In mathematics, the Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–Seidenberg projection property—is named after Alfred Tarski and Abraham Seidenberg.[1] It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectives ∨ (or), ∧ (and), ¬ (not) and quantifiers ∀ (for all), ∃ (exists) is equivalent to a similar formula without quantifiers. An important consequence is the decidability of the theory of real-closed fields.
Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a computer. George E. Collins introduced the algorithm of cylindrical algebraic decomposition, which allows quantifier elimination over the reals in double exponential time. This complexity is optimal, as there are examples where the output has a double exponential number of connected components. This algorithm is therefore fundamental, and it is widely used in computational algebraic geometry.
A semialgebraic set in Rn is a finite union of sets defined by a finite number of polynomial equations and inequalities, that is by a finite number of statements of the form
and
for polynomials p and q. We define a projection map π : Rn +1 → Rn by sending a point (x1, ..., xn, xn +1) to (x1, ..., xn). Then the Tarski–Seidenberg theorem states that if X is a semialgebraic set in Rn +1 for some n ≥ 1, then π(X) is a semialgebraic set in Rn.
If we only define sets using polynomial equations and not inequalities then we define algebraic sets rather than semialgebraic sets. For these sets the theorem fails, i.e. projections of algebraic sets need not be algebraic. As a simple example consider the hyperbola in R2 defined by the equation
This is a perfectly good algebraic set, but projecting it down by sending (x, y) in R2 to x in R produces the set of points satisfying x ≠ 0. This is a semialgebraic set, but it is not an algebraic set as the algebraic sets in R are R itself, the empty set and the finite sets.
This example shows also that, over the complex numbers, the projection of an algebraic set may be non-algebraic. Thus the existence of real algebraic sets with non-algebraic projections does not rely on the fact that the field of real numbers is not algebraically closed.
Another example is the parabola in R2, which is defined by the equation
Its projection onto the x-axis is the half-line [0, ∞), a semialgebraic set that cannot be obtained from algebraic sets by (finite) intersections, unions, and set complements.
This result confirmed that semialgebraic sets in Rn form what is now known as an o-minimal structure on R. These are collections of subsets Sn of Rn for each n ≥ 1 such that we can take finite unions and complements of the subsets in Sn and the result will still be in Sn, moreover the elements of S1 are simply finite unions of intervals and points. The final condition for such a collection to be an o-minimal structure is that the projection map on the first n coordinates from Rn +1 to Rn must send subsets in Sn +1 to subsets in Sn. The Tarski–Seidenberg theorem tells us that this holds if Sn is the set of semialgebraic sets in Rn.
San Pietro CapofiumeStato Italia Regione Emilia-Romagna Provincia Bologna Comune Molinella Altitudine10 m s.l.m. Codice WMO16144 T. media gennaio2,4 °C T. media luglio23,1 °C T. media annua13,0 °C T. max. assoluta04/08/2017: +39,7 °C T. min. assoluta11/01/1985: -24,8 °C Prec. medie annue611,2 mm Coordinate44°39′13.63″N 11°37′22.28″E / 44.653785°N 11.622856°E44.653785; 11.622856Coordinate: 44°39′13.63
American comedian Jason WolinerWoliner in July 2011BornJason Woliner (1980-06-01) June 1, 1980 (age 43)New York City, U.S.Occupation(s)Director, writer, actorYears active1989–present Jason Woliner is an American director,[1] writer and a former child actor. He was the non-performing member of the comedy group Human Giant and directed the bulk of their output. After that, he directed, co-wrote, and acted as showrunner on the cult Adult Swim series Eagleheart starring Chris Ell…
Опис файлу Опис постер фільму «Не мине і року...» Джерело https://ru.kinorium.com/738873/ Час створення 1973 Автор зображення Кіностудія ім. О. Довженка Ліцензія див. нижче Обґрунтування добропорядного використання Обґрунтування добропорядного використання не вказано назву статті [?]…
Sceptridium Sceptridium dissectum, onvruchtbaar en vruchtbaar blad Taxonomische indeling Rijk:Plantae (Planten)Stam:Embryophyta (Landplanten)Clade:TracheophytaClade:EuphyllophytaClade:MonilophytaKlasse:PsilotopsidaOrde:OphioglossalesFamilie:Ophioglossaceae (Addertongfamilie) Geslacht SceptridiumLyon (1905) Portaal Biologie Sceptridium is een geslacht van varens uit de addertongfamilie (Ophioglossaceae), die ook weleens tot het geslacht Botrychium worden gerekend. Naamgeving en etymo…
موتور عشاير محمد سفينة تاج موتورعشايرمحمدسفينه تاج - قرية - تقسيم إداري البلد إيران المحافظة كرمان المقاطعة أرزوئیة الناحية الناحية المركزية القسم الريفي Arzuiyeh السكان التعداد السكاني 169 نسمة (إحصاء 2006) معلومات أخرى التوقيت توقيت إيران (+3:30 غرينيتش) توقيت صيفي ت…
1954 Italian filmWhere Is Freedom?(Dov'è la libertà?)Directed byRoberto RosselliniWritten byVitaliano Brancati Ennio Flaiano Antonio PietrangeliVincenzo TalaricoProduced byCarlo PontiDino De LaurentiisStarringTotòCinematographyAldo TontiTonino Delli ColliMusic byRenzo RosselliniRelease date1954Running time84 minCountryItalyLanguageItalian Where Is Freedom? (Italian: Dov'è la libertà?) is a 1954 Italian comedy-drama film directed by Roberto Rossellini.[1] The film had a troubled prod…
Gereja Kalimantan Evangelis (Kalimantan Evangelical Church)PenggolonganProtestanOrientasiCalvinisKitab suciBiblePemimpinPdt. Dr. Simpon F. Lion, M.ThWilayahKalimantan, IndonesiaKantor pusatBanjarmasin, Kalimantan SelatanPendiriRheinische Missionsgezelschaft zu Barmen (RMG), Basler Misssionsgezellschaft (BM)Didirikan10 April 1839 (Baptisan Pertama bagi orang Dayak) KalimantanJemaat1.280Umat330.735 jiwa (Terus bertambah)Rohaniwan1.056 Pendeta (Terus Bertambah)Sekolah menengah1 (SMK GKE Mandomai)Pe…
Frauen & Macht. Ein Manifest (englisch Women & Power: A Manifesto) wurde von der britischen Althistorikerin Mary Beard verfasst. Sie vertritt in ihrem Buch die These, dass Ausgrenzung von Frauen aus dem öffentlichen Raum eine fast 3000-jährige Kulturgeschichte habe, deren Mechanismen bis heute nachwirken und Frauen nach wie vor von Machtpositionen ausschließen.[1] Inhaltsverzeichnis 1 Inhalt 2 Entstehungsgeschichte 3 Rezeption 4 Weblinks 5 Einzelnachweise Inhalt Bebilderung de…
1949 Indian filmAndazFilm posterDirected byMehboob KhanWritten byS. Ali RazaScreenplay byS. Ali RazaStory byShums LucknaviProduced byMehboob KhanStarringNargisDilip KumarRaj KapoorCinematographyFaredoon IraniEdited byShamsudin KadriMusic byNaushadProductioncompanyMehboob StudioDistributed byMehboob ProductionsRelease date 21 March 1949 (1949-03-21) Running time147 minsCountryIndiaLanguageHindi Andaz (transl. Style) is a 1949 Indian Hindi language romantic drama film, directe…
2013 Indian Tamil action film This article is about Indian tamil movie. For region in southern India, see Pandya Nadu. Pandiya NaaduTheatrical posterDirected bySuseenthiranWritten bySuseenthiranBhaskar Sakthi(Dialogue)Screenplay bySuseenthiranStory bySuseenthiranProduced byVishalStarringVishalBharathirajaLakshmi MenonCinematographyR. MadhiEdited byAnthonyMusic byD. ImmanProductioncompanyVishal Film FactoryDistributed byVendhar MoviesRelease date 2 November 2013 (2013-11-02) Runnin…
Chinese diplomat and politician In this Chinese name, the family name is Jiang. Jiang Enzhu姜恩柱Jiang in 2007Chairperson of the Foreign Affairs Committee of National People's CongressIn officeMarch 2003 – March 2008Preceded byZeng JianhuiSucceeded byLi ZhaoxingDirector of the Liaison Office of the Central People's Government in the Hong Kong Special Administrative RegionIn officeJanuary 2000 – August 2002Preceded byZhou NanSucceeded byGao SirenChinese Ambassador to the …
Abdelkader Bensalahعبد القـادر بن صالحAbdelkader Bensalah pada tahun 2019Penjabat Kepala Negara AljazairMasa jabatan9 April – 19 Desember 2019Perdana MenteriNoureddine BedouiPendahuluAbdelaziz Bouteflika (Presiden)PenggantiAbdelmadjid Tebboune (Presiden)Presiden Dewan NegaraMasa jabatan2 Juli 2002 – 9 April 2019PendahuluBachir BoumazaPenggantiSalah Goudjil (Penjabat)Presiden Majelis Rakyat NasionalMasa jabatan14 Juni 1997 – 10 Juni 2002PendahuluDi…
Boxing competition Bad BloodDate29 November 2014VenueExCel, London, EnglandTitle(s) on the lineEuropean, WBO International and vacant British heavyweight titlesTale of the tapeBoxer Derek Chisora Tyson FuryNickname Del Boy The Gypsy KingHometown London, England Manchester, EnglandPre-fight record 20–4 (13 KOs) 22–0 (16 KOs)Height 6 ft 2 in (188 cm) 6 ft 9 in (206 cm)Weight 241+1⁄2 lb (110 kg) 264 lb (120 kg)Style Orthodox OrthodoxRecogn…
This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: The Little Mermaid Jr. – news · newspapers · books · scholar · JSTOR (April 2016) (Learn how and when to remove this template message) The Little Mermaid JrMusicAlan MenkenLyricsGlenn Slater Howard AshmanBookDoug WrightBasisThe Little Mermaid by Hans Christian AndersenThe Little Me…
PizundaПиҵундаПицунда ბიჭვინთა Staat: Abchasien Abchasien (de facto)Georgien Georgien (de jure) Rajon: Rajon Gagra Koordinaten: 43° 10′ N, 40° 20′ O43.16666666666740.333333333333Koordinaten: 43° 10′ N, 40° 20′ O Einwohner: 7000 (2008) Zeitzone: Moscow Time (UTC+3) Telefonvorwahl: (+7 840) 23 Kfz-Kennzeichen: ABH Gemeindeart: Stadt Bürgermeister: Beslan Ardsinba Webpräsenz: www.pitsu…
Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Siklus lisogenik – berita · surat kabar · buku · cendekiawan · JSTOR Siklus lisogenik dalam virologi merupakan siklus reproduksi virus selain siklus litik. Tahapan dari siklus ini hampir sama dengan siklus …
Paddle for spanking For other uses, see Paddle (disambiguation). A spanking paddle A spanking paddle is an implement used to strike a person on the buttocks. The act of spanking a person with a paddle is known as paddling. A paddling may be for punishment (normally of a student at school in the United States), or as an initiation or hazing ritual. Description A paddle has two parts: a handle and a blade. Most paddles are designed to be held with one hand, but a giant paddle may be designed to be…
Ayub 42Kitab Ayub lengkap pada Kodeks Leningrad, dibuat tahun 1008.KitabKitab AyubKategoriKetuvimBagian Alkitab KristenPerjanjian LamaUrutan dalamKitab Kristen18← pasal 41 Mazmur 1 → Ayub 42 (disingkat Ayb 42) adalah bagian terakhir dari Kitab Ayub di Alkitab Ibrani dan Perjanjian Lama dalam Alkitab Kristen. Kitab ini menceritakan riwayat Ayub, seorang yang saleh, dan pencobaan yang dialaminya.[1] Pasal ini berisi penyesalan Ayub dan pemulihan keadaannya oleh Allah.[2]…
Films Donald Trump has produced or hosted Trump in 2015 Trump's star on the Hollywood Walk of Fame This article is part of a series aboutDonald Trump Business and personal Business career The Trump Organization wealth tax returns Media career The Apprentice books filmography Eponyms Family Foundation American football Golf Honors Public image in popular culture SNL parodies handshakes Legal affairs Sexual misconduct allegations Nicknames pseudonyms Racial views Comments on John McCain Conspiracy…
fusionar en|t=20190126|Folclore de Chile}} Este artículo o sección tiene referencias, pero necesita más para complementar su verificabilidad.Este aviso fue puesto el 20 de diciembre de 2016. Violeta Parra, folclorista chilena. La música folclórica de Chile se caracteriza por la mezcla de sonidos tradicionales aborígenes con aquellos traídos desde España. La cueca, danza nacional desde 1979,[1] es un buen ejemplo de ello: tiene características propias dependiendo de la zona del pa…
Lokasi Pengunjung: 3.142.40.44