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Mathematical induction

Mathematical induction is a mathematical proof technique. It is used to prove that a property P holds for every natural number n, i.e. for n = 0, 1, 2, 3, so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one; the method of induction requires two cases to be proved. The first case, called the base case, proves that the property holds for the number 0; the second case, called the induction step, proves that if the property holds for one natural number n it holds for the next natural number n + 1. These two steps establish the property P for every natural number n = 0, 1, 2, 3, … The base case does not begin with n = 0. In fact, it begins with the number one, it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number.

The method can be extended to prove statements about more general well-founded structures, such as trees. Mathematical induction in this extended sense is related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs. Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning as used in philosophy. Mathematical induction is an inference rule used in formal proofs. Proofs by mathematical induction are, in fact, examples of deductive reasoning. In 370 BC, Plato's Parmenides may have contained an early example of an implicit inductive proof; the earliest implicit traces of mathematical induction may be found in Euclid's proof that the number of primes is infinite and in Bhaskara's "cyclic method". An opposite iterated technique, counting down rather than up, is found in the Sorites paradox, where it was argued that if 1,000,000 grains of sand formed a heap, removing one grain from a heap left it a heap a single grain of sand forms a heap.

An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle. None of these ancient mathematicians, explicitly stated the induction hypothesis. Another similar case was that of Francesco Maurolico in his Arithmeticorum libri duo, who used the technique to prove that the sum of the first n odd integers is n 2; the earliest rigorous use of induction was by Gersonides. The first explicit formulation of the principle of induction was given by Pascal in his Traité du triangle arithmétique. Another Frenchman, made ample use of a related principle: indirect proof by infinite descent; the induction hypothesis was employed by the Swiss Jakob Bernoulli, from on it became more or less well known. The modern rigorous and systematic treatment of the principle came only in the 19th century, with George Boole, Augustus de Morgan, Charles Sanders Peirce,Giuseppe Peano, Richard Dedekind.

The simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n. The proof consists of two steps: The initial or base case: prove that the statement holds for 0, or 1; the induction step, inductive step, or step case: prove that for every n, if the statement holds for n it holds for n + 1. In other words, assume that the statement holds for some arbitrary natural number n, prove that the statement holds for n + 1; the hypothesis in the inductive step, that the statement holds for a particular n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis for n and uses this assumption, involving n, to prove that the statement holds for n + 1. Authors who prefer to define natural numbers to begin at 0 use that value in the base case. Authors who prefer to define natural numbers to begin at 1 use that value in the base case. Mathematical induction can be used to prove that the following statement, P, holds for all natural numbers n. 0 + 1 + 2 + ⋯ + n = n 2.

P gives a formula for the sum of the natural numbe

Marion Greeves

Marion Janet Greeves, MBE was the first one of only two female members of the Senate of Northern Ireland, having been elected to serve as an independent member on 20 June 1950, retiring on 10 June 1969. Greeves, born in England, was the daughter of George Cadbury, a Quaker philanthropist, his second wife, Elizabeth Mary Taylor, she married linen manufacturer William Edward Greeves, Deputy Lieutenant and High Sheriff of County Armagh, a Quaker, on 14 February 1918, in Bournville. The couple had five children, she lived a normal life at Ardeevin House, County Armagh. A Girl Guides centre is named in her honour, she was President of the organisation in Ulster. In 1948, Marion Greeves was awarded the MBE, she died in 1979, less than two weeks before her 85th birthday

Battle of Gaixia

The Battle of Gaixia was fought in 202 BC during the Chu–Han Contention between the forces of Liu Bang and Xiang Yu. The battle concluded with victory for Liu Bang, who proclaimed himself Emperor of China and founded the Han Dynasty; the Han forces had earned many major victories against Chu, but they did still not control most of the country. Most of eastern China was still under Chu control. Xiang Yu was able to reorganize his forces, strike back at Liu Bang. At this point, major disagreements had occurred between Han Xin; the primary reasons were because Liu Bang refused to give Han Xin too much control over the Han army, his refusal to use many of Han Xin's suggestions. As a result, Han Xin withheld his forces in Qi. Liu Bang was only able to hang on because of the assistance from another excellent military strategist, Zhang Liang. In most battles, Xiang Yu paid sufficient attention to resource logistics. In this battle, Zhang Liang was successful in assaulting Xiang Yu's supply lines, which Xiang Yu did not care much about, this hurt the Chu army's effectiveness.

On the other hand, Zhang Liang was successful in keeping the Han supply lines open. During a conversation, Xiang Yu's archer hit Liu Bang with an arrow. However, Liu Bang was able to hide this fact from his own army and Xiang Yu, hence keeping the morale of the Han troops high. Sensing the tension between Liu Bang and Han Xin, Xiang Yu tried to persuade Han Xin to ally with him, or at least to stay neutral in this war. Xiang Yu warned Han Xin. Xiang Yu offered to accept Han Xin's Qi as a third nation after Chu and Han. However, due to the past history between the two, Han Xin refused any diplomatic relationship with Xiang Yu. Xiang Yu threatened to kill Liu Bang's captured father and wife and have them cooked over a fire, to force Liu Bang to surrender. Liu Bang replied that since the two had been named'brothers' he would be cooking his own father, that Xiang Yu should not forget to send him a cup of'their' father's flesh to share as good brothers, but Xiang Yu still did not kill them. At one point, Xiang Yu was about to capture Liu Bang.

Liu Bang agreed with everything that Han Xin requested, Han Xin agreed to help. With the arrival of Han Xin, Liu Bang was able to convince Xiang Yu to agree to a peace treaty. At the end, both parties agreed that the two countries of Chu could co-exist peacefully. Liu Bang's father and wife were returned to Liu Bang. In October 202 BC, Xiang Yu started to move his forces back east. Unbeknownst to Xiang Yu, this was a tactic from Han Xin; the Chu forces had sieged the Han fortress for a long time. On top of that, Chu troops were getting less food; when news reached them that the war was over, that there would be long term peace ahead, they were overjoyed. Han Xin led many attacks against the joyful Chu forces. In anger, Xiang Yu tried to fight his way back to Liu Bang's fortress, but Han Xin had set up many traps and ambushes along the way. Xiang Yu decided that a quick victory was no longer possible with his low morale troops, decided to temporarily retreat back to the Chu capital, regroup there.

Han Xin knew. Han Xin ordered his forces to increase the number of ambushes, in order to force the Chu troops into a canyon area near Gaixia, where Xiang Yu could not move at will; as the ambushes increased, Xiang Yu became more and more certain that the main traps would await him inside the canyon. So although his troops were ambushed, he insisted that his troops head straight back to the capital city through the main road as fast as they could, avoiding the side paths through the canyon. For Xiang Yu, fortune turned against him. In one of the ambushes, Xiang Yu's beloved wife, Consort Yu, who always traveled with his forces, was captured by Han troops. Han Xin ordered that she be taken into the canyon. Xiang Yu, without a choice, sent most of his tired forces back to the capital on the main road, while he himself led a smaller force of 100,000 soldiers into the canyon to save his wife. Xiang Yu hoped to save his wife and to get out before becoming entrapped. However, the Han forces, under Han Xin's orders, moved his wife deep into the canyon.

By the time he reached them and saved his wife, he and his army were too deep into the canyon to retreat safely. Han Xin proceeded with his master plan: "Ambush from Ten Sides". Han Xin first fought Xiang Yu face to face, retreated. Xiang Yu gave chase, but soon found himself trapped among the numerous Han army. Everywhere Xiang Yu led his forces, more ambushes and traps awaited them. With the repeated ambushes and encirclements, the Han troops began to elongate Xiang Yu's columns and disrupt their formation, allowing their decimation piecemeal; this not only caused heavy casualties for Chu, but crushed the Chu army's morale, since escaping alive seemed impossible. By December 202 BC, the Chu troops were trapped without supply in the canyon. To further break the Chu army's spirit, Han Xin employed the "Chu Song from Four Sides" tactic, he captured Chu troops to sing Chu songs. Xiang Yu thought that the Western Chu had been conquered while he had been trapped there, his cause was lost; the Chu soldiers started to desert their camps and escape on their own.

Xiang Yu tried with force to stop his troops from leaving the ranks. Bu

Tariq Mahmood (detainee)

Tariq Mahmood is a British Pakistani man, captured in Islamabad by Pakistani security forces in October 2003. His family reports that Tariq was tortured, while in Pakistani custody, with the knowledge or cooperation of UK and American security officials. Tariq Mahmood is a married father of two from Birmingham; the former taxi driver flew to Pakistan in 2001 to settle a land dispute over a family home there. In October 2003, Mahmood was held on suspicion of being associated with a "banned organization" under the Security of Pakistan Act, Section 10, was not given immediate access to courts despite his British citizenship. Mahmood was assigned a 10 November 2003 court date in Islamabad, made court appearances over the following four weeks. However, despite the ongoing legal process, his whereabouts became unclear by early 2004. Pakistani security turned him over to American forces, prompting fears he would be sent to Guantanamo Bay Naval Base. In February 2004, Pakistani intelligence sources indicated Tariq Mahmood had been transported to Bagram Airfield, Afghanistan, a "stepping stone" to Guantanamo Bay.

Human Rights Watch listed him as one of 39 ghost detainees in 2005, who are not given any legal rights or access to counsel, who are not reported to or seen by the International Committee of the Red Cross. On February 19, 2004, The Guardian listed the nine UK citizens known to have been held in Guantanamo, they listed him as a possible 10th UK citizen held in Guantanamo. His presence in Guantanamo has never been confirmed. According to articles from The Guardian quoted in a report by a committee of the UK Parliament, Tariq is believed to have made his home in Dubai following his release in 2004. List of people who disappeared

Red Riding Hoodlum

Red Riding Hoodlum is the 74th animated cartoon short subject in the Woody Woodpecker series. Released theatrically on February 11, 1957, the film was produced by Walter Lantz Productions and distributed by Universal International, based in fairy tale Little Red Riding Hood, by written by Charles Perrault and the Brothers Grimm. Woody Woodpecker's nephew Knothead and niece Splinter are reading the story of Little Red Riding Hood when Woody sends them on an errand to deliver a basket of goodies to their grandmother's house, they encounter a wolf and soon realize that their trip is occurring just like the original Red Riding Hood story. While the wolf takes a shortcut, Knothead & Splinter take a "short-shortcut" to get to Granny's house first, but along the way, Knothead & Splinter encounter the homes of The Three Little Pigs, The Three Bears and the Old Woman Who Lived In A Shoe, before Smokey Bear informs the two that the next house is Granny's. Knothead & Splinter convince Granny to read them the story of Little Red Riding Hood to distract her while they deal with the wolf.

Granny encounters the wolf, but instead of being frightened, Granny puts on a red wig and make-up and gives the wolf a big kiss. The cartoon ends with Granny and the wolf getting married by a dog minister while Knothead & Splinter hold Granny's wedding dress train. Though this short is a part of the Woody Woodpecker series, its main stars are Woody's nephew and niece Knothead and Splinter, making their second appearance since the previous year's Get Lost. Smokey Bear, advertising mascot for the United States Forest Service, makes three appearances in the cartoon. First Smokey tells Knothead and Splinter where Granny's house is he says that he hopes party attendees are "careful with their cigarettes", warns the wolf about being careful with matches; the ending is similar to A Fine Feathered Frenzy. In that cartoon the elderly yet rich and well manicured widow Gorgeous Gal falls in love with Woody the second she lays eyes on him. Gorgeous Gal makes not one costume change but several while winking and flirting with her "Baby!"

She wastes no time trying to put her arms around the Woodpecker. Much like Granny she too manages to marry the object of her heart's desire against his wishes; the film ends with her becoming Mrs. Gorgeous Gal Woodpecker. Cooke, Komorowski, Shakarian and Tatay, Jack. "1957". The Walter Lantz Cartune Encyclopedia

Epirus water frog

The Epirus water frog is a species of frog in the family Ranidae. It is found in western Greece, including Kerkyra, the southern areas of Albania; the species is collected from the wild for human consumption. Like most frogs, Epirus water frogs show sexual dimorphism. Males can grow with females growing larger to 3.3 inches. The dorsal side is green with irregular black spots; the underside is pale. Male vocal sacs are olive aside from mating season; the species occurs in Mediterranean-type shrubby vegetation, swamps, freshwater lakes and marshes, plantations. It is threatened by habitat loss, is classified as vulnerable as populations within its small range are fragmented