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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] [Amazon] |
Fair Cake-cutting
Fair cake-cutting is a kind of fair division problem. The problem involves a ''heterogeneous'' resource, such as a cake with different toppings, that is assumed to be ''divisible'' – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences over different parts of the cake, i.e., some people prefer the chocolate toppings, some prefer the cherries, some just want as large a piece as possible. The division should be ''unanimously'' fair – each person should receive a piece believed to be a fair share. The "cake" is only a metaphor; procedures for fair cake-cutting can be used to divide various kinds of resources, such as land estates, advertisement space or broadcast time. The prototypical procedure for fair cake-cutting is divide and choose, which is mentioned in the book of Book of Genesis, Genesis to resolve Abraham and Lot's conflict. This procedure solves the fa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Hugo Steinhaus
Hugo Dyonizy Steinhaus ( , ; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów (now Lviv, Ukraine), where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war. Author of around 170 scientific articles and books, Steinhaus has left his legacy and contribution in many branches of mathematics, such as functional analysis, geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Weller's Theorem
Weller's theorem is a theorem in economics. It says that a heterogeneous resource ("cake") can be divided among ''n'' partners with different valuations in a way that is both Pareto-efficient (PE) and envy-free (EF). Thus, it is possible to divide a cake fairly without compromising on economic efficiency. Moreover, Weller's theorem says that there exists a price such that the allocation and the price are a competitive equilibrium (CE) with equal incomes (EI). Thus, it connects two research fields which were previously unrelated: fair cake-cutting and general equilibrium. Background Fair cake-cutting has been studied since the 1940s. There is a heterogeneous divisible resource, such as a cake or a land-estate. There are ''n'' partners, each of whom has a personal value-density function over the cake. The value of a piece to a partner is the integral of his value-density over that piece (this means that the value is a nonatomic measure over the cake). The envy-free cake-cutting pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lyapunov Vector-measure Theorem
Lyapunov (, in old-Russian often written Лепунов) is a Russian surname that is sometimes also romanized as Ljapunov, Liapunov or Ljapunow. Notable people with the surname include: * Alexey Lyapunov (1911–1973), Russian mathematician * Aleksandr Lyapunov (1857–1918), son of Mikhail (1820–1868), Russian mathematician and mechanician, after whom the following are named: ** Lyapunov dimension ** Lyapunov equation ** Lyapunov exponent ** Lyapunov function ** Lyapunov fractal ** Lyapunov stability ** Lyapunov's central limit theorem ** Lyapunov time ** Lyapunov vector ** Lyapunov (crater) * Boris Lyapunov (1862–1943), son of Mikhail (1820–1868), Russian expert in Slavic studies * Mikhail Lyapunov (1820–1868), Russian astronomer * Mikhail Nikolaevich Lyapunov (1848–1909), Russian military officer and lawyer * Prokopy Lyapunov (d. 1611), Russian statesman * Sergei Lyapunov (1859–1924), son of Mikhail (1820–1868), Russian composer * Zakhary Lyapunov Zakhary Petro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
