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Language education

Language education – the process and practice of teaching a second or foreign language – is primarily a branch of applied linguistics, but can be an interdisciplinary field. There are four main learning categories for language education: communicative competencies, proficiencies, cross-cultural experiences, and multiple literacies.

Need
Increasing globalization has created a great need for people in the workforce who can communicate in multiple languages. Common languages are used in areas such as trade, tourism, diplomacy, technology, media, translation, interpretation and science. Many countries such as Korea (Kim Yeong-seo, 2009), Japan (Kubota, 1998) and China (Kirkpatrick & Zhichang, 2002) frame education policies to teach at least one foreign language at the primary and secondary school levels. However, some countries such as India, Singapore, Malaysia, Pakistan, and the Philippines use a second official language in their governments. According to GAO (2010), China has recently been putting enormous importance on foreign language learning, especially the English language.

Ancient to medieval period
Ancient learners seem to have started by reading, memorizing and reciting little stories and dialogues that provided basic vocabulary and grammar in naturalistic contexts. These texts (and they seem to have always been coherent texts, never isolated sentences such as those modern learners often practise on) covered topics such as getting dressed in the morning (and how to manage the slaves who helped with that task), going to school (and evading punishment for not having been there yesterday), visiting a sick friend (and how to find an individual unit in a Roman apartment block), trading insults (and how to concede a fight gracefully), or getting a new job (a piece of cake if you have studied with me, an ancient teacher assured his students mendaciously). The texts were presented bilingually in two narrow columns, the language you were learning on the left and the one you already knew on the right, with the columns matching line for line: each line was effectively a glossary, while each column was a text.

Although the need to learn foreign languages is almost as old as human history itself, the origins of modern language education are in the study and teaching of Latin in the 17th century. In the Ancient Near East, Akkadian was the language of diplomacy, as in the Amarna letters. For many centuries, Latin had been the dominant language of education, commerce, religion, and government in much of the Western world. By the end of the 16th century, it had largely been displaced by French, Italian, and English. John Amos Comenius was one of many people who tried to reverse this trend. He composed a complete course for learning Latin, covering the entire school curriculum, culminating in his Opera Didactica Omnia, 1657.

In this work, Comenius also outlined his theory of language acquisition. He is one of the first theorists to write systematically about how languages are learned and about pedagogical methodology for language acquisition. He held that language acquisition must be allied with sensation and experience. Teaching must be oral. The schoolroom should have models of things, and failing that, pictures of them. As a result, he also published the world's first illustrated children's book, Orbis sensualium pictus. The study of Latin diminished from the study of a living language to be used in the real world to a subject in the school curriculum. Such decline brought about a new justification for its study. It was then claimed that its study of Latin developed intellectual ability, and the study of Latin grammar became an end in and of itself.

"Grammar schools" from the 16th to 18th centuries focused on teaching the grammatical aspects of Classical Latin. Advanced students continued grammar study with the addition of rhetoric.

18th century
The study of modern languages did not become part of the curriculum of European schools until the 18th century. Based on the purely academic study of Latin, students of modern languages did much of the same exercises, studying grammatical rules and translating abstract sentences. Oral work was minimal, and students were instead required to memorize grammatical rules and apply these to decode written texts in the target language. This tradition-inspired method became known as the grammar-translation method.

19th and 20th centuries
Innovation in foreign language teaching began in the 19th century and became very rapid in the 20th century. It led to a number of different and sometimes conflicting methods, each claiming to be a major improvement over the previous or contemporary methods. The earliest applied linguists included Jean Manesca, Heinrich Gottfried Ollendorff (1803–1865), Henry Sweet (1845–1912), Otto Jespersen (1860–1943), and Harold Palmer (1877–1949). They worked on setting language teaching principles and approaches based on linguistic and psychological theories, but they left many of the specific practical details for others to devise.

The history of foreign-language education in the 20th century and the methods of teaching (such as those related below) might appear to be a history of failure. Very few students in U.S. universities who have a foreign language as a major attain "minimum professional proficiency". Even the "reading knowledge" required for a PhD degree is comparable only to what second-year language students read, and only very few researchers who are native English speakers can read and assess information written in languages other than English. Even a number of famous linguists are monolingual.

However, anecdotal evidence for successful second or foreign language learning is easy to find, leading to a discrepancy between these cases and the failure of most language programs. This tends to make the research of second-language acquisition emotionally charged. Older methods and approaches such as the grammar translation method and the direct method are dismissed and even ridiculed, as newer methods and approaches are invented and promoted as the only and complete solution to the problem of the high failure rates of foreign language students.

Most books on language teaching list the various methods that have been used in the past, often ending with the author's new method. These new methods are usually presented as coming only from the author's mind, as the authors generally give no credence to what was done before and do not explain how it relates to the new method. For example, descriptive linguists seem to claim unhesitatingly that there were no scientifically based language teaching methods before their work (which led to the audio-lingual method developed for the U.S. Army in World War II). However, there is significant evidence to the contrary. It is also often inferred or even stated that older methods were completely ineffective or have died out completely, though in reality, even the oldest methods are still in use (e.g. the Berlitz version of the direct method). Proponents of new methods have been so sure that their ideas are so new and so correct that they could not conceive that the older ones have enough validity to cause controversy. This was in turn caused by emphasis on new scientific advances, which has tended to blind researchers to precedents in older work.

There have been two major branches in the field of language learning, the empirical and theoretical, and these have almost completely separate histories, with each gaining ground over the other at one time or another. Examples of researchers on the empiricist side are Jesperson, Palmer, and Leonard Bloomfield, who promote mimicry and memorization with pattern drills. These methods follow from the basic empiricist position that language acquisition results from habits formed by conditioning and drilling. In its most extreme form, language learning is seen as much the same as any other learning in any other species, human language being essentially the same as communication behaviors seen in other species.

On the theoretical side are, for example, Francois Gouin, M.D. Berlitz, and Emile B. De Sauzé, whose rationalist theories of language acquisition dovetail with linguistic work done by Noam Chomsky and others. These have led to a wider variety of teaching methods, ranging from the grammar-translation method and Gouin's "series method" to the direct methods of Berlitz and De Sauzé. With these methods, students generate original and meaningful sentences to gain a functional knowledge of the rules of grammar. This follows from the rationalist position that man is born to think and that language use is a uniquely human trait impossible in other species. Given that human languages share many common traits, the idea is that humans share a universal grammar which is built into our brain structure. This allows us to create sentences that we have never heard before but that can still be immediately understood by anyone who understands the specific language being spoken. The rivalry between the two camps is intense, with little communication or cooperation between them.

21st century
Over time, language education has developed in schools and has become a part of the education curriculum around the world. In some countries, such as the United States, language education (also referred to as World Languages) has become a core subject along with main subjects such as English, Maths and Science.

In some countries, such as Australia, it is so common nowadays for a foreign language to be taught in schools that the subject of language education is referred to LOTE or Language Other Than English. In the majority of English-speaking education centers, French, Spanish, and German are the most popular languages to study and learn. English as a Second Language (ESL) is also available for students whose first language is not English and they are unable to speak it to the required standard.

Teaching foreign language in classrooms
Language education may take place as a general school subject or in a specialized language school. There are many methods of teaching languages. Some have fallen into relative obscurity and others are widely used; still others have a small following, but offer useful insights.

While sometimes confused, the terms "approach", "method" and "technique" are hierarchical concepts.

An approach is a set of assumptions about the nature of language and language learning, but does not involve procedure or provide any details about how such assumptions should be implemented into the classroom setting. Such can be related to second-language acquisition theory.

There are three principal "approaches":
 * 1) The structural view treats language as a system of structurally related elements to code meaning (e.g. grammar).
 * 2) The functional view sees language as a vehicle to express or accomplish a certain function, such as requesting something.
 * 3) The interactive view sees language as a vehicle for the creation and maintenance of social relations, focusing on patterns of moves, acts, negotiation and interaction found in conversational exchanges. This approach has been fairly dominant since the 1980s.

A method is a plan for presenting the language material to be learned, and should be based upon a selected approach. In order for an approach to be translated into a method, an instructional system must be designed considering the objectives of the teaching/learning, how the content is to be selected and organized, the types of tasks to be performed, the roles of students, and the roles of teachers.
 * 1) Examples of structural methods are grammar translation and the audio-lingual method.
 * 2) Examples of functional methods include the oral approach / situational language teaching.
 * 3) Examples of interactive methods include the direct method, the series method, communicative language teaching, language immersion, the Silent Way, suggestopedia, the natural approach, tandem language learning, total physical response, Teaching Proficiency through Reading and Storytelling and Dogme language teaching.

A technique (or strategy) is a very specific, concrete stratagem or trick designed to accomplish an immediate objective. Such are derived from the controlling method, and less directly, from the approach.

Online and self-study courses
Hundreds of languages are available for self-study, from scores of publishers, for a range of costs, using a variety of methods. The course itself acts as a teacher and has to choose a methodology, just as classroom teachers do.

Audio recordings and books
Audio recordings use native speakers, and one strength is helping learners improve their accent. Some recordings have pauses for the learner to speak. Others are continuous so the learner speaks along with the recorded voice, similar to learning a song.

Audio recordings for self-study use many of the methods used in classroom teaching, and have been produced on records, tapes, CDs, DVDs and websites.

Most audio recordings teach words in the target language by using explanations in the learner's own language. An alternative is to use sound effects to show meaning of words in the target language. The only language in such recordings is the target language, and they are comprehensible regardless of the learner's native language.

Language books have been published for centuries, teaching vocabulary and grammar. The simplest books are phrasebooks to give useful short phrases for travelers, cooks, receptionists, or others who need specific vocabulary. More complete books include more vocabulary, grammar, exercises, translation, and writing practice.

Also, various other "language learning tools" have been entering the market in recent years.

Internet and software
Software can interact with learners in ways that books and audio cannot:

 * 1) Some software records the learner, analyzes the pronunciation, and gives feedback.
 * 2) Software can present additional exercises in areas where a particular learner has difficulty, until the concepts are mastered.
 * 3) Software can pronounce words in the target language and show their meaning by using pictures instead of oral explanations. The only language in such software is the target language. It is comprehensible regardless of the learner's native language.

Websites provide various services geared toward language education. Some sites are designed specifically for learning languages:

 * 1) Some software runs on the web itself, with the advantage of avoiding downloads, and the disadvantage of requiring an internet connection.
 * 2) Some publishers use the web to distribute audio, texts and software, for use offline. For example, various travel guides, for example Lonely Planet, offer software supporting language education.
 * 3) Some websites offer learning activities such as quizzes or puzzles to practice language concepts.
 * 4) Language exchange sites connect users with complementary language skills, such as a native Spanish speaker who wants to learn English with a native English speaker who wants to learn Spanish. Language exchange websites essentially treat knowledge of a language as a commodity, and provide a marketlike environment for the commodity to be exchanged. Users typically contact each other via chat, VoIP, or email. Language exchanges have also been viewed as a helpful tool to aid language learning at language schools. Language exchanges tend to benefit oral proficiency, fluency, colloquial vocabulary acquisition, and vernacular usage, rather than formal grammar or writing skills. Across Australasia, 'Education Perfect' – an online learning site- is frequently used as it enables teachers to monitor students' progress as students gain a "point" for every new word remembered. There is an annual international Education Perfect languages contest held in May.

Many other websites are helpful for learning languages, even though they are designed, maintained and marketed for other purposes:

Some Internet content is free, often from government and nonprofit sites such as BBC Online, Book2, Foreign Service Institute, with no or minimal ads. Some are ad-supported, such as newspapers and YouTube. Some require a payment.
 * 1) All countries have websites in their own languages, which learners elsewhere can use as primary material for study: news, fiction, videos, songs, etc. In a study conducted by the Center for Applied Linguistics, it was noted that the use of technology and media has begun to play a heavy role in facilitating language learning in the classroom. With the help of the internet, students are readily exposed to foreign media (music videos, television shows, films) and as a result, teachers are taking heed of the internet's influence and are searching for ways to combine this exposure into their classroom teaching.
 * 2) Translation sites let learners find the meaning of foreign text or create foreign translations of text from their native language.
 * 3) Speech synthesis or text to speech (TTS) sites and software let learners hear pronunciation of arbitrary written text, with pronunciation similar to a native speaker.
 * 4) Course development and learning management systems such as Moodle are used by teachers, including language teachers.
 * 5) Web conferencing tools can bring remote learners together.
 * 6) Players of computer games can practice a target language when interacting in massively multiplayer online games and virtual worlds, commonly English. In 2005, the virtual world Second Life started to be used for foreign language tuition, sometimes with entire businesses being developed.  In addition, Spain's language and cultural institute Instituto Cervantes has an "island" on Second Life.

Learning strategies
Language learning strategies have attracted increasing focus as a way of understanding the process of language acquisition.

Listening as a way to learn
Clearly listening is used to learn, but not all language learners employ it consciously. Listening to understand is one level of listening but focused listening is not something that most learners employ as a strategy. Focused listening is a strategy in listening that helps students listen attentively with no distractions. Focused listening is very important when learning a foreign language as the slightest accent on a word can change the meaning completely.

Reading as a way to learn
Many people read to understand but the strategy of reading text to learn grammar and discourse styles can also be employed. Parallel texts may be used to improve comprehension.

Speaking as a way to learn
Alongside listening and reading exercises, practicing conversation skills is an important aspect of language acquisition. Language learners can gain experience in speaking foreign languages through in-person language classes, language meet-ups, university language exchange programs, joining online language learning communities, and traveling to a country where the language is spoken.

Learning vocabulary
Translation and rote memorization have been the two strategies that have been employed traditionally. There are other strategies that also can be used such as guessing, based on looking for contextual clues, spaced repetition with a use of various apps, games and tools (e.g. Duolingo and Anki). Knowledge about how the brain works can be utilized in creating strategies for how to remember words.

Learning Esperanto as a propaedeutic strategy for learning other languages
Esperanto, the most widely used international auxiliary language, was founded by L. L. Zamenhof, a Polish-Jewish ophthalmologist, in 1887, aimed to eliminate language barriers in the international contacts. Esperanto is an constructed language created on the basis of the Indo-European languages, absorbing the reasonable factors of commonality of the Germanic languages. Esperanto is completely consistent in its speech and writing. By learning twenty-eight letters and mastering the phonetic rules, one can read and write any words. With further simplification and standardization, Esperanto becomes more easily mastered than other languages. Ease of learning helps one build the confidence and learning Esperanto, as a learning strategy, constitutes a good introduction to foreign language study.

Blended learning
Blended learning combines face-to-face teaching with distance education, frequently electronic, either computer-based or web-based. It has been a major growth point in the ELT (English Language Teaching) industry over the last ten years.

Some people, though, use the phrase 'Blended Learning' to refer to learning taking place while the focus is on other activities. For example, playing a card game that requires calling for cards may allow blended learning of numbers (1 to 10).

Skill teaching
When talking about language skills, the four basic ones are: listening, speaking, reading and writing. However, other, more socially based skills have been identified more recently such as summarizing, describing, narrating etc. In addition, more general learning skills such as study skills and knowing how one learns have been applied to language classrooms.

In the 1970s and 1980s, the four basic skills were generally taught in isolation in a very rigid order, such as listening before speaking. However, since then, it has been recognized that we generally use more than one skill at a time, leading to more integrated exercises. Speaking is a skill that often is underrepresented in the traditional classroom. This is due to the fact that it is considered harder to teach and test. There are numerous texts on teaching and testing writing but relatively few on speaking.

More recent textbooks stress the importance of students working with other students in pairs and groups, sometimes the entire class. Pair and group work give opportunities for more students to participate more actively. However, supervision of pairs and groups is important to make sure everyone participates as equally as possible. Such activities also provide opportunities for peer teaching, where weaker learners can find support from stronger classmates.

Sandwich technique
In foreign language teaching, the sandwich technique is the oral insertion of an idiomatic translation in the mother tongue between an unknown phrase in the learned language and its repetition, in order to convey meaning as rapidly and completely as possible. The mother tongue equivalent can be given almost as an aside, with a slight break in the flow of speech to mark it as an intruder.

When modeling a dialogue sentence for students to repeat, the teacher not only gives an oral mother tongue equivalent for unknown words or phrases, but repeats the foreign language phrase before students imitate it: L2 => L1 => L2. For example, a German teacher of English might engage in the following exchange with the students:

 * Teacher: "Let me try – lass mich versuchen – let me try."
 * Students: "Let me try."

Mother tongue mirroring
Mother tongue mirroring is the adaptation of the time-honoured technique of literal translation or word-for word translation for pedagogical purposes. The aim is to make foreign constructions salient and transparent to learners and, in many cases, spare them the technical jargon of grammatical analysis. It differs from literal translation and interlinear text as used in the past since it takes the progress learners have made into account and only focuses upon a specific structure at a time. As a didactic device, it can only be used to the extent that it remains intelligible to the learner, unless it is combined with a normal idiomatic translation. This technique is seldom referred to or used these days.

Back-chaining
Back-chaining is a technique used in teaching oral language skills, especially with polysyllabic or difficult words. The teacher pronounces the last syllable, the student repeats, and then the teacher continues, working backwards from the end of the word to the beginning.

For example, to teach the name 'Mussorgsky' a teacher will pronounce the last syllable: -sky, and have the student repeat it. Then the teacher will repeat it with -sorg- attached before: -sorg-sky, and all that remains is the first syllable: Mus-sorg-sky.

Code switching
Code switching is a special linguistic phenomenon that the speaker consciously alternates two or more languages according to different time, places, contents, objects and other factors. Code switching shows its functions while one is in the environment that mother tongue are not playing a dominant role in students' life and study, such as the children in the bilingual family or in the immigrant family. That is to say, the capability of using code switching, relating to the transformation of phonetics, words, language structure, expression mode, thinking mode, cultural differences and so on, is needed to be guided and developed in the daily communication environment. Most people learn foreign language in the circumstance filled with the using of their native language so that their ability of code switching cannot be stimulated, and thus the efficiency of foreign language acquisition would decrease. Therefore, as a teaching strategy, code switching is used to help students better gain conceptual competences and to provide rich semantic context for them to understand some specific vocabularies.

By region
Practices in language education may vary by region however the underlying understandings which drive it are fundamentally similar. Rote repetition, drilling, memorisation and grammar conjugating are used the world over. Sometimes there are different preferences teaching methods by region. Language immersion is popular in some European countries, but is not used very much in the United States, in Asia or in Australia.

Early childhood education
Early childhood is the fastest and most critical period for one to master language in their life. Children's language communication is transformed from non-verbal communication to verbal communication from ages of one to five. Their mastery of language is largely acquired naturally by living in a verbal communication environment. As long as we are good at guiding and creating opportunities for children, children's language ability is easy to be developed and cultivated.

Compulsory education
Compulsory education, for most people, is the period that they have access to a second or foreign language for the first time. In this period, the most professional foreign language education and academic atmosphere are provided to the students. They can get help and motivation from teachers and be activated by the peers at any time. One would be able to undergo a lot of specialized learning in order to truly master a great number of rules of vocabulary, grammar and verbal communication.

Adult education
Learning a foreign language during adulthood means one is pursuing a higher value of themself by obtaining a new skill. At this stage, individuals have already developed the ability to supervise themself learning a language. However, at the same time, the pressure is also an obstacle for adults.

Elderly education
Compared to other life stages, this period is the hardest to learn a new language due to gradual brain deterioration and memory loss. Notwithstanding its difficulty, language education for seniors can slow this brain degeneration and active ageing.

Language study holidays
An increasing number of people are now combining holidays with language study in the native country. This enables the student to experience the target culture by meeting local people. Such a holiday often combines formal lessons, cultural excursions, leisure activities, and a homestay, perhaps with time to travel in the country afterwards. Language study holidays are popular across Europe (Malta & UK being the most popular) and Asia due to the ease of transportation and variety of nearby countries. These holidays have become increasingly more popular in Central and South America in such countries as Guatemala, Ecuador and Peru. As a consequence of this increasing popularity, several international language education agencies have flourished in recent years. Though education systems around the world invest enormous sums of money into language teaching the outcomes in terms of getting students to actually speak the language(s) they are learning outside the classroom are often unclear.

With the increasing prevalence of international business transactions, it is now important to have multiple languages at one's disposal. This is also evident in businesses outsourcing their departments to Eastern Europe.

Minority language education policy
The principal policy arguments in favor of promoting minority language education are the need for multilingual workforces, intellectual and cultural benefits and greater inclusion in global information society. Access to education in a minority language is also seen as a human right as granted by the European Convention on Human Rights and Fundamental Freedoms, the European Charter for Regional or Minority Languages and the UN Human Rights Committee. Bilingual Education has been implemented in many countries including the United States, in order to promote both the use and appreciation of the minority language, as well as the majority language concerned.

Materials and e-learning for minority language education
Suitable resources for teaching and learning minority languages can be difficult to find and access, which has led to calls for the increased development of materials for minority language teaching. The internet offers opportunities to access a wider range of texts, audios and videos. Language learning 2.0 (the use of web 2.0 tools for language education) offers opportunities for material development for lesser-taught languages and to bring together geographically dispersed teachers and learners.

Acronyms and abbreviations

 * ALL: Apprenticeship Language Learning
 * CALL: computer-assisted language learning
 * CLIL: content and language integrated learning
 * CELI: Certificato di Conoscenza della Lingua Italiana
 * CLL: community language learning
 * DELE: Diploma de Español como Lengua Extranjera
 * DELF: diplôme d'études en langue française
 * EFL: English as a foreign language
 * EAL/D: English as an additional language or dialect
 * EAP: English for academic purposes
 * ELL: English language learning
 * ELT: English language teaching
 * ESL: English as a second language
 * ESP: English for specific purposes English for specific purposes
 * FLL: foreign language learning
 * FLT: foreign language teaching
 * HLL: heritage language learning
 * IATEFL: International Association of Teachers of English as a Foreign Language International Association of Teachers of English as a Foreign Language
 * L1: first language, native language, mother tongue
 * L2: second language (or any additional language)
 * LDL: Lernen durch Lehren (German for learning by teaching)
 * LOTE: Languages Other Than English
 * MFL: modern foreign languages
 * SLA: second-language acquisition
 * TELL: technology-enhanced language learning
 * TEFL: teaching English as a foreign language
 * TEFLA: teaching English as a foreign language to adults
 * TESOL: teaching English to speakers of other languages
 * TEYL: teaching English to young learners
 * TPR: Total Physical Response
 * TPRS: Teaching Proficiency through Reading and Storytelling
 * UNIcert is a European language education system of many universities based on the Common European Framework of Reference for Languages.

title: Language education


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Wilkinson Microwave Anisotropy Probe

The Wilkinson Microwave Anisotropy Probe (WMAP), originally known as the Microwave Anisotropy Probe (MAP and Explorer 80), was a NASA spacecraft operating from 2001 to 2010 which measured temperature differences across the sky in the cosmic microwave background (CMB) – the radiant heat remaining from the Big Bang. Headed by Professor Charles L. Bennett of Johns Hopkins University, the mission was developed in a joint partnership between the NASA Goddard Space Flight Center and Princeton University. The WMAP spacecraft was launched on 30 June 2001 from Florida. The WMAP mission succeeded the COBE space mission and was the second medium-class (MIDEX) spacecraft in the NASA Explorer program. In 2003, MAP was renamed WMAP in honor of cosmologist David Todd Wilkinson (1935–2002), who had been a member of the mission's science team. After nine years of operations, WMAP was switched off in 2010, following the launch of the more advanced Planck spacecraft by European Space Agency (ESA) in 2009.

WMAP's measurements played a key role in establishing the current Standard Model of Cosmology: the Lambda-CDM model. The WMAP data are very well fit by a universe that is dominated by dark energy in the form of a cosmological constant. Other cosmological data are also consistent, and together tightly constrain the Model. In the Lambda-CDM model of the universe, the age of the universe is $13.772$ billion years. The WMAP mission's determination of the age of the universe is to better than 1% precision. The current expansion rate of the universe is (see Hubble constant) $69.32 km·s^{−1}·Mpc^{−1}$. The content of the universe currently consists of $4.628 %$ ordinary baryonic matter; $24.02 %$ cold dark matter (CDM) that neither emits nor absorbs light; and $71.35 %$ of dark energy in the form of a cosmological constant that accelerates the expansion of the universe. Less than 1% of the current content of the universe is in neutrinos, but WMAP's measurements have found, for the first time in 2008, that the data prefer the existence of a cosmic neutrino background with an effective number of neutrino species of $3.26$. The contents point to a Euclidean flat geometry, with curvature ($$\Omega_{k}$$) of $-0.003$. The WMAP measurements also support the cosmic inflation paradigm in several ways, including the flatness measurement.

The mission has won various awards: according to Science magazine, the WMAP was the Breakthrough of the Year for 2003. This mission's results papers were first and second in the "Super Hot Papers in Science Since 2003" list. Of the all-time most referenced papers in physics and astronomy in the INSPIRE-HEP database, only three have been published since 2000, and all three are WMAP publications. Bennett, Lyman A. Page Jr., and David N. Spergel, the latter both of Princeton University, shared the 2010 Shaw Prize in astronomy for their work on WMAP. Bennett and the WMAP science team were awarded the 2012 Gruber Prize in cosmology. The 2018 Breakthrough Prize in Fundamental Physics was awarded to Bennett, Gary Hinshaw, Norman Jarosik, Page, Spergel, and the WMAP science team.

In October 2010, the WMAP spacecraft was derelict in a heliocentric graveyard orbit after completing nine years of operations. All WMAP data are released to the public and have been subject to careful scrutiny. The final official data release was the nine-year release in 2012.

Some aspects of the data are statistically unusual for the Standard Model of Cosmology. For example, the largest angular-scale measurement, the quadrupole moment, is somewhat smaller than the Model would predict, but this discrepancy is not highly significant. A large cold spot and other features of the data are more statistically significant, and research continues into these.

Objectives

The WMAP objective was to measure the temperature differences in the Cosmic Microwave Background (CMB) radiation. The anisotropies then were used to measure the universe's geometry, content, and evolution; and to test the Big Bang model, and the cosmic inflation theory. For that, the mission created a full-sky map of the CMB, with a 13 arcminutes resolution via multi-frequency observation. The map required the fewest systematic errors, no correlated pixel noise, and accurate calibration, to ensure angular-scale accuracy greater than its resolution. The map contains 3,145,728 pixels, and uses the HEALPix scheme to pixelize the sphere. The telescope also measured the CMB's E-mode polarization, and foreground polarization. Its service life was 27 months; 3 to reach the position, and 2 years of observation.

Development
The MAP mission was proposed to NASA in 1995, selected for definition study in 1996, and approved for development in 1997.

The WMAP was preceded by two missions to observe the CMB; (i) the Soviet RELIKT-1 that reported the upper-limit measurements of CMB anisotropies, and (ii) the U.S. COBE satellite that first reported large-scale CMB fluctuations. The WMAP was 45 times more sensitive, with 33 times the angular resolution of its COBE satellite predecessor. The successor European Planck mission (operational 2009–2013) had a higher resolution and higher sensitivity than WMAP and observed in 9 frequency bands rather than WMAP's 5, allowing improved astrophysical foreground models.

Spacecraft

The telescope's primary reflecting mirrors are a pair of Gregorian 1.4 xx dishes (facing opposite directions), that focus the signal onto a pair of 0.9 xx secondary reflecting mirrors. They are shaped for optimal performance: a carbon fibre shell upon a Korex core, thinly-coated with aluminium and silicon oxide. The secondary reflectors transmit the signals to the corrugated feedhorns that sit on a focal plane array box beneath the primary reflectors.

The receivers are polarization-sensitive differential radiometers measuring the difference between two telescope beams. The signal is amplified with High-electron-mobility transistor (HEMT) low-noise amplifiers, built by the National Radio Astronomy Observatory (NRAO). There are 20 feeds, 10 in each direction, from which a radiometer collects a signal; the measure is the difference in the sky signal from opposite directions. The directional separation azimuth is 180°; the total angle is 141°. To improve subtraction of foreground signals from our Milky Way galaxy, the WMAP used five discrete radio frequency bands, from 23 GHz to 94 GHz.

The WMAP's base is a 5.0 m-diameter solar panel array that keeps the instruments in shadow during CMB observations, (by keeping the craft constantly angled at 22°, relative to the Sun). Upon the array sit a bottom deck (supporting the warm components) and a top deck. The telescope's cold components: the focal-plane array and the mirrors, are separated from the warm components with a cylindrical, 33 cm-long thermal isolation shell atop the deck.

Passive thermal radiators cool the WMAP to approximately 90 K; they are connected to the low-noise amplifiers. The telescope consumes 419 W of power. The available telescope heaters are emergency-survival heaters, and there is a transmitter heater, used to warm them when off. The WMAP spacecraft's temperature is monitored with platinum resistance thermometers.

The WMAP's calibration is effected with the CMB dipole and measurements of Jupiter; the beam patterns are measured against Jupiter. The telescope's data are relayed daily via a 2-GHz transponder providing a 667 kbit/s downlink to a 70 m Deep Space Network station. The spacecraft has two transponders, one a redundant backup; they are minimally active – about 40 minutes daily – to minimize radio frequency interference. The telescope's position is maintained, in its three axes, with three reaction wheels, gyroscopes, two star trackers and Sun sensors, and is steered with eight hydrazine thrusters.

Launch, trajectory, and orbit
The WMAP spacecraft arrived at the Kennedy Space Center on 20 April 2001. After being tested for two months, it was launched via Delta II 7425 launch vehicle on 30 June 2001. It began operating on its internal power five minutes before its launching, and continued so operating until the solar panel array deployed. The WMAP was activated and monitored while it cooled. On 2 July 2001, it began working, first with in-flight testing (from launching until 17 August 2001), then began constant, formal work. Afterwards, it effected three Earth-Moon phase loops, measuring its sidelobes, then flew by the Moon on 30 July 2001, en route to the Sun-Earth Lagrange point, arriving there on 1 October 2001, becoming the first CMB observation mission posted there.

Locating the spacecraft at Lagrange 2, (1500000 km from Earth) thermally stabilizes it and minimizes the contaminating solar, terrestrial, and lunar emissions registered. To view the entire sky, without looking to the Sun, the WMAP traces a path around in a Lissajous orbit ca. 1.0° to 10°, with a 6-month period. The telescope rotates once every 2 minutes 9 seconds (0.464 rpm) and precesses at the rate of 1 revolution per hour. WMAP measured the entire sky every six months, and completed its first, full-sky observation in April 2002.

Pseudo-Correlation Radiometer
The WMAP instrument consists of pseudo-correlation differential radiometers fed by two back-to-back 1.5 m primary Gregorian reflectors. This instrument uses five frequency bands from 22 GHz to 90 GHz to facilitate rejection of foreground signals from our own Galaxy. The WMAP instrument has a 3.5° x 3.5° field of view (FoV).

Foreground radiation subtraction
The WMAP observed in five frequencies, permitting the measurement and subtraction of foreground contamination (from the Milky Way and extra-galactic sources) of the CMB. The main emission mechanisms are synchrotron radiation and free-free emission (dominating the lower frequencies), and astrophysical dust emissions (dominating the higher frequencies). The spectral properties of these emissions contribute different amounts to the five frequencies, thus permitting their identification and subtraction.

Foreground contamination is removed in several ways. First, subtract extant emission maps from the WMAP's measurements; second, use the components' known spectral values to identify them; third, simultaneously fit the position and spectra data of the foreground emission, using extra data sets. Foreground contamination was reduced by using only the full-sky map portions with the least foreground contamination, while masking the remaining map portions.

One-year data release

On 11 February 2003, NASA published the first-year's worth of WMAP data. The latest calculated age and composition of the early universe were presented. In addition, an image of the early universe, that "contains such stunning detail, that it may be one of the most important scientific results of recent years" was presented. The newly released data surpass previous CMB measurements.

Based upon the Lambda-CDM model, the WMAP team produced cosmological parameters from the WMAP's first-year results. Three sets are given below; the first and second sets are WMAP data; the difference is the addition of spectral indices, predictions of some inflationary models. The third data set combines the WMAP constraints with those from other CMB experiments (ACBAR and CBI), and constraints from the 2dF Galaxy Redshift Survey and Lyman alpha forest measurements. There are degenerations among the parameters, the most significant is between $$n_s$$ and $$\tau$$; the errors given are at 68% confidence.

Using the best-fit data and theoretical models, the WMAP team determined the times of important universal events, including the redshift of reionization, $13.4$; the redshift of decoupling, $13.7$ (and the universe's age at decoupling, $km/Mpc·s$); and the redshift of matter/radiation equality, $72$. They determined the thickness of the surface of last scattering to be $70$ in redshift, or $71$. They determined the current density of baryons, $0.024$, and the ratio of baryons to photons, $0.023$. The WMAP's detection of an early reionization excluded warm dark matter.

The team also examined Milky Way emissions at the WMAP frequencies, producing a 208-point source catalogue.

Three-year data release

The three-year WMAP data were released on 17 March 2006. The data included temperature and polarization measurements of the CMB, which provided further confirmation of the standard flat Lambda-CDM model and new evidence in support of inflation.

The 3-year WMAP data alone shows that the universe must have dark matter. Results were computed both only using WMAP data, and also with a mix of parameter constraints from other instruments, including other CMB experiments (Arcminute Cosmology Bolometer Array Receiver (ACBAR), Cosmic Background Imager (CBI) and BOOMERANG), Sloan Digital Sky Survey (SDSS), the 2dF Galaxy Redshift Survey, the Supernova Legacy Survey and constraints on the Hubble constant from the Hubble Space Telescope.

[a] Optical depth to reionization improved due to polarization measurements. [b] <0.30 when combined with SDSS data. No indication of non-gaussianity.

Five-year data release

The five-year WMAP data were released on 28 February 2008. The data included new evidence for the cosmic neutrino background, evidence that it took over half billion years for the first stars to reionize the universe, and new constraints on cosmic inflation.

The improvement in the results came from both having an extra two years of measurements (the data set runs between midnight on 10 August 2001 to midnight of 9 August 2006), as well as using improved data processing techniques and a better characterization of the instrument, most notably of the beam shapes. They also make use of the 33-GHz observations for estimating cosmological parameters; previously only the 41-GHz and 61-GHz channels had been used.

Improved masks were used to remove foregrounds. Improvements to the spectra were in the 3rd acoustic peak, and the polarization spectra.

The measurements put constraints on the content of the universe at the time that the CMB was emitted; at the time 10% of the universe was made up of neutrinos, 12% of atoms, 15% of photons and 63% dark matter. The contribution of dark energy at the time was negligible. It also constrained the content of the present-day universe; 4.6% atoms, 23% dark matter and 72% dark energy.

The WMAP five-year data was combined with measurements from Type Ia supernova (SNe) and Baryon acoustic oscillations (BAO).

The elliptical shape of the WMAP skymap is the result of a Mollweide projection.

The data puts limits on the value of the tensor-to-scalar ratio, r <0.22 (95% certainty), which determines the level at which gravitational waves affect the polarization of the CMB, and also puts limits on the amount of primordial non-gaussianity. Improved constraints were put on the redshift of reionization, which is $0.022$, the redshift of decoupling, $0.14$ (as well as age of universe at decoupling, $0.14$) and the redshift of matter/radiation equality, $0.135$.

The extragalactic source catalogue was expanded to include 390 sources, and variability was detected in the emission from Mars and Saturn.

Seven-year data release

The seven-year WMAP data were released on 26 January 2010. As part of this release, claims for inconsistencies with the standard model were investigated. Most were shown not to be statistically significant, and likely due to a posteriori selection (where one sees a weird deviation, but fails to consider properly how hard one has been looking; a deviation with 1:1000 likelihood will typically be found if one tries one thousand times). For the deviations that do remain, there are no alternative cosmological ideas (for instance, there seem to be correlations with the ecliptic pole). It seems most likely these are due to other effects, with the report mentioning uncertainties in the precise beam shape and other possible small remaining instrumental and analysis issues.

The other confirmation of major significance is of the total amount of matter/energy in the universe in the form of dark energy – 72.8% (within 1.6%) as non 'particle' background, and dark matter – 22.7% (within 1.4%) of non baryonic (sub-atomic) 'particle' energy. This leaves matter, or baryonic particles (atoms) at only 4.56% (within 0.16%).

Nine-year data release

On 29 December 2012, the nine-year WMAP data and related images were released. $0.166$ billion-year-old temperature fluctuations and a temperature range of ± 200 microkelvins are shown in the image. In addition, the study found that 95% of the early universe is composed of dark matter and dark energy, the curvature of space is less than 0.4% of "flat" and the universe emerged from the cosmic Dark Ages "about 400 million years" after the Big Bang.

Main result

The main result of the mission is contained in the various oval maps of the CMB temperature differences. These oval images present the temperature distribution derived by the WMAP team from the observations by the telescope during the mission. Measured is the temperature obtained from a Planck's law interpretation of the microwave background. The oval map covers the whole sky. The results are a snapshot of the universe around 375,000 years after the Big Bang, which happened about 13.8 billion years ago. The microwave background is very homogeneous in temperature (the relative variations from the mean, which presently is still 2.7 kelvins, are only of the order of $0.2$). The temperature variations corresponding to the local directions are presented through different colors (the "red" directions are hotter, the "blue" directions cooler than the average).

Follow-on missions and future measurements

The original timeline for WMAP gave it two years of observations; these were completed by September 2003. Mission extensions were granted in 2002, 2004, 2006, and 2008 giving the spacecraft a total of 9 observing years, which ended August 2010 and in October 2010 the spacecraft was moved to a heliocentric "graveyard" orbit.

The Planck spacecraft also measured the CMB from 2009 to 2013 and aims to refine the measurements made by WMAP, both in total intensity and polarization. Various ground- and balloon-based instruments have also made CMB contributions, and others are being constructed to do so. Many are aimed at searching for the B-mode polarization expected from the simplest models of inflation, including The E and B Experiment (EBEX), Spider, BICEP and Keck Array (BICEP2), Keck, QUIET, Cosmology Large Angular Scale Surveyor (CLASS), South Pole Telescope (SPTpol) and others.

On 21 March 2013, the European-led research team behind the Planck spacecraft released the mission's all-sky map of the cosmic microwave background. The map suggests the universe is slightly older than previously thought. According to the map, subtle fluctuations in temperature were imprinted on the deep sky when the cosmos was about 370,000 years old. The imprint reflects ripples that arose as early, in the existence of the universe, as the first nonillionth (10−30) of a second. Apparently, these ripples gave rise to the present vast cosmic web of galaxy clusters and dark matter. Based on the 2013 data, the universe contains 4.9% ordinary matter, 26.8% dark matter and 68.3% dark energy. On 5 February 2015, new data was released by the Planck mission, according to which the age of the universe is 13.799 ± 0.021 billion years and the Hubble constant is 67.74 ± 0.46 (km/s)/Mpc.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Wilkinson Microwave Anisotropy Probe


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Friend of a friend

In sociology, a friend of a friend is a human contact that exists because of a mutual friend. Person C is a friend of a friend of person A when there is a person B that is a friend of both A and C. Thus the human relation "friend of a friend" is a compound relation among friends, similar to the uncle and aunt relations of kinship. Though friendship is a reciprocal relation, the relation of a friend of a friend may not be a friendship, though it holds potential for coalition building and dissemination of information.

Balance theory
The tendency of a friend of a friend to become a friend was noted by Fritz Heider, though he also considered the possibility that one of the friendships might breakdown, according to balance theory, which his view of human triangles is called. According to Heider, the friend of a friend contact could be stressful enough to undermine one or another of the friendships. Extending the study of social dynamics caused by such friend-of-a-friend tensions to social networks beyond triangles, Dorwin Cartwright and Frank Harary used signed graphs to indicate positive or negative sentiments between persons. In 1963 Anatol Rapoport summarized the theory: "The hypothesis implies roughly that attitudes of the group members will change is such a way that one's friends' friends will tend to become one's friends, ..." In September 1975 Dartmouth College offered a symposium on these dynamics.

Bo Anderson made an analysis of the friend-of-a-friend relationship in connection with his criticism of balance theory.
 * We have all encountered cases in which somebody has said, "You should meet so-and-so", only to find that we have little in common with that person, even though he or she was introduced to us by a mutual friend...In some friendships the persons value the exclusiveness of their relationship and are therefore not likely to let others into it. Friends differ from acquaintances in that they are not merely slots in a grid of social network relationships, but are valued for their personal, unique qualities. Hence, when I relate to a friend of a friend, I need to know something about the perceptions and exchanges that make up this friendship. My reaction to my friend's friend (or spouse) may even be unfavorable, although I may also well understand and sympathize with my friend’s affection for her, given his needs, perceptions, interests and so on.

Considering friendship between people to be a binary relation, the connection to a friend of a friend is a composition of the relationship with itself. Composed relations are used to describe kinship, so it may be natural to apply composition to friendship. One consequence is that frequently a person's friends have more friends than him (the friendship paradox), which accents the reach of the compound connection. But the fact that friendship is not automatically a transitive relation produces some social dynamics.

Other usage
In some social sciences, the phrase is used as a half-joking shorthand for the fact that much of the information on which people act comes from distant sources (as in "it happened to a friend of a friend of mine") and cannot be confirmed. It is probably best known from urban legend studies, where it was popularized by Jan Harold Brunvand.

The acronym FOAF was coined by Rodney Dale and used in his 1978 book The Tumour in the Whale: A Collection of Modern Myths.

Other languages

 * "Dúirt bean liom go ndúirt bean léi" (Irish proverb) – similar Irish language term literally meaning a woman told me that a woman told her that...
 * "L'homme qui a vu l'homme qui a vu l'ours" (French proverb) – similar French language proverb literally meaning The man who saw the man who saw the bear, in which the bear is never seen, only heard of.
 * "Un amigo me dijo que un amigo le dijo..." (Spanish proverb) – meaning literally A friend told me that a friend told him that...
 * "Jedna paní povídala..." (Czech proverb) – similar Czech language proverb literally meaning One lady said...
 * "Teman kepada teman saya..." Bahasa Indonesia; literally meaning friend of my friend.
 * "Babaturana babaturan urang..." Basa Sunda; literally meaning friend of my friend. There is another version of this phrase in Sundanese language, "Babaturan dulur urang", which means "friend of my relatives".
 * "카더라..." Korean; Gyeongsang dialect word literally meaning Who said that...
 * "Diz-se que..." or "Dizem que..." Portuguese; literally meaning It is said that... or They say that...
 * "Freundesfreund" German; literally meaning a friend's friend
 * "Umgani womgani wami" IsiZulu language; meaning my friend's friend
 * "朋友的朋友" or "我的friend的friend" Cantonese / Hong Kong English; meaning my friend's friend
 * "Itshomi ye tshomi yam" isiXhosa language; meaning "friend of a friend"

Web ontology language
In information science, an ontology describes categories, properties and relations between concepts, data and entities. The phrase "Friend Of A Friend", converted to the acronym FOAF, has been adopted in Web Ontology Language. It has been used in WebID for identifying correspondents, and to designate a secure authentication protocol.

title: Friend of a friend


text:
Arithmetical hierarchy

In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define them. Any set that receives a classification is called arithmetical. The arithmetical hierarchy was invented independently by Kleene (1943) and Mostowski (1946).

The arithmetical hierarchy is important in computability theory, effective descriptive set theory, and the study of formal theories such as Peano arithmetic.

The Tarski–Kuratowski algorithm provides an easy way to get an upper bound on the classifications assigned to a formula and the set it defines.

The hyperarithmetical hierarchy and the analytical hierarchy extend the arithmetical hierarchy to classify additional formulas and sets.

The arithmetical hierarchy of formulas
The arithmetical hierarchy assigns classifications to the formulas in the language of first-order arithmetic. The classifications are denoted $$\Sigma^0_n$$ and $$\Pi^0_n$$ for natural numbers n (including 0). The Greek letters here are lightface symbols, which indicates that the formulas do not contain set parameters.

If a formula $$\phi$$ is logically equivalent to a formula having no unbounded quantifiers, i.e. in which all quantifiers are bounded quantifiers then $$\phi$$ is assigned the classifications $$\Sigma^0_0$$ and $$\Pi^0_0$$.

The classifications $$\Sigma^0_n$$ and $$\Pi^0_n$$ are defined inductively for every natural number n using the following rules:
 * If $$\phi$$ is logically equivalent to a formula of the form $$\exists m_1 \exists m_2\cdots \exists m_k \psi$$, where $$\psi$$ is $$\Pi^0_n$$, then $$\phi$$ is assigned the classification $$\Sigma^0_{n+1}$$.
 * If $$\phi$$ is logically equivalent to a formula of the form $$\forall m_1 \forall m_2\cdots \forall m_k \psi$$, where $$\psi$$ is $$\Sigma^0_n$$, then $$\phi$$ is assigned the classification $$\Pi^0_{n+1}$$.

A $$\Sigma^0_n$$ formula is equivalent to a formula that begins with some existential quantifiers and alternates $$n-1$$ times between series of existential and universal quantifiers; while a $$\Pi^0_n$$ formula is equivalent to a formula that begins with some universal quantifiers and alternates analogously.

Because every first-order formula has a prenex normal form, every formula is assigned at least one classification. Because redundant quantifiers can be added to any formula, once a formula is assigned the classification $$\Sigma^0_n$$ or $$\Pi^0_n$$ it will be assigned the classifications $$\Sigma^0_m$$ and $$\Pi^0_m$$ for every m > n. The only relevant classification assigned to a formula is thus the one with the least n; all the other classifications can be determined from it.

Meaning of the notation
The following meanings can be attached to the notation for the arithmetical hierarchy on formulas.

The subscript $$n$$ in the symbols $$\Sigma^0_n$$ and $$\Pi^0_n$$ indicates the number of alternations of blocks of universal and existential first-order quantifiers that are used in a formula. Moreover, the outermost block is existential in $$\Sigma^0_n$$ formulas and universal in $$\Pi^0_n$$ formulas.

The superscript $$0$$ in the symbols $$\Sigma^0_n$$, $$\Pi^0_n$$, and $$\Delta^0_n$$ indicates the type of the objects being quantified over. Type 0 objects are natural numbers, and objects of type $$i+1$$ are functions that map the set of objects of type $$i$$ to the natural numbers. Quantification over higher type objects, such as functions from natural numbers to natural numbers, is described by a superscript greater than 0, as in the analytical hierarchy. The superscript 0 indicates quantifiers over numbers, the superscript 1 would indicate quantification over functions from numbers to numbers (type 1 objects), the superscript 2 would correspond to quantification over functions that take a type 1 object and return a number, and so on.

Examples

 * The $$\Sigma^0_1$$ sets of numbers are those definable by a formula of the form $$\exists n_1 \cdots \exists n_k \psi(n_1,\ldots,n_k,m)$$ where $$\psi$$ has only bounded quantifiers. These are exactly the recursively enumerable sets.
 * The set of natural numbers that are indices for Turing machines that compute total functions is $$\Pi^0_2$$. Intuitively, an index $$e$$ falls into this set if and only if for every $$m$$ "there is an $$s$$ such that the Turing machine with index $$e$$ halts on input $$m$$ after $$s$$ steps".  A complete proof would show that the property displayed in quotes in the previous sentence is definable in the language of Peano arithmetic by a $$\Sigma^0_1$$ formula.
 * Every $$\Sigma^0_1$$ subset of Baire space or Cantor space is an open set in the usual topology on the space. Moreover, for any such set there is a computable enumeration of Gödel numbers of basic open sets whose union is the original set. For this reason, $$\Sigma^0_1$$ sets are sometimes called effectively open.  Similarly, every $$\Pi^0_1$$ set is closed and the $$\Pi^0_1$$ sets are sometimes called effectively closed.
 * Every arithmetical subset of Cantor space or Baire space is a Borel set. The lightface Borel hierarchy extends the arithmetical hierarchy to include additional Borel sets.  For example, every $$\Pi^0_2$$ subset of Cantor or Baire space is a $G_\delta$ set, that is, a set that equals the intersection of countably many open sets.   Moreover, each of these open sets is $$\Sigma^0_1$$ and the list of Gödel numbers of these open sets has a computable enumeration.  If $$\phi(X,n,m)$$ is a $$\Sigma^0_0$$ formula with a free set variable $$X$$ and free number variables $$n,m$$ then the $$\Pi^0_2$$ set $$\{X \mid \forall n \exists m \phi(X,n,m)\}$$ is the intersection of the $$\Sigma^0_1$$ sets of the form $$\{ X \mid \exists m \phi(X,n,m)\}$$ as $$n$$ ranges over the set of natural numbers.
 * The $$\Sigma^0_0=\Pi^0_0=\Delta^0_0$$ formulas can be checked by going over all cases one by one, which is possible because all their quantifiers are bounded. The time for this is polynomial in their arguments (e.g. polynomial in $$n$$ for $$\varphi(n)$$); thus their corresponding decision problems are included in E (as $$n$$ is exponential in its number of bits). This no longer holds under alternative definitions of $$\Sigma^0_0=\Pi^0_0=\Delta^0_0$$ that allow the use of primitive recursive functions, as now the quantifiers may be bound by any primitive recursive function of the arguments.
 * The $$\Sigma^0_0=\Pi^0_0=\Delta^0_0$$ formulas under an alternative definition, that allows the use of primitive recursive functions with bounded quantifiers, correspond to sets of natural numbers of the form $$\{n: f(n) = 0\}$$ for a primitive recursive function $$f$$. This is because allowing bounded quantifier adds nothing to the definition: for a primitive recursive $$f$$, $$\forall k<n: f(k)=0$$ is the same as $$f(0)+f(1)+...f(n)=0$$, and $$\exists k<n: f(k)=0$$ is the same as  $$ f(0)\cdot f(1)\cdot\ldots\cdot f(n)=0$$; with course-of-values recursion each of these can be defined by a single primitive recursive function.

The arithmetical hierarchy of sets of natural numbers
A set X of natural numbers is defined by a formula φ in the language of Peano arithmetic (the first-order language with symbols "0" for zero, "S" for the successor function, "+" for addition, "×" for multiplication, and "=" for equality), if the elements of X are exactly the numbers that satisfy φ. That is, for all natural numbers n,
 * $$n \in X \Leftrightarrow \mathbb{N} \models \varphi(\underline n),$$

where $$\underline n$$ is the numeral in the language of arithmetic corresponding to $$n$$. A set is definable in first-order arithmetic if it is defined by some formula in the language of Peano arithmetic.

Each set X of natural numbers that is definable in first-order arithmetic is assigned classifications of the form $$\Sigma^0_n$$, $$\Pi^0_n$$, and $$\Delta^0_n$$, where $$n$$ is a natural number, as follows. If X is definable by a $$\Sigma^0_n$$ formula then X is assigned the classification $$\Sigma^0_n$$. If X is definable by a $$\Pi^0_n$$ formula then X is assigned the classification $$\Pi^0_n$$. If X is both $$\Sigma^0_n$$ and $$\Pi^0_n$$ then $$X$$ is assigned the additional classification $$\Delta^0_n$$.

Note that it rarely makes sense to speak of $$\Delta^0_n$$ formulas; the first quantifier of a formula is either existential or universal. So a $$\Delta^0_n$$ set is not necessarily defined by a $$\Delta^0_n$$ formula in the sense of a formula that is both $$\Sigma^0_n$$ and $$\Pi^0_n$$; rather, there are both $$\Sigma^0_n$$ and  $$\Pi^0_n$$ formulas that define the set. For example, the set of odd natural numbers $$n$$ is definable by either $$\forall k(n\neq 2\times k)$$ or $$\exists k(n=2\times k+1)$$.

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of the set of natural numbers. Instead of formulas with one free variable, formulas with k free first-order variables are used to define the arithmetical hierarchy on sets of k-tuples of natural numbers. These are in fact related by the use of a pairing function.

Relativized arithmetical hierarchies
Just as we can define what it means for a set X to be recursive relative to another set Y by allowing the computation defining X to consult Y as an oracle we can extend this notion to the whole arithmetic hierarchy and define what it means for X to be $$\Sigma^0_n$$, $$\Delta^0_n$$ or $$\Pi^0_n$$ in Y, denoted respectively $$\Sigma^{0,Y}_n$$, $$\Delta^{0,Y}_n$$ and $$\Pi^{0,Y}_n$$. To do so, fix a set of natural numbers Y and add a predicate for membership of Y to the language of Peano arithmetic. We then say that X is in $$\Sigma^{0,Y}_n$$ if it is defined by a $$\Sigma^0_n$$ formula in this expanded language. In other words, X is $$\Sigma^{0,Y}_n$$ if it is defined by a $$\Sigma^{0}_n$$ formula allowed to ask questions about membership of Y. Alternatively one can view the $$\Sigma^{0,Y}_n$$ sets as those sets that can be built starting with sets recursive in Y and alternately taking unions and intersections of these sets up to n times.

For example, let Y be a set of natural numbers. Let X be the set of numbers divisible by an element of Y. Then X is defined by the formula $$\phi(n)=\exists m \exists t (Y(m)\land m\times t = n)$$ so X is in $$\Sigma^{0,Y}_1$$ (actually it is in $$\Delta^{0,Y}_0$$ as well, since we could bound both quantifiers by n).

Arithmetic reducibility and degrees
Arithmetical reducibility is an intermediate notion between Turing reducibility and hyperarithmetic reducibility.

A set is arithmetical (also arithmetic and arithmetically definable) if it is defined by some formula in the language of Peano arithmetic. Equivalently X is arithmetical if X is $$\Sigma^0_n$$ or $$\Pi^0_n$$ for some natural number n. A set X is arithmetical in a set Y, denoted $$X \leq_A Y$$, if X is definable as some formula in the language of Peano arithmetic extended by a predicate for membership of Y. Equivalently, X is arithmetical in Y if X is in $$\Sigma^{0,Y}_n$$ or $$\Pi^{0,Y}_n$$ for some natural number n. A synonym for $$X \leq_A Y$$ is: X is arithmetically reducible to Y.

The relation $$X \leq_A Y$$ is reflexive and transitive, and thus the relation $$\equiv_A$$ defined by the rule

 * $$ X \equiv_A Y \iff X \leq_A Y \land Y \leq_A X$$

is an equivalence relation. The equivalence classes of this relation are called the arithmetic degrees; they are partially ordered under $$\leq_A$$.

The arithmetical hierarchy of subsets of Cantor and Baire space
The Cantor space, denoted $$2^{\omega}$$, is the set of all infinite sequences of 0s and 1s; the Baire space, denoted $$\omega^{\omega}$$ or $$\mathcal{N}$$, is the set of all infinite sequences of natural numbers. Note that elements of the Cantor space can be identified with sets of natural numbers and elements of the Baire space with functions from natural numbers to natural numbers.

The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers can naturally be viewed as quantifying over Cantor space. A subset of Cantor space is assigned the classification $$\Sigma^0_n$$ if it is definable by a $$\Sigma^0_n$$ formula. The set is assigned the classification $$\Pi^0_n$$ if it is definable by a $$\Pi^0_n$$ formula. If the set is both $$\Sigma^0_n$$ and $$\Pi^0_n$$ then it is given the additional classification $$\Delta^0_n$$. For example, let $$O\subseteq 2^{\omega}$$ be the set of all infinite binary strings that aren't all 0 (or equivalently the set of all non-empty sets of natural numbers). As $$O=\{ X\in 2^\omega | \exists n (X(n)=1) \} $$ we see that $$O$$ is defined by a $$\Sigma^0_1$$ formula and hence is a $$\Sigma^0_1$$ set.

Note that while both the elements of the Cantor space (regarded as sets of natural numbers) and subsets of the Cantor space are classified in arithmetic hierarchies, these are not the same hierarchy. In fact the relationship between the two hierarchies is interesting and non-trivial. For instance the $$\Pi^0_n$$ elements of the Cantor space are not (in general) the same as the elements $$X$$ of the Cantor space so that $$\{X\}$$ is a $$\Pi^0_n$$ subset of the Cantor space. However, many interesting results relate the two hierarchies.

There are two ways that a subset of Baire space can be classified in the arithmetical hierarchy.
 * A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from $$\omega$$ to $$\omega$$ to the characteristic function of its graph.  A subset of Baire space is given the classification $$\Sigma^0_n$$, $$\Pi^0_n$$, or $$\Delta^0_n$$ if and only if the corresponding subset of Cantor space has the same classification.
 * An equivalent definition of the arithmetical hierarchy on Baire space is given by defining the arithmetical hierarchy of formulas using a functional version of second-order arithmetic; then the arithmetical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space. This alternate definition gives exactly the same classifications as the first definition.

A parallel definition is used to define the arithmetical hierarchy on finite Cartesian powers of Baire space or Cantor space, using formulas with several free variables. The arithmetical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor space and Baire space because they fit with the language of ordinary second-order arithmetic.

Note that we can also define the arithmetic hierarchy of subsets of the Cantor and Baire spaces relative to some set of natural numbers. In fact boldface $$\mathbf{\Sigma}^0_n$$ is just the union of $$\Sigma^{0,Y}_n$$ for all sets of natural numbers Y. Note that the boldface hierarchy is just the standard hierarchy of Borel sets.

Extensions and variations
It is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol for each primitive recursive function. This variation slightly changes the classification of $$\Sigma^0_0=\Pi^0_0=\Delta^0_0$$, since using primitive recursive functions in first-order Peano arithmetic requires, in general, an unbounded existential quantifier, and thus some sets that are in $$\Sigma^0_0$$ by this definition are strictly in $$\Sigma^0_1$$ by the definition given in the beginning of this article. The class $$\Sigma^0_1$$ and thus all higher classes in the hierarchy remain unaffected.

A more semantic variation of the hierarchy can be defined on all finitary relations on the natural numbers; the following definition is used. Every computable relation is defined to be $$\Sigma^0_0=\Pi^0_0=\Delta^0_0$$. The classifications $$\Sigma^0_n$$ and $$\Pi^0_n$$ are defined inductively with the following rules. This variation slightly changes the classification of some sets. In particular, $$\Sigma^0_0=\Pi^0_0=\Delta^0_0$$, as a class of sets (definable by the relations in the class), is identical to $$\Delta^0_1$$ as the latter was formerly defined. It can be extended to cover finitary relations on the natural numbers, Baire space, and Cantor space.
 * If the relation $$R(n_1,\ldots,n_l,m_1,\ldots, m_k)\,$$ is $$\Sigma^0_n$$ then the relation $$S(n_1,\ldots, n_l) = \forall m_1\cdots \forall m_k R(n_1,\ldots,n_l,m_1,\ldots,m_k)$$ is defined to be $$\Pi^0_{n+1}$$
 * If the relation $$R(n_1,\ldots,n_l,m_1,\ldots, m_k)\,$$ is $$\Pi^0_n$$ then the relation $$S(n_1,\ldots,n_l) = \exists m_1\cdots \exists m_k R(n_1,\ldots,n_l,m_1,\ldots,m_k)$$ is defined to be $$\Sigma^0_{n+1}$$

Properties
The following properties hold for the arithmetical hierarchy of sets of natural numbers and the arithmetical hierarchy of subsets of Cantor or Baire space.

 * The collections $$\Pi^0_n$$ and $$\Sigma^0_n$$ are closed under finite unions and finite intersections of their respective elements.
 * A set is $$\Sigma^0_n$$ if and only if its complement is $$\Pi^0_n$$. A set is $$\Delta^0_n$$ if and only if the set is both $$\Sigma^0_n$$ and $$\Pi^0_n$$, in which case its complement will also be $$\Delta^0_n$$.
 * The inclusions $$\Pi^0_n \subsetneq \Pi^0_{n+1}$$ and $$\Sigma^0_n \subsetneq \Sigma^0_{n+1}$$ hold for all $$n$$. Thus the hierarchy does not collapse. This is a direct consequence of Post's theorem.
 * The inclusions $$\Delta^0_n \subsetneq \Pi^0_n$$, $$\Delta^0_n \subsetneq \Sigma^0_n$$ and $$\Sigma^0_n \cup \Pi^0_n \subsetneq \Delta^0_{n+1}$$ hold for $$n \geq 1$$.
 * For example, for a universal Turing machine T, the set of pairs (n,m) such that T halts on n but not on m, is in $$\Delta^0_2$$ (being computable with an oracle to the halting problem) but not in $$\Sigma^0_1 \cup \Pi^0_1$$.
 * $$\Sigma^0_0 = \Pi^0_0 = \Delta^0_0 = \Sigma^0_0 \cup \Pi^0_0 \subseteq \Delta^0_1$$. The inclusion is strict by the definition given in this article, but an identity with $$\Delta^0_1$$ holds under one of the variations of the definition given above.

Computable sets
If S is a Turing computable set, then both S and its complement are recursively enumerable (if T is a Turing machine giving 1 for inputs in S and 0 otherwise, we may build a Turing machine halting only on the former, and another halting only on the latter).

By Post's theorem, both S and its complement are in $$\Sigma^0_1$$. This means that S is both in $$\Sigma^0_1$$ and in $$\Pi^0_1$$, and hence it is in $$\Delta^0_1$$.

Similarly, for every set S in $$\Delta^0_1$$, both S and its complement are in $$\Sigma^0_1$$ and are therefore (by Post's theorem) recursively enumerable by some Turing machines T1 and T2, respectively. For every number n, exactly one of these halts. We may therefore construct a Turing machine T that alternates between T1 and T2, halting and returning 1 when the former halts or halting and returning 0 when the latter halts. Thus T halts on every n and returns whether it is in S; so S is computable.

Summary of main results
The Turing computable sets of natural numbers are exactly the sets at level $$\Delta^0_1$$ of the arithmetical hierarchy. The recursively enumerable sets are exactly the sets at level $$\Sigma^0_1$$.

No oracle machine is capable of solving its own halting problem (a variation of Turing's proof applies). The halting problem for a $$\Delta^{0,Y}_n$$ oracle in fact sits in $$\Sigma^{0,Y}_{n+1}$$.

Post's theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees. In particular, it establishes the following facts for all n ≥ 1:
 * The set $$\emptyset^{(n)}$$ (the nth Turing jump of the empty set) is many-one complete in $$\Sigma^0_n$$.
 * The set $$\mathbb{N} \setminus \emptyset^{(n)}$$ is many-one complete in $$\Pi^0_n$$.
 * The set $$\emptyset^{(n-1)}$$ is Turing complete in $$\Delta^0_n$$.

The polynomial hierarchy is a "feasible resource-bounded" version of the arithmetical hierarchy in which polynomial length bounds are placed on the numbers involved (or, equivalently, polynomial time bounds are placed on the Turing machines involved). It gives a finer classification of some sets of natural numbers that are at level $$\Delta^0_1$$ of the arithmetical hierarchy.

title: Arithmetical hierarchy


text:
Central

Central is an adjective usually referring to being in the center of some place or (mathematical) object.

Central may also refer to:

Directions and generalised locations

 * Central Africa, a region in the centre of Africa continent, also known as Middle Africa
 * Central America, a region in the centre of America continent
 * Central Asia, a region in the centre of Eurasian continent
 * Central Australia, a region of the Australian continent
 * Central Belt, an area in the centre of Scotland
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Countries

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Jamaica

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United Kingdom

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Jamaica

 * Kingston Central (Jamaica Parliament constituency)
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United Kingdom

 * Aberdeen Central (UK Parliament constituency)
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 * Stoke-on-Trent Central (UK Parliament constituency)
 * Sunderland Central (UK Parliament constituency)
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Other countries

 * Central Italy (European Parliament constituency)
 * Central constituency (disambiguation), Russoa

Companies

 * Central (home improvement store), a home improvement store chain located in Nova Scotia, Canada
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 * Central Trains, a former UK train operating company owned by National Express Group
 * ITV Central (formerly Central Independent Television), the ITV company formed from ATV
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Education

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Railways

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 * Central line (London Underground), on the London Underground
 * Central Railway (India)
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 * Central Córdoba de Santiago del Estero, a football (soccer) club in Argentina
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 * Rosario Central, an important football club in Argentina
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Other uses

 * Central (Department Store), a retail chain of the Pantaloon Retail India
 * Central (TV channel), a defunct TV channel in Singapore
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 * Central School of Speech and Drama, London
 * Central Time Zone, a time zone in the Americas
 * The Central, a commercial and residential building on Eu Tong Sen Street, Singapore
 * Central, a fictional in the American television sitcom My Name Is Earl
 * Cochrane Central Register of Controlled Trials (CENTRAL), see

title: Central