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Presentation of a group

In mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We say G has presentation ⟨ S ∣ R ⟩. Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R; as a simple example, the cyclic group of order n has the presentation ⟨ a ∣ a n = 1 ⟩, where 1 is the group identity. This may be written equivalently as ⟨ a ∣ a n ⟩, thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity; such terms are called relators, distinguishing them from the relations that do include an equals sign. Every group has a presentation, in fact many different presentations.

A related but different concept is that of an absolute presentation of a group. A free group on a set S is a group where each element can be uniquely described as a finite length product of the form: s 1 a 1 s 2 a 2 ⋯ s n a n where the si are elements of S, adjacent si are distinct, ai are non-zero integers. In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse. If G is any group, S is a generating subset of G every element of G is of the above form. For example, the dihedral group D8 of order sixteen can be generated by a rotation, r, of order 8. However, we have, for example, rfr = f, r7 = r−1, etc. so such products are not unique in D8. Each such product equivalence can be expressed as an equality to the identity, such as rfrf = 1 r8 = 1 f2 = 1. Informally, we can consider these products on the left hand side as being elements of the free group F = <r, f>, can consider the subgroup R of F, generated by these strings.

If we let N be the subgroup of F generated by all conjugates x−1Rx of R it follows by definition that every element of N is a finite product x1−1r1x1 ⋯ xm−1rm xm of members of such conjugates. It follows that each element of N, when considered as a product in D8, will evaluate to 1, thus D8 is isomorphic to the quotient group F/N. We say that D8 has presentation ⟨ r, f ∣ r 8 = 1, f 2 = 1, 2 = 1 ⟩. Here the set of generators is S =, the set of relations is R =. We see R abbreviated, giving the presentation ⟨ r, f ∣ r 8 = f 2 = 2 = 1 ⟩. An shorter form drops the equality and identity signs, to list just the set of relators, which is. Doing this gives the presentation ⟨ r, f ∣ r 8, f 2, 2 ⟩. All three presentations are equivalent. Although the notation ⟨S|R⟩ used in this article for a presentation is now the most common, earlier writers used different variations on the same format; such notations include: ⟨S|R⟩ ⟨S. Let R be a set of words on S, so R gives a subset of F S. To form a group with presentation ⟨ S ∣ R ⟩, the idea is to take F S quotient by the smallest normal subgroup such that each element of R gets identified with the identity.

Note that R might not be a subgroup, let alone a normal subgroup of F S, so we cannot take a quotient by R. The solution is to take the normal closure N of R in F S; the group ⟨ S ∣ R ⟩ is defined as the quotient group ⟨ S ∣ R ⟩ = F S /

Shoreland Lutheran High School

Shoreland Lutheran High School is a Lutheran High School in Somers, affiliated with the Wisconsin Evangelical Lutheran Synod and operated by a federation of 24 area congregations. Shoreland offers a variety of courses, including music, foreign languages, small engine repair, computer science courses, business course designed to jump start the students for college level business courses. Further, a pre-ministry program for students wishing to become teachers or pastors in the synod is available, it has five Advanced Placement classes: AP Biology, AP Calculus, AP Chemistry, AP History, AP Literature and Composition. Honors Spanish & Honors Physics. In the 2014-2015 school year Shoreland opened a STEM Academy. Through a partnership with MSOE it offers students transcript credit. Shoreland opened in 1971 at Friedens Lutheran Church in Wisconsin, its current building was constructed in 1979, with an addition to the west end and major internal renovations completed in 1997. The Science wing addition added three labs, a STEM classroom and another room in 2014.

A final addition in the near future will include renovating the lunch area, office complex, adding an auxiliary gymnasium and an auditorium. Shoreland offers the following activity in conjunction with the Metro Classic Conference Honors Band Shoreland offers the following Festival Choir Band Drama Forensics Robotics Club National Honor Society Ministry Club Student Ambassadors Student Council Students for Life Shoreland is a member of the Metro Classic Conference, offers the following sports: Junior varsity, varsity football Junior varsity, varsity soccer Freshman, junior varsity, varsity boys' and girls' basketball Freshman, junior varsity, varsity girls' volleyball Junior varsity, varsity softball Junior varsity, varsity baseball Junior varsity, varsity girls' soccer Track and field Cross country Boys' and girls' golf Junior varsity, varsity wrestling Cheerleading Official website School Review website

Jong Hyun-wook

Jong Hyun-Wook, nicknamed "Slave Jong", is a former South Korean relief pitcher in the Korea Baseball Organization. Jong attended Dongdaemoon Commerce High School in South Korea. In 1995, he was selected to the South Korea national junior team, participated in the 1995 World Junior Baseball Championship in Boston, United States, along with future Major League pitchers Kim Sun-Woo, Seo Jae-Weong and Kim Byung-Hyun. Signed by the Samsung Lions on January 1998, he temporarily left the Samsung Lions for the two-year military service In the 2005 season, returned to the team in August 2007. Many baseball fans describe him as a "whatever-is-needed utility pitcher" because of his ability to fill in as a starting pitcher or relief pitcher as needed. In the 2008 season, Jong took a role as a utility pitcher for the Lions, pitching 127 innings in 53 games, starting 7, going 10-4 with an ERA of 3.40 and 11 holds in the process. His 3.40 ERA ranked 9th in the league, pitching over the required innings to qualify for the ERA title.

After the 2008 season, Jong was called up to the South Korea national baseball team for the 2009 World Baseball Classic. Although Jong was expected to garner nothing but mopup duties for the team, he was brought up in the five critical games as a long reliever because of the multiple injuries to the South Korea's bullpen, he emerged as a WBC relief hero, going 1–0 with a 1.74 ERA and 13 strikeouts in 10.1 innings pitched. He tied for third in the Classic in strikeouts. In Round 2, he silenced the Mexican bats, earning the win with two strikeouts and allowing only a hit in 2.2 innings pitched. Jong appeared in the semifinal game against Venezuela, pitched a 1.1 inning no-hitter, striking out MLB All-Star sluggers Melvin Mora, Carlos Guillén and Magglio Ordóñez. In the 2009 KBO season, Jong appeared in 62 games, taking a role as a utility pitcher and posting 8 wins, 6 saves and 16 holds, he was pitching a 2.20 ERA for a 5-4 record in 65.1 innings before the All-Star break, but his stats dipped in the second half of the season due to arm fatigue and he finished the year with a 3.42 ERA.

He is known as "Slave Jong" because of him being used as a starter and a long reliever in his team. Due to his good performance in the 2009 World Baseball Classic, many netizens in South Korea called him "Guk-no" which means "national slave". Career statistics and player information from Baseball-Reference Career statistics and player information from Korea Baseball Organization

Trade barrier

Trade barriers are government-induced restrictions on international trade. Economists agree that trade barriers are detrimental and decrease overall economic efficiency. Most trade barriers work on the same principle: the imposition of some sort of cost on trade that raises the price or availability of the traded products. If two or more nations use trade barriers against each other a trade war results. Barriers take the form of non-tariff barriers to trade. In theory, free trade involves the removal of all such barriers, except those considered necessary for health or national security. In practice, however those countries promoting free trade subsidize certain industries, such as agriculture and steel. High income countries tend to have less trade barriers than middle income countries which, in turn, tend to have less trade barriers than low income countries. Small states tend to have lower trade barriers than large states; the most common trade barriers are on agricultural goods. Textiles and footwear are the manufactured goods which are most protected by trade barriers.

Tariffs have been declining in the last twenty years as the influence of the World Trade Organization has grown, but states have increased their use of non-tariff barriers. According to Chad Bown and Meredith Crowley, world trade is "probably" vastly more liberal in current times than was the case historically. According to Ronald Findlay and Kevin H. O’Rourke, "for the nineteenth and twentieth centuries trade barriers and transport costs were the most important barriers to trade", they write, "during the mercantilist era price gaps were as to be due to trade monopolies and wars as to transport costs and tariffs, which are more quantifiable."Georgetown University Professor Marc L. Busch and McGill University Professor Krzysztof J. Pelc note that modern trade deals are long and complex because they tackle non-tariff barriers to trade, such as different standards and regulations, in addition to tariffs. Due to decreasing tariff barriers since World War II, countries have become likely to enact trade barriers in forms other than tariffs.

National firms lobby their own governments to enact regulations that are designed to keep out foreign firms, modern trade deals are one way to do away with such regulations. The barriers can take many forms, including the following: Tariffs Non-tariff barriers to trade include: Import licenses Export control / licenses Import quotas Subsidies Voluntary Export Restraints Local content requirements Embargo Currency devaluation Trade restriction Trade barriers are criticized for the effect they have on the developing world; because rich-country players call most of the shots and set trade policies, goods such as crops that developing countries are best at producing still face high barriers. Trade barriers such as taxes on food imports or subsidies for farmers in developed economies lead to overproduction and dumping on world markets, thus lowering prices and hurting poor-country farmers. Tariffs tend to be anti-poor, with low rates for raw commodities and high rates for labor-intensive processed goods.

The Commitment to Development Index measures the effect that rich country trade policies have on the developing world. Trade barriers are a combination of conformity and per-shipment requirements requested abroad, weak inspection or certification procedures at home; the impact of trade barriers on companies and countries is uneven. One particular study showed. Another negative aspect of trade barriers is that they result in a limited choice of products and would therefore force customers to pay higher prices and accept inferior quality. Trade barriers obstruct free trade. Before exporting or importing to other countries, they must be aware of restrictions that the government imposes on the trade. Subsequently, they need to make sure that they are not violating the restrictions by checking related regulations on tax or duty, they need a license in order to ensure a smooth export or import business and reduce the risk of penalty or violation. Sometimes the situation becomes more complicated with the changing of policy and restrictions of a country.

North American Free Trade Agreement South Asia Free Trade Agreement European Free Trade Association Union of South American Nations New West Partnership Gulf Cooperation Council common market Agricultural policy Customs union Denied trade screening End-use certificate Free trade International free trade agreement Region lock Free economic zone. ITC's Market Access Map, an online database of customs tariffs and market requirements ITC's Non-Tariff Measures Business Survey database, including regulatory and procedural obstacles that trading companies face both at home and abroad


Mufumbwe is a town of Mufumbwe District in the North-Western Province of Zambia. It is located at 13°41'0S 24°48'0E with an altitude of 1069 metres; the main economic activity of the town is Agriculture with most of the population being involved in small scale farming. The main crop grown in the area is maize. Most households in Mufumbwe maintain a year round stock of maize; the maize meal "Bunga" in Kikaonde the main Language of the Town is the staple food in Mufumbwe. The District has two chiefdoms namely Mushima. There are a total of 16 wards i.e Kashima west Kashima East Matushi Munyambala Kamabuta Kalambo Chizela Shukwe Kikonge Kalengwa Kabipupu Musonweji Kaminzekenzeke Mushima Lalafuta Miluji Area Member of Parliament - Hon Elliot Kamondo Council chairperson- Hon Elijah Munyompe Banking services are available through a Cavmont Bank Mufumbwe branch and Zanaco Express services. The District is centrally located in North-Western Province. 13°41′0″S 24°48′0″E

John Shaw Rennie

Sir John Shaw Rennie was a British civil servant and the United Nations Relief and Works Agency for Palestine Refugees in the Near East's Commissioner-General from 1971 to 1977. John Rennie was born in Glasgow and educated at Hillhead High School, Glasgow University and Balliol College, Oxford. In 1951 Rennie was appointed Britain's deputy colonial secretary for Mauritius, he was the British Resident in Vanuatu from 1955 to 1962. From 1962 to 1968 he was Governor of Mauritius. From 1968 to 1971 Rennie was UNRWA deputy Commissioner-General under Laurence Michelmore, who persuaded then-U. N. Secretary-General U Thant to appoint Rennie as his successor