# kb

id: list-of-linguistic-example-sentences-20260505-0
file: data/en.wikipedia.org/wiki/List_of_linguistic_example_sentences-0.md
category: reference
title: List of linguistic example sentences
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguistic_example_sentences
date_saved: 2026-05-05T08:11:34.058624+00:00
instance: kb-cron
parent: list-of-linguistic-example-sentences-20260505-0
summary:
The following is a partial list of linguistic example sentences illustrating various linguistic phenomena. == Ambiguity == Different types of ambiguity which are possible in language. === Lexical ambiguity === Demonstrations of words which have multiple meanings dependent on context. Will, will Will will Will Will's will? – Will (a person), will (future tense auxiliary verb) Will (a second person) will (bequeath) [to] Will (a third person) Will's (the second person) will (a document)? (Someone asked Will 1 directly if Will 2 plans to bequeath his own will, the document, to Will 3.) Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo. – Bison (the plural of "buffalos" or "buffalo" is also accepted) from Buffalo, New York, whom bison from Buffalo bully, bully bison from Buffalo. Police police Police police police police Police police. – Police officers from Police, Poland, whom police officers from Police patrol, patrol police officers from Police. Rose rose to put rose roes on her rows of roses. (Robert J. Baran) – Rose [a person] rose [stood] to put rose [pink-colored] roes [fish eggs as fertilizer] on her rows of roses [flower]. James while John had had had had had had had had had had had a


id: list-of-linguistic-example-sentences-20260505-1
file: data/en.wikipedia.org/wiki/List_of_linguistic_example_sentences-1.md
category: reference
title: List of linguistic example sentences
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguistic_example_sentences
date_saved: 2026-05-05T08:11:34.058624+00:00
instance: kb-cron
parent: list-of-linguistic-example-sentences-20260505-0
summary:
Colorless green ideas sleep furiously (Noam Chomsky): example that is grammatically correct but based on semantic combinations that are contradictory and therefore would not normally occur. Hold the news reader's nose squarely, waiter, or friendly milk will countermand my trousers. == Semantics and context == Demonstrations of sentences where the semantic interpretation is bound to context or knowledge of the world. The large ball crashed right through the table because it was made of Styrofoam: ambiguous use of a pronoun: The word "it" refers to the table being made of Styrofoam; but "it" would immediately refer to the large ball if we replaced "Styrofoam" with "steel" without any other change in its syntactic parse. The bee landed on the flower because it had nectar: The pronoun "it" refers to the "flower" but changes to the "bee" if we replace "had" with "wanted". We bought the boys apples because they were so hungry: "they" refers to the boys, but if "hungry" is replaced with "cheap", with no grammatical change, it refers to the apples. St Paul’s Cathedral is amusing, awful, and artificial: The adjectives are contronyms and their main, contemporary meanings conflict with minority meanings of "amazing, awe-inspiring, and artistic." ===


id: list-of-linguistic-example-sentences-20260505-2
file: data/en.wikipedia.org/wiki/List_of_linguistic_example_sentences-2.md
category: reference
title: List of linguistic example sentences
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguistic_example_sentences
date_saved: 2026-05-05T08:11:34.058624+00:00
instance: kb-cron
parent: list-of-linguistic-example-sentences-20260505-0
summary:
=== German === A famous example of lexical ambiguity is the following sentence: "Wenn hinter Fliegen Fliegen fliegen, fliegen Fliegen Fliegen hinterher.", meaning "When flies fly behind flies, then flies fly in pursuit of flies." It takes advantage of some German nouns and corresponding verbs being homonymous. While not noticeable in spoken language, in written language the difference shows: "Fliegen" ("flies"), being a noun, is written with a capital "F", whereas "fliegen" ("to fly"), being a verb, is not. The comma can be left out without changing the meaning. There are several variations of this sentence pattern, although they do not work as smoothly as the original. Dutch language shares this same example, with the noticeable difference of not capitalising the initials of nouns, making it "Als achter vliegen vliegen vliegen, vliegen vliegen vliegen achterna." === Dutch === Kees Torn expanded on the example given in the German section ("Als achter vliegen vliegen vliegen, vliegen vliegen vliegen achterna."), from which he created: "Als, in de plaats waar van de makkelijk te zeven zeven zeven zeven zeven zeven zeven, Zeven, zeven zeven zeven zeven zeven, zeven zeven zeven zeven zeven." which uses the fact that "zeven" has multiple roles: it is


id: list-of-family-name-affixes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_family_name_affixes-0.md
category: reference
title: List of family name affixes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_family_name_affixes
date_saved: 2026-05-05T08:11:35.382968+00:00
instance: kb-cron
parent: list-of-family-name-affixes-20260505-0
summary:
Family name affixes are a clue for surname etymology and can sometimes determine the ethnic origin of a person. This is a partial list of affixes. == Prefixes == === Arabic === Abu – (Arabic) "father of"; Al – (Arabic) "Family of" or "House of" (in conjunction with name of ancestor) Bet – (Arabic from "Beyt") "house of" Bint – (Arabic) "daughter of"; Binti, Binte (Malaysian version) El – (Arabic see Al) Ibn – (Arabic) "son of" === Armenian === Ter – (Eastern Armenian) "son/daughter of a Priest" Der – (Western Armenian) "son/daughter of a priest"; (German) "the" (masculine nominative), "of the" (feminine genitive) === Berber === Ayt/At/Ag/Id u- – (Berber) "of" At/n Ath – (Berber) "(son[s]) of" U - (Berber) "son of"/"descendent of" Ult - (Berber) "daughter of" === Dutch === de – (Dutch) "the" 's – (Dutch) "of the"; contraction of des, genitive case of the definite article de. Example: 's Gravesande. 't – (Dutch) "the"; contraction of the neuter definite article het. ter – (Dutch) "at the" van – (Dutch) "of", "from" van de, van den, van der, van 't – (Dutch) "of the", "from the" === French === "of", "of the", "from"; often a sign of


id: list-of-family-name-affixes-20260505-1
file: data/en.wikipedia.org/wiki/List_of_family_name_affixes-1.md
category: reference
title: List of family name affixes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_family_name_affixes
date_saved: 2026-05-05T08:11:35.382968+00:00
instance: kb-cron
parent: list-of-family-name-affixes-20260505-0
summary:
== Suffixes == -a, -ya Kurdish means "of" (female) (by two surnames) -à (Catalan) feminine -ana "of or from [a locality or place]" (Català -Catalan); and also the name of a job (Manyà -ironsmith), from Latin -ānus, -āna -ac (Croatian, Serbian, Slovenian, Southern French) -ach (Ukrainian, Belarusian /Belarusian Latin: -ač/) -acz (Polish) -aei (Persian) (See -i) for words that end in the long vowel A -aitis (Lithuanian) "son of" -aitė (Lithuanian) signifies an unmarried female -aty Americanized form -aj (Albanian) (pronounced AY; meaning “of the") It denotes the name of the family, which mostly comes from the male founder of the family, but also from a place, as in, Lash-aj (from the village Lashaj of Kastrat, MM, Shkodër). It is likely that its ancient form, still found in MM, was an [i] in front of the last name, as in ‘Déda i Lékajve’ (Déd of Lekës). For ease of use, the [i] in front of the last name, and the ending _ve, were dropped. If the last name ends in [a], then removing the [j] would give the name of the patriarch or the place, as in, Grudaj - j = Gruda (place in MM). Otherwise, removing the whole ending


id: list-of-family-name-affixes-20260505-2
file: data/en.wikipedia.org/wiki/List_of_family_name_affixes-2.md
category: reference
title: List of family name affixes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_family_name_affixes
date_saved: 2026-05-05T08:11:35.382968+00:00
instance: kb-cron
parent: list-of-family-name-affixes-20260505-0
summary:
-ik (Estonian) if it follows a tree name, has a meaning "grove" -ikh, -ykh (Russian) -in (Russian (all Eastern Slavic languages), Bulgarian) possessive -ina (female equivalent of -in; especially rare for male names, but the suffix alone is an actual female name) -yn (Russian, Belarusian, Ukrainian) possessive -in (French) diminutive -in (Dutch, German) suffix attached to old Germanic female surnames (e.g. female surname "Mayerin", the wife of "Mayer") -ge (Sinhalese) "From the house of" -ing, ink (Anglo-Saxon, Dutch, German) "descendant" -ino (a common suffix for male Latino and Italian names) -ipa (Abkhazian) "son of" -ipha (Abkhazian) "girl of" -is (Greek, /male/ Lithuanian) -ienė (Lithuanian) female version -ytė (Lithuanian) unmarried female version -ishin, -yshyn (Ukrainian) possessive (e.g. Romanishin = son of wife of Roman) -ishina, -yshyna (female equivalent of -ishin, -yshyn) -iu (Romanian) -ius (Lithuanian) "son of" -iv (Ukrainian) possessive. -iz (Spanish) including Spanish-speaking countries "son of", very unfrequent compared to -ez (Muñiz < Muño) -j (Adygean)"old" -ja (Sindhi, Punjabi) "from, of", often denoting ancestral hometowns and villages. Mostly with the combinations -ija, -uja and -eja (eg. Makhija, Ahuja, Taneja) -jerhin/-jerin (Kyrghyz) "place (of origin)" Usually, this form of the surname is assigned to kairylmans who do not have a surname.


id: list-of-family-name-affixes-20260505-3
file: data/en.wikipedia.org/wiki/List_of_family_name_affixes-3.md
category: reference
title: List of family name affixes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_family_name_affixes
date_saved: 2026-05-05T08:11:35.382968+00:00
instance: kb-cron
parent: list-of-family-name-affixes-20260505-0
summary:
-vich (Belarusian /Belarusian Latin: -vič/, occasionally a respelling of original Serbian, Croatian -vić) "son of" -vych (Ukrainian) -wicz (Polish), -wic (Polish) -vić (Serbian, Croatian, Bosnian, Montenegrin) -vič (Slovenian, Slovak, Czech [rarely]), -vic (Slovak, Czech [rarely]) -vici (Romanianized respellings) -vics, -vits (Hungarianized respellings) -vitz, -witz, -witch, -witsch (Germanized or Anglicized respellings) -vičius (Lithuanian) Lithuanianized version of the Belarusian -vich (Belarusian Latin: -vič) and Polish -wicz -vičiutė (Lithuanian) signifies an unmarried female -vičs (Latvian) Latvianized version of the Belarusian -vich (Belarusian Latin: -vič) and Polish –wicz -vedi (Sanskrit, Hindi) "learned in _ Vedas" (e.g. Trivedi = "learned in 3 Vedas") -wala, -wallah, wali, vala, vali (Hindustani, Gujarati, Bengali, Marathi) denotes the occupation or place of origin (Occupation example: Batliwala – one who deals with bottles. Place example: Suratwala – one from Surat) -wan (Indonesian) denotes a male name -war (Marathi), often used to signify members of the Arya Vyshya community, residing in central and western India. -wati (Indonesian) denotes a female name -wi (Arabic) "from" -y (Arabic/Persian) Means descendant of. -y (See -i) -yal -ycz (Polish) -yk (Polish, Belarusian, Ukrainian) -ynas (Lithuanian) "son of" -ys (English) representing i. the archaic plural form, or ii. a diminutive form. Variant forms not limited to


id: father-honorific-20260505-0
file: data/en.wikipedia.org/wiki/Father_(honorific)-0.md
category: reference
title: Father (honorific)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Father_(honorific)
date_saved: 2026-05-05T08:11:36.654088+00:00
instance: kb-cron
parent: father-honorific-20260505-0
summary:
Father has been used as both title and honorific in various languages, synonyms and historical contexts. It may sometimes denote a title of authority or of honour. == List of uses of "father" in various languages == === By culture and/or language === Ab (Semitic) Bwana ("our father"), from Swahili, meaning an important person or safari leader Abu in Kunya (Arabic), used as epithet for "father of X" Baba, mark of respect in: Indian honorific Hindu and Sikh Baba (honorific) in Persian language In Malaysia as an honorific of respect to address Chinese people born in the British Straits Settlements Batko, a Ukrainian honorific meaning "father" List of people considered father or mother of a field Founding father Father of the Nation/Father of the Country Pater Patriae Fathers of Confederation Founding fathers of the European Union Founding Fathers of the United States Pater familias (Latin), title for head of household in Ancient Rome === Personifications === Father Time Father Christmas Ded Moroz ("Father Frost") === By religion === God the Father ==== Buddhism ==== Abbot (Buddhism) a title for a monk who holds the position of administrator of a Buddhist monastery or Buddhist temple ==== Christianity ==== Patriarch (Greek, literally


id: list-of-formal-language-and-literal-string-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formal_language_and_literal_string_topics-0.md
category: reference
title: List of formal language and literal string topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formal_language_and_literal_string_topics
date_saved: 2026-05-05T08:11:37.827966+00:00
instance: kb-cron
parent: list-of-formal-language-and-literal-string-topics-20260505-0
summary:
This is a list of formal language and literal string topics, by Wikipedia page. == Formal languages == Abstract syntax tree Backus-Naur form Categorial grammar Chomsky hierarchy Concatenation Context-free grammar Context-sensitive grammar Context-sensitive language Decidable language ECLR-attributed grammar Finite language Formal grammar Formal language Formal system Generalized star height problem Kleene algebra Kleene star L-attributed grammar LR-attributed grammar Myhill–Nerode theorem Parsing expression grammar Prefix grammar Pumping lemma Recursively enumerable language Regular expression Regular grammar Regular language S-attributed grammar Star height Star height problem Syntactic monoid Syntax (logic) Tree-adjoining grammar == Literal strings == Anagram Case sensitivity Infinite monkey theorem Lexical analysis Lexeme Lexicography Lexicon Lipogram The Library of Babel Palindrome Pangram Sequence alignment == Classical cryptography == Atbash cipher Autokey cipher Bazeries cylinder Bible code Bifid cipher Caesar cipher Cardan grille Enigma machine Frequency analysis Index of coincidence Playfair cipher Polyalphabetic substitution Polybius square ROT13, ROT47 Scytale Steganography Substitution cipher Tabula recta Transposition cipher Vigenère cipher


id: gardiner-s-sign-list-20260505-0
file: data/en.wikipedia.org/wiki/Gardiner's_sign_list-0.md
category: reference
title: Gardiner's sign list
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Gardiner's_sign_list
date_saved: 2026-05-05T08:11:39.245476+00:00
instance: kb-cron
parent: gardiner-s-sign-list-20260505-0
summary:
Gardiner's sign list is a list of common Egyptian hieroglyphs compiled by Sir Alan Gardiner. It is considered a standard reference in the study of ancient Egyptian hieroglyphs. Gardiner lists only the common forms of Egyptian hieroglyphs, but he includes extensive subcategories, and also both vertical and horizontal forms for many hieroglyphs. He includes size-variation forms to aid with the reading of hieroglyphs in running blocks of text. In contrast, for example, the Budge Reference has about 1,000 hieroglyphs listed in 50 pages, but with no size variations. Gardiner does not cross-index signs; once a sign is put on one of his lists, other significant uses may be overlooked. One example of this is 𓅒 G16, nbtỉ, the ideogram for the Two Ladies, goddesses Wadjet as the cobra and Nekhbet as the white vulture. These are the protective and patron goddesses of the separate Egyptian kingdoms that joined into ancient Egypt, who were both then displayed on the uraeus of Wadjet when the unification occurred and afterward considered jointly to be the protectors of Egypt and the pharaohs. This ideogram is listed only in the bird list (G), and overlooked on the deity list (C) and the reptile list (I).


id: list-of-grammatical-cases-20260505-0
file: data/en.wikipedia.org/wiki/List_of_grammatical_cases-0.md
category: reference
title: List of grammatical cases
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_grammatical_cases
date_saved: 2026-05-05T08:11:40.445790+00:00
instance: kb-cron
parent: list-of-grammatical-cases-20260505-0
summary:
This is a list of grammatical cases as they are used by various inflectional languages that have declension. This list will mark the case, when it is used, an example of it, and then finally what language(s) the case is used in. == Location and movement == Note: Most cases used for location and motion can be used for time as well. === Location === === Motion from === === Motion to === === Motion via === === Time === === Chart for review for the basic cases === == Morphosyntactic alignment == For meanings of the terms agent, patient, experiencer, and instrument, see thematic relation. == Relation == == Semantics == == State == == References ==


id: grammatical-mood-20260505-0
file: data/en.wikipedia.org/wiki/Grammatical_mood-0.md
category: reference
title: Grammatical mood
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Grammatical_mood
date_saved: 2026-05-05T08:11:41.638208+00:00
instance: kb-cron
parent: grammatical-mood-20260505-0
summary:
In linguistics, grammatical mood is a grammatical feature of verbs, used for signaling modality. In other words, it is the use of verbal inflections that allow speakers to express their attitude toward what they are saying (for example, a statement of fact, of desire, of command, etc.). The term is also used more broadly to describe the syntactic expression of modality – that is, the use of verb phrases that do not involve inflection of the verb itself. Mood is distinct from grammatical tense or grammatical aspect, although the same word patterns are used for expressing more than one of these meanings at the same time in many languages, including English and most other modern Indo-European languages. (See tense–aspect–mood for a discussion of this.) Some examples of moods are indicative, interrogative, imperative, subjunctive, injunctive, optative, and irrealis/potential. These are all finite forms of the verb. Infinitives, gerunds, and participles, which are non-finite forms of the verb, are not considered to be examples of moods. Some Uralic Samoyedic languages have more than ten moods; Nenets has as many as sixteen. The original Indo-European inventory of moods consisted of indicative, subjunctive, optative, and imperative. Not every Indo-European language has all of these


id: grammatical-mood-20260505-1
file: data/en.wikipedia.org/wiki/Grammatical_mood-1.md
category: reference
title: Grammatical mood
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Grammatical_mood
date_saved: 2026-05-05T08:11:41.638208+00:00
instance: kb-cron
parent: grammatical-mood-20260505-0
summary:
The conditional mood is used for speaking of an event whose realization is dependent upon another condition, particularly, but not exclusively, in conditional sentences. In Modern English, this type of modality is expressed via a periphrastic construction, with the form would + infinitive, (for example, I would buy), and thus is a mood only in the broad sense and not in the more common narrow sense of the term "mood" requiring morphological changes in the verb. In other languages, verbs have a specific conditional inflection. In German, the conditional mood is identical to one of the two subjunctive moods (Konjunktiv II, see above). In the Romance languages, the conditional form is used primarily in the apodosis (main clause) of conditional sentences, and in a few set phrases where it expresses courtesy or doubt. The main verb in the protasis (dependent clause) is usually in the subjunctive or in the indicative mood. However, this is not a universal trait and among others in German (as above), Finnish, and Romanian (even though the last is a Romance language), the conditional mood is used in both the apodosis and the protasis. A further example is a sentence "I would buy a house if


id: grammatical-mood-20260505-2
file: data/en.wikipedia.org/wiki/Grammatical_mood-2.md
category: reference
title: Grammatical mood
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Grammatical_mood
date_saved: 2026-05-05T08:11:41.638208+00:00
instance: kb-cron
parent: grammatical-mood-20260505-0
summary:
The presumptive mood is used to express presupposition or hypothesis, regardless of the fact denoted by the verb, as well as other more or less similar attitudes: doubt, curiosity, concern, condition, indifference, and inevitability. It is used in Romanian, Hindi, Gujarati, and Punjabi. In Romanian, the presumptive mood conjugations of the verb vrea are used with the infinitive form of verbs. The present tense and the past tense infinitives are respectively used to form the present and the past tense of the presumptive mood. In Hindi, the presumptive mood conjugations of the verb honā (to be) are used with the perfective, habitual, and progressive aspectual participles to form the perfective presumptive, habitual presumptive, and the progressive presumptive moods. The same presumptive mood conjugations are used for present, future, and past tenses. Note: The translations are just the closest possible English approximations and not exact. Only masculine conjugations are shown for Hindi. === Hypothetical === A few languages use a hypothetical mood, which is used in sentences such as "you could have cut yourself", representing something that might have happened but did not. === Inferential === The inferential mood is used to report unwitnessed events without confirming them. Often, there is


id: greenberg-s-linguistic-universals-20260505-0
file: data/en.wikipedia.org/wiki/Greenberg's_linguistic_universals-0.md
category: reference
title: Greenberg's linguistic universals
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Greenberg's_linguistic_universals
date_saved: 2026-05-05T08:11:42.812249+00:00
instance: kb-cron
parent: greenberg-s-linguistic-universals-20260505-0
summary:
The American linguist Joseph Greenberg (1915–2001) proposed a set of linguistic universals based primarily on a set of 30 languages. The following list is verbatim from the list printed in the appendix of Greenberg's Universals of Language and "Universals Restated", sorted by context. The numbering is fixed to keep Greenberg's number associations as these are commonly referenced by number; e.g.: "Greenberg's linguistic universal number 12". == Typology == "In declarative sentences with nominal subject and object, the dominant order is almost always one in which the subject precedes the object." "In languages with prepositions, the genitive almost always follows the governing noun, while in languages with postpositions it almost always precedes." "Languages with dominant VSO order are always prepositional." "With overwhelmingly greater than chance frequency, languages with normal SOV order are postpositional." "If a language has dominant SOV order and the genitive follows the governing noun, then the adjective likewise follows the noun." "All languages with dominant VSO order have SVO as an alternative or as the only alternative basic order." == Syntax == == Morphology == == References ==


id: index-of-linguistics-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_linguistics_articles-0.md
category: reference
title: Index of linguistics articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_linguistics_articles
date_saved: 2026-05-05T08:11:57.009286+00:00
instance: kb-cron
parent: index-of-linguistics-articles-20260505-0
summary:
Linguistics is the scientific study of human language. Someone who engages in this study is called a linguist. == A == Abbreviation - Abessive case - Ablaut - Absolutive case - Abugida - Accusative case - Acute accent - Accent (phonetics) - Accent (sociolinguistics) - Acronym - Adessive case - Adjective - Adjunct - Adposition - Adpositional phrase - Adverb - Adverbial - Adverbial phrase - Affix - Affricate consonant - Agglutination - Agglutinative language - Allative case - Allomorph - Allophone - Alphabet - Analytic language - Anaphora - Animacy - Anthropological linguistics - Alveolar consonant - Antonym - Aorist - Applied linguistics - Approximant - Areal feature - Article - Articulatory gestures - Articulatory phonetics - Aspect - Asterisk - Attrition - Attraction - Augment (Bantu languages) - Augment (Indo-European) - Auxiliary verb == B == Back-formation - Backronym - Bilabial consonant - Breathy voice - Breve == C == Calque - Capitalization - Capitonym - Cardinal vowel - Case - Case in tiers - Cedilla - Chiasmus - Circumfix - Circumflex - Clefting - Click consonant - Closed-class word - Cognate - Cognitive science - Coherence - Colloquialism - Comitative case - Comparative - Comparative linguistics -


id: iron-metaphor-20260505-0
file: data/en.wikipedia.org/wiki/Iron_(metaphor)-0.md
category: reference
title: Iron (metaphor)
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Iron_(metaphor)
date_saved: 2026-05-05T08:11:58.214234+00:00
instance: kb-cron
parent: iron-metaphor-20260505-0
summary:
Iron, when used metaphorically, refers to certain traits of the metal iron. Used as an adjective and sometimes as a noun, it refers to something stern, harsh, strong, unyielding, inflexible, rigid, sturdy, strong, robust, hard. == List of iron metaphors == === Persons and characters === Iron Duke (disambiguation) Iron Felix (disambiguation) Iron Lady (disambiguation) Iron Man (disambiguation) Iron Mike (disambiguation) Princess Iron Fan (disambiguation) Otto von Bismarck, known as the Iron Chancellor Iron Chef, a Japanese cooking show Joseph LaFlesche, known as Iron Eye Iron Eyes Cody, Sicilian-American actor The Iron Heinrich, a rough translation of the German fairy tale The Frog Prince Iron Heroes, a variant book of rules for the game Dungeons & Dragons Iron John, a German fairy tale Eon Kid, English title of the TV series Iron Kid Iron Lad, a fictional superhero Iron Maniac, an evil alternative universe character of fictional Marvel superhero Iron Man Iron Munro, a fictional superhero Persephone, known as the Iron Queen Hossein Khosrow Ali Vaziri, known as the Iron Sheik Iron Shell, Brule Sioux chief Iron Tail, Oglala Sioux warrior Tigran Petrosian, known as Iron Tigran Timur as a name, meaning iron in Turkic languages, widespread in Western and


id: list-of-language-creators-20260505-0
file: data/en.wikipedia.org/wiki/List_of_language_creators-0.md
category: reference
title: List of language creators
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_language_creators
date_saved: 2026-05-05T08:11:59.391484+00:00
instance: kb-cron
parent: list-of-language-creators-20260505-0
summary:
A language creator, sometimes referred to as a conlanger (), is a person who invents constructed languages (or "conlangs"). == Professional language creators == Individuals who have been hired to create languages. Victoria Fromkin - Paku (a.k.a. Pakuni) Paul Frommer - Na'vi, Barsoomian Madhan Karky - Kiliki Marc Okrand - Klingon, Vulcan, Atlantean Matt Pearson - Thhtmaa David J. Peterson - Dothraki, Valyrian, Kastithanu (Castithan), L'Irathi (Irathient), Indojisnen, Sondiv, Shiväisith, Lishepus, Trigedasleng, Noalath, Inha, Munja'kin John H. G. Scott - Baldung (for Archive 81) Britton Watkins - Yautja (for Predator: Badlands) Wolf Wikeley - Tho Fan == Published international auxiliary language creators == Individuals who have created languages intended for international communication. George Boeree: Lingua Franca Nova Léon Bollack: Bolak Louis Couturat and Louis de Beaufront: Ido language Alexander Gode: Interlingua Lancelot Hogben: Interglossa Otto Jespersen: Novial Francis Lodwick: Common Writing Jackson Moore: Moss François Sudre: Solresol Charles Kay Ogden: Basic English Giuseppe Peano: Latino sine flexione Waldemar Rosenberger: Idiom Neutral Johann Martin Schleyer: Volapük Kenneth Searight: Sona Edgar de Wahl: Interlingue John Wilkins: unnamed universal language L. L. Zamenhof: Esperanto == Published zonal auxiliary language creators == Individuals who have created zonal auxiliary languages intended to facilitate communication speakers


id: list-of-latin-script-alphabets-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Latin-script_alphabets-0.md
category: reference
title: List of Latin-script alphabets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Latin-script_alphabets
date_saved: 2026-05-05T08:12:01.086550+00:00
instance: kb-cron
parent: list-of-latin-script-alphabets-20260505-0
summary:
The lists and tables below summarize and compare the letter inventories of some of the Latin-script alphabets. In this article, the scope of the word "alphabet" is broadened to include letters with tone marks, and other diacritics used to represent a wide range of orthographic traditions, without regard to whether or how they are sequenced in their alphabet or the table. Parentheses indicate characters not used in modern standard orthographies of the languages, but used in obsolete and/or dialectal forms. == Letters contained in the ISO basic Latin alphabet == === Alphabets that contain only ISO basic Latin letters === Among alphabets for natural languages the English,[36] Indonesian, and Malay alphabets only use the 26 letters in both cases. Among alphabets for constructed languages the Ido and Interlingua alphabets only use the 26 letters, while Toki Pona uses a 14-letter subset. ==== Extended by ligatures ==== German (ß), Scandinavian (æ) ==== Extended by diacritical marks ==== Spanish (ñ), German (ä, ö, and ü) ==== Extended by multigraphs ==== Filipino (ng) Old French (ch) === Alphabets that contain all ISO basic Latin letters === Among alphabets for natural languages the Afrikaans,[54] Aromanian, Azerbaijani (some dialects),[53] Basque,[4] Celtic British, Catalan,[6] Cornish, Czech,[8]


id: list-of-latin-script-alphabets-20260505-1
file: data/en.wikipedia.org/wiki/List_of_Latin-script_alphabets-1.md
category: reference
title: List of Latin-script alphabets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Latin-script_alphabets
date_saved: 2026-05-05T08:12:01.086550+00:00
instance: kb-cron
parent: list-of-latin-script-alphabets-20260505-0
summary:
=== Notes === ↑↑↑↑ In classical Latin, the digraphs ⟨ch⟩, ⟨ph⟩, ⟨rh⟩, ⟨th⟩ were used in loanwords from Greek, but they were not included in the alphabet. The ligatures ⟨æ⟩, ⟨œ⟩ and ⟨w⟩, as well as lowercase letters, were added to the alphabet only in Middle Ages. The letters ⟨j⟩ and ⟨u⟩ were used as typographical variants of ⟨i⟩ and ⟨v⟩, respectively, roughly until the Enlightenment. ↑↑↑↑ In Afrikaans, ⟨c⟩ and ⟨q⟩ are only (and ⟨x⟩ and ⟨z⟩ almost only) used in loanwords. ↑↑↑↑ Albanian officially has the digraphs ⟨dh, gj, ll, nj, rr, sh, th, xh, zh⟩, which is sufficient to represent the Tosk dialect. The Gheg dialect supplements the official alphabet with 6 nasal vowels, namely ⟨â, ê, î, ô, û, ŷ⟩. ↑↑↑↑ Arbëresh officially has the digraphs ⟨dh, gj, hj, ll, nj, rr, sh, th, xh, zh⟩. Arbëresh has the distinctive ⟨hj⟩, which is considered as a letter in its own right. ↑↑ Achomi also has the digraph ⟨a'⟩. ↑↑↑↑ Azeri only uses the letter ⟨ä⟩ as a substitute for ⟨ə⟩ if the latter cannot be used (it was replaced by the schwa one year later because it is the most common letter). These cases should be


id: list-of-latin-script-alphabets-20260505-2
file: data/en.wikipedia.org/wiki/List_of_Latin-script_alphabets-2.md
category: reference
title: List of Latin-script alphabets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Latin-script_alphabets
date_saved: 2026-05-05T08:12:01.086550+00:00
instance: kb-cron
parent: list-of-latin-script-alphabets-20260505-0
summary:
Many pairs or triplets are read as digraphs or trigraphs depending on context, but are not treated as such lexicographically: consonants ⟨ph, (ng), th, gu/gü, qu, ce, ch/(sh/sch), rh⟩; vocal vowels ⟨(ee), ai/ay, ei/ey, eu, au/eau, ou⟩; nasal vowels ⟨ain/aim, in/im/ein, un/um/eun, an/am, en/em, om/on⟩; the half-consonant -⟨(i)ll⟩-; half-consonant and vowel pairs ⟨oi, oin/ouin, ien, ion⟩. When rules that govern the French orthography are not observed, they are read as separate letters, or using an approximating phonology of a foreign language for loan words, and there are many exceptions. In addition, most final consonants are mute (including those consonants that are part of feminine, plural, and conjugation endings). ⟨ÿ⟩ and ⟨ü⟩ are only used in certain geographical names and proper names plus their derivatives, or, in the case of ⟨ü⟩ with diaeresis, newly proposed reforms, e.g. capharnaüm 'shambles' is derived from the proper name Capharnaüm. ⟨æ⟩ occurs only in Latin or Greek loanwords. ↑ Fula has ⟨x⟩ as part of the alphabet in all countries except Guinea, Guinea-Bissau, Liberia, and Sierra Leone (used only in loanwords in these countries). ⟨ɠ⟩, which is used only in loanwords (but still part of the alphabet), is used in Guinea only. Fula also uses


id: list-of-latin-script-alphabets-20260505-3
file: data/en.wikipedia.org/wiki/List_of_Latin-script_alphabets-3.md
category: reference
title: List of Latin-script alphabets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Latin-script_alphabets
date_saved: 2026-05-05T08:12:01.086550+00:00
instance: kb-cron
parent: list-of-latin-script-alphabets-20260505-0
summary:
Y is used only in certain dialects and not in the standard language. F and H are only used in loanwords. ↑↑↑↑ A nearby language, Pite Sami, uses Lule Sami orthography but also includes the letters ⟨đ⟩ and ⟨ŧ⟩, which are not in Lule Sami. ↑↑↑↑ Lithuanian also has the digraphs: ⟨ch, dz, dž, ie, uo⟩. However, these are not considered separate letters of the alphabet. F, H, and the digraph CH are only used in loanwords. Demanding publications such as dictionaries, maps, schoolbooks etc. need additional diacritical marks to differentiate homographs. Using grave accent on A, E, I, O, U, acute accent on all vowels, and tilde accent on all vowels and on L, M, N and R. Small E and I (also with ogonek) must retain the dot when additional accent mark is added to the character; the use of ì and í (with missing dot) is considered unacceptable. ↑↑↑↑ In Livonian, the letters Ö, Ȫ, Y, Ȳ were used by the older generation, but the younger generation merged these sounds; Around the late 1990s, these letters were removed from the alphabet. ↑↑↑↑ Maltese also has the digraphs: ⟨ie, għ⟩. ↑ Māori only uses ⟨g⟩ in ⟨ng⟩ digraph.


id: list-of-latin-script-alphabets-20260505-4
file: data/en.wikipedia.org/wiki/List_of_Latin-script_alphabets-4.md
category: reference
title: List of Latin-script alphabets
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Latin-script_alphabets
date_saved: 2026-05-05T08:12:01.086550+00:00
instance: kb-cron
parent: list-of-latin-script-alphabets-20260505-0
summary:
⟨i⟩ or ⟨u⟩] and, when both are closed, generally over the first". In this context, the use of ⟨ï⟩ is rare, but part of the normative orthography. ↑ Swedish uses ⟨é⟩ in well integrated loan words like idé and armé, although ⟨é⟩ is considered a modified ⟨e⟩, while ⟨å⟩, ⟨ä⟩, ⟨ö⟩ are letters. ⟨á⟩ and ⟨à⟩ are rarely used words. ⟨w⟩ and ⟨z⟩ are used in some integrated words like webb and zon. ⟨q⟩, ⟨ü⟩, ⟨è⟩, and ⟨ë⟩ are used for names only, but exist in Swedish names. For foreign names ⟨ó⟩, ⟨ç⟩, ⟨ñ⟩ and more are sometimes used, but usually not. Swedish has many digraphs and some trigraphs. ⟨ch, dj, lj, rl, rn, rs, sj, sk, si, ti, sch, skj, stj⟩ and others are usually pronounced as one sound. ↑↑↑↑ Tswana also has the digraphs: ⟨kg, kh, ng, ph, th, tl, tlh, ts, tsh, tš, tšh⟩. The letters ⟨c⟩, ⟨q⟩, and ⟨x⟩ only appear in onomatopoeic and loanwords. The letters ⟨v⟩ and ⟨z⟩ only appear in loanwords. ↑↑↑↑ Turkmen had a slightly different alphabet in 1993–1995 (which used some rare letters) ⟨ý⟩ was written as ⟨ÿ⟩ (capital ⟨¥⟩), ⟨ň⟩ as ⟨ñ⟩, ⟨ş⟩ as ⟨¢⟩ (capital ⟨$⟩), and ⟨ž⟩ as


id: leipzig-jakarta-list-20260505-0
file: data/en.wikipedia.org/wiki/Leipzig–Jakarta_list-0.md
category: reference
title: Leipzig–Jakarta list
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Leipzig–Jakarta_list
date_saved: 2026-05-05T08:12:02.231607+00:00
instance: kb-cron
parent: leipzig-jakarta-list-20260505-0
summary:
The Leipzig–Jakarta list of 100 words is used by linguists to test the degree of chronological separation of languages by comparing words that are resistant to borrowing. The Leipzig–Jakarta list became available in 2009. The word list is named after the cities of Leipzig, Germany, and Jakarta, Indonesia, the places where the list was conceived and created. In the 1950s, the linguist Morris Swadesh published a list of 200 words called the Swadesh list, allegedly the 200 lexical concepts found in all languages that were least likely to be borrowed from other languages. Swadesh later whittled his list down to 100 items. The Swadesh list, however, was based mainly on intuition, according to Martin Haspelmath and Uri Tadmor. In origin, the words in the Swadesh lists were chosen for their universal, culturally independent availability in as many languages as possible, regardless of their "stability". Nevertheless, the stability of the resulting list of "universal" vocabulary under language change and the potential use of this fact for purposes of glottochronology have been analyzed by numerous authors, including Marisa Lohr 1999, 2000. The Swadesh list was put together by Morris Swadesh on the basis of his intuition. Similar more recent lists, such as


id: list-of-language-reforms-of-english-20260505-0
file: data/en.wikipedia.org/wiki/List_of_language_reforms_of_English-0.md
category: reference
title: List of language reforms of English
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_language_reforms_of_English
date_saved: 2026-05-05T08:12:03.341301+00:00
instance: kb-cron
parent: list-of-language-reforms-of-english-20260505-0
summary:
For centuries, many people have called for language reforms of English, which vary in approach from the radical (completely overhauling existing conventions) to the conservative (preserving most while removing irregularities). Phonetic alphabets limited to English do not belong here. See Category:Phonetic alphabets. == Spelling reforms == Spelling reforms are attempts to regularize English spelling either by reducing the number of irregularities or by making it completely phonemic. This may be done using the existing basic English alphabet more uniformly (basic), by adding diacritics (diacriticized), by adding new letters (extended), by removing letters in the script or writing (squeezed), by removing letters and replacing it by new ones (altered) or by replacing it entirely (replaced). Such proposals include: == Subsets == Subsets are reforms that use a restricted wordlist and grammar. English subsets include: Attempto Controlled English Aviation English Basic English ClearTalk Common Logic Controlled English E-Prime Gellish Formal English Globish Learning English Logical English ModeLang Newspeak (fictional) Plain English Processable English (PENG) Seaspeak SBVR Structured English Simplified Technical English Specialized English == Vocabulary reforms == Vocabulary reforms seek to reform English by changing or restricting its words without changing its grammar. Anglish: the use of native (Germanic) words only, and


id: list-of-linguists-20260505-0
file: data/en.wikipedia.org/wiki/List_of_linguists-0.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
A linguist in the academic sense is a person who studies natural language (an academic discipline known as linguistics). Ambiguously, the word is sometimes also used to refer to a polyglot (one who knows several languages), a translator/interpreter (especially in the military), or a grammarian (a scholar of grammar), but these uses of the word are distinct (and one does not have to be multilingual in order to be an academic linguist). The following is a list of notable academic linguists. In the list, the description is presented as: surname, given name (country, year of birth–year of death), main achievement == A == Abbott, Barbara (United States, 1943–), pragmatics, semantics Abel, Carl (Germany, 1837–1906), comparative lexicography Abdul Haq, Maulvi (India, 1870–1961), Urdu language Abrams, Lise (United States), psycholinguistics Abramson, Arthur S. (United States, 1925–2017), phonetics Adamou, Evangelia (France), language contact Adams, Douglas Q. (United States), English language, comparative linguistics, Tocharian language Adler, George J. (Germany/United States, 1821–1868), lexicography, German language, English language Aijmer, Karin (Sweden, 1939–), pragmatics, semantics Aikhenvald, Alexandra Yurievna (Russia, 1957–), syntax, typology, Amazonian languages, Papuan languages, Hebrew language, Russian language Aitken, Adam Jack (UK, 1921–1998), lexicography Ajduković, Jovan (Serbia, 1968–), Slavic languages, sociolinguistics, contact linguistics, Russian language,


id: list-of-linguists-20260505-1
file: data/en.wikipedia.org/wiki/List_of_linguists-1.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== C == Caland, Willem (Netherlands, 1859–1932), historical linguistics, Sanskrit Campbell, Lyle (United States, 1942–), Native American languages, historical linguistics Canger, Una (Denmark, 1938–), Mesoamerican languages Canonge, Elliott D. (United States, 1921–1971), Comanche, Inuit Capell, Arthur (Australia, 1902–1986), Australian languages, Austronesian languages, Papuan languages Cardona, George (United States, 1936–), Indo-European studies Carnap, Rudolf (Germany, 1891–1970), syntax, constructed languages Carnie, Andrew (Canada, 1969–), syntax Caro Baroja, Julio (1914–1995), Basque language Caro, Miguel A. (Colombia, 1843–1909), Spanish language, Colombian Spanish Carpenter, William Henry (United States, 1853–1936), Icelandic language Cart, Théophile (France, 1855–1931), Esperanto Carter, Hazel (UK, 1928–2016), Bantu languages Catford, J. C. (UK, 1917–2009), phonetics Chadwick, John (UK, 1920–1998), Linear B Chafik, Mohamed (Morocco, 1926–), writer and specialist in the Berber or Amazigh languages, dialects, and literature Chafe, Wallace (United States, 1927–2019), cognitive linguistics, semantics Chao Yuen Ren (China, 1892–1982), Chinese language Chakrabarti, Byomkes (India, 1923–1981), Santali language, Bengali language, comparative linguistics Champollion, Jean-François (France, 1790–1832), Egyptian hieroglyphs Chambers, Jack (Canada, 1938–), sociolinguistics Chatterji, Suniti Kumar (India, 1890–1977), Bengali language Chelliah, Shobhana, Tibeto-Burman languages Chierchia, Gennaro (Italy, 1953–), semantics, pragmatics Choijinzhab (PR China, 1931–2022), Mongolian language Chomsky, Noam (United States, 1928–), syntax, universal grammar Choueiri, Lina (Lebanon), syntax, Lebanese Arabic Chyet, Michael


id: list-of-linguists-20260505-2
file: data/en.wikipedia.org/wiki/List_of_linguists-2.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== F == Fairuzabadi (Iran, 1329–1414), lexicography, Arabic grammar Farhady, Hossein (Iran, born 1947), Iranian applied linguist Fassi Fehri, Abdelkader (Morocco, 1947–), Arabic syntax Fierz-David, Linda (Germany, 1891–1955), philology Fillmore, Charles J. (United States, 1929–2014), syntax, lexical semantics, cognitive linguistics, lexicography Firbas, Jan (Czech Republic, 1921–2000), English language, functional linguistics Firth, John Rupert (UK, 1890–1960), phonetics, phonology, prosody Fischer-Jørgensen, Eli (Denmark 1911–2010), phonetics, phonology, Danish language Fishman, Joshua (United States, 1926–2015), sociology of language Fiske, Willard (United States, 1831–1904), Northern European languages, Icelandic language Fodor, Janet Dean (United States, 1942–2023), psycholinguistics, semantics, syntax Fodor, Jerry Alan (United States, 1935–2017), psycholinguistics, language of thought Foley, William A. (Australia, 1949–), Papuan languages, Austronesian languages Ford, Jeremiah Denis Mathias (United States, 1873–1958), Spanish language Fowler, Carol A. (United States), phonetics, phonology François, Alexandre (France), Austronesian languages, historical linguistics, language contact Freiman, Aleksandr Arnoldovich (Poland/Russia, 1879–1968), Iranian languages French, David Heath (United States, 1918–1994), Native American languages Friedrich, Johannes (Germany, 1893–1972), Hittite language Fromkin, Victoria (United States, 1923–2000), theoretical linguistics, constructed languages Fujitani Nariakira (Japan, 1738–1779), Japanese language == G == Galloway, Brent D. (United States, 1944–2014), Amerindian languages, Halkomelem language Gamkrelidze, Thomas V. (Georgia, 1929–2021), Indo-European studies, Georgian language Gans, Eric (United States,


id: list-of-linguists-20260505-3
file: data/en.wikipedia.org/wiki/List_of_linguists-3.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== H == Haarmann, Harald (Germany, 1946–), evolutionary linguistics, language contact Haas, Mary Rosamund (United States, 1910–1996), Native American languages, Thai language, historical linguistics Haase, Martin (Germany, 1962–), Romance languages Hagberg, Carl August (Sweden, 1810–1864), Scandinavian languages Hajič, Jan (Czech Republic), computational linguistics Hajičová, Eva (Czech Republic, 1935–), corpus linguistics Hale, Kenneth Locke (United States, 1934–2001), syntax, phonology Hall, Kira (United States, 1962–), sociocultural linguistics Hall, Robert A., Jr. (United States, 1911–1997), Romance languages, Pidgins and Creoles Halle, Morris (Latvia/United States, 1923–2018), phonology, morphology Halliday, Michael Alexander Kirkwood (UK/Australia, 1925–2018), systemic functional grammar, ecolinguistics, applied linguistics Hammarström, Harald (Sweden, 1977–), computational linguistics, historical linguistics, linguistic typology Hammond, Michael (United States, 1957–), phonology, computational linguistics, syntax Hamp, Eric P. (United States, 1920–2019), Indo-European languages, Native American languages Hanzeli, Victor (United States, 1925–1991), Romance languages Haq, Mehr Abdul (Pakistan, 1915–1995), Saraiki language Harder, Peter (Denmark, 1950–), English language, functional linguistics Harkavy, Alexander (Belarus/United States, 1863–1939), Yiddish language, lexicography Harley, Heidi B. (United States, 1969–), distributed morphology, syntax Harrington, John Peabody (United States, 1884–1961), Native American languages, phonetics Alice C. Harris (United States, 1947-), Kartvelian languages, historical linguistics Harris, Roy (UK, 1931–2015), semiology, integrationism Harris, Zellig Sabbetai (Ukraine/United States, 1909–1992), structural linguistics, discourse


id: list-of-linguists-20260505-4
file: data/en.wikipedia.org/wiki/List_of_linguists-4.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== K == Kaplan, Ronald M. (United States, 1946—), computational linguistics Karadžić, Vuk Stefanović (Serbia, 1787—1864), Serbian language, lexicography Kari, James (United States), Native American languages Kasravi, Ahmad (Iran, 1890—1946), ancient languages, Iranian languages Katz, Jerrold J. (United States, 1932—2002), semantics, generative grammar Kaufman, Terrence (United States, 1937—2022), historical linguistics, contact linguistics, Mesoamerican languages Kay, Martin (UK, United States, 1935–2021), computational linguistics Kay, Paul (United States, 1934—), construction grammar Kayne, Richard S. (United States, 1944—), syntax, transformational grammar Kazama Kiyozō (Japan, 1928–), Japanese language Keating, Patricia (United States, 1952—), phonetics Kellogg, Samuel H. (United States, 1839–1899), Hindi language Kenyon, John Samuel (United States, 1874—1959), English language, lexicography, phonology Keyser, Samuel Jay (United States, 1935–), phonology, English language Kiesling, Scott Fabius (United States), sociolinguistics Kindaichi Haruhiko (Japan, 1913–2004), Japanese language Kindaichi Kyōsuke (Japan, 1882–1971), Ainu language Kinkade, M. Dale (United States, 1933—2004), Salishan languages Kiparsky, Paul (Finland/United States, 1941–), phonology, morphology Kirby, Simon (UK), computational linguistics, evolutionary linguistics Klima, Edward (United States, 1931—2008), sign language Klinkenberg, Jean-Marie (Belgium, 1944–), rhetorics, semiotics, stylistics Kloekhorst, Alwin (Netherlands, 1978–), Anatolian languages Knechtges, David R. (United States, 1942–), East Asian languages, Chinese language Knorozov, Yuri Valentinovich (Russia, 1922—1999), Maya hieroglyphics, writing systems Kober, Alice (UK/United


id: list-of-linguists-20260505-5
file: data/en.wikipedia.org/wiki/List_of_linguists-5.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== M == MacWhinney, Brian (United States, 1945–), language acquisition, second-language acquisition, corpus linguistics Maddieson, Ian (United States, 1942–), phonetics Fray Francisco Maldonado (1571– c. 1640), Guatemalan Franciscan linguist and historian Manzini, Rita (Italy), generative syntax, Romance languages Malkiel, Yakov (United States, 1914–1998), etymology, philology Manaster Ramer, Alexis (United States/Poland, 1956–), phonology, syntax, poetics, etymology Marantz, Alec (United States, 1959–), distributed morphology March, Francis Andrew (United States, 1825–1911), comparative linguistics, lexicography, Old English language, English language Margolis, Max Leopold (Lithuania/United States, 1866–1932), Semitic languages Marr, Nikolay Yakovlevich (Georgia/Russia, 1865–1934), historical linguistics, comparative linguistics, origin of language Martin, James (Sydney, Australia, 1950–), genre Martin, Samuel Elmo (United States, 1924–2009), Korean language, Japanese language Martinet, André (France, 1908–1999), structuralism, historical linguistics, constructed languages Martinet, Jeanne (France, 1920–2018), semiotics, constructed languages Massam, Diane (Canada), syntax, Austronesian languages Matasović, Ranko (Croatia, 1968–), historical linguistics, Celtic languages Mathesius, Vilém (Czech Republic, 1882– 1945), phonology, syntax, English language, Czech language Matisoff, James A. (United States, 1937–), Tibeto-Burman languages, phonology Matthews, Peter Hugoe (UK, 1934–2023), morphology, syntax Matthews, Stephen (UK/PR China), typology, syntax, semantics, Cantonese language Mattingly, Ignatius G. (United States, 1927–2004), phonetics, speech synthesis, speech perception Matveyev, Aleksandr (Russia, 1926–2010), onomastics, etymology McArthur, Tom (UK, 1938–2020),


id: list-of-linguists-20260505-6
file: data/en.wikipedia.org/wiki/List_of_linguists-6.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== P == Pāṇini (India, ca. 520–460 BC), Sanskrit, morphology, descriptive linguistics, generative linguistics Frank R. Palmer (England, 1922–2019), semantics, English language, Ethiopian languages Partee, Barbara Hall (United States, 1940–), semantics Paul, Hermann Otto Theodor (Germany, 1846–1921), lexicography, German language Pawley, Andrew Kenneth (Australia/New Zealand, 1941–2026), Austronesian languages, Papuan languages, lexicography, phraseology Peacock, Dmitri Rudolf (Britain, 1842–1892), Kartvelian languages Pedersen, Holger (Denmark, 1867–1953), Celtic languages, historical linguistics, Nostratic languages Pedersen, Johannes (Denmark, 1883–1977), Hebrew language Pei, Mario Andrew (Italy/United States, 1901–1978), Italian language, Indo-European languages Pesetsky, David Michael (United States, 1957–), transformational grammar Keith Malcolm Petyt (UK, 1941–), sociolinguistics Phillipson, Robert (UK/Denmark, 1942–), language policy Pierrehumbert, Janet (United States, 1954–), phonetics, phonology Piller, Ingrid (Australia, 1967–), applied linguistics, sociolinguistics, intercultural communication Pinault, Georges-Jean (France, 1955–), Tokharian, Indo-European linguistics Pike, Kenneth Lee (United States, 1912–2000), English language, constructed languages, tagmemics Pilch, Herbert (Germany, 1927–2018), Old English, Celtic languages, phonetics Pimsleur, Paul (United States, 1927–1976), language acquisition, French language, phonetics Pinker, Steven (Canada/United States, 1954–), language acquisition, syntax, semantics Piron, Claude (Switzerland, 1931–2008), Esperanto, psycholinguistics Polinsky, Maria (United States), syntax, Austronesian languages Pollard, Carl Jesse (United States, 1947–), syntax, semantics Pollock, Jean-Yves (France, 1946–), syntax Poplack, Shana (United States), sociolinguistics Poppe, Nicholas


id: list-of-linguists-20260505-7
file: data/en.wikipedia.org/wiki/List_of_linguists-7.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== S == Sacks, Harvey (United States, 1935–1975), conversation analysis Sadock, Jerrold (United States), syntax, morphology, pragmatics, Greenlandic language, Yiddish language Sag, Ivan (United States, 1949–2013), syntax, construction grammar Sagart, Laurent (France, 1951–), Chinese linguistics and Austronesian languages Sakaguchi, Alicja (Poland/Germany, 1954–), interlinguistics, Esperanto Shakirova, Liya (Soviet Union/Russia, 1921–2015), Russian language Salo, David (United States, 1969–), constructed languages, Tocharian languages, Elvish languages Sampson, Geoffrey (UK, 1944–), philosophy of language Sánchez Carrión, José María (Spain, 1952–), Basque language, sociolinguistics, historical linguistics Sanders, Ted (Netherlands, 1963–), Coherence in text, discourse Sankoff, Gillian (Canada, 1943–), sociolinguistics Sankrityayan, Rahul (India, 1893–1963), Tibetan language, Hindi language Sapir, Edward (Germany/United States, 1884–1939), Native American languages, constructed languages, semantics Sarkar, Prabhat Ranjan (India, 1921–1990), Transliteration, Science of Letters Saunders, Irene (United States/PR China), lexicography, Chinese language de Saussure, Ferdinand (Switzerland/France, 1857–1913), semantics, Indo-European studies, structural linguistics Saxon, Leslie (Canada), First Nations languages Sayce, Archibald Henry (UK, 1846–1933), Akkadian language Schegloff, Emanuel (United States, 1937–), conversation analysis Schleicher, August (Germany, 1821–1868), Indo-European studies, language development, historical linguistics Schmidt, Johannes (Germany, 1843–1901), historical linguistics, Indo-European studies Schmidt, Richard (United States, 1941–2017), second-language acquisition Schmidt, Wilhelm (Germany/Austria/Switzerland, 1868–1954), Mon–Khmer languages Searle, John Rogers (United States, 1932–), philosophy of language, pragmatics


id: list-of-linguists-20260505-8
file: data/en.wikipedia.org/wiki/List_of_linguists-8.md
category: reference
title: List of linguists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_linguists
date_saved: 2026-05-05T08:12:04.693680+00:00
instance: kb-cron
parent: list-of-linguists-20260505-0
summary:
== W == Wackernagel, Jacob (Switzerland, 1853–1938), Indo-European studies, Sanskrit Wang Li (PR China, 1900–1986), Chinese language Watanabe Shōichi (Japan, 1930–2017), Japanese language Watkins, Calvert (United States, 1933–2013), comparative linguistics, Indo-European languages Weeks, Raymond (United States, 1863–1954), phonetics, French language Weinreich, Max (Latvia/United States, 1893– 1969), Yiddish language Weinreich, Uriel (Poland/United States, 1926–1967), sociolinguistics, dialectology, semantics, Yiddish language Wells, John Christopher (UK, 1939–), phonetics, Esperanto Westermann, Diedrich Hermann (Germany, 1875–1956), languages of Africa, typology Westphal, Ernst Oswald Johannes (South Africa/UK, 1919–1990), Bantu languages, Khoisan languages Whalen, Douglas H. (United States), phonology, endangered languages Wheeler, Benjamin Ide (United States, 1854–1927), historical linguistics, comparative linguistics, Greek language White, Lydia (UK/Canada, 1946–), second-language acquisition Whitney, William Dwight (United States, 1827–1894), lexicography, Sanskrit, English language Wilks, Yorick (UK, 1939–2023), artificial intelligence, computational linguistics, natural language processing, semantics Whorf, Benjamin Lee (United States, 1897–1941), Native American languages, Maya script, linguistic relativity Wichmann, Søren (Denmark, 1964–), Mesoamerican languages, Mixe–Zoque languages, Mayan languages, Maya script Widdowson, Henry G. (UK, 1935–), English language, discourse analysis Wierzbicka, Anna (Poland/Australia, 1938–), semantics, pragmatics Wiese, Richard (Germany, 1953–), phonology, morphology, neurolinguistics Williams, Nicholas Jonathan Anselm (UK/Ireland, 1942–), Cornish language, Irish language, Manx language, phonology Williams, Samuel Wells (United States/China, 1812–1884), Chinese


id: lists-of-countries-and-territories-by-official-language-20260505-0
file: data/en.wikipedia.org/wiki/Lists_of_countries_and_territories_by_official_language-0.md
category: reference
title: Lists of countries and territories by official language
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lists_of_countries_and_territories_by_official_language
date_saved: 2026-05-05T08:12:05.950216+00:00
instance: kb-cron
parent: lists-of-countries-and-territories-by-official-language-20260505-0
summary:
This is a list of lists of countries and territories by official language. List of countries and territories where Afrikaans or Dutch are official languages List of countries and territories where Arabic is an official language List of countries and territories where Chinese is an official language List of countries and territories where English is an official language List of countries and territories where French is an official language List of countries and territories where German is an official language List of countries and territories where Persian is an official language List of countries and territories where Romanian is an official language List of countries and territories where Russian is an official language List of countries and territories where Spanish is an official language List of countries and territories where Tamil is an official language == Other official languages == === Greek === === Hindustani === === Italian === === Malay === === Portuguese === === Turkish === == See also == List of official languages by country and territory De facto == References ==


id: list-of-multilingual-countries-and-regions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-0.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
This is an incomplete list of areas with either multilingualism at the community level or at the personal level. There is a distinction between social and personal bilingualism. Many countries, such as Belarus, Belgium, Canada, Finland, India, Ireland, South Africa and Switzerland, which are officially multilingual, may have many monolinguals in their population (Canada exemplifies this situation, as it is a country in which western provinces are predominantly anglophone while the province of Quebec, along with the bilingual New Brunswick, emphasize French prominence in the eastern side). Officially monolingual countries, on the other hand, such as France, can have sizable multilingual populations. Some countries have official languages but also have regional and local official languages, notably Brazil, China, Indonesia, Mexico, Philippines, Russia, Spain and Taiwan. == Africa == === Central Africa === Cameroon: French and English (both official), as well as Cameroonian Pidgin. Many ethnic and tribal languages including Basaa, Duala, Manenguba, Bikya, Bung, Fula, Kanuri, Ngumba, Yeni, Bamum, Bafia, Bakweri language and many others. Some also have fluency in the German, Portuguese and Spanish languages. Central African Republic: French & Sango (both official) and 50 other African languages. Chad: French and Arabic (both official) + more than 100 African


id: list-of-multilingual-countries-and-regions-20260505-1
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-1.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== West Africa === Benin: French (official) and many indigenous languages including Fon, Yoruba & Songhay (specifically Dendi). Burkina Faso: French (official), Moore and Jula (regional languages) and indigenous Sudanic languages. Cape Verde: Portuguese and Cape Verdean Creole. Côte d'Ivoire: French (official), Baule, Jula, and 60 other indigenous languages. Gambia: English (official), Mandinka, Wolof, Fula and others. Ghana: English (official), Akan, Dagaare/Wale, Dagbane, Dangme, Ewe, Ga, Gonja, Kasem, Nzema and 70 others. Guinea: French (official), Maninka, Fula and Susu. Guinea-Bissau: Portuguese (official), Kriol and indigenous languages. Liberia: English (official) and 20 African languages. Mali: French (official), Bambara (most widely spoken), Fula and Songhay (specifically Dendi). 11 languages are used as mediums of instruction in primary schools Niger: French (official) plus ten other languages recognised as national ones, including Hausa (spoken by half the population) and Songhay (specifically Zarma) Nigeria: English (official), Yoruba, Hausa and Igbo are the four languages of the parliament (each of which has over 20 million speakers) plus 529 other African languages (some of which have over a million speakers) and Pidgin. Senegal: French (official), Wolof (widely spoken), Fula (specifically Pulaar), Diola, Malinké, Sérère, Soninké (national languages) plus other African languages Sierra Leone: English (official), Krio (most


id: list-of-multilingual-countries-and-regions-20260505-2
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-2.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== North America === Canada is officially bilingual under the Official Languages Act and the Constitution of Canada that require the federal government to deliver services in both official languages: English and French. As well, minority language rights are guaranteed where numbers warrant. 56.9% of the population speak English as their first language while 22.9% are native speakers of French. The remaining population belong to some of Canada's many immigrant populations or to the indigenous population. See Bilingualism in Canada Alberta has a specific French policy since 2017. The Canadian province of British Columbia has a sizable population that speaks Mandarin or Cantonese, particularly in the city of Vancouver and its satellite town of Richmond. There is a provincial law on First Nations languages. The Canadian province of New Brunswick, with a large Acadian population (33% French-speaking) is officially bilingual. The Canadian province of Quebec, (7.9% English-speaking) Note: Quebec's largest city, Montreal, is a multilingual city with half the population having French as their mother tongue, and the other half having other languages (including English) as their mother tongue (see Language demographics of Quebec). A majority of Montrealers, whether they call themselves francophone, anglophone or allophone, know both French and


id: list-of-multilingual-countries-and-regions-20260505-4
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-4.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== South America === Argentina has several ethnic communities of European, Asian and indigenous origins (the Andean and northeast regions), who speak their own languages, but uses de facto Spanish as the official language of the country. In most of the country, there is a sizable but bilingual Italian-speaking population. Chaco Province recognises Qom, Moqoit y Wichi as official languages of the province along with Spanish Corrientes Province Guarani is recognised as the second official language Santiago del Estero Province gives official recognition to Quechua in its constitution Bolivia is officially multilingual, supporting Spanish and 36 native languages. Brazil, Portuguese (official) and upwards to 100 languages spoken mainly in the urban areas (European and Asian) and indigenous languages in the Amazon. The use of indigenous languages in primary education is enshrined in the constitution. Espírito Santo – German and East Pomeranian are recognized by constitution as part of the state's cultural heritage Rio de Janeiro – Yoruba, Bantu and Jeje (African Languages) are recognized by constitution as part of the state's cultural heritage Rio Grande do Sul – Talian (Italian dialect) and Riograndense Hünsrick (German dialect) are recognized by constitution as part of the state's cultural heritage Santa Catarina –


id: list-of-multilingual-countries-and-regions-20260505-5
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-5.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== East Asia === In Mainland China, Standard Mandarin (Putonghua) is the official language and is spoken in all regions. It is used for official and formal purposes, by the media, and in education as the language of instruction. However, on money notes, there are texts both in Mandarin (Han) and in Mongolian, Tibetan, Uyghur, and Zhuang. In every locality and region, local varieties of Chinese are spoken in daily life. These range from being quite similar to Putonghua, such as Tianjin dialect, to those that are mutually unintelligible with Putonghua such as Jiangsu, Zhejiang and Shanghai dialect (Wu) or Guangzhou dialect (Cantonese). In the autonomous regions, minority languages are used (such as Tibetan in Tibet or Mongolian in Inner Mongolia, Uyghur, Kazakh and others in Xinjiang). Japan: a special law provides for promotion of Ainu in Hokkaido Taiwan: A national language in Taiwan is legally defined as "a natural language used by an original people group of Taiwan and the Taiwan Sign Language". This includes Formosan languages, Hakka, Mandarin and Taiwanese. In Hong Kong, English and Chinese are official languages. All road signs are written in both languages. English is the dominant language in the judiciary and in higher


id: list-of-multilingual-countries-and-regions-20260505-6
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-6.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
Afghanistan: Pashto and Dari (Afghan Persian) are the official and most widely spoken languages in Afghanistan, with the former serving as a national language and the latter spoken as a lingua franca. Other minor languages include Uzbek and Turkmen, Balochi and Pashayi, Nuristani (Askunu, Kamkata-viri, Vasi-vari, Tregami and Kalasha-ala), Pamiri (Shughni, Munji, Ishkashimi and Wakhi), Brahui, Hindko, Kyrgyz. Bangladesh: Noidien Bengali is the official and most widely spoken language. However, there are many local languages (some of which are considered Bengali dialects, like Seletino, Nueklikos and Chetgonien) spoken in different regions of Bangladesh, as well as minority languages. Speakers of these languages are often bilingual in their local language and Standard Bangla. Additionally, the use of English is widespread in education and the judiciary in the country. In the CHT region, tribal languages like Genmanoos, Mouge/Reccen, Cucbarooco, Meru, Tensankês, Bomsche/Bomiano, Shö, Xumi, Sac, Penkaans, Lušei/Mizo, et cetera are spoken. Chetgonien and Nodien Bengali are the main dialects spoken in lower Chittagong. Chetgonien has a noticeable influence from Portuguese due to the historical presence of Portuguese traders and settlers in the area, which is much more pronounced in Chetgonien Bengali than Nodien Bengali. Chetgonien, as well as Seletino and other Bengali


id: list-of-multilingual-countries-and-regions-20260505-7
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-7.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== Southeast Asia === Brunei: Malay (official) and English Cambodia: Khmer is the official language, but French is spoken by a minority and sometimes used in government and education. Mandarin is spoken in business and commerce. East Timor – Tetum and Portuguese are the official languages; English and Indonesian "shall be working languages within the public administration side by side with official languages as long as it is deemed necessary" Indonesia is the largest bilingual country in the world, with approximately 200 million people speak more than one language. Indonesians speak about 746 different languages. Javanese has the most users in terms of native speakers (about 80 million). However, the sole official (or so-called "unity language") is Indonesian which has only 30 million L1 speakers (compared to Indonesia 260 million population). The role of Indonesian is important to glue together different ethnics and languages in Indonesia. Though Indonesian is considered the nation's only official language, regional governments have rights to conduct regional languages study at schools. Many people in Indonesia are bilingual at an early age. They speak a local native language with their families whereas the official Indonesian language is used to communicate with people from other regions and


id: list-of-multilingual-countries-and-regions-20260505-8
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-8.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== Western Asia === Bahrain: Arabic is the official language, and English is the most commonly used and studied second language, especially in education, international relations, and the media. In addition, Persian and Urdu are widely spoken. Cyprus has two official languages: Greek and Turkish. Both languages were spoken throughout the island before 1974. After 1974, and the partition of the island, Turkish became the sole official language in the Turkish-Cypriot-controlled north whereas the internationally recognized Republic of Cyprus retains both languages as official. English is also widely spoken and understood throughout the island. Armenian and Arabic are the two official minority language. In Iran, Persian is the only official language, but Azerbaijani (along with related varieties such as Qashqa'i and Kalaj) has upwards of 20 million speakers. Other minority languages include Kurdish, Turkmen, and Balochi. Assyrian is spoken by a Christian minority in the vicinity of Urmia. In the southwestern Iranian province of Khuzestan, most people speak Khuzestani Persian, Khuzestani Arabic, and Standard Persian, sometimes in addition to their own community languages such as Lur, Qashqa'i, Domari or Mandaic where applicable. None of the non-Persian languages are taught at schools. In Iraq, Arabic is the official language of the


id: list-of-multilingual-countries-and-regions-20260505-9
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-9.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== Central Europe === Austria has one official language: German. However, it also has Croatian and Slovenian minorities, all of whose languages are protected under federal laws. Certain functions are also guaranteed for Romany, Hungarian, Czech and Slovak in Vienna and Burgenland, under the European Charter for Regional or Minority Languages. In the Czech Republic, several municipalities of Trans-Olza area have official bilingualism (Czech and Polish). Bilingual signs are permitted if a minority constitutes at least a 10% of the population of the municipality. German is recognized as a minority language because of the previous visible German presence in Bohemia. Germany has German as its official national language. Low German is recognized as a regional language in at least five north German states. Lower Sorbian is an official minority language in Brandenburg, Upper Sorbian in Saxony, Sater Frisian in a part of Lower Saxony, and North Frisian varieties and Danish in Schleswig-Holstein. A language without its own territory, Romany (including the language of the Sinte people) is an official minority language as well. Germany is home to large numbers of people from other regions, and some of their languages, such as Turkish, Russian, and Polish, are widely used throughout the


id: list-of-multilingual-countries-and-regions-20260505-10
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-10.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== Northern Europe === Denmark has one official language, Danish, but in South Jutland, use of German for certain functions is provided for. In Greenland, Greenlandic is the principal language, while Danish must be thoroughly taught, and all Greenlanders are Danish-Greenlandic bilinguals. Faroe Islands has two official languages: Faroese and Danish. The other Scandinavian languages, Norwegian and Swedish, are understood by most without much difficulty. English is taught in schools, often as a third language. Finland is constitutionally bilingual and has therefore two equally national languages, Finnish and Swedish, and the minority languages Sami (Northern Sami, Inari Sami and Skolt Sami), Romani and Finnish Sign Language are recognized by the constitution. The Swedish-speaking population of Finland comprises about 5.5% of the population, mostly in Svenskfinland. Municipalities are bilingual if the Swedish or Finnish minority is at least 6–8%. Åland is monolingually Swedish by law. Sami is an official language (besides Finnish) in the municipalities of northern Finland. In Norway six municipalities of Troms and Finnmark county, Sami is used officially along Norwegian. In addition, kvensk, romani and romanes have status as minority languages. Sweden has Swedish as its official language. Finnish, Meänkieli, Romani, Sami and Yiddish are recognized as minority


id: list-of-multilingual-countries-and-regions-20260505-11
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-11.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== Southeastern Europe === Albania has one official language, Albanian. In some regions of southern Albania, Greek serves as co-official. Other languages such as Italian and Greek are widely spoken throughout the country, and are considered minority languages. Recognised minority languages include: Aromanian, Romanian, Serbo-Croatian, Macedonian, Bulgarian, French, Italian and Greek. The majority of Albanians are multilingual, speaking more than 3 languages, which is due to the large number of Albanian immigrants in Europe and elsewhere, as well as political and socio-cultural relations with their neighbours. As a consequence, Albanians are considered one of the most linguistically diverse peoples in Europe. During Albania's Italian occupation and the subsequent communist period, Italian television and radio were a source of education and entertainment for many Albanians; as a result, 60–70% of Albania's population has a command of Italian. Albania's Greek communities, as well as returning migrants from Greece and Greek national arrivals, continue to raise the status of Greek in the country. Albania is also part of the Francophonie, with 320,000 French speakers. In Bulgaria, the official language is Bulgarian, but significant minority languages are recognized at a local level, with commitments made in respect of use of Romanian, Czech, Croatian, German,


id: list-of-multilingual-countries-and-regions-20260505-12
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-12.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
=== Western Europe === Belgium has three official languages: Dutch (59%) in the north, French (40%) in the south and a small minority speaks German (1%). Its bilingual capital, Brussels is mainly French-speaking, with Dutch speakers as a minority. These languages have the status of 'official language' only in specified language areas as defined by the constitution. In Flanders, 59% and 53% of the Flemings know French and English respectively; in Wallonia, only 19% and 17% know Dutch or English. In each region, Belgium's third official language, German, is notably less known than Dutch, French or English. Wallonia recognises all of its vernacular dialect groups as regional languages, Flanders does not. France has a monolingual policy for the republic to conduct government business only in French. There are, however, levels of fluency in regional languages: Alsatian, Basque (the department of Pyrénées-Atlantiques), Breton (the regional government of Brittany has adopted some policies to promote the teaching of Breton), Catalan (the department of Pyrénées-Orientales has a particular charter for supporting Catalan), Corsican (Corsican teaching in the island's schools is provided for by law), Flemish, Franco-Provençal, Lorraine Franconian and Occitan (sometimes called Provençal). The country as whole is linguistically dominated by French. Ireland,


id: list-of-multilingual-countries-and-regions-20260505-13
file: data/en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions-13.md
category: reference
title: List of multilingual countries and regions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_multilingual_countries_and_regions
date_saved: 2026-05-05T08:12:07.199573+00:00
instance: kb-cron
parent: list-of-multilingual-countries-and-regions-20260505-0
summary:
== Oceania == Australia – English is the de facto official language of Australia. Auslan is recognised by the Australian Government and is spoken by many Australian deaf people. Australian Aboriginal languages, of which there are approximately 290–363, have recognition – though only a few are popularly spoken by Australian Indigenous people, including: Wiradjuri, Pitjantjatjara, Kalaw Lagaw Ya, Ngaanyatjarra, Warlpiri and Australian Kriol. Some government departments and agencies provide translations of documents in multiple languages for non-English speakers. Fiji – Fijian, English and Fiji Hindi (All official) Tongan and Tuvaluan are also spoken. Kiribati – Gilbertese and English (official); Gilbertese is the majority language, English language is the prevailing language for constitutional text Marshall Islands – Marshallese and English (both official) Micronesia – English (official) but each state has its own regional language: Chuukese (Chuuk), Kosraean (Kosrae), Pohnpeian (Pohnpei), and Yapese (Yap). In addition other language such as Pingelapese, Ngatikese, Satawalese, Puluwatese, Mortlockese, Mokilese, Ulithian, Woleaian, Nukuoro, and Kapingamarangi are recognized. Nauru – Nauruan is the official language. English is also spoken along with it. New Caledonia, a special collectivity of France – French and Kanak languages (primarily Drehu, Nengone, Paicî, Ajië and Xârâcùù) New Zealand – a small percentage


id: index-of-phonetics-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_phonetics_articles-0.md
category: reference
title: Index of phonetics articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_phonetics_articles
date_saved: 2026-05-05T08:12:09.282005+00:00
instance: kb-cron
parent: index-of-phonetics-articles-20260505-0
summary:
== A == Acoustic phonetics Active articulator Affricate Airstream mechanism Alexander John Ellis Alexander Melville Bell Alfred C. Gimson Allophone Alveolar approximant (ɹ) Alveolar click (ǃ) Alveolar consonant Alveolar ejective (tʼ) Alveolar ejective affricate (tsʼ) Alveolar ejective fricative (sʼ) Alveolar flap (ɾ) Alveolar lateral approximant (l, l̥) Alveolar lateral ejective affricate (tɬʼ) Alveolar lateral ejective fricative (ɬʼ) Alveolar lateral flap (ɺ) Alveolar nasal (n) Alveolar ridge Alveolar trill (r, r̥) Alveolo-palatal consonant Alveolo-palatal ejective fricative (ɕʼ) Apical consonant Approximant consonant Articulatory phonetics Aspirated consonant (◌ʰ) Auditory phonetics == B == Back vowel Basis of articulation Bernd J. Kröger Bilabial click (ʘ) Bilabial consonant Bilabial ejective (pʼ) Bilabial flap (ⱱ̟) Bilabial nasal (m) Bilabial trill (ʙ) Breathy voice Bidental consonant == C == Cardinal vowel Central vowel Checked vowel Click consonant Close back rounded vowel (u) Close back unrounded vowel (ɯ) Close central rounded vowel (ʉ) Close central unrounded vowel (ɨ) Close front rounded vowel (y) Close front unrounded vowel (i) Close vowel Close-mid back rounded vowel (o) Close-mid back unrounded vowel (ɤ) Close-mid central rounded vowel (ɵ) Close-mid central unrounded vowel (ɘ) Close-mid front rounded vowel (ø) Close-mid front unrounded vowel (e) Close-mid vowel Co-articulated consonant Coarticulation Comparison of ASCII


id: list-of-polyglots-20260505-0
file: data/en.wikipedia.org/wiki/List_of_polyglots-0.md
category: reference
title: List of polyglots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyglots
date_saved: 2026-05-05T08:12:10.749388+00:00
instance: kb-cron
parent: list-of-polyglots-20260505-0
summary:
This is a list of notable people with a knowledge of six or more languages. == Deceased == === Antiquity and Middle Ages === Mithridates VI (135–63 BC), King of Pontus. According to Pliny the Elder, Mithridates could speak the languages of all of the twenty-two nations that he ruled. Cleopatra (69–30 BC), Queen of Egypt. According to Plutarch, Cleopatra spoke many languages in addition to her native language, Greek, including Latin, Egyptian, Ethiopian, Trogodyte, and the languages of the Hebraioi, Arabes, Syrians, Medes, and Parthians. Al-Farabi (870–950), Islamic philosopher. He was reputed to know seventy languages. Frederick II (1194–1250), King of Sicily and Holy Roman Emperor. He knew Sicilian, French, Latin, Greek, German, and Arabic. === Modern age, pre-18th century === Mehmed II (1432–1481), Sultan of the Ottoman Empire. In addition to his native language, Turkish, he learnt Arabic, Hebrew, Persian, Latin, and Greek. Elizabeth I (1533–1603), Queen of England and Ireland. She is thought to have known English, Welsh, French, Spanish, Italian, Latin, Greek, and some German. Athanasius Kircher (1602–1680), German Jesuit scholar. He was said to know twelve languages. John Milton (1608–1674), English poet. He knew English, Italian, Latin, Greek, and Hebrew, and to a lesser extent


id: list-of-polyglots-20260505-1
file: data/en.wikipedia.org/wiki/List_of_polyglots-1.md
category: reference
title: List of polyglots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyglots
date_saved: 2026-05-05T08:12:10.749388+00:00
instance: kb-cron
parent: list-of-polyglots-20260505-0
summary:
=== 19th century === William Rowan Hamilton (1805–1865), Irish mathematician. Encouraged by his uncle, he learnt many languages while still young, including Latin, Greek, Hindustani, Bengali, Marathi, Sanskrit, Persian, Arabic, Hebrew, Aramaic, Syriac, and Malay. Henry Wadsworth Longfellow (1807–1882), American poet. He knew Latin, Italian, French, Spanish, Portuguese, and German. He also had some knowledge of Dutch, Danish, Swedish, Icelandic, and Finnish. Jan Prosper Witkiewicz (1808–1839), Polish-Lithuanian explorer and diplomat. He spoke Polish, Russian, English, German, French, Persian, Pashto, Kazakh, and Chagatai Turkish. Solomon Caesar Malan (1812–1894), British clergyman. He was conversant with around eighty languages, including German, Spanish, Italian, Latin, Sanskrit, Hebrew, Arabic, Chinese, and Tibetan. Otto von Bismarck (1815–1898), Prussian statesman. In addition to his native language, German, he spoke English and French, and to a lesser extent Italian, Russian, Polish, and Plattdeutsch. He also had some knowledge of Latin. Herschell Filipowski (1816–1872), Lithuanian-born British actuary and editor. He was conversant with Polish, Russian, German, English, French, Spanish, Latin, Hebrew, Arabic, and Chinese. Karl Marx (1818–1883), German political theorist. He spoke German, English, and French, and could read Greek, Latin, Italian, Spanish, Romanian, and Russian. C. V. Runganada Sastri (1819–1881), Indian civil servant. The Dictionary of Indian Biography


id: list-of-polyglots-20260505-2
file: data/en.wikipedia.org/wiki/List_of_polyglots-2.md
category: reference
title: List of polyglots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyglots
date_saved: 2026-05-05T08:12:10.749388+00:00
instance: kb-cron
parent: list-of-polyglots-20260505-0
summary:
=== 20th century === ==== 1900s ==== Sukarno (1901–1970), Indonesian politician. He spoke Javanese, Sundanese, Balinese, Indonesian, Dutch, German, English, French, Arabic, and Japanese. Dora Bloch (1902–1976), Israeli hostage. She spoke Hebrew, Arabic, Russian, German, English, and Italian. Steven Runciman (1903–2000), British historian. He began learning French, Latin, Greek, and Russian at a young age, and later learnt Bulgarian, Old Church Slavonic, Armenian, and some Turkish. He may also have had some knowledge of Arabic and a number of other languages. John von Neumann (1903–1957), Hungarian mathematician. He spoke Hungarian, English, German, and French, and also knew Latin and Greek. Syed Mujtaba Ali (1904–1974), Bangladeshi writer. He knew Bengali, Urdu, Hindi, Gujarati, Marathi, Sanskrit, Persian, Pashto, English, German, French, Italian, and Arabic. Pent Nurmekund (1906–1996), Estonian linguist. He could read over eighty languages and could speak a great many of them. H. S. David (1907–1981) Sri Lankan priest. He reportedly knew thirty-three languages: Tamil, Telugu, Kannada, Malayalam, Tulu, Kui, Brahui, Sinhalese, Hindi, Vedic, Sanskrit, Prakrit, Pali, Avestan, English, Dutch, German, Danish, Norwegian, Swedish, French, Spanish, Portuguese, Italian, Latin, Greek, Lithuanian, Russian, Malay, Arabic, Hebrew, Assyrian, and Sumerian. Muhammad Hamidullah (1908–2002), stateless scholar. He knew ten languages and published works in


id: list-of-polyglots-20260505-3
file: data/en.wikipedia.org/wiki/List_of_polyglots-3.md
category: reference
title: List of polyglots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyglots
date_saved: 2026-05-05T08:12:10.749388+00:00
instance: kb-cron
parent: list-of-polyglots-20260505-0
summary:
==== 1930s ==== Barry Farber (1930–2020), American radio host. When inducted into the army, he qualified for work in fourteen languages. He had some knowledge of twenty-five languages in total. George Fernandes (1930–2019), Indian politician. In addition to his mother tongue, Konkani, he spoke Marathi, Hindi, Urdu, English, Tamil, Malayalam, Kannada, and Tulu. He also knew Latin. P. B. Sreenivas (1930–2013), Indian singer. He spoke eight languages, including Kannada, Urdu, and English. Abdul Rahman Ghassemlou (1930–1989), Iranian politician. He spoke Kurdish, Persian, English, German, French, Russian, Czech, Slovak, Turkish, Azerbaijani, and Arabic. Jon Elia (1931–2002), Pakistani poet. He knew Urdu, Sanskrit, Persian, English, Arabic, and Hebrew. Corazon Aquino (1933–2009), former president of the Philippines. She spoke Kapampangan, Tagalog, English, Spanish, French, and Japanese. Kenneth L. Hale (1934–2001), American linguist. He spoke over fifty languages, including Spanish, Portuguese, French, Dutch, Irish, Polish, Basque, Turkish, Hebrew, Chinese, Japanese, Navajo, Jemez, Hopi, Oʼodham, Wômpanâak, Ulwa, Miskitu, Warlpiri, and Lardil. Jeong Su-il (1934–2025), Chinese-born North Korean spy. He spoke Korean, Chinese, Japanese, Tagalog, Malay, Arabic, Persian, Russian, French, Spanish, German, and English. William Fulco (1936–2021), American Jesuit priest and scholar. He knew nine languages, including Latin, Greek, German, Aramaic, and Hebrew. Pope Francis (1936–2025),


id: list-of-polyglots-20260505-4
file: data/en.wikipedia.org/wiki/List_of_polyglots-4.md
category: reference
title: List of polyglots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyglots
date_saved: 2026-05-05T08:12:10.749388+00:00
instance: kb-cron
parent: list-of-polyglots-20260505-0
summary:
=== Europe === John C. Wells (1939–), British phonetician. He studied Latin and Ancient Greek at university, and speaks English, German, Welsh, French, Spanish, Italian, Modern Greek, and Esperanto with varying degrees of fluency. He also has some knowledge of Polish, Russian, Swedish, Danish, Norwegian, and Japanese. Werner Herzog (1942–), German filmmaker. In addition to his native Bavarian, he knows German, English, Ancient Greek, Modern Greek, Latin, Spanish, French, and some Italian. Queen Silvia of Sweden (1943–), spouse of King Carl XVI Gustaf. The daughter of a German father and a Brazilian mother, she speaks German and Portuguese natively. She also knows Spanish, French, English, and Swedish, and has some knowledge of Swedish sign language. André Rieu (1949–), Dutch violinist and conductor. He speaks Dutch, English, German, French, Spanish, and Italian. Arsène Wenger (1949–), French football manager. He speaks French, German, English, Spanish, and Italian, and has some knowledge of Japanese. Noel Malcolm (1956–), British historian. He speaks most Western and Eastern European languages. Ranga Yogeshwar (1959–), Luxembourgish physicist and science journalist. He speaks Luxembourgish, German, English, French, Hindi, Tamil, Kannada, and Malayalam. Johan Vandewalle (1960–), Belgian linguist. In 1987, at the age of twenty-six, he won the Polyglot of


id: list-of-polyglots-20260505-5
file: data/en.wikipedia.org/wiki/List_of_polyglots-5.md
category: reference
title: List of polyglots
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_polyglots
date_saved: 2026-05-05T08:12:10.749388+00:00
instance: kb-cron
parent: list-of-polyglots-20260505-0
summary:
=== North America === Powell Janulus (1939–), Canadian court interpreter. According to the 1986 Guinness Book of World Records, he "has worked with 41 languages in the Provincial Court of British Columbia, Vancouver, Canada." Stephen Krashen (1941–), American linguist. He knows English, German, Yiddish, Spanish, French, Hebrew, and Amharic. Steve Kaufmann (1945–), Canadian businessman. He has some understanding of twenty languages, although his ability to speak and write these languages varies to a considerable degree. Some of the languages he has studied are French, Spanish, Portuguese, German, Swedish, Russian, Japanese, Mandarin, and Cantonese. Pope Leo XIV (1955–), leader of the Catholic Church. He speaks English, Italian, Spanish, Portuguese, and French, and can read Latin and German. Viggo Mortensen (1958–), Danish-American actor. He grew up bilingual in English and Spanish, and later learnt Danish, Italian, and French. He also has some knowledge of Arabic. Julie Payette (1963–), former governor general of Canada. She speaks French and English natively, and can converse in Spanish, Italian, Russian, and German. Alexander Argüelles (1964–), American linguist. He speaks most of the Germanic and Romance languages (in particular, German, Dutch, Swedish, French, Spanish, Portuguese, and Italian), as well as Russian, Korean, and Arabic, and he has


id: list-of-russian-language-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Russian_language_topics-0.md
category: reference
title: List of Russian language topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Russian_language_topics
date_saved: 2026-05-05T08:12:15.976545+00:00
instance: kb-cron
parent: list-of-russian-language-topics-20260505-0
summary:
The list of Russian language topics stores articles on grammar and other language-related topics that discuss (or should discuss) peculiarities of the Russian language (as well as of other languages) or provide examples from Russian language for these topics. The list complements the Category:Russian language and does not overlap with it. The "—" marks articles where the information about Russian language is inadequate or missing. == Categories == Category:Cyrillic alphabets Category:Cyrillic letters Category:Cyrillization Category:Language comparison Category:Russian language == Articles == Adposition Assimilation (linguistics) (also reasonably covered in Russian phonology) Barbarism (linguistics) Capitalization Clitic Code-switching Compound (linguistics) Consonant mutation Continuous and progressive aspects Copula Cyrillic script Diacritic Diminutive Double negative Expressive loan False cognate Filler (linguistics) — Foreign language influences in English — Four-letter word Frequentative Gemination Gender-neutrality in languages with grammatical gender Gobbledygook Grammatical aspect Grammatical cases: Accusative case Dative case Genitive case Instrumental case Locative case Nominative case Prepositional case Vocative case Grammatical mood Hypothetical mood Hypocorism Iotation Khalyava Language game — Letter Zyu List of ethnic slurs List of offensive terms per nationality List of Russian federal subject name etymologies Malapropism#Examples in Russian language Measure word Metasyntactic variable Minimal pair Minced oath — Mondegreen Mojibake Morse code Morse


id: list-of-russian-linguists-and-philologists-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Russian_linguists_and_philologists-0.md
category: reference
title: List of Russian linguists and philologists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Russian_linguists_and_philologists
date_saved: 2026-05-05T08:12:17.360181+00:00
instance: kb-cron
parent: list-of-russian-linguists-and-philologists-20260505-0
summary:
This list of Russian linguists and philologists includes notable linguists from the Russian Federation, the Soviet Union, the Russian Empire and other predecessor states of Russia. == A == Vasily Abaev, prominent researcher of Iranian languages Solomon Adlivankin, Soviet linguist, the founder of Perm derivatology school, took part in compiling Akchim dialect dictionary Vladimir Admoni, linguist, literary critic, translator and poet, worked on the theory of grammar, historic and modern German syntax, defended Joseph Brodsky in court in 1964 Alexander Afanasyev, leading Russian folklorist, recorded and published over 600 Russian fairy tales, by far the largest folktale collection by any one man in the world == B == Ivan Baudouin de Courtenay, co-inventor of the concept of phoneme and the systematic treatment of alternations, pioneer of synchronic analysis and mathematical linguistics Victor Bayda, linguist specializing in Celtic and Germanic languages Alexander Belskiy, Soviet specialist in literary criticism, famous Anglicist, founder of philological faculty at Perm State University, founder of Foreign literature Department at PSU Otto von Böhtlingk, prominent Indologist and Sanskrit grammarian Fyodor Buslaev, philologist and folklorist, representative of the Mythological school of comparative literature Yakov Brandt, Sinologist == D == Vladimir Dal, greatest Russian language lexicographer of the 19th


id: list-of-schools-of-linguistics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_schools_of_linguistics-0.md
category: reference
title: List of schools of linguistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_schools_of_linguistics
date_saved: 2026-05-05T08:12:18.588746+00:00
instance: kb-cron
parent: list-of-schools-of-linguistics-20260505-0
summary:
This is a list of schools in linguistics. Columbia School of Linguistics Copenhagen School Formal linguistics Functional linguistics Systemic functional linguistics Sydney School Kharkiv Linguistic School Leiden school Leiden University Centre for Linguistics (LUCL) London School of Linguistics Moscow School of Comparative Linguistics Prague linguistic circle Structural linguistics == References ==


id: list-of-language-self-study-programs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_language_self-study_programs-0.md
category: reference
title: List of language self-study programs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_language_self-study_programs
date_saved: 2026-05-05T08:12:19.872679+00:00
instance: kb-cron
parent: list-of-language-self-study-programs-20260505-0
summary:
Self-study language acquisition programs allow learning without having a teacher present, and the courses can supplement or replace classroom instruction. Universities use self-study programs for less-commonly taught languages, where having professors is not feasible. Self-study programs are available on paper, audio files, video files, smartphone apps, computers, or any combination. This list is limited to programs that teach four or more languages. There are many others that teach one language. Alphabetical lists of languages show the courses available to learn each language, at All Language Resources, Lang1234, Martindale's Language Center, Omniglot, and Rüdiger Köppe. (UCLA Language Materials Project has ended.) For the thousands of languages not listed on those sites, for which no course exists, Global Recordings Network has recorded a standard set of Bible stories in 6,000 languages. With effort, learners can study any language by comparing their recordings to the same story in a language they know. The list of self-study programs, below, shows the number of languages taught by each program, the name of the program, and the number of different languages used for instruction. Multiple languages of instruction may be available for some but not all courses. For example, Reise Know-How uses six languages to teach


id: list-of-serbo-croatian-words-of-greek-origin-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Serbo-Croatian_words_of_Greek_origin-0.md
category: reference
title: List of Serbo-Croatian words of Greek origin
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Serbo-Croatian_words_of_Greek_origin
date_saved: 2026-05-05T08:12:21.339660+00:00
instance: kb-cron
parent: list-of-serbo-croatian-words-of-greek-origin-20260505-0
summary:
Greek influence was widespread throughout the Balkans during the Middle Ages, influencing the languages within it, including Serbo-Croatian. Many words of Greek origin were borrowed from other languages, while most others came via contact with the Greeks. Some words are present and common in the modern vernaculars of Serbo-Croatian: hiljada (хиљада), tiganj (тигањ), patos (патос). Almost every word of the Serbian Orthodox ceremonies are of Greek origin: parastos (парастос). AG stands for Ancient Greek origin. MG stands for Modern Greek origin. C stands for Cyrillic (script). L stands for Latin (script). == See also == Arabic-Persian-Greek-Serbian Conversation Textbook == References == "Grecizmi u srpskom jeziku". ЈУЖНОСЛОВЕНСКИ филолог. Srpska akademija nauka i umetnosti: 375–403. GGKEY:00RD5D429DG.


id: list-of-serbo-croatian-words-of-turkish-origin-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Serbo-Croatian_words_of_Turkish_origin-0.md
category: reference
title: List of Serbo-Croatian words of Turkish origin
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Serbo-Croatian_words_of_Turkish_origin
date_saved: 2026-05-05T08:12:22.585909+00:00
instance: kb-cron
parent: list-of-serbo-croatian-words-of-turkish-origin-20260505-0
summary:
Serbo-Croatian vernacular has over time borrowed and adopted a lot of words of Turkish origin. The Ottoman conquest of the Balkans began a linguistical contact between Ottoman Turkish and South Slavic languages, a period of influence since at least the late 14th up until the 20th century, when large territories of Shtokavian-speaking areas became conquered and made into provinces of the Ottoman Empire, into what was collectively known as Rumelia. As the Ottoman Turkish language itself extensively borrowed from Arabic, Persian and Central Asian Turkic languages other than itself, many words of such origins also entered Serbo-Croatian via Turkish. Numerous migrations in the war-torn Western Balkans helped spread Shtokavian and its enriched vernacular. Some Turkisms in Serbo-Croatian have entered the language through other languages, such as Italian, while some words of Greek origin have been adopted in their Turkish forms. Over the passage of time, many Turkish loanwords were completely adopted into standardized varieties of Serbo-Croatian and are no longer considered loanwords, if not for lack of true synonyms. Numerous such Turkisms (e.g. bakar, alat, sat, čarape, šećer, or boja) are often preferred to later introduced Germanisms and Latinisms (farba, kolur, tinta, pigment). This is mostly the case with the


id: list-of-summer-schools-of-linguistics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_summer_schools_of_linguistics-0.md
category: reference
title: List of summer schools of linguistics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_summer_schools_of_linguistics
date_saved: 2026-05-05T08:12:23.827484+00:00
instance: kb-cron
parent: list-of-summer-schools-of-linguistics-20260505-0
summary:
This is a list of summer schools of linguistics. Eastern Generative Grammar (The EGG): Romania: Brašov, Romania (2024) Serbia: Novi Sad, Serbia (2023) Czech Republic: Brno, Czech Republic (2022) Online: (2021) Poland: Wrocław, Poland (2019) Hungary: European Summer School in Generative Grammar (EGG) - Debrecen (2014) European Summer School in Logic, Language and Information (ESSLLI) Belgium: Leuven, Belgium (2024) Slovenia: Ljubljana, Slovenia (2023) Ireland: Galway, Ireland (2022) Netherlands: Utrecht, Netherlands (2021) Germany: Forensic Linguistics Short Course (FLsc) - Düsseldorf (2018–2019, 2021–2024) Germany: Summer School on Corpus Phonology - Augsburg (2008) Greece: The Crete Summer School of Linguistics (CreteLing) - Rethymno (2017, 2018, 2019, 2022, 2023, 2024) Poland: Summer Institute "Languages and Cultures in Contact / in Contrast" - Zakopane (2008) Portugal: NOVA - Lisbon Summer School and Graduate Conference in Linguistics - Lisbon Portugal: U. Minho - APL Summer School of Linguistics 2015: Experimental Methods in Syntax - Braga United Kingdom: International Summer School in Forensic Linguistic Analysis - Birmingham (2000–2010) United States: Summer Institute of Linguistics - University of North Dakota (every summer) United States: Linguistics Society of America Summer Institutes, held in odd-numbered years, including: Empirical Foundations for Theories of Language - Stanford (2007) Linguistic Structure and Language


id: swadesh-list-20260505-0
file: data/en.wikipedia.org/wiki/Swadesh_list-0.md
category: reference
title: Swadesh list
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Swadesh_list
date_saved: 2026-05-05T08:12:25.180092+00:00
instance: kb-cron
parent: swadesh-list-20260505-0
summary:
A Swadesh list () is a compilation of tentatively universal concepts for the purposes of lexicostatistics. That is, a Swadesh list is a list of forms and concepts which all languages, without exception, have terms for, such as star, hand, water, kill, sleep, and so forth. The number of such terms is small – a few hundred at most, or possibly less than a hundred. The inclusion or exclusion of many terms is subject to debate among linguists; thus, there are several different lists, and some authors may refer to "Swadesh lists." The Swadesh list is named after linguist Morris Swadesh. Translations of a Swadesh list into a set of languages allow for researchers to quantify the interrelatedness of those languages. Swadesh lists are used in lexicostatistics (the quantitative assessment of the genealogical relatedness of languages) and glottochronology (the dating of language divergence). For instance, the terms on a Swadesh list can be compared between two languages (since both languages will have them) to see if they are related and how closely, thus giving useful information that can be further applied to comparison of the languages. (Actual lexicostatistics is quite complicated, and usually sets of languages are compared.) == Versions


id: swadesh-list-20260505-1
file: data/en.wikipedia.org/wiki/Swadesh_list-1.md
category: reference
title: Swadesh list
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Swadesh_list
date_saved: 2026-05-05T08:12:25.180092+00:00
instance: kb-cron
parent: swadesh-list-20260505-0
summary:
Holman et al. (2008) found that in identifying the relationships between Chinese dialects the Swadesh–Yakhontov list was less accurate than the original Swadesh-100 list. Further they found that a different (40-word) list (also known as the ASJP list) was just as accurate as the Swadesh-100 list. However, they calculated the relative stability of the words by comparing retentions between languages in established language families. They found no statistically significant difference in the correlations in the families of the Old versus the New World. The ranked Swadesh-100 list, with Swadesh numbers and relative stability, is as follows (Holman et al., Appendix. Asterisked words appear on the 40-word list): === Sign languages === In studying the sign languages of Vietnam and Thailand, linguist James Woodward noted that the traditional Swadesh list applied to spoken languages was unsuited for sign languages. The Swadesh list results in overestimation of the relationships between sign languages, due to indexical signs such as pronouns and parts of the body. The modified list is as follows, in mostly alphabetical order: == See also == Other lists A General Service List of English Words — roughly 2,000 of the most common English words Dolgopolsky list — the 15 words


id: list-of-syntactic-phenomena-20260505-0
file: data/en.wikipedia.org/wiki/List_of_syntactic_phenomena-0.md
category: reference
title: List of syntactic phenomena
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_syntactic_phenomena
date_saved: 2026-05-05T08:12:26.508678+00:00
instance: kb-cron
parent: list-of-syntactic-phenomena-20260505-0
summary:
A list of phenomena in syntax. Anaphora Agreement Answer ellipsis Antecedent-contained deletion Binding Case Clitics Control Coreference Differential Object Marking Discontinuity Do-support Dummy pronouns Ellipsis Ergative verb Exceptional case-marking Existential clauses Expletives Extraposition Gapping Heavy NP shift Inverse copula sentences Movement paradoxes Negative inversion Non-configurational language Parasitic gaps Pied-piping Pro-drop Pseudogapping Raising (linguistics) Reciprocal (grammar) Reflexive pronouns Reflexive verbs Right node raising Scrambling Shifting Sluicing Small clause Stripping Subject-auxiliary inversion Subject-verb inversion Topicalization Tough movement Unaccusative verbs V2 word order Verb phrase ellipsis Wh-movement


id: list-of-text-corpora-20260505-0
file: data/en.wikipedia.org/wiki/List_of_text_corpora-0.md
category: reference
title: List of text corpora
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_text_corpora
date_saved: 2026-05-05T08:12:27.684351+00:00
instance: kb-cron
parent: list-of-text-corpora-20260505-0
summary:
Text corpora (singular: text corpus) are large and structured sets of texts, which have been systematically collected. Text corpora are used by both AI developers to train large language models and corpus linguists and within other branches of linguistics for statistical analysis, hypothesis testing, finding patterns of language use, investigating language change and variation, and teaching language proficiency. == English language == American National Corpus Bank of English BookCorpus British National Corpus Bergen Corpus of London Teenage Language (COLT) Brown Corpus, forming part of the "Brown Family" of corpora, together with LOB, Frown and F-LOB COCA: see below at English-Corpora.org COHA: see below at English-Corpora.org Corpus Resource Database (CoRD), more than 80 English language corpora. Coruña Corpus, a corpus of late Modern English scientific writing covering the period 1700–1900, developed by the Muste research group at the University of A Coruña DBLP Discovery Dataset (D3), a corpus of computer science publications with sentient metadata. English-Corpora.org, which contains (among others): iWeb, the Intelligent Web-based Corpus: 14 billion words, 6 countries, 2017 COCA, the Corpus of Contemporary American English: 1.0 billion words, American, 1990-2019 COHA, the Corpus of Historical American English: 475 million words, American, 1820-2019 NOW, News on the Web: 23.2


id: list-of-text-corpora-20260505-1
file: data/en.wikipedia.org/wiki/List_of_text_corpora-1.md
category: reference
title: List of text corpora
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_text_corpora
date_saved: 2026-05-05T08:12:27.684351+00:00
instance: kb-cron
parent: list-of-text-corpora-20260505-0
summary:
== Parallel corpora of diverse languages == Chinese/English Political Interpreting Corpus (CEPIC) consists of transcripts of speeches delivered by top political figures from Hong Kong, Beijing, Washington DC and London, as well as their translated/interpreted texts. Developed by Jun Pan and HKBU Library. Europarl Corpus - proceedings of the European Parliament from 1996 to 2012 EUR-Lex corpus - collection of all official languages of the European Union, created from the EUR-Lex database OPUS: Open source Parallel Corpus in many many languages Tatoeba A parallel corpus which contains over 8.9 million sentences in multiple languages; 107 languages have more than 1,000 sentences each; a further 81 languages have from 100 to 1,000 sentences each. NTU-Multilingual Corpus in 7 languages (ara, eng, ind, jpn, kor, mcn, vie) (legacy repo) SeedLing corpus - A Seed Corpus for the Human Language Project with 1000+ languages from various sources. GRALIS parallel texts for various Slavic languages, compiled by the institute for Slavic languages at Graz University (Branko Tošović et al.) The ACTRES Parallel Corpus (P-ACTRES 2.0) is a bidirectional English-Spanish corpus consisting of original texts in one language and their translation into the other. P-ACTRES 2.0 contains over 6 million words considering both directions together.


id: list-of-radicals-in-unicode-20260505-0
file: data/en.wikipedia.org/wiki/List_of_radicals_in_Unicode-0.md
category: reference
title: List of radicals in Unicode
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_radicals_in_Unicode
date_saved: 2026-05-05T08:12:28.926560+00:00
instance: kb-cron
parent: list-of-radicals-in-unicode-20260505-0
summary:
The List of Unicode radicals comprises those Unicode characters that represent radical components of CJK characters, Tangut characters or Yi syllables. These are used primarily for indexing characters in dictionaries. There are two CJK radicals blocks: the "Kangxi Radicals" block that includes the 214 standard radicals used in the Kangxi Dictionary; and the "CJK Radicals Supplement" block that includes 115 radical components used in other modern dictionaries, including simplified Chinese and Japanese radicals forms. There is one "Tangut Components" block that includes 768 radicals and components that are used to index Tangut characters in dictionaries of the Tangut script or to describe the structure of Tangut characters. There is one "Yi Radicals" block that includes 55 radicals used to index Yi characters in dictionaries of the standardized Yi script used for writing the Nuosu language in Southern Sichuan and Northern Yunnan. Sets of radicals for other sinoform scripts, such as Jurchen, have also been proposed for encoding in Unicode. == Unicode tables == == See also == Section headers of a Chinese dictionary List of Shuowen Jiezi radicals, a system of 540 components used by Xu Shen (d. ~147AD) in his Shuowen Jiezi List of Kangxi radicals, a system of


id: table-of-vowels-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_vowels-0.md
category: reference
title: Table of vowels
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_vowels
date_saved: 2026-05-05T08:12:30.290922+00:00
instance: kb-cron
parent: table-of-vowels-20260505-0
summary:
This table lists the vowel letters of the International Phonetic Alphabet. == See also == List of consonants Index of phonetics articles == References ==


id: list-of-words-having-different-meanings-in-american-and-british-english-a-l-20260505-0
category: reference
title: List of words having different meanings in American and British English (A–L)
tags: science, encyclopedia
date_saved: 2026-05-05T08:12:31.390086+00:00
instance: kb-cron
parent: list-of-words-having-different-meanings-in-american-and-british-english-a-l-20260505-0
file:
data/en.wikipedia.org/wiki/List_of_words_having_different_meanings_in_American_and_British_English_(A–L)-0.md

summary:
This is the List of words having different meanings in British and American English: A–L. For the second portion of the list, see List of words having different meanings in American and British English: M–Z. Asterisked (*) meanings, though found chiefly in the specified region, also have some currency in the other region; other definitions may be viewed by the other as Briticisms or Americanisms respectively. Additional usage notes are provided where useful. == A == == B == == C == == D == == E == == F == == G == == H == == I == == J == == K == == L == == See also == List of words having different meanings in British and American English: M–Z List of American words not widely used in the United Kingdom List of British words not widely used in the United States == References == == Further reading == Note: the below are general references on this topic. Individual entries have not yet been audited against the references below and readers looking for verifiable information should consult the works below unless individual entries in the article's table are properly sourced. Bickerton, Anthea (1985). American-English, English-American

source_url:
https://en.wikipedia.org/wiki/List_of_words_having_different_meanings_in_American_and_British_English_(A–L)


id: lists-of-mathematics-topics-20260505-0
file: data/en.wikipedia.org/wiki/Lists_of_mathematics_topics-0.md
category: reference
title: Lists of mathematics topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lists_of_mathematics_topics
date_saved: 2026-05-05T08:12:33.059777+00:00
instance: kb-cron
parent: lists-of-mathematics-topics-20260505-0
summary:
Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link to only a few. The template below includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference tables. They also cover equations named after people, societies, mathematicians, journals, and meta-lists. The purpose of this list is not similar to that of the Mathematics Subject Classification formulated by the American Mathematical Society. Many mathematics journals ask authors of research papers and expository articles to list subject codes from the Mathematics Subject Classification in their papers. The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. This list has some items that would not fit in such a classification, such as list of exponential topics and list of factorial and binomial topics, which may surprise the reader with the diversity of their coverage. == Basic mathematics == This branch is typically taught in secondary education or in the first


id: lists-of-mathematics-topics-20260505-1
file: data/en.wikipedia.org/wiki/Lists_of_mathematics_topics-1.md
category: reference
title: Lists of mathematics topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lists_of_mathematics_topics
date_saved: 2026-05-05T08:12:33.059777+00:00
instance: kb-cron
parent: lists-of-mathematics-topics-20260505-0
summary:
A differential equation is an equation involving an unknown function and its derivatives. In a dynamical system, a fixed rule describes the time dependence of a point in a geometrical space. The mathematical models used to describe the swinging of a clock pendulum, the flow of water in a pipe, or the number of fish each spring in a lake are examples of dynamical systems. List of dynamical systems and differential equations topics List of nonlinear partial differential equations List of partial differential equation topics ==== Mathematical physics ==== Mathematical physics is concerned with "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories".1 List of mathematical topics in classical mechanics List of mathematical topics in quantum theory List of mathematical topics in relativity List of string theory topics Index of wave articles ==== Theory of computation ==== The fields of mathematics and computing intersect both in computer science, the study of algorithms and data structures, and in scientific computing, the study of algorithmic methods for solving problems in mathematics, science, and engineering. List of algorithm general topics List of computability and complexity topics Lists


id: lists-of-mathematics-topics-20260505-2
file: data/en.wikipedia.org/wiki/Lists_of_mathematics_topics-2.md
category: reference
title: Lists of mathematics topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Lists_of_mathematics_topics
date_saved: 2026-05-05T08:12:33.059777+00:00
instance: kb-cron
parent: lists-of-mathematics-topics-20260505-0
summary:
List of things named after Niels Henrik Abel List of things named after George Airy List of things named after Jean d'Alembert List of things named after Archimedes List of things named after Vladimir Arnold List of things named after Emil Artin List of things named after Stefan Banach List of things named after Thomas Bayes List of things named after members of the Bernoulli family List of things named after Jakob Bernoulli List of things named after Friedrich Bessel List of things named after Élie Cartan List of things named after Augustin-Louis Cauchy List of things named after Arthur Cayley List of things named after Pafnuty Chebyshev List of things named after John Horton Conway List of things named after Richard Dedekind List of things named after Pierre Deligne List of things named after Peter Gustav Lejeune Dirichlet List of things named after Albert Einstein List of things named after Euclid List of things named after Leonhard Euler List of things named after Paul Erdős List of things named after Pierre de Fermat List of things named after Fibonacci List of things named after Joseph Fourier List of things named after Erik Fredholm List of things named after Ferdinand


id: list-of-mathematical-abbreviations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_abbreviations-0.md
category: reference
title: List of mathematical abbreviations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_abbreviations
date_saved: 2026-05-05T08:12:34.336401+00:00
instance: kb-cron
parent: list-of-mathematical-abbreviations-20260505-0
summary:
This following list features abbreviated names of mathematical functions, function-like operators and other mathematical terminology. This list is limited to abbreviations of two or more letters (excluding number sets). The capitalization of some of these abbreviations is not standardized – different authors might use different capitalizations. == A == A – adele ring or algebraic numbers. a.a.s. – asymptotically almost surely. AC – Axiom of Choice, or set of absolutely continuous functions. a.c. – absolutely continuous. acrd – inverse chord function. ad – adjoint representation (or adjoint action) of a Lie group. adj – adjugate of a matrix. a.e. – almost everywhere. AFSOC - Assume for the sake of contradiction Ai – Airy function. AL – Action limit. Alt – alternating group (Alt(n) is also written as An.) A.M. – arithmetic mean. AP – arithmetic progression. arccos – inverse cosine function. arccosec – inverse cosecant function. (Also written as arccsc.) arccot – inverse cotangent function. arccsc – inverse cosecant function. (Also written as arccosec.) arcexsec – inverse exsecant function. arcosech – inverse hyperbolic cosecant function. (Also written as arcsch.) arcosh – inverse hyperbolic cosine function. arcoth – inverse hyperbolic cotangent function. arcsch – inverse hyperbolic cosecant function. (Also written as


id: list-of-mathematical-abbreviations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_mathematical_abbreviations-1.md
category: reference
title: List of mathematical abbreviations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_abbreviations
date_saved: 2026-05-05T08:12:34.336401+00:00
instance: kb-cron
parent: list-of-mathematical-abbreviations-20260505-0
summary:
== H == H – quaternion numbers. hacover – hacoversine function. (Also written as hacovers, hcv.) hacovercos – hacovercosine function. (Also written as hcc.) hacovers – hacoversine function. (Also written as hacover, hcv.) hav – haversine function. (Also written as sem.) havercos – havercosine function. (Also written as hvc.) h.c. – Hermitian conjugate, often used as part of + h.c. (Also written as H.c.) hcc – hacovercosine function. (Also written as hacovercos.) hcv – hacoversine function. (Also written as hacover, hacovers.) hcf – highest common factor of two numbers. (Also written as gcd.) H.M. – harmonic mean. HOL – higher-order logic. Hom – Hom functor. hom – hom-class. hot – higher order term. HOTPO – half or triple plus one. hvc – havercosine function. (Also written as havercos.) hyp – hypograph of a function. == I == iff – if and only if. IH – induction hypothesis. iid – independent and identically distributed random variables. Im – imaginary part of a complex number. (Also written as ℑ {\displaystyle \Im } .) im – image. inf – infimum of a set. (Also written as glb.) int – interior. I.o. – Infinitely often. == K == ker – kernel. == L ==


id: list-of-mathematical-abbreviations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_mathematical_abbreviations-2.md
category: reference
title: List of mathematical abbreviations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_abbreviations
date_saved: 2026-05-05T08:12:34.336401+00:00
instance: kb-cron
parent: list-of-mathematical-abbreviations-20260505-0
summary:
== U == undef – a function or expression is undefined. usc – upper semi-continuity. == V == V – volume. var – variance of a random variable. vcs – vercosine function. (Also written as vercos.) ver – versine function. (Also written as vers, siv.) vercos – vercosine function. (Also written as vcs.) vers – versine function. (Also written as ver, siv.) == W == W^5 – which was what we wanted. Synonym of Q.E.D. walog – without any loss of generality. wff – well-formed formula. whp – with high probability. wlog – without loss of generality. WMA – we may assume. WO – well-ordered set. WOP – well-ordered principle. w.p. – with probability. wp1 – with probability 1. wrt – with respect to or with regard to. WTP – want to prove. WTS – want to show. == X == XOR – exclusive or in logic. == Z == Z – integer numbers. ZF – Zermelo–Fraenkel axioms of set theory. ZFC – Zermelo–Fraenkel axioms (with the Axiom of Choice) of set theory. == See also == List of letters used in mathematics, science, and engineering ISO 31-11 Language of mathematics List of mathematical jargon Mathematical notation Notation in probability


id: list-of-abstract-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_abstract_algebra_topics-0.md
category: reference
title: List of abstract algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_abstract_algebra_topics
date_saved: 2026-05-05T08:12:35.563780+00:00
instance: kb-cron
parent: list-of-abstract-algebra-topics-20260505-0
summary:
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in pedagogy. == Basic language == Algebraic structures are defined primarily as sets with operations. Algebraic structure Subobjects: subgroup, subring, subalgebra, submodule etc. Binary operation Closure of an operation Associative property Distributive property Commutative property Unary operator Additive inverse, multiplicative inverse, inverse element Identity element Cancellation property Finitary operation Arity Structure preserving maps called homomorphisms are vital in the study of algebraic objects. Homomorphisms Kernels and cokernels Image and coimage Epimorphisms and monomorphisms Isomorphisms Isomorphism theorems There are several basic ways to combine algebraic objects of the same type to produce a third object of the


id: list-of-abstract-algebra-topics-20260505-1
file: data/en.wikipedia.org/wiki/List_of_abstract_algebra_topics-1.md
category: reference
title: List of abstract algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_abstract_algebra_topics
date_saved: 2026-05-05T08:12:35.563780+00:00
instance: kb-cron
parent: list-of-abstract-algebra-topics-20260505-0
summary:
=== Concepts and theorems === Composition series Length of a module Structure theorem for finitely generated modules over a principal ideal domain Homological dimension Projective dimension Injective dimension Flat dimension Global dimension Weak global dimension Cohomological dimension Krull dimension Regular sequence (algebra), depth (algebra) Fitting lemma Schur's lemma Nakayama's lemma Krull–Schmidt theorem Steinitz exchange lemma Jordan–Hölder theorem Artin–Rees lemma Schanuel's lemma Morita equivalence Progenerator == Representation theory == Representation theory (& outline) Algebra representation Group representation Lie algebra representation Maschke's theorem Schur's lemma Equivariant map Frobenius reciprocity Induced representation Restricted representation Affine representation Projective representation Modular representation theory Quiver (mathematics) Representation theory of Hopf algebras == Non-associative systems == === General === Associative property, Associator Heap (mathematics) Magma (algebra) Loop (algebra), Quasigroup Nonassociative ring, Non-associative algebra Universal enveloping algebra Lie algebra (see also list of Lie group topics and list of representation theory topics) Jordan algebra Alternative algebra Power associativity Flexible algebra === Examples === Cayley–Dickson construction Octonions Sedenions Trigintaduonions Hyperbolic quaternions Virasoro algebra == Generalities == Algebraic structure Universal algebra Variety (universal algebra) Congruence relation Free object Generating set (universal algebra) Clone (algebra) Kernel of a function Kernel (algebra) Isomorphism class Isomorphism theorem Fundamental theorem on homomorphisms Universal property


id: index-of-accounting-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_accounting_articles-0.md
category: reference
title: Index of accounting articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_accounting_articles
date_saved: 2026-05-05T08:12:36.783155+00:00
instance: kb-cron
parent: index-of-accounting-articles-20260505-0
summary:
This page is an index of accounting topics. == A == Accounting ethics - Accounting information system - Accounting research - Activity-Based Costing - Assets == B == Balance sheet - Big Four auditors - Bond - Bookkeeping - Book value == C == Cash-basis accounting - Cash-basis versus accrual-basis accounting - Cash flow statement - Certified General Accountant - Certified Management Accountants - Certified Public Accountant - Chartered accountant - Chart of accounts - Common stock - Comprehensive income - Construction accounting - Convention of conservatism - Convention of disclosure - Cost accounting - Cost of capital - Cost of goods sold - Creative accounting - Credit - Credit note - Current asset - Current liability == D == Debitcapital reserve - Debit note - Debt - Deficit (disambiguation) - Depreciation - Diluted earnings per share - Dividend - Double-entry bookkeeping system - Dual aspect == E == E-accounting - EBIT - EBITDA - Earnings per share - Engagement Letter - Entity concept - Environmental accounting - Expense - Equity - Equivalent Annual Cost == F == Financial Accounting Standards Board - Financial accountancy - Financial audit - Financial reports - Financial statements - Fixed assets - Fixed assets


id: outline-of-algebra-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_algebra-0.md
category: reference
title: Outline of algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_algebra
date_saved: 2026-05-05T08:12:38.054609+00:00
instance: kb-cron
parent: outline-of-algebra-20260505-0
summary:
The following outline is provided as an overview of and topical guide to algebra: Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. == Branches == Pre-algebra Elementary algebra Boolean algebra Abstract algebra Linear algebra Universal algebra == Algebraic equations == An algebraic equation is an equation involving only algebraic expressions in the unknowns. These are further classified by degree. Linear equation – algebraic equation of degree one. Polynomial equation – equation in


id: list-of-algebraic-constructions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algebraic_constructions-0.md
category: reference
title: List of algebraic constructions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algebraic_constructions
date_saved: 2026-05-05T08:12:40.458341+00:00
instance: kb-cron
parent: list-of-algebraic-constructions-20260505-0
summary:
An algebraic construction is a method by which an algebraic entity is defined or derived from another. Instances include: Cayley–Dickson construction Proj construction Grothendieck group Gelfand–Naimark–Segal construction Ultraproduct ADHM construction Burnside ring Simplicial set Fox derivative Mapping cone (homological algebra) Prym variety Todd class Adjunction (field theory) Vaughan Jones construction Strähle construction Coset construction Plus construction Algebraic K-theory Gelfand–Naimark–Segal construction Stanley–Reisner ring construction Quotient ring construction Ward's twistor construction Hilbert symbol Hilbert's arithmetic of ends Colombeau's construction Vector bundle Integral monoid ring construction Integral group ring construction Category of Eilenberg–Moore algebras Kleisli category Adjunction (field theory) Lindenbaum–Tarski algebra construction Freudenthal magic square Stone–Čech compactification


id: list-of-algebraic-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algebraic_geometry_topics-0.md
category: reference
title: List of algebraic geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algebraic_geometry_topics
date_saved: 2026-05-05T08:12:41.639499+00:00
instance: kb-cron
parent: list-of-algebraic-geometry-topics-20260505-0
summary:
This is a list of algebraic geometry topics, by Wikipedia page. == Classical topics in projective geometry == Affine space Projective space Projective line, cross-ratio Projective plane Line at infinity Complex projective plane Complex projective space Plane at infinity, hyperplane at infinity Projective frame Projective transformation Fundamental theorem of projective geometry Duality (projective geometry) Real projective plane Real projective space Segre embedding of a product of projective spaces Rational normal curve == Algebraic curves == Conics, Pascal's theorem, Brianchon's theorem Twisted cubic Elliptic curve, cubic curve Elliptic function, Jacobi's elliptic functions, Weierstrass's elliptic functions Elliptic integral Complex multiplication Weil pairing Hyperelliptic curve Klein quartic Modular curve Modular equation Modular function Modular group Supersingular primes Fermat curve Bézout's theorem Brill–Noether theory Genus (mathematics) Riemann surface Riemann–Hurwitz formula Riemann–Roch theorem Abelian integral Differential of the first kind Jacobian variety Generalized Jacobian Moduli of algebraic curves Hurwitz's theorem on automorphisms of a curve Clifford's theorem on special divisors Gonality of an algebraic curve Weil reciprocity law Algebraic geometry codes == Algebraic surfaces == Enriques–Kodaira classification List of algebraic surfaces Ruled surface Cubic surface Veronese surface Del Pezzo surface Rational surface Enriques surface K3 surface Hodge index theorem Elliptic surface Surface of general type


id: list-of-algebraic-number-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algebraic_number_theory_topics-0.md
category: reference
title: List of algebraic number theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algebraic_number_theory_topics
date_saved: 2026-05-05T08:12:42.947013+00:00
instance: kb-cron
parent: list-of-algebraic-number-theory-topics-20260505-0
summary:
This is a list of algebraic number theory topics. == Basic topics == These topics are basic to the field, either as prototypical examples, or as basic objects of study. Algebraic number field Gaussian integer, Gaussian rational Quadratic field Cyclotomic field Cubic field Biquadratic field Quadratic reciprocity Ideal class group Dirichlet's unit theorem Discriminant of an algebraic number field Ramification (mathematics) Root of unity Gaussian period == Important problems == Fermat's Last Theorem Class number problem for imaginary quadratic fields Stark–Heegner theorem Heegner number Langlands program == General aspects == Different ideal Dedekind domain Splitting of prime ideals in Galois extensions Decomposition group Inertia group Frobenius automorphism Chebotarev's density theorem Totally real field Local field p-adic number p-adic analysis Adele ring Idele group Idele class group Adelic algebraic group Global field Hasse principle Hasse–Minkowski theorem Galois module Galois cohomology Brauer group == Class field theory == Class field theory Abelian extension Kronecker–Weber theorem Hilbert class field Takagi existence theorem Hasse norm theorem Artin reciprocity Local class field theory == Iwasawa theory == Iwasawa theory Herbrand–Ribet theorem Vandiver's conjecture Stickelberger's theorem Euler system p-adic L-function == Arithmetic geometry == Arithmetic geometry Complex multiplication Abelian variety of CM-type Chowla–Selberg formula Hasse–Weil zeta


id: outline-of-algebraic-structures-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_algebraic_structures-0.md
category: reference
title: Outline of algebraic structures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
date_saved: 2026-05-05T08:12:44.195968+00:00
instance: kb-cron
parent: outline-of-algebraic-structures-20260505-0
summary:
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms. Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition. Concrete examples of each structure will be found in the articles listed. Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath Archived 2007-11-13 at the


id: outline-of-algebraic-structures-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_algebraic_structures-1.md
category: reference
title: Outline of algebraic structures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
date_saved: 2026-05-05T08:12:44.195968+00:00
instance: kb-cron
parent: outline-of-algebraic-structures-20260505-0
summary:
Semiring: a ringoid such that S is a monoid under each operation. Addition is typically assumed to be commutative and associative, and the monoid product is assumed to distribute over the addition on both sides, and the additive identity 0 is an absorbing element in the sense that 0 x = 0 for all x. Near-ring: a semiring whose additive monoid is a (not necessarily abelian) group. Ring: a semiring whose additive monoid is an abelian group. Commutative ring: a ring in which the multiplication operation is commutative. Division ring: a nontrivial ring in which division by nonzero elements is defined. Integral domain: A nontrivial commutative ring in which the product of any two nonzero elements is nonzero. Field: a commutative division ring (i.e. a commutative ring which contains a multiplicative inverse for every nonzero element). Nonassociative rings: These are like rings, but the multiplication operation need not be associative. Lie ring: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the Jacobi identity rather than associativity. Jordan ring: a commutative nonassociative ring that respects the Jordan identity Boolean ring: a commutative ring with idempotent multiplication operation. Kleene algebras: a semiring with idempotent addition and


id: outline-of-algebraic-structures-20260505-2
file: data/en.wikipedia.org/wiki/Outline_of_algebraic_structures-2.md
category: reference
title: Outline of algebraic structures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_algebraic_structures
date_saved: 2026-05-05T08:12:44.195968+00:00
instance: kb-cron
parent: outline-of-algebraic-structures-20260505-0
summary:
Algebra over a ring (also R-algebra): a module over a commutative ring R, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication by elements of R. Algebra over a field: This is a ring which is also a vector space over a field. Multiplication is usually assumed to be associative. The theory is especially well developed. Associative algebra: an algebra over a ring such that the multiplication is associative. Nonassociative algebra: a module over a commutative ring, equipped with a ring multiplication operation that is not necessarily associative. Often associativity is replaced with a different identity, such as alternation, the Jacobi identity, or the Jordan identity. Lie algebra: a special type of nonassociative algebra whose product satisfies the Jacobi identity. Jordan algebra: a special type of nonassociative algebra whose product satisfies the Jordan identity. Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras. Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras. Graded algebra: a graded vector space with an algebra structure compatible with the grading. The idea is that if the grades of


id: list-of-complex-and-algebraic-surfaces-20260505-0
file: data/en.wikipedia.org/wiki/List_of_complex_and_algebraic_surfaces-0.md
category: reference
title: List of complex and algebraic surfaces
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_complex_and_algebraic_surfaces
date_saved: 2026-05-05T08:12:45.547440+00:00
instance: kb-cron
parent: list-of-complex-and-algebraic-surfaces-20260505-0
summary:
This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification. == Kodaira dimension −∞ == === Rational surfaces === Projective plane ==== Quadric surfaces ==== Cone (geometry) Cylinder Ellipsoid Hyperboloid Paraboloid Sphere Spheroid ==== Rational cubic surfaces ==== Cayley nodal cubic surface, a certain cubic surface with 4 nodes Cayley's ruled cubic surface Clebsch surface or Klein icosahedral surface Fermat cubic Monkey saddle Parabolic conoid Plücker's conoid Whitney umbrella ==== Rational quartic surfaces ==== Châtelet surfaces Dupin cyclides, inversions of a cylinder, torus, or double cone in a sphere Gabriel's horn Right circular conoid Roman surface or Steiner surface, a realization of the real projective plane in real affine space Tori, surfaces of revolution generated by a circle about a coplanar axis ==== Other rational surfaces in space ==== Boy's surface, a sextic realization of the real projective plane in real affine space Enneper surface, a nonic minimal surface Henneberg surface, a minimal surface of degree 15 Bour's minimal surface, a surface of degree 16 Richmond surfaces, a family of minimal surfaces of variable degree ==== Other families of rational surfaces ==== Coble surfaces Del Pezzo surfaces,


id: list-of-algebras-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algebras-0.md
category: reference
title: List of algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algebras
date_saved: 2026-05-05T08:12:46.737962+00:00
instance: kb-cron
parent: list-of-algebras-20260505-0
summary:
This is a list of possibly nonassociative algebras. An algebra is a module, wherein you can also multiply two module elements. (The multiplication in the module is compatible with multiplication-by-scalars from the base ring). *-algebra Affine Lie algebra Akivis algebra Algebra for a monad Albert algebra Alternative algebra AW*-algebra Azumaya algebra Banach algebra Birman–Wenzl algebra Boolean algebra Borcherds algebra Brauer algebra C*-algebra Central simple algebra Clifford algebra Cluster algebra Dendriform algebra Differential graded algebra Differential graded Lie algebra Exterior algebra F-algebra Filtered algebra Flexible algebra Freudenthal algebra Functional-theoretic algebra Factorization algebra Genetic algebra Geometric algebra Gerstenhaber algebra Graded algebra Griess algebra Group algebra Group algebra of a locally compact group Hall algebra Hecke algebra of a locally compact group Heyting algebra Hopf algebra Hurwitz algebra Hypercomplex algebra Incidence algebra Iwahori–Hecke algebra Jordan algebra Kac–Moody algebra Kleene algebra Leibniz algebra Lie algebra Lie superalgebra Malcev algebra Matrix algebra Non-associative algebra Octonion algebra Pre-Lie algebra Poisson algebra Process algebra Quadratic algebra Quadric geometric algebra Quaternion algebra Rees algebra Relation algebra Relational algebra Rota–Baxter algebra Schur algebra Semisimple algebra Separable algebra Shuffle algebra Sigma-algebra Simple algebra Structurable algebra Supercommutative algebra Symmetric algebra Tensor algebra Universal enveloping algebra Vertex operator algebra von Neumann algebra


id: list-of-algorithm-general-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algorithm_general_topics-0.md
category: reference
title: List of algorithm general topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithm_general_topics
date_saved: 2026-05-05T08:12:47.939703+00:00
instance: kb-cron
parent: list-of-algorithm-general-topics-20260505-0
summary:
This is a list of algorithm general topics. Analysis of algorithms Ant colony algorithm Approximation algorithm Best and worst cases Big O notation Combinatorial search Competitive analysis Computability theory Computational complexity theory Embarrassingly parallel problem Emergent algorithm Evolutionary algorithm Fast Fourier transform Genetic algorithm Graph exploration algorithm Heuristic Hill climbing Implementation Las Vegas algorithm Lock-free and wait-free algorithms Monte Carlo algorithm Numerical analysis Online algorithm Polynomial time approximation scheme Problem size Pseudorandom number generator Quantum algorithm Random-restart hill climbing Randomized algorithm Running time Sorting algorithm Search algorithm Stable algorithm (disambiguation) Super-recursive algorithm Tree search algorithm == See also == List of algorithms for specific algorithms List of computability and complexity topics for more abstract theory List of complexity classes, complexity class List of data structures.


id: list-of-algorithms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_algorithms-0.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
An algorithm is a fundamental set of rules or defined procedures that are typically designed and used to be a simpler way to solve a specific problem or a broad set of problems. Simply speaking, algorithms define different processes, sets of rules and regulations, or methodologies that are to be followed through in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are risk assessments, anticipatory policing, and pattern recognition technology. The following is a list of well-known algorithms. == Automated planning == == Combinatorial algorithms == === General combinatorial algorithms === Brent's algorithm: finds a cycle in function value iterations using only two iterators Floyd's cycle-finding algorithm: finds a cycle in function value iterations Gale–Shapley algorithm: solves the stable matching problem Pseudorandom number generators (uniformly distributed—see also List of pseudorandom number generators for other PRNGs with varying degrees of convergence and varying statistical quality): ACORN generator Blum Blum Shub Lagged Fibonacci generator Linear congruential generator Mersenne Twister === Graph algorithms === Blossom algorithm: algorithm for constructing maximum-cardinality matching on graphs. Coloring algorithm: algorithms for graph (vertex or


id: list-of-algorithms-20260505-1
file: data/en.wikipedia.org/wiki/List_of_algorithms-1.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
==== Approximate sequence matching ==== Bitap algorithm: fuzzy algorithm that determines if strings are approximately equal. Phonetic algorithms Daitch–Mokotoff Soundex: a Soundex refinement which allows matching of Slavic and Germanic surnames Double Metaphone: an improvement on Metaphone Match rating approach: a phonetic algorithm developed by Western Airlines Metaphone: an algorithm for indexing words by their sound, when pronounced in English NYSIIS: phonetic algorithm, improves on Soundex Soundex: a phonetic algorithm for indexing names by sound, as pronounced in English String metrics: computes a similarity or dissimilarity (distance) score between two pairs of text strings Damerau–Levenshtein distance: computes a distance measure between two strings, improves on Levenshtein distance Dice's coefficient (also known as the Dice coefficient): a similarity measure related to the Jaccard index Hamming distance: sum number of positions which are different Jaro–Winkler distance: is a measure of similarity between two strings Levenshtein edit distance: computes a metric for the amount of difference between two sequences Trigram search: search for text when the exact syntax or spelling of the target object is not precisely known ==== Selection algorithms ==== Introselect Quickselect ==== Sequence search ==== Linear search: locates an item in an unsorted sequence Selection algorithm: finds the kth largest


id: list-of-algorithms-20260505-2
file: data/en.wikipedia.org/wiki/List_of_algorithms-2.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Chien search: a recursive algorithm for determining roots of polynomials defined over a finite field Schreier–Sims algorithm: computing a base and strong generating set (BSGS) of a permutation group Todd–Coxeter algorithm: Procedure for generating cosets. === Computer algebra === Buchberger's algorithm: finds a Gröbner basis Cantor–Zassenhaus algorithm: factor polynomials over finite fields Faugère F4 algorithm: finds a Gröbner basis (also mentions the F5 algorithm) Gosper's algorithm: find sums of hypergeometric terms that are themselves hypergeometric terms Knuth–Bendix completion algorithm: for rewriting rule systems Multivariate division algorithm: for polynomials in several indeterminates Pollard's kangaroo algorithm (also known as Pollard's lambda algorithm): an algorithm for solving the discrete logarithm problem Polynomial long division: an algorithm for dividing a polynomial by another polynomial of the same or lower degree Risch algorithm: an algorithm for the calculus operation of indefinite integration (i.e. finding antiderivatives) === Geometry === Closest pair problem: find the pair of points (from a set of points) with the smallest distance between them Collision detection algorithms: check for the collision or intersection of two given solids Cone algorithm: identify surface points Convex hull algorithms: determining the convex hull of a set of points Chan's algorithm Gift wrapping algorithm or Jarvis march


id: list-of-algorithms-20260505-3
file: data/en.wikipedia.org/wiki/List_of_algorithms-3.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Computation of π: Bailey–Borwein–Plouffe formula: (BBP formula) a spigot algorithm for the computation of the nth binary digit of π Borwein's algorithm: an algorithm to calculate the value of 1/π Chudnovsky algorithm: a fast method for calculating the digits of π Gauss–Legendre algorithm: computes the digits of pi Division algorithms: for computing quotient and/or remainder of two numbers Goldschmidt division Long division Newton–Raphson division: uses Newton's method to find the reciprocal of D, and multiply that reciprocal by N to find the final quotient Q. Non-restoring division Restoring division SRT division Exponentiation: Addition-chain exponentiation: exponentiation by positive integer powers that requires a minimal number of multiplications Exponentiating by squaring: an algorithm used for the fast computation of large integer powers of a number Hyperbolic and Trigonometric Functions: BKM algorithm: computes elementary functions using a table of logarithms CORDIC: computes hyperbolic and trigonometric functions using a table of arctangents Montgomery reduction: an algorithm that allows modular arithmetic to be performed efficiently when the modulus is large Multiplication algorithms: fast multiplication of two numbers Booth's multiplication algorithm: a multiplication algorithm that multiplies two signed binary numbers in two's complement notation Fürer's algorithm: an integer multiplication algorithm for very large numbers possessing a


id: list-of-algorithms-20260505-4
file: data/en.wikipedia.org/wiki/List_of_algorithms-4.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
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parent: list-of-algorithms-20260505-0
summary:
=== Optimization algorithms === Hybrid Algorithms Alpha–beta pruning: search to reduce number of nodes in minimax algorithm A hybrid BFGS-Like method (see more https://doi.org/10.1016/j.cam.2024.115857) Branch and bound Bruss algorithm: see odds algorithm Chain matrix multiplication Combinatorial optimization: optimization problems where the set of feasible solutions is discrete Greedy randomized adaptive search procedure (GRASP): successive constructions of a greedy randomized solution and subsequent iterative improvements of it through a local search Hungarian method: a combinatorial optimization algorithm which solves the assignment problem in polynomial time Conjugate gradient methods (see more https://doi.org/10.1016/j.jksus.2022.101923) Constraint satisfaction AC-3 algorithm general algorithms for the constraint satisfaction Chaff algorithm: an algorithm for solving instances of the Boolean satisfiability problem Davis–Putnam algorithm: check the validity of a first-order logic formula Difference map algorithm general algorithms for the constraint satisfaction Davis–Putnam–Logemann–Loveland algorithm (DPLL): an algorithm for deciding the satisfiability of propositional logic formula in conjunctive normal form, i.e. for solving the CNF-SAT problem Exact cover problem Min conflicts algorithm general algorithms for the constraint satisfaction Algorithm X: a nondeterministic algorithm Dancing Links: an efficient implementation of Algorithm X Cross-entropy method: a general Monte Carlo approach to combinatorial and continuous multi-extremal optimization and importance sampling Differential evolution Dynamic Programming: problems


id: list-of-algorithms-20260505-5
file: data/en.wikipedia.org/wiki/List_of_algorithms-5.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
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parent: list-of-algorithms-20260505-0
summary:
Algorithms for calculating variance: avoiding instability and numerical overflow Approximate counting algorithm: allows counting large number of events in a small register Bayesian statistics Nested sampling algorithm: a computational approach to the problem of comparing models in Bayesian statistics Clustering algorithms Average-linkage clustering: a simple agglomerative clustering algorithm Canopy clustering algorithm: an unsupervised pre-clustering algorithm related to the K-means algorithm Chinese whispers Complete-linkage clustering: a simple agglomerative clustering algorithm DBSCAN: a density based clustering algorithm Expectation-maximization algorithm Fuzzy clustering: a class of clustering algorithms where each point has a degree of belonging to clusters FLAME clustering (Fuzzy clustering by Local Approximation of MEmberships): define clusters in the dense parts of a dataset and perform cluster assignment solely based on the neighborhood relationships among objects Fuzzy c-means k-means clustering: cluster objects based on attributes into partitions k-means++: a variation of this, using modified random seeds k-medoids: similar to k-means, but chooses datapoints or medoids as centers KHOPCA clustering algorithm: a local clustering algorithm, which produces hierarchical multi-hop clusters in static and mobile environments. Linde–Buzo–Gray algorithm: a vector quantization algorithm to derive a good codebook Lloyd's algorithm (Voronoi iteration or relaxation): group data points into a given number of categories, a popular


id: list-of-algorithms-20260505-6
file: data/en.wikipedia.org/wiki/List_of_algorithms-6.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Asymmetric (public key) encryption: ElGamal Elliptic curve cryptography MAE1 NTRUEncrypt RSA Digital signatures (asymmetric authentication): DSA, and its variants: ECDSA and Deterministic ECDSA EdDSA (Ed25519) RSA Cryptographic hash functions (see also the section on message authentication codes): BLAKE MD5 – Note that there is now a method of generating collisions for MD5 RIPEMD-160 SHA-1 – Note that there is now a method of generating collisions for SHA-1 SHA-2 (SHA-224, SHA-256, SHA-384, SHA-512) SHA-3 (SHA3-224, SHA3-256, SHA3-384, SHA3-512, SHAKE128, SHAKE256) Tiger (TTH), usually used in Tiger tree hashes WHIRLPOOL Cryptographically secure pseudo-random number generators Blum Blum Shub – based on the hardness of factorization Fortuna, intended as an improvement on Yarrow algorithm Linear-feedback shift register (note: many LFSR-based algorithms are weak or have been broken) Yarrow algorithm Key exchange Diffie–Hellman key exchange Elliptic-curve Diffie–Hellman (ECDH) Key derivation functions, often used for password hashing and key stretching Argon2 bcrypt PBKDF2 scrypt Message authentication codes (symmetric authentication algorithms, which take a key as a parameter): HMAC: keyed-hash message authentication Poly1305 SipHash Secret sharing, secret splitting, key splitting, M of N algorithms Blakey's scheme Shamir's secret sharing Symmetric (secret key) encryption: Advanced Encryption Standard (AES), winner of NIST competition, also known as Rijndael Blowfish


id: list-of-algorithms-20260505-7
file: data/en.wikipedia.org/wiki/List_of_algorithms-7.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Hopcroft's algorithm, Moore's algorithm, and Brzozowski's algorithm: algorithms for minimizing the number of states in a deterministic finite automaton Powerset construction: algorithm to convert nondeterministic automaton to deterministic automaton. Tarski–Kuratowski algorithm: a non-deterministic algorithm which provides an upper bound for the complexity of formulas in the arithmetical hierarchy and analytical hierarchy == Information theory and signal processing == === Coding theory === ==== Error detection and correction ==== BCH Codes Berlekamp–Massey algorithm Peterson–Gorenstein–Zierler algorithm Reed–Solomon error correction BCJR algorithm: decoding of error correcting codes defined on trellises (principally convolutional codes) Forward error correction Gray code Hamming codes Hamming(7,4): a Hamming code that encodes 4 bits of data into 7 bits by adding 3 parity bits Hamming distance: sum number of positions which are different Hamming weight (population count): find the number of 1 bits in a binary word Redundancy checks Adler-32 Cyclic redundancy check Damm algorithm Fletcher's checksum Longitudinal redundancy check (LRC) Luhn algorithm: a method of validating identification numbers Luhn mod N algorithm: extension of Luhn to non-numeric characters Parity: simple/fast error detection technique Verhoeff algorithm ==== Lossless compression algorithms ==== Burrows–Wheeler transform: preprocessing useful for improving lossless compression Context tree weighting Delta encoding: aid to compression of data


id: list-of-algorithms-20260505-8
file: data/en.wikipedia.org/wiki/List_of_algorithms-8.md
category: reference
title: List of algorithms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_algorithms
date_saved: 2026-05-05T08:12:49.096714+00:00
instance: kb-cron
parent: list-of-algorithms-20260505-0
summary:
Cache algorithms CHS conversion: converting between disk addressing systems Double dabble: convert binary numbers to BCD Hash function: convert a large, possibly variable-sized amount of data into a small datum, usually a single integer that may serve as an index into an array Fowler–Noll–Vo hash function: fast with low collision rate Pearson hashing: computes 8-bit value only, optimized for 8-bit computers Zobrist hashing: used in the implementation of transposition tables Unicode collation algorithm Xor swap algorithm: swaps the values of two variables without using a buffer == Database algorithms == Algorithms for Recovery and Isolation Exploiting Semantics (ARIES): transaction recovery Join algorithms Block nested loop Hash join Nested loop join Sort-Merge Join The Chase == Distributed systems algorithms == Clock synchronization Berkeley algorithm Cristian's algorithm Intersection algorithm Marzullo's algorithm Consensus (computer science): agreeing on a single value or history among unreliable processors Chandra–Toueg consensus algorithm Paxos algorithm Raft (computer science) Detection of Process Termination Dijkstra-Scholten algorithm Huang's algorithm Lamport ordering: a partial ordering of events based on the happened-before relation Leader election: a method for dynamically selecting a coordinator Bully algorithm Mutual exclusion Lamport's Distributed Mutual Exclusion Algorithm Naimi-Trehel's log(n) Algorithm Maekawa's Algorithm Raymond's Algorithm Ricart–Agrawala Algorithm Snapshot algorithm: record


id: list-of-aperiodic-sets-of-tiles-20260505-0
file: data/en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles-0.md
category: reference
title: List of aperiodic sets of tiles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_aperiodic_sets_of_tiles
date_saved: 2026-05-05T08:12:50.348730+00:00
instance: kb-cron
parent: list-of-aperiodic-sets-of-tiles-20260505-0
summary:
In geometry, a tiling is a partition of the plane (or any other geometric setting) into closed sets (called tiles), without gaps or overlaps (other than the boundaries of the tiles). A tiling is considered periodic if there exist translations in two independent directions which map the tiling onto itself. Such a tiling is composed of a single fundamental unit or primitive cell which repeats endlessly and regularly in two independent directions. An example of such a tiling is shown in the adjacent diagram (see the image description for more information). A tiling that cannot be constructed from a single primitive cell is called nonperiodic. If a given set of tiles allows only nonperiodic tilings, then this set of tiles is called aperiodic. The tilings obtained from an aperiodic set of tiles are often called aperiodic tilings, though strictly speaking it is the tiles themselves that are aperiodic. (The tiling itself is said to be "nonperiodic".) The first table explains the abbreviations used in the second table. The second table contains all known aperiodic sets of tiles and gives some additional basic information about each set. This list of tiles is still incomplete. == Explanations == == List == ==


id: outline-of-arithmetic-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_arithmetic-0.md
category: reference
title: Outline of arithmetic
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_arithmetic
date_saved: 2026-05-05T08:12:51.611767+00:00
instance: kb-cron
parent: outline-of-arithmetic-20260505-0
summary:
The following outline is provided as an overview of and topical guide to arithmetic: Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms, and the use of such operations to perform more complicated calculations. Arithmetic is fundamental for both pure mathematics and its practical application to science and daily life. In pure mathematics, arithmetic is defined formally for various types of numbers, such as integers, fractions of integers (rational numbers), and real numbers. In practical application, arithmetic is usually approximate, operating on decimal or binary numbers, with rounding. == Essence of arithmetic == Elementary arithmetic Decimal arithmetic Decimal point Numeral Place value == History of arithmetic == == Arithmetic operations and related concepts == Order of operations Addition Summation – Answer after adding a sequence of numbers Additive inverse Subtraction – Taking away numbers Multiplication – Repeated addition Multiple – Product of multiplication Least common multiple Multiplicative inverse Division – Repeated subtraction Modulo – The remainder of division Quotient – Result of division Quotition and partition – How many parts are there, and what is the size


id: list-of-mathematical-artists-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_artists-0.md
category: reference
title: List of mathematical artists
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_artists
date_saved: 2026-05-05T08:12:52.917714+00:00
instance: kb-cron
parent: list-of-mathematical-artists-20260505-0
summary:
This is a list of artists who actively explored mathematics in their artworks. Art forms practised by these artists include painting, sculpture, architecture, textiles and origami. Some artists such as Piero della Francesca and Luca Pacioli went so far as to write books on mathematics in art. Della Francesca wrote books on solid geometry and the emerging field of perspective, including De Prospectiva Pingendi (On Perspective for Painting), Trattato d’Abaco (Abacus Treatise), and De corporibus regularibus (Regular Solids), while Pacioli wrote De divina proportione (On Divine Proportion), with illustrations by Leonardo da Vinci, at the end of the fifteenth century. Merely making accepted use of some aspect of mathematics such as perspective does not qualify an artist for admission to this list. The term "fine art" is used conventionally to cover the output of artists who produce a combination of paintings, drawings and sculptures. == List == == References == == External links == Saint Louis University: List of mathematical artists, by field


id: list-of-axioms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_axioms-0.md
category: reference
title: List of axioms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_axioms
date_saved: 2026-05-05T08:12:54.221468+00:00
instance: kb-cron
parent: list-of-axioms-20260505-0
summary:
This is a list of axioms as that term is understood in mathematics. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system. == ZF (the Zermelo–Fraenkel axioms without the axiom of choice) == Together with the axiom of choice (see below), these are the de facto standard axioms for contemporary mathematics or set theory. They can be easily adapted to analogous theories, such as mereology. Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity Axiom schema of specification See also Zermelo set theory. == Axiom of choice == With the Zermelo–Fraenkel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable. === Equivalents of AC === Hausdorff maximality theorem Well-ordering theorem Zorn's lemma === Stronger than AC === Axiom of global choice === Weaker than AC === Axiom of countable choice Axiom of dependent choice Boolean prime ideal theorem Axiom of uniformization === Alternates incompatible with AC === Axiom of real determinacy == Other axioms of mathematical logic == Von Neumann–Bernays–Gödel axioms Continuum


id: list-of-banach-spaces-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Banach_spaces-0.md
category: reference
title: List of Banach spaces
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Banach_spaces
date_saved: 2026-05-05T08:12:55.569651+00:00
instance: kb-cron
parent: list-of-banach-spaces-20260505-0
summary:
In the mathematical field of functional analysis, Banach spaces are among the most important objects of study. In other areas of mathematical analysis, most spaces which arise in practice turn out to be Banach spaces as well. == Classical Banach spaces == According to Diestel (1984, Chapter VII), the classical Banach spaces are those defined by Dunford & Schwartz (1958), which is the source for the following table. Glossary of symbols for the table below: F {\displaystyle \mathbb {F} } denotes the field of real numbers R {\displaystyle \mathbb {R} } or complex numbers C . {\displaystyle \mathbb {C} .} K {\displaystyle K} is a compact Hausdorff space. p , q ∈ R {\displaystyle p,q\in \mathbb {R} } are real numbers with 1 < p , q < ∞ {\displaystyle 1<p,q<\infty } that are Hölder conjugates, meaning that they satisfy 1 q + 1 p = 1 {\displaystyle {\frac {1}{q}}+{\frac {1}{p}}=1} and thus also q = p p − 1 . {\displaystyle q={\frac {p}{p-1}}.} Σ {\displaystyle \Sigma } is a σ {\displaystyle \sigma } -algebra of sets. Ξ {\displaystyle \Xi } is an algebra of sets (for spaces only requiring finite additivity, such as the ba space). μ {\displaystyle \mu


id: list-of-works-by-nikolay-bogolyubov-20260505-0
file: data/en.wikipedia.org/wiki/List_of_works_by_Nikolay_Bogolyubov-0.md
category: reference
title: List of works by Nikolay Bogolyubov
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_works_by_Nikolay_Bogolyubov
date_saved: 2026-05-05T08:12:56.878536+00:00
instance: kb-cron
parent: list-of-works-by-nikolay-bogolyubov-20260505-0
summary:
List of some published works of Nikolay Bogolyubov in chronological order: == 1924 == N. N. Bogolyubov (1924). On the behavior of solutions of linear differential equations at infinity (Russian: О поведении решений линейных дифференциальных уравнений на бесконечности). == 1934 == N. M. Krylov and N. N. Bogolyubov (1934). On various formal expansions of non-linear mechanics (in Ukrainian). Kiev: Izdat. Zagal'noukr. Akad. Nauk. == 1937 == N. N. Bogoliubov and N. M. Krylov (1937). "La theorie generalie de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire" (in French). Ann. Math. II 38: 65–113. Zbl. 16.86. == 1945 == Боголюбов Н. Н. (1945). О некоторых статистических методах в математической физике (in Russian). Киев: Изд-во АН УССР. N. N. Bogoliubov (1945). On Some Statistical Methods in Mathematical Physics (in Russian). Kyiv: Izdatelstvo AN USSR. == 1946 == N. N. Bogoliubov (1946). "Kinetic Equations" (in Russian). Journal of Experimental and Theoretical Physics 16 (8): 691–702. N. N. Bogoliubov (1946). "Kinetic Equations" (in English). Journal of Physics 10 (3): 265–274. == 1947 == N. N. Bogoliubov, K. P. Gurov (1947). "Kinetic Equations in Quantum Mechanics" (in Russian). Journal of Experimental and Theoretical Physics 17 (7): 614–628. N. N.


id: list-of-boolean-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Boolean_algebra_topics-0.md
category: reference
title: List of Boolean algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Boolean_algebra_topics
date_saved: 2026-05-05T08:12:58.151075+00:00
instance: kb-cron
parent: list-of-boolean-algebra-topics-20260505-0
summary:
This is a list of topics around Boolean algebra and propositional logic. == Articles with a wide scope and introductions == Algebra of sets Boolean algebra (structure) Boolean algebra Field of sets Logical connective Propositional calculus == Boolean functions and connectives == Ampheck Analysis of Boolean functions Balanced Boolean function Bent function Boolean algebras canonically defined Boolean function Boolean matrix Boolean-valued function Conditioned disjunction Evasive Boolean function Exclusive or Functional completeness Logical biconditional Logical conjunction Logical disjunction Logical equality Logical implication Logical negation Logical NOR Majority function Material conditional Minimal axioms for Boolean algebra Peirce arrow Read-once function Sheffer stroke Sole sufficient operator Symmetric Boolean function Symmetric difference Zhegalkin polynomial == Examples of Boolean algebras == Boolean domain Complete Boolean algebra Interior algebra Two-element Boolean algebra == Extensions of Boolean algebras == Derivative algebra (abstract algebra) Free Boolean algebra Monadic Boolean algebra == Generalizations of Boolean algebras == De Morgan algebra First-order logic Heyting algebra Lindenbaum–Tarski algebra Skew Boolean algebra == Syntax == Algebraic normal form Boolean conjunctive query Canonical form (Boolean algebra) Conjunctive normal form Disjunctive normal form Formal system == Technical applications == And-inverter graph Logic gate Boolean analysis == Theorems and specific laws == Boolean prime ideal


id: list-of-calculus-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_calculus_topics-0.md
category: reference
title: List of calculus topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_calculus_topics
date_saved: 2026-05-05T08:12:59.314130+00:00
instance: kb-cron
parent: list-of-calculus-topics-20260505-0
summary:
This is a list of calculus topics. == Limits == Limit (mathematics) Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous function == Differential calculus == Derivative Notation Newton's notation for differentiation Leibniz's notation for differentiation Simplest rules Derivative of a constant Sum rule in differentiation Constant factor rule in differentiation Linearity of differentiation Power rule Chain rule Local linearization Product rule Quotient rule Inverse functions and differentiation Implicit differentiation Stationary point Maxima and minima First derivative test Second derivative test Extreme value theorem Differential equation Differential operator Newton's method Taylor's theorem L'Hôpital's rule General Leibniz rule Mean value theorem Logarithmic derivative Differential (calculus) Related rates Regiomontanus' angle maximization problem Rolle's theorem == Integral calculus == Antiderivative/Indefinite integral Simplest rules Sum rule in integration Constant factor rule in integration Linearity of integration Arbitrary constant of integration Cavalieri's quadrature formula Fundamental theorem of calculus Integration by parts Inverse chain rule method Integration by substitution Tangent half-angle substitution Differentiation under the integral sign Trigonometric substitution Partial fractions in integration Quadratic integral Proof that 22/7 exceeds π Trapezium rule Integral of the secant function Integral of secant cubed Arclength Solid of revolution


id: outline-of-calculus-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_calculus-0.md
category: reference
title: Outline of calculus
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_calculus
date_saved: 2026-05-05T08:13:00.609028+00:00
instance: kb-cron
parent: outline-of-calculus-20260505-0
summary:
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of contemporary mathematics education. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient. == Branches of calculus == Differential calculus Integral calculus Multivariable calculus Fractional calculus Differential Geometry == History of calculus == History of calculus Important publications in calculus == General calculus concepts == Continuous function Derivative Fundamental theorem of calculus Integral Limit Non-standard analysis Partial derivative Infinite Series == Calculus scholars == Sir Isaac Newton Gottfried Leibniz Leonhard Euler == Calculus lists == List of calculus topics == See also == Glossary of calculus Table of mathematical symbols == References == == External links == Weisstein, Eric W. "Calculus". MathWorld. Topics on Calculus at PlanetMath. Calculus Made Easy (1914) by Silvanus P. Thompson Full text in PDF Calculus.org: The Calculus page at University of California, Davis – contains resources and links to other sites COW: Calculus on the Web Archived 2007-10-14 at the Wayback Machine at Temple University - contains resources ranging from pre-calculus and associated algebra Online Integrator (WebMathematica) from Wolfram Research The Role of


id: outline-of-category-theory-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_category_theory-0.md
category: reference
title: Outline of category theory
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_category_theory
date_saved: 2026-05-05T08:13:01.794317+00:00
instance: kb-cron
parent: outline-of-category-theory-20260505-0
summary:
The following outline is provided as an overview of and guide to category theory: Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the mid-20th century in their foundational work on algebraic topology. Category theory can be used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. Many areas of computer science also rely on category theory, such as functional programming and semantics. == Essence of category theory == Category Functor Natural transformation == Branches of category theory == Homological algebra Diagram chasing Topos theory Enriched category theory Higher category theory Categorical logic Applied category theory == Specific categories == Category of sets Concrete category Category of small categories Category of vector spaces Category of graded vector spaces Category of chain complexes Category of finite dimensional Hilbert spaces Category of sets and relations Category of topological spaces Category of metric spaces Category of preordered sets Category of groups Category of abelian groups Category of rings


id: list-of-chaotic-maps-20260505-0
file: data/en.wikipedia.org/wiki/List_of_chaotic_maps-0.md
category: reference
title: List of chaotic maps
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_chaotic_maps
date_saved: 2026-05-05T08:13:03.039151+00:00
instance: kb-cron
parent: list-of-chaotic-maps-20260505-0
summary:
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems. Chaotic maps and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems). == List of chaotic maps == == List of fractals == Cantor set de Rham curve Gravity set, or Mitchell-Green gravity set Julia set - derived from complex quadratic map Koch snowflake - special case of de Rham curve Lyapunov fractal Mandelbrot set - derived from complex quadratic map Menger sponge Newton fractal Nova fractal - derived from Newton fractal Quaternionic fractal - three dimensional complex quadratic map Sierpinski carpet Sierpinski triangle == References ==


id: list-of-mathematical-topics-in-classical-mechanics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_topics_in_classical_mechanics-0.md
category: reference
title: List of mathematical topics in classical mechanics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_topics_in_classical_mechanics
date_saved: 2026-05-05T08:13:04.549140+00:00
instance: kb-cron
parent: list-of-mathematical-topics-in-classical-mechanics-20260505-0
summary:
This is a list of mathematical topics in classical mechanics, by Wikipedia page. See also list of variational topics, correspondence principle. == Newtonian physics == Newton's laws of motion Inertia, Kinematics, rigid body Momentum, kinetic energy Parallelogram of force Circular motion Rotational speed Angular speed Angular momentum torque angular acceleration moment of inertia parallel axes rule perpendicular axes rule stretch rule centripetal force, centrifugal force, Reactive centrifugal force Laplace–Runge–Lenz vector Euler's disk elastic potential energy Mechanical equilibrium D'Alembert's principle Degrees of freedom (physics and chemistry) Frame of reference Inertial frame of reference Galilean transformation Principle of relativity == Conservation laws == Conservation of momentum Conservation of linear momentum Conservation of angular momentum Conservation of energy Potential energy Conservative force Conservation of mass == Law of universal gravitation == Projectile motion Kepler's laws of planetary motion Escape velocity Potential well Weightlessness Lagrangian point N-body problem Kolmogorov–Arnold–Moser theorem Virial theorem Gravitational binding energy Speed of gravity Newtonian limit Hill sphere Roche lobe Roche limit == Hamiltonian mechanics == Phase space Symplectic manifold Liouville's theorem (Hamiltonian) Poisson bracket Poisson algebra Poisson manifold Antibracket algebra Hamiltonian constraint Moment map Contact geometry Analysis of flows Nambu mechanics == Lagrangian mechanics == Action (physics) Lagrangian Euler–Lagrange


id: list-of-cohomology-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_cohomology_theories-0.md
category: reference
title: List of cohomology theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_cohomology_theories
date_saved: 2026-05-05T08:13:25.127193+00:00
instance: kb-cron
parent: list-of-cohomology-theories-20260505-0
summary:
This is a list of some of the ordinary and generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectra. For other sorts of homology theories see the links at the end of this article. == Notation == S = π = S 0 {\displaystyle S=\pi =S^{0}} is the sphere spectrum. S n {\displaystyle S^{n}} is the spectrum of the n {\displaystyle n} -dimensional sphere S n Y = S n ∧ Y {\displaystyle S^{n}Y=S^{n}\land Y} is the n {\displaystyle n} th suspension of a spectrum Y {\displaystyle Y} . [ X , Y ] {\displaystyle [X,Y]} is the abelian group of morphisms from the spectrum X {\displaystyle X} to the spectrum Y {\displaystyle Y} , given (roughly) as homotopy classes of maps. [ X , Y ] n = [ S n X , Y ] {\displaystyle [X,Y]_{n}=[S^{n}X,Y]} [ X , Y ] ∗ {\displaystyle [X,Y]_{*}} is the graded abelian group given as the sum of the groups [ X , Y ] n {\displaystyle [X,Y]_{n}} . π n ( X ) = [ S n , X ] = [ S , X ] n {\displaystyle \pi


id: list-of-cohomology-theories-20260505-1
file: data/en.wikipedia.org/wiki/List_of_cohomology_theories-1.md
category: reference
title: List of cohomology theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_cohomology_theories
date_saved: 2026-05-05T08:13:25.127193+00:00
instance: kb-cron
parent: list-of-cohomology-theories-20260505-0
summary:
=== K theory with coefficients === Spectrum: KG G is some abelian group; for example the localization Z(p) at the prime p. Other K-theories can also be given coefficients. === Self conjugate K-theory === Spectrum: KSC Coefficient ring: to be written... The coefficient groups π i {\displaystyle \pi _{i}} (KSC) have period 4 in i, given by the sequence Z, Z2, 0, Z, repeated. Introduced by Donald W. Anderson in his unpublished 1964 University of California, Berkeley Ph.D. dissertation, "A new cohomology theory". === Connective K-theories === Spectrum: ku for connective K-theory, ko for connective real K-theory. Coefficient ring: For ku, the coefficient ring is the ring of polynomials over Z on a single class v1 in dimension 2. For ko, the coefficient ring is the quotient of a polynomial ring on three generators, η in dimension 1, x4 in dimension 4, and v14 in dimension 8, the periodicity generator, modulo the relations that 2η = 0, x42 = 4v14, η3 = 0, and ηx = 0. Roughly speaking, this is K-theory with the negative dimensional parts killed off. === KR-theory === This is a cohomology theory defined for spaces with involution, from which many of the other K-theories can


id: list-of-combinatorial-computational-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_combinatorial_computational_geometry_topics-0.md
category: reference
title: List of combinatorial computational geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_combinatorial_computational_geometry_topics
date_saved: 2026-05-05T08:13:26.374691+00:00
instance: kb-cron
parent: list-of-combinatorial-computational-geometry-topics-20260505-0
summary:
List of combinatorial computational geometry topics enumerates the topics of computational geometry that states problems in terms of geometric objects as discrete entities and hence the methods of their solution are mostly theories and algorithms of combinatorial character. See List of numerical computational geometry topics for another flavor of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. == Construction/representation == Boolean operations on polygons Convex hull Hyperplane arrangement Polygon decomposition Polygon triangulation Minimal convex decomposition Minimal convex cover problem (NP-hard) Minimal rectangular decomposition Tessellation problems Shape dissection problems Straight skeleton Stabbing line problem Triangulation Delaunay triangulation Point-set triangulation Polygon triangulation Voronoi diagram == Extremal shapes == Minimum bounding box (Smallest enclosing box, Smallest bounding box) 2-D case: Smallest bounding rectangle (Smallest enclosing rectangle) There are two common variants of this problem. In many areas of computer graphics, the bounding box (often abbreviated to bbox) is understood to be the smallest box delimited by sides parallel to coordinate axes which encloses the objects in question. In other applications, such as packaging, the problem is to find the smallest box the object (or objects) may fit in ("packaged"). Here


id: index-of-combinatorics-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_combinatorics_articles-0.md
category: reference
title: Index of combinatorics articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_combinatorics_articles
date_saved: 2026-05-05T08:13:27.553086+00:00
instance: kb-cron
parent: index-of-combinatorics-articles-20260505-0
summary:
== A == Abstract simplicial complex Addition chain Scholz conjecture Algebraic combinatorics Alternating sign matrix Almost disjoint sets Antichain Arrangement of hyperplanes Assignment problem Quadratic assignment problem Audioactive decay == B == Barcode Matrix code QR Code Universal Product Code Bell polynomials Bertrand's ballot theorem Binary matrix Binomial theorem Block design Balanced incomplete block design(BIBD) Symmetric balanced incomplete block design (SBIBD) Partially balanced incomplete block designs (PBIBDs) Block walking Boolean satisfiability problem 2-satisfiability 3-satisfiability Bracelet (combinatorics) Bruck–Chowla–Ryser theorem == C == Catalan number Cellular automaton Collatz conjecture Combination Combinatorial design Combinatorial number system Combinatorial optimization Combinatorial search Constraint satisfaction problem Conway's Game of Life Cycles and fixed points Cyclic order Cyclic permutation Cyclotomic identity == D == Data integrity Alternating bit protocol Checksum Cyclic redundancy check Luhn formula Error detection Error-detecting code Error-detecting system Message digest Redundancy check Summation check De Bruijn sequence Deadlock Delannoy number Dining philosophers problem Mutual exclusion Rendezvous problem Derangement Dickson's lemma Dinitz conjecture Discrete optimization Dobinski's formula == E == Eight queens puzzle Entropy coding Enumeration Algebraic enumeration Combinatorial enumeration Burnside's lemma Erdős–Ko–Rado theorem Euler number == F == Faà di Bruno's formula Factorial number system Family of sets Faulhaber's formula Fifteen puzzle Finite geometry


id: outline-of-combinatorics-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_combinatorics-0.md
category: reference
title: Outline of combinatorics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_combinatorics
date_saved: 2026-05-05T08:13:28.713632+00:00
instance: kb-cron
parent: outline-of-combinatorics-20260505-0
summary:
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. == Essence of combinatorics == Matroid Greedoid Ramsey theory Van der Waerden's theorem Hales–Jewett theorem Umbral calculus, binomial type polynomial sequences Combinatorial species == Branches of combinatorics == Algebraic combinatorics Analytic combinatorics Arithmetic combinatorics Combinatorics on words Combinatorial design theory Enumerative combinatorics Extremal combinatorics Geometric combinatorics Graph theory Infinitary combinatorics Matroid theory Order theory Partition theory Probabilistic combinatorics Topological combinatorics === Multi-disciplinary fields that include combinatorics === Coding theory Combinatorial optimization Combinatorics and dynamical systems Combinatorics and physics Discrete geometry Finite geometry Phylogenetics == History of combinatorics == History of combinatorics == General combinatorial principles and methods == Combinatorial principles Trial and error, brute-force search, bogosort, British Museum algorithm Pigeonhole principle Method of distinguished element Mathematical induction Recurrence relation, telescoping series Generating functions as an application of formal power series Cyclic sieving Schrödinger method Exponential generating function Stanley's reciprocity theorem Binomial coefficients and their properties Combinatorial proof Double counting (proof technique) Bijective proof Inclusion–exclusion principle Möbius inversion formula Parity, even and odd permutations Combinatorial Nullstellensatz Incidence algebra Greedy algorithm Divide and conquer algorithm Akra–Bazzi method Dynamic programming Branch and bound Birthday attack, birthday paradox Floyd's


id: list-of-commutative-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_commutative_algebra_topics-0.md
category: reference
title: List of commutative algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_commutative_algebra_topics
date_saved: 2026-05-05T08:13:29.965551+00:00
instance: kb-cron
parent: list-of-commutative-algebra-topics-20260505-0
summary:
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers Z {\displaystyle \mathbb {Z} } ; and p-adic integers. Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. == Research fields == Combinatorial commutative algebra Invariant theory === Active research areas === Serre's multiplicity conjectures Homological conjectures == Basic notions == Commutative ring Module (mathematics) Ring ideal, maximal ideal, prime ideal Ring homomorphism Ring monomorphism Ring epimorphism Ring isomorphism Zero divisor Chinese remainder theorem == Classes of rings == Field (mathematics) Algebraic number field Polynomial ring Integral domain Boolean algebra (structure) Principal ideal domain Euclidean domain Unique factorization domain Dedekind domain Nilpotent elements and reduced rings Dual numbers Tensor product of fields Tensor product of R-algebras == Constructions with commutative rings == Quotient ring Field of fractions Product of rings Annihilator (ring theory) Integral closure == Localization and completion == Completion (ring theory) Formal power


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-0.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
Geometric algebra is an extension of vector algebra, providing additional algebraic structures on vector spaces, with geometric interpretations. Vector algebra uses all dimensions and signatures, as does geometric algebra, notably 3+1 spacetime as well as 2 dimensions. == Basic concepts and operations == Geometric algebra (GA) is an extension or completion of vector algebra (VA). The reader is herein assumed to be familiar with the basic concepts and operations of VA and this article will mainly concern itself with operations in G 3 {\displaystyle {\mathcal {G}}_{3}} the GA of 3D space (nor is this article intended to be mathematically rigorous). In GA, vectors are not normally written boldface as the meaning is usually clear from the context. The fundamental difference is that GA provides a new product of vectors called the "geometric product". Elements of GA are graded multivectors: scalars are grade 0, usual vectors are grade 1, bivectors are grade 2 and the highest grade (3 in the 3D case) is traditionally called the pseudoscalar and designated I {\displaystyle I} . The ungeneralized 3D vector form of the geometric product is: a b = a ⋅ b + a ∧ b {\displaystyle ab=a\cdot b+a\wedge b} that is the sum


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-1
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-1.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
The pseudovector/bivector subalgebra of the geometric algebra of Euclidean 3-dimensional space form a 3-dimensional vector space themselves. Let the standard unit pseudovectors/bivectors of the subalgebra be i = e 2 e 3 {\displaystyle \mathbf {i} =\mathbf {e_{2}} \mathbf {e_{3}} } , j = e 1 e 3 {\displaystyle \mathbf {j} =\mathbf {e_{1}} \mathbf {e_{3}} } , and k = e 1 e 2 {\displaystyle \mathbf {k} =\mathbf {e_{1}} \mathbf {e_{2}} } , and the anti-commutative commutator product be defined as A × B = 1 2 ( A B − B A ) {\displaystyle A\times B={\tfrac {1}{2}}(AB-BA)} , where A B {\displaystyle AB} is the geometric product. The commutator product is distributive over addition and linear, as the geometric product is distributive over addition and linear. From the definition of the commutator product, i {\displaystyle \mathbf {i} } , j {\displaystyle \mathbf {j} } and k {\displaystyle \mathbf {k} } satisfy the following equalities: i × j = 1 2 ( i j − j i ) = 1 2 ( ( e 2 e 3 e 1 e 3 − e 1 e 3 e 2 e 3 ) = 1 2 ( − e 2 e 3 e


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-2
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-2.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
A × B = ( A 1 i + A 2 j + A 3 k ) × ( B 1 i + B 2 j + B 3 k ) = A 1 B 1 i × i + A 1 B 2 i × j + A 1 B 3 i × k + A 2 B 1 j × i + A 2 B 2 j × j + A 2 B 3 j × k + A 3 B 1 k × i + A 3 B 2 k × j + A 3 B 3 k × k = A 1 B 2 k − A 1 B 3 j − A 2 B 1 k + A 2 B 3 i + A 3 B 1 j − A 3 B 2 i = ( A 2 B 3 − A 3 B 2 ) i + ( A 3 B 1 − A 1 B 3 ) j + ( A 1 B 2 − A 2 B 1 ) k {\displaystyle {\begin{aligned}\mathbf {A} \times \mathbf {B} &=(A_{1}\mathbf {i} +A_{2}\mathbf {j} +A_{3}\mathbf {k} )\times (B_{1}\mathbf {i} +B_{2}\mathbf {j} +B_{3}\mathbf {k} )\\&=A_{1}B_{1}\mathbf {i} \times


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-3
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-3.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
This is a useful theoretic result. For numerical problems row reduction with pivots and other methods are more stable and efficient. When the wedge product is coupled with the Clifford product and put into a natural geometric context, the fact that the determinants are used in the expression of R N {\displaystyle {\mathbb {R} }^{N}} parallelogram area and parallelepiped volumes (and higher-dimensional generalizations thereof) also comes as a nice side-effect. As is also shown below, results such as Cramer's rule also follow directly from the wedge product's selection of non-identical elements. The result is then simple enough that it could be derived easily if required instead of having to remember or look up a rule. Two variables example [ a b ] [ x y ] = a x + b y = c . {\displaystyle {\begin{bmatrix}a&b\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=ax+by=c.} Pre- and post-multiplying by a {\displaystyle a} and b {\displaystyle b} , ( a x + b y ) ∧ b = ( a ∧ b ) x = c ∧ b {\displaystyle (ax+by)\wedge b=(a\wedge b)x=c\wedge b} a ∧ ( a x + b y ) = ( a ∧ b ) y = a ∧ c {\displaystyle a\wedge (ax+by)=(a\wedge b)y=a\wedge c} Provided


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-4
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-4.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
v − u ^ ( u ^ ⋅ v ) = 1 ‖ u ‖ 2 ( ‖ u ‖ 2 v − u ( u ⋅ v ) ) {\displaystyle v-{\hat {u}}({\hat {u}}\cdot v)={\frac {1}{{\Vert u\Vert }^{2}}}({\Vert u\Vert }^{2}v-u(u\cdot v))} The rejection can be expressed as a single geometric algebraic product in a few different ways u u 2 ( u v − u ⋅ v ) = 1 u ( u ∧ v ) = u ^ ( u ^ ∧ v ) = ( v ∧ u ^ ) u ^ {\displaystyle {\frac {u}{{u}^{2}}}(uv-u\cdot v)={\frac {1}{u}}(u\wedge v)={\hat {u}}({\hat {u}}\wedge v)=(v\wedge {\hat {u}}){\hat {u}}} The similarity in form between the projection and the rejection is notable. The sum of these recovers the original vector v = u ^ ( u ^ ⋅ v ) + u ^ ( u ^ ∧ v ) {\displaystyle v={\hat {u}}({\hat {u}}\cdot v)+{\hat {u}}({\hat {u}}\wedge v)} Here the projection is in its customary vector form. An alternate formulation is possible that puts the projection in a form that differs from the usual vector formulation v = 1 u ( u ⋅ v ) + 1 u ( u ∧ v ) = (


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-5
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-5.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
Notice the similarities between this planar rejection result and the vector rejection result. To calculate the component of a vector outside of a plane we take the volume spanned by three vectors (trivector) and "divide out" the plane. Independent of any use of the geometric product it can be shown that this rejection in terms of the standard basis is x = 1 ( A u , v ) 2 ∑ i < j < k | w i w j w k u i u j u k v i v j v k | | u i u j u k v i v j v k e i e j e k | {\displaystyle x={\frac {1}{(A_{u,v})^{2}}}\sum _{i<j<k}{\begin{vmatrix}w_{i}&w_{j}&w_{k}\\u_{i}&u_{j}&u_{k}\\v_{i}&v_{j}&v_{k}\\\end{vmatrix}}{\begin{vmatrix}u_{i}&u_{j}&u_{k}\\v_{i}&v_{j}&v_{k}\\{e}_{i}&{e}_{j}&{e}_{k}\\\end{vmatrix}}} where ( A u , v ) 2 = ∑ i < j | u i u j v i v j | = − ( u ∧ v ) 2 {\displaystyle (A_{u,v})^{2}=\sum _{i<j}{\begin{vmatrix}u_{i}&u_{j}\\v_{i}&v_{j}\end{vmatrix}}=-(u\wedge v)^{2}} is the squared area of the parallelogram formed by u {\displaystyle u} , and v {\displaystyle v} . The (squared) magnitude of x {\displaystyle x} is ‖ x ‖ 2 = x ⋅ w = 1 ( A u , v ) 2 ∑ i


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-6
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-6.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
A 2 = − ( u ∧ v ) 2 = ∑ i < j | u i u j v i v j | 2 . {\displaystyle A^{2}=-(\mathbf {u} \wedge \mathbf {v} )^{2}=\sum _{i<j}{\begin{vmatrix}u_{i}&u_{j}\\v_{i}&v_{j}\end{vmatrix}}^{2}.} Note that this squared bivector is a geometric multiplication; this computation can alternatively be stated as the Gram determinant of the two vectors. === Angle between two vectors === ( sin ⁡ θ ) 2 = ‖ u × v ‖ 2 ‖ u ‖ 2 ‖ v ‖ 2 {\displaystyle ({\sin \theta })^{2}={\frac {{\Vert \mathbf {u} \times \mathbf {v} \Vert }^{2}}{{\Vert \mathbf {u} \Vert }^{2}{\Vert \mathbf {v} \Vert }^{2}}}} ( sin ⁡ θ ) 2 = − ( u ∧ v ) 2 u 2 v 2 {\displaystyle ({\sin \theta })^{2}=-{\frac {(\mathbf {u} \wedge \mathbf {v} )^{2}}{{\mathbf {u} }^{2}{\mathbf {v} }^{2}}}} === Volume of the parallelopiped formed by three vectors === In vector algebra, the volume of a parallelopiped is given by the square root of the squared norm of the scalar triple product: V 2 = ‖ ( u × v ) ⋅ w ‖ 2 = | u 1 u 2 u 3 v 1 v 2 v 3 w 1 w


id: comparison-of-vector-algebra-and-geometric-algebra-20260505-7
file: data/en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra-7.md
category: reference
title: Comparison of vector algebra and geometric algebra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Comparison_of_vector_algebra_and_geometric_algebra
date_saved: 2026-05-05T08:13:31.235748+00:00
instance: kb-cron
parent: comparison-of-vector-algebra-and-geometric-algebra-20260505-0
summary:
d d t ( r ‖ r ‖ ) = 1 ‖ r ‖ 3 ( r × d r d t ) × r = ( r ^ × 1 ‖ r ‖ d r d t ) × r ^ {\displaystyle {\frac {d}{dt}}\left({\frac {\mathbf {r} }{\Vert \mathbf {r} \Vert }}\right)={\frac {1}{{\Vert \mathbf {r} \Vert }^{3}}}\left(\mathbf {r} \times {\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {r} =\left({\hat {\mathbf {r} }}\times {\frac {1}{\Vert \mathbf {r} \Vert }}{\frac {d\mathbf {r} }{dt}}\right)\times {\hat {\mathbf {r} }}} The equivalent geometric product generalization is d d t ( r ‖ r ‖ ) = 1 ‖ r ‖ 3 r ( r ∧ d r d t ) = 1 r ( r ^ ∧ d r d t ) {\displaystyle {\frac {d}{dt}}\left({\frac {\mathbf {r} }{\Vert \mathbf {r} \Vert }}\right)={\frac {1}{{\Vert \mathbf {r} \Vert }^{3}}}\mathbf {r} \left(\mathbf {r} \wedge {\frac {d\mathbf {r} }{dt}}\right)={\frac {1}{\mathbf {r} }}\left({\hat {\mathbf {r} }}\wedge {\frac {d\mathbf {r} }{dt}}\right)} Thus this derivative is the component of 1 ‖ r ‖ d r d t {\displaystyle {\frac {1}{\Vert \mathbf {r} \Vert }}{\frac {d\mathbf {r} }{dt}}} in the direction perpendicular to r {\displaystyle \mathbf {r} } . In other words, this is 1 ‖


id: list-of-mathematics-competitions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_competitions-0.md
category: reference
title: List of mathematics competitions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_competitions
date_saved: 2026-05-05T08:13:32.568377+00:00
instance: kb-cron
parent: list-of-mathematics-competitions-20260505-0
summary:
Mathematics competitions or mathematical olympiads are competitive events where participants complete a math test. These tests may require multiple choice or numeric answers, or a detailed written solution or proof. == International mathematics competitions == World Mathematics Invitational — for all grade levels K-12, Taiwanese based international mathematics competition, with English and Chinese as the official languages of the papers. Championnat International de Jeux Mathématiques et Logiques — for all ages, mainly for French-speaking countries, but participation is not limited by language. China Girls Mathematical Olympiad (CGMO) — held annually for teams of girls representing different regions within China and a few other countries. European Girls' Mathematical Olympiad (EGMO) — since April 2012 Integration Bee — competition in integral calculus held in various institutions of higher learning in the United States and some other countries International Mathematical Modeling Challenge — team contest for high school students International Mathematical Olympiad (IMO) — the oldest international Olympiad, occurring annually since 1959. International Mathematics Competition for University Students (IMC) — international competition for undergraduate students. Mathematical Contest in Modeling (MCM) — team contest for undergraduates Mathematical Kangaroo — worldwide competition. Mental Calculation World Cup — contest for the best mental calculators Primary Mathematics


id: list-of-complex-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_complex_analysis_topics-0.md
category: reference
title: List of complex analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_complex_analysis_topics
date_saved: 2026-05-05T08:13:33.858629+00:00
instance: kb-cron
parent: list-of-complex-analysis-topics-20260505-0
summary:
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering. See also: glossary of real and complex analysis. == Overview == Complex numbers Complex plane Complex functions Complex derivative Holomorphic functions Harmonic functions Elementary functions Polynomial functions Exponential functions Trigonometric functions Hyperbolic functions Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions Residue theory Isometries in the complex plane == Related fields == Number theory Hydrodynamics Thermodynamics Electrical engineering == Local theory == Holomorphic function Antiholomorphic function Cauchy–Riemann equations Conformal mapping Conformal welding Power series Radius of convergence Laurent series Meromorphic function Entire function Pole (complex analysis) Zero (complex analysis) Residue (complex analysis) Isolated singularity Removable singularity Essential singularity Branch point Principal branch Weierstrass–Casorati theorem Landau's constants Holomorphic functions are analytic Schwarzian derivative Analytic capacity Disk algebra Univalent function == Growth and distribution of values == Ahlfors theory Bieberbach conjecture Borel–Carathéodory theorem Corona theorem Hadamard three-circle theorem Hardy space Hardy's theorem Maximum modulus principle Nevanlinna theory Paley–Wiener theorem Phragmén-Lindelöf principle Progressive function


id: list-of-complexity-classes-20260505-0
file: data/en.wikipedia.org/wiki/List_of_complexity_classes-0.md
category: reference
title: List of complexity classes
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_complexity_classes
date_saved: 2026-05-05T08:13:35.107427+00:00
instance: kb-cron
parent: list-of-complexity-classes-20260505-0
summary:
This is a list of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics. Many of these classes have a 'co' partner which consists of the complements of all languages in the original class. For example, if a language L is in NP then the complement of L is in co-NP. (This does not mean that the complement of NP is co-NP—there are languages which are known to be in both, and other languages which are known to be in neither.) "The hardest problems" of a class refer to problems which belong to the class such that every other problem of that class can be reduced to it. == References == == External links == Complexity Zoo - list of over 500 complexity classes and their properties


id: list-of-computability-and-complexity-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_computability_and_complexity_topics-0.md
category: reference
title: List of computability and complexity topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_computability_and_complexity_topics
date_saved: 2026-05-05T08:13:36.334916+00:00
instance: kb-cron
parent: list-of-computability-and-complexity-topics-20260505-0
summary:
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with what can be computed, in principle. Computational complexity theory deals with how hard computations are, in quantitative terms, both with upper bounds (algorithms whose complexity in the worst cases, as use of computing resources, can be estimated), and from below (proofs that no procedure to carry out some task can be very fast). == Calculation == Lookup table Mathematical table Multiplication table Generating trigonometric tables History of computers Multiplication algorithm Peasant multiplication Division by two Exponentiating by squaring Addition chain Scholz conjecture Presburger arithmetic == Computability theory: models of computation == Arithmetic circuits Algorithm Procedure, recursion Finite-state automaton Mealy machine Minsky register machine Moore machine State diagram State transition system Deterministic finite automaton Nondeterministic finite automaton Generalized nondeterministic finite automaton Regular language Pumping lemma Myhill–Nerode theorem Regular expression Regular grammar Prefix grammar Tree automaton Pushdown automaton Context-free grammar Büchi automaton Chomsky hierarchy Context-sensitive language, context-sensitive grammar Recursively enumerable language Register machine Stack machine Petri net Post machine Rewriting Markov algorithm Term rewriting String rewriting system L-system Knuth–Bendix completion algorithm Star height Star height problem Generalized


id: computational-complexity-of-mathematical-operations-20260505-0
file: data/en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations-0.md
category: reference
title: Computational complexity of mathematical operations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Computational_complexity_of_mathematical_operations
date_saved: 2026-05-05T08:13:37.584907+00:00
instance: kb-cron
parent: computational-complexity-of-mathematical-operations-20260505-0
summary:
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. See big O notation for an explanation of the notation used. Note: Due to the variety of multiplication algorithms, M ( n ) {\displaystyle M(n)} below stands in for the complexity of the chosen multiplication algorithm. == Arithmetic functions == This table lists the complexity of mathematical operations on integers. On stronger computational models, specifically a pointer machine and consequently also a unit-cost random-access machine it is possible to multiply two n-bit numbers in time O(n). == Algebraic functions == Here we consider operations over polynomials and n denotes their degree; for the coefficients we use a unit-cost model, ignoring the number of bits in a number. In practice this means that we assume them to be machine integers. For this section M ( n ) {\displaystyle M(n)} indicates the time needed for multiplying two polynomials of degree at most n {\displaystyle n} . == Special functions == Many of the methods in this section are given in Borwein & Borwein. === Elementary functions === The elementary functions are constructed by


id: list-of-computer-algebra-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_computer_algebra_systems-0.md
category: reference
title: List of computer algebra systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_computer_algebra_systems
date_saved: 2026-05-05T08:13:38.747454+00:00
instance: kb-cron
parent: list-of-computer-algebra-systems-20260505-0
summary:
The following tables provide a comparison of computer algebra systems (CAS). A CAS is a package comprising a set of algorithms for performing symbolic manipulations on algebraic objects, a language to implement them, and an environment in which to use the language. A CAS may include a user interface and graphics capability, and to be effective may require a large library of algorithms, efficient data structures, and a fast kernel. == General == These computer algebra systems are sometimes combined with "front end" programs that provide a better user interface, such as the general-purpose GNU TeXmacs. === Functionality === Below is a summary of significantly developed symbolic functionality in each of the systems. ^ via SymPy ^ via qepcad optional package Those which do not "edit equations" may have a GUI, plotting, ASCII graphic formulae and math font printing. The ability to generate plaintext files is also a sought-after feature because it allows a work to be understood by people who do not have a computer algebra system installed. === Operating system support === The software can run under their respective operating systems natively without emulation. Some systems must be compiled first using an appropriate compiler for the source language


id: list-of-computer-graphics-and-descriptive-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_computer_graphics_and_descriptive_geometry_topics-0.md
category: reference
title: List of computer graphics and descriptive geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_computer_graphics_and_descriptive_geometry_topics
date_saved: 2026-05-05T08:13:40.076467+00:00
instance: kb-cron
parent: list-of-computer-graphics-and-descriptive-geometry-topics-20260505-0
summary:
This is a list of computer graphics and descriptive geometry topics, by article name. == !-9 == 2D computer graphics 2D geometric model 3D computer graphics 3D modeling 3D projection 3D rendering == A == A-buffer Algorithmic art Aliasing Alpha compositing Alpha mapping Alpha to coverage Ambient occlusion Anamorphosis Anisotropic filtering Anti-aliasing Asymptotic decider Augmented reality Axis-aligned bounding box Axonometric projection == B == B-spline Back-face culling Barycentric coordinate system Beam tracing Bézier curve Bézier surface Bicubic interpolation Bidirectional reflectance distribution function Bidirectional scattering distribution function Bidirectional texture function Bilateral filter Bilinear interpolation Bin (computational geometry) Binary space partitioning Bit blit Bit plane Bitmap Bitmap textures Blend modes Blinn–Phong reflection model Bloom (shader effect) Bounding interval hierarchy Bounding sphere Bounding volume Bounding volume hierarchy Bresenham's line algorithm Bump mapping == C == Calligraphic projection Cel shading Channel (digital image) Checkerboard rendering Circular thresholding Clip coordinates Clipmap Clipping (computer graphics) Clipping path Collision detection Color depth Color gradient Color space Colour banding Color bleeding (computer graphics) Color cycling Composite Bézier curve Compositing Computational geometry Compute kernel Computer animation Computer art Computer graphics Computer graphics (computer science) Computer graphics lighting Computer-generated imagery Cone tracing Constructive solid geometry Control point (mathematics) Convex hull


id: list-of-conjectures-by-paul-erd-s-20260505-0
file: data/en.wikipedia.org/wiki/List_of_conjectures_by_Paul_Erdős-0.md
category: reference
title: List of conjectures by Paul Erdős
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_conjectures_by_Paul_Erdős
date_saved: 2026-05-05T08:13:41.334180+00:00
instance: kb-cron
parent: list-of-conjectures-by-paul-erd-s-20260505-0
summary:
The prolific mathematician Paul Erdős and his various collaborators made many famous mathematical conjectures, over a wide field of subjects, and in many cases Erdős offered monetary rewards for solving them. The unsolved problems are commonly known as Erdős problems. == Unsolved == The Erdős–Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3. The Erdős–Hajnal conjecture that in a family of graphs defined by an excluded induced subgraph, every graph has either a large clique or a large independent set. The Erdős–Mollin–Walsh conjecture on consecutive triples of powerful numbers. The Erdős–Selfridge conjecture that a covering system with distinct moduli contains at least one even modulus. The Erdős–Straus conjecture on the Diophantine equation 4/n = 1/x + 1/y + 1/z. The Erdős conjecture on arithmetic progressions in sequences with divergent sums of reciprocals. The Erdős–Szekeres conjecture on the number of points needed to ensure that a point set contains a large convex polygon. The Erdős–Turán conjecture on additive bases of natural numbers. A conjecture on quickly growing integer sequences with rational reciprocal series. A conjecture with Norman Oler on circle packing in an equilateral triangle with a number of circles one less


id: list-of-conjectures-20260505-0
file: data/en.wikipedia.org/wiki/List_of_conjectures-0.md
category: reference
title: List of conjectures
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_conjectures
date_saved: 2026-05-05T08:13:42.537850+00:00
instance: kb-cron
parent: list-of-conjectures-20260505-0
summary:
This is a list of notable mathematical conjectures. == Open problems == The following conjectures remain open. The (incomplete) column "cites" lists the number of results for a Google Scholar search for the term, in double quotes as of September 2022. == Conjectures now proved (theorems) == The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic names. Deligne's conjecture on 1-motives == Disproved (no longer conjectures) == The conjectures in following list were not necessarily generally accepted as true before being disproved. Aharoni-Korman conjecture also known as the fishbone conjecture Atiyah conjecture (not a conjecture to start with) Borsuk's conjecture Chinese hypothesis (not a conjecture to start with) Doomsday conjecture Euler's sum of powers conjecture Ganea conjecture Generalized Smith conjecture Hauptvermutung Hedetniemi's conjecture, counterexample announced 2019 Hirsch conjecture (disproved 2010) Kaplansky unit conjecture Intersection graph conjecture Kelvin's conjecture Kouchnirenko's conjecture Mertens conjecture Pólya conjecture, 1919 (1958) Ragsdale conjecture Schoenflies conjecture (disproved 1910) Tait's conjecture Von Neumann conjecture Weyl–Berry conjecture Williamson conjecture == Refutations of conventional wisdom == In contemporary mathematics, ideas are supposedly not accepted absent rigorous proof. Historically, some statements that were generally accepted were then shown to be false.


id: list-of-continuity-related-mathematical-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics-0.md
category: reference
title: List of continuity-related mathematical topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_continuity-related_mathematical_topics
date_saved: 2026-05-05T08:13:43.792858+00:00
instance: kb-cron
parent: list-of-continuity-related-mathematical-topics-20260505-0
summary:
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways. == Continuity of functions and measures == Continuous function Absolutely continuous function Absolute continuity of a measure with respect to another measure Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is absolutely continuous with respect to Lebesgue measure. This less inclusive sense is equivalent to the condition that every set whose Lebesgue measure is 0 has probability 0. Geometric continuity Parametric continuity Smoothness == Continuum == Continuum (set theory), the real line or the corresponding cardinal number Linear continuum, any ordered set that shares certain properties of the real line Continuum (topology), a nonempty compact connected metric space (sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets and the cardinality of the set of all real numbers. The latter cardinality is equal to the cardinality of the set of all subsets of a countably infinite set. Cardinality of the continuum, a cardinal number that represents the


id: list-of-convexity-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_convexity_topics-0.md
category: reference
title: List of convexity topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_convexity_topics
date_saved: 2026-05-05T08:13:45.110385+00:00
instance: kb-cron
parent: list-of-convexity-topics-20260505-0
summary:
This is a list of convexity topics, by Wikipedia page. Alpha blending - the process of combining a translucent foreground color with a background color, thereby producing a new blended color. This is a convex combination of two colors allowing for transparency effects in computer graphics. Barycentric coordinates - a coordinate system in which the location of a point of a simplex (a triangle, tetrahedron, etc.) is specified as the center of mass, or barycenter, of masses placed at its vertices. The coordinates are non-negative for points in the convex hull. Borsuk's conjecture - a conjecture about the number of pieces required to cover a body with a larger diameter. Solved by Hadwiger for the case of smooth convex bodies. Bond convexity - a measure of the non-linear relationship between price and yield duration of a bond to changes in interest rates, the second derivative of the price of the bond with respect to interest rates. A basic form of convexity in finance. Carathéodory's theorem (convex hull) - If a point x of Rd lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its


id: list-of-coordinate-charts-20260505-0
file: data/en.wikipedia.org/wiki/List_of_coordinate_charts-0.md
category: reference
title: List of coordinate charts
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_coordinate_charts
date_saved: 2026-05-05T08:13:46.208272+00:00
instance: kb-cron
parent: list-of-coordinate-charts-20260505-0
summary:
This article contains a non-exhaustive list of coordinate charts for Riemannian manifolds and pseudo-Riemannian manifolds. Coordinate charts are mathematical objects of topological manifolds, and they have multiple applications in theoretical and applied mathematics. When a differentiable structure and a metric are defined, greater structure exists, and this allows the definition of constructs such as integration and geodesics. == Charts for Riemannian and pseudo-Riemannian surfaces == The following charts (with appropriate metric tensors) can be used in the stated classes of Riemannian and pseudo-Riemannian surfaces: == See also == Coordinate chart Coordinate system Metric tensor List of mathematics lists, particularly: List of multivariable calculus topics List of Fourier analysis topics List of differential geometry topics == References ==


id: list-of-common-coordinate-transformations-20260505-0
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-0.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
This is a list of some of the most commonly used coordinate transformations. == 2-dimensional == Let ( x , y ) {\displaystyle (x,y)} be the standard Cartesian coordinates, and ( r , θ ) {\displaystyle (r,\theta )} the standard polar coordinates. === To Cartesian coordinates === ==== From polar coordinates ==== x = r cos ⁡ θ y = r sin ⁡ θ ∂ ( x , y ) ∂ ( r , θ ) = [ cos ⁡ θ − r sin ⁡ θ sin ⁡ θ − r cos ⁡ θ ] Jacobian = det ∂ ( x , y ) ∂ ( r , θ ) = r {\displaystyle {\begin{aligned}x&=r\cos \theta \\y&=r\sin \theta \\[5pt]{\frac {\partial (x,y)}{\partial (r,\theta )}}&={\begin{bmatrix}\cos \theta &-r\sin \theta \\\sin \theta &{\phantom {-}}r\cos \theta \end{bmatrix}}\\[5pt]{\text{Jacobian}}=\det {\frac {\partial (x,y)}{\partial (r,\theta )}}&=r\end{aligned}}} ==== From log-polar coordinates ==== x = e ρ cos ⁡ θ , y = e ρ sin ⁡ θ . {\displaystyle {\begin{aligned}x&=e^{\rho }\cos \theta ,\\y&=e^{\rho }\sin \theta .\end{aligned}}} By using complex numbers ( x , y ) = x + i y ′ {\displaystyle (x,y)=x+iy'} , the transformation can be written as x + i y = e ρ + i θ {\displaystyle


id: list-of-common-coordinate-transformations-20260505-1
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-1.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as illustrated). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent. If, in the alternative definition, θ is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged. All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation. === To Cartesian coordinates === ==== From spherical coordinates ==== x = ρ sin ⁡ θ cos ⁡ φ y = ρ sin ⁡ θ sin ⁡ φ z = ρ cos ⁡ θ ∂


id: list-of-common-coordinate-transformations-20260505-2
file: data/en.wikipedia.org/wiki/List_of_common_coordinate_transformations-2.md
category: reference
title: List of common coordinate transformations
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations
date_saved: 2026-05-05T08:13:47.536234+00:00
instance: kb-cron
parent: list-of-common-coordinate-transformations-20260505-0
summary:
∂ ( r , θ , h ) ∂ ( x , y , z ) = ( x x 2 + y 2 y x 2 + y 2 0 − y x 2 + y 2 x x 2 + y 2 0 0 0 1 ) {\displaystyle {\frac {\partial (r,\theta ,h)}{\partial (x,y,z)}}={\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}&0\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}&0\\0&0&1\end{pmatrix}}} ==== From spherical coordinates ==== r = ρ sin ⁡ φ h = ρ cos ⁡ φ θ = θ ∂ ( r , h , θ ) ∂ ( ρ , φ , θ ) = ( sin ⁡ φ ρ cos ⁡ φ 0 cos ⁡ φ − ρ sin ⁡ φ 0 0 0 1 ) det ∂ ( r , h , θ ) ∂ ( ρ , φ , θ ) = − ρ {\displaystyle {\begin{aligned}r&=\rho \sin \varphi \\h&=\rho \cos \varphi \\\theta &=\theta \\{\frac {\partial (r,h,\theta )}{\partial (\rho ,\varphi ,\theta )}}&={\begin{pmatrix}\sin \varphi &\rho \cos \varphi &0\\\cos \varphi &-\rho \sin \varphi &0\\0&0&1\\\end{pmatrix}}\\\det {\frac {\partial (r,h,\theta )}{\partial (\rho ,\varphi ,\theta )}}&=-\rho \end{aligned}}} === Arc-length, curvature and torsion from Cartesian coordinates === s = ∫ 0 t x ′ 2 + y ′ 2 + z ′


id: index-of-cryptography-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-0.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
Articles related to cryptography include: == A == A5/1 • A5/2 • ABA digital signature guidelines • ABC (stream cipher) • Abraham Sinkov • Acoustic cryptanalysis • Adaptive chosen-ciphertext attack • Adaptive chosen plaintext and chosen ciphertext attack • Advantage (cryptography) • ADFGVX cipher • Adi Shamir • Advanced Access Content System • Advanced Encryption Standard • Advanced Encryption Standard process • Adversary • AEAD block cipher modes of operation • Affine cipher • Agnes Meyer Driscoll • AKA (security) • Akelarre (cipher) • Alan Turing • Alastair Denniston • Al Bhed language • Alex Biryukov • Alfred Menezes • Algebraic Eraser • Algorithmically random sequence • Alice and Bob • All-or-nothing transform • Alphabetum Kaldeorum • Alternating step generator • American Cryptogram Association • AN/CYZ-10 • Anonymous publication • Anonymous remailer • Antoni Palluth • Anubis (cipher) • Argon2 • ARIA (cipher) • Arlington Hall • Arne Beurling • Arnold Cipher • Array controller based encryption • Arthur Scherbius • Arvid Gerhard Damm • Asiacrypt • Atbash • Attribute-based encryption • Attack model • Auguste Kerckhoffs • Authenticated encryption • Authentication • Authorization certificate • Autokey cipher • Avalanche effect == B == B-Dienst • Babington Plot • Baby-step


id: index-of-cryptography-articles-20260505-1
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-1.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== D == D'Agapeyeff cipher • Daniel J. Bernstein • Data Authentication Algorithm • Data Encryption Standard • Datagram Transport Layer Security • David Chaum • David Kahn • David Naccache • David Wagner • David Wheeler (computer scientist) • Davies attack • Davies–Meyer hash • DEAL • Decipherment • Decisional Diffie–Hellman assumption • Decorrelation theory • Decrypt • DeCSS • Defence Signals Directorate • Degree of anonymity • Delegated Path Discovery • Delegated Path Validation • Deniable encryption • Derek Taunt • Derived unique key per transaction • DES Challenges • DES supplementary material • DES-X • Deterministic encryption • DFC (cipher) • Dictionary attack • Differential cryptanalysis • Differential-linear attack • Differential power analysis • Diffie–Hellman key exchange • Diffie–Hellman problem • DigiCipher 2 • Digital Fortress • Digital rights management • Digital signature • Digital Signature Algorithm • Digital signature forgery • Digital timestamping • Digital watermarking • Dilly Knox • Dining cryptographers problem • Diplomatic bag • Direct Anonymous Attestation • Discrete logarithm • Disk encryption • Disk encryption hardware • Disk encryption software • Distance-bounding protocol • Distinguishing attack • Distributed.net • DMA attack • dm-crypt • Dmitry Sklyarov • DomainKeys • Don Coppersmith •


id: index-of-cryptography-articles-20260505-2
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-2.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== K == Kaisa Nyberg • Kalyna (cipher) • Kasiski examination • KASUMI • KCDSA • KeePass • Kerberos (protocol) • Kerckhoffs's principle • Kevin McCurley (cryptographer) • Key-agreement protocol • Key authentication • Key clustering • Key (cryptography) • Key derivation function • Key distribution center • Key escrow • Key exchange • Keyfile • Key generation • Key generator • Key management • Key-recovery attack • Key schedule • Key server (cryptographic) • Key signature (cryptography) • Keysigning • Key signing party • Key size • Key space (cryptography) • Keystream • Key stretching • Key whitening • KG-84 • KHAZAD • Khufu and Khafre • Kiss (cryptanalysis) • KL-43 • KL-51 • KL-7 • Kleptography • KN-Cipher • Knapsack problem • Known-key distinguishing attack • Known-plaintext attack • KOI-18 • KOV-14 • Kryha • Kryptos • KSD-64 • Kupyna • Kuznyechik • KW-26 • KW-37 • KY-3 • KY-58 • KY-68 • KYK-13 == L == Lacida • Ladder-DES • Lamport signature • Lars Knudsen • Lattice-based cryptography • Laurance Safford • Lawrie Brown • LCS35 • Leo Marks • Leonard Adleman • Leon Battista Alberti • Leo Rosen • Leslie Yoxall • LEVIATHAN (cipher) • LEX


id: index-of-cryptography-articles-20260505-3
file: data/en.wikipedia.org/wiki/Index_of_cryptography_articles-3.md
category: reference
title: Index of cryptography articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_cryptography_articles
date_saved: 2026-05-05T08:13:48.739443+00:00
instance: kb-cron
parent: index-of-cryptography-articles-20260505-0
summary:
== R == Rabbit (cipher) • Rabin cryptosystem • Rabin signature • RadioGatún • Rail fence cipher • Rainbow table • Ralph Merkle • Rambutan (cryptography) • Random function • Randomness tests • Random number generator attack • Random oracle • RC2 • RC4 • RC5 • RC6 • Rebound attack • Reciprocal cipher • Red/black concept • REDOC • Red Pike (cipher) • Reflector (cipher machine) • Regulation of Investigatory Powers Act 2000 • Reihenschieber • Rekeying (cryptography) • Related-key attack • Replay attack • Reservehandverfahren • Residual block termination • Rijndael key schedule • Rijndael S-box • Ring signature • RIPEMD • Rip van Winkle cipher • Robert Morris (cryptographer) • Rockex • Rolf Noskwith • Ron Rivest • Room 40 • Root certificate • Ross J. Anderson • Rossignols • ROT13 • Rotor machine • RSA RSA • RSA-100 • RSA-1024 • RSA-110 • RSA-120 • RSA-129 • RSA-130 • RSA-140 • RSA-150 • RSA-1536 • RSA-155 • RSA-160 • RSA-170 • RSA-180 • RSA-190 • RSA-200 • RSA-2048 • RSA-210 • RSA-220 • RSA-230 • RSA-232 • RSA-240 • RSA-250 • RSA-260 • RSA-270 • RSA-280 • RSA-290 • RSA-300 • RSA-309 • RSA-310 • RSA-320


id: list-of-curves-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_curves_topics-0.md
category: reference
title: List of curves topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_curves_topics
date_saved: 2026-05-05T08:13:49.986081+00:00
instance: kb-cron
parent: list-of-curves-topics-20260505-0
summary:
This is an alphabetical index of articles related to curves used in mathematics. Acnode Algebraic curve Arc Asymptote Asymptotic curve Barbier's theorem Bézier curve Bézout's theorem Birch and Swinnerton-Dyer conjecture Bitangent Bitangents of a quartic Cartesian coordinate system Caustic Cesàro equation Chord (geometry) Cissoid Circumference Closed timelike curve concavity Conchoid (mathematics) Confocal Contact (mathematics) Contour line Crunode Cubic Hermite curve Curvature Curve orientation Curve fitting Curve-fitting compaction Curve of constant width Curve of pursuit Curves in differential geometry Cusp Cyclogon De Boor algorithm Differential geometry of curves Eccentricity (mathematics) Elliptic curve cryptography Envelope (mathematics) Fenchel's theorem Genus (mathematics) Geodesic Geometric genus Great-circle distance Harmonograph Hedgehog (curve) [1] Hilbert's sixteenth problem Hyperelliptic curve cryptography Inflection point Inscribed square problem intercept, y-intercept, x-intercept Intersection number Intrinsic equation Isoperimetric inequality Jordan curve Jordan curve theorem Knot Limit cycle Linking coefficient List of circle topics Loop (knot) M-curve Mannheim curve[2] Meander (mathematics) Mordell conjecture Natural representation Opisometer Orbital elements Osculating circle Osculating plane Osgood curve Parallel (curve) Parallel transport Parametric curve Bézier curve Spline (mathematics) Hermite spline Beta spline B-spline Higher-order spline NURBS Perimeter Pi Plane curve Pochhammer contour Polar coordinate system Prime geodesic Projective line Ray Regular parametric representation Reuleaux triangle Ribaucour curve[3][4]


id: differentiation-rules-20260505-0
file: data/en.wikipedia.org/wiki/Differentiation_rules-0.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
This article is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus. == Elementary rules of differentiation == Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb {R} } ) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers ( C {\textstyle \mathbb {C} } ). === Constant term rule === For any value of c {\textstyle c} , where c ∈ R {\textstyle c\in \mathbb {R} } , if f ( x ) {\textstyle f(x)} is the constant function given by f ( x ) = c {\textstyle f(x)=c} , then d f d x = 0 {\textstyle {\frac {df}{dx}}=0} . ==== Proof ==== Let c ∈ R {\textstyle c\in \mathbb {R} } and f ( x ) = c {\textstyle f(x)=c} . By the definition of the derivative: f ′ ( x ) = lim h → 0 f ( x + h ) − f ( x ) h = lim h → 0 ( c ) − ( c ) h = lim h → 0 0 h = lim


id: differentiation-rules-20260505-1
file: data/en.wikipedia.org/wiki/Differentiation_rules-1.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
h ′ ( x ) = − f ′ ( x ) ( f ( x ) ) 2 , {\displaystyle h'(x)=-{\frac {f'(x)}{(f(x))^{2}}},} wherever f {\textstyle f} is nonzero. In Leibniz's notation, this formula is written: d ( 1 f ) d x = − 1 f 2 d f d x . {\displaystyle {\frac {d\left({\frac {1}{f}}\right)}{dx}}=-{\frac {1}{f^{2}}}{\frac {df}{dx}}.} The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule. === Quotient rule === If f {\textstyle f} and g {\textstyle g} are functions, then: ( f g ) ′ = f ′ g − g ′ f g 2 , {\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-g'f}{g^{2}}},} wherever g {\textstyle g} is nonzero. This can be derived from the product rule and the reciprocal rule. === Generalized power rule === The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f {\textstyle f} and g {\textstyle g} , ( f g ) ′ = ( e g ln ⁡ f ) ′ = f g ( f ′ g f + g ′ ln ⁡ f ) , {\displaystyle (f^{g})'=\left(e^{g\ln f}\right)'=f^{g}\left(f'{g \over f}+g'\ln


id: differentiation-rules-20260505-2
file: data/en.wikipedia.org/wiki/Differentiation_rules-2.md
category: reference
title: Differentiation rules
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Differentiation_rules
date_saved: 2026-05-05T08:13:51.299386+00:00
instance: kb-cron
parent: differentiation-rules-20260505-0
summary:
Γ ′ ( x ) = ∫ 0 ∞ t x − 1 e − t ln ⁡ t d t = Γ ( x ) ( ∑ n = 1 ∞ ( ln ⁡ ( 1 + 1 n ) − 1 x + n ) − 1 x ) = Γ ( x ) ψ ( x ) , {\displaystyle {\begin{aligned}\Gamma '(x)&=\int _{0}^{\infty }t^{x-1}e^{-t}\ln t\,dt\\&=\Gamma (x)\left(\sum _{n=1}^{\infty }\left(\ln \left(1+{\dfrac {1}{n}}\right)-{\dfrac {1}{x+n}}\right)-{\dfrac {1}{x}}\right)\\&=\Gamma (x)\psi (x),\end{aligned}}} with ψ ( x ) {\textstyle \psi (x)} being the digamma function, expressed by the parenthesized expression to the right of Γ ( x ) {\textstyle \Gamma (x)} in the line above. === Riemann zeta function === ζ ( x ) = ∑ n = 1 ∞ 1 n x {\displaystyle \zeta (x)=\sum _{n=1}^{\infty }{\frac {1}{n^{x}}}} ζ ′ ( x ) = − ∑ n = 1 ∞ ln ⁡ n n x = − ln ⁡ 2 2 x − ln ⁡ 3 3 x − ln ⁡ 4 4 x − ⋯ = − ∑ p prime p − x ln ⁡ p ( 1 − p − x ) 2 ∏ q prime , q ≠ p 1 1 − q


id: list-of-derivatives-and-integrals-in-alternative-calculi-20260505-0
file: data/en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi-0.md
category: reference
title: List of derivatives and integrals in alternative calculi
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_derivatives_and_integrals_in_alternative_calculi
date_saved: 2026-05-05T08:13:52.505671+00:00
instance: kb-cron
parent: list-of-derivatives-and-integrals-in-alternative-calculi-20260505-0
summary:
There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea. The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). == Table == In the following table; ψ ( x ) = Γ ′ ( x ) Γ ( x ) {\displaystyle \psi (x)={\frac {\Gamma '(x)}{\Gamma (x)}}} is the digamma function, K ⁡ ( x ) = e ζ ′ ( − 1 , x ) − ζ ′ ( − 1 ) = e z − z 2 2 + z 2 ln ⁡ ( 2 π ) − ψ ( − 2 ) ( z ) {\displaystyle \operatorname {K} (x)=e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=e^{{\frac {z-z^{2}}{2}}+{\frac {z}{2}}\ln(2\pi )-\psi ^{(-2)}(z)}} is the K-function, ( ! x ) = Γ ( x + 1 , − 1 ) e {\displaystyle (!x)={\frac {\Gamma (x+1,-1)}{e}}} is subfactorial, B a ( x ) = − a ζ ( − a + 1 , x ) {\displaystyle B_{a}(x)=-a\zeta (-a+1,x)} are the generalized to real


id: list-of-differential-geometry-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_differential_geometry_topics-0.md
category: reference
title: List of differential geometry topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_differential_geometry_topics
date_saved: 2026-05-05T08:13:53.815515+00:00
instance: kb-cron
parent: list-of-differential-geometry-topics-20260505-0
summary:
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. == Differential geometry of curves and surfaces == === Differential geometry of curves === List of curves topics Frenet–Serret formulas Curves in differential geometry Line element Curvature Radius of curvature Osculating circle Curve Fenchel's theorem === Differential geometry of surfaces === Theorema egregium Gauss–Bonnet theorem First fundamental form Second fundamental form Gauss–Codazzi–Mainardi equations Dupin indicatrix Asymptotic curve Curvature Principal curvatures Mean curvature Gauss curvature Elliptic point Types of surfaces Minimal surface Ruled surface Conical surface Developable surface Nadirashvili surface == Foundations == === Calculus on manifolds === See also multivariable calculus, list of multivariable calculus topics Manifold Differentiable manifold Smooth manifold Banach manifold Fréchet manifold Tensor analysis Tangent vector Tangent space Tangent bundle Cotangent space Cotangent bundle Tensor Tensor bundle Vector field Tensor field Differential form Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle Frobenius theorem (differential topology) Integral curve === Differential topology === Diffeomorphism Large diffeomorphism Orientability characteristic class Chern class Pontrjagin class spin structure differentiable map submersion immersion Embedding Whitney embedding theorem Critical value Sard's theorem Saddle point Morse


id: outline-of-discrete-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_discrete_mathematics-0.md
category: reference
title: Outline of discrete mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_discrete_mathematics
date_saved: 2026-05-05T08:13:55.045428+00:00
instance: kb-cron
parent: outline-of-discrete-mathematics-20260505-0
summary:
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics, therefore, excludes topics in "continuous mathematics" such as calculus and analysis. Included below are many of the standard terms used routinely in university-level courses and in research papers. This is not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may be encountered. Logic – Study of correct reasoning Modal logic – Type of formal logic Set theory – Branch of mathematics that studies sets Number theory – Branch of pure mathematics Combinatorics – Branch of discrete mathematics Finite mathematics – Syllabus in college and university mathematics Graph theory – Area of discrete mathematics Digital geometry – Deals with digitized models or images of objects of the 2D or 3D Euclidean space Digital topology – Properties of 2D or 3D digital images that correspond to classic topological properties Algorithmics – Sequence of operations for


id: outline-of-discrete-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Outline_of_discrete_mathematics-1.md
category: reference
title: Outline of discrete mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_discrete_mathematics
date_saved: 2026-05-05T08:13:55.045428+00:00
instance: kb-cron
parent: outline-of-discrete-mathematics-20260505-0
summary:
Left-hand side and right-hand side of an equation – Mathematical nomenclaturePages displaying short descriptions of redirect targets Linear equation – Equation that does not involve powers or products of variables Quadratic equation – Polynomial equation of degree two Solution point – Mathematical formula expressing equalityPages displaying short descriptions of redirect targets Arithmetic progression – Sequence of equally spaced numbers Recurrence relation – Pattern defining an infinite sequence of numbers Finite difference – Discrete analog of a derivative Difference operator – Pattern defining an infinite sequence of numbersPages displaying short descriptions of redirect targets Groups – Set with associative invertible operation Group isomorphism – Bijective group homomorphism Subgroups – Subset of a group that forms a group itselfPages displaying short descriptions of redirect targets Fermat's little theorem – A prime p divides a^p–a for any integer a Cryptography – Practice and study of secure communication techniques Faulhaber's formula – Expression for sums of powers === Mathematical relations === Binary relation – Relationship between elements of two sets Heterogeneous relation – Relationship between elements of two setsPages displaying short descriptions of redirect targets Reflexive relation – Binary relation that relates every element to itself Reflexive property of equality – Basic notion of


id: divisibility-rule-20260505-0
file: data/en.wikipedia.org/wiki/Divisibility_rule-0.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American. == Divisibility rules for numbers 1−30 == The rules given below transform a given number into a generally smaller number, while preserving divisibility by the divisor of interest. Therefore, unless otherwise noted, the resulting number should be evaluated for divisibility by the same divisor. In some cases the process can be iterated until the divisibility is obvious; for others (such as examining the last n digits) the result must be examined by other means. For divisors with multiple rules, the rules are generally ordered first for those appropriate for numbers with many digits, then those useful for numbers with fewer digits. == Step-by-step examples == === Divisibility by 2 === First, take any number (for this example it will be 376) and note


id: divisibility-rule-20260505-1
file: data/en.wikipedia.org/wiki/Divisibility_rule-1.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Divisibility by 5 === Divisibility by 5 is easily determined by checking the last digit in the number (475), and seeing if it is either 0 or 5. If the last number is either 0 or 5, the entire number is divisible by 5. If the last digit in the number is 0, then the result will be the remaining digits multiplied by 2. For example, the number 40 ends in a zero, so take the remaining digits (4) and multiply that by two (4 × 2 = 8). The result is the same as the result of 40 divided by 5(40/5 = 8). If the last digit in the number is 5, then the result will be the remaining digits multiplied by two, plus one. For example, the number 125 ends in a 5, so take the remaining digits (12), multiply them by two (12 × 2 = 24), then add one (24 + 1 = 25). The result is the same as the result of 125 divided by 5 (125/5=25). Example. If the last digit is 0 110 (The original number) 11 0 (Take the last digit of the number, and check if it is 0 or


id: divisibility-rule-20260505-2
file: data/en.wikipedia.org/wiki/Divisibility_rule-2.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Take for instance the number 371 Change all occurrences of 7, 8 or 9 into 0, 1 and 2, respectively. In this example, we get: 301. This second step may be skipped, except for the left most digit, but following it may facilitate calculations later on. Now convert the first digit (3) into the following digit in the sequence 13264513... In our example, 3 becomes 2. Add the result in the previous step (2) to the second digit of the number, and substitute the result for both digits, leaving all remaining digits unmodified: 2 + 0 = 2. So 301 becomes 21. Repeat the procedure until you have a recognizable multiple of 7, or to make sure, a number between 0 and 6. So, starting from 21 (which is a recognizable multiple of 7), take the first digit (2) and convert it into the following in the sequence above: 2 becomes 6. Then add this to the second digit: 6 + 1 = 7. If at any point the first digit is 8 or 9, these become 1 or 2, respectively. But if it is a 7 it should become 0, only if no other digits follow. Otherwise, it should


id: divisibility-rule-20260505-3
file: data/en.wikipedia.org/wiki/Divisibility_rule-3.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
For all of the above examples, subtracting the first three digits from the last three results in a multiple of seven. Notice that leading zeros are permitted to form a 6-digit pattern. This phenomenon forms the basis for Steps B and C. Step B: If the integer is between 1001 and one million, find a repeating pattern of 1, 2, or 3 digits that forms a 6-digit number that is close to the integer (leading zeros are allowed and can help you visualize the pattern). If the positive difference is less than 1000, apply Step A. This can be done by subtracting the first three digits from the last three digits. For example: 341,355 − 341,341 = 14 -> 1 − (4×2) = 1 − 8 = −7 YES 67,326 − 067,067 = 259 -> 25 − (9×2) = 25 − 18 = 7 YES The fact that 999,999 is a multiple of 7 can be used for determining divisibility of integers larger than one million by reducing the integer to a 6-digit number that can be determined using Step B. This can be done easily by adding the digits left of the first six to the last six and


id: divisibility-rule-20260505-4
file: data/en.wikipedia.org/wiki/Divisibility_rule-4.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Finding remainder of a number when divided by 7 7 − (1, 3, 2, −1, −3, −2, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, −1, −3, −2 Minimum magnitude sequence (1, 3, 2, 6, 4, 5, cycle repeats for the next six digits) Period: 6 digits. Recurring numbers: 1, 3, 2, 6, 4, 5 Positive sequence Multiply the right most digit by the left most digit in the sequence and multiply the second right most digit by the second left most digit in the sequence and so on and so for. Next, compute the sum of all the values and take the modulus of 7. Example: What is the remainder when 1036125837 is divided by 7? Multiplication of the rightmost digit = 1 × 7 = 7 Multiplication of the second rightmost digit = 3 × 3 = 9 Third rightmost digit = 8 × 2 = 16 Fourth rightmost digit = 5 × −1 = −5 Fifth rightmost digit = 2 × −3 = −6 Sixth rightmost digit = 1 × −2 = −2 Seventh rightmost digit = 6 × 1 = 6 Eighth rightmost digit = 3 × 3 =


id: divisibility-rule-20260505-5
file: data/en.wikipedia.org/wiki/Divisibility_rule-5.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
Another digit grouping method of divisibility by 7 Method This method determines whether an integer is divisible by 7 by grouping digits in sets of three: Starting from the right (units place), divide the number into groups of three digits. If necessary, prepend leading zeros to complete the leftmost group. Starting from the rightmost group, alternately add and subtract the groups moving left. That is, subtract the second group from the first, add the third, subtract the fourth, and so on. Evaluate the resulting sum. If the result is divisible by 7, then the original number is divisible by 7; otherwise, it is not. Example To test the number 10,517,815: Group the digits: 010 | 517 | 815 Apply alternating operations: 815 − 517 + 010 = 308 Since 308 is divisible by 7, the original number is also divisible by 7. Proof of correctness This method is based on modular arithmetic. Observe that: 1000 ≡ −1 (mod 7) since 1000 leaves a remainder of 6 when divided by 7, and 6 ≡ −1 (mod 7). Any integer can be expressed as a sum of multiples of powers of 1000: N = a₀ + a₁·1000 + a₂·1000² + a₃·1000³ +


id: divisibility-rule-20260505-6
file: data/en.wikipedia.org/wiki/Divisibility_rule-6.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Composite divisors === A number is divisible by a given divisor if it is divisible by the highest power of each of its prime factors. For example, to determine divisibility by 36, check divisibility by 4 and by 9. Note that checking 3 and 12, or 2 and 18, would not be sufficient. A table of prime factors may be useful. A composite divisor may also have a rule formed using the same procedure as for a prime divisor, given below, with the caveat that the manipulations involved may not introduce any factor which is present in the divisor. For instance, one cannot make a rule for 14 that involves multiplying the equation by 7. This is not an issue for prime divisors because they have no smaller factors. To test the divisibility of a number by a power of 2 or a power of 5 (2n or 5n, in which n is a positive integer), one only need to look at the last n digits of that number. To test divisibility by any number expressed as the product of prime factors p 1 n p 2 m p 3 q {\displaystyle p_{1}^{n}p_{2}^{m}p_{3}^{q}} , we can separately test for


id: divisibility-rule-20260505-7
file: data/en.wikipedia.org/wiki/Divisibility_rule-7.md
category: reference
title: Divisibility rule
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Divisibility_rule
date_saved: 2026-05-05T08:13:56.367542+00:00
instance: kb-cron
parent: divisibility-rule-20260505-0
summary:
=== Proof using basic algebra === Many of the simpler rules can be produced using only algebraic manipulation, creating binomials and rearranging them. By writing a number as the sum of each digit times a power of 10 each digit's power can be manipulated individually. ==== Case where all digits are summed ==== This method works for divisors that are factors of 10 − 1 = 9. Using 3 as an example, 3 divides 9 = 10 − 1. That means 10 ≡ 1 ( mod 3 ) {\displaystyle 10\equiv 1{\pmod {3}}} (see modular arithmetic). The same for all the higher powers of 10: 10 n ≡ 1 n ≡ 1 ( mod 3 ) . {\displaystyle 10^{n}\equiv 1^{n}\equiv 1{\pmod {3}}.} They are all congruent to 1 modulo 3. Since two things that are congruent modulo 3 are either both divisible by 3 or both not, we can interchange values that are congruent modulo 3. So, in a number such as the following, we can replace all the powers of 10 by 1: 100 ⋅ a + 10 ⋅ b + 1 ⋅ c ≡ ( 1 ) a + ( 1 ) b + ( 1 ) c (


id: list-of-dynamical-systems-and-differential-equations-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics-0.md
category: reference
title: List of dynamical systems and differential equations topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_dynamical_systems_and_differential_equations_topics
date_saved: 2026-05-05T08:14:00.321935+00:00
instance: kb-cron
parent: list-of-dynamical-systems-and-differential-equations-topics-20260505-0
summary:
This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. == Dynamical systems, in general == Deterministic system (mathematics) Linear system Partial differential equation Dynamical systems and chaos theory Chaos theory Chaos argument Butterfly effect 0-1 test for chaos Bifurcation diagram Feigenbaum constant Sharkovskii's theorem Attractor Strange nonchaotic attractor Stability theory Mechanical equilibrium Astable Monostable Bistability Metastability Feedback Negative feedback Positive feedback Homeostasis Damping ratio Dissipative system Spontaneous symmetry breaking Turbulence Perturbation theory Control theory Non-linear control Adaptive control Hierarchical control Intelligent control Optimal control Dynamic programming Robust control Stochastic control System dynamics, system analysis Takens' theorem Exponential dichotomy Liénard's theorem Krylov–Bogolyubov theorem Krylov-Bogoliubov averaging method == Abstract dynamical systems == Measure-preserving dynamical system Ergodic theory Mixing (mathematics) Almost periodic function Symbolic dynamics Time scale calculus Arithmetic dynamics Sequential dynamical system Graph dynamical system Topological dynamical system == Dynamical systems, examples == List of chaotic maps Logistic map Lorenz attractor Lorenz-96 Iterated function system Tetration Ackermann function Horseshoe map Hénon map Arnold's cat map Population dynamics == Complex dynamics == Fatou set Julia set Mandelbrot set == Difference equations == Recurrence relation Matrix difference equation


id: list-of-eponyms-of-special-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_eponyms_of_special_functions-0.md
category: reference
title: List of eponyms of special functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_eponyms_of_special_functions
date_saved: 2026-05-05T08:14:02.125990+00:00
instance: kb-cron
parent: list-of-eponyms-of-special-functions-20260505-0
summary:
This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym). Named symmetric functions, and other special polynomials, are included. == A == Niels Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial Sir George Biddell Airy: Airy function Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chihara polynomial C. T. Anger: Anger–Weber function Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized Appell polynomials Richard Askey: Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson) == B == Ernest William Barnes: Barnes G-function E. T. Bell: Bell polynomials Bender–Dunne polynomial Jacob Bernoulli: Bernoulli polynomial Friedrich Bessel: Bessel function, Bessel–Clifford function H. Blasius: Blasius functions R. P. Boas, R. C. Buck: Boas–Buck polynomial Böhmer integral Erland Samuel Bring: Bring radical de Bruijn function Buchstab function Burchnall, Chaundy: Burchnall–Chaundy polynomial == C == Leonard Carlitz: Carlitz polynomial Arthur Cayley, Capelli: Cayley–Capelli operator Celine's polynomial Charlier polynomial Pafnuty Chebyshev: Chebyshev polynomials Elwin Bruno Christoffel, Darboux: Christoffel–Darboux relation Cyclotomic polynomials == D == H. G. Dawson:


id: list-of-mathematical-examples-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_examples-0.md
category: reference
title: List of mathematical examples
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_examples
date_saved: 2026-05-05T08:14:04.638441+00:00
instance: kb-cron
parent: list-of-mathematical-examples-20260505-0
summary:
This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be an "example" as the term should be understood here (an elegant proof of an isolated but particularly striking fact, as opposed to a proof of a general theorem, could perhaps be considered an "example"). The discussion page for list of mathematical topics has some comments on this. Eventually this page may have its own discussion page. This page links to itself in order that edits to this page will be included among related changes when the user clicks on that button. The concrete example within the article titled Rao–Blackwell theorem is perhaps one of the best ways for a probabilist ignorant of statistical inference to get a quick impression of the flavor of that subject. == Uncategorized examples, alphabetized == Alexander horned sphere All horses are the same color Cantor function Cantor set Checking if a coin is biased Concrete illustration of the central limit theorem Differential equations of mathematical physics Dirichlet function Discontinuous linear map Efron's non-transitive


id: list-of-exceptional-set-concepts-20260505-0
file: data/en.wikipedia.org/wiki/List_of_exceptional_set_concepts-0.md
category: reference
title: List of exceptional set concepts
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_exceptional_set_concepts
date_saved: 2026-05-05T08:14:06.643357+00:00
instance: kb-cron
parent: list-of-exceptional-set-concepts-20260505-0
summary:
This is a list of exceptional set concepts. In mathematics, and in particular in mathematical analysis, it is very useful to be able to characterise subsets of a given set X as 'small', in some definite sense, or 'large' if their complement in X is small. There are numerous concepts that have been introduced to study 'small' or 'exceptional' subsets. In the case of sets of natural numbers, it is possible to define more than one concept of 'density', for example. See also list of properties of sets of reals. Almost all Almost always Almost everywhere Almost never Almost surely Analytic capacity Closed unbounded set Cofinal (mathematics) Cofinite Dense set IP set 2-large Large set (Ramsey theory) Meagre set Measure zero Natural density Negligible set Nowhere dense set Null set, conull set Partition regular Piecewise syndetic set Schnirelmann density Small set (combinatorics) Stationary set Syndetic set Thick set Thin set (Serre)


id: list-of-exponential-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_exponential_topics-0.md
category: reference
title: List of exponential topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_exponential_topics
date_saved: 2026-05-05T08:14:08.848554+00:00
instance: kb-cron
parent: list-of-exponential-topics-20260505-0
summary:
This is a list of exponential topics, by Wikipedia page. See also list of logarithm topics. Accelerating change Approximating natural exponents (log base e) Artin–Hasse exponential Bacterial growth Baker–Campbell–Hausdorff formula Carlitz exponential Cell growth Barometric formula Beer–Lambert law Characterizations of the exponential function Catenary Compound interest C0-semigroup De Moivre's formula Derivative of the exponential map Doléans-Dade exponential Double exponential function Doubling time e-folding Elimination half-life Error exponent Euler's formula Euler's identity e (mathematical constant) Exponent Exponent bias Exponential (disambiguation) Exponential backoff Exponential decay Exponential dichotomy Exponential discounting Exponential diophantine equation Exponential dispersion model Exponential distribution Exponential error Exponential factorial Exponential family Exponential field Ordered exponential field Exponential formula Exponential function Exponential generating function Exponential-Golomb coding Exponential growth Exponential hierarchy Exponential integral Exponential integrator Exponential map (Lie theory) Exponential map (Riemannian geometry) Exponential map (discrete dynamical systems) Exponential notation Exponential object (category theory) Exponential polynomials—see also Touchard polynomials (combinatorics) Exponential response formula Exponential sheaf sequence Exponential smoothing Exponential stability Exponential sum Exponential time Sub-exponential time Exponential tree Exponential type Exponentially equivalent measures Exponentiating by squaring Exponentiation Forgetting curve Four exponentials conjecture Fourier analysis List of Fourier analysis topics Gaussian function Gudermannian function Half-exponential function Half-life Hyperbolic function Inflation, inflation rate Interest


id: exterior-calculus-identities-20260505-0
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-0.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
This article summarizes several identities in exterior calculus, a mathematical calculus used in differential geometry. == Notation == The following notation is used in this article. === Manifold === M {\displaystyle M} , N {\displaystyle N} are n {\displaystyle n} -dimensional smooth manifolds, where n ∈ N {\displaystyle n\in \mathbb {N} } . That is, differentiable manifolds that can be differentiated enough times for the purposes on this page. p ∈ M {\displaystyle p\in M} , q ∈ N {\displaystyle q\in N} denote one point on each of the manifolds. The boundary of a manifold M {\displaystyle M} is a manifold ∂ M {\displaystyle \partial M} , which has dimension n − 1 {\displaystyle n-1} . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} . === Tangent and cotangent bundles === T M {\displaystyle TM} , T ∗ M {\displaystyle T^{*}M} denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold M {\displaystyle M} . T p M {\displaystyle T_{p}M} , T q N {\displaystyle T_{q}N} denote the tangent spaces of M {\displaystyle M} , N {\displaystyle


id: exterior-calculus-identities-20260505-1
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-1.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Exterior derivative === The exterior derivative d k : Ω k ( M ) → Ω k + 1 ( M ) {\displaystyle d_{k}:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)} is defined for all 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . We generally omit the subscript when it is clear from the context. For a 0 {\displaystyle 0} -form f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} we have d 0 f ∈ Ω 1 ( M ) {\displaystyle d_{0}f\in \Omega ^{1}(M)} as the 1 {\displaystyle 1} -form that gives the directional derivative, i.e., for the section X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} we have ( d 0 f ) ( X ) = ∂ X f {\displaystyle (d_{0}f)(X)=\partial _{X}f} , the directional derivative of f {\displaystyle f} along X {\displaystyle X} . For 0 < k ≤ n {\displaystyle 0<k\leq n} , ( d k ω ) ( X 0 , … , X k ) = ∑ 0 ≤ j ≤ k ( − 1 ) j d 0 ( ω ( X 0 , … , X ^ j , … , X k ) ) ( X


id: exterior-calculus-identities-20260505-2
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-2.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
A one-form α ∈ Ω 1 ( M ) {\displaystyle \alpha \in \Omega ^{1}(M)} corresponds to the unique vector field α ♯ ∈ Γ ( T M ) {\displaystyle \alpha ^{\sharp }\in \Gamma (TM)} such that for all X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} , we have: α ( X ) = g ( α ♯ , X ) . {\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).} These mappings extend via multilinearity to mappings from k {\displaystyle k} -vector fields to k {\displaystyle k} -forms and k {\displaystyle k} -forms to k {\displaystyle k} -vector fields through ( A 1 ∧ A 2 ∧ ⋯ ∧ A k ) ♭ = A 1 ♭ ∧ A 2 ♭ ∧ ⋯ ∧ A k ♭ {\displaystyle (A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }} ( α 1 ∧ α 2 ∧ ⋯ ∧ α k ) ♯ = α 1 ♯ ∧ α 2 ♯ ∧ ⋯ ∧ α k ♯ . {\displaystyle (\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.} === Hodge star === For an n-manifold M, the Hodge star


id: exterior-calculus-identities-20260505-3
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-3.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Cohomology === The k {\displaystyle k} -th cohomology of a manifold M {\displaystyle M} and its exterior derivative operators d 0 , … , d n − 1 {\displaystyle d_{0},\ldots ,d_{n-1}} is given by H k ( M ) := ker ( d k ) im ( d k − 1 ) {\displaystyle H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}} Two closed k {\displaystyle k} -forms α , β ∈ Ω k ( M ) {\displaystyle \alpha ,\beta \in \Omega ^{k}(M)} are in the same cohomology class if their difference is an exact form i.e. [ α ] = [ β ] ⟺ α − β = d η for some η ∈ Ω k − 1 ( M ) {\displaystyle [\alpha ]=[\beta ]\ \ \Longleftrightarrow \ \ \alpha {-}\beta =d\eta \ {\text{ for some }}\eta \in \Omega ^{k-1}(M)} A closed surface of genus g {\displaystyle g} will have 2 g {\displaystyle 2g} generators which are harmonic. === Dirichlet energy === Given α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , its Dirichlet energy is E D ( α ) := 1 2 ⟨ ⟨ d α , d α ⟩ ⟩ + 1 2 ⟨ ⟨ δ α ,


id: exterior-calculus-identities-20260505-4
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-4.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
⋆ ( λ 1 α + λ 2 β ) = λ 1 ( ⋆ α ) + λ 2 ( ⋆ β ) {\displaystyle {\star }(\lambda _{1}\alpha +\lambda _{2}\beta )=\lambda _{1}({\star }\alpha )+\lambda _{2}({\star }\beta )} for λ 1 , λ 2 ∈ R {\displaystyle \lambda _{1},\lambda _{2}\in \mathbb {R} } ( linearity ) ⋆ ⋆ α = s ( − 1 ) k ( n − k ) α {\displaystyle {\star }{\star }\alpha =s(-1)^{k(n-k)}\alpha } for α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , n = dim ⁡ ( M ) {\displaystyle n=\dim(M)} , and s = sign ⁡ ( g ) {\displaystyle s=\operatorname {sign} (g)} the sign of the metric ⋆ ( − 1 ) = s ( − 1 ) k ( n − k ) ⋆ {\displaystyle {\star }^{(-1)}=s(-1)^{k(n-k)}{\star }} ( inversion ) ⋆ ( f α ) = f ( ⋆ α ) {\displaystyle {\star }(f\alpha )=f({\star }\alpha )} for f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} ( commutative with 0 {\displaystyle 0} -forms ) ⟨ ⟨ α , α ⟩ ⟩ = ⟨ ⟨ ⋆ α , ⋆ α ⟩ ⟩ {\displaystyle \langle \!\langle


id: exterior-calculus-identities-20260505-5
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-5.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
( α ∧ β ∧ γ ) ( X , Y , Z ) = det [ α ( X ) α ( Y ) α ( Z ) β ( X ) β ( Y ) β ( Z ) γ ( X ) γ ( Y ) γ ( Z ) ] {\displaystyle (\alpha \wedge \beta \wedge \gamma )(X,Y,Z)=\det {\begin{bmatrix}\alpha (X)&\alpha (Y)&\alpha (Z)\\\beta (X)&\beta (Y)&\beta (Z)\\\gamma (X)&\gamma (Y)&\gamma (Z)\end{bmatrix}}} ( α 1 ∧ … ∧ α l ) ( X 1 , … , X l ) = det [ α 1 ( X 1 ) α 1 ( X 2 ) … α 1 ( X l ) α 2 ( X 1 ) α 2 ( X 2 ) … α 2 ( X l ) ⋮ ⋮ ⋱ ⋮ α l ( X 1 ) α l ( X 2 ) … α l ( X l ) ] {\displaystyle (\alpha _{1}\wedge \ldots \wedge \alpha _{l})(X_{1},\ldots ,X_{l})=\det {\begin{bmatrix}\alpha _{1}(X_{1})&\alpha _{1}(X_{2})&\dots &\alpha _{1}(X_{l})\\\alpha _{2}(X_{1})&\alpha _{2}(X_{2})&\dots &\alpha _{2}(X_{l})\\\vdots &\vdots &\ddots &\vdots \\\alpha _{l}(X_{1})&\alpha _{l}(X_{2})&\dots &\alpha _{l}(X_{l})\end{bmatrix}}} === Projection and rejection === ( − 1 ) k ι X ⋆ α = ⋆ ( X ♭ ∧ α


id: exterior-calculus-identities-20260505-6
file: data/en.wikipedia.org/wiki/Exterior_calculus_identities-6.md
category: reference
title: Exterior calculus identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Exterior_calculus_identities
date_saved: 2026-05-05T08:14:13.652508+00:00
instance: kb-cron
parent: exterior-calculus-identities-20260505-0
summary:
=== Identities in Euclidean 3-space === Let Euclidean metric g ( X , Y ) := ⟨ X , Y ⟩ = X ⋅ Y {\displaystyle g(X,Y):=\langle X,Y\rangle =X\cdot Y} . We use ∇ = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) {\displaystyle \nabla =\left({\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right)} differential operator R 3 {\displaystyle \mathbb {R} ^{3}} ι X α = g ( X , α ♯ ) = X ⋅ α ♯ {\displaystyle \iota _{X}\alpha =g(X,\alpha ^{\sharp })=X\cdot \alpha ^{\sharp }} for α ∈ Ω 1 ( M ) {\displaystyle \alpha \in \Omega ^{1}(M)} . d e t ( X , Y , Z ) = ⟨ X , Y × Z ⟩ = ⟨ X × Y , Z ⟩ {\displaystyle \mathbf {det} (X,Y,Z)=\langle X,Y\times Z\rangle =\langle X\times Y,Z\rangle } ( scalar triple product ) X × Y = ( ⋆ ( X ♭ ∧ Y ♭ ) ) ♯ {\displaystyle X\times Y=({\star }(X^{\flat }\wedge Y^{\flat }))^{\sharp }} ( cross product ) ι X α = − ( X × A ) ♭ {\displaystyle \iota _{X}\alpha =-(X\times A)^{\flat }} if α ∈ Ω 2 ( M )


id: list-of-factorial-and-binomial-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_factorial_and_binomial_topics-0.md
category: reference
title: List of factorial and binomial topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_factorial_and_binomial_topics
date_saved: 2026-05-05T08:14:15.007034+00:00
instance: kb-cron
parent: list-of-factorial-and-binomial-topics-20260505-0
summary:
This is a list of factorial and binomial topics in mathematics. See also binomial (disambiguation). Abel's binomial theorem Alternating factorial Antichain Beta function Bhargava factorial Binomial coefficient Pascal's triangle Binomial distribution Binomial proportion confidence interval Binomial-QMF (Daubechies wavelet filters) Binomial series Binomial theorem Binomial transform Binomial type Carlson's theorem Catalan number Fuss–Catalan number Central binomial coefficient Combination Combinatorial number system De Polignac's formula Difference operator Difference polynomials Digamma function Egorychev method Erdős–Ko–Rado theorem Euler–Mascheroni constant Faà di Bruno's formula Factorial Factorial moment Factorial number system Factorial prime Factoriangular number Gamma distribution Gamma function Gaussian binomial coefficient Gould's sequence Hyperfactorial Hypergeometric distribution Hypergeometric function identities Hypergeometric series Incomplete beta function Incomplete gamma function Jordan–Pólya number Kempner function Lah number Lanczos approximation Lozanić's triangle Macaulay representation of an integer Mahler's theorem Multinomial distribution Multinomial coefficient, Multinomial formula, Multinomial theorem Multiplicities of entries in Pascal's triangle Multiset Multivariate gamma function Narayana numbers Negative binomial distribution Nörlund–Rice integral Pascal matrix Pascal's pyramid Pascal's simplex Pascal's triangle Permutation List of permutation topics Pochhammer symbol (also falling, lower, rising, upper factorials) Poisson distribution Polygamma function Primorial Proof of Bertrand's postulate Sierpinski triangle Star of David theorem Stirling number Stirling transform Stirling's approximation Subfactorial Table of Newtonian


id: list-of-finite-simple-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_finite_simple_groups-0.md
category: reference
title: List of finite simple groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite_simple_groups
date_saved: 2026-05-05T08:14:16.205745+00:00
instance: kb-cron
parent: list-of-finite-simple-groups-20260505-0
summary:
In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups. The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates. == Summary == The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = A3(2) and A2(4) both have order 20160, and that the group Bn(q) has the same order as Cn(q) for q odd, n > 2. The smallest of the latter pairs of groups are B3(3) and C3(3) which both have order 4585351680.) There is an unfortunate conflict between the notations for the alternating groups An and the groups of Lie


id: list-of-finite-simple-groups-20260505-1
file: data/en.wikipedia.org/wiki/List_of_finite_simple_groups-1.md
category: reference
title: List of finite simple groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite_simple_groups
date_saved: 2026-05-05T08:14:16.205745+00:00
instance: kb-cron
parent: list-of-finite-simple-groups-20260505-0
summary:
=== Ree groups, 2G2(32n+1) === Simplicity: Simple for n ≥ 1. The group 2G2(3) is not simple, but its derived group 2G2(3)′ is a simple subgroup of index 3. Order: q3 (q3 + 1) (q − 1), where q = 32n+1 Schur multiplier: Trivial for n ≥ 1 and for 2G2(3)′. Outer automorphism group: 1⋅f⋅1, where f = 2n + 1. Other names: Ree(32n+1), R(32n+1), E2∗(32n+1) . Isomorphisms: The derived group 2G2(3)′ is isomorphic to A1(8). Remarks: 2G2(32n+1) has a doubly transitive permutation representation on 33(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 32n+1 elements. == Sporadic groups == === Mathieu groups, M11, M12, M22, M23, M24 === === Janko groups, J1, J2, J3, J4 === === Conway groups, Co1, Co2, Co3 === === Fischer groups, Fi22, Fi23, Fi24′ === === Higman–Sims group, HS === Order: 29 ⋅ 32 ⋅ 53 ⋅ 7 ⋅ 11 = 44352000 Schur multiplier: Order 2. Outer automorphism group: Order 2. Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co2 and in Co3. === McLaughlin group, McL === Order: 27 ⋅ 36 ⋅ 53 ⋅


id: list-of-finite-dimensional-nichols-algebras-20260505-0
file: data/en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras-0.md
category: reference
title: List of finite-dimensional Nichols algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
date_saved: 2026-05-05T08:14:17.618854+00:00
instance: kb-cron
parent: list-of-finite-dimensional-nichols-algebras-20260505-0
summary:
In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts U q ( g ) + {\displaystyle U_{q}({\mathfrak {g}})^{+}} of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts u q ( g ) + {\displaystyle u_{q}({\mathfrak {g}})^{+}} of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity. The following article lists all known finite-dimensional Nichols algebras B ( V ) {\displaystyle {\mathfrak {B}}(V)} where V {\displaystyle V} is a Yetter–Drinfel'd module over a finite group G {\displaystyle G} , where the group is generated by the support of V {\displaystyle V} . For more details on Nichols algebras see Nichols algebra. There are two major cases: G {\displaystyle G} abelian, which implies V {\displaystyle V} is


id: list-of-finite-dimensional-nichols-algebras-20260505-1
file: data/en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras-1.md
category: reference
title: List of finite-dimensional Nichols algebras
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
date_saved: 2026-05-05T08:14:17.618854+00:00
instance: kb-cron
parent: list-of-finite-dimensional-nichols-algebras-20260505-0
summary:
=== Nichols algebras from Coxeter groups === For every finite coxeter system ( W , S ) {\displaystyle (W,S)} the Nichols algebra over the conjugacy class(es) of reflections was studied in (reflections on roots of different length are not conjugate, see fourth example fellow). They discovered in this way the following first Nichols algebras over nonabelian groups : The case S 2 ≅ Z 2 {\displaystyle \mathbb {S} _{2}\cong \mathbb {Z} _{2}} is the rank 1 diagonal Nichols algebra u i ( A 1 ) + {\displaystyle u_{i}(A_{1})^{+}} of dimension 2. === Other Nichols algebras of rank 1 === === Nichols algebras of rank 2, type Gamma-3 === These Nichols algebras were discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 2 type Gamma-4 === This Nichols algebra was discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 2, type T === This Nichols algebra was discovered during the classification of Heckenberger and Vendramin. === The Nichols algebra of rank 3 involving Gamma-3 === This Nichols algebra was the last Nichols algebra discovered during the classification of Heckenberger and Vendramin. === Nichols algebras from diagram folding === The following families


id: list-of-first-order-theories-20260505-0
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-0.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
In first-order logic, a first-order theory is given by a set of axioms in some language. This entry lists some of the more common examples used in model theory and some of their properties. == Preliminaries == For every natural mathematical structure there is a signature σ listing the constants, functions, and relations of the theory together with their arities, so that the object is naturally a σ-structure. Given a signature σ there is a unique first-order language Lσ that can be used to capture the first-order expressible facts about the σ-structure. There are two common ways to specify theories: List or describe a set of sentences in the language Lσ, called the axioms of the theory. Give a set of σ-structures, and define a theory to be the set of sentences in Lσ holding in all these models. For example, the "theory of finite fields" consists of all sentences in the language of fields that are true in all finite fields. An Lσ theory may: be consistent: no proof of contradiction exists; be satisfiable: there exists a σ-structure for which the sentences of the theory are all true (by the completeness theorem, satisfiability is equivalent to consistency); be complete:


id: list-of-first-order-theories-20260505-1
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-1.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
~ has an infinite number of equivalence classes; ~ has exactly n equivalence classes (for any fixed positive integer n); All equivalence classes are infinite; All equivalence classes have size exactly n (for any fixed positive integer n). The theory of an equivalence relation with exactly 2 infinite equivalence classes is an easy example of a theory which is ω-categorical but not categorical for any larger cardinal. The equivalence relation ~ should not be confused with the identity symbol '=': if x=y then x~y, but the converse is not necessarily true. Theories of equivalence relations are not all that difficult or interesting, but often give easy examples or counterexamples for various statements. The following constructions are sometimes used to produce examples of theories with certain spectra; in fact by applying them to a small number of explicit theories T one gets examples of complete countable theories with all possible uncountable spectra. If T is a theory in some language, we define a new theory 2T by adding a new binary relation to the language, and adding axioms stating that it is an equivalence relation, such that there are an infinite number of equivalence classes all of which are models of


id: list-of-first-order-theories-20260505-2
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-2.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
== Boolean algebras == There are several different signatures and conventions used for Boolean algebras: The signature has two constants, 0 and 1, and two binary functions ∧ and ∨ ("and" and "or"), and one unary function ¬ ("not"). This can be confusing as the functions use the same symbols as the propositional functions of first-order logic. In set theory, a common convention is that the language has two constants, 0 and 1, and two binary functions · and +, and one unary function −. The three functions have the same interpretation as the functions in the first convention. Unfortunately, this convention clashes badly with the next convention: In algebra, the usual convention is that the language has two constants, 0 and 1, and two binary functions · and +. The function · has the same meaning as ∧, but a+b means a∨b∧¬(a∧b). The reason for this is that the axioms for a Boolean algebra are then just the axioms for a ring with 1 plus ∀x x2 = x. Unfortunately this clashes with the standard convention in set theory given above. The axioms are: The axioms for a distributive lattice (see above) ∀a a∧¬a = 0, ∀a a∨¬a =


id: list-of-first-order-theories-20260505-3
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-3.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
== Rings and fields == The signature of (unital) rings has two constants 0 and 1, two binary functions + and ×, and, optionally, one unary negation function −. Rings Axioms: Addition makes the ring into an abelian group, multiplication is associative and has an identity 1, and multiplication is left and right distributive. Commutative rings The axioms for rings plus ∀x ∀y xy = yx. Fields The axioms for commutative rings plus ∀x (¬ x = 0 → ∃y xy = 1) and ¬ 1 = 0. Many of the examples given here have only universal, or algebraic axioms. The class of structures satisfying such a theory has the property of being closed under substructure. For example, a subset of a group closed under the group actions of multiplication and inverse is again a group. Since the signature of fields does not usually include multiplicative and additive inverse, the axioms for inverses are not universal, and therefore a substructure of a field closed under addition and multiplication is not always a field. This can be remedied by adding unary inverse functions to the language. For any positive integer n the property that all equations of degree n have a


id: list-of-first-order-theories-20260505-4
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-4.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
∀ A ∃ b ∃ c ∃ d b A ∧ c A ∧ d A ∧ ¬ b = c ∧ ¬ b = d ∧ ¬ c = d {\displaystyle \forall A\exists b\exists c\exists d\;bA\land cA\land dA\land \lnot b=c\land \lnot b=d\land \lnot c=d} (Every line has at least 3 points) Euclid did not state all the axioms for Euclidean geometry explicitly, and the first complete list was given by Hilbert in Hilbert's axioms. This is not a first-order axiomatization as one of Hilbert's axioms is a second order completeness axiom. Tarski's axioms are a first-order axiomatization of Euclidean geometry. Tarski showed this axiom system is complete and decidable by relating it to the complete and decidable theory of real closed fields. == Differential algebra == The theory DF of differential fields. The signature is that of fields (0, 1, +, −, ×) together with a unary function ∂, the derivation. The axioms are those for fields together with ∀ u ∀ v ∂ ( u v ) = u ∂ v + v ∂ u {\displaystyle \forall u\forall v\,\partial (uv)=u\,\partial v+v\,\partial u} ∀ u ∀ v ∂ ( u + v ) = ∂ u + ∂ v


id: list-of-first-order-theories-20260505-5
file: data/en.wikipedia.org/wiki/List_of_first-order_theories-5.md
category: reference
title: List of first-order theories
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_first-order_theories
date_saved: 2026-05-05T08:14:20.032260+00:00
instance: kb-cron
parent: list-of-first-order-theories-20260505-0
summary:
∀x ¬ Sx = 0 ∀x ¬ x = 0 → ∃y Sy = x ∀x∀y Sx = Sy → x = y ∀x x + 0 = x ∀x∀y x + Sy = S(x + y) ∀x x × 0 = 0 ∀x∀y x × Sy = (x × y) + x. IΣn is first-order Peano arithmetic with induction restricted to Σn formulas (for n = 0, 1, 2, ...). The theory IΣ0 is often denoted by IΔ0. This is a series of more and more powerful fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0 with an axiom stating that xy exists for all x and y (with the usual properties). First-order Peano arithmetic, PA. The "standard" theory of arithmetic. The axioms are the axioms of Robinson arithmetic above, together with the axiom scheme of induction: ϕ ( 0 ) ∧ ( ∀ x ϕ ( x ) → ϕ ( S x ) ) → ( ∀ x ϕ ( x ) ) {\displaystyle \phi (0)\wedge (\forall x\phi (x)\rightarrow \phi (Sx))\rightarrow (\forall x\phi (x))} for any formula φ in the language


id: list-of-formal-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formal_systems-0.md
category: reference
title: List of formal systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formal_systems
date_saved: 2026-05-05T08:14:21.897478+00:00
instance: kb-cron
parent: list-of-formal-systems-20260505-0
summary:
This is a list of formal systems, also known as logical calculi. == Mathematical == Functional calculus, a way to apply various types of functions to operators Matrix calculus, a specialized notation for multivariable calculus over spaces of matrices Umbral calculus, the combinatorics of certain operations on polynomials Vector calculus (also called vector analysis), comprising specialized notations for multivariable analysis of vectors in an inner-product space === Logical === Predicate calculus, specifies the rules of inference governing the logic of predicates Propositional calculus, specifies the rules of inference governing the logic of propositions === In theoretical computer science (Formal language) === Modal μ-calculus, a common temporal logic used by formal verification methods such as model checking Lambda calculus, a formulation of the theory of reflexive functions that has deep connections to computational theory Kappa calculus, a reformulation of the first-order fragment of typed lambda calculus Rho calculus, introduced as a general means to uniformly integrate rewriting into lambda calculus Process calculus, a set of approaches to formulating formal models of concurrent systems Ambient calculus, a family of models for concurrent systems based on the concept of agent mobility Join calculus, a theoretical model for the design of distributed programming languages


id: list-of-formulas-in-elementary-geometry-20260505-0
file: data/en.wikipedia.org/wiki/List_of_formulas_in_elementary_geometry-0.md
category: reference
title: List of formulas in elementary geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_formulas_in_elementary_geometry
date_saved: 2026-05-05T08:14:23.334016+00:00
instance: kb-cron
parent: list-of-formulas-in-elementary-geometry-20260505-0
summary:
This is a short list of some common mathematical shapes and figures and the formulas that describe them. == Two-dimensional shapes == Sources: == Three-dimensional shapes == This is a list of volume formulas of basic shapes: Cone – 1 3 π r 2 h {\textstyle {\frac {1}{3}}\pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cone's height; Cube – a 3 {\textstyle a^{3}} , where a {\textstyle a} is the side's length; Cuboid – a b c {\textstyle abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the sides' length; Cylinder – π r 2 h {\textstyle \pi r^{2}h} , where r {\textstyle r} is the base's radius and h {\textstyle h} is the cylinder's height; Ellipsoid – 4 3 π a b c {\textstyle {\frac {4}{3}}\pi abc} , where a {\textstyle a} , b {\textstyle b} , and c {\textstyle c} are the semi-major and semi-minor axes' length; Sphere – 4 3 π r 3 {\textstyle {\frac {4}{3}}\pi r^{3}} , where r {\textstyle r} is the radius; Parallelepiped – a b c K {\textstyle abc{\sqrt {K}}} , where a {\textstyle a} , b


id: list-of-fourier-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Fourier_analysis_topics-0.md
category: reference
title: List of Fourier analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Fourier_analysis_topics
date_saved: 2026-05-05T08:14:24.703918+00:00
instance: kb-cron
parent: list-of-fourier-analysis-topics-20260505-0
summary:
This is a list of Fourier analysis topics. == Fourier analysis == Multiplier (Fourier analysis) Fourier shell correlation Pinsky phenomenon === Fourier series === Generalized Fourier series Regressive discrete Fourier series Gibbs phenomenon Sigma approximation Dini test Poisson summation formula Spectrum continuation analysis Convergence of Fourier series Half-range Fourier series === Fourier transforms === List of Fourier-related transforms Fourier transform on finite groups Fractional Fourier transform Continuous Fourier transform Fourier operator Fourier inversion theorem Sine and cosine transforms Parseval's theorem Paley–Wiener theorem Projection-slice theorem Frequency spectrum === Discrete Fourier transforms === Discrete Fourier series Non-uniform discrete Fourier transform DFT matrix Entropy influence conjecture Fast cosine transform Fast Fourier transform Cyclotomic fast Fourier transform Sparse Fourier transform === Applications of Fourier analysis === Fourier amplitude sensitivity testing Fourier optics Quantum mechanics, for Wavefunctions Uncertainty principle Quantum Fourier transform == Algebra and analysis == Periodic function Almost periodic function ATS theorem Modulus of continuity Banach algebra Compact group Haar measure Hardy space Sobolev space Topological group === Harmonic analysis === Set of uniqueness Pontryagin duality Plancherel theorem Peter–Weyl theorem Fourier integral operator Oscillatory integral operator === Harmonic functions === Laplace operator Laplace equation Dirichlet problem Unit circle Unit disc Spherical harmonic ===


id: list-of-fourier-related-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Fourier-related_transforms-0.md
category: reference
title: List of Fourier-related transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Fourier-related_transforms
date_saved: 2026-05-05T08:14:26.129378+00:00
instance: kb-cron
parent: list-of-fourier-related-transforms-20260505-0
summary:
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. == Continuous transforms == Applied to functions of continuous arguments, Fourier-related transforms include: Two-sided Laplace transform Mellin transform, another closely related integral transform Laplace transform: the Fourier transform may be considered a special case of the imaginary axis of the bilateral Laplace transform Fourier transform, with special cases: Fourier series When the input function/waveform is periodic, the Fourier transform output is a Dirac comb function, modulated by a discrete sequence of finite-valued coefficients that are complex-valued in general. These are called Fourier series coefficients. The term Fourier series actually refers to the inverse Fourier transform, which is a sum of sinusoids at discrete frequencies, weighted by the Fourier series coefficients. When the non-zero portion of the input function has finite duration, the Fourier transform is continuous and finite-valued. But a discrete subset of its values


id: index-of-fractal-related-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_fractal-related_articles-0.md
category: reference
title: Index of fractal-related articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_fractal-related_articles
date_saved: 2026-05-05T08:14:27.328439+00:00
instance: kb-cron
parent: index-of-fractal-related-articles-20260505-0
summary:
This is a list of fractal topics, by Wikipedia page, See also list of dynamical systems and differential equations topics. 1/f noise Apollonian gasket Attractor Box-counting dimension Cantor distribution Cantor dust Cantor function Cantor set Cantor space Chaos theory Coastline Constructal theory Dimension Dimension theory Dragon curve Fatou set Fractal Fractal antenna Fractal art Fractal compression Fractal flame Fractal landscape Fractal transform Fractint Graftal Iterated function system Horseshoe map How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Julia set Koch snowflake L-system Lebesgue covering dimension Lévy C curve Lévy flight List of fractals by Hausdorff dimension Lorenz attractor Lyapunov fractal Mandelbrot set Menger sponge Minkowski–Bouligand dimension Multifractal analysis Olbers' paradox Perlin noise Power law Rectifiable curve Scale-free network Self-similarity Sierpinski carpet Sierpiński curve Sierpinski triangle Space-filling curve T-square (fractal) Topological dimension


id: list-of-fractals-by-hausdorff-dimension-20260505-0
file: data/en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension-0.md
category: reference
title: List of fractals by Hausdorff dimension
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension
date_saved: 2026-05-05T08:14:28.578190+00:00
instance: kb-cron
parent: list-of-fractals-by-hausdorff-dimension-20260505-0
summary:
According to Benoit Mandelbrot, "A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." Presented here is a list of fractals, ordered by increasing Hausdorff dimension, to illustrate what it means for a fractal to have a low or a high dimension. == Deterministic fractals == == Random and natural fractals == == See also == Fractal dimension Hausdorff dimension Scale invariance == Notes and references == == Further reading == Mandelbrot, Benoît (1982). The Fractal Geometry of Nature. W.H. Freeman. ISBN 0-7167-1186-9. Peitgen, Heinz-Otto (1988). Saupe, Dietmar (ed.). The Science of Fractal Images. Springer Verlag. ISBN 0-387-96608-0. Barnsley, Michael F. (1 January 1993). Fractals Everywhere. Morgan Kaufmann. ISBN 0-12-079061-0. Sapoval, Bernard; Mandelbrot, Benoît B. (2001). Universalités et fractales: jeux d'enfant ou délits d'initié?. Flammarion-Champs. ISBN 2-08-081466-4. == External links == The fractals on Mathworld Other fractals on Paul Bourke's website Soler's Gallery Fractals on mathcurve.com 1000fractales.free.fr - Project gathering fractals created with various software Fractals unleashed IFStile - software that computes the dimension of the boundary of self-affine tiles


id: list-of-functional-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_functional_analysis_topics-0.md
category: reference
title: List of functional analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_functional_analysis_topics
date_saved: 2026-05-05T08:14:29.761801+00:00
instance: kb-cron
parent: list-of-functional-analysis-topics-20260505-0
summary:
This is a list of functional analysis topics. See also: Glossary of functional analysis. == Hilbert space == == Functional analysis, classic results == == Operator theory == == Banach space examples == Lp space Hardy space Sobolev space Tsirelson space ba space == Real and complex algebras == == Topological vector spaces == == Amenability == Amenable group Von Neumann conjecture == Wavelets == == Quantum theory == See also list of mathematical topics in quantum theory == Probability == Free probability Bernstein's theorem == Non-linear == Fixed-point theorems in infinite-dimensional spaces == History == Stefan Banach (1892–1945) Hugo Steinhaus (1887–1972) John von Neumann (1903-1957) Alain Connes (born 1947) Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis Earliest Known Uses of Some of the Words of Mathematics: Matrices and Linear Algebra


id: list-of-mathematical-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_functions-0.md
category: reference
title: List of mathematical functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_functions
date_saved: 2026-05-05T08:14:30.991878+00:00
instance: kb-cron
parent: list-of-mathematical-functions-20260505-0
summary:
In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are "anonymous", with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations. See also List of types of functions == Elementary functions == Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) === Algebraic functions === Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer (or equivalently rational) coefficients. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line Linear function: First degree polynomial, graph is a straight line. Quadratic function: Second degree polynomial, graph is a parabola. Cubic function: Third degree polynomial. Quartic function: Fourth degree polynomial. Quintic function: Fifth degree polynomial. Rational functions: A ratio


id: list-of-types-of-functions-20260505-0
file: data/en.wikipedia.org/wiki/List_of_types_of_functions-0.md
category: reference
title: List of types of functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_functions
date_saved: 2026-05-05T08:14:32.235773+00:00
instance: kb-cron
parent: list-of-types-of-functions-20260505-0
summary:
In mathematics, functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. == Relative to set theory == These properties concern the domain, the codomain and the image of functions. Injective function: has a distinct value for each distinct input. Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain. Surjective function: has a preimage for every element of the codomain, that is, the codomain equals the image. Also called a surjection or onto function. Bijective function: is both an injection and a surjection, and thus invertible. Identity function: maps any given element to itself. Constant function: has a fixed value regardless of its input. Empty function: whose domain equals the empty set. Set function: whose input is a set. Set-valued function: whose values are sets. Choice function called also selector or uniformizing function: assigns to each set one of its elements. == Relative to an operator (c.q. a group or other structure) == These properties concern how the function is affected by arithmetic operations on its argument. The following are special examples


id: list-of-types-of-functions-20260505-1
file: data/en.wikipedia.org/wiki/List_of_types_of_functions-1.md
category: reference
title: List of types of functions
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_types_of_functions
date_saved: 2026-05-05T08:14:32.235773+00:00
instance: kb-cron
parent: list-of-types-of-functions-20260505-0
summary:
== Ways of defining functions/relation to type theory == Polynomial function: defined by evaluating a polynomial. Linear function; also affine function. Quadratic function Cubic function Quartic function Quintic function Rational function: ratio of two polynomial functions. In particular, Möbius transformation called also linear fractional function. Algebraic function: defined as the root of a polynomial equation. Transcendental function: analytic but not algebraic. Also hypertranscendental function. Composite function: is formed by the composition of two functions f and g, by mapping x to f (g(x)). Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine). Implicit function: defined implicitly by a relation between the argument(s) and the value. Piecewise function: is defined by different expressions on different intervals. Computable function: an algorithm can do the job of the function. Also semicomputable function; primitive recursive function; partial recursive function. In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating what it should map to. For this purpose, the ↦ {\displaystyle \mapsto } symbol or Church's λ {\displaystyle \lambda } is often used. Also, sometimes mathematicians notate a function's domain and codomain by writing e.g. f


id: list-of-examples-in-general-topology-20260505-0
file: data/en.wikipedia.org/wiki/List_of_examples_in_general_topology-0.md
category: reference
title: List of examples in general topology
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_examples_in_general_topology
date_saved: 2026-05-05T08:14:33.479770+00:00
instance: kb-cron
parent: list-of-examples-in-general-topology-20260505-0
summary:
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology Cantor space Co-kappa topology Cocountable topology Cofinite topology Compact-open topology Compactification Discrete topology Double-pointed cofinite topology Extended real number line Finite topological space Hawaiian earring Hilbert cube Irrational cable on a torus Lakes of Wada Long line Order topology Lexicographical/dictionary order Ordinal number topology Real line Split interval Overlapping interval topology Moore plane Sierpiński space Sorgenfrey line Sorgenfrey plane Space-filling curve Topologist's sine curve Trivial topology Unit interval Zariski topology == See also == Counterexamples in Topology π-Base: An Interactive Encyclopedia of Topological Spaces


id: list-of-genetic-algorithm-applications-20260505-0
file: data/en.wikipedia.org/wiki/List_of_genetic_algorithm_applications-0.md
category: reference
title: List of genetic algorithm applications
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_genetic_algorithm_applications
date_saved: 2026-05-05T08:14:34.797128+00:00
instance: kb-cron
parent: list-of-genetic-algorithm-applications-20260505-0
summary:
This is a list of genetic algorithm (GA) applications. == Natural Sciences, Mathematics and Computer Science == Bayesian inference links to particle methods in Bayesian statistics and hidden Markov chain models Artificial creativity Chemical kinetics (gas and solid phases) Calculation of bound states and local-density approximations Code-breaking, using the GA to search large solution spaces of ciphers for the one correct decryption. Computer architecture: using GA to find out weak links in approximate computing such as lookahead. Configuration applications, particularly physics applications of optimal molecule configurations for particular systems like C60 (buckyballs) Construction of facial composites of suspects by eyewitnesses in forensic science. Data Center/Server Farm. Distributed computer network topologies Electronic circuit design, known as evolvable hardware Evolutionary image processing Feature selection for Machine Learning Feynman-Kac models File allocation for a distributed system Filtering and signal processing Finding hardware bugs. Game theory equilibrium resolution Genetic Algorithm for Rule Set Production Scheduling applications, including job-shop scheduling and scheduling in printed circuit board assembly. The objective being to schedule jobs in a sequence-dependent or non-sequence-dependent setup environment in order to maximize the volume of production while minimizing penalties such as tardiness. Satellite communication scheduling for the NASA Deep Space Network was shown


id: list-of-geodesic-polyhedra-and-goldberg-polyhedra-20260505-0
file: data/en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra-0.md
category: reference
title: List of geodesic polyhedra and Goldberg polyhedra
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra
date_saved: 2026-05-05T08:14:36.120154+00:00
instance: kb-cron
parent: list-of-geodesic-polyhedra-and-goldberg-polyhedra-20260505-0
summary:
This is a list of selected geodesic polyhedra and Goldberg polyhedra, two infinite classes of polyhedra. Geodesic polyhedra and Goldberg polyhedra are duals of each other. The geodesic and Goldberg polyhedra are parameterized by integers m and n, with m > 0 {\displaystyle m>0} and n ≥ 0 {\displaystyle n\geq 0} . T is the triangulation number, which is equal to T = m 2 + m n + n 2 {\displaystyle T=m^{2}+mn+n^{2}} . == Icosahedral == == Octahedral == == Tetrahedral == == References == Wenninger, Magnus (1979), Spherical Models, Cambridge University Press, ISBN 978-0-521-29432-4, MR 0552023, archived from the original on July 4, 2008 Reprinted by Dover 1999 ISBN 978-0-486-40921-4


id: outline-of-geometry-20260505-0
file: data/en.wikipedia.org/wiki/Outline_of_geometry-0.md
category: reference
title: Outline of geometry
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Outline_of_geometry
date_saved: 2026-05-05T08:14:37.296179+00:00
instance: kb-cron
parent: outline-of-geometry-20260505-0
summary:
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. Geometry is one of the oldest mathematical sciences. Modern geometry also extends into non-Euclidean spaces, topology, and fractal dimensions, bridging pure mathematics with applications in physics, computer science, and data visualization. == Types, methodologies, and terminologies of geometry. == Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive solid geometry Contact geometry Convex geometry Descriptive geometry Differential geometry Digital geometry Discrete geometry Distance geometry Elliptic geometry Enumerative geometry Epipolar geometry Euclidean geometry Finite geometry Fractal geometry Geometry of numbers Hyperbolic geometry Incidence geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Noncommutative algebraic geometry Noncommutative geometry Ordered geometry Parabolic geometry Plane geometry Projective geometry Quantum geometry Riemannian geometry Ruppeiner geometry Solid geometry Spherical geometry Symplectic geometry Synthetic geometry Systolic geometry Taxicab geometry Toric geometry Transformation geometry Tropical geometry == General geometry concepts == === General concepts === Chirality Handedness Relative direction Mirror image Coordinate-free treatment Four-dimensional space Infinitesimal transformation Geometric progression Geometric shape Pi Angular velocity Linear velocity De Moivre's theorem Similar triangles


id: graduate-studies-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Graduate_Studies_in_Mathematics-0.md
category: reference
title: Graduate Studies in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Studies_in_Mathematics
date_saved: 2026-05-05T08:14:38.533221+00:00
instance: kb-cron
parent: graduate-studies-in-mathematics-20260505-0
summary:
Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published in hardcover and e-book formats. == List of books == == See also == Graduate Texts in Mathematics == Notes == == External links == Graduate Studies in Mathematics on AMS Bookstore ISSN 1065-7339


id: graduate-texts-in-mathematics-20260505-0
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-0.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.


id: graduate-texts-in-mathematics-20260505-1
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-1.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
== List of books == Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ISBN 978-1-4613-8170-9) Measure and Category – A Survey of the Analogies between Topological and Measure Spaces, John C. Oxtoby (1980, 2nd ed., ISBN 978-0-387-90508-2) Topological Vector Spaces, H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ISBN 978-0-387-98726-2) A Course in Homological Algebra, Peter Hilton, Urs Stammbach (1997, 2nd ed., ISBN 978-0-387-94823-2) Categories for the Working Mathematician, Saunders Mac Lane (1998, 2nd ed., ISBN 978-0-387-98403-2) Projective Planes, Daniel R. Hughes, Fred C. Piper, (1982, ISBN 978-3-540-90043-6) A Course in Arithmetic, Jean-Pierre Serre (1996, ISBN 978-0-387-90040-7) Axiomatic Set Theory, Gaisi Takeuti, Wilson M. Zaring, (1973, ISBN 978-3-540-90050-4) Introduction to Lie Algebras and Representation Theory, James E. Humphreys (1997, ISBN 978-0-387-90053-7) A Course in Simple-Homotopy Theory, Marshall. M. Cohen, (1973, ISBN 0-387-90056-X) Functions of One Complex Variable I, John B. Conway (1978, 2nd ed., ISBN 978-0-387-90328-6) Advanced Mathematical Analysis, Richard Beals (1973, ISBN 978-0-387-90065-0) Rings and Categories of Modules, Frank W. Anderson, Kent R. Fuller (1992, 2nd ed., ISBN 978-0-387-97845-1) Stable Mappings and Their Singularities, Martin Golubitsky, Victor Guillemin, (1974, ISBN 978-0-387-90072-8) Lectures in Functional Analysis and Operator Theory, Sterling K. Berberian, (1974, ISBN


id: graduate-texts-in-mathematics-20260505-2
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-2.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Koblitz (1993, 2nd ed., ISBN 978-0-387-97966-3) Representations of Compact Lie Groups, Theodor Bröcker, Tammo tom Dieck (1985, ISBN 978-3-540-13678-1) Finite Reflection Groups, L.C. Grove, C.T. Benson (1985, 2nd ed., ISBN 978-0-387-96082-1) Harmonic Analysis on Semigroups – Theory of Positive Definite and Related Functions, Christian Berg, Jens Peter Reus Christensen, Paul Ressel (1984, ISBN 978-0-387-90925-7) Galois Theory, Harold M. Edwards (1984, ISBN 978-0-387-90980-6) Lie Groups, Lie Algebras, and Their Representations, V. S. Varadarajan (1984, ISBN 978-0-387-90969-1) Complex Analysis, Serge Lang (1999, 4th ed., ISBN 978-0-387-98592-3) Modern Geometry — Methods and Applications Part II: The Geometry and Topology of Manifolds, B. A. Dubrovin, Anatoly Timofeevich Fomenko, Sergei Novikov (1985, ISBN 978-0-387-96162-0) SL2(R), Serge Lang (1985, ISBN 978-0-387-96198-9) The Arithmetic of Elliptic Curves, Joseph H. Silverman (2009, 2nd ed., ISBN 978-0-387-09493-9) Applications of Lie Groups to Differential Equations, Peter J. Olver (2nd ed., 1993, ISBN 978-1-4684-0276-6) Holomorphic Functions and Integral Representations in Several Complex Variables, R. Michael Range (1986, ISBN 978-0-387-96259-7) Univalent Functions and Teichmüller Spaces, O. Lehto (1987, ISBN 978-1-4613-8654-4) Algebraic Number Theory, Serge Lang (1994, 2nd ed., ISBN 978-0-387-94225-4) Elliptic Curves, Dale Husemöller (2004, 2nd ed., ISBN 978-0-387-95490-5) Elliptic Functions, Serge Lang (1987, 2nd ed., ISBN 978-0-387-96508-6) Brownian Motion and Stochastic


id: graduate-texts-in-mathematics-20260505-3
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-3.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Robert (2000, ISBN 978-0-387-98669-2) Theory of Bergman Spaces, Hakan Hedenmalm, Boris Korenblum, Kehe Zhu (2000, ISBN 978-0-387-98791-0) An Introduction to Riemann–Finsler Geometry, David Bao, Shiing-Shen Chern, Zhongmin Shen (2000, ISBN 978-1-4612-7070-6) Diophantine Geometry, Marc Hindry, Joseph H. Silverman (2000, ISBN 978-0-387-98975-4) Introduction to Topological Manifolds, John M. Lee (2011, 2nd ed., ISBN 978-1-4419-7939-1) The Symmetric Group — Representations, Combinatorial Algorithms, and Symmetric Functions, Bruce E. Sagan (2001, 2nd ed., ISBN 978-0-387-95067-9) Galois Theory, Jean-Pierre Escofier (2001, ISBN 978-0-387-98765-1) Rational Homotopy Theory, Yves Félix, Stephen Halperin, Jean-Claude Thomas (2000, ISBN 978-0-387-95068-6) Problems in Analytic Number Theory, M. Ram Murty (2007, 2nd ed., ISBN 978-0-387-95143-0) Algebraic Graph Theory, Chris Godsil, Gordon Royle (2001, ISBN 978-0-387-95241-3) Analysis for Applied Mathematics, Ward Cheney (2001, ISBN 978-0-387-95279-6) A Short Course on Spectral Theory, William Arveson (2002, ISBN 978-0-387-95300-7) Number Theory in Function Fields, Michael Rosen (2002, ISBN 978-0-387-95335-9) Algebra, Serge Lang (2002, Revised 3rd ed., ISBN 978-0-387-95385-4) Lectures on Discrete Geometry, Jiří Matoušek (2002, ISBN 978-0-387-95374-8) From Holomorphic Functions to Complex Manifolds, Klaus Fritzsche, Hans Grauert (2002, ISBN 978-0-387-95395-3) Partial Differential Equations, Jürgen Jost, (2013, 3rd ed., ISBN 978-1-4614-4808-2) Algebraic Functions and Projective Curves, David M. Goldschmidt, (2003, ISBN 978-0-387-95432-5) Matrices — Theory and Applications,


id: graduate-texts-in-mathematics-20260505-4
file: data/en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics-4.md
category: reference
title: Graduate Texts in Mathematics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Graduate_Texts_in_Mathematics
date_saved: 2026-05-05T08:14:39.833032+00:00
instance: kb-cron
parent: graduate-texts-in-mathematics-20260505-0
summary:
Getz, Heekyoung Hahn (2024, ISBN 978-3-031-41151-9) An Invitation to Mathematical Logic, David Marker (2024, ISBN 978-3-031-55367-7) Fundamentals of Fourier Analysis, Loukas Grafakos (2024, ISBN 978-3-031-56499-4) A Course in Real Algebraic Geometry, Claus Scheiderer (2024, ISBN 978-3-031-69212-3) An Introduction to Microlocal Analysis, Peter Hintz (2025, ISBN 978-3-031-90705-0) Complex Analysis in One Variable and Riemann Surfaces, Mei-Chi Shaw, Charles M. Stanton (2025, ISBN 978-3-031-93641-8) Metric Lie Groups: Carnot-Carathéodory Spaces from the Homogeneous Viewpoint, Enrico Le Donne (2025, ISBN 978-3-031-98831-8) From Classical L-Functions to Modern Reciprocity Laws, Claus Sorensen (2025, ISBN 978-3-032-03034-4) Unitary Representations and Unitary Duals, Manfred Einsiedler, Thomas Ward (2025, ISBN 978-3-032-03898-2) Commutative Algebra, Shiro Goto, Kei-ichi Watanabe (2025, ISBN 978-981-95-4226-0) == See also == Graduate Studies in Mathematics == Notes == == External links == Springer-Verlag's Summary of Graduate Texts in Mathematics


id: list-of-graph-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graph_theory_topics-0.md
category: reference
title: List of graph theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graph_theory_topics
date_saved: 2026-05-05T08:14:41.041813+00:00
instance: kb-cron
parent: list-of-graph-theory-topics-20260505-0
summary:
This is a list of graph theory topics, by Wikipedia page. See glossary of graph theory for basic terminology. == Examples and types of graphs == == Graph coloring == == Paths and cycles == == Trees == === Terminology === Node Child node Parent node Leaf node Root node Root (graph theory) === Operations === Tree structure Tree data structure Cayley's formula Kőnig's lemma Tree (set theory) (need not be a tree in the graph-theory sense, because there may not be a unique path between two vertices) Tree (descriptive set theory) Euler tour technique == Graph limits == Graphon == Graphs in logic == Conceptual graph Entitative graph Existential graph Laws of Form Logical graph == Mazes and labyrinths == Labyrinth Maze Maze generation algorithm == Algorithms == Ant colony algorithm Breadth-first search Depth-first search Depth-limited search FKT algorithm Flood fill Graph exploration algorithm Matching (graph theory) Max flow min cut theorem Maximum-cardinality search Shortest path Dijkstra's algorithm Bellman–Ford algorithm A* algorithm Floyd–Warshall algorithm Topological sorting Pre-topological order == Other topics == == Networks, network theory == See list of network theory topics == Hypergraphs == Helly family Intersection (Line) Graphs of hypergraphs


id: list-of-graphs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_graphs-0.md
category: reference
title: List of graphs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_graphs
date_saved: 2026-05-05T08:14:43.462282+00:00
instance: kb-cron
parent: list-of-graphs-20260505-0
summary:
This partial list of graphs contains definitions of graphs and graph families. For collected definitions of graph theory terms that do not refer to individual graph types, such as vertex and path, see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Category:Graphs. Some of the finite structures considered in graph theory have names, sometimes inspired by the graph's topology, and sometimes after their discoverer. A famous example is the Petersen graph, a concrete graph on 10 vertices that appears as a minimal example or counterexample in many different contexts. == Individual graphs == == Highly symmetric graphs == === Strongly regular graphs === The strongly regular graph on v vertices and rank k is usually denoted srg(v,k,λ,μ). === Symmetric graphs === A symmetric graph is one in which there is a symmetry (graph automorphism) taking any ordered pair of adjacent vertices to any other ordered pair; the Foster census lists all small symmetric 3-regular graphs. Every strongly regular graph is symmetric, but not vice versa. === Semi-symmetric graphs === == Graph families == === Complete graphs === The complete graph on n {\displaystyle n} vertices is often called the n {\displaystyle n} -clique


id: list-of-group-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_group_theory_topics-0.md
category: reference
title: List of group theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_group_theory_topics
date_saved: 2026-05-05T08:14:44.910624+00:00
instance: kb-cron
parent: list-of-group-theory-topics-20260505-0
summary:
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography. == Structures and operations == == Basic properties of groups == === Group homomorphisms === == Basic types of groups == === Simple groups and their classification === Classification of finite simple groups === Permutation and symmetry groups === == Concepts groups share with other mathematics == == Mathematical objects making use of a group operation == == Mathematical


id: list-of-small-groups-20260505-0
file: data/en.wikipedia.org/wiki/List_of_small_groups-0.md
category: reference
title: List of small groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_small_groups
date_saved: 2026-05-05T08:14:46.116505+00:00
instance: kb-cron
parent: list-of-small-groups-20260505-0
summary:
The following list in mathematics contains the finite groups of small order up to group isomorphism. == Counts == For n = 1, 2, … the number of nonisomorphic groups of order n is 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, ... (sequence A000001 in the OEIS) For labeled groups, see (sequence A034383 in the OEIS). == Glossary == Each group is named by Small Groups Library as Goi, where o is the order of the group, and i is the index used to label the group within that order. Common group names: Zn: the cyclic group of order n (the notation Cn is also used; it is isomorphic to the additive group of Z/nZ) Dihn: the dihedral group of order 2n (often the notation Dn or D2n is used) Dn: the dihedral group of order 2n, the same as Dihn (notation used in section List of small non-abelian groups) K4: the Klein four-group of order 4, isomorphic to Z2 × Z2 and Dih2 Sn: the symmetric group of degree n, containing the n! permutations of n elements An: the alternating group of degree n, containing the


id: list-of-harmonic-analysis-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_harmonic_analysis_topics-0.md
category: reference
title: List of harmonic analysis topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_harmonic_analysis_topics
date_saved: 2026-05-05T08:14:47.331318+00:00
instance: kb-cron
parent: list-of-harmonic-analysis-topics-20260505-0
summary:
This is a list of harmonic analysis topics. See also list of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of mathematical analysis, mathematical physics and engineering. == Fourier analysis == === Fourier series === Periodic function Trigonometric function Trigonometric polynomial Exponential sum Dirichlet kernel Fejér kernel Gibbs phenomenon Parseval's identity Parseval's theorem Weyl differintegral Generalized Fourier series Orthogonal functions Orthogonal polynomials Empirical orthogonal functions Set of uniqueness === Fourier transform === Continuous Fourier transform Fourier inversion theorem Plancherel's theorem Convolution Convolution theorem Positive-definite function Poisson summation formula Paley–Wiener theorem Sobolev space Time–frequency representation Quantum Fourier transform == Topological groups == Topological abelian group Haar measure Discrete Fourier transform Dirichlet character Amenable group Von Neumann's conjecture Pontryagin duality Kronecker's theorem on diophantine approximation Almost periodic function Bohr compactification Wiener's tauberian theorem == Representation theory == Representation of a Lie group Unitary representation Irreducible representation Harish-Chandra character Restricted representation Induced representation Peter–Weyl theorem Spherical harmonic Casimir operator Hecke operator Stone–von Neumann theorem Discrete series representation Tempered representation Langlands program == Fast Fourier transform == Bluestein's FFT algorithm Cooley–Tukey FFT algorithm Rader's FFT algorithm Number-theoretic transform Irrational base discrete weighted


id: highly-composite-number-20260505-0
file: data/en.wikipedia.org/wiki/Highly_composite_number-0.md
category: reference
title: Highly composite number
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Highly_composite_number
date_saved: 2026-05-05T08:15:00.548879+00:00
instance: kb-cron
parent: highly-composite-number-20260505-0
summary:
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If d(n) denotes the number of divisors of a positive integer n, then a positive integer N is highly composite if d(N) > d(n) for all n < N. For example, 6 is highly composite because d(6) = 4, and for n = 1,2,3,4,5, you get d(n) = 1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. Ramanujan wrote a paper on highly composite numbers in 1915. The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040 (= 7!), as the ideal number of citizens in a city. Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040. == Examples == The first 41 highly composite numbers are listed in the table below (sequence A002182 in the


id: highly-composite-number-20260505-1
file: data/en.wikipedia.org/wiki/Highly_composite_number-1.md
category: reference
title: Highly composite number
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Highly_composite_number
date_saved: 2026-05-05T08:15:00.548879+00:00
instance: kb-cron
parent: highly-composite-number-20260505-0
summary:
lim sup x → ∞ log ⁡ Q ( x ) log ⁡ log ⁡ x ≤ 1.71 . {\displaystyle \limsup _{x\,\to \,\infty }{\frac {\log Q(x)}{\log \log x}}\leq 1.71\ .} == Related sequences == Highly composite numbers greater than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800. 10 of the first 38 highly composite numbers are superior highly composite numbers. The sequence of highly composite numbers (sequence A002182 in the OEIS) is a subset of the sequence of smallest numbers k with exactly n divisors (sequence A005179 in the OEIS). Highly composite numbers whose number of divisors is also a highly composite number are 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200 (sequence A189394 in the OEIS). It is known that this sequence is complete. A positive integer n is a


id: list-of-mathematics-history-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematics_history_topics-0.md
category: reference
title: List of mathematics history topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematics_history_topics
date_saved: 2026-05-05T08:15:01.857681+00:00
instance: kb-cron
parent: list-of-mathematics-history-topics-20260505-0
summary:
This is a list of mathematics history topics, by Wikipedia page. See also list of mathematicians, timeline of mathematics, history of mathematics, list of publications in mathematics. 1729 (anecdote) Adequality Archimedes Palimpsest Archimedes' use of infinitesimals Arithmetization of analysis Brachistochrone curve Chinese mathematics Cours d'Analyse Edinburgh Mathematical Society Erlangen programme Fermat's Last Theorem Greek mathematics Thomas Little Heath Hilbert's problems History of topos theory Hyperbolic quaternion Indian mathematics Islamic mathematics Italian school of algebraic geometry Kraków School of Mathematics Law of Continuity Lwów School of Mathematics Nicolas Bourbaki Non-Euclidean geometry Scottish Café Seven bridges of Königsberg Spectral theory Synthetic geometry Tautochrone curve Unifying theories in mathematics Waring's problem Warsaw School of Mathematics == Academic positions == Lowndean Professor of Astronomy and Geometry Lucasian professor Rouse Ball Professor of Mathematics Sadleirian Chair == See also == Mathematics portal


id: list-of-homological-algebra-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_homological_algebra_topics-0.md
category: reference
title: List of homological algebra topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_homological_algebra_topics
date_saved: 2026-05-05T08:15:03.084460+00:00
instance: kb-cron
parent: list-of-homological-algebra-topics-20260505-0
summary:
This is a list of homological algebra topics, by Wikipedia page. Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. == Basic techniques == Cokernel Exact sequence Chain complex Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective resolution Injective resolution Koszul complex Exact functor Derived functor Ext functor Tor functor Filtration (abstract algebra) Spectral sequence Abelian category Triangulated category Derived category == Applications == Group cohomology Galois cohomology Lie algebra cohomology Sheaf cohomology Whitehead problem Homological conjectures in commutative algebra


id: list-of-mathematical-identities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_identities-0.md
category: reference
title: List of mathematical identities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_identities
date_saved: 2026-05-05T08:15:04.285513+00:00
instance: kb-cron
parent: list-of-mathematical-identities-20260505-0
summary:
This article lists mathematical identities, that is, identically true relations holding in mathematics. Binet-cauchy identity Binomial inverse theorem Binomial identity Brahmagupta–Fibonacci two-square identity Candido's identity Cassini and Catalan identities Degen's eight-square identity Difference of two squares Euler's four-square identity Euler's identity Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities Heine's identity Hermite's identity Lagrange's identity Lagrange's trigonometric identities List of logarithmic identities MacWilliams identity Matrix determinant lemma Newton's identity Parseval's identity Pfister's sixteen-square identity Sherman–Morrison formula Sophie Germain identity Sun's curious identity Sylvester's determinant identity Vandermonde's identity Woodbury matrix identity == Identities for classes of functions == Exterior calculus identities Fibonacci identities: Combinatorial Fibonacci identities and Other Fibonacci identities Hypergeometric function identities List of integrals of logarithmic functions List of topics related to π List of trigonometric identities Inverse trigonometric functions Logarithmic identities Summation identities Vector calculus identities == See also == List of inequalities List of set identities and relations – Equalities for combinations of sets == External links == A Collection of Algebraic Identities Matrix Identities


id: list-of-impossible-puzzles-20260505-0
file: data/en.wikipedia.org/wiki/List_of_impossible_puzzles-0.md
category: reference
title: List of impossible puzzles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_impossible_puzzles
date_saved: 2026-05-05T08:15:05.485545+00:00
instance: kb-cron
parent: list-of-impossible-puzzles-20260505-0
summary:
This is a list of puzzles that cannot be solved. An impossible puzzle is a puzzle that cannot be resolved, either due to lack of sufficient information, or any number of logical impossibilities. 15 Puzzle – Slide fifteen numbered tiles into numerical order. It is impossible to solve in half of the starting positions. Five room puzzle – Cross each wall of a diagram exactly once with a continuous line. MU puzzle – Transform the string MI to MU according to a set of rules. Mutilated chessboard problem – Place 31 dominoes of size 2×1 on a chessboard with two opposite corners removed. Coloring the edges of the Petersen graph with three colors. Seven Bridges of Königsberg – Walk through a city while crossing each of seven bridges exactly once. Squaring the circle, the impossible problem of constructing a square with the same area as a given circle, using only a compass and straightedge. Three cups problem – Turn three cups right-side up after starting with one wrong and turning two at a time. Three utilities problem – Connect three cottages to gas, water, and electricity without crossing lines. Thirty-six officers problem – Arrange six regiments consisting of six officers


id: list-of-incomplete-proofs-20260505-0
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-0.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
This page lists notable examples of incomplete or incorrect published mathematical proofs. Most of these were accepted as complete or correct for several years but later discovered to contain gaps or errors. There are both examples where a complete proof was later found, or where the alleged result turned out to be false. == Results later proved rigorously == Euclid's Elements. Euclid's proofs are essentially correct, but strictly speaking sometimes contain gaps because he tacitly uses some unstated assumptions, such as the existence of intersection points. In 1899 David Hilbert gave a complete set of (second order) axioms for Euclidean geometry, called Hilbert's axioms, and between 1926 and 1959 Tarski gave some complete sets of first order axioms, called Tarski's axioms. Isoperimetric inequality. For three dimensions it states that the shape enclosing the maximum volume for its surface area is the sphere. It was formulated by Archimedes but not proved rigorously until the 19th century, by Hermann Schwarz. Infinitesimals. In the 18th century there was widespread use of infinitesimals in calculus, though these were not really well defined. Calculus was put on firm foundations in the 19th century, and Robinson put infinitesimals in a rigorous basis with the introduction of


id: list-of-incomplete-proofs-20260505-1
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-1.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
== Incorrect results == In 1759 Euler claimed that there were no closed knight tours on a chess board with 3 rows, but in 1917 Ernest Bergholt found tours on 3 × 10 and 3 × 12 boards. Euler's conjecture on Graeco-Latin squares. In the 1780s Euler conjectured that no such squares exist for any oddly even number n ≡ 2 (mod 4). In 1959, R. C. Bose and S. S. Shrikhande constructed counterexamples of order 22. Then E. T. Parker found a counterexample of order 10 using a one-hour computer search. Finally Parker, Bose, and Shrikhande showed this conjecture to be false for all n ≥ 10. In 1798 A. M. Legendre claimed that 6 is not the sum of 2 rational cubes, which as Lamé pointed out in 1865 is false as 6 = (37/21)3 + (17/21)3. In 1803, Gian Francesco Malfatti claimed to prove that a certain arrangement of three circles would cover the maximum possible area inside a right triangle. However, to do so he made certain unwarranted assumptions about the configuration of the circles. It was shown in 1930 that circles in a different configuration could cover a greater area, and in 1967 that Malfatti's


id: list-of-incomplete-proofs-20260505-2
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-2.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
This caused an error in the title of Janko's paper A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroup on J4: it does not have the full covering group as a subgroup, as the full covering group is larger than was realized at the time. The original statement of the classification of N-groups by Thompson in 1968 accidentally omitted the Tits group, though he soon fixed this. In 1975, Leitzel, Madan, and Queen incorrectly claimed that there are only 7 function fields over finite fields with genus > 0 and class number 1, but in 2013 Stirpe found another; there are in fact exactly 8. Busemann–Petty problem. Zhang published two papers in the Annals of Mathematics in 1994 and 1999, in the first of which he proved that the Busemann–Petty problem in R4 has a negative solution, and in the second of which he proved that it has a positive solution. Algebraic stacks. The book Laumon & Moret-Bailly (2000) on algebraic stacks mistakenly claimed that morphisms of algebraic stacks induce morphisms of lisse-étale topoi. The results depending on this were repaired by Olsson (2007).


id: list-of-incomplete-proofs-20260505-3
file: data/en.wikipedia.org/wiki/List_of_incomplete_proofs-3.md
category: reference
title: List of incomplete proofs
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_incomplete_proofs
date_saved: 2026-05-05T08:15:06.686663+00:00
instance: kb-cron
parent: list-of-incomplete-proofs-20260505-0
summary:
== Theories proven inconsistent == Frege's foundations of mathematics in his 1879 book Begriffsschrift turned out to be inconsistent because of Russell's paradox, found in 1901. Church's original published attempt in 1932 to define a formal system was inconsistent, as was his correction in 1933. The consistent part of his system later became the lambda calculus. Quine published his original description of the system Mathematical Logic in 1940, but in 1942 Rosser showed it was inconsistent. Wang found a correction in 1950; the consistency of this revised system is still unclear. In 1967 Reinhardt proposed Reinhardt cardinals, which Kunen showed to be inconsistent with ZFC in 1971, though they are not known to be inconsistent with ZF. Per Martin-Löf's original version of intuitionistic type theory proposed in 1971 was shown to be inconsistent by Jean-Yves Girard in 1972, and was replaced by a corrected version. == Status unclear == Uniform convergence. In his Cours d'Analyse of 1821, Cauchy "proved" that if a sum of continuous functions converges pointwise, then its limit is also continuous. However, Abel observed in 1826 that this is not the case. For the conclusion to hold, "pointwise convergence" must be replaced with "uniform convergence". It is


id: indefinite-product-20260505-0
file: data/en.wikipedia.org/wiki/Indefinite_product-0.md
category: reference
title: Indefinite product
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Indefinite_product
date_saved: 2026-05-05T08:15:08.101960+00:00
instance: kb-cron
parent: indefinite-product-20260505-0
summary:
In mathematics, the indefinite product operator is the inverse operator of Q ( f ( x ) ) = f ( x + 1 ) f ( x ) {\textstyle Q(f(x))={\frac {f(x+1)}{f(x)}}} . It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Thus Q ( ∏ x f ( x ) ) = f ( x ) . {\displaystyle Q\left(\prod _{x}f(x)\right)=f(x)\,.} More explicitly, if ∏ x f ( x ) = F ( x ) {\textstyle \prod _{x}f(x)=F(x)} , then F ( x + 1 ) F ( x ) = f ( x ) . {\displaystyle {\frac {F(x+1)}{F(x)}}=f(x)\,.} If F(x) is a solution of this functional equation for a given f(x), then so is CF(x) for any constant C. Therefore, each indefinite product actually represents a family of functions, differing by a multiplicative constant. == Period rule == If T {\displaystyle T} is a period of function f ( x ) {\displaystyle f(x)} then ∏ x f ( T x ) = C f ( T x ) x − 1 {\displaystyle \prod _{x}f(Tx)=Cf(Tx)^{x-1}} == Connection to indefinite sum == Indefinite product can be expressed in terms of indefinite sum: ∏


id: list-of-inequalities-20260505-0
file: data/en.wikipedia.org/wiki/List_of_inequalities-0.md
category: reference
title: List of inequalities
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_inequalities
date_saved: 2026-05-05T08:15:09.363624+00:00
instance: kb-cron
parent: list-of-inequalities-20260505-0
summary:
This article lists Wikipedia articles about named mathematical inequalities. == Inequalities in pure mathematics == === Analysis === Agmon's inequality Askey–Gasper inequality Babenko–Beckner inequality Bernoulli's inequality Bernstein's inequality (mathematical analysis) Bessel's inequality Bihari–LaSalle inequality Bohnenblust–Hille inequality Borell–Brascamp–Lieb inequality Brezis–Gallouet inequality Carleman's inequality Carlson's inequality Chebyshev–Markov–Stieltjes inequalities Chebyshev's sum inequality Clarkson's inequalities Eilenberg's inequality Fekete–Szegő inequality Fenchel's inequality Friedrichs' inequality Gagliardo–Nirenberg interpolation inequality Gårding's inequality Grothendieck inequality Grunsky's inequalities Hanner's inequalities Hardy's inequality Hardy–Littlewood inequality Hardy–Littlewood–Sobolev inequality Harnack's inequality Hausdorff–Young inequality Hermite–Hadamard inequality Hilbert's inequality Hölder's inequality Jackson's inequality Jensen's inequality Khabibullin's conjecture on integral inequalities Kantorovich inequality Karamata's inequality Korn's inequality Ladyzhenskaya's inequality Landau–Kolmogorov inequality Lebedev–Milin inequality Lieb–Thirring inequality Littlewood's 4/3 inequality Markov brothers' inequality Mashreghi–Ransford inequality Max–min inequality Minkowski's inequality Poincaré inequality Popoviciu's inequality Prékopa–Leindler inequality Rayleigh–Faber–Krahn inequality Remez inequality Riesz rearrangement inequality Schur test Shapiro inequality Sobolev inequality Steffensen's inequality Szegő inequality Three spheres inequality Trace inequalities Trudinger's theorem Turán's inequalities Von Neumann's inequality Wirtinger's inequality for functions Young's convolution inequality Young's inequality for products ==== Inequalities relating to means ==== Hardy–Littlewood maximal inequality Inequality of arithmetic and geometric means Ky Fan inequality Levinson's inequality Maclaurin's inequality Mahler's inequality Muirhead's inequality Newton's inequalities Stein–Strömberg theorem === Combinatorics ===


id: index-of-information-theory-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_information_theory_articles-0.md
category: reference
title: Index of information theory articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_information_theory_articles
date_saved: 2026-05-05T08:15:10.528092+00:00
instance: kb-cron
parent: index-of-information-theory-articles-20260505-0
summary:
This is a list of information theory topics. A Mathematical Theory of Communication algorithmic information theory arithmetic coding channel capacity Communication Theory of Secrecy Systems conditional entropy conditional quantum entropy confusion and diffusion cross-entropy data compression entropic uncertainty (Hirchman uncertainty) entropy encoding entropy (information theory) Fisher information Hick's law Huffman coding information bottleneck method information theoretic security information theory joint entropy Kullback–Leibler divergence lossless compression negentropy noisy-channel coding theorem (Shannon's theorem) principle of maximum entropy quantum information science range encoding redundancy (information theory) Rényi entropy self-information Shannon–Hartley theorem


id: list-of-integration-and-measure-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics-0.md
category: reference
title: List of integration and measure theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_integration_and_measure_theory_topics
date_saved: 2026-05-05T08:15:12.959879+00:00
instance: kb-cron
parent: list-of-integration-and-measure-theory-topics-20260505-0
summary:
This is a list of integration and measure theory topics, by Wikipedia page. == Intuitive foundations == Length Area Volume Probability Moving average == Riemann integral == Riemann sum Riemann–Stieltjes integral Bounded variation Jordan content == Improper integrals == Cauchy principal value == Measure theory and the Lebesgue integral == Measure (mathematics) Sigma algebra Separable sigma algebra Filtration (abstract algebra) Borel algebra Borel measure Indicator function Lebesgue measure Lebesgue integration Lebesgue's density theorem Counting measure Complete measure Haar measure Outer measure Borel regular measure Radon measure Measurable function Null set, negligible set Almost everywhere, conull set Lp space Borel–Cantelli lemma Lebesgue's monotone convergence theorem Fatou's lemma Absolutely continuous Uniform absolute continuity Total variation Radon–Nikodym theorem Fubini's theorem Double integral Vitali set, non-measurable set == Extensions == Henstock–Kurzweil integral Amenable group Banach–Tarski paradox Hausdorff paradox == Integral equations == Fredholm equation Fredholm operator Liouville–Neumann series == Integral transforms == See also list of transforms, list of Fourier-related transforms Kernel (integral operator) Convolution Radon transform == Integral geometry == Buffon's needle Hadwiger's theorem mean width intrinsic volumes == Other == Stokes theorem Differentiation under the integral sign Contour integration Examples of contour integration == See also == List of calculus topics List


id: list-of-international-mathematical-olympiads-20260505-0
file: data/en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiads-0.md
category: reference
title: List of International Mathematical Olympiads
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_International_Mathematical_Olympiads
date_saved: 2026-05-05T08:15:14.252706+00:00
instance: kb-cron
parent: list-of-international-mathematical-olympiads-20260505-0
summary:
The first of the International Mathematical Olympiads (IMOs) was held in Romania in 1959. The oldest of the International Science Olympiads, the IMO has since been held annually, except in 1980. That year, the competition initially planned to be held in Mongolia was cancelled due to the Soviet invasion of Afghanistan. Because the competition was initially founded for Eastern European countries participating in the Warsaw Pact, under the influence of the Eastern Bloc, the earlier IMOs were hosted only in Eastern European countries, gradually spreading to other nations. The first IMO was held in Romania in 1959. Seven countries entered – Bulgaria, Czechoslovakia, East Germany, Hungary, Poland, Romania and the Soviet Union – with the hosts finishing as the top-ranked nation. The number of participating countries has since risen: 14 countries took part in 1969, 50 in 1989, and 104 in 2009. North Korea is the only country whose entire team has been caught cheating, resulting in its disqualification at the 32nd IMO in 1991 and the 51st IMO in 2010. (However, the 2010 case was controversial.) There have been other disqualifications of contestants due to cheating, but such cases are not officially made public. In January 2011, Google gave


id: list-of-countries-by-medal-count-at-international-mathematical-olympiad-20260505-0
category: reference
title: List of countries by medal count at International Mathematical Olympiad
tags: science, encyclopedia
date_saved: 2026-05-05T08:15:15.567788+00:00
instance: kb-cron
parent: list-of-countries-by-medal-count-at-international-mathematical-olympiad-20260505-0
file:
data/en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad-0.md

summary:
The following is the top 100 list of countries by medal count at the International Mathematical Olympiad: == Notes == ^ This team is now defunct. == References ==

source_url:
https://en.wikipedia.org/wiki/List_of_countries_by_medal_count_at_International_Mathematical_Olympiad


id: list-of-irreducible-tits-indices-20260505-0
file: data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-0.md
category: reference
title: List of irreducible Tits indices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices
date_saved: 2026-05-05T08:15:16.864015+00:00
instance: kb-cron
parent: list-of-irreducible-tits-indices-20260505-0
summary:
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field k, not assumed to be algebraically closed. The possible irreducible indices were classified by Jacques Tits, and this classification is reproduced below. (Because every index is a direct sum of irreducible indices, classifying all indices amounts to classifying irreducible indices.) == Organization of the list == An index can be represented as a Dynkin diagram with certain vertices drawn close to each other (the orbit of the vertices under the *-action of the Galois group of k) and with certain sets of vertices circled (the orbits of the non-distinguished vertices under the *-action). This representation captures the full information of the index except when the underlying Dynkin diagram is D4, in which case one must distinguish between an action by the cyclic group C3 or the permutation group S3. Alternatively, an index can be represented using the name of the underlying Dynkin diagram together with additional superscripts and subscripts, to be explained momentarily. This representation, together with the labeled Dynkin diagram described in the previous paragraph, captures the full information of the index.


id: list-of-irreducible-tits-indices-20260505-1
file: data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-1.md
category: reference
title: List of irreducible Tits indices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices
date_saved: 2026-05-05T08:15:16.864015+00:00
instance: kb-cron
parent: list-of-irreducible-tits-indices-20260505-0
summary:
Conditions d = 2a | 2n, d ≥ 1, rd ≤ n, n ≠ rd + 1. Distinguished vertices d, 2d, ..., rd. Description Algebraic group : If k has characteristic ≠ 2, the special unitary group SU2n/d(D,h), where D is a division algebra of degree d over k and h is a hermitian form of discriminant 1 and index r, relative to a k-linear involution σ of D, an "involution of the first kind such that the subring Dσ has dimension d(d + 1)/2; or equivalently, when d > 1 and char k ≠ 2, the group SU2n/d where D and h are as above except that h is anti-hermitian form of discriminant 1 and index r, and Dσ has dimension d(d − 1)/2. Special fields Over a finite field, d = 1 and n = r; over the reals, d = 1 and n − r = 2m, or d = 2 and n = 2r; over a p-adic field or local function field, d = 1 and r = n or n − 2, or d = 2 and n = 2r or 2r + 3; over a number field, d = 1 and n − r


id: list-of-irreducible-tits-indices-20260505-2
file: data/en.wikipedia.org/wiki/List_of_irreducible_Tits_indices-2.md
category: reference
title: List of irreducible Tits indices
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_irreducible_Tits_indices
date_saved: 2026-05-05T08:15:16.864015+00:00
instance: kb-cron
parent: list-of-irreducible-tits-indices-20260505-0
summary:
=== E1337,0 === Image: Special fields: This type exists only over the reals and over some number fields; does not exist over any finite field nor over any local non-archimedean nor global function field. === E787,1 === Image: Special fields: This type does not exist over any finite field nor any local nor global field. === E667,1 === Image: Special fields: This type does not exist over any finite field nor over any local nor global field. === E487,1 === Image: Special fields: This type does not exist over any finite field nor any local nor global fields. === E317,2 === Image: Special fields: This type exists only over some number fields; does not exists over any finite field, nor any local nor global field. === E287,3 === Image: Special fields: This type exists only over the reals and over some number fields; does not exists over any finite field, nor local non-archimedean nor global function fields. === E97,4 === Image: Special fields: This type does not exist over any finite field; it exists over any local and global field. === E07,7 === Image: Special fields: This type exists over any field. == E8 == === E2488,0 === Image: Special


id: karp-s-21-np-complete-problems-20260505-0
file: data/en.wikipedia.org/wiki/Karp's_21_NP-complete_problems-0.md
category: reference
title: Karp's 21 NP-complete problems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Karp's_21_NP-complete_problems
date_saved: 2026-05-05T08:15:18.156275+00:00
instance: kb-cron
parent: karp-s-21-np-complete-problems-20260505-0
summary:
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete. In his 1972 paper, "Reducibility Among Combinatorial Problems", Richard Karp used Stephen Cook's 1971 theorem that the Boolean satisfiability problem is NP-complete (also called the Cook–Levin theorem) to show that there is a polynomial time many-one reduction from the Boolean satisfiability problem to each of 21 combinatorial and graph theoretical computational problems, thereby showing that they are all NP-complete. This was one of the first demonstrations that many natural computational problems occurring throughout computer science are computationally intractable, and it drove interest in the study of NP-completeness and the P versus NP problem. == The problems == Karp's 21 problems are shown below, many with their original names. The nesting indicates the direction of the reductions used. For example, Knapsack was shown to be NP-complete by reducing Exact cover to Knapsack. Satisfiability: the Boolean satisfiability problem for formulas in conjunctive normal form (often referred to as SAT) 0–1 integer programming (A variation in which only the restrictions must be satisfied, with no optimization) Clique (see also independent set problem) Set packing Vertex cover Set covering Feedback node set Feedback arc set Directed Hamilton


id: list-of-knot-theory-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_knot_theory_topics-0.md
category: reference
title: List of knot theory topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_knot_theory_topics
date_saved: 2026-05-05T08:15:19.329939+00:00
instance: kb-cron
parent: list-of-knot-theory-topics-20260505-0
summary:
Knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself. == History == == Knots, links, braids == Knot (mathematics) gives a general introduction to the concept of a knot. Two classes of knots: torus knots and pretzel knots Cinquefoil knot also known as a (5, 2) torus knot. Figure-eight knot (mathematics) the only 4-crossing knot Granny knot (mathematics) and Square knot (mathematics) are a connected sum of two Trefoil knots Perko pair, two entries in a knot table that were later shown to be identical. Stevedore knot (mathematics), a prime knot with crossing number 6 Three-twist knot is the twist knot with three-half twists, also known as the


id: list-of-laplace-transforms-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Laplace_transforms-0.md
category: reference
title: List of Laplace transforms
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Laplace_transforms
date_saved: 2026-05-05T08:15:20.618665+00:00
instance: kb-cron
parent: list-of-laplace-transforms-20260505-0
summary:
The following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable t (often time) to a function of a complex variable s (complex angular frequency). == Properties == The Laplace transform of a function f ( t ) {\displaystyle f(t)} can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily. === Linearity === For functions f {\displaystyle f} and g {\displaystyle g} and for scalar a {\displaystyle a} , the Laplace transform satisfies L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } {\displaystyle {\mathcal {L}}\{af(t)+bg(t)\}=a{\mathcal {L}}\{f(t)\}+b{\mathcal {L}}\{g(t)\}} and is, therefore, regarded as a linear operator. === Time shifting === The Laplace transform of f ( t − a ) u ( t − a ) {\displaystyle f(t-a)u(t-a)} , where u {\displaystyle u} is the Heaviside step function, is e − a s F ( s


id: list-of-large-cardinal-properties-20260505-0
file: data/en.wikipedia.org/wiki/List_of_large_cardinal_properties-0.md
category: reference
title: List of large cardinal properties
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_large_cardinal_properties
date_saved: 2026-05-05T08:15:21.962706+00:00
instance: kb-cron
parent: list-of-large-cardinal-properties-20260505-0
summary:
This page includes a list of large cardinal properties in the mathematical field of set theory. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a given type implies the existence of cardinals of most of the types listed above that type, and for most listed cardinal descriptions φ of lesser consistency strength, Vκ satisfies "there is an unbounded class of cardinals satisfying φ". The following table usually arranges cardinals in order of consistency strength, with size of the cardinal used as a tiebreaker. In a few cases (such as strongly compact cardinals) the exact consistency strength is not known and the table uses the current best guess. "Small" cardinals: 0, 1, 2, ..., ℵ 0 , ℵ 1 {\displaystyle \aleph _{0},\aleph _{1}} ,..., κ = ℵ κ {\displaystyle \kappa =\aleph _{\kappa }} , ... (see Aleph number) the height of the minimal transitive model of ZFC worldly cardinals weakly and strongly inaccessible, α-inaccessible, and hyper inaccessible cardinals weakly and strongly Mahlo, α-Mahlo, and hyper Mahlo cardinals. reflecting cardinals pseudo uplifting cardinals, uplifting cardinals weakly compact (= Π11-indescribable), Πmn-indescribable, totally


id: list-of-lemmas-20260505-0
file: data/en.wikipedia.org/wiki/List_of_lemmas-0.md
category: reference
title: List of lemmas
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_lemmas
date_saved: 2026-05-05T08:15:23.303385+00:00
instance: kb-cron
parent: list-of-lemmas-20260505-0
summary:
This following is a list of lemmas (or, "lemmata", i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. == Algebra == Abhyankar's lemma Aubin–Lions lemma Bergman's diamond lemma Fitting lemma Injective test lemma Hua's lemma (exponential sums) Krull's separation lemma Schanuel's lemma (projective modules) Schwartz–Zippel lemma Shapiro's lemma Stewart–Walker lemma (tensors) Whitehead's lemma (Lie algebras) Zariski's lemma === Algebraic geometry === Abhyankar's lemma Fundamental lemma (Langlands program) === Category theory === Five lemma Horseshoe lemma Nine lemma Short five lemma Snake lemma Splitting lemma Yoneda lemma === Linear algebra === Matrix determinant lemma Matrix inversion lemma === Group theory === Burnside's lemma also known as the Cauchy–Frobenius lemma Frattini's lemma (finite groups) Goursat's lemma Mautner's lemma (representation theory) Ping-pong lemma (geometric group theory) Schreier's subgroup lemma Schur's lemma (representation theory) Zassenhaus lemma === Polynomials === Gauss's lemma (polynomials) Schwartz–Zippel lemma === Ring theory and commutative algebra === Artin–Rees lemma Hensel's lemma (commutative rings) Nakayama lemma Noether's normalization lemma Prime avoidance lemma === Universal algebra === Jónsson's lemma == Analysis == Fekete's lemma Fundamental lemma of the calculus of variations Hopf lemma Sard's lemma (singularity theory)


id: list-of-lie-groups-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_Lie_groups_topics-0.md
category: reference
title: List of Lie groups topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_Lie_groups_topics
date_saved: 2026-05-05T08:15:25.912263+00:00
instance: kb-cron
parent: list-of-lie-groups-topics-20260505-0
summary:
This is a list of Lie group topics, by Wikipedia page. == Examples == See Table of Lie groups for a list General linear group, special linear group SL2(R) SL2(C) Unitary group, special unitary group SU(2) SU(3) Orthogonal group, special orthogonal group Rotation group SO(3) SO(8) Generalized orthogonal group, generalized special orthogonal group The special unitary group SU(1,1) is the unit sphere in the ring of coquaternions. It is the group of hyperbolic motions of the Poincaré disk model of the Hyperbolic plane. Lorentz group Spinor group Symplectic group Exceptional groups G2 F4 E6 E7 E8 Affine group Euclidean group Poincaré group Heisenberg group == Lie algebras == Commutator Jacobi identity Universal enveloping algebra Baker–Campbell–Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop algebra Graded Lie algebra == Foundational results == One-parameter group, One-parameter subgroup Matrix exponential Infinitesimal transformation Lie's third theorem Maurer–Cartan form Cartan's theorem Cartan's criterion Local Lie group Formal group law Hilbert's fifth problem Hilbert–Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification (Lie group) == Semisimple theory == Simple Lie group Compact Lie group, Compact real form Semisimple Lie algebra Root system Simply laced group ADE classification Maximal torus


id: table-of-lie-groups-20260505-0
file: data/en.wikipedia.org/wiki/Table_of_Lie_groups-0.md
category: reference
title: Table of Lie groups
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Table_of_Lie_groups
date_saved: 2026-05-05T08:15:27.102675+00:00
instance: kb-cron
parent: table-of-lie-groups-20260505-0
summary:
This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics. == Real Lie groups and their algebras == Column legend Cpt: Is this group G compact? (Yes or No) π 0 {\displaystyle \pi _{0}} : Gives the group of components of G. The order of the component group gives the number of connected components. The group is connected if and only if the component group is trivial (denoted by 0). π 1 {\displaystyle \pi _{1}} : Gives the fundamental group of G whenever G is connected. The group is simply connected if and only if the fundamental group is trivial (denoted by 0). UC: If G is not simply connected, gives the


id: list-of-limits-20260505-0
file: data/en.wikipedia.org/wiki/List_of_limits-0.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x. == Limits for general functions == === Definitions of limits and related concepts === lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} if and only if ∀ ε > 0 ∃ δ > 0 : 0 < | x − c | < δ ⟹ | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon } . This is the (ε, δ)-definition of limit. The limit superior and limit inferior of a sequence are defined as lim sup n → ∞ x n = lim n → ∞ ( sup m ≥ n x m ) {\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)} and lim inf n → ∞ x n = lim n → ∞ ( inf m ≥ n x m ) {\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)} . A function, f ( x ) {\displaystyle f(x)} , is said to be continuous at a point, c, if


id: list-of-limits-20260505-1
file: data/en.wikipedia.org/wiki/List_of_limits-1.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
If lim x → c f ( x ) = lim x → c h ( x ) = L {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L} and f ( x ) ≤ g ( x ) ≤ h ( x ) {\displaystyle f(x)\leq g(x)\leq h(x)} for all x in an open interval that contains c, except possibly c itself, lim x → c g ( x ) = L . {\displaystyle \lim _{x\to c}g(x)=L.} This is known as the squeeze theorem. This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c. == Polynomials and functions of the form xa == lim x → c a = a {\displaystyle \lim _{x\to c}a=a} === Polynomials in x === lim x → c x = c {\displaystyle \lim _{x\to c}x=c} lim x → c ( a x + b ) = a c + b {\displaystyle \lim _{x\to c}(ax+b)=ac+b} lim x → c x n = c n {\displaystyle \lim _{x\to c}x^{n}=c^{n}} if n is a positive integer lim x → ∞ x / a = { ∞ , a > 0 does not exist , a = 0 − ∞ ,


id: list-of-limits-20260505-2
file: data/en.wikipedia.org/wiki/List_of_limits-2.md
category: reference
title: List of limits
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_limits
date_saved: 2026-05-05T08:15:28.347754+00:00
instance: kb-cron
parent: list-of-limits-20260505-0
summary:
lim x → 1 ln ⁡ ( x ) x − 1 = 1 {\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1} lim x → 0 ln ⁡ ( x + 1 ) x = 1 {\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1} lim x → 0 − ln ⁡ ( 1 + a ( e − x − 1 ) ) x = a {\displaystyle \lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a} . This limit follows from L'Hôpital's rule. lim x → 0 x ln ⁡ x = 0 {\displaystyle \lim _{x\to 0}x\ln x=0} , hence lim x → 0 x x = 1 {\displaystyle \lim _{x\to 0}x^{x}=1} lim x → ∞ ln ⁡ x x = 0 {\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x}}=0} === Logarithms to arbitrary bases === For b > 1, lim x → 0 + log b ⁡ x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty } lim x → ∞ log b ⁡ x = ∞ {\displaystyle \lim _{x\to \infty }\log _{b}x=\infty } For b < 1, lim x → 0 + log b ⁡ x = ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty } lim x → ∞ log b ⁡ x = − ∞ {\displaystyle \lim _{x\to \infty


id: list-of-books-on-history-of-number-systems-20260505-0
file: data/en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems-0.md
category: reference
title: List of books on history of number systems
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_books_on_history_of_number_systems
date_saved: 2026-05-05T08:15:29.597547+00:00
instance: kb-cron
parent: list-of-books-on-history-of-number-systems-20260505-0
summary:
This list compiles notable works that explore the history and development of number systems across various civilizations and time periods. These works cover topics ranging from ancient numeral systems and arithmetic methods to the evolution of mathematical notations and the impact of numerals on science, trade, and culture. == Overview == Number systems have been central to the development of human civilization, enabling record-keeping, commerce, astronomy, and scientific advancement. Early systems such as tally marks and Roman numerals gradually gave way to more abstract and efficient representations like the Babylonian base-60 system and the Hindu–Arabic numerals, now standard worldwide. The invention of zero, positional notation, and symbolic mathematics has had profound philosophical and technological implications. == Notable works on the history of number systems == == Works on the history of zero == == Children's books on the history of numbers == == Historical texts == == References ==


id: list-of-regular-polytope-compounds-20260505-0
file: data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-0.md
category: reference
title: List of regular polytope compounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds
date_saved: 2026-05-05T08:15:32.110805+00:00
instance: kb-cron
parent: list-of-regular-polytope-compounds-20260505-0
summary:
This article lists the regular polytope compounds in Euclidean, spherical and hyperbolic spaces. == Two dimensional compounds == For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with n/m sides. A new figure is obtained by rotating these regular n/m-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/m minus one, and combining these figures. An extreme case of this is where n/m is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon. In other cases where n and m have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures, improper star polygons or compound polygons. The same notation {n/m} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form k{n} as being more correct, where usually k = m. A further


id: list-of-regular-polytope-compounds-20260505-1
file: data/en.wikipedia.org/wiki/List_of_regular_polytope_compounds-1.md
category: reference
title: List of regular polytope compounds
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_regular_polytope_compounds
date_saved: 2026-05-05T08:15:32.110805+00:00
instance: kb-cron
parent: list-of-regular-polytope-compounds-20260505-0
summary:
Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive. === Euclidean 3-space compounds === The only regular Euclidean compound honeycombs are an infinite family of compounds of cubic honeycombs, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[d{4,3,4}]{4,3,4}. === Hyperbolic 3-space compounds === C. W. L. Garner described two dual pairs of regular hyperbolic compound honeycombs in 1970: the compact pair 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5}, and the paracompact pair {6,3,3}[5{6,3,4}] and [5{4,3,6}]{3,3,6}. He did not consider vertex-regular compounds where the vertices are at infinity, or (reciprocally) cell-regular compounds where the cells are centred at infinity. In 2019, Peter McMullen (who focused only on the compact case) pointed out and filled a gap in Garner's proof of completeness, so that it is now proven that 2{5,3,4}[5{4,3,5}] and [5{5,3,4}]2{4,3,5} are the only compact regular


id: list-of-things-named-after-felix-hausdorff-20260505-0
file: data/en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff-0.md
category: reference
title: List of things named after Felix Hausdorff
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_things_named_after_Felix_Hausdorff
date_saved: 2026-05-05T08:15:33.394906+00:00
instance: kb-cron
parent: list-of-things-named-after-felix-hausdorff-20260505-0
summary:
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician. He made significant contributions to mathematics, specifically in the fields of set theory and topology. == Mathematics == Baker–Campbell–Hausdorff formula Gromov–Hausdorff convergence Hausdorff–Young inequality Hausdorff completion Hausdorff dimension Hausdorff distance Hausdorff gap Hausdorff maximal principle Hausdorff measure Hausdorff Medal Hausdorff moment problem Hausdorff paradox Hausdorff space == Places == Felix Hausdorff International Meeting Centre at the University of Greifswald Hausdorff Center for Mathematics == Space == 24947 Hausdorff == References ==


id: index-of-logarithm-articles-20260505-0
file: data/en.wikipedia.org/wiki/Index_of_logarithm_articles-0.md
category: reference
title: Index of logarithm articles
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/Index_of_logarithm_articles
date_saved: 2026-05-05T08:15:34.685156+00:00
instance: kb-cron
parent: index-of-logarithm-articles-20260505-0
summary:
This is a list of logarithm topics, by Wikipedia page. See also the list of exponential topics. Acoustic power Amoeba (mathematics) Antilogarithm Apparent magnitude Baker's theorem Bel Benford's law Binary logarithm Bode plot Henry Briggs Bygrave slide rule Cologarithm Common logarithm Complex logarithm Discrete logarithm Discrete logarithm records e Representations of e El Gamal discrete log cryptosystem Harmonic series History of logarithms Hyperbolic sector Iterated logarithm Otis King Law of the iterated logarithm Linear form in logarithms Linearithmic List of integrals of logarithmic functions Log canonical singularity Log-likelihood ratio Log-log graph Log-normal distribution Log-periodic antenna Log semiring Log structure Log-Weibull distribution Logarithmic algorithm Logarithmic convolution Logarithmic decrement Logarithmic derivative Logarithmic differential Logarithmic differentiation Logarithmic distribution Logarithmic form Logarithmic graph paper Logarithmic growth Logarithmic identities Logarithmic mean Logarithmic number system Logarithmic scale Logarithmic spiral Logit LogSumExp Mantissa is a disambiguation page; see common logarithm for the traditional concept of mantissa; see significand for the modern concept used in computing. Matrix logarithm Mel scale Mercator projection Mercator series Moment magnitude scale John Napier Napierian logarithm Natural logarithm Natural logarithm of 2 Neper Offset logarithmic integral pH Plethystic logarithm Pollard's kangaroo algorithm Pollard's rho algorithm for logarithms Polylogarithm Polylogarithmic function Prime number theorem


id: list-of-mathematical-logic-topics-20260505-0
file: data/en.wikipedia.org/wiki/List_of_mathematical_logic_topics-0.md
category: reference
title: List of mathematical logic topics
tags: science, encyclopedia
source_url: https://en.wikipedia.org/wiki/List_of_mathematical_logic_topics
date_saved: 2026-05-05T08:15:35.913586+00:00
instance: kb-cron
parent: list-of-mathematical-logic-topics-20260505-0
summary:
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms. == Working foundations == Peano axioms Giuseppe Peano Mathematical induction Structural induction Recursive definition Naive set theory Element (mathematics) Ur-element Singleton (mathematics) Simple theorems in the algebra of sets Algebra of sets Power set Empty set Non-empty set Empty function Universe (mathematics) Axiomatization Axiomatic system Axiom schema Axiomatic method Formal system Mathematical proof Direct proof Reductio ad absurdum Proof by exhaustion Constructive proof Nonconstructive proof Tautology Consistency proof Arithmetization of analysis Foundations of mathematics Formal language Principia Mathematica Hilbert's program Impredicative Definable real number Algebraic logic Boolean algebra (logic) Dialectica space categorical logic == Model theory == Finite model theory Descriptive complexity theory Model checking Trakhtenbrot's theorem Computable model theory Tarski's exponential function problem Undecidable problem Institutional model theory Institution (computer science) Non-standard analysis Non-standard calculus Hyperinteger Hyperreal number Transfer principle Overspill Elementary Calculus: An Infinitesimal Approach Criticism of non-standard analysis Standard part function Set theory Forcing (mathematics) Boolean-valued model Kripke semantics General frame Predicate logic First-order logic Infinitary logic Many-sorted logic Higher-order logic Lindström quantifier Second-order


