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Weyl algebra

In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients, namely expressions of the form f m ∂ X m + f m − 1 ∂ X m − 1 + ⋯ + f 1 ∂ X + f 0. More let F be the underlying field, let F be the ring of polynomials in one variable, X, with coefficients in F; each fi lies in F. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X; the Weyl algebra is an example of a simple ring, not a matrix ring over a division ring. It is a noncommutative example of a domain, an example of an Ore extension; the Weyl algebra is isomorphic to the quotient of the free algebra on two generators, X and Y, by the ideal generated by the element Y X − X Y − 1. The Weyl algebra is the first in an infinite family of algebras known as Weyl algebras; the n-th Weyl algebra, An, is the ring of differential operators with polynomial coefficients in n variables. It is generated by ∂ Xi, i = 1... n. Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics.

It is a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra equal to the unit of the universal enveloping algebra. The Weyl algebra is referred to as the symplectic Clifford algebra. Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms. One may give an abstract construction of the algebras An in terms of relations. Start with an abstract vector space V equipped with a symplectic form ω. Define the Weyl algebra W to be W:= T /, where T is the tensor algebra on V, the notation means "the ideal generated by". In other words, W is the algebra generated by V subject only to the relation vu − uv = ω. W is isomorphic to An via the choice of a Darboux basis for ω; the algebra W is a quantization of the symmetric algebra Sym. If V is over a field of characteristic zero W is isomorphic to the underlying vector space of the symmetric algebra Sym equipped with a deformed product – called the Groenewold–Moyal product.

The isomorphism is given by the symmetrization map from Sym to W a 1 ⋯ a n ↦ 1 n! ∑ σ ∈ S n a σ ⊗ ⋯ ⊗ a σ. If one prefers to have the iħ and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by Xi and iħ∂Xi. Thus, the Weyl algebra is a quantization of the symmetric algebra, the same as the Moyal quantization, but the former is in terms of generators and relations and the latter is in terms of a deformed multiplication. In the case of exterior algebras, the analogous quantization to the Weyl one is the Clifford algebra, referred to as the orthogonal Clifford algebra. In the case that the ground field F has characteristic zero, the nth Weyl algebra is a simple Noetherian domain, it has global dimension n, in contrast to the ring it deforms, which has global dimension 2n. It has no finite-dimensional representations. Although this follows from simplicity, it can be more directly shown by taking the trace σ and σ for some finite-dimensional representation σ.

T r = t r. Since the trace of a commutator is zero, the trace of the identity is the dimension of the matrix, the representation must be zero dimensional. In fact, there are stronger statements than the absence of finite-dimensional representations. To any finitely generated An-module M, there is a cor

1968 NCAA University Division Cross Country Championships

The 1968 NCAA University Division Cross Country Championships were the 30th annual cross country meet to determine the team and individual national champions of men's collegiate cross country running in the United States. Held on November 25, 1968, the meet was hosted by Manhattan College at Van Cortlandt Park in the Bronx, New York City, New York; the distance for this race was 6 miles. All NCAA University Division members were eligible to qualify. In total, 24 teams and 217 individual runners contested this championship; the team national championship was again retained by their third title. The individual championship was won by Michael Ryan, from Air Force, with a time of 29:16.8. Distance: 6 miles NCAA Men's Division II Cross Country Championship

Daylighting (streams)

Daylighting is the term for restoration of an open-air watercourse which had at some point been diverted below ground back into an above-ground channel. The rationale behind returning the riparian environment of a stream, wash, or river to a more natural state is to reduce runoff, create habitat for species in need of it, or improve an area's aesthetics. In the UK, the practice is known as deculverting. In addition to its use in urban design and planning the term refers to the public process of advancing such projects. According to the Planning and Development Department of the City of Berkeley, "A general consensus has developed that protecting and restoring natural creeks' functions is achievable over time in an urban environment while recognizing the importance of property rights." Natural drainage systems help manage stormwater by infiltrating and slowing the flow of stormwater and bioremediating pollutants by soils and plants, reducing impervious surfaces, using porous paving, increasing vegetation, improving related pedestrian amenities.

Natural features — open, vegetated swales, stormwater cascades, small wetland ponds — mimic the functions of nature lost to urbanization. At the heart are plants and the deep, healthy soils that support them. All three combine to form a "living infrastructure" that, unlike pipes and vaults, increase in functional value over time; some efforts to blend urban development with natural systems use innovative drainage design and landscaping instead of traditional curbs and gutters and vaults. One such demonstration project in the Pipers Creek watershed reduced imperviousness by more than 18 percent; the project built bioswales, landscape elements intended to remove silt and pollution from surface runoff water and planted 100 evergreen trees and 1,100 shrubs. From 2001 to 2003, the project reduced the volume of stormwater leaving the street in a 2-year storm event by 98%; such a reduction can reduce storm damage to water quality and habitats for species such as the iconic salmon. The engineering alternatives have a expensive initial price, since they are replacing existing structures, albeit life-limited ones.

Further, conventional systems do not consider full cost accounting. The NDS alternatives can provide returns on investment by improving urban environments; the Street Edge Alternatives Street breaks most of the conventions of 150 years of standard American street design. Narrow, curved streets, open drainage swales, an abundance of diverse plants and trees welcome pedestrians as well as diverse species. Adjacent residents maintain city infrastructure in the form of street "gardens" in front of their homes, visually integrating the neighborhood along the street; the Natural Drainage System united the community visually and socially. The 110th Cascades SEA are a creek-like cascade of stair-stepped natural, seasonal pools that intercept, infiltrate and filter over 21 acres of stormwater draining through the project. Viable, daylighted streams exist only where neighbourhoods are intimately connected to restoration and stewardship values in their watersheds, since the health of an urban stream can not long survive carelessness or neglect.

With impervious surfaces having replaced most of the natural ground cover in urban environments, habitat for wildlife is reduced compared to historic baselines. Hydrologic changes have resulted, impervious waterways directly carry non-point pollution through urban creeks. One effective solution is to restore riparian habitat; this improves the entire urban watershed, far beyond the riparian channel itself. In the 1880s there were over 50 wild salmon streams in Vancouver alone. However, as Vancouver grew, these streams were lost to urbanization, they were covered by roads and businesses. They were lost when they were buried beneath sewers or culverts; the City of Vancouver and its residents are now making an effort to uncover these lost streams and restore them back to their natural state. The Hastings Creek Stream Daylighting Project was proposed in 1994 as a way to manage storm water and for aesthetic purposes; the idea was to bring the stream back to its once natural formation which would improve the surrounding habitat for wildlife as well as the proposed purposes.

This project's plan was finalized in 1997, work began the same year. The stream had existed in Hastings Park until 1935 when the Park became focused on entertainment rather than its original purpose when it was given to the city in 1889, to be a retreat for those with a passion for the outdoors; as the Pacific Nation Exhibition grounds continued to expand there was a continued loss of natural woodlands and waterways. It was not until the 1980s when the surrounding community began to look at continuing to uphold its original purpose; the daylighting project made major progress in 2013 in the area located in the Creekway Park, a parking lot. The daylighted stream will one day connect the Sanctuary in Hastings Park to the Burrard Inlet; the progress made in Creekway Park is a major step towards this goal. This daylighting project improved pedestrian and bikeway transit; this stream is now able to obtain the stormwater from the surrounding area, which reduces the load, felt by the municipality's storm sewers.

It is the storms in early autumn which provide the water flow for the creek, meaning that there is variable flow throughout the year. During the late summer months the moist soil is relied upon to maintain the vegetation of the area; this variation in flow does not allow for salmon migration through the creek.

Boomba music

Boomba music referred to as kapuka, is a form of hip pop music popular in Kenya. It incorporates hip hop and African traditional musical styles; the lyrics are in Sheng or local dialects. It is associated with the Ogopa Deejays and is believed to have originated in the late 1990s with artists such as Redsan, Bebe Cool and Chameleone, it went on to dominate East African airwaves after the release of the Ogopa Deejays' first album in 2001, which included artists such as the late E-Sir, Nameless, Mr. Lenny, Amani, Mr. Googz and Vinnie Banton amongst others; the over-saturation of playlists with this music style caused some to criticise it, including the artists K-South who had a hit titled "Kapuka This, Kapuka That". This led to other styles such as genge music being formed by producer Clemo to diversify the market. Despite the critics the music continues to be popular in Kenya although more so now in Uganda. Of late, "genge" has come to refer more to boomba and "kapuka" music. Riftsyde flava/NAXVEGAS MUZIK is a genre of hip hop/Afro Dancehall music that originated from the town of Nakuru or colloquially Naks/NaxVegas as many youths call it.

The music is a blend of indigenous kapuka mixed with a reggae baseline. It is characterised by flashy Musicians who rap about their sexual escapades and the number of women that they keep, but cipher their language to avoid being explicit. Riftsyde Flava has evolved in recent times. Artists in this genre include award-winning artist Teferah who has numerous hits having worked with Nyanda from Brick and Lace on the song "Put It On Me" produced in Romania by Tommo records, getting signed to an American music label Mpact Muzik in 2017. Teferah is seen as a break-through artist in this genre. Other Riftsyde Flava artists are Kimbado, Hush BK, Avril, Zae de Kim, Tomsy and many more. Kapuka rap is a genre of hip hop music, derived from Boomba music, but incorporates more dance-pop and electronic music than reggae and ragga but still incorporates some hip hop. Artists such as Collo and Camp Mulla incorporate kapuka rap in some of their songs. "KArE" - P-Unit "Party Don't Stop" - Camp Mulla "Feel No Pain" - Camp Mulla "Chini ya Maji" -Collo Kenyan hip hop Genge, another form of Kenyan hip-hop

Jessica Mindich

Jessica Mindich is a former lawyer and the founder and CEO of the Caliber Collection, which takes guns off streets and transforms them with bullet casings into jewelry and nonprofit donations. The Caliber Collection began in January 2012 in collaboration with the Mayor of Newark, NJ, Cory Booker, as a way to turn illegal and unwanted guns into jewelry. 20 % of the net proceeds are to fund voluntary gun amnesty programs. Since Caliber has partnered additionally with San Francisco, Detroit and most North Miami Beach. To date, the programs have taken 2,500 illegal guns off the streets of those cities and has raised over $165,000 for police departments. In 2013, Jessica Mindich was able to present the Mayor Booker of Newark, NJ with a check for $40,000, as a result of nine weeks of sales, to fund her first gun buyback and amnesty program in Newark, NJ. A second buyback was funded in Newark by Mindich’s company in 2014 and collected 185 guns ranging from sniper rifles to assault weapons. In 2016, Mindich donated $25,000 to the Wayne County Sheriff’s Office in Michigan to buyback 500 guns.

The Caliber Foundation is a registered 501 that supports victims and communities affected by illegal gun violence. In 2014, Serena Williams and the Serena Williams Fund announced a grant to the Caliber Foundation to prevent young people from experiencing the same pain Williams' had through the murder of her sister in 2003. In 2016, street artist Shepard Fairey donated 20% of sales from his art exhibit at the Library Street Collective in Detroit, MI. In July 2016, Fairey dedicated a portion of the proceeds of the sale from his gun control American Civics collaboration with photographer Jim Marshall to the Caliber Foundation. Mindich is the founder of the Raise The Caliber initiative, a national advocacy campaign to end illegal gun violence. Proceeds from partnerships under Raise The Caliber are donated to the Caliber Foundation. In 2016, Mindich became a frequent Huffington Post journalist creating a series of interviews under the title YOU CAN DO SOMETHING ABOUT IT. Interviews included men and women who are leaders and influencers on the subject of gun violence in America.

Caliber Collection website