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Infimum and supremum

In mathematics, the infimum of a subset S of a ordered set T is the greatest element in T, less than or equal to all elements of S, if such an element exists. The term greatest lower bound is commonly used; the supremum of a subset S of a ordered set T is the least element in T, greater than or equal to all elements of S, if such an element exists. The supremum is referred to as the least upper bound; the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary ordered sets are considered; the concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the positive real numbers ℝ+ does not have a minimum, because any given element of ℝ+ could be divided in half resulting in a smaller number, still in ℝ+.

There is, however one infimum of the positive real numbers: 0, smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound. A lower bound of a subset S of a ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S. An upper bound of a subset S of a ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b. Infima and suprema do not exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, or if the set of lower bounds does not contain a greatest element. However, if an infimum or supremum does exist, it is unique. Ordered sets for which certain infima are known to exist become interesting. For instance, a lattice is a ordered set in which all nonempty finite subsets have both a supremum and an infimum, a complete lattice is a ordered set in which all subsets have both a supremum and an infimum.

More information on the various classes of ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique. If S contains a greatest element that element is the supremum. If S contains a least element that element is the infimum; the infimum of a subset S of a ordered set P, assuming it exists, does not belong to S. If it does, it is a minimum or least element of S. Similarly, if the supremum of S belongs to S, it is a maximum or greatest element of S. For example, consider the set of negative real numbers; this set has no greatest element, since for every element of the set, there is another, element. For instance, for any negative real number x, there is another negative real number x 2, greater. On the other hand, every real number greater than or equal to zero is an upper bound on this set. Hence, 0 is the least upper bound of the negative reals, so the supremum is 0; this set has a supremum but no greatest element.

However, the definition of maximal and minimal elements is more general. In particular, a set can have many maximal and minimal elements, whereas infima and suprema are unique. Whereas maxima and minima must be members of the subset, under consideration, the infimum and supremum of a subset need not be members of that subset themselves. A ordered set may have many minimal upper bounds without having a least upper bound. Minimal upper bounds are those upper bounds for which there is no smaller element, an upper bound; this does not say that each minimal upper bound is smaller than all other upper bounds, it is not greater. The distinction between "minimal" and "least" is only possible when the given order is not a total one. In a ordered set, like the real numbers, the concepts are the same; as an example, let S be the set of all finite subsets of natural numbers and consider the ordered set obtained by taking all sets from S together with the set of integers ℤ and the set of positive real numbers ℝ+, ordered by subset inclusion as above.

Both ℤ and ℝ+ are greater than all finite sets of natural numbers. Yet, neither is ℝ+ smaller than ℤ nor is the converse true: both sets are minimal upper bounds but none is a supremum; the least-upper-bound property is an example of the aforementioned completeness properties, typical for the set of real numbers. This property is sometimes called Dedekind completeness. If an ordered set S has the property that every nonempty subset of S having an upper bound has a least upper bound S is said to have the least-upper-bound property; as noted above, the set ℝ of all real numbers has the least-upper-bound property. The set ℤ of integers has the least-upper-bound property.

102.2 Smooth FM

102.2 Smooth FM was an Independent Local Radio station for Greater London. It replaced 102.2 Jazz FM on 7 June 2005 at 10 am, with the help of R&B singer Lemar and the breakfast show host Jon Scragg. The first track played, keeping with the name of the newly launched radio station was Sade Adu's "Smooth Operator", was owned by the radio division of the Guardian Media Group, GMG Radio. Following disappointing audience figures, the station was closed on 23 March 2007 and relaunched as 102.2 Smooth Radio the following Monday, following a successful format change request to Ofcom to play music oriented at listeners aged 50 and above. 102.2 Smooth FM was available on DAB across London, Central Scotland, the North East of England, South Wales and the Severn Estuary and the West Midlands, as well as on Freeview channel 718. In 2005, the Guardian Media Group made the decision to drop the jazz name from the Jazz FM brand and relaunch the station as Smooth FM; the London version of Jazz FM closed on 27 May 2005 to prepare for the launch of Smooth FM on 7 June.

102.2 Smooth FM played middle of the road music, soul and R&B during the day and, as part of its licence requirements, focused on jazz music at night. Smooth FM played specialist jazz and soul shows at weekends, details of which are listed below; the station was launched on the premise of a'clutter-free' listen, offering 40 minutes of non-stop music every hour without commercial interruptions, deliberately posed as a direct challenge to the'might' of the BBC and a tactic aimed at increasing the total number of hours listeners stayed with the station. The'Smooth 40' became the'9-5 Smooth 40', with off-peak shows introducing more commercial breaks into their output, before the concept was dropped altogether in mid-2006. Weekday programming featured an'all-request' feature entitled'Smooth on Demand' at 2 pm and 7 pm, Monday to Friday, where listeners were invited to'demand' their favourite song by calling a local-rate phone number. Smooth FM ran a number of'big money' promotions to entice listeners to trial the station - the'Smooth £10K Tripleplay' ran for several weeks in late 2005, giving £10,000 to a listener for identifying three consecutive songs in a particular order.

Another large prize was awarded to Dawn Muggleton on 19 April 2006, however - who identified the'Smooth Secret Song', winning £118,454 at the end of a contest that had run for several months, which was, at that time, the biggest cash prize awarded on UK radio since 1999. The station held exclusive live commentary rights for Chelsea FC soccer matches in London for the 2005/06 and 2006/07 seasons. Commentary was provided by Kerry Dixon; the station's jingle package was voiced by Mitch Johnson. 102.2 Smooth FM's presenters included: Mike Allen Dave Brown Campbell Burnap Jim Colvin Jenni Costello Kevin Greening Rosie Kendrick Dave Koz Ramsey Lewis Gavin McCoy Howard Pearce Russell Pockett David Prever Jon Scragg Mark Walker Sarah Ward Nigel Williams Paul Wisdom Peter Young Motown Sunday: Four hours of Motown classics hosted by programme director Mark Walker and by Dave Brown. Legends of Jazz with Ramsey Lewis: A show which featured classic jazz recordings from major and influential jazz artists.

Mainstem with Campbell Burnap: A two-hour programme which included many forms of jazz from classic to Latin as well as a mix of jazz from the younger players of the day. Peter Young: Three hours of funk and jazz music. Smooth Selection: A selection of mellow jazz through the evening, first presented by Sarah Ward by Mark Walker and by Dave Brown. Smooth Edge: Four hours of soul music on in the early hours of Saturday and Sunday morning; the Dave Koz Radio Show: The jazz saxophonist presented a two-hour syndicated radio show on Sunday nights featuring the latest tracks in the world of smooth jazz, alongside interviews with well-known smooth jazz artists. The Late Lounge with Rosie Kendrick: A two-hour show featuring chillout grooves and jazz; the Saturday Night Experience: Broadcast on Saturday evenings from 9pm and presented by Mike Chadwick, the show specialised in music with a distinct cutting edge. On 20 October 2006, GMG Radio announced that it was requesting a change of format for Smooth FM from Ofcom, moving the station away from its daytime soul and R&B remit, instead offering easy listening music and speech for the over 50s and an improved local news service.

Ofcom approved the changes on 8 December 2006, with the condition that GMG retained the 45 hours of jazz per week that constituted part of the former licence requirement. Smooth FM trailed the forthcoming changes from the beginning of March 2007, promising'more of London's smooth favourites mixed with the best songs from the past five decades'; the station closed at 6:02pm on Friday 23 March 2007 with newsreader Sam Gudger uttering the final words - "that was Smooth FM" - followed by a weekend of preview music, before the station's replacement, 102.2 Smooth Radio, launched on Monday 26 March. Timeline of Smooth Radio Official website Press Release about the rebrand GMG Radio article about the launch of 102.2 Smooth FM

Storm Sanders

Storm Sanders is an Australian tennis player. Sanders has won one singles title and twelve doubles titles on the ITF Women's Circuit. On 10 February 2014, she reached her best singles ranking of world No. 202. On 9 October 2017, she peaked at No. 63 in the doubles rankings. Sanders debuted on the ITF junior circuit in December 2007, on the senior circuit in November 2008, she won her first professional tournament in February 2013. Sanders began, her first tournament was the Apia International Sydney, where she received a wildcard into qualifying. She stunned Eugenie Bouchard in the first round in two tiebreak sets, but lost in the second round against Misaki Doi. Sanders received a wildcard into qualifying at the Australian Open where she lost in the first round against Yuliya Beygelzimer. In February, after failing to qualify for the McDonald's Burnie International, Sanders celebrated a breakthrough victory, winning the $25,000 Launceston Tennis International tournament, she won without dropping a set.

She achieved the rare feat of defeating the top seeds in both the qualifying draw and the main draw en route to victory. Sanders reached the top 500 in the WTA rankings for the first time. A month she reached the final of the $25,000 event in Ipswich, losing to Jelena Pandžić in three sets. In July, together with her British partner Naomi Broady, Sanders won the $50,000 Gold River Women's Challenger, defeating Robin Anderson and Lauren Embree in straight sets. In US Open qualifying, Sanders lost in the first round to Uzbekistan's Nigina Abduraimova. Sanders began the season after receiving a wild card into qualifying, she opened with a three set win over Irina-Camelia Begu. Although taking the opening set, Sanders lost against third seed Hsieh Su-wei in three sets in the second round; the following week, Sanders was awarded a wild card to the main draw of the Hobart International. A first-round win over Peng Shuai saw her match up with second seed Kirsten Flipkens in the second round. Pushing the top-20 ranked Belgian to the brink, Sanders lost in a tough three set match, lasting over two and a half hours.

Despite the close loss, it was announced that Sanders had been given a wildcard into the singles main draw of the Australian Open, having been given wild cards for the doubles draw the previous two years. She played Camila Giorgi in round one, she lost in the first round of women's and mixed doubles. Given a wild card for the Hobart International, Sanders lost in round one to Camila Giorgi in three sets, she was given a wild card for the Australian Open, but lost at the first stage again, this time to world No. 46 Klára Koukalová in straight sets. In July, she qualified for the Jiangxi International - the first time Sanders has come through qualifying at a WTA-level event. Storm Sanders at the Women's Tennis Association Storm Sanders at the International Tennis Federation Storm Sanders at Tennis Australia

Cactus Air Force

Cactus Air Force refers to the ensemble of Allied air power assigned to the island of Guadalcanal from August 1942 until December 1942 during the early stages of the Guadalcanal Campaign those operating from Henderson Field. The term "Cactus" comes from the Allied code name for the island. In 1943 the Cactus Air Force was subsumed into, a joint command of Allied air units in the Solomon Islands. On 7 December 1941, the Japanese attacked the U. S. Pacific fleet at Pearl Harbor, Hawaii; the attack crippled much of the U. S. led to a state of war between the two nations. In launching this war, Japanese leaders sought to neutralize the American fleet, seize possessions rich in natural resources, obtain strategic military bases to defend their far-flung empire. Japanese forces attacked and took control of Hong Kong, the Philippines, Malaya, the Dutch East Indies, Wake Island, New Britain, Guam. Two attempts by the Japanese to extend their defensive perimeter in the south and central Pacific were thwarted in the Battle of the Coral Sea and the Battle of Midway.

These two strategic victories for the Allies provided an opportunity to take the initiative and launch a counter-offensive against the Japanese somewhere in the Pacific. The Allies chose the Solomon Islands the southern Solomon Islands of Guadalcanal and Florida. Allied strategists knew that the Japanese Navy had occupied Tulagi in May 1942 and had constructed a seaplane base near there. Concern grew when in early July 1942 the Japanese Navy began constructing a significant airfield near Lunga Point on nearby Guadalcanal island; these bases, when complete, would protect Japan's major base at Rabaul, threaten Allied supply and communication lines across the South Pacific to Australia and New Zealand, establish a staging base for possible future offensives against the New Hebrides, Fiji and New Caledonia. The Allied plan to attack the southern Solomons was conceived by U. S. Navy Admiral Ernest King, the Commander in Chief of the United States Fleet. King proposed the counter-offensive to deny the use of the southern Solomon Islands by the Japanese as bases to threaten the supply routes between the United States and Australia, to use them as starting points for a campaign with the goal of isolating the new and major Japanese base at Rabaul while supporting the Allied New Guinea campaign.

All of this had the eventual goal of opening the way for the U. S. to retake the Philippines. The American Admiral Chester Nimitz, the Allied Commander-in-Chief for all forces in the Pacific, created the South Pacific theater of operations, with Vice Admiral Robert L. Ghormley placed in command on 19 June 1942, to direct the Allied offensive in the Solomons. On 7 August 1942, the First Marine Division landed on Tulagi and Guadalcanal at Lunga Point, capturing the completed Japanese airfield and marking the first counter-offensive taken by the Allies in the Pacific Theater. More construction work began on the airfield mainly using captured Japanese equipment. On 12 August, the airfield was renamed Henderson Field, for Major Lofton R. Henderson, killed during the Battle of Midway and, the first Marine Corps pilot killed during the battle. By 18 August, Henderson Field was ready for operation; when the first planes began arriving, Henderson Field could be described as an airfield. It was an irregularly shaped blob cut out of the island growth, half in and half out of a coconut grove, with a runway, too short and few revetments to protect the aircraft from shrapnel.

Upon landing on Henderson Field on 4 September, the Commanding Officer of Marine Aircraft Group 25, Colonel W. Fiske Marshall described the field by stating it "looked like a Doré drawing of hell."The runway was a northwest to southeast running, 2,400-foot long gravel surface with an extra 1,000 feet of Marston Matting, pockmarked with craters from Japanese artillery and naval gunfire. The strip was in such poor condition. In the heat, the field was a bowl of black dust; when it rained, the airfield turned muddy, miring planes in liquid muck. Major Marion Carl described it as "...the only place on Earth where you could stand up to your knees in mud and still get dust in your eyes." The heavier SBD dive bombers had it the worst, since their hard rubber tires, designed for aircraft carrier landings and take-offs, ripped up the runways like plowshares. Wooden wheels were experimented with; the runway was extended and widened several times during the long Guadalcanal campaign, it was 3,800 feet long and 150 wide by 4 September.

Henderson Field was very close to the thinly-held lines of the U. S. First Marine Division, so security was always a concern. There were aircraft hangars, or repair buildings. Damaged aircraft were cannibalized for spare parts, with no bomb hoists, all aircraft munitions had to be hand-loaded onto the warplanes. Fuel, always critically low, had to be hand pumped out of 55 gallon drums. After the arrival of fuel trucks, aviation gasoline still had to be hand-pumped into the trucks. On 9 September 1942, the U. S. 6th Naval Construction Battalion opened up a second runway about one mile to the east of Henderson Field's original runway. This new runway, called "Fighter 1", consisted of tamped-down sod, it was about 4,600 feet long and 300 feet wide; the Marine fighter squadrons began operating out of Fighter 1, with the rest of the aircraft operating out of Henderson Field continued to use the original runway – thereafter was referred to as "Bomb

Corral, Chile

Corral is a town and sea port in Los Ríos Region, Chile. It is located south of Corral Bay. Corral is best known for the forts of Corral Bay, a system of defensive batteries and forts made to protect Valdivia during colonial times. Corral was the headquarters of the system. Economic activities in Corral revolve around forestry, fishing, port services and both heritage and eco tourism; the settlement of Corral grew out from the headquarters of the forts of Corral Bay that were built in 1645 to protect the city of Valdivia. By that time Spanish ships sailed through Valdivia River all the way to Valdivia but Corral soon took over the role of receiving major ships. With the Spanish king's Decree of Free Trade of 1778 cities in Spanish America were allowed to have direct commerce with Spain, which stimulated commerce in Corral that benefited from trade routes across Cape Horn. During the Chilean Independence War Corral and Valdivia functioned together with Chiloé as a prominent royalist strongholds.

In 1820 Thomas Cochrane, commanding the newly created Chilean Navy, took Corral and Valdivia by an amphibious attack. From the times of Chile's independence until the early 20th century Corral gained prominence as an important port for traffic between the Pacific and the Atlantic Ocean, as Chilean port were declared open to ships sailing under any state flag. After being incorporated into Chile the forts in Corral fell into disuse. Since late colonial times Corral became not only the main port of entrance to the interior of Valdivia but of Osorno, founded in 1796; the economic activity of Corral suffered several setbacks in the 20th century that led it to become a minor port by international and Chilean standards. The Panama Canal opened in 1914, Valdivia and Osorno became connected to central Chile by train in 1906 and 1905 respectively; this changed the trade and human traffic circuits in the region, resulting in Corral losing most of the international traffic that anchored there, but of the Chilean goods and human traffic, diverted by the railroads.

In 1910 after a huge investment Altos Hornos y Acerías de Corral opened in Corral the largest steel mill at that time in South America. It produced high-cost pig iron using charcoal, was labour-intensive; the steel mill proved to be an economical failure and was closed in 1958. The decline of Corral culminated with the 1960 Valdivia earthquake that destroyed houses and port facilities. There is only one pier and it is owned by a local company and used for shipping wood chips; the maximum permissible draught is 12.20 metres. According to the 2002 census of the National Statistics Institute, Corral spans an area of 766.7 km2 and has 5,463 inhabitants. Of these, 3,670 lived in 1,793 in rural areas; the population fell by 5.2 % between the 2002 censuses. As a commune, Corral is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde, directly elected every four years; the 2008-2012 alcalde is Gaston Peréz González. Within the electoral divisions of Chile, Corral is represented in the Chamber of Deputies by Alfonso De Urresti and Roberto Delmastro as part of the 53rd electoral district.

The commune is represented in the Senate by Andrés Allamand and Eduardo Frei Ruiz-Tagle as part of the 16th senatorial constituency. Altos Hornos y Acerías de Corral Valdivian fort system Chaihuín Guape Municipality of Corral

Kadhem Sharif

Kadhem Sharif al-Jabouri is an Iraqi wrestler and weightlifter. He is most famous for attempting to use a sledgehammer to bring down the statue of Saddam Hussein at the Firdos Square in Baghdad. Kadhem Sharif was born like his father became obsessed with motorcycles. "I was 12. I bought my first one eight years a 1966 Fatboy," Sharif recalls; as he grew up he developed a passion for wrestling and weightlifting, becoming a world-class athlete. Many of his sporting adventures held danger and he remembers that every time the Iraqi team did badly the leader of Iraqi sports, Saddam's son Uday Hussein, would order that everyone have their heads and eyebrows shaved and on at least one occasion, they were put in prison. Sharif's association did not end there. Sharif developed a weightlifting program for Uday. Sharif supplied some supplements but Uday instead started abusing steroids Steroids affected him, he became an addict; the doctors said he should not mix alcohol and steroids. He was trying to be a hero by taking more tablets.

But he failed. Uday took up collecting motorcycles like Sharif stealing some from him, it was at this point that they had Sharif refused to fix any of Uday's bikes. It was that he was thrown in jail on trumped up charges and spent two years in Iraqi prisons. On April 9, 2003 Sharif had his chance to take revenge. Hearing that the Saddam regime had fallen he took a 10 kg sledgehammer and went to work on the statue of Saddam Hussein that stood in Firdos Square in Baghdad; the square is directly in front of the Palestine Hotel where the world's journalists had been staying. Seeing a crowd and a story, the reporters and their cameras streamed out of the hotel and video taped the falling of Saddam Hussein's statue. While Sharif's actions with the sledgehammer only resulted in a small dent in the statue base and bloody hands, he does claim to have handed over the Iraqi flag, placed on the statue; the legality of Sharif's motorcycle business and collection has always been murky and he has admitted that some of his bikes were stolen in neighboring countries and smuggled into Iraq along with dealing with the bikes looted from Kuwait when Saddam Hussein invaded the country.

His business caught up with him when in 2005 he was arrested in connection with dealing stolen bikes. In a 2016 interview Kadhem said that he expressed his regret over his role in the Firdos Square statue destruction in which he attacked the statue with a sledgehammer, he said "Saddam has gone, but now in his place we have a 1,000 Saddam's." and "When I go past that statue, I feel pain and shame. I ask myself. I'd like to put it back up. I'd put it back up but I'm afraid I would be killed." He looked back to Saddam's time with nostalgia. He was as solid as a wall. There was no corruption or looting, it was safe. You could be safe." He went on to say that he hopes the two leaders viewed as responsible for the overthrow of Saddam, Tony Blair and George Bush, should be brought to justice. Kadhem had been a bike collector all his life but after the war his collection expanded until he was arrested. In his collection there were a number of notable bike including: 1914 - This Norton was part of the escort of King Faisal, Iraq's first monarch.

1947 - British-made BSA that he found rusting in a farm in "Anbar province", an area of western Iraq notorious for insurgent violence since the 2003 US-led invasion. 1957 British Norton motorcycle that Saddam Hussein the revolutionary rode when on the run after the assassination of Iraqi President Abd al-Karim Qasim half a century ago. 1957 Rolls-Royce Silver Cloud the property of the Emir of Kuwait. Saddam's eldest son, Uday Hussein, became an invalid after he was injured in a 1996 assassination attempt. Uday a collector of bikes, had his collection made into three-wheelers so he could still drive them. After Saddam's fall Kadhem was able to get his hands on all of Uday's monster three-wheeled invalid motorcycles. Picture of Sharif with his motorcycles Bowen, Jeremy. "Iraq Chilcot inquiry: Bitterness in Baghdad". BBC News. Retrieved July 7, 2016