, non categor... | https://en.wikipedia.org/wiki/List_of_category_theory_topics |
This is alist ofLie grouptopics, by Wikipedia page.
SeeTable of Lie groupsfor a list | https://en.wikipedia.org/wiki/List_of_Lie_group_topics |
Geometric group theoryis an area inmathematicsdevoted to the study offinitely generated groupsvia exploring the connections betweenalgebraicproperties of suchgroupsandtopologicalandgeometricproperties of spaces on which these groups canactnon-trivially (that is, when the groups in question are realized as geometric sym... | https://en.wikipedia.org/wiki/Geometric_group_theory |
Inmathematics, asystem of parametersfor alocalNoetherian ringofKrull dimensiondwithmaximal idealmis a set of elementsx1, ...,xdthat satisfies any of the following equivalent conditions:
Every local Noetherian ring admits a system of parameters.[1]
It is not possible for fewer thandelements to generate an ideal whose ... | https://en.wikipedia.org/wiki/System_of_parameters |
Inmathematics, afilterororder filteris a specialsubsetof apartially ordered set(poset), describing "large" or "eventual" elements. Filters appear inorderandlattice theory, but alsotopology, whence they originate. The notiondualto a filter is anorder ideal.
Special cases of filters includeultrafilters, which are filter... | https://en.wikipedia.org/wiki/Filter_(mathematics) |
In the mathematical field ofset theory, anidealis apartially orderedcollection ofsetsthat are considered to be "small" or "negligible". Everysubsetof an element of the ideal must also be in the ideal (this codifies the idea that an ideal is a notion of smallness), and theunionof any two elements of the ideal must also ... | https://en.wikipedia.org/wiki/Ideal_(set_theory) |
In mathematics, asemigroupis analgebraic structureconsisting of asettogether with anassociativeinternalbinary operationon it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmeticmultiplication):x⋅y, or simplyxy, denotes the result of apply... | https://en.wikipedia.org/wiki/Semigroup_ideal |
Inmathematics, the(field) normis a particular mapping defined infield theory, which maps elements of a larger field into asubfield.
LetKbe afieldandLafiniteextension(and hence analgebraic extension) ofK.
The fieldLis then afinite-dimensionalvector spaceoverK.
Multiplication byα, an element ofL,
is aK-linear transfo... | https://en.wikipedia.org/wiki/Field_norm |
Inmathematics, theDedekind zeta functionof analgebraic number fieldK, generally denoted ζK(s), is a generalization of theRiemann zeta function(which is obtained in the case whereKis thefield of rational numbersQ). It can be defined as aDirichlet series, it has anEuler productexpansion, it satisfies afunctional equation... | https://en.wikipedia.org/wiki/Dedekind_zeta_function |
TheChebotarev density theoreminalgebraic number theorydescribes statistically the splitting ofprimesin a givenGalois extensionKof the fieldQ{\displaystyle \mathbb {Q} }ofrational numbers. Generally speaking, a prime integer will factor into severalideal primesin the ring ofalgebraic integersofK. There are only finitely... | https://en.wikipedia.org/wiki/Chebotarev%27s_density_theorem |
The following list inmathematicscontains thefinite groupsof smallorderup togroup isomorphism.
Forn= 1, 2, … the number of nonisomorphic groups of ordernis
For labeled groups, seeOEIS:A034383.
Each group is named bySmall Groups libraryas Goi, whereois the order of the group, andiis the index used to label the group ... | https://en.wikipedia.org/wiki/List_of_small_groups |
Inmathematics, aCayley graph, also known as aCayley color graph,Cayley diagram,group diagram, orcolor group,[1]is agraphthat encodes the abstract structure of agroup. Its definition is suggested byCayley's theorem(named afterArthur Cayley), and uses a specifiedset of generatorsfor the group. It is a central tool incomb... | https://en.wikipedia.org/wiki/Cayley_graph |
Inmathematics, afinitely generated moduleis amodulethat has afinitegenerating set. A finitely generated module over aringRmay also be called afiniteR-module,finite overR,[1]or amodule of finite type.
Related concepts includefinitely cogenerated modules,finitely presented modules,finitely related modulesandcoherent mod... | https://en.wikipedia.org/wiki/Finitely_generated_module |
Innumber theory, given aprime numberp,[note 1]thep-adic numbersform an extension of therational numberswhich is distinct from thereal numbers, though with some similar properties;p-adic numbers can be written in a form similar to (possiblyinfinite)decimals, but with digits based on a prime numberprather than ten, and e... | https://en.wikipedia.org/wiki/P-adic_integers |
Inmathematics, theinverse limit(also called theprojective limit) is a construction that allows one to "glue together" several relatedobjects, the precise gluing process being specified bymorphismsbetween the objects. Thus, inverse limits can be defined in anycategoryalthough their existence depends on the category that... | https://en.wikipedia.org/wiki/Inverse_limit |
In mathematics, adyadic rationalorbinary rationalis a number that can be expressed as afractionwhosedenominatoris apower of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important incomputer sciencebecause they are the only ones with finitebinary representations. Dyadic rat... | https://en.wikipedia.org/wiki/Dyadic_rational |
Inmathematics, particularlyset theory, afinite setis asetthat has afinitenumber ofelements. Informally, a finite set is a set which one could in principle count and finish counting. For example,
is a finite set with five elements. The number of elements of a finite set is anatural number(possibly zero) and is called ... | https://en.wikipedia.org/wiki/Finite_set |
This article listsmathematicalproperties and laws ofsets, involving the set-theoreticoperationsofunion,intersection, andcomplementationand therelationsof setequalityand setinclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.... | https://en.wikipedia.org/wiki/List_of_set_identities_and_relations |
Infunctional analysisand related areas ofmathematics, analmost open mapbetweentopological spacesis amapthat satisfies a condition similar to, but weaker than, the condition of being anopen map.
As described below, for certain broad categories oftopological vector spaces,allsurjectivelinear operators are necessarily al... | https://en.wikipedia.org/wiki/Almost_open_map |
Inmathematics, particularly infunctional analysisandtopology,closed graphis a property offunctions.[1][2]A functionf:X→Ybetweentopological spaceshas aclosed graphif itsgraphis aclosed subsetof theproduct spaceX×Y.
A related property isopen graph.[3]
This property is studied because there are many theorems, known ascl... | https://en.wikipedia.org/wiki/Closed_graph |
Infunctional analysis, a branch of mathematics, aclosed linear operatoror often aclosed operatoris alinear operatorwhose graph is closed (seeclosed graph property). It is a basic example of anunbounded operator.
Theclosed graph theoremsays a linear operatorf:X→Y{\displaystyle f:X\to Y}betweenBanach spacesis a closed o... | https://en.wikipedia.org/wiki/Closed_linear_operator |
Inmathematics, more specificallytopology, alocal homeomorphismis afunctionbetweentopological spacesthat, intuitively, preserves local (though not necessarily global) structure.
Iff:X→Y{\displaystyle f:X\to Y}is a local homeomorphism,X{\displaystyle X}is said to be anétale spaceoverY.{\displaystyle Y.}Local homeomorphi... | https://en.wikipedia.org/wiki/Local_homeomorphism |
Intopologya branch of mathematics, aquasi-open maporquasi-interior mapis afunctionwhich has similar properties tocontinuous maps.
However, continuous maps and quasi-open maps are not related.[1]
A functionf:X→Ybetweentopological spacesXandYis quasi-open if, for any non-emptyopen setU⊆X, theinterioroff('U)inYis non-em... | https://en.wikipedia.org/wiki/Quasi-open_map |
Intopologyand related areas ofmathematics, thequotient spaceof atopological spaceunder a givenequivalence relationis a new topological space constructed by endowing thequotient setof the original topological space with thequotient topology, that is, with thefinest topologythat makescontinuousthecanonical projection map... | https://en.wikipedia.org/wiki/Quotient_map_(topology) |
Inmathematics, especiallytopology, aperfect mapis a particular kind ofcontinuous functionbetweentopological spaces. Perfect maps are weaker thanhomeomorphisms, but strong enough to preserve some topological properties such aslocal compactnessthat are not always preserved by continuous maps.
LetX{\displaystyle X}andY{\... | https://en.wikipedia.org/wiki/Perfect_map |
Inmathematics, afunctionbetweentopological spacesis calledproperifinverse imagesofcompact subsetsare compact.[1]Inalgebraic geometry, theanalogousconcept is called aproper morphism.
There are several competing definitions of a "properfunction".
Some authors call a functionf:X→Y{\displaystyle f:X\to Y}between twotopol... | https://en.wikipedia.org/wiki/Proper_map |
Inmathematics, specificallytopology, asequence covering mapis any of a class ofmapsbetweentopological spaceswhose definitions all somehow relate sequences in thecodomainwith sequences in thedomain. Examples includesequentiallyquotientmaps,sequence coverings,1-sequence coverings, and2-sequence coverings.[1][2][3][4]Thes... | https://en.wikipedia.org/wiki/Sequence_covering_map |
Inmathematics, especiallyseveral complex variables, ananalytic polyhedronis a subset of thecomplex spaceCnof the form
whereDis a bounded connected open subset ofCn,fj{\displaystyle f_{j}}areholomorphiconDandPis assumed to berelatively compactinD.[1]Iffj{\displaystyle f_{j}}above are polynomials, then the set is called... | https://en.wikipedia.org/wiki/Analytic_polyhedron |
Inmathematics, aCaccioppoli setis a subset ofRn{\displaystyle \mathbb {R} ^{n}}whoseboundaryis (in a suitable sense)measurableand has (at leastlocally) afinitemeasure. A synonym isset of (locally) finite perimeter. Basically, a set is a Caccioppoli set if itscharacteristic functionis afunction of bounded variation, and... | https://en.wikipedia.org/wiki/Caccioppoli_set |
Inmathematics, aHermitian symmetric spaceis aHermitian manifoldwhich at every point has an inversion symmetry preserving the Hermitian structure. First studied byÉlie Cartan, they form a natural generalization of the notion ofRiemannian symmetric spacefromreal manifoldstocomplex manifolds.
Every Hermitian symmetric sp... | https://en.wikipedia.org/wiki/Hermitian_symmetric_space#Classical_domains |
Inmathematics, areal intervalis thesetof allreal numberslying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negativeinfinity, indicating the interval extends without abound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any... | https://en.wikipedia.org/wiki/Interval_(mathematics) |
Inmathematics, aLipschitz domain(ordomain with Lipschitz boundary) is adomaininEuclidean spacewhoseboundaryis "sufficiently regular" in the sense that it can be thought of as locally being the graph of aLipschitz continuous function. The term is named after theGermanmathematicianRudolf Lipschitz.
Letn∈N{\displaystyle ... | https://en.wikipedia.org/wiki/Lipschitz_domain |
Inmathematics,point-free geometryis ageometrywhose primitiveontologicalnotion isregionrather thanpoint. Twoaxiomatic systemsare set out below, one grounded inmereology, the other inmereotopologyand known asconnection theory.
Point-free geometry was first formulated byAlfred North Whitehead,[1]not as a theory ofgeometr... | https://en.wikipedia.org/wiki/Whitehead%27s_point-free_geometry |
Inmathematics, abase(orbasis;pl.:bases) for thetopologyτof atopological space(X, τ)is afamilyB{\displaystyle {\mathcal {B}}}ofopen subsetsofXsuch that every open set of the topology is equal to theunionof somesub-familyofB{\displaystyle {\mathcal {B}}}. For example, the set of allopen intervalsin thereal number lineR{... | https://en.wikipedia.org/wiki/Base_(topology) |
Inmathematical analysis, adomainorregionis anon-empty,connected, andopen setin atopological space. In particular, it is any non-empty connected opensubsetof thereal coordinate spaceRnor thecomplex coordinate spaceCn. A connected open subset ofcoordinate spaceis frequently used for thedomain of a function.[1]
The basic... | https://en.wikipedia.org/wiki/Domain_(mathematical_analysis) |
Inmathematics, more specifically intopology, anopen mapis afunctionbetween twotopological spacesthat mapsopen setsto open sets.[1][2][3]That is, a functionf:X→Y{\displaystyle f:X\to Y}is open if for any open setU{\displaystyle U}inX,{\displaystyle X,}theimagef(U){\displaystyle f(U)}is open inY.{\displaystyle Y.}Likewis... | https://en.wikipedia.org/wiki/Open_map |
Intopology, asubbase(orsubbasis,prebase,prebasis) for thetopologyτof atopological space(X, τ)is a subcollectionB{\displaystyle B}ofτ{\displaystyle \tau }that generatesτ,{\displaystyle \tau ,}in the sense thatτ{\displaystyle \tau }is the smallest topology containingB{\displaystyle B}as open sets. A slightly different de... | https://en.wikipedia.org/wiki/Subbase |
Inmathematics, apointxis called anisolated pointof a subsetS(in atopological spaceX) ifxis an element ofSand there exists aneighborhoodofxthat does not contain any other points ofS. This is equivalent to saying that thesingleton{x}is anopen setin the topological spaceS(considered as asubspaceofX). Another equivalent fo... | https://en.wikipedia.org/wiki/Isolated_point |
Intopologyand related areas ofmathematics, theneighbourhood system,complete system of neighbourhoods,[1]orneighbourhood filterN(x){\displaystyle {\mathcal {N}}(x)}for a pointx{\displaystyle x}in atopological spaceis the collection of allneighbourhoodsofx.{\displaystyle x.}
Neighbourhood of a point or set
Anopen neigh... | https://en.wikipedia.org/wiki/Neighbourhood_system |
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