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# Normal closure (group theory)

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In group theory, the normal closure of a subset $S$ of a group $G$ is the smallest normal subgroup of $G$ containing $S.$

## Properties and description

Formally, if $G$ is a group and $S$ is a subset of $G,$ the normal closure $\operatorname {ncl} _{G}(S)$ of $S$ is the intersection of all normal subgroups of $G$ containing $S$: $\operatorname {ncl} _{G}(S)=\bigcap _{S\subseteq N\triangleleft G}N.$
The normal closure $\operatorname {ncl} _{G}(S)$ is the smallest normal subgroup of $G$ containing $S,$ in the sense that $\operatorname {ncl} _{G}(S)$ is a subset of every normal subgroup of $G$ that contains $S.$
The subgroup $\operatorname {ncl} _{G}(S)$ is generated by the set $S^{G}=\{s^{g}:g\in G\}=\{g^{-1}sg:g\in G\}$ of all conjugates of elements of $S$ in $G.$
Therefore one can also write $\operatorname {ncl} _{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\dots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}:n\geq 0,\epsilon _{i}=\pm 1,s_{i}\in S,g_{i}\in G\}.$
Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set $\varnothing$ is the trivial subgroup.
A variety of other notations are used for the normal closure in the literature, including $\langle S^{G}\rangle ,$ $\langle S\rangle ^{G},$ $\langle \langle S\rangle \rangle _{G},$ and $\langle \langle S\rangle \rangle ^{G}.$
Dual to the concept of normal closure is that of normal interior or normal core, defined as the join of all normal subgroups contained in $S.$

## Group presentations

For a group $G$ given by a presentation $G=\langle S\mid R\rangle$ with generators $S$ and defining relators $R,$ the presentation notation means that $G$ is the quotient group $G=F(S)/\operatorname {ncl} _{F(S)}(R),$ where $F(S)$ is a free group on $S.$
# Xanthophyllum adenotus

Xanthophyllum adenotus is a plant in the family Polygalaceae. The specific epithet adenotus is from the Greek meaning 'gland', referring to the leaf glands.

## Description

Xanthophyllum adenotus grows as a shrub or tree up to 15 metres (50 ft) tall with a trunk diameter of up to 25 centimetres (10 in). The flowers are pinkish to pale violet, drying dark red. The pale or reddish-brown fruits are round and measure up to 1.8 cm (1 in) in diameter.

## Distribution and habitat

Xanthophyllum adenotus grows naturally in Sumatra and Borneo. Its habitat is mixed dipterocarp forests from sea-level to 700 metres (2,300 ft) altitude.

## Varieties

One variety is currently recognised:
* X. adenotus var. arsatii
# Xanthophyllum beccarianum

Xanthophyllum beccarianum is a tree in the family Polygalaceae. It is named for the Italian botanist Odoardo Beccari.

## Description

Xanthophyllum beccarianum grows up to 12 metres (40 ft) tall with a trunk diameter of up to 17 centimetres (7 in). The bark is grey-green and smooth. The flowers are orange-red when dry. The round fruits measure up to 1.5 cm (1 in) in diameter.

## Distribution and habitat

Xanthophyllum beccarianum is endemic to Borneo. Its habitat is mixed dipterocarp forests from sea-level to 200 metres (700 ft) altitude.
# Władysław Aleksander Łubieński

Władysław Aleksander Łubieński (1703–1767) was archbishop of Lwów (1758–59) and primate of Poland (1759–1767). He was an ally of the Czartoryski Familia and of the Russian Empire and an opponent of religious tolerance. He acted as Interrex in 1763–1764, after the death of King Augustus III of Poland and prior to the election of Stanisław August Poniatowski as king of the Polish–Lithuanian Commonwealth.
# Xingan salamander

The Xingan salamander (Hynobius maoershanensis) is a species of salamander in the family Hynobiidae, endemic to China: it is only known from its type locality, Mao'ershan (Mount Mao'er) in the Xing'an County, Guangxi. Its natural habitats are marshes and the surrounding forests. It is threatened by habitat loss and, living close to the summit of Mao'ershan at around 2,000 m (6,600 ft) asl, by climate change. It occurs within the Mao'ershan National Nature Reserve.
# Museum of Islamic Art, Ghazni

The Museum of Islamic Art is a museum located in Ghazni, Afghanistan. It is located in Rauza, a suburb of Ghazni. The museum was first opened by the Italian Archaeological Mission in 1966 in the restored sixteenth-century Mausoleum of Abd al-Razzaq to display artifacts of the Islamic period. Work was halted during the war with the Soviet Union after 1979 in which several of its artifacts were damaged. It was since restored in 2004–2007. A number of artifacts unearthed in the Ghazni area are also found in museums in Kabul.
# Xtrac Limited

Xtrac Limited, also known as Xtrac Transmission Technology, is a British engineering company founded in 1984 by the former Hewland engineer Mike Endean to make 4WD systems and gearboxes for rallycross and later rally and racing cars. Endean, together with Chris Goddard, who had been working on the electronic components essential to the system, in 1983 developed the first Xtrac 4WD system, for Norwegian rallycross star Martin Schanche. Schanche had thought up the idea of a changeable hydraulic 4WD system (his 1984 ERC winning Ford Escort Mk3 Xtrac-Zakspeed had a stepless FWD:RWD ratio of 28:72 to 50:50) and financed its development. This 560bhp so-called Xtrac No. 1 was bought by Endean, in the mid-1990s, who raced it himself for many a year in carsprints and hillclimbs.
"Xtrac started making gearboxes in the mid to late 1980s for the then-new Group A rally cars from their original premises in Wokingham and then continued to do so after moving in 1986 to a new 20,000 sq ft factory in Finchampstead."
In 2000, Xtrac built a bespoke 8,200 m$^2$ (88,000 sq ft) high tech factory in Thatcham, Berkshire, England, and is currently supplying many top level sectors of motor sport, including F1, prototype and GT sportscars, IndyCar, Grand-AM, rally cars and touring cars. In 2010 the company supplied its complete 1044 gearbox, designed and developed in only six months to three F1 teams: Lotus, Virgin and HRT. This is mated to the Cosworth CA2010 engine.
In addition to supplying many high-profile motorsport customers, Xtrac has expanded in other areas such as automotive, marine and aerospace. The Pagani Huayra was a vehicle which used the Xtrac 1007 AMT transmission.
In 2019 it was announced that Xtrac would be the sole supplier of gearboxes for the US based NASCAR Championship, this was due to start in 2021 but with Covid-19 restrictions being in place this was delayed until 2022
In 2020 it was announce that Xtrac would supply gearboxes as part of the hybrid power unit for the Le Mans Daytona h endurance hybrid sport car with Bosch supplying the motor generator unit and Williams Advanced Engineering supplying the batteries.

## Footnotes

* ↑ Motor Sport, April 2010, Pages 108–109
* ↑ Racecar Engineering, Vol 20 No 3, March 2010, Pages 31–36.
* ↑ "Xtrac - Xtrac Announced as Sole Gearbox and Driveline Supplier for NASCAR's 2022 Next Gen Cup Car". Xtrac. 2021-05-05. Retrieved 2022-06-16.
* ↑ "Xtrac appointed gearbox supplier for New Hybrid sports car endurance racing". Newbury Weekly News Business Today section. 8 October 2020.
# Xtra Ottawa

Xtra Ottawa (formerly Capital Xtra) was a lesbian, gay, bisexual and transgender community newspaper published in Ottawa, Ontario, Canada. It was launched in 1993. Unlike its biweekly sister publications Xtra in Toronto and Xtra Vancouver in Vancouver, Xtra Ottawa started as a monthly, and was later published 17 times a year, with a publication schedule of every three weeks. The Ottawa edition had a circulation of 20,000 copies which reached 36,000 readers.
Printed on newsprint in tabloid format from its establishment in 1993, Pink Triangle Press announced on January 14, 2015, that the paper edition would be discontinued and the publication would continue in an exclusively digital media format.
Xtra Ottawa also published a gay tourism map, Out in Ottawa, every spring; the Ultimate Pride Guide in mid-July; and the Index gay business directory in December.

## History

Xtra Ottawa was launched by Pink Triangle Press as Capital Xtra in 1993, shortly after the organization learned that Ottawa's existing LGBT publication, GO Info, had largely collapsed and was publishing irregularly with only a skeleton staff. Brandon Matheson, then a freelance journalist working in the Parliament Hill Press Gallery, was chosen to launch the project in conjunction with George Hartsgrove, then an administrator at the University of Ottawa. GO Info briefly tried to beef up its content in the face of its new competition, but folded in early 1994.
In 2007, Ottawa City Council attempted to pass restrictions on the paper's distribution in public facilities after a complaint was filed by Greg Evans.
In March 2010, Capital Xtra was renamed Xtra Ottawa with the launch of a redesign of the Xtra papers in the Toronto, Ottawa and Vancouver markets.
The paper's final print edition was published on February 12, 2015.

## Contributors

Early contributors to the magazine included Irshad Manji, Alex Munter, David Pepper, Andrew Griffin and Don McLean. Other contributors have included Glenn Crawford, Suki Lee, Blaine Marchand, Ariel Troster and Bradley Turcotte. Photographic contributors have included Pat Croteau, Eric Faure, Phillip Hannan, Shawn Scallen and Ben Welland.

## Community involvement

Xtra Ottawa put on a series of events every year for the Ottawa queer community: the LGX business & consumer expo each spring, the Transgress Festival as part of the Ottawa International Writer's Festival each October, and the Capital Xtra! Community Achievement Awards ceremony to acknowledge local volunteers and activists.
# Xylocopa bombylans

Xylocopa bombylans, the peacock carpenter bee, is a species of carpenter bee found in Australia. It gets its common name by its habit of burrowing into wood.
It was originally described by Danish naturalist Johan Christian Fabricius in 1775. Its specific epithet is the Latin bombylans "like a bumblebee". The name is derived from the Ancient Greek (ξυλοκὀπος / ksu-lo-KO-pos) "wood-cutter".
As its name implies, the peacock carpenter bee is a metallic blue-green or green in colour, although it may even appear purplish from some angles. A large stocky bee, it is often heard by its loud low-pitched buzzing while flying between flowers. The male has white face markings. The bee does have a sting which is potentially painful, although no stings have been recorded.
The natural distribution is from the Cape York Peninsula in north Queensland south along the eastern seaboard to the Sydney region (where it overlaps with X. aerata). They can be seen from spring to autumn, commonly feeding at pea flowers of the family Fabaceae, such as Gompholobium species, including Gompholobium latifolium in spring, and Pultenaea elliptica in autumn. It is a pollinator of Melastoma affine. Other flowers visited include those of Banksia, Xanthorrhoea, Hardenbergia violacea, Tristania, Leptospermum, Aotus, Cassia and Leucopogon.
The Peacock carpenter bee nests by hollowing out stalks of grasstrees (Xanthorrhoea), or soft wood such as Banksia, Casuarina, Melaleuca and Leptospermum. The female excavates a tunnel with her jaws and picks up and dumps the wood shavings outside. The hollow can reach 30 centimetres (12 in) long by 11–14 millimetres (0.43–0.55 in) in diameter. Larger pieces of wood may allow for multiple tunnels. Several female bees may use a nest, one breeding and the others guarding. A bee defends the 7–10-millimetre (0.3–0.4 in) wide entrance by blocking it with its abdomen (compare Allodapula). Both male and female bees may overwinter within the tunnels. The tunnels are partitioned into several cells, where the mother bee lays an egg in each accompanied by provisions of nectar and pollen.
# Xylaria hypoxylon

Xylaria hypoxylon is a species of bioluminescent fungus in the family Xylariaceae. It is known by a variety of common names, such as the candlestick fungus, the candlesnuff fungus, carbon antlers, or the stag's horn fungus. The fruit bodies, characterized by erect, elongated black branches with whitened tips, typically grow in clusters on decaying hardwood. The fungus can cause a root rot in hawthorn and gooseberry plants.

## Taxonomy

Xylaria hypoxylon was first described by Carl Linnaeus in 1745, and then later mentioned by him in his Species Plantarum II.
The specific epithet is derived from the Greek words hypo meaning "below", and xylon, meaning "wood".
Genetic evidence has created the Xylaria hypoxylon complex. The complex was created when taxonomists and amateurs noticed large variation in appearance and spore shape within the main species and decided that the differences were large enough to warrant a complex formation. This complex has been debated between different experts as valid and invalid for the past three years and has remained in tension.
Though they are similar, the complex does not include Xylaria polymorpha and Xylaria longipes which look similar but are not closely enough related to Xylaria hypoxylon to apply.

## Description

Fruit bodies (ascocarps) are cylindrical or flattened with dimensions of 3–8 centimetres (1+1⁄8–3+1⁄8 in) tall × 2–8 mm thick. The erect ascocarps are often twisted or bent, and typically sparsely branched, often in a shape resembling a stag's antlers. Specimens found earlier in the season, in spring, may be covered completely in asexual spores (conidia), which manifests itself as a white to grayish powdery deposit. Later in the season, mature ascocarps are charcoal-black, and have minute pimple-like bumps called perithecia on the surface. These are minute rounded spore bearing structures with tiny holes, or ostioles, for the release of sexual spores (ascospores). The perithecia are embedded in the flesh of the ascocarp, the stroma, which is tough, elastic, and white. Within the perithecia, the asci are 100 × 8 μm.
Ascospores are kidney-shaped, black, and smooth, with dimensions of 10–14 × 4–6 μm. The asexual spores (mitospores) are ellipsoid in shape, smooth, and hyaline.

### Edibility

Although not poisonous, the small size and tough texture of this fungus deter consumption. It is considered inedible by some guides.

### Similar species

Many species of the genus resemble X. hypoxylon, including Xylaria polymorpha, which is thicker and not as branched.

## Chemical compounds

A variety of chemical compounds with in vitro properties have been identified in this fungus. The compounds xylarial A and B both have moderate cytotoxic activity against the human hepatocellular carcinoma cell line Hep G2. The pyrone derivative compounds named xylarone and 8,9-dehydroxylarone also have cytotoxic activity. Several cytochalasins, compounds that bind to actin in muscle tissue, have been found in the fungus. X. hypoxylon also contains a carbohydrate-binding protein, a lectin, with a unique sugar specificity, and which has potent anti-tumor effects in various tumor cell lines.
# Xylariaceae

The Xylariaceae are a family of mostly small ascomycetous fungi. It is one of the most commonly encountered groups of ascomycetes and is found throughout the temperate and tropical regions of the world. They are typically found on wood, seeds, fruits, or plant leaves, some even associated with insect nests. Most decay wood and many are plant pathogens.
One example of this family is King Alfred's Cake (Daldinia concentrica).
Phylogenetic analyses published in 2009 suggest that there are two main lineages in this family, Hypoxyloideae and Xylarioideae.

## Genera

This is a complete list of genera in the Xylariaceae, based on the 2007 Outline of Ascomycota. A question mark before the genus name indicates that the placement of that taxon in this family is uncertain.
Amphirosellinia — Annulohypoxylon — Anthostomella — Appendixia — Areolospora — ?Ascotricha — Ascovirgaria — Astrocystis — Barrmaelia — Biscogniauxia — Calceomyces — Camillea — Chaenocarpus — Collodiscula — Creosphaeria — Cyanopulvis — Daldinia — Discoxylaria — ?Emarcea — Engleromyces — Entoleuca — Entonaema — Euepixylon — Fasciatispora — Fassia — Gigantospora — Guestia — Halorosellinia — Helicogermslita — Holttumia — Hypocopra — Hypoxylon — Induratia — Jumillera — Kretzschmaria — Kretzschmariella — Leprieuria — ?Leptomassaria — Lopadostoma — Muscodor — Myconeesia — Nemania — Nipicola — Obolarina — Occultitheca — Ophiorosellinia — Pandanicola — Paramphisphaeria — ?Paucithecium — Phylacia — Pidoplitchkoviella — Podosordaria — Poroleprieuria — Poronia — Pyrenomyxa (=Pulveria) — Rhopalostroma — Rosellinia — Sabalicola — Sarcoxylon — ?Sclerodermatopsis — ?Seynesia — Spirodecospora — Stereosphaeria — Stilbohypoxylon — Striatodecospora — Stromatoneurospora — Thamnomyces — Theissenia — Thuemenella — Vivantia — Wawelia — Whalleya — Xylaria — Xylocoremium — Xylotumulus
# Xylariopsis esakii

Xylariopsis esakii is a species of beetle in the family Cerambycidae. It was described by Mitono in 1943. It is known from Taiwan.
# Mvumi Makulu

Mvumi Makulu is an administrative ward in the Dodoma Rural district of the Dodoma Region of Tanzania. According to the 2002 census, the ward has a total population of 6,549.
# Mvzzle

Gift Selaelo Morukhuladi (born 11 November 1994), professionally known as Mvzzle (/ˈmʌˈzl/) is a South African DJ, singer-songwriter, and producer born in Musina and raised in Westenburg, Polokwane. He is famous for producing the single "Umlilo" by DJ Zinhle.

## Personal life

Born Gift S. Morukhuladi, in Musina and raised in a village called Mountain View. His mother is a businesswoman and his father was the Conductor of the military band. At age 8, he moved to Polokwane International Gateway to be with his parents. At this point, his dad had gone from Conductor to Band Master at the SA Army Band.

## Early life
### 2014–2015

Before producing House music, Mvzzle was a full time Hip Hop producer and rapper in high school. He was with a musical trio called Cyclops alongside his teammates Septic Rooger (Kagiso Mokwena) and Gran-T (Thabiso Ledwaba), which Mvzzle produced his first singles and mixtape with. After the trio disbanded, Mvzzle started his solo career producing both house and rap music, releasing singles titled Carpe Diem and Thick Madam, which he entered the SABC1's 1s and 2s competition with.

### 2016

Initially producing Hip-Hop and R&B, he started producing house music in 2016. He dropped out of college to pursue music and the assistance of South African Arts and Culture Youth Forum who gave him opportunities to play at events such as Major League Gardens, Mapungubwe Arts Festival, and Motsepe Foundation.

### 2018

He entered SABC 1's Dj competition, 1's and 2's – where he placed third. After the competition, MVZZLE contacted DJ Zinhle with two beats, one of which is now the song "Umlilo" featuring Rethabile – who had earlier been in contact with DJ Zinhle on Instagram proposing a collaboration with her.

## Career

DJ Zinhle's Umlilo has been certified multi-platinum and within its first three months, it reached over 5.1 million streams.
In celebration of the song's success, DJ Zinhle released a second video for Umlilo in December.
The hit single went on to receive triple-platinum plaques, and multiple SAMAs nominations. The single won a NAMA award. The female producer DJ Zinhle was accused of buying nominations for the song.

## Discography
### Singles and EPs

| Title | Type | Release details | Certification |
| Uvalo | Single | Release date: 2020Label: Warner Music South AfricaFormats: Digital download | TBA |
| Sonke | EP | Release date: 2021Label: Warner Music South AfricaFormats: Digital download | TBA |

## Filmography
### Television

| Year | Title | Role | Notes | Results | Ref. |
| 2018 | 1's and 2's | Himself | Contestant | Runner-up | |

## Awards and nominations

| Year | Award Ceremony | Prize | Recipient/Nominated work | Result | Ref. |
| 2020 | South African Music Awards | Record of the Year (fan-voted) | "Umlilo" | Nominated | |
| 2021 | South African Music Awards | Best Music Video of the Year | "Umlilo" | Nominated | |
# YPCC

YPCC may refer to:
* Yupiit Piciryarait Cultural Center, Bethel, Alaska, United States
* Cocos (Keeling) Islands Airport, Australia
# Music for Mechanics

Music for Mechanics is the first sequential collection of the American comic book series Love and Rockets by the Hernandez brothers and published in 1985.
The cover of the book is by Jaime Hernandez and the preface by Carter Scholz.

## Contents

These first stories, dated 1981–1985, are science-fiction oriented, adding monsters and dinosaurs and extra-terrestrials to the first latino themes, and some stories and themes may or may not be considered canon with the universes developed by the brothers in later books.
| No. | Title | Author | Pages | Comments |
| 1. | Mechan-X | Jaime Hernandez | 8 | Signed "Izzy Ruebens"; introducing characters Maggie and Hopey. |
| 2. | Locas Tambien | Jaime | 4 | |
| 3. | How to Kill a | Jaime | 4 | Signed "Isabel Ruebens"; not listed in the "Contents" of the book. |
| 4. | BEM | Gilbert Hernandez | 40 | 5 parts; introducing character Luba). |
| 5. | Barrio Huerta | Jaime | 1 | |
| 6. | Penny Century, You're Fired | Jaime | 1 | |
| 7. | Music for Monsters I | Gilbert | 6 | |
| 8. | Mecanicos | Jaime | 6 | |
| 9. | Mechanics | Jaime | 41 | Introducing wrestling. |
| 10. | Radio Zero | Gilbert | 10 | |
| 11. | Music for Monsters II | Gilbert | 4 | |
| 12. | Somewhere in California | Mario Hernandez | 10 | |
| 13. | A Little Story | Gilbert | 6 | Village of Palomar first appearance, not cited though. |
| | Cover gallery | Los Bros Hernandez | 4 | |

## Chronology

Next album: Chelo's Burden
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