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Inequality (mathematics)

In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: The notation a < b means that a is less than b. The notation a > b means. In either case, a is not equal to b; these relations are known as strict inequalities, meaning that a is less than b. In contrast to strict inequalities, there are two types of inequality relations that are not strict: The notation a ≤ b or a ⩽ b means that a is less than or equal to b; the notation a ≥ b or a ⩾ b means that a is equal to b. If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size. On the other hand, the notation a ≠ b means that a is not equal to b, is sometimes considered a form of strict inequality, it does not say that one is greater than the other, or that they can be compared in size.

In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another by several orders of magnitude. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation; the notation a ≪ b means. The notation a ≫ b means. In all of the cases above, any two symbols mirroring. Inequalities are governed by the following properties. All of these properties hold if all of the non-strict inequalities are replaced by their corresponding strict inequalities and — in the case of applying a function — monotonic functions are limited to monotonic functions; the relations ≤ and ≥ are each other's converse, meaning that for any real numbers a and b: a ≤ b and b ≥ a are equivalent. The transitive property of inequality states that for any real numbers a, b, c: If a ≤ b and b ≤ c a ≤ c. If either of the premises is a strict inequality the conclusion is a strict inequality: If a ≤ b and b < c a < c. If a < b and b ≤ c a < c.

A common constant c may be subtracted from both sides of an inequality. So, for any real numbers a, b, c: If a ≤ b a + c ≤ b + c and a − c ≤ b − c. In other words, the inequality relation is preserved under addition and the real numbers are an ordered group under addition; the properties that deal with multiplication and division state that for any real numbers, a, b and non-zero c: If a ≤ b and c > 0 ac ≤ bc and a/c ≤ b/c. If a ≤ b and c < 0 ac ≥ bc and a/c ≥ b/c. In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More this applies for an ordered field. For more information, see § Ordered fields; the property for the additive inverse states that for any real numbers a and b: If a ≤ b −a ≥ −b. If both numbers are positive the inequality relation between the multiplicative inverses is opposite of that between the original numbers. More for any non-zero real numbers a and b that are both positive: If a ≤ b 1/a ≥ 1/b.

All of the cases for the signs of a and b can be written in chained notation, as follows: If 0 < a ≤ b 1/a ≥ 1/b > 0. If a ≤ b < 0 0 > 1/a ≥ 1/b. If a < 0 < b 1/a < 0 < 1/b. Any monotonically increasing function, by its definition, may be applied to both sides of an inequality without breaking the inequality relation. However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed; the rules for the additive inverse, the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function. If the inequality is strict and the function is monotonic the inequality remains strict. If only one of these conditions is strict the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function. A few examples of this rule are: Raising both sides of an inequality to a power n > 0, when a and b are positive real numbers:0 ≤ a ≤ b ⇔ 0 ≤ an ≤ bn. 0 ≤ a ≤ b ⇔ a−n ≥ b−n ≥ 0.

Taking the natural logarithm on both sides of an inequality, when a and b are positive real numbers:0 < a ≤ b ⇔ ln ≤ ln. 0 < a < b ⇔ ln < ln. A partial order is a binary relation ≤ over a set P, reflexive and transitive; that is, for all a, b, c in P, it must satisfy the three following clauses: a ≤ a if a ≤ b and b ≤ a a = b if a ≤ b and b ≤ c a ≤ c A set with a partial order is called a ordered set. Those are the basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set P include: For every a and b in P, a ≤ b or b ≤ a. For all a and b in P fo

Federico Recalde

Federico Nicolás Recalde is an Argentine professional footballer who plays as a midfielder for Villa Dálmine. Recalde's career began with Villa Dálmine, he was an unused substitute in the Copa Argentina against Tristán Suárez on 7 February 2015, with his professional debut arriving a month in a Primera B Nacional defeat to Atlético Tucumán. His first senior goal came in his penultimate match of 2016, netting in a four-goal win away to Douglas Haig on 12 June 2016. In total, Recalde scored once in his opening four campaigns; as of 1 April 2019. Federico Recalde at Soccerway

Finley, New South Wales

Finley is a town in the Riverina region of New South Wales, Australia. It is the largest town in the Berrigan Shire local government area. At the 2016 census, Finley had a population of 2,519 people; the town is located 140 kilometres west of Albury on the intersection of the Newell Highway and Riverina Highways. The first permanent residence in the town was built in 1878; the post office opened on 1 January 1881 but was known as Murray Hut until 1893. Europeans first settled the area around Finley in the early 1840s, with wheat becoming the main crop; the Finley Agricultural & Pastoral Association was formed in 1912 and held its first show on 17 September 1913. The agricultural show is still held annually on the first Sunday in September. Periods of severe drought, combined with the Great Depression of the early 1930s, forced many farmers to abandon their holdings. In 1935, construction on the Mulwala Canal began in order to provide employment and bring water to the area’s rich farmland, with irrigation reaching the area in 1939, celebrated with a'Back-To-Finley' event.

This enabled the region to prosper with beef and dairy cattle, wheat, barley and canola. Finley has a number of heritage-listed sites, including: Narrandera-Tocumwal railway: Finley Railway Precinct Finley has two primary schools, St Joseph's School, Finley Public School. Finley High School attracts students from a wide catchment including the towns of Berrigan, Tocumwal and Blighty. Finley is home to a campus of Riverina TAFE. Australian rules football and netball are all popular in the town. Sporting teams include the Finley Football Club; the town offers soccer, touch rugby, tennis and a Pony Club. The Finley Rodeo Committee holds an annual rodeo every January and Finley Apex Club hosts a tractor pull every February. Finley has two bowling green locations. Allan Jeans - Former St Kilda Australian rules footballer & coach, Hawthorn Australian rules football coach Jack Hawkins - Former Geelong Australian rules footballer Shane Crawford - Former Hawthorn Australian rules footballer and 1999 Brownlow Medallist, 2019 I’m A Celebrity...

Get Me Out Of Here contestant Mark Whiley - Former GWS Giants & Carlton Blues Australian rules footballer Tom Hawkins - Current Geelong Australian rules footballer Kram and Janet - members of rock band Spiderbait Craig Giles - musician Maicie Close - 2018 State Finalist of The Land Sydney Royal Showgirl Competition Andrew Richardson - 2016 Six Medal Winner - Australian Baking Association's Annual Best Pie and Pastie competition Community website Berrigan Shire - Official Site Finley Railway Station Finley Show Society Inc

Battle of Ceva

In the Battle of Ceva on 16 April 1796, troops of the First French Republic under Pierre Augereau fought against part of the army of the Kingdom of Sardinia-Piedmont led by General Giuseppe Felice, Count Vital. Augereau assaulted the strong defensive position without success. At the direction of the Sardinian army commander, Feldmarschal-Leutnant Michelangelo Colli, Vital withdrew on the 17th in order to avoid being trapped by a second French division; the Montenotte Campaign began on 10 April when Feldzeugmeister Johann Beaulieu's Austrian army attacked the extreme right flank of General of Division Napoleon Bonaparte's army near Genoa. Bonaparte launched a successful counterattack on 12 April at the Battle of Montenotte. On 13 April, MG Augereau's reinforced division defeated part of Colli's Sardinian army at the Battle of Millesimo. In the Second Battle of Dego on 14 and 15 April, the Austrians were defeated again. While Beaulieu reorganized his badly shaken army at Acqui Terme to the northeast, Bonaparte prepared to increase the separation between the Piedmontese from their Austrian allies by driving Colli farther to the west.

On 15 April, Colli assembled a force on high ground at Montezemolo in order to cover the fortress of Ceva. Meanwhile, MG Jean Sérurier's division advanced from Ormea north toward Ceva along the Tanaro River valley. Fearing he might be cut off from Ceva, the Piedmontese commander fell back to the fortress. Marching from the east, Augereau occupied Montezemolo early on 16 April his division moved north and west in an attempt to outflank Ceva. See the Montenotte 1796 Campaign Order of Battle for a list of French and Sardinian units and organizations. In 1796, the fortress loomed over the north side of Ceva; the Sardinians deployed atop a ridge that runs from the fortress north about seven kilometers to the hamlet of La Pedaggera, where the modern SP 661 and SP 32 intersect. Sardinian and Austrian engineers had fortified the ridge, which overlooks the Bovina stream, with a series of redoubts. General Brempt held the north end of the line with several Piedmontese battalions plus the Austrian Belgioso Infantry Regiment # 44.

Vital defended the south end of the line with nine battalions, General Count di Tornaforte commanded the three battalions in the fortress. Augereau's division formed several columns under General of Brigade Martial Beyrand and BG Barthélemy Joubert. Two French columns pressed home their attacks on Brempt's left flank while one column assaulted Vital's position near Mondoni in the center; the Sardinians repelled all attacks on the ridge. That evening Sérurier camped within sight of Ceva, threatening to turn the southern flank of the Sardinian line. On the northern flank, Brempt reported. Though they had won a defensive success, the mood of the Piedmontese generals was gloomy and they recommended a retreat; that night, Colli held a council of war in which he decided to withdraw most of the army west behind the Corsaglia River, leaving Tornaforte with one battalion to hold Ceva fortress. Some units were sent northwest to Cherasco to prevent the French from cutting between Colli and the Piedmontese capital of Turin.

Augereau occupied the abandoned Sardinian positions on 17 April. Bonaparte decided to mask the fortress with a force under BG Jean Rusca and continue to press the Piedmontese back on Cuneo; the French lost about 600 men wounded. Brempt reported a loss of 150; the Sardinians won another rear guard action at San Michele Mondovi on 19 April. This was followed by the decisive French victory at the Battle of Mondovì on 21 April. Boycott-Brown, Martin; the Road to Rivoli. London: Cassell & Co. 2001. ISBN 0-304-35305-1 Chandler, David; the Campaigns of Napoleon. New York: Macmillan, 1966. Fiebeger, G. J.. The Campaigns of Napoleon Bonaparte of 1796–1797. West Point, New York: US Military Academy Printing Office. Historyofwar.org Battle of Ceva by J. Rickard, 27 January 2009 Fort de la Ceva le 16 avril 1796 — Giuseppe Pietro Bagetti watercolour of the battle, painted 1803

Andrew Perlman

Andrew Perlman is an American entrepreneur who has co-founded nine venture-backed companies in the telecom, high-tech, energy and biotechnology industries. He is General Partner of GreatPoint Ventures, the former Chairman and CEO of GreatPoint Energy, a company based in Chicago, Illinois which develops technology to produce clean natural gas from coal. Perlman has been featured on the MIT Technology Review’s list of the world's top 35 innovators under the age of 35 and Crain’s Chicago Business’s list of 40 leaders under 40 in Chicago. Perlman and GreatPoint Energy have been profiled by the Wall Street Journal, NPR, Fast Company. Perlman grew up in the cities of Newton and Cohassett in Massachusetts. At age 12, he began tracking down the owners of dormant bank accounts, taking a 20% commission in exchange for leading them to their forgotten money. In his youth, he applied for a federal license to construct an ethanol still in his parents’ house, though he was unsuccessful. Perlman began college at Washington University in St. Louis, as a sophomore he attempted to license and commercialize a university-owned technology to prevent credit card fraud.

The university declined the request because he was still a student, so Perlman dropped out. Perlman and a friend dropped out of Washington University and moved to Washington D. C. where they “hung around business and government offices, knocking on doors, asking anyone they could find about some kind of new technology they could turn into a business.” The two had little technical education, but through research and trial and error, they built a device that converts voice calls into a data format. By age 22, they signed a deal for the first $14 million in startup financing for their new company Cignal Global Communications. Three years Perlman sold the company for $200 million. Perlman went on to “launch five successful startups before he turned 30.” On its list of the world’s top 35 innovators under 35, the MIT Technology Review cites Perlman’s former project GreatPoint Energy as well as other disruptive technology ventures in important areas: “cheaper desalination plants, anti-obesity medicines, drugs that fight diseases of old age, and.”To date, Perlman has helped launch the following companies: Cignal Global Communications – A pioneer in integrated voice/data communications technology.

Coatue Corporation – Developer of high speed flash memory chips. Coskata, Inc. – Commercializing technology to produce low cost liquid fuels and chemicals from natural gas or gasified biomass Sirtris Pharmaceuticals – Commercializing Sirtuin activating compounds to prevent and treat diseases of old age, such as cancer, heart disease, Alzheimer’s, diabetes. Zafgen – Developing novel drugs for the treatment of obesity. Zafgen executed an IPO on the NASDAQ in June 2014 and trades as AltaRock Energy – Developed and commercialized enhanced geothermal technology for 100% renewable power production. Foro Energy – Commercializing high power lasers for the oil, natural gas and mining industries. Oasys Water – Provider of advanced seawater desalination and water treatment technology GreatPoint Energy – Developed a catalyst to convert coal into clean-burning methane gas, using much less water and chemicals than hydrofracking and producing much less pollution than coal combustion power plantsPerlman is on the board of directors for both Coskata Energy and Oasys Water.

He sits on the board of AMP Americas, an integrated transportation company working with compressed natural gas. As the former CEO, Perlman was responsible for developing the company's strategy, attracting commercialization partners and the development and execution of capital projects; the company has raised $562 million to date and is backed by leading strategic investors including The Dow Chemical Company, Suncor Energy, AES Corporation, Peabody Energy, as well as major financial institutions and venture capital firms, including Kleiner Perkins Caufield & Byers, Khosla Ventures, Draper Fisher Jurvetson, Advanced Technology Ventures, Citi’s Sustainable Development Investments. GreatPoint Energy official site

Shidler, Oklahoma

Shidler is a city in Osage County, United States. The population was 441 at the 2010 census, a 15.2 percent decrease from 520 at the 2000 census. Shidler was founded in December 1921 and named for Eugene S. Shidler, a Pawhuska, Oklahoma banker and rancher; the town grew to a population of about 5,000 due to the discovery of petroleum nearby and the arrival of the Osage Railway in February 1922. In that year, Shidler had six plants making gasoline. Shidler had a reputation for lawlessness with highway robberies common. By the late 1920s, the oil boom had subsided and Shidler began to lose population. Shidler's population in the 1930 census was the downward trend continued. Shidler today is a quiet farming and ranching community although there is still some petroleum production in the area. During World War II, citizens of Shidler lobbied to prevent the internment of the Yamamoto family from Shidler. Thanks to the efforts of US Senator Elmer Thomas, the internment order was lifted. Shidler is located at 36°46′49″N 96°39′43″W.

It is 29 miles northwest of Pawhuska, the seat of Osage County. According to the United States Census Bureau, the city has a total area of 0.8 square miles, all of it land. As of the census of 2000, there were 520 people, 231 households, 148 families residing in the city; the population density was 678.8 people per square mile. There were 278 housing units at an average density of 362.9 per square mile. The racial makeup of the city was 81.35% White, 14.04% Native American, 0.96% from other races, 3.65% from two or more races. Hispanic or Latino of any race were 2.50% of the population. There were 231 households out of which 25.5% had children under the age of 18 living with them, 49.8% were married couples living together, 9.5% had a female householder with no husband present, 35.9% were non-families. 31.6% of all households were made up of individuals and 17.3% had someone living alone, 65 years of age or older. The average household size was 2.25 and the average family size was 2.78. In the city, the population was spread out with 23.5% under the age of 18, 9.6% from 18 to 24, 22.5% from 25 to 44, 23.1% from 45 to 64, 21.3% who were 65 years of age or older.

The median age was 41 years. For every 100 females, there were 92.6 males. For every 100 females age 18 and over, there were 86.0 males. The median income for a household in the city was $29,732, the median income for a family was $35,156. Males had a median income of $31,932 versus $17,143 for females; the per capita income for the city was $16,245. About 11.0% of families and 15.9% of the population were below the poverty line, including 23.1% of those under age 18 and 13.5% of those age 65 or over. The town is home of the Shidler Public Schools Fighting Tigers; the school has about 250 students in kindergarten through 12th grade. The school brings in students from the surrounding towns of Grainola, Webb City, Kaw City, Burbank. Shidler Public Schools