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Problem Of The Nile
The problem of the Nile is a mathematical problem related to equal partitions of measures. The problem was first presented by Ronald Fisher in 1936–1938. It is presented by Dubins and Spanier in the following words:"Each year, the Nile would flood, thereby irrigating or perhaps devastating parts of the agricultural land of a predynastic Egyptian village. The value of different portions of the land would depend upon the height of the flood. In question was the possibility of giving to each of the ''k'' residents, piece of land whose value would be 1/''k'' of the total land value, no matter what the height of the flood."Formally, for each height ''h'', there is a nonatomic measure ''vh'' on the land, which represents the land values when the height of the Nile is ''h''. In general, there can be infinitely many different heights, and hence, infinitely many different measures. William Feller showed in 1938 that a solution for the general case might not exist. When the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Fisher
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who almost single-handedly created the foundations for modern statistical science" and "the single most important figure in 20th century statistics". In genetics, his work used mathematics to combine Mendelian genetics and natural selection; this contributed to the revival of Darwinism in the early 20th-century revision of the theory of evolution known as the modern synthesis. For his contributions to biology, Fisher has been called "the greatest of Darwin’s successors". Fisher held strong views on race and eugenics, insisting on racial differences. Although he was clearly a eugenist and advocated for the legalization of voluntary sterilization of those with heritable mental disabilities, there is some debate as to whether Fisher supported sc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nile
The Nile, , Bohairic , lg, Kiira , Nobiin language, Nobiin: Áman Dawū is a major north-flowing river in northeastern Africa. It flows into the Mediterranean Sea. The Nile is the longest river in Africa and has historically been considered the List of rivers by length, longest river in the world, though this has been contested by research suggesting that the Amazon River is slightly longer.Amazon Longer Than Nile River, Scientists Say Of the world's major rivers, the Nile is one of the smallest, as measured by annual flow in cubic metres of water. About long, its drainage basin covers eleven countries: the Democratic Republic of the Congo, Tanzania, Burundi, Rwanda, Uganda, Kenya, Ethiopia, Erit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prehistoric Egypt
Prehistoric Egypt and Predynastic Egypt span the period from the earliest human settlement to the beginning of the Early Dynastic Period around 3100 BC, starting with the first Pharaoh, Narmer for some Egyptologists, Hor-Aha for others, with the name Menes also possibly used for one of these kings. At the end of prehistory, "Predynastic Egypt" is traditionally defined as the period from the final part of the Neolithic period beginning c. 6000 BC to the end of the Naqada III period c. 3000 BC. The dates of the Predynastic period were first defined before widespread archaeological excavation of Egypt took place, and recent finds indicating very gradual Predynastic development have led to controversy over when exactly the Predynastic period ended. Thus, various terms such as " Protodynastic period", "Zero Dynasty" or "Dynasty 0" are used to name the part of the period which might be characterized as Predynastic by some and Early Dynastic by others. The Predynastic period is genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonatomic Measure
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless. Definition Given a measurable space (X, \Sigma) and a measure \mu on that space, a set A\subset X in \Sigma is called an atom if \mu(A) > 0 and for any measurable subset B \subset A with \mu(B) of A are atoms, and /math> is called an atomic class. If \mu is a \sigma-finite measure, there are countably many atomic classes. Examples * Consider the set ''X'' = and let the sigma-algebra \Sigma be the power set of ''X''. Define the measure \mu of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons , for ''i'' = 1, 2, ..., 9, 10 is an atom. * Consider the Lebesgue measure on the real line. This measure has no atoms. Atomic measures A \sigma-finite measure \mu on a measurable space (X, \Sigma) is called atomic or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian-American mathematician specializing in probability theory. Early life and education Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian-Austrian Catholic, and Eugen Viktor Feller, son of a Polish-Jewish father (David Feller) and an Austrian mother (Elsa Holzer). Eugen Feller was a famous chemist and created ''Elsa fluid'' named after his mother. According to Gian-Carlo Rota, Eugen Feller's surname was a "Slavic tongue twister", which William changed at the age of twenty. This claim appears to be false. His forename, Vilibald, was chosen by his Catholic mother for the saint day of his birthday. Work Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath, he fled the Nazis and went to Copenhagen, Denmark in 1933. He also lectured in Sweden (Stockholm and Lund). As a refugee in Sweden, Feller reported being tro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerzy Neyman
Jerzy Neyman (April 16, 1894 – August 5, 1981; born Jerzy Spława-Neyman; ) was a Polish mathematician and statistician who spent the first part of his professional career at various institutions in Warsaw, Poland and then at University College London, and the second part at the University of California, Berkeley. Neyman first introduced the modern concept of a confidence interval into statistical hypothesis testing and co-revised Ronald Fisher's null hypothesis testing (in collaboration with Egon Pearson). Life and career He was born into a Polish family in Bendery, in the Bessarabia Governorate of the Russian Empire, the fourth of four children of Czesław Spława-Neyman and Kazimiera Lutosławska. His family was Roman Catholic and Neyman served as an altar boy during his early childhood. Later, Neyman would become an agnostic. Neyman's family descended from a long line of Polish nobles and military heroes. He graduated from the Kamieniec Podolski gubernial gymnasium for boys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dubins–Spanier Theorems
The Dubins–Spanier theorems are several theorems in the theory of fair cake-cutting. They were published by Lester Dubins and Edwin Spanier in 1961. Although the original motivation for these theorems is fair division, they are in fact general theorems in measure theory. Setting There is a set U, and a set \mathbb which is a sigma-algebra of subsets of U. There are n partners. Every partner i has a personal value measure V_i: \mathbb \to \mathbb. This function determines how much each subset of U is worth to that partner. Let X a partition of U to k measurable sets: U = X_1 \sqcup \cdots \sqcup X_k. Define the matrix M_X as the following n\times k matrix: :M_X ,j= V_i(X_j) This matrix contains the valuations of all players to all pieces of the partition. Let \mathbb be the collection of all such matrices (for the same value measures, the same k, and different partitions): :\mathbb = \ The Dubins–Spanier theorems deal with the topological properties of \mathbb. State ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exact Division
Exact division, also called consensus division, is a partition of a continuous resource (" cake") into some ''k'' pieces, such that each of ''n'' people with different tastes agree on the value of each of the pieces. For example, consider a cake which is half chocolate and half vanilla. Alice values only the chocolate and George values only the vanilla. The cake is divided into three pieces: one piece contains 20% of the chocolate and 20% of the vanilla, the second contains 50% of the chocolate and 50% of the vanilla, and the third contains the rest of the cake. This is an exact division (with ''k''=3 and ''n''=2), as both Alice and George value the three pieces as 20%, 50% and 30% respectively. Several common variants and special cases are known by different terms: * Consensus halving – the cake should be partitioned into two pieces (''k''=2), and all agents agree that the pieces have equal values. *Consensus 1/''k''-division, for any constant ''k''>1 - the cake should be partition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Egon Pearson
Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson and Maria, née Sharpe, and, like his father, a leading British statistician. Career He was educated at Winchester College and Trinity College, Cambridge, and succeeded his father as professor of statistics at University College London and as editor of the journal ''Biometrika''. Pearson is best known for development of the Neyman–Pearson lemma of statistical hypothesis testing. He was elected a Fellow of the Econometric Society in 1948. He was President of the Royal Statistical Society in 1955–56, and was awarded its Guy Medal in gold in 1955. He was appointed a CBE in 1946. He was elected a Fellow of the Royal Society in March 1966. His candidacy citation read: Family life Pearson married Eileen Jolly in 1934 and the couple had two daughters, Judith and Sarah. Eileen died of pneumonia in 1949. Pearson subsequently married Margaret Theodosia Scott in 1967 and the couple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georges Darmois
Georges Darmois (24 June 1888 – 3 January 1960) was a French mathematician and statistician. He pioneered in the theory of sufficiency, in stellar statistics, and in factor analysis. He was also one of the first French mathematicians to teach British mathematical statistics. He is one of the eponyms of the Koopman–Pitman–Darmois theorem and sufficient statistics and exponential families. Biography Darmois was born on 24 June 1888 in Éply. He was admitted to École normale supérieure in 1906 and passed subsequently the agrégation de mathematiques in 1909. From 1911 to 1914, he was a qualified assistant (agrégé préparateur) at the École normale supérieure, where his scientific activities were directed by Émile Borel who rapidly appreciated his talent. Darmois earned his doctorate from the University of Paris in 1921. He defended a thesis on algebraic curves and partial differential equations before the jury consisting of Émile Picard and Édouard Gours ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stromquist–Woodall Theorem
The Stromquist–Woodall theorem is a theorem in fair division and measure theory. Informally, it says that, for any cake, for any ''n'' people with different tastes, and for any fraction ''w'', there exists a subset of the cake that all people value at exactly a fraction ''w'' of the total cake value, and it can be cut using at most 2n-2 cuts. The theorem is about a circular 1-dimensional cake (a "pie"). Formally, it can be described as the interval ,1in which the two endpoints are identified. There are ''n'' continuous measures over the cake: V_1,\ldots,V_n; each measure represents the valuations of a different person over subsets of the cake. The theorem says that, for every weight w \in ,1/math>, there is a subset C_w, which all people value at exactly w: : \forall i = 1,\ldots,n: \,\,\,\,\, V_i(C_w)=w, where C_w is a union of at most n-1 intervals. This means that 2n-2 cuts are sufficient for cutting the subset C_w. If the cake is not circular (that is, the endpoints are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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