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XYZ Inequality
In combinatorial mathematics, the XYZ inequality, also called the Fishburn–Shepp inequality, is an inequality for the number of linear extensions of finite partial orders. The inequality was conjectured by Ivan Rival and Bill Sands in 1981. It was proved by Lawrence Shepp in . An extension was given by Peter Fishburn in . It states that if ''x'', ''y'', and ''z'' are incomparable elements of a finite poset, then : P(x\prec y)P(x\prec z) \leqslant P((x\prec y) \wedge (x\prec z)), where ''P''(A) is the probability that a linear order extending the partial order \prec has the property A. In other words, the probability that x\prec z increases if one adds the condition that x\prec y. In the language of conditional probability, : P(x \prec z) \leqslant P(x \prec z \mid x \prec y). The proof uses the Ahlswede–Daykin inequality. See also * FKG inequality In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Extension
In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. Definitions Given any partial orders \,\leq\, and \,\leq^*\, on a set X, \,\leq^*\, is a linear extension of \,\leq\, exactly when (1) \,\leq^*\, is a total order and (2) for every x, y \in X, if x \leq y, then x \leq^* y. It is that second property that leads mathematicians to describe \,\leq^*\, as extending \,\leq. Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set P to a chain C on the same ground set. Order-extension principle The statement that every partial order can be extended to a total order is known as the order-extension principle. A proof using the axiom of choice was first published by Edward Marczewski in 1930. Marczewski write ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''incompar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ivan Rival
Ivan Rival (March 15, 1947 – January 22, 2002 in Ottawa, Ontario, Canada) was a Canadian mathematician and computer scientist, a professor of mathematics at the University of Calgary and of computer science at the University of Ottawa.... Rival's Ph.D. thesis concerned lattice theory. After moving to Calgary he began to work more generally with partially ordered sets, and to study fixed point theorems for partially ordered structures. He was a frequent organizer of conferences in order theory, and in 1984 he founded the journal ''Order''. As a computer scientist at Ottawa, he shifted research topics, applying his expertise in order theory to the study of data structures, computational geometry, and graph drawing. Rival grew up in Hamilton, Ontario. He earned a bachelor's degree at McMaster University in 1969, and received his Ph.D. from the University of Manitoba in 1974 under the supervision of George Gratzer.. After postdoctoral stints visiting Robert Dilworth at Caltech ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence Shepp
Lawrence Alan Shepp (September 9, 1936 Brooklyn, NY – April 23, 2013, Tucson, AZ) was an American mathematician, specializing in statistics and computational tomography. Shepp obtained his PhD from Princeton University in 1961 with a dissertation entitled ''Recurrent Sums of Random Variables''. His advisor was William Feller. He joined Bell Laboratories in 1962. He joined Rutgers University in 1997. He joined University of Pennsylvania in 2010. His work in tomography has had biomedical imaging applications, and he has also worked as professor of radiology at Columbia University (1973–1996), as a mathematician in the radiology service of Columbia Presbyterian Hospital. Awards and honors * 2014: IEEE Marie Sklodowska-Curie Award * 2012: Became a fellow of the American Mathematical Society. * 1992: Elected member of the Institute of Medicine * 1989: Elected member of the National Academy of Sciences * 1979: IEEE Distinguished Scientist Award in 1979 * 1979: Lester R. For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter C
Peter may refer to: People * List of people named Peter, a list of people and fictional characters with the given name * Peter (given name) ** Saint Peter (died 60s), apostle of Jesus, leader of the early Christian Church * Peter (surname), a surname (including a list of people with the name) Culture * Peter (actor) (born 1952), stage name Shinnosuke Ikehata, Japanese dancer and actor * ''Peter'' (album), a 1993 EP by Canadian band Eric's Trip * ''Peter'' (1934 film), a 1934 film directed by Henry Koster * ''Peter'' (2021 film), Marathi language film * "Peter" (''Fringe'' episode), an episode of the television series ''Fringe'' * ''Peter'' (novel), a 1908 book by Francis Hopkinson Smith * "Peter" (short story), an 1892 short story by Willa Cather Animals * Peter, the Lord's cat, cat at Lord's Cricket Ground in London * Peter (chief mouser), Chief Mouser between 1929 and 1946 * Peter II (cat), Chief Mouser between 1946 and 1947 * Peter III (cat), Chief Mouser between 1947 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of Comparability, comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is and the event is known or assumed to have occurred, "the conditional probability of given ", or "the probability of under the condition ", is usually written as or occasionally . This can also be understood as the fraction of probability B that intersects with A: P(A \mid B) = \frac. For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone unwell (sick) is coughing might be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ahlswede–Daykin Inequality
The Ahlswede–Daykin inequality , also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects fr ...). The inequality states that if f_1,f_2,f_3,f_4 are nonnegative functions on a finite distributive lattice such that :f_1(x)f_2(y)\le f_3(x\vee y)f_4(x\wedge y) for all ''x'', ''y'' in the lattice, then :f_1(X)f_2(Y)\le f_3(X\vee Y)f_4(X\wedge Y) for all subsets ''X'', ''Y'' of the lattice, where :f(X) = \sum_f(x) and :X\vee Y = \ :X\wedge Y = \. The Ahlswede–Daykin inequality can be used to provide a short ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FKG Inequality
In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to . Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model. An earlier version, for the special case of i.i.d. variables, called Harris inequality, is due to , see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has the same conclusion as the Griffiths inequalities, but the hypotheses are different. The inequality Let X be a finite distributive lattice, and ''μ'' a nonnegative function on it, that is assumed to satisfy the (FKG) lattice condit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequalities
Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * Spatial inequality, the unequal distribution of income and resources across geographical regions * Income inequality metrics, used to measure income and economic inequality among participants in a particular economy * International inequality, economic differences between countries Healthcare * Health equity, the study of differences in the quality of health and healthcare across different populations Mathematics * Inequality (mathematics), a relation between two values when they are different Social sciences * Educational inequality, the unequal distribution of academic resources to socially excluded communities * Gender inequality, unequal treatment or perceptions of individuals due to their gender * Participation inequality, the pheno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Combinatorics
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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