text:
Province of Belluno

The province of Belluno (provincia di Belluno; provinz Belluno; provinzia de Belum) is a province in the Veneto region of Italy. Its capital is the city of Belluno.

It has an area of 3610 km2 and a population of about 205,000.

Geography
Situated in the Alps, the province of Belluno consists almost entirely of mountainous terrain. It encompasses the natural and historical regions of Cadore, Feltrino, Alpago, Val di Zoldo, Agordino, Comelico and Ampezzano. The province is home to the Dolomites, including Tofane, Marmolada, Tre Cime di Lavaredo, and Antelao. For much of its course, the river Piave, runs through Belluno, as do its affluents the Boite and the Cordevole.

The southern part is called Valbelluna, the widest and most populous valley of the province, which is bordered by the Venetian Prealps. The National Park of Belluno Dolomites is located in the province.

Climate
The province of Belluno's climate is among the most severe in the Alps. It is mostly influenced by the continental, provided by the Dolomites and it is very similar to the eastern Tyrol's climate, or even more severe in the Prealps of the southern side.

The province may be divided in the following climatic zones:

 * The lowest valleys, at approximately 700 m, that usually have an almost mild climate, at the boundary line between the Oceanic climate Cfb and the humid continental climate Dfb: the coldest month's average is usually between 0 °C and −3 °C. The summers are hot but not as hot as the valleys of the rivers Po and Adige. The hottest month's average temperature is between 18 °C and 21 °C. This area, as most of the oriental Alps, is very wet. Belluno's precipitation average is above 1300 mm per year, snow is usual during all the winter months, and may occur even in March or November. Some years the winter may be milder due to the humidity, with averages around the freezing point, and big snowfalls with frequent snowbound. Main cities in this area: Belluno, Feltre, Seren del Grappa.
 * The midland's valleys, between 700 m and 1500 m (or until 1300 meters in the Prealps area), which have a typical humid continental climate Dfb. This climate is more severe in the extreme northern or southwestern regions of the province, while the central region (near Arabba or Cortina d’Ampezzo) is usually milder. Winter's average temperature is between −7 °C and −5 °C in the North and Southwest during January, and between −5 °C and −3 °C around the central region. The coldest town, Santo Stefano di Cadore, at an elevation of only 900 meters, has temperature averages in January between −7 °C and −6 °C. Other towns of relatively low elevations have really low averages, which would be found only above 1700 meters in other Italians provinces, as well as in any French or western Swiss provinces. The snow season depends on the altitude. Between 700 and 1200 meters, from early November until the middle of March, between 1200 and 1500 (or 1600 in Arabba), from late October until late April. The summer is usually wet and mild, but hotter than areas at the same altitude in the occidental valleys. In the lower lands (around 900 meters) temperature average around 16 °C or 17 °C. In higher lands (around 1400 meters) average around 14 °C. Rain is usual from March until November. Some towns and villages with this climate: Santo Stefano di Cadore (severe), Cortina d'Ampezzo (mild – sometimes Cfb), Arabba (mild if considered the altitude), Auronzo di Cadore (severe), Danta di Cadore, Sappada.
 * The valleys around 1500 to 1900 meters, with cold continental climate Dfc. It is the wettest climate of the province. Some areas on the southern side of the province may receive more than 2000 mm of precipitation per year, especially near the lakes, which can not make the climate mild in this region, as in other alpine regions. Winters are very cold and snowy, with temperature averages between −9 °C and −5 °C. The snow season usually starts in late September and lasts until late May Snows are not frequents, but can occur even during the summer, in early September or late August. The summer is rainy, sometimes foggy, and cool. Temperatures average in July between 10 °C and 12 °C. This area has a very few numbers of sunny days per year, and the average sunshine hours per day is also quite low, around 1 hour or less in January and around 4 or 5 hours in June and July. Due to the severity of the climate, only a few villages are found in this area, usually fractions of other towns. One of the most inhabited places in this area is Misurina, a village near the lake with the same name.
 * The high lands, above 1900 meters, with mountain's tundra climate ET. There are no inhabited places in this area. All the months have temperature averages below 10 °C. Above 2000 meters the precipitation rate starts to decrease. The higher places are drier, and sunnier. The winter temperature averages in the mountain areas vary according to the altitude, and are always milder than the valley's temperatures at the same quote. In January, the average temperature in mountains areas are around −3 °C at 1500 m (while they can reach −6 °C in valleys), −6 °C at 2000 m, −8 °C at 2500 m and −10 °C at 3000 meters. Summers are cold above 2000 meters. The average temperature in July is around 12 °C at 1500 m, 8 °C at 2000 m, 6 °C at 2500 and 3 °C at 3000 meters. In Belluno province, the perpetual frost climate EF is not found, because the highest point, the Marmolada, at a few more than 3200 meters, has summer's average temperature of approximately 20 °C.

History
The oldest archaeological find in the province is that of Lagole di Calalzo, in Cadore, belonging to a c. 5th century BC sanctuary dedicated to a health god of Paleoveneti. A larger site, a necropolis, has been excavated in the commune of Mel. Findings from these sites are now housed in the Museums of Pieve di Cadore and Mel.

From the 10th to 12th century the area that became Belluno was part of the March of Verona, a march of the Holy Roman Empire. During the 14th century much of the area came under the control of the Carraresi lords of Padua. In 1404 the Carraresi territories, including Feltre and Belluno, were conquered by the Republic of Venice, becoming part of the Venetian Domini di Terraferma. Cadore in the north also became part of the Domini di Terraferma following the Venetian conquest of Friuli in 1420. In 1511 Maximilian I, Holy Roman Emperor conquered the town of Cortina d'Ampezzo, detaching it from Cadore and incorporating it into the County of Tyrol.

Venetian rule lasted until 1797 when Venice was conquered by the French First Republic during the Italian campaigns of the French Revolutionary Wars. Rather than being annexed by France or its satellites, the Venetian territory east of Lake Garda, including Belluno, was ceded to the Habsburg monarchy (Austria) in the Treaty of Campo Formio, thereby becoming the Venetian Province. This state of affairs only lasted until 1805 however, as Austria (the Austrian Empire since 1804) ceded the Venetian Province to the Napoleonic Kingdom of Italy in the Treaty of Pressburg. Within the Kingdom of Italy, modern Belluno roughly corresponded to the Piave department department.

After the Napoleonic Wars the former Venetian Province was returned to Austria by the 1815 Congress of Vienna, becoming part of the newly formed Kingdom of Lombardy–Venetia. Under Austrian rule Lombardy–Venetia was organised into provinces roughly corresponding to the Napoleonic departments; the Piave department was succeeded by the province of Belluno, which broadly coincides with the modern province.

During the unification of Italy Austria lost Lombardy–Venetia to what became the Kingdom of Italy (the western part (i.e. Lombardy) in 1859, the eastern part (i.e. Venetia and Mantua) including Belluno in 1866).

In 1919, following World War I, Italy obtained what had been the southern part of the County of Tyrol from Austria-Hungary: Trentino and South Tyrol. In 1923 Cortina_d'Ampezzo, Colle Santa Lucia and Livinallongo del Col di Lana were detached from this territory and added to Belluno.

In 1943, when the Italian government signed an armistice with the Allies, the province was occupied by Nazi Germany, which reorganised it (along with the region of Trentino-Südtirol) as the Operation Zone of the Alpine Foothills and put it under the administration of Gauleiter Franz Hofer. The region was de facto annexed to the German Reich until the end of the war. This status ended along with the Nazi regime and Italian rule was restored in 1945.

Economy
Until recently, the province's economy was based on poor mountain agriculture, now less important, with the exception of the cultivation of beans in the Lamon highlands and the production of Piave cheese in the Dolomites valleys. Today, Belluno has one of the most important industrial sectors of northern Italy, the production of eyeglasses (Luxottica has its headquarters in Agordo, for example). Also important are the manufacturing sectors of home appliances (Zanussi and others) and bathroom fixtures (Ceramica Dolomite, IdealStandard).

One of the most important factors of the Bellunese economy is tourism. Cortina d'Ampezzo, Alleghe, Arabba, Sappada, and other locations are renowned in Italy and abroad.

Linguistic minorities
In the province, language minorities are present and recognized by law:
 * a Ladin minority, recognized in 42 municipalities, is present in the regions of Ladinia (Colle Santa Lucia-Col, Livinallongo del Col di Lana-Fodom and Cortina d'Ampezzo-Anpezo), Cadore, Agordino and Zoldo.
 * a German-speaking minority was present in the municipalities of Farra d'Alpago and Tambre (now extinct).

Inhabitants with foreign citizenship
The top ten countries of origin of the inhabitants of Belluno with foreign citizenship at December 31, 2018 were:
 * 🇷🇴 Romania 2041
 * 🇺🇦 Ukraine 1397
 * 🇲🇦 Morocco 1383
 * 🇦🇱 Albania 954
 * China 922
 * 🇲🇰 North Macedonia 713
 * 🇲🇩 Moldova 597
 * Kosovo 546
 * Croatia 328
 * 🇧🇷 Brazil 231

Comuni
There are 60 comuni (: comune) in the province. As of December 31, 2018, the main comuni (municipalities) by population are:

title: Province of Belluno


text:
Talk:Monday demonstrations in GDR

Censorship
Why did the East German state-controlled media report the demonstrations? If they hadn't, people in other cities would only have heard about them through rumour, making them much more unsure about emulating the protests and much more unsure about the potential consequences. 86.136.88.187 04:37, 15 April 2006 (UTC)
 * West Germany had radio and TV transmissions sent into East Germany, so that most people except the Southeastern rim had "West TV" reception. Also, gossip (even via telephone) should not be underestimated. Finally, Leipzig was in the centre of the most populated part of East Germany, making it a sufficient pressure on the SED government. FlammingoParliament 21:50, 18 January 2007 (UTC)

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

title: Talk:Monday demonstrations in GDR


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Museum of Chinese in America

The Museum of Chinese in America (abbreviated MOCA) is a museum in New York City which exhibits Chinese American history. It is a nonprofit 501(c)(3) education and cultural institution that presents the living history, heritage, culture, and diverse experiences of Chinese Americans through exhibitions, educational services and public programs. Much of its collection was damaged or destroyed in a fire in January 2020. After being closed for more than a year following the fire, the museum reopened to the public on July 15, 2021.

History
Founded in 1980 in Manhattan's Chinatown, the museum began as the New York Chinatown History Project by historian John Kuo Wei Tchen and community resident and activist Charles Lai to promote understanding of the Chinese American experience and to address the concern that "the memories and experiences of aging older generations would perish without oral history, photo documentation, research and collecting efforts." From 1997 to 2006, Fay Chew Matsuda served as director of the museum.

In 2005, the museum received part of a $20 million grant from the Carnegie Corporation, made possible through a donation by then-New York City mayor Michael Bloomberg.

The museum moved to a new site at 215 Centre Street in 2009. The space was designed by architect Maya Lin and was six times as large as the old site. The permanent exhibition, With a Single Step, was designed by Matter Practice. In May 2011, Herb Tam became curator and director of exhibitions.

In 2019, the museum relaunched their gift store with a new partner, the Asian American retail brand Pearl River Mart. Called MOCA Shop by Pearl River, the store is a "curated collection of items that hold great meaning in Chinese American culture."

In January 2020, a fire damaged the building at 70 Mulberry Street, where the museum's collection was held, with about 85,000 items potentially affected by water damage. While it was initially believed that nearly all of them might have been lost, a substantial part was determined to be "very much salvageable" several days after the incident. Around 35,000 items had been digitized and backed up before the fire. Much of the collection was restored by disaster-relief specialists who worked to prevent mold growth and preserve structure.

The museum bought its main building at 215 Centre Street in 2024 for $51.1 million.

Exhibitions
The core exhibition With a Single Step: Stories in the Making of America discusses over 160 years of Chinese American history and is augmented annually with two to four rotating exhibitions on thematic, historic, and artistic subjects.

The museum in January 2015 presented Waves of Identity: 35 Years of Archiving. The title of the exhibit was inspired by a Chinese proverb, "Each wave of the Yangtze River pushes at the wave ahead."

Current

 * With a Single Step: Stories in the Making of America (September 1, 2009 – December 31, 2023)

Former

 * Sour, Sweet, Bitter, Spicy: Stories of Chinese Food and Identity in America (March 17, 2023 – November 12, 2023)
 * Responses: Asian American Voices Resisting the Tides of Racism (July 15, 2021 – October 29, 2023)
 * an unlikely photojournalist: Emile Bocian in Chinatown (December 14, 2020 – December 31, 2021)
 * Gathering: Collecting and Documenting Chinese American History (October 17, 2019 – March 22, 2020)
 * The Chinese Helped Build the Railroad – The Railroad Helped Build America (October 17, 2019 – March 22, 2020)
 * Interior Lives: Photographs of Chinese Americans in the 1980s by Bud Glick (October 18, 2018 – March 24, 2019)
 * FOLD: Golden Venture Paper Sculptures (October 5, 2017 - March 25, 2018)
 * Sour, Sweet, Bitter, Spicy: Stories of Chinese Food and Identity in America (October 6, 2016 – September 10, 2017)
 * Stage Design by Ming Cho Lee (April 28, 2016 – September 11, 2016)
 * Chinese Style: Rediscovering the Architecture of Poy Gum Lee, 1923–1968 (September 24, 2015 – March 27, 2016)
 * SUB URBANISMS: Casino Urbanization, Chinatowns, and the Contested American Landscape (September 24, 2015 – March 27, 2016)
 * Yu Lik Wai: It's a Bright Guilty World (October 8, 2015 – August 11, 2016)
 * Water to Paper, Paint to Sky: The Art of Tyrus Wong (March 26, 2015 – September 13, 2015)
 * Memory Prints: The Story World of Phillip Chen (September 25, 2014 – January 3, 2015)
 * Waves of Identity: 35 Years of Archiving (September 25, 2014 – January 3, 2015)
 * Oil and Water: Reinterpreting Ink (April 24, 2014 – September 14, 2014)
 * The Lee Family of New York Chinatown Since 1888 (October 23, 2013 – June 6, 2014)
 * A Floating Population (December 13, 2013 – April 13, 2014)
 * Portraits of New York Chinatown (December 13, 2013 – April 13, 2014)
 * Front Row: Chinese American Designers (April 26, 2013 – December 1, 2013)
 * Shanghai Glamour: New Women 1910s–40s (April 26, 2013 – November 3, 2013)
 * Marvels and Monsters: Unmasking Asian Images in U.S. Comics, 1942–1986 and Alt. Comics: Asian American Artists Reinvent the Comic (September 27, 2012 – February 24, 2013)
 * America through a Chinese Lens (April 26, 2012 – September 9, 2012)
 * Lee Mingwei: The Travelers and The Quartet Project (October 20, 2011 – March 26, 2012)
 * Unearthing (May 5, 2011 – September 19, 2011)
 * Chinese Puzzles (November 6, 2010 – April 9, 2011)
 * Chinatown POV: Reflections on September 11 (September 11, 2010 – November 29, 2010)
 * Both Here and There: Yale-China and a Century of Transformative Encounters (September 2, 2010 – October 11, 2010)
 * Here & Now: Chapter III Towards Transculturalism (February 11, 2010 – March 28, 2010)
 * Here & Now: Chapter II Crossing Boundaries (November 19, 2009 – January 4, 2010)
 * Here & Now: Chinese Artists in New York Chapter 1 (September 22, 2009 – November 2, 2009)

Collections
As of early 2020, MOCA's Collections and Research Center contained more than 85,000 artifacts, photos, memorabilia, documents, oral histories, and art work. The collection covers a timespan of 160 years and includes e.g. historical Chinese restaurant menus, boat tickets, family photographs, and wedding dresses.

Research Center
The museum's previous gallery space on 70 Mulberry Street is used as an archival center and serves as a research center. The Research Center contains applications, such as Web-based versions of gallery exhibitions and an interactive timeline of Chinese American history. The center also provides resources on topics such as immigration and diversity. The Research Center was damaged by fire in 2020.

Special collections
Special collections include:

Legacy Awards Gala
At MOCA's 2015 Legacy Awards Gala, the museum honored several people and organizations for their roles in Chinese-American culture. Honorees were the C.V. Starr Scholars, Hong Kong-born American actor Nancy Kwan, and Kohn Pedersen Fox Associates founding design partner William C. Louie.

2014 honorees included: Tyrus Wong, Theodore T. Wang, and Victor and William Fung Foundation Ltd.

2013 honorees included:
 * Michael Bloomberg
 * Pei-Yuan Chia
 * Ming Tsai
 * Wang Yannan

2012 honorees included:
 * Angelica Cheung
 * Silas Chou
 * Calvin Tsao
 * Humberto Leon and Carol Lim

2011 honorees included:
 * Oscar L. Tang
 * David Liu
 * Dominic Ng
 * Pichet Ong

2010 honorees included:
 * The Chao Family
 * Maurice R. Greenberg & The Starr Foundation
 * HSBC Bank USA
 * Anita Lo
 * Bill and Judith Davidson Moyers
 * Ben Wang
 * Major General John L. Fugh *(posthumous award)

2009 honorees included: Previous awardees:
 * Ronnie C. Chan
 * Charles B. Wang Community Health Center
 * Anla Cheng
 * Mark E. Kingdon
 * Lucy Liu
 * Wan-go Weng
 * Jerry Yang
 * William Y. Chang (2000)

title: Museum of Chinese in America


text:
Old Irish

Old Irish, also called Old Gaelic (Goídelc, Ogham script : ᚌᚑᚔᚇᚓᚂᚉ; Sean-Ghaeilge; Seann-Ghàidhlig; Shenn Yernish or Shenn Ghaelg), is the oldest form of the Goidelic/Gaelic language for which there are extensive written texts. It was used from c. 600 to c.  900. The main contemporary texts are dated c. 700–850; by 900 the language had already transitioned into early Middle Irish. Some Old Irish texts date from the 10th century, although these are presumably copies of texts written at an earlier time. Old Irish is thus forebear to Modern Irish, Manx and Scottish Gaelic.

Old Irish is known for having a particularly complex system of morphology and especially of allomorphy (more or less unpredictable variations in stems and suffixes in differing circumstances), as well as a complex sound system involving grammatically significant consonant mutations to the initial consonant of a word. Apparently, neither characteristic was present in the preceding Primitive Irish period, though initial mutations likely existed in a non-grammaticalised form in the prehistoric era.

Contemporary Old Irish scholarship is still greatly influenced by the works of a small number of scholars active in the late 19th and early 20th centuries, such as Rudolf Thurneysen (1857–1940) and Osborn Bergin (1873–1950).

Notable characteristics
Notable characteristics of Old Irish compared with other old Indo-European languages, are:
 * Initial mutations, including lenition, nasalisation and aspiration/gemination.
 * A complex system of verbal allomorphy.
 * A system of conjugated prepositions that is unusual in Indo-European languages but common to Celtic languages. There is a great deal of allomorphy here, as well.
 * Infixed or prefixed object prepositions, which are inserted between the verb stem and its initial prefix(es). If a verb lacks any such prefixes, a dummy prefix is normally added.
 * Special verbal conjugations are used to signal the beginning of a relative clause.

Old Irish also preserves most aspects of the complicated Proto-Indo-European (PIE) system of morphology. Nouns and adjectives are declined in three genders (masculine, feminine, neuter); three numbers (singular, dual, plural); and five cases (nominative, vocative, accusative, dative and genitive). Most PIE noun stem classes are maintained (o-, yo-, ā-, yā-, i-, u-, r-, n-, s-, and consonant stems). Most of the complexities of PIE verbal conjugation are also maintained, and there are new complexities introduced by various sound changes (see below).

Classification
Old Irish was the only known member of the Goidelic branch of the Celtic languages, which is, in turn, a subfamily of the wider Indo-European language family that also includes the Slavonic, Italic/Romance, Indo-Aryan and Germanic subfamilies, along with several others. Old Irish is the ancestor of all modern Goidelic languages: Modern Irish, Scottish Gaelic and Manx.

A still older form of Irish is known as Primitive Irish. Fragments of Primitive Irish, mainly personal names, are known from inscriptions on stone written in the Ogham alphabet. The inscriptions date from about the 4th to the 6th centuries. Primitive Irish appears to have been very close to Common Celtic, the ancestor of all Celtic languages, and it had a lot of the characteristics of other archaic Indo-European languages.

Consonants
The consonant inventory of Old Irish is shown in the chart below. The complexity of Old Irish phonology is from a four-way split of phonemes inherited from Primitive Irish, with both a fortis–lenis and a "broad–slender" (velarised vs. palatalised) distinction arising from historical changes. The sounds are the broad lenis equivalents of broad fortis ; likewise for the slender (palatalised) equivalents. (However, most sounds actually derive historically from, since  was relatively rare in Old Irish, being a recent import from other languages such as Latin.)

 * {| class="wikitable" style="text-align: center;"

! colspan="2" | ! Labial ! Dental ! Alveolar ! Velar ! Glottal ! rowspan="2" | Nasal ! broad ! slender ! rowspan="2" | Plosive ! broad ! slender ! rowspan="2" | Fricative ! broad ! slender ! rowspan="2" | Nasalized fricative ! broad ! slender ! rowspan="2" | Approximant ! broad ! slender ! rowspan="2" | Lateral ! broad ! slender
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2" |
 * colspan="2"|
 * colspan="2"|
 * colspan="2" |
 * }
 * }
 * }
 * }

Some details of Old Irish phonetics are not known. may have been pronounced or, as in Modern Irish. may have been the same sound as or. The precise articulation of the fortis sonorants is unknown, but they were probably longer, tenser and generally more strongly articulated than their lenis counterparts, as in the Modern Irish and Scottish dialects that still possess a four-way distinction in the coronal nasals and laterals. and may have been pronounced  and  respectively. The difference between and  may have been that the former were trills while the latter were flaps. and were derived from an original fortis–lenis pair.

Vowels
Old Irish had distinctive vowel length in both monophthongs and diphthongs. Short diphthongs were monomoraic, taking up the same amount of time as short vowels, while long diphthongs were bimoraic, the same as long vowels. (This is much like the situation in Old English but different from Ancient Greek whose shorter and longer diphthongs were bimoraic and trimoraic, respectively: vs. .) The inventory of Old Irish long vowels changed significantly over the Old Irish period, but the short vowels changed much less.

The following short vowels existed:

 * {| class="wikitable" style="text-align: center;"

! ! colspan="2" | Monophthongs ! colspan="2" | Diphthongs !Close !Mid !Open
 * 1
 * 1
 * 1
 * 1
 * colspan="2" align="center" |, (?)
 * colspan="2" align="center" |
 * }

1The short diphthong likely existed very early in the Old Irish period, but merged with  later on and in many instances was replaced with  due to paradigmatic levelling. It is attested once in the phrase i routh by the prima manus of the Würzburg Glosses.

arose from the u-infection of stressed by a  that preceded a palatalized consonant. This vowel faced much inconsistency in spelling, often detectable by a word containing it being variably spelled with across attestations. Tulach "hill, mound" is the most commonly cited example of this vowel, with the spelling of its inflections including tulach itself, telaig, telocho, tilchaib, taulich and tailaig. This special vowel also ran rampant in many words starting with the stressed prefix air- (from Proto-Celtic *ɸare).

Archaic Old Irish (before about 750) had the following inventory of long vowels:

 * {|class="wikitable" style="text-align: center;"

! ! colspan="2" | Monophthongs ! colspan="2" | Diphthongs ! Close ! Mid ! Open
 * 1
 * , (?)2
 * , 3
 * , 3
 * colspan="2" |
 * colspan="2" | 3
 * }

1Both and  were normally written  but must have been pronounced differently because they have different origins and distinct outcomes in later Old Irish. stems from Proto-Celtic *ē (< PIE *ei), or from ē in words borrowed from Latin. generally stems from compensatory lengthening of short *e because of loss of the following consonant (in certain clusters) or a directly following vowel in hiatus. It is generally thought that was higher than. Perhaps was  while  was. They are clearly distinguished in later Old Irish, in which becomes  (but  before a palatal consonant). becomes in all circumstances. Furthermore, is subject to u-affection, becoming  or, while  is not.

2A similar distinction may have existed between and, both written , and stemming respectively from former diphthongs (*eu, *au, *ou) and from compensatory lengthening. However, in later Old Irish both sounds appear usually as, sometimes as , and it is unclear whether existed as a separate sound any time in the Old Irish period.

3 existed only in early archaic Old Irish (c. 700 or earlier); afterwards it merged into. Neither sound occurred before another consonant, and both sounds became in later Old Irish (often  or  before another vowel). The late does not develop into, suggesting that  >  postdated  >.

Later Old Irish had the following inventory of long vowels:

 * {| class="wikitable" style="text-align: center;"

! ! colspan="2" | Monophthongs ! colspan="2" | Diphthongs ! Close ! Mid ! Open
 * 1
 * 1
 * 1
 * 1
 * colspan="2" |
 * colspan="2" |
 * }

1Early Old Irish and  merged in later Old Irish. It is unclear what the resulting sound was, as scribes continued to use both and  to indicate the merged sound. The choice of in the table above is somewhat arbitrary.

The distribution of short vowels in unstressed syllables is a little complicated. All short vowels may appear in absolutely final position (at the very end of a word) after both broad and slender consonants. The front vowels and  are often spelled  and  after broad consonants, which might indicate a retracted pronunciation here, perhaps something like  and. All ten possibilities are shown in the following examples:

 * {| class="wikitable"

! Old Irish ! Pronunciation ! English ! Annotations
 * marba
 * kill
 * 1 sg. subj.
 * léicea
 * leave
 * 1 sg. subj.
 * marbae
 * kill
 * 2 sg. subj.
 * léice
 * leave
 * 2 sg. subj.
 * marbai
 * kill
 * 2 sg. indic.
 * léici
 * leave
 * 2 sg. indic.
 * súlo
 * eye
 * gen.
 * doirseo
 * door
 * gen.
 * marbu
 * kill
 * 1 sg. indic.
 * léiciu
 * leave
 * 1 sg. indic.
 * }
 * súlo
 * eye
 * gen.
 * doirseo
 * door
 * gen.
 * marbu
 * kill
 * 1 sg. indic.
 * léiciu
 * leave
 * 1 sg. indic.
 * }
 * 1 sg. indic.
 * léiciu
 * leave
 * 1 sg. indic.
 * }
 * 1 sg. indic.
 * }

The distribution of short vowels in unstressed syllables, other than when absolutely final, was quite restricted. It is usually thought that there were only two allowed phonemes: (written  depending on the quality of surrounding consonants) and  (written  or ). The phoneme tended to occur when the following syllable contained an *ū in Proto-Celtic (for example, dligud  "law" (dat.) < PC *dligedū), or after a broad labial (for example, lebor  "book"; domun  "world"). The phoneme occurred in other circumstances. The occurrence of the two phonemes was generally unrelated to the nature of the corresponding Proto-Celtic vowel, which could be any monophthong: long or short.

Long vowels also occur in unstressed syllables. However, they rarely reflect Proto-Celtic long vowels, which were shortened prior to the deletion (syncope) of inner syllables. Rather, they originate in one of the following ways:
 * from the late resolution of a hiatus of two adjacent vowels (usually as a result of loss of *s between vowels);
 * from compensatory lengthening in response to loss of a consonant (cenél "kindred, gender" < *cenethl; du·air-chér "I have purchased" < *-chechr, preterite of crenaid "buys");
 * from assimilation of an unstressed vowel to a corresponding long stressed vowel;
 * from late compounding;
 * from lengthening of short vowels before unlenited, still in progress in Old Irish (compare erríndem "highest" vs. rind "peak").

Stress
Stress is generally on the first syllable of a word. However, in verbs it occurs on the second syllable when the first syllable is a clitic (the verbal prefix as- in as·beir "he says"). In such cases, the unstressed prefix is indicated in grammatical works with a following centre dot.

Orthography
As with most medieval languages, the orthography of Old Irish is not fixed, so the following statements are to be taken as generalisations only. Individual manuscripts may vary greatly from these guidelines.

The Old Irish alphabet consists of the following eighteen letters of the Latin alphabet:
 * a, b, c, d, e, f, g, h, i, l, m, n, o, p, r, s, t, u,

in addition to the five long vowels, shown by an acute accent (´):
 * á, é, í, ó, ú,

the lenited consonants denoted with a superdot (◌̇):
 * ḟ, ṡ,

and the eclipsis consonants also denoted with a superdot:
 * ṁ, ṅ.

Old Irish digraphs include the lenition consonants:
 * ch, fh, th, ph, sh,

the eclipsis consonants:
 * mb, nd, ng; ṁb, ṅd, ṅg,

the geminatives:
 * bb, cc, ll, mm, nn, pp, rr, tt,

and the diphthongs:
 * aé/áe/aí/ái, oé/óe/oí/ói,
 * uí, ía, áu, úa, éu, óu, iu, au, eu,
 * ai, ei, oi, ui; ái, éi, ói, úi.

The following table indicates the broad pronunciation of various consonant letters in various environments:

 * {| class="wikitable" style="text-align: center;"

!colspan=6|Broad consonant phonemes !rowspan=2|Letter !colspan=3|Word-initial !colspan=2|Non-initial !unmutated !eclipsed !lenited !single !geminate
 * b
 * colspan=2|
 * c
 * d
 * colspan=2|
 * —
 * f
 * —
 * g
 * colspan=2|
 * —
 * h
 * colspan=5|See explanation below
 * l
 * —
 * colspan=2|
 * m
 * —
 * colspan=2|
 * n
 * —
 * colspan=2|
 * p
 * r
 * —
 * colspan=2|
 * s₁
 * —
 * —
 * s₂1
 * —
 * colspan=2| $⟨mb⟩$
 * —
 * t
 * }
 * Angle brackets ⟨⟩ here indicate graphemic differences to the unmutated consonant.
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s
 * —
 * colspan=2|
 * m
 * —
 * colspan=2|
 * n
 * —
 * colspan=2|
 * p
 * r
 * —
 * colspan=2|
 * s₁
 * —
 * —
 * s₂1
 * —
 * colspan=2| $⟨bb⟩$
 * —
 * t
 * }
 * Angle brackets ⟨⟩ here indicate graphemic differences to the unmutated consonant.
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s
 * r
 * —
 * colspan=2|
 * s₁
 * —
 * —
 * s₂1
 * —
 * colspan=2| $⟨ch⟩$
 * —
 * t
 * }
 * Angle brackets ⟨⟩ here indicate graphemic differences to the unmutated consonant.
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s
 * s₂1
 * —
 * colspan=2| $⟨cc⟩$
 * —
 * t
 * }
 * Angle brackets ⟨⟩ here indicate graphemic differences to the unmutated consonant.
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s
 * }
 * Angle brackets ⟨⟩ here indicate graphemic differences to the unmutated consonant.
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s
 * }
 * Angle brackets ⟨⟩ here indicate graphemic differences to the unmutated consonant.
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s
 * A dash (—) here indicates that the respective consonant is not subject to eclipsis. These consonants are: r, l, n, s

1The s₂ arises from older *sw or *sɸ which is lenited to. In Old Irish there are only several words containing s₂: sïur :: fïur, phïur, sister; sesser :: mórfesser, six persons / seven (lit. great six) persons; or in reduplicated verbs do·seinn :: do·sephainn, pursue.

When the consonants b, d, g are eclipsed by the preceding word (always from a word-initial position), their spelling and pronunciation change to: $⟨nd⟩$, $⟨ḟ/fh⟩$ , $⟨ng⟩$

Generally, geminating a consonant ensures its unmutated sound. While the letter $⟨ll⟩$ may be voiced at the end of some words, but when it is written double $⟨mm⟩$ it is always voiceless  in regularised texts; however, even final  was often written "cc", as in bec / becc "small, little" (Modern Irish and Scottish beag, Manx beg).

In later Irish manuscripts, lenited f and s are denoted with the letter h $⟨nn⟩$, $⟨ph⟩$, instead of using a superdot $⟨pp⟩$, $⟨rr⟩$.

When initial s stemmed from Primitive Irish *sw-, its lenited version is $⟨ṡ/sh⟩$.

The slender (palatalised) variants of the 13 consonants are denoted with marking the letter. They occur in the following environments:
 * Before a written e, é, i, í
 * After a written i, when not followed by a vowel letter (but not after the diphthongs aí, oí, uí)

Although Old Irish has both a sound and a letter h, there is no consistent relationship between the two. Vowel-initial words are sometimes written with an unpronounced h, especially if they are very short (the Old Irish preposition i "in" was sometimes written hi) or if they need to be emphasised (the name of Ireland, Ériu, was sometimes written Hériu). On the other hand, words that begin with the sound are usually written without it: a ór  "her gold". If the sound and the spelling co-occur, it is by coincidence, as ní hed "it is not".

Stops following vowels
The voiceless stops of Old Irish are c, p, t. They contrast with the voiced stops g, b, d. Additionally, the letter m can behave similarly to a stop following vowels. These seven consonants often mutate when not in the word-initial position.

In non-initial positions, the single-letter voiceless stops c, p, and t become the voiced stops, , and respectively unless they are written double. Ambiguity in these letters' pronunciations arises when a single consonant follows an l, n, or r. The lenited stops ch, ph, and th become, , and respectively.
 * {| class="wikitable"

!colspan=3|Non-initial voiceless stops $⟨f/ph⟩$, $⟨th⟩$, $⟨tt⟩$ !Old Irish !Pronunciation !English !colspan=3|Lenited consonants $⟨mb⟩$, $⟨nd⟩$, $⟨ng⟩$
 * macc
 * son
 * bec or becc
 * small
 * op or opp
 * refuse
 * bratt
 * mantle
 * brot or brott
 * goad
 * refuse
 * bratt
 * mantle
 * brot or brott
 * goad
 * mantle
 * brot or brott
 * goad
 * goad
 * goad
 * ech
 * horse
 * oíph
 * beauty
 * áth
 * ford
 * }
 * beauty
 * áth
 * ford
 * }
 * ford
 * }

The voiced stops b, d, and g become fricative, , and , respectively—identical sounds to their word-initial lenitions.
 * {| class="wikitable"

!colspan=3|Non-initial voiced stops $⟨c⟩$, $⟨cc⟩$, $⟨fh⟩$ !Old Irish !Pronunciation !English
 * dub
 * black
 * mod
 * work
 * mug
 * slave
 * claideb
 * sword
 * claidib
 * swords
 * }
 * slave
 * claideb
 * sword
 * claidib
 * swords
 * }
 * claidib
 * swords
 * }
 * }

In non-initial positions, the letter m usually becomes the nasal fricative, but in some cases it becomes a nasal stop, denoted as. In cases in which it becomes a stop, m is often written double to avoid ambiguity.
 * {| class="wikitable"

!colspan=3|Non-initial consonant $⟨sh⟩$ !Old Irish !Pronunciation !English
 * dám
 * company
 * lom or lomm
 * bare
 * }
 * lom or lomm
 * bare
 * }
 * }

Stops following other consonants
Ambiguity arises in the pronunciation of the stop consonants (c, g, t, d, p, b) when they follow l, n, or r:
 * {| class="wikitable"

!colspan=3|Homographs involving $⟨ḟ⟩$, $⟨ṡ⟩$, $⟨f⟩$ !Old Irish !Pronunciation !English
 * derc
 * hole
 * derc
 * red
 * daltae
 * fosterling
 * celtae
 * who hide
 * anta
 * of remaining
 * antae
 * who remain
 * }
 * celtae
 * who hide
 * anta
 * of remaining
 * antae
 * who remain
 * }
 * antae
 * who remain
 * }
 * who remain
 * }

After m, the letter b is naturally a stop. After d, l, r, the letter b is fricative :
 * {| class="wikitable"

!colspan=3|Consonant $⟨c⟩$ !Old Irish !Pronunciation !English
 * imb
 * butter
 * odb
 * knot (in a tree)
 * delb
 * image
 * marb
 * dead
 * }
 * delb
 * image
 * marb
 * dead
 * }
 * dead
 * }
 * }

After n or r, the letter d is a stop :
 * {| class="wikitable"

!colspan=3|Consonant $⟨p⟩$ !Old Irish !Pronunciation !English
 * bind
 * melodious
 * cerd
 * art, skill
 * }
 * cerd
 * art, skill
 * }
 * }

After n, l, or r, the letter g is usually a stop, but it becomes a fricative in a few words:
 * {| class="wikitable"

!colspan=3|Consonant $⟨t⟩$ !Old Irish !Pronunciation !English
 * long
 * ship
 * delg or delc
 * thorn
 * argat or arggat
 * silver
 * ingen
 * daughter
 * ingen
 * nail, claw
 * bairgen
 * loaf of bread
 * }
 * ingen
 * daughter
 * ingen
 * nail, claw
 * bairgen
 * loaf of bread
 * }
 * bairgen
 * loaf of bread
 * }
 * loaf of bread
 * }

The consonants l, n, r
The letters l, n, r are generally written double when they indicate tense sonorants and single when they indicate lax sonorants. Originally, it reflected an actual difference between single and geminate consonants, as tense sonorants in many positions (such as between vowels or word-finally) developed from geminates. As the gemination was lost, the use of written double consonants was repurposed to indicate tense sonorants. Doubly written consonants of this sort do not occur in positions where tense sonorants developed from non-geminated Proto-Celtic sonorants (such as word-initially or before a consonant).
 * {| class="wikitable"

!Old Irish !Pronunciation !English
 * corr
 * crane
 * cor
 * putting
 * coll
 * hazel
 * col
 * sin
 * sonn
 * stake
 * son
 * sound
 * ingen
 * daughter
 * ingen
 * nail, claw
 * }
 * sonn
 * stake
 * son
 * sound
 * ingen
 * daughter
 * ingen
 * nail, claw
 * }
 * daughter
 * ingen
 * nail, claw
 * }
 * nail, claw
 * }
 * }

Geminate consonants appear to have existed since the beginning of the Old Irish period, but they were simplified by the end, as is generally reflected by the spelling. Eventually, however, ll, mm, nn, rr were repurposed to indicate nonlenited variants of those sounds in certain positions.

Vowels
Written vowels a, ai, e, i in poststressed syllables (except when absolutely word-final) all seem to represent phonemic. The particular vowel that appears is determined by the quality (broad vs. slender) of the surrounding consonants and has no relation to the etymological vowel quality:
 * {|class="wikitable"

! Preceding consonant !! Following consonant !! Spelling !! Example
 * broad || broad || $⟨ch⟩$ || dígal "vengeance" (nom.)
 * broad || slender (in open syllable) || $⟨ph⟩$ ||
 * broad || slender (in closed syllable) || $⟨th⟩$ || dígail "vengeance" (acc./dat.)
 * slender || broad || $⟨g⟩$ || dliged "law" (acc.)
 * slender || slender || $⟨b⟩$ || dligid "law" (gen.)
 * }
 * slender || broad || $⟨d⟩$ || dliged "law" (acc.)
 * slender || slender || $⟨m⟩$ || dligid "law" (gen.)
 * }
 * slender || slender || $⟨l⟩$ || dligid "law" (gen.)
 * }

It seems likely that spelling variations reflected allophonic variations in the pronunciation of.

History
Old Irish underwent extensive phonological changes from Proto-Celtic in both consonants and vowels. Final syllables were lost or transphonologized as grammatical mutations on the following word. In addition, unstressed syllables faced various reductions and deletions of their vowels.

Grammar
Old Irish is a fusional, nominative-accusative, and VSO language.

Nouns decline for 5 cases: nominative, accusative, genitive, prepositional, vocative; 3 genders: masculine, feminine, neuter; 3 numbers: singular, dual, plural. Adjectives agree with nouns in case, gender, and number. The prepositional case is called the dative by convention.

Verbs conjugate for 3 tenses: past, present, future; 3 aspects: simple, perfective, imperfective; 4 moods: indicative, subjunctive, conditional, imperative; 2 voices: active, and passive; independent, and dependent forms; and simple, and complex forms. Verbs display tense, aspect, mood, voice, and sometimes portmanteau forms through suffixes, or stem vowel changes for the former four. Proclitics form a verbal complex with the core verb, and the verbal complex is often preceded by preverbal particles such as ní (negative marker), in (interrogative marker), ro (perfective marker). Direct object personal pronouns occur between the preverb and the verbal stem. Verbs agree with their subject in person and number. A single verb can stand as an entire sentence. Emphatic particles such as -sa and -se are affixed to the end of the verb.

Prepositions inflect for person and number, and different prepositions govern different cases, sometimes depending on the semantics intended.

title: Old Irish


text:
Wikipedia:Articles for deletion/High schools in New Jersey

 This page is an archive of the discussion about the proposed deletion of the article below. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was ambiguous. I count 8 clear "delete" votes, 4 clear "keep" votes and one "move". Failing to reach a clear concensus to delete, the article is kept for now. Rossami (talk) 23:25, 16 Feb 2005 (UTC)

High schools in New Jersey
One in a series. This is what categories are for. --BM 19:33, 2 Feb 2005 (UTC)

 * delete; an article on "education in New Jersey" would be welcome however Mozzerati 21:52, 2005 Feb 2 (UTC)
 * Keep. There are lists of prisons, universitis, an god knows what on Wikipedia. There's no reason for deletion. (Note: There are many High Schools that are proposed for deletion also) Beta_M talk, |contrib ( Ë-Mail )
 * Delete. Worthless unless complete; far too long if complete. —Korath (Talk) 13:45, Feb 3, 2005 (UTC)
 * Keep. Lists are good for things that are not very notable and hard to create keepable articles about. bbx 17:20, 3 Feb 2005 (UTC)
 * Keep all lists of schools. Agree with bbx.  —RaD Man (talk) 20:07, 3 Feb 2005 (UTC)
 * Delete. My comments are the same as in the current VfD on High schools in Connecticut.
 * At 21:03, 3 Feb 2005, User:BM forgot to type four tildes. dbenbenn | talk 14:42, 14 Feb 2005 (UTC)
 * Delete unmaintainable lists. As pointed out, we have categories for this.Jayjg (talk) 22:53, 3 Feb 2005 (UTC)
 * Delete. Unwieldly and unencyclopedic.  Wikipedia is not a general knowledge base.  Even if all schools are determined at some point to be notable, a better system than this would be needed to manage the information. Indrian 19:17, Feb 4, 2005 (UTC)
 * Delete. There are categories for a reason. Carrp | Talk 18:21, 5 Feb 2005 (UTC)
 * Delete. Use a category. Wile E. Heresiarch 08:31, 6 Feb 2005 (UTC)
 * As per the lengthy reasoning given in Votes for deletion/High schools in Connecticut: Move to the correct List of high schools in X form, Keep with the cautions that Wikipedia is not a repository of links and Wikipedia is not a directory, and send to Cleanup for all but one of the articles which do not meet the standard of ("List of") High schools in Connecticut and simply do not contain enough data to meet the purpose of lists. Uncle G 13:41, 2005 Feb 7 (UTC)
 * Delete. If someone wants to make a useful list at List of high schools in New Jersey, let them start from scratch.  dbenbenn | talk 15:35, 14 Feb 2005 (UTC)
 * Keep and allow for organic growth. GRider\talk 21:48, 14 Feb 2005 (UTC)

This page is now preserved as an archive of the debate and, like some other VfD subpages, is no longer 'live'. Subsequent comments on the issue, the deletion, or the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.


text:
Chromatic polynomial

The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem. It was generalised to the Tutte polynomial by Hassler Whitney and W. T. Tutte, linking it to the Potts model of statistical physics.

History
George David Birkhoff introduced the chromatic polynomial in 1912, defining it only for planar graphs, in an attempt to prove the four color theorem. If $$P(G, k)$$ denotes the number of proper colorings of G with k colors then one could establish the four color theorem by showing $$P(G, 4)>0$$ for all planar graphs G. In this way he hoped to apply the powerful tools of analysis and algebra for studying the roots of polynomials to the combinatorial coloring problem.

Hassler Whitney generalised Birkhoff’s polynomial from the planar case to general graphs in 1932. In 1968, Ronald C. Read asked which polynomials are the chromatic polynomials of some graph, a question that remains open, and introduced the concept of chromatically equivalent graphs. Today, chromatic polynomials are one of the central objects of algebraic graph theory.

Definition

For a graph G, $$P(G,k)$$ counts the number of its (proper) vertex k-colorings. Other commonly used notations include $$P_G(k)$$, $$\chi_G(k)$$, or $$\pi_G(k)$$. There is a unique polynomial $$P(G,x)$$ which evaluated at any integer k ≥ 0 coincides with $$P(G,k)$$; it is called the chromatic polynomial of G.

For example, to color the path graph $$P_3$$ on 3 vertices with k colors, one may choose any of the k colors for the first vertex, any of the $$k - 1$$ remaining colors for the second vertex, and lastly for the third vertex, any of the $$k - 1$$ colors that are different from the second vertex's choice. Therefore, $$P(P_3,k) = k \cdot (k-1) \cdot (k-1)$$ is the number of k-colorings of $$P_3$$. For a variable x (not necessarily integer), we thus have $$P(P_3,x)=x(x-1)^2=x^3-2x^2+x$$. (Colorings which differ only by permuting colors or by automorphisms of G are still counted as different.)

Deletion–contraction
The fact that the number of k-colorings is a polynomial in k follows from a recurrence relation called the deletion–contraction recurrence or Fundamental Reduction Theorem. It is based on edge contraction: for a pair of vertices $$u$$ and $$v$$ the graph $$G/uv$$ is obtained by merging the two vertices and removing any edges between them. If $$u$$ and $$v$$ are adjacent in G, let $$G-uv$$ denote the graph obtained by removing the edge $$uv$$. Then the numbers of k-colorings of these graphs satisfy:
 * $$P(G,k)=P(G-uv, k)- P(G/uv,k)$$

Equivalently, if $$u$$ and $$v$$ are not adjacent in G and $$G+uv$$ is the graph with the edge $$uv$$ added, then
 * $$P(G,k)= P(G+uv, k) + P(G/uv,k)$$

This follows from the observation that every k-coloring of G either gives different colors to $$u$$ and $$v$$, or the same colors. In the first case this gives a (proper) k-coloring of $$G+uv$$, while in the second case it gives a coloring of $$G/uv$$. Conversely, every k-coloring of G can be uniquely obtained from a k-coloring of $$G+uv$$ or $$G/uv$$ (if $$u$$ and $$v$$ are not adjacent in G).

The chromatic polynomial can hence be recursively defined as
 * $$P(G,x)=x^n$$ for the edgeless graph on n vertices, and
 * $$P(G,x)=P(G-uv, x)- P(G/uv,x)$$ for a graph G with an edge $$uv$$ (arbitrarily chosen).

Since the number of k-colorings of the edgeless graph is indeed $$k^n$$, it follows by induction on the number of edges that for all G, the polynomial $$P(G,x)$$ coincides with the number of k-colorings at every integer point x = k. In particular, the chromatic polynomial is the unique interpolating polynomial of degree at most n through the points
 * $$\left \{ (0, P(G, 0)), (1, P(G, 1)), \ldots, (n, P(G, n)) \right \}.$$

Tutte’s curiosity about which other graph invariants satisfied such recurrences led him to discover a bivariate generalization of the chromatic polynomial, the Tutte polynomial $$T_G(x,y)$$.

Properties
For fixed G on n vertices, the chromatic polynomial $$P(G, x)$$ is a monic polynomial of degree exactly n, with integer coefficients.

The chromatic polynomial includes at least as much information about the colorability of G as does the chromatic number. Indeed, the chromatic number is the smallest positive integer that is not a zero of the chromatic polynomial,
 * $$\chi (G)=\min\{ k\in\mathbb{N} : P(G, k) > 0 \}.$$

The polynomial evaluated at $$-1$$, that is $$P(G,-1)$$, yields $$(-1)^{|V(G)|}$$ times the number of acyclic orientations of G.

The derivative evaluated at 1, $$P'(G, 1)$$ equals the chromatic invariant $$\theta(G)$$ up to sign. If G has n vertices and c components $$G_1, \ldots, G_c$$, then
 * The coefficients of $$ x^0, \ldots, x^{c-1}$$ are zeros.
 * The coefficients of $$ x^c, \ldots, x^n$$ are all non-zero and alternate in signs.
 * The coefficient of $$x^n$$ is 1 (the polynomial is monic).
 * The coefficient of $$x^{n-1}$$ is $$-|E(G)|. $$

We prove this via induction on the number of edges on a simple graph G with $$n$$ vertices and $$k$$ edges. When $$k = 0$$, G is an empty graph. Hence per definition $$P(G, x)= x^n$$. So the coefficient of $$x^{n-1}$$ is $$0$$, which implies the statement is true for an empty graph. When $$k = 1$$, as in G has just a single edge, $$P(G, x) = x^n - x^{n-1}$$. Thus coefficient of $$x^{n-1}$$ is $$-1 = -|E(G)|$$. So the statement holds for k = 1. Using strong induction assume the statement is true for $$k = 0,1,2,\ldots,(k-1)$$. Let G have $$k$$ edges. By the contraction-deletion principle, "$ P(G, x) = P(G-e, x) - P(G/e, x),$"Let $$P(G-e, x) = x^n - a_{n-1}x^{n-1} + a_{n-2}x^{n-2}-\cdots,$$ and $$P(G/e, x) = x^{n-1} - b_{n-2} x^{n-2} + b_{n-3}x^{n-3}-\cdots.$$ Hence $$P(G, x) = x^n - (a_{n-1} +1)x^{n-1} +\cdots$$. Since $$G-e$$ is obtained from G by removal of just one edge e, $$a_{n-1} = k - 1$$, so $$a_{n-1} + 1 = k$$ and thus the statement is true for k.

 * The coefficient of $$x^1$$ is $$(-1)^{n-1}$$ times the number of acyclic orientations that have a unique sink, at a specified, arbitrarily chosen vertex.
 * The absolute values of coefficients of every chromatic polynomial form a log-concave sequence.
 * $$\scriptstyle P(G, x) = P(G_1, x)P(G_2,x) \cdots P(G_c,x)$$

The last property is generalized by the fact that if G is a k-clique-sum of $$G_1$$ and $$G_2$$ (i.e., a graph obtained by gluing the two at a clique on k vertices), then
 * $$P(G, x) = \frac{P(G_1,x)P(G_2,x)}{x(x-1)\cdots(x-k+1)}.$$

A graph G with n vertices is a tree if and only if
 * $$P(G, x) = x(x-1)^{n-1}.$$

Chromatic equivalence

Two graphs are said to be chromatically equivalent if they have the same chromatic polynomial. Isomorphic graphs have the same chromatic polynomial, but non-isomorphic graphs can be chromatically equivalent. For example, all trees on n vertices have the same chromatic polynomial. In particular, $$(x-1)^3x$$ is the chromatic polynomial of both the claw graph and the path graph on 4 vertices.

A graph is chromatically unique if it is determined by its chromatic polynomial, up to isomorphism. In other words, G is chromatically unique, then $$P(G, x) = P(H, x)$$ would imply that G and H are isomorphic. All cycle graphs are chromatically unique.

Chromatic roots
A root (or zero) of a chromatic polynomial, called a “chromatic root”, is a value x where $$P(G, x)=0$$. Chromatic roots have been very well studied, in fact, Birkhoff’s original motivation for defining the chromatic polynomial was to show that for planar graphs, $$P(G, x)>0$$ for x ≥ 4. This would have established the four color theorem.

No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root of every graph with at least one edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27. A result of Tutte connects the golden ratio $$\varphi$$ with the study of chromatic roots, showing that chromatic roots exist very close to $$\varphi^2$$: If $$G_n$$ is a planar triangulation of a sphere then

 * $$P(G_n,\varphi^2) \leq \varphi^{5-n}.$$

While the real line thus has large parts that contain no chromatic roots for any graph, every point in the complex plane is arbitrarily close to a chromatic root in the sense that there exists an infinite family of graphs whose chromatic roots are dense in the complex plane.

Colorings using all colors
For a graph G on n vertices, let $$e_k$$ denote the number of colorings using exactly k colors up to renaming colors (so colorings that can be obtained from one another by permuting colors are counted as one; colorings obtained by automorphisms of G are still counted separately). In other words, $$e_k$$ counts the number of partitions of the vertex set into k (non-empty) independent sets. Then $$k! \cdot e_k$$ counts the number of colorings using exactly k colors (with distinguishable colors). For an integer x, all x-colorings of G can be uniquely obtained by choosing an integer k ≤ x, choosing k colors to be used out of x available, and a coloring using exactly those k (distinguishable) colors. Therefore:
 * $$P(G,x) = \sum_{k=0}^x \binom{x}{k} k! \cdot e_k = \sum_{k=0}^x (x)_k \cdot e_k,$$

where $$(x)_k = x(x-1)(x-2)\cdots(x-k+1)$$ denotes the falling factorial. Thus the numbers $$e_k$$ are the coefficients of the polynomial $$P(G,x)$$ in the basis $$1,(x)_1,(x)_2,(x)_3,\ldots$$ of falling factorials.

Let $$a_k$$ be the k-th coefficient of $$P(G,x)$$ in the standard basis $$1,x,x^2,x^3,\ldots$$, that is:
 * $$P(G,x) = \sum_{k=0}^n a_k x^k$$

Stirling numbers give a change of basis between the standard basis and the basis of falling factorials. This implies:
 * $$a_k = \sum_{j=0}^n (-1)^{j-k} \begin{bmatrix}j\\k\end{bmatrix} e_j$$  and   $$e_k = \sum_{j=0}^n \begin{Bmatrix}j\\k\end{Bmatrix} a_j. $$

Categorification
The chromatic polynomial is categorified by a homology theory closely related to Khovanov homology.

Algorithms
Computational problems associated with the chromatic polynomial include

 * finding the chromatic polynomial $$P(G, x)$$ of a given graph G;
 * evaluating $$P(G, x)$$ at a fixed x for given G.

The first problem is more general because if we knew the coefficients of $$P(G, x)$$ we could evaluate it at any point in polynomial time because the degree is n. The difficulty of the second type of problem depends strongly on the value of x and has been intensively studied in computational complexity. When x is a natural number, this problem is normally viewed as computing the number of x-colorings of a given graph. For example, this includes the problem #3-coloring of counting the number of 3-colorings, a canonical problem in the study of complexity of counting, complete for the counting class #P.

Efficient algorithms
For some basic graph classes, closed formulas for the chromatic polynomial are known. For instance this is true for trees and cliques, as listed in the table above.

Polynomial time algorithms are known for computing the chromatic polynomial for wider classes of graphs, including chordal graphs and graphs of bounded clique-width. The latter class includes cographs and graphs of bounded tree-width, such as outerplanar graphs.

Deletion–contraction
The deletion-contraction recurrence gives a way of computing the chromatic polynomial, called the deletion–contraction algorithm. In the first form (with a minus), the recurrence terminates in a collection of empty graphs. In the second form (with a plus), it terminates in a collection of complete graphs. This forms the basis of many algorithms for graph coloring. The ChromaticPolynomial function in the Combinatorica package of the computer algebra system Mathematica uses the second recurrence if the graph is dense, and the first recurrence if the graph is sparse. The worst case running time of either formula satisfies the same recurrence relation as the Fibonacci numbers, so in the worst case, the algorithm runs in time within a polynomial factor of

 * $$\varphi^{n+m}=\left (\frac{1+\sqrt{5}}{2} \right)^{n+m}\in O\left(1.62^{n+m}\right),$$

on a graph with n vertices and m edges. The analysis can be improved to within a polynomial factor of the number $$t(G)$$ of spanning trees of the input graph. In practice, branch and bound strategies and graph isomorphism rejection are employed to avoid some recursive calls, the running time depends on the heuristic used to pick the vertex pair.

Cube method
There is a natural geometric perspective on graph colorings by observing that, as an assignment of natural numbers to each vertex, a graph coloring is a vector in the integer lattice. Since two vertices $$i$$ and $$j$$ being given the same color is equivalent to the $$i$$’th and $$j$$’th coordinate in the coloring vector being equal, each edge can be associated with a hyperplane of the form $$\{x\in\mathbb R^d:x_i=x_j\}$$. The collection of such hyperplanes for a given graph is called its graphic arrangement. The proper colorings of a graph are those lattice points which avoid forbidden hyperplanes. Restricting to a set of $$k$$ colors, the lattice points are contained in the cube $$[0,k]^n$$. In this context the chromatic polynomial counts the number of lattice points in the $$[0,k]$$-cube that avoid the graphic arrangement.

Computational complexity
The problem of computing the number of 3-colorings of a given graph is a canonical example of a #P-complete problem, so the problem of computing the coefficients of the chromatic polynomial is #P-hard. Similarly, evaluating $$P(G, 3)$$ for given G is #P-complete. On the other hand, for $$k=0,1,2$$ it is easy to compute $$P(G, k)$$, so the corresponding problems are polynomial-time computable. For integers $$k>3$$ the problem is #P-hard, which is established similar to the case $$k=3$$. In fact, it is known that $$P(G, x)$$ is #P-hard for all x (including negative integers and even all complex numbers) except for the three “easy points”. Thus, from the perspective of #P-hardness, the complexity of computing the chromatic polynomial is completely understood.

In the expansion

 * $$P(G, x)= a_1 x + a_2x^2+\cdots +a_nx^n,$$

the coefficient $$a_n$$ is always equal to 1, and several other properties of the coefficients are known. This raises the question if some of the coefficients are easy to compute. However the computational problem of computing ar for a fixed r ≥ 1 and a given graph G is #P-hard, even for bipartite planar graphs.

No approximation algorithms for computing $$P(G, x)$$ are known for any x except for the three easy points. At the integer points $$k=3,4,\ldots$$, the corresponding decision problem of deciding if a given graph can be k-colored is NP-hard. Such problems cannot be approximated to any multiplicative factor by a bounded-error probabilistic algorithm unless NP = RP, because any multiplicative approximation would distinguish the values 0 and 1, effectively solving the decision version in bounded-error probabilistic polynomial time. In particular, under the same assumption, this rules out the possibility of a fully polynomial time randomised approximation scheme (FPRAS). There is no FPRAS for computing $$P(G, x)$$ for any x > 2, unless NP = RP holds.

title: Chromatic polynomial


text:
Melbourne, Quebec

Melbourne is a township municipality located in Le Val-Saint-François Regional County Municipality in the Estrie region of Quebec, Canada.

Demographics
In the 2021 Census of Population conducted by Statistics Canada, Melbourne had a population of $1,096$ living in $439$ of its $494$ total private dwellings, a change of NaN% from its 2016 population of $1,063$. With a land area of 173.93 km2, it had a population density of in 2021.

Mother tongue (2011)

Notable people

 * Marie-Claire Blais (1939–2021), writer and novelist

title: Melbourne, Quebec


text:
W. W. Phelps (Mormon)

William Wines Phelps (February 17, 1792 – March 7, 1872) was an early leader of the Latter Day Saint movement. He printed the first edition of the Book of Commandments that became a standard work of the church and wrote numerous hymns, some of which are included in the current version of the Church of Jesus Christ of Latter-day Saints' (LDS Church) hymnal. He was at times both close to and at odds with church leadership. He testified against Joseph Smith, providing evidence that helped persuade authorities to arrest Smith. He was excommunicated three times and rejoined the church each time. He was a ghostwriter for Smith. Phelps was called by Smith to serve as assistant president of the church in Missouri and as a member of the Council of Fifty. After Smith's death, Phelps supported Brigham Young, who was the church's new president.

Early life
Phelps was born in Hanover Township, New Jersey on February 17, 1792. He was named after American Revolutionary War general William Wines (also spelled "Winds"). His father, Enon Phelps, and mother, Mehitable Goldsmith, moved the family to Homer, New York, in 1800. Phelps was a descendant of the Puritan leader William Phelps. He was mostly self-taught, acquiring knowledge in various areas such as theology, meteorology, and history. He once sought the office of lieutenant governor of New York. He worked as an apprentice to a printer.

On April 28, 1815, he married Sally Waterman in Smyrna, New York. The pair had ten children: eight daughters and two sons. After marrying Sally, Phelps began publishing the Western Courier in Homer in 1820. In this capacity, "he verbally attacked his foes and they him." He next moved to Trumansburgh, Tompkins County, New York, where he edited the anti-Masonic newspaper Lake Light. In 1827, he relocated to Canandaigua, New York, where he published and edited another anti-Masonic newspaper, the Ontario Phoenix. Phelps was described by Dean Jessee as "one of [the] founders" of the anti-Masonic movement in New York.

Involvement in Latter Day Saint movement
Phelps purchased a copy of the Book of Mormon from Parley P. Pratt on April 9, 1830, just three days after Church of Christ was organized. He and his wife Sally Waterman read the book and "became converted to its truth." Phelps then met Joseph Smith on December 24, 1830, and became convinced that Smith was a prophet. On April 29, 1831, Phelps was imprisoned at Lyons, New York, by a "couple of Presbyterian traders, for a small debt, for the purpose, as [he] was informed, of 'keeping [him] from joining the Mormons.'"

Kirtland, Ohio
Phelps moved to Kirtland, Ohio in 1831. He soon visited Joseph Smith and asked him to ask God for an answer as to what Phelps should do. Smith delivered a revelation (now known as section 55 of the Doctrine and Covenants) in which Phelps was encouraged to join the church, preach as a missionary, and be the printer for the church. He was then baptized on June 16, 1831. He was subsequently chosen "to head printing and publishing for the Church" a few days later, on June 20. He moved to Jackson County, Missouri in 1832 and spearheaded the church's publishing arm, W. W. Phelps & Co., in Independence, Missouri, where he edited the Evening and Morning Star from June 1832 to 1833. He was also superintendent of the local schools. On July 20, 1833, while working to publish the church's Book of Commandments, a mob of vigilantes attacked Phelps's home, seizing printing materials, destroying the press, and throwing Phelps's family and furniture out-of-doors. Most of the copies of the Book of Commandments were destroyed in the raid. He fled to Clay County, Missouri, where he was called as Assistant President of the Church in Missouri on July 3, 1834.

In the early part of 1835, Phelps and his son, Waterman, were called to Kirtland, arriving on May 16, 1835, and departing on April 9, 1836. They resided with Joseph Smith's family temporarily. During his stay in Kirtland, Phelps acted as "co-steward over the modern revelations" alongside Joseph Smith, editing the sections of the Doctrine and Covenants for publication. He was also tasked with editing and publishing the Latter Day Saints' Messenger and Advocate. Phelps donated US$500 towards the erection of the Kirtland Temple and was present for the establishment of the School of the Prophets. In Kirtland, he helped print the first Latter Day Saint hymnal in 1835, which included his own hymn, "The Spirit of God Like a Fire Is Burning", sung at the dedication of the Kirtland Temple. Phelps wrote "at least thirty-five of the ninety hymns" included in the first LDS hymnal.

In late June or early July 1835, Joseph Smith acquired Egyptian papyri from Michael Chandler, and Phelps began assisting with the translation of what would become the Book of Abraham in the Pearl of Great Price, acting as Smith's scribe. On January 13, 1836, Phelps was tasked with compiling the "rules and regulations" of the Kirtland Temple.

Far West, Missouri
From 1834, Phelps was a counselor to David Whitmer in the presidency of the church in Missouri, and, in that capacity, he helped found the town of Far West, Missouri, purchasing the land for the town using church funds alongside John Whitmer. He served as postmaster in Far West. Phelps was called before the High Council on March 10, 1838, and was accused of profiting from Far West land deals and reneging on a $2,000 contribution to "the house of the Lord" that was not paid. On March 10, 1838, he was excommunicated from the church. In June 1838, Phelps, Oliver Cowdery, David Whitmer, John Whitmer, and Lyman E. Johnson were warned to leave Far West, "or a more fatal calamity shall befall you." In November 1838, Phelps was summoned to be tried a witness at the treason hearing of Smith in Missouri. According to The Joseph Smith Papers, "his testimony helped lead to Smith's incarceration in the Liberty, Missouri, jail in winter 1838–1839."

Excommunication and rebaptism
Unlike Cowdery and the Whitmers, Phelps remained in Far West after "the dissenters" were warned to leave in June 1838. He appears to have had a short-lived détente with church leadership. On July 8, 1838, Smith received a revelation saying that Phelps and fellow dissenter, Frederick G. Williams, could be ordained as elders and serve missions abroad. At the time of the Mormon surrender of Far West, Phelps was one of the Mormon negotiators. But during the Richmond hearings of November 1838, Phelps was one of several who bore witness against Smith and other leaders, aiding in their imprisonment in Missouri until April 1839. This led to Phelps's excommunication in Quincy, Illinois on March 17, 1839. He then moved to Dayton, Ohio. In June 1840, Phelps pleaded for forgiveness in a letter to Smith. Smith replied with an offer of full fellowship, and ended with a variant of Charles Wesley's couplet, "'Come on, dear brother, since the war is past, For friends at first are friends again at last.'"  Phelps thus reunited with the church through rebaptism sixteen months after his excommunication. He moved back to Kirtland in May 1841.

Nauvoo years
Phelps served a brief mission in the eastern United States in 1841. He then moved to Nauvoo, Illinois, where he donated $1,000 to the construction of the Nauvoo Temple and worked there in as an ordinance worker. On August 27, 1841, he replaced Robert B. Thompson as Smith's clerk. Beginning in February 1843, Phelps became the ghostwriter of many of Smith's important written works of the Nauvoo period, including "General Joseph Smith's Appeal to the Green Mountain Boys" of November 1843; Smith's theodemocratic presidential platform of January 1844; and "The Voice of Innocence", which was presented to and unanimously approved by the Relief Society in February 1844 to rebut claims of polygamy in Nauvoo. Phelps also worked alongside John Taylor in editing the Times and Seasons and Nauvoo Neighbor and Willard Richards in compiling Joseph Smith's personal history. The latter effort eventually became History of the Church. Phelps was endowed on December 9, 1843 and received his "second anointing" on February 2, 1844, promising him exaltation. He was also made a member of the Council of Fifty and the Nauvoo City Council. In Nauvoo, Phelps spoke out in favor of the destruction of an opposition newspaper, the Nauvoo Expositor. He believed that the city charter gave the church leaders power to declare the newspaper a nuisance. Shortly afterwards, the press and type were carried into the street and destroyed. After the death of Joseph Smith, Phelps gave the eulogy at his funeral.

During the succession crisis in 1844, Phelps sided with Brigham Young and the Quorum of the Twelve. In an effort to maintain order in the church, "he used his considerable influence in August and September 1844 to sustain the Twelve Apostles as leaders during the succession crisis." In 1846, he entered into plural marriage, marrying Laura Stowell and Elizabeth Dunn on February 2, 1846, in Nauvoo. He was excommunicated for the third time on December 9, 1847, for entering into an unauthorized polygamous marriage, but he was rebaptized two days later.

Westward exodus, death, and legacy
Phelps took part in the Mormon Exodus across the Great Plains and settled in Salt Lake City in 1848. In November 1849, he left Salt Lake to explore southern Utah Territory with Parley P. Pratt. Phelps also served in the Utah territorial legislature from 1851 to 1857 and on the board of regents for the University of Deseret (now the University of Utah). He participated in the creation of the Provisional State of Deseret's constitution and wrote an almanac documenting the activities of the Latter-day Saints in Utah for fourteen years. He also helped develop the Deseret alphabet and obtained the first printing press used to print the Deseret News. He wrote poems and articles for the Deseret News, as well as essays on religious topics such as the Second Coming, the priesthood, and Joseph Smith's revelations. He began working as a lawyer in 1851 and "defended numerous Saints in the courts." Phelps also joined the Deseret Horticultural Society and Deseret Theological Institute. In 1856, he wrote the LDS hymn "If You Could Hie to Kolob" for Brigham Young. He died on March 7, 1872, in Salt Lake City, Utah Territory and is buried in the Salt Lake City Cemetery. According to attorney and author George M. McCune, "He died a faithful and zealous disciple of the restoration."

Hymns
Phelps is probably best known for his legacy of Mormon hymns, many of which appear in the current edition of the LDS Church's hymnal.

 * Adam-ondi-Ahman*
 * Come, All Ye Saints of Zion*
 * Come, All Ye Saints Who Dwell on Earth*
 * Come, Let Us Sing an Evening Hymn*
 * Gently Raise the Sacred Strain*
 * Glorious Things Are Sung of Zion
 * Hosanna Anthem
 * If You Could Hie to Kolob
 * Now Let Us Rejoice*
 * Now We'll Sing with One Accord*
 * O God, the Eternal Father*
 * O Stop and Tell Me, Red Man*
 * Praise to the Man
 * The Spirit of God Like a Fire Is Burning*
 * We're Not Ashamed to Own Our Lord*

Phelps also reworded popular hymns turning them into uniquely Latter Day Saint hymns.
 * Joy to the World! the Lord will Come*
 * Redeemer of Israel*

* Included in the first Latter Day Saint hymnal in 1835.

title: W. W. Phelps (Mormon)


text:
Meteor Music Awards

A Meteor Ireland Music Award was an accolade bestowed upon professionals in the music industry in Ireland and further afield. They had been bestowed each year since 2001, replacing the IRMA Ireland Music Awards held in the 1990s. Promoted by MCD Productions, the ceremony at which these accolades were bestowed upon worthy recipients was referred to colloquially as The Meteors, though occasionally also by its full title.

Event organisers confirmed in January 2011 that there would be no awards ceremony that year, with Meteor's cancellation of its sponsorship of the event widely blamed for this abrupt occurrence.

History
The Meteor Ireland Music Awards were the equivalent to the Canadian Juno Awards, the American Grammy Awards, the Echo Awards in Germany, and the United Kingdom's BRIT Awards. The awards take their name from their sponsors, Meteor.

Each year there was a mix of live performances and award presentations at a ceremony conducted in the Point Theatre, Dublin (2001–2007) and the Royal Dublin Society's Simmonscourt, Dublin (2008–2010). Irish artists to have showcased their music included Snow Patrol, Sinéad O'Connor, U2, Bell X1, Aslan, Westlife, the Blizzards, the Frames, the Coronas, Director, Hothouse Flowers, Cathy Davey, the Devlins, the Thrills, Paddy Casey, and the Immediate, whilst previous live performances by international artists have included the Pussycat Dolls, Amy Winehouse, Sugababes, Counting Crows, the Darkness, Kaiser Chiefs, Lionel Richie, and Tom Jones.

Presenters included both Irish and international figures from music, sport, film, television, and beauty: Joe Elliott, Denis Hickie, Colin Farrell, Alex Zane, and Rosanna Davison. The award ceremony was hosted by a number of different personalities throughout its history: Ed Byrne, Patrick Kielty, Amanda Byram, Podge and Rodge alongside Deirdre O'Kane, and Dara Ó Briain.

Originally held in the Point Theatre in Dublin, in 2008 the award ceremony moved to the RDS Simmonscourt of the Royal Dublin Society until its cancellation in 2011.

Recipients by year
A list of winners is to be found on the Meteor website.

title: Meteor Music Awards


text:
Sixpence

Sixpence may refer to:

Currency

 * Sixpence (British coin)
 * Sixpence (Irish coin)
 * Sixpence (Australian)
 * Sixpence (New Zealand coin)

Other uses

 * Flat cap, a rounded cap with a brim, also called a sixpence
 * Collin Sixpence (born 1974), Zimbabwean sculptor

title: Sixpence