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https://en.wikipedia.org/wiki/Lindel%C3%B6f%27s%20theorem
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In mathematics, Lindelöf's theorem is a result in complex analysis named after the Finnish mathematician Ernst Leonard Lindelöf. It states that a holomorphic function on a half-strip in the complex plane that is bounded on the boundary of the strip and does not grow "too fast" in the unbounded direction of the strip must remain bounded on the whole strip. The result is useful in the study of the Riemann zeta function, and is a special case of the Phragmén–Lindelöf principle. Also, see Hadamard three-lines theorem.
Statement of the theorem
Let be a half-strip in the complex plane:
Suppose that is holomorphic (i.e. analytic) on and that there are constants
, , and such that
and
Then is bounded by on all of :
Proof
Fix a point inside . Choose , an integer and large enough such that
. Applying maximum modulus principle to the function and
the rectangular area we obtain , that is, . Letting yields
as required.
References
Theorems in complex analysis
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https://en.wikipedia.org/wiki/Monadic%20predicate%20calculus
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In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic in which all relation symbols in the signature are monadic (that is, they take only one argument), and there are no function symbols. All atomic formulas are thus of the form , where is a relation symbol and is a variable.
Monadic predicate calculus can be contrasted with polyadic predicate calculus, which allows relation symbols that take two or more arguments.
Expressiveness
The absence of polyadic relation symbols severely restricts what can be expressed in the monadic predicate calculus. It is so weak that, unlike the full predicate calculus, it is decidable—there is a decision procedure that determines whether a given formula of monadic predicate calculus is logically valid (true for all nonempty domains). Adding a single binary relation symbol to monadic logic, however, results in an undecidable logic.
Relationship with term logic
The need to go beyond monadic logic was not appreciated until the work on the logic of relations, by Augustus De Morgan and Charles Sanders Peirce in the nineteenth century, and by Frege in his 1879 Begriffsschrifft. Prior to the work of these three men, term logic (syllogistic logic) was widely considered adequate for formal deductive reasoning.
Inferences in term logic can all be represented in the monadic predicate calculus. For example the argument
All dogs are mammals.
No mammal is a bird.
Thus, no dog is a bird.
can be notated in the language of monadic predicate calculus as
where , and denote the predicates of being, respectively, a dog, a mammal, and a bird.
Conversely, monadic predicate calculus is not significantly more expressive than term logic. Each formula in the monadic predicate calculus is equivalent to a formula in which quantifiers appear only in closed subformulas of the form
or
These formulas slightly generalize the basic judgements considered in term logic. For example, this form allows statements such as "Every mammal is either a herbivore or a carnivore (or both)", . Reasoning about such statements can, however, still be handled within the framework of term logic, although not by the 19 classical Aristotelian syllogisms alone.
Taking propositional logic as given, every formula in the monadic predicate calculus expresses something that can likewise be formulated in term logic. On the other hand, a modern view of the problem of multiple generality in traditional logic concludes that quantifiers cannot nest usefully if there are no polyadic predicates to relate the bound variables.
Variants
The formal system described above is sometimes called the pure monadic predicate calculus, where "pure" signifies the absence of function letters. Allowing monadic function letters changes the logic only superficially, whereas admitting even a single binary function letter results in an undecidable logic.
Monadic second-order logic allows predicates of higher arity i
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https://en.wikipedia.org/wiki/George%20B.%20Thomas
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George Brinton Thomas Jr. (January 11, 1914 – October 31, 2006) was an American mathematician and professor of mathematics at MIT. Internationally, he is best known for being the author of the widely used calculus textbook Calculus and Analytical Geometry, known today as Thomas' Calculus.
Early life
Born in Boise, Idaho, Thomas' early years were difficult. His father, George Brinton Thomas Sr., was a bank employee, and his mother, Georgia Fay Thomas (née Goin), died in the 1919 Influenza Epidemic, just eight days before his fifth birthday. His father remarried shortly thereafter, to Lena Steward. They lived in a tent with a wooden floor and a coal stove.
After his stepmother Lena died from complications due to childbirth, the father and son moved to the Spokane Valley in Washington State, where they both attended Spokane University. George Thomas Sr. married again, to Gertrude Alice Johnson. Thomas began attending Washington State College (now Washington State University), after Spokane University went bankrupt. There, he earned a B.A. in 1934 and an M.A. in 1936, both in mathematics and mathematics education.
On August 15, 1936, Thomas married Jane Heath at her family's home in South Bend, Washington. The couple lived in Pullman, Washington for a year; Thomas worked at a local shoe store to save money for further graduate education.
In 1937, Thomas was accepted into the graduate mathematics program at Cornell University. At Cornell, Thomas worked as an instructor while pursuing his research in number theory.
Academic career
Thomas finished his doctoral work in 1940 and was immediately hired by MIT for a one-year teaching appointment. He was well liked at MIT, and was invited to join the faculty after his teaching fellowship ended.
During the Second World War, Thomas was involved in early computation systems and programmed the differential analyzer to calculate firing tables for the Navy.
In 1952, George and Jane Thomas moved into the Conantum community in Concord, Massachusetts, where many younger Harvard and MIT faculty members lived.
Calculus and Analytic Geometry
In 1951, Addison-Wesley was then a new publishing company specializing in textbooks and technical literature. The management was unhappy with the calculus textbook they were then publishing, so they approached Thomas, asking if he could revise the book. Instead, he went ahead with an entirely new book. The first edition came out in 1952; Calculus and Analytic Geometry became one of the most famous and widely used texts on the subject. For many of the later editions (from the 5th onwards), Thomas was assisted by co-author Ross L. Finney, which gave rise to the text's metonym Thomas & Finney such was its ubiquity in calculus teaching.
Thomas' Calculus
Following Ross Finney's death in 2000, the text has simply been known as Thomas' Calculus from the 10th edition onward. The 14th edition, now edited by contemporary authors, is the most recent version of the te
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https://en.wikipedia.org/wiki/Hedgehog%20space
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In mathematics, a hedgehog space is a topological space consisting of a set of spines joined at a point.
For any cardinal number , the -hedgehog space is formed by taking the disjoint union of real unit intervals identified at the origin (though its topology is not the quotient topology, but that defined by the metric below). Each unit interval is referred to as one of the hedgehog's spines. A -hedgehog space is sometimes called a hedgehog space of spininess .
The hedgehog space is a metric space, when endowed with the hedgehog metric if and lie in the same spine, and by if and lie in different spines. Although their disjoint union makes the origins of the intervals distinct, the metric makes them equivalent by assigning them 0 distance.
Hedgehog spaces are examples of real trees.
Paris metric
The metric on the plane in which the distance between any two points is their Euclidean distance when the two points belong to a ray though the origin, and is otherwise the sum of the distances of the two points from the origin, is sometimes called the Paris metric because navigation in this metric resembles that in the radial street plan of Paris: for almost all pairs of points, the shortest path passes through the center. The Paris metric, restricted to the unit disk, is a hedgehog space where K is the cardinality of the continuum.
Kowalsky's theorem
Kowalsky's theorem, named after Hans-Joachim Kowalsky, states that any metrizable space of weight can be represented as a topological subspace of the product of countably many spaces.
See also
Comb space
Long line (topology)
Rose (topology)
References
Other sources
General topology
Topological spaces
Trees (topology)
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https://en.wikipedia.org/wiki/Moore%20space%20%28topology%29
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In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold:
Any two distinct points can be separated by neighbourhoods, and any closed set and any point in its complement can be separated by neighbourhoods. (X is a regular Hausdorff space.)
There is a countable collection of open covers of X, such that for any closed set C and any point p in its complement there exists a cover in the collection such that every neighbourhood of p in the cover is disjoint from C. (X is a developable space.)
Moore spaces are generally interesting in mathematics because they may be applied to prove interesting metrization theorems. The concept of a Moore space was formulated by R. L. Moore in the earlier part of the 20th century.
Examples and properties
Every metrizable space, X, is a Moore space. If {A(n)x} is the open cover of X (indexed by x in X) by all balls of radius 1/n, then the collection of all such open covers as n varies over the positive integers is a development of X. Since all metrizable spaces are normal, all metric spaces are Moore spaces.
Moore spaces are a lot like regular spaces and different from normal spaces in the sense that every subspace of a Moore space is also a Moore space.
The image of a Moore space under an injective, continuous open map is always a Moore space. (The image of a regular space under an injective, continuous open map is always regular.)
Both examples 2 and 3 suggest that Moore spaces are similar to regular spaces.
Neither the Sorgenfrey line nor the Sorgenfrey plane are Moore spaces because they are normal and not second countable.
The Moore plane (also known as the Niemytski space) is an example of a non-metrizable Moore space.
Every metacompact, separable, normal Moore space is metrizable. This theorem is known as Traylor’s theorem.
Every locally compact, locally connected normal Moore space is metrizable. This theorem was proved by Reed and Zenor.
If , then every separable normal Moore space is metrizable. This theorem is known as Jones’ theorem.
Normal Moore space conjecture
For a long time, topologists were trying to prove the so-called normal Moore space conjecture: every normal Moore space is metrizable. This was inspired by the fact that all known Moore spaces that were not metrizable were also not normal. This would have been a nice metrization theorem. There were some nice partial results at first; namely properties 7, 8 and 9 as given in the previous section.
With property 9, we see that we can drop metacompactness from Traylor's theorem, but at the cost of a set-theoretic assumption. Another example of this is Fleissner's theorem that the axiom of constructibility implies that locally compact, normal Moore spaces are metrizable.
On the other hand, under the continuum hypothesis (CH) and also under Martin's axiom and not CH, there are several examples of non-metrizable norm
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https://en.wikipedia.org/wiki/Moore%20space
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In mathematics, Moore space may refer to:
Moore space (algebraic topology)
Moore space (topology), a regular, developable topological space.
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https://en.wikipedia.org/wiki/Development%20%28topology%29
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In the mathematical field of topology, a development is a countable collection of open covers of a topological space that satisfies certain separation axioms.
Let be a topological space. A development for is a countable collection of open coverings of , such that for any closed subset and any point in the complement of , there exists a cover such that no element of which contains intersects . A space with a development is called developable.
A development such that for all is called a nested development. A theorem from Vickery states that every developable space in fact has a nested development. If is a refinement of , for all , then the development is called a refined development.
Vickery's theorem implies that a topological space is a Moore space if and only if it is regular and developable.
References
General topology
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https://en.wikipedia.org/wiki/Fuzzy%20subalgebra
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Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure.
Definition
Consider a first order language for algebraic structures with a monadic predicate symbol S. Then a fuzzy subalgebra is a fuzzy model of a theory containing, for any n-ary operation h, the axioms
and, for any constant c, S(c).
The first axiom expresses the closure of S with respect to the operation h, and the second expresses the fact that c is an element in S. As an example, assume that the valuation structure is defined in [0,1] and denote by the operation in [0,1] used to interpret the conjunction. Then a fuzzy subalgebra of an algebraic structure whose domain is D is defined by a fuzzy subset of D such that, for every d1,...,dn in D, if h is the interpretation of the n-ary operation symbol h, then
Moreover, if c is the interpretation of a constant c such that s(c) = 1.
A largely studied class of fuzzy subalgebras is the one in which the operation coincides with the minimum. In such a case it is immediate to prove the following proposition.
Proposition. A fuzzy subset s of an algebraic structure defines a fuzzy subalgebra if and only if for every λ in [0,1], the closed cut {x ∈ D : s(x)≥ λ} of s is a subalgebra.
Fuzzy subgroups and submonoids
The fuzzy subgroups and the fuzzy submonoids are particularly interesting classes of fuzzy subalgebras. In such a case a fuzzy subset s of a monoid (M,•,u) is a fuzzy submonoid if and only if
where u is the neutral element in A.
Given a group G, a fuzzy subgroup of G is a fuzzy submonoid s of G such that
s(x) ≤ s(x−1).
It is possible to prove that the notion of fuzzy subgroup is strictly related with the notions of fuzzy equivalence. In fact, assume that S is a set, G a group of transformations in S and (G,s) a fuzzy subgroup of G. Then, by setting
e(x,y) = Sup{s(h) : h is an element in G such that h(x) = y}
we obtain a fuzzy equivalence. Conversely, let e be a fuzzy equivalence in S and, for every transformation h of S, set
s(h)= Inf{e(x,h(x)): x∈S}.
Then s defines a fuzzy subgroup of transformation in S. In a similar way we can relate the fuzzy submonoids with the fuzzy orders.
Bibliography
Klir, G. and Bo Yuan, Fuzzy Sets and Fuzzy Logic (1995)
Zimmermann H., Fuzzy Set Theory and its Applications (2001), .
Chakraborty H. and Das S., On fuzzy equivalence 1, Fuzzy Sets and Systems, 11 (1983), 185-193.
Demirci M., Recasens J., Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets and Systems, 144 (2004), 441-458.
Di Nola A., Gerla G., Lattice valued algebras, Stochastica, 11 (1987), 137-150.
Hájek P., Metamathematics of fuzzy logic. Kluwer 1998.
Klir G., UTE H. St.Clair and Bo Yuan Fuzzy Set Theory Foundations and Applications,1997.
Gerla G., Scarpati M., Similarities, Fuzzy Groups: a Galois Connection, J. Math. Anal. Appl., 29
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https://en.wikipedia.org/wiki/Nash%27s%20theorem
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In mathematics, Nash's theorem may refer to one of the following:
the Nash embedding theorems in differential geometry
Nash's theorem on the existence of Nash equilibria in game theory
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https://en.wikipedia.org/wiki/United%20States%20men%27s%20national%20soccer%20team%20records%20and%20statistics
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This is a comprehensive list of the United States national soccer team's competitive, individual, team, and head-to-head records.
Individual records
Player records
.
Players in bold are still active for selection for the national team.
Coaching records
Most coaching appearances
Bruce Arena: 148
Team records
Biggest victory
8–0 vs. Barbados, June 15, 2008
Competition records
The U.S. regularly competes at the FIFA World Cup, the CONCACAF Gold Cup, and the Summer Olympics. The U.S. has also played in the FIFA Confederations Cup, Copa América by invitation, as well as several minor tournaments.
The best result for the United States in a World Cup tournament came in 1930 when the team reached the semi-finals. The team was composed of six naturalized internationals, five of them from Scotland and one from England. The best result in the modern era is the 2002 World Cup, when the U.S. reached the quarter-finals. The worst world Cup tournament results in the modern era were group stage eliminations in 1990, 1998, and 2006, although the country failed to even qualify for the final tournament in 2018.
In the Confederations Cup, the United States finished in third place in both 1992 and 1999, and were runner-up in 2009. The United States appeared in their first intercontinental tournament final at the 2009 Confederations Cup. In the semi-finals, the United States upset top ranked Spain 2–0, to advance to the final. In the final, the United States lost 3–2 to Brazil after leading 2–0 at halftime.
The U.S. men's soccer team have played in the Summer Olympics since 1924. From that tournament to 1980, only amateur and state-sponsored Eastern European players were allowed on Olympic teams. The Olympics became a full international tournament in 1984 after the IOC allowed full national teams from outside FIFA CONMEBOL & UEFA confederations. Ever since 1992 the men's Olympic event has been age-restricted, under 23 plus three overage players, and participation has been by the United States men's national under-23 soccer team.
In regional competitions, the United States has won the CONCACAF Gold Cup seven times, with their most recent title in 2021. Their best ever finish at the Copa América was fourth-place at the 1995 and 2016 editions.
FIFA World Cup
CONCACAF Gold Cup
CONCACAF Championship 1963–1989, CONCACAF Gold Cup 1991–present
Summer Olympics
Copa América
South American Championship 1916–1967, Copa América 1975–present
FIFA Confederations Cup
CONCACAF Nations League
Head-to-head record
The following tables summarizes the all-time record for the United States men's national soccer team, first broken down by confederation and then the team's head-to-head record by decade. The United States has played matches against 105 current and former national teams, with the latest result, a win, coming against Ghana on October 17, 2023.
Through United States vs. Ghana on October 17, 2023.
AFC (18–11–8)
CAF (12–6–2)
CONCACAF (221–90–82)
CONM
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https://en.wikipedia.org/wiki/Offense%20efficiency%20rating
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In basketball statistics, Offensive Efficiency Rating (OER) is the average number of points scored by a basketball player per shot taken.
This includes missed field goals as well as free throws. The statistic stems from the previously created Player Efficiency Rating (PER). The per-minute rating was created by John Hollinger. Hollinger states, "The PER sums up all a player's positive accomplishments, subtracts the negative accomplishments, and returns a per-minute rating of a player's performance."
References
Basketball terminology
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https://en.wikipedia.org/wiki/Factor%20base
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In computational number theory, a factor base is a small set of prime numbers commonly used as a mathematical tool in algorithms involving extensive sieving for potential factors of a given integer.
Usage in factoring algorithms
A factor base is a relatively small set of distinct prime numbers P, sometimes together with -1. Say we want to factorize an integer n. We generate, in some way, a large number of integer pairs (x, y) for which , , and can be completely factorized over the chosen factor base—that is, all their prime factors are in P.
In practice, several integers x are found such that has all of its prime factors in the pre-chosen factor base. We represent each expression as a vector of a matrix with integer entries being the exponents of factors in the factor base. Linear combinations of the rows corresponds to multiplication of these expressions. A linear dependence relation mod 2 among the rows leads to a desired congruence . This essentially reformulates the problem into a system of linear equations, which can be solved using numerous methods such as Gaussian elimination; in practice advanced methods like the block Lanczos algorithm are used, that take advantage of certain properties of the system.
This congruence may generate the trivial ; in this case we try to find another suitable congruence. If repeated attempts to factor fail we can try again using a different factor base.
Algorithms
Factor bases are used in, for example, Dixon's factorization, the quadratic sieve, and the number field sieve. The difference between these algorithms is essentially the methods used to generate (x, y) candidates. Factor bases are also used in the Index calculus algorithm for computing discrete logarithms.
References
Integer factorization algorithms
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https://en.wikipedia.org/wiki/South%20Carolina%20Governor%27s%20School%20for%20Science%20and%20Mathematics
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The South Carolina Governor's School for Science and Mathematics (GSSM) is a public, boarding high school for students in grades 11 and 12, located in Hartsville, South Carolina. The school concentrates on science and mathematics, but offers the full spectrum of the humanities as well.
Academics
Students at GSSM select from a wide range of STEM courses during their two years on campus. Typically, 18 AP courses are offered and 45% of the STEM courses are listed as "Above AP". Students can conduct semester or year-long scientific investigations, in addition to the required Summer Program for Research Interns (SPRI). During SPRI, students conduct six weeks of mentored scientific, business or economics research at university or corporate R&D labs across South Carolina or in locations across the United States and other countries. In 2009, GSSM began the Research Exchange Scholars Program (later renamed the Research Experience Scholars Program) with exchange students from Pforzheim, Germany and Daejeon, Korea. The program has grown to include more locations in Germany and China, and plans are underway to include sites in a number of countries. In 2017, the RESP program included sites in Germany, Korea, and China.
In addition to a rigorous STEM curriculum, GSSM also offers a wealth of humanities courses and a January Interim mini-mester of experiential courses and national and international trips. Courses and trips vary from year to year.
Every student is assigned a college counselor that has years of experience working in college admission offices. GSSM helps students get into the colleges they want and helps them apply for scholarships.
Outreach
Beyond the education provided to students in residence, the school delivers outreach programs for middle schoolers in satellite locations statewide, as well as for rising 8th through 10th graders at its annual, residential summer program, GoSciTech.
GoSciTech, originally named Summer Science Program (SSP), began as a one-week program in 1990. As of 2012, the camp expanded to offer up to four weeks of courses and was renamed GoSciTech. In 2016, nearly 30 courses were offered to over 500 students.
Satellite camps include iTEAMS Xtreme and iTEAMS Xtreme: Next Generation, computer science, technology and robotics day camps. CREATEng, an engineering and design thinking day camp, was added in 2014. More than 1300 students and teachers from across the state participated in GSSM's summer outreach programs during the 2016 summer.
History
The school was founded in 1988 by Governor Carroll Campbell and Charles W. Coker on the grounds of Coker College. It moved to a purpose-built campus nearby in 2003. In 2010, two new wings were added: the Academic Wing, containing classrooms and laboratories, and the Student Activities Center, which includes an engineering projects center and also a gym, weight room, game room, and kitchen. In 2015, the school reached maximum capacity of 288 students.
Admissions
Admission to GS
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https://en.wikipedia.org/wiki/Personal%20income%20in%20the%20United%20States
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Personal income is an individual's total earnings from wages, investment interest, and other sources. The Bureau of Labor Statistics reported a median weekly personal income of $1,037 for full-time workers in the United States in Q1 2022. For the year 2020, the U.S. Census Bureau estimates that the median annual earnings for all workers (aged 15 and over) was $41,535; and more specifically estimates that median annual earnings for those who worked full-time, year round, was $56,287.
Income patterns are evident on the basis of age, sex, ethnicity and educational characteristics. In 2005 roughly half of all those with graduate degrees were among the nation's top 15% of income earners. Among different demographics (gender, marital status, ethnicity) for those over the age of 18, median personal income ranged from $3,317 for an unemployed, married Asian American female to $55,935 for a full-time, year-round employed Asian American male. According to the US Census, men tended to have higher income than women, while Asians and Whites earned more than African Americans and Hispanics.
Income statistics
In the United States the most widely cited personal income statistics are the Bureau of Economic Analysis's personal income and the Census Bureau's per capita money income. The two statistics spring from different traditions of measurement—personal income from national economic accounts and money income from household surveys. BEA's statistics relate personal income to measures of production, including GDP, and is considered an indicator of consumer spending. The Census Bureau's statistics provide detail on income distribution and demographics and are used to produce the nation's official poverty statistics.
Personal income and disposable personal income
BEA's personal income measures the income received by persons from participation in production, from government and business transfers, and from holding interest-bearing securities and corporate stocks. Personal income also includes income received by nonprofit institutions serving households, by private non-insured welfare funds, and by private trust funds. BEA also publishes disposable personal income, which measures the income available to households after paying federal and state and local government income taxes.
Income from production is generated both by the labor of individuals (for example, in the form of wages and salaries and of proprietors' income) and by the capital that they own (in the form of rental income of persons). Income that is not earned from production in the current period—such as capital gains, which relate to changes in the price of assets over time—is excluded.
BEA's monthly personal income estimates are one of several key macroeconomic indicators that the National Bureau of Economic Research considers when dating the business cycle.
Personal income and disposable personal income are provided both as aggregate and as per capita statistics. BEA produces monthly estimates o
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https://en.wikipedia.org/wiki/James%20Beckett%20%28statistician%29
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James Beckett III is an American statistician, author, editor, and publisher. His publications are well known in the hobby of sports card collecting.
Beckett earned a Ph.D. degree in statistics at Southern Methodist University in 1975 and then joined the faculty of Bowling Green State University as an associate professor. While at Bowling Green, Beckett began preparing baseball card price guides, which he offered free upon request.
Beckett price guides rely upon information from sellers throughout the United States, who supply information on customer interest and sales of products. Price guides typically carry two value labels, one based upon a high value, the other denoting low values. As the condition of collectibles is important in ascertaining their value, Beckett price guides also typically include a series of definitions for estimating condition.
In November 1984, Beckett began publishing Beckett Baseball Card Monthly. (Rival publication Tuff Stuff was also founded that year.) Beckett Baseball Card Monthly grew in popularity and became the basis for the success of Beckett Media, now based in Dallas, Texas. Beckett Publications produces price guides for a variety of sports collectibles (Beckett's Football, Basketball, and Hockey guides would start in the early 1990s, with Beckett's monthly Racing Guide following in 1996). Market values for non-sports card collectibles such as Pokémon Cards and related products are also tracked. Beckett retains a position as Senior Advisor for Beckett Media, but is no longer Editor/Publisher as he was during the 1980s and 1990s.
The publishing company became the leading publisher of sports and entertainment market collectible guides and was acquired by Apprise Media in 2005.
Personal life
Beckett has three children from his first marriage. He is married to the former Diane Burgdorf, daughter of a Dallas car dealer and ex-wife of Sir Mark Thatcher. From time to time, Beckett attends some of the most prominent sports card and collectibles conventions held in various U.S. cities and is considered a celebrity within the industry.
References
Sources
Beckett Fact Sheet
"About the Author," Dr. James Beckett, et al., eds., Beckett Almanac of Baseball Cards and Collectibles, Number 8 (Dallas: Beckett Publications, 2003), pg. 10.
American publishers (people)
Southern Methodist University alumni
Bowling Green State University faculty
Trading cards
Year of birth missing (living people)
Living people
American statisticians
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https://en.wikipedia.org/wiki/Rank%20test
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In statistics, a rank test is any test involving ranks. Examples include:
Wilcoxon signed-rank test
Kruskal–Wallis one-way analysis of variance
Mann–Whitney U (special case)
Page's trend test
Friedman test
Rank products
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https://en.wikipedia.org/wiki/Mediated%20VPN
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A mediated VPN is a virtual private network topology where two or more participants connect to a central switchboard server managed typically by a third party in order to create a virtual private network between them, as distinct from a typical VPN arrangement whereby clients of an organisation connect to a VPN concentrator managed by the same organization.
Typically a switchboard server (referred to as a mediator) will manage several VPNs, identifying each individually by authentication credentials (such as username, network name and passwords). The mediator's role is to assign IP addresses to each participant in a VPN, and to encrypt data through the switchboard server in order to keep it secure from other participants in other VPNs.
See also
Virtual private network
Virtual Private LAN Service
Point-to-point (telecommunications)
References
Network architecture
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https://en.wikipedia.org/wiki/Bilinear%20program
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In mathematics, a bilinear program is a nonlinear optimization problem whose objective or constraint functions are bilinear. An example is the pooling problem.
References
Bilinear program at the Mathematical Programming Glossary.
Mathematical optimization
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https://en.wikipedia.org/wiki/Gamma%20test
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Gamma test may refer to:
Gamma test (statistics)
An alternate name for "release candidate" in the software release life cycle
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https://en.wikipedia.org/wiki/Hamming%20graph
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Hamming graphs are a special class of graphs named after Richard Hamming and used in several branches of mathematics (graph theory) and computer science. Let be a set of elements and a positive integer. The Hamming graph has vertex set , the set of ordered -tuples of elements of , or sequences of length from . Two vertices are adjacent if they differ in precisely one coordinate; that is, if their Hamming distance is one. The Hamming graph is, equivalently, the Cartesian product of complete graphs .
In some cases, Hamming graphs may be considered more generally as the Cartesian products of complete graphs that may be of varying sizes. Unlike the Hamming graphs , the graphs in this more general class are not necessarily distance-regular, but they continue to be regular and vertex-transitive.
Special cases
, which is the generalized quadrangle
, which is the complete graph
, which is the lattice graph and also the rook's graph
, which is the singleton graph
, which is the hypercube graph . Hamiltonian paths in these graphs form Gray codes.
Because Cartesian products of graphs preserve the property of being a unit distance graph, the Hamming graphs and are all unit distance graphs.
Applications
The Hamming graphs are interesting in connection with error-correcting codes and association schemes, to name two areas. They have also been considered as a communications network topology in distributed computing.
Computational complexity
It is possible in linear time to test whether a graph is a Hamming graph, and in the case that it is, find a labeling of it with tuples that realizes it as a Hamming graph.
References
External links
Parametric families of graphs
Regular graphs
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https://en.wikipedia.org/wiki/Jad%C3%ADlson%20%28footballer%2C%20born%201977%29
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José Jadílson dos Santos Silva (born December 4, 1977 in Maceió, Alagoas), most commonly known as Jadílson, is a Brazilian football defender.
Club statistics
Honours
Goiás
Campeonato Goiano: 2006
São Paulo
Campeonato Brasileiro Série A: 2007
Cruzeiro
Campeonato Mineiro: 2008
Personal Honours
Brazilian Silver Ball (Placar) - Best Left Back: 2005
References
External links
globoesporte.globo.com
CBF
sambafoot
saopaulofc.net
1977 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
Clube de Regatas Brasil players
Associação Portuguesa de Desportos players
Botafogo Futebol Clube (SP) players
Guarani FC players
Hokkaido Consadole Sapporo players
Fluminense FC players
Paraná Clube players
Goiás Esporte Clube players
São Paulo FC players
Cruzeiro Esporte Clube players
Grêmio Foot-Ball Porto Alegrense players
Grêmio Barueri Futebol players
Men's association football defenders
Footballers from Maceió
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https://en.wikipedia.org/wiki/Al%C3%AA%20%28footballer%2C%20born%201986%29
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Alexandre Luiz Fernandes (born 21 January 1986), commonly known as Alê, is a Brazilian footballer who plays as a defensive midfielder for CA Juventus.
Club statistics
Honours
São Paulo
Campeonato Paulista: 2005
Copa Libertadores: 2005
Campeonato Brasileiro: 2006
Contract
Atlético Mineiro: 14 September 2010 to 14 September 2014
Americana Futebol Ltda.: 2 May 2011 to 31 December 2011 (on loan)
External links
1986 births
Living people
Footballers from São Paulo
Brazilian men's footballers
Men's association football midfielders
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Campeonato Brasileiro Série C players
São Paulo FC players
Clube Atlético Juventus players
Botafogo de Futebol e Regatas players
Esporte Clube Santo André players
Club Athletico Paranaense players
Clube Atlético Mineiro players
Avaí FC players
Associação Portuguesa de Desportos players
Rio Claro Futebol Clube players
J2 League players
Cerezo Osaka players
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in Japan
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https://en.wikipedia.org/wiki/Kannan%20Soundararajan
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Kannan Soundararajan (born December 27, 1973) is an Indian-born American mathematician and a professor of mathematics at Stanford University. Before moving to Stanford in 2006, he was a faculty member at University of Michigan, where he had also pursued his undergraduate studies. His main research interest is in analytic number theory, particularly in the subfields of automorphic L-functions, and multiplicative number theory.
Early life
Soundararajan grew up in Madras and was a student at Padma Seshadri High School in Nungambakkam in Madras. In 1989, he attended the prestigious Research Science Institute. He represented India at the International Mathematical Olympiad in 1991 and won a Silver Medal.
Education
Soundararajan joined the University of Michigan, Ann Arbor, in 1991 for undergraduate studies, and graduated with highest honours in 1995. Soundararajan won the inaugural Morgan Prize in 1995 for his work in analytic number theory while an undergraduate at the University of Michigan, where he later served as professor. He joined Princeton University in 1995 and did his Ph.D under the guidance of Professor Peter Sarnak.
Career
After his Ph.D. he received the first five-year fellowship from the American Institute of Mathematics, and held positions at Princeton University, the Institute for Advanced Study, and the University of Michigan. He moved to Stanford University in 2006 where he is, as of November 2022, the Anne T. and Robert M. Bass Professor of Mathematics.
He provided a proof of a conjecture of Ron Graham in combinatorial number theory jointly with Ramachandran Balasubramanian. He made important contributions in settling the arithmetic Quantum Unique Ergodicity conjecture for Maass wave forms and modular forms.
Awards
He received the Salem Prize in 2003 "for contributions to the area of Dirichlet L-functions and related character sums". In 2005, he won the $10,000 SASTRA Ramanujan Prize, shared with Manjul Bhargava, awarded by SASTRA in Thanjavur, India, for his outstanding contributions to number theory. In 2011, he was awarded the Infosys science foundation prize. He was awarded the Ostrowski prize in 2011, shared with lb Madsen and David Preiss, for a cornucopia of fundamental results in the last five years to go along with his brilliant earlier work.
He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory". In July 2017, Soundararajan was a plenary lecturer in the Mathematical Congress of the Americas. He was elected to the 2018 class of fellows of the American Mathematical Society. Kannan Soundararajan was invited as a plenary speaker of the 2022 International Congress of Mathematicians, scheduled to take place in Saint Petersburg, but moved to Helsinki and online because of the 2022 Russian invasion of Ukraine.
Selected publications
R. Holowinsky and K. Soundararajan, "Mass equidistribution for Hecke eigenforms," arXiv:0809.1636v1
K. Soundararajan, "Nonvanishing o
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https://en.wikipedia.org/wiki/Differential%20ideal
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In the theory of differential forms, a differential ideal I is an algebraic ideal in the ring of smooth differential forms on a smooth manifold, in other words a graded ideal in the sense of ring theory, that is further closed under exterior differentiation d, meaning that for any form α in I, the exterior derivative dα is also in I.
In the theory of differential algebra, a differential ideal I in a differential ring R is an ideal which is mapped to itself by each differential operator.
Exterior differential systems and partial differential equations
An exterior differential system consists of a smooth manifold and a differential ideal
.
An integral manifold of an exterior differential system consists of a submanifold having the property that the pullback to of all differential forms contained in vanishes identically.
One can express any partial differential equation system as an exterior differential system with independence condition. Suppose that we have a kth order partial differential equation system for maps , given by
.
The graph of the -jet of any solution of this partial differential equation system is a submanifold of the jet space, and is an integral manifold of the contact system on the -jet bundle.
This idea allows one to analyze the properties of partial differential equations with methods of differential geometry. For instance, we can apply the Cartan–Kähler_theorem to a system of partial differential equations by writing down the associated exterior differential system. We can frequently apply Cartan's equivalence method to exterior differential systems to study their symmetries and their diffeomorphism invariants.
Perfect differential ideals
A differential ideal is perfect if it has the property that if it contains an element then it contains any element such that for some .
References
Robert Bryant, Phillip Griffiths and Lucas Hsu, Toward a geometry of differential equations(DVI file), in Geometry, Topology, & Physics, Conf. Proc. Lecture Notes Geom. Topology, edited by S.-T. Yau, vol. IV (1995), pp. 1–76, Internat. Press, Cambridge, MA
Robert Bryant, Shiing-Shen Chern, Robert Gardner, Phillip Griffiths, Hubert Goldschmidt, Exterior Differential Systems, Springer--Verlag, Heidelberg, 1991.
Thomas A. Ivey, J. M. Landsberg, Cartan for beginners. Differential geometry via moving frames and exterior differential systems. Second edition. Graduate Studies in Mathematics, 175. American Mathematical Society, Providence, RI, 2016.
H. W. Raudenbush, Jr. "Ideal Theory and Algebraic Differential Equations", Transactions of the American Mathematical Society, Vol. 36, No. 2. (Apr., 1934), pp. 361–368. Stable URL:
J. F. Ritt, Differential Algebra, Dover, New York, 1950.
Differential forms
Differential algebra
Differential systems
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https://en.wikipedia.org/wiki/Hypercone
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In geometry, a hypercone (or spherical cone) is the figure in the 4-dimensional Euclidean space represented by the equation
It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is also named "spherical cone" because its intersections with hyperplanes perpendicular to the w-axis are spheres. A four-dimensional right hypercone can be thought of as a sphere which expands with time, starting its expansion from a single point source, such that the center of the expanding sphere remains fixed. An oblique hypercone would be a sphere which expands with time, again starting its expansion from a point source, but such that the center of the expanding sphere moves with a uniform velocity.
Parametric form
A right spherical hypercone can be described by the function
with vertex at the origin and expansion speed s.
A right spherical hypercone with radius r and height h can be described by the function
An oblique spherical hypercone could then be described by the function
where is the 3-velocity of the center of the expanding sphere.
An example of such a cone would be an expanding sound wave as seen from the point of view of a moving reference frame: e.g. the sound wave of a jet aircraft as seen from the jet's own reference frame.
Note that the 3D-surfaces above enclose 4D-hypervolumes, which are the 4-cones proper.
Geometrical interpretation
The spherical cone consists of two unbounded nappes, which meet at the origin and are the analogues of the nappes of the 3-dimensional conical surface. The upper nappe corresponds with the half with positive w-coordinates, and the lower nappe corresponds with the half with negative w-coordinates.
If it is restricted between the hyperplanes w = 0 and w = r for some nonzero r, then it may be closed by a 3-ball of radius r, centered at (0,0,0,r), so that it bounds a finite 4-dimensional volume. This volume is given by the formula r4, and is the 4-dimensional equivalent of the solid cone. The ball may be thought of as the 'lid' at the base of the 4-dimensional cone's nappe, and the origin becomes its 'apex'.
This shape may be projected into 3-dimensional space in various ways. If projected onto the xyz hyperplane, its image is a ball. If projected onto the xyw, xzw, or yzw hyperplanes, its image is a solid cone. If projected onto an oblique hyperplane, its image is either an ellipsoid or a solid cone with an ellipsoidal base (resembling an ice cream cone). These images are the analogues of the possible images of the solid cone projected to 2 dimensions.
Construction
The (half) hypercone may be constructed in a manner analogous to the construction of a 3D cone. A 3D cone may be thought of as the result of stacking progressively smaller discs on top of each other until they taper to a point. Alternatively, a 3D cone may be regarded as the volume swept out by an upright isosceles triangle as it rotates about its base.
A 4D hype
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https://en.wikipedia.org/wiki/Quasi-Frobenius%20Lie%20algebra
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In mathematics, a quasi-Frobenius Lie algebra
over a field is a Lie algebra
equipped with a nondegenerate skew-symmetric bilinear form
, which is a Lie algebra 2-cocycle of with values in . In other words,
for all , , in .
If is a coboundary, which means that there exists a linear form such that
then
is called a Frobenius Lie algebra.
Equivalence with pre-Lie algebras with nondegenerate invariant skew-symmetric bilinear form
If is a quasi-Frobenius Lie algebra, one can define on another bilinear product by the formula
.
Then one has
and
is a pre-Lie algebra.
See also
Lie coalgebra
Lie bialgebra
Lie algebra cohomology
Frobenius algebra
Quasi-Frobenius ring
References
Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979.
Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .
Lie algebras
Coalgebras
Symplectic topology
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https://en.wikipedia.org/wiki/Karl%20Sigmund
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Karl Sigmund (born July 26, 1945) is a Professor of Mathematics at the University of Vienna and one of the pioneers of evolutionary game theory.
Career
Sigmund was schooled in the Lycée Francais de Vienne. From 1963 to 1968 he studied at the Institute of Mathematics at the University of Vienna, and obtained his Ph.D. under the supervision of Leopold Schmetterer. He spent his postdoctorate years (1968 to 1973) at Manchester ('68-'69), the Institut des hautes études scientifiques in Bures-sur-Yvette near Paris ('69-'70), the Hebrew University in Jerusalem (1970-'71), the University of Vienna (1971-'72) and the Austrian Academy of Sciences (1972-'73). In 1972 he received habilitation.
In 1973, Sigmund was appointed C3-professor at the University of Göttingen, and in 1974 became a full professor at the Institute of Mathematics in Vienna. His main scientific interest during these years was in ergodic theory and dynamical systems. From 1977 on, Sigmund became increasingly interested in different fields of biomathematics, and collaborated with Peter Schuster and Josef Hofbauer on mathematical ecology, chemical kinetics and population genetics, but especially on the new field of evolutionary game dynamics and replicator equations. Together with Martin Nowak, Christoph Hauert and Hannelore Brandt, he worked on game dynamical approaches to questions related with the evolution of cooperation in biological and human populations.
Since 1984, Sigmund has also worked as a part-time scientist at the International Institute for Applied Systems Analysis (IIASA) in Laxenburg, Lower Austria.
Honours and recognition
Sigmund was head of the Institute of Mathematics at the University of Vienna from 1983 to 1985, managing editor of the scientific journal Monatshefte für Mathematik from 1991 to 2001, vice-president (1995 to 1997) and president (1997 to 2001) of the Austrian Mathematical Society, corresponding member (1996) and full member (1999) of the Austrian Academy of Sciences, and member of the Leopoldina (2003). He has also given many plenary lectures, for instance at the International Congress of Mathematicians in 1998. He was awarded the Gauss Lectureship in 2003.
In 2010 he received an honorary doctorate (Doctor Philosophiae Honoris Causa) from the University of Helsinki. In 2012 he received the Isaacs Award.
Other details
During the last decade, Sigmund became increasingly interested in the history of mathematics and in particular, the Vienna Circle. He co-edited the mathematical works of Hans Hahn and Karl Menger and organised in 2001 an exhibition on the exodus of Austrian mathematicians fleeing the Nazis and in 2006 an exhibition on Kurt Gödel. From 2003 to 2005 he was vice-president of the Austrian Science Fund (FWF).
Because of his intimate knowledge of the Vienna Circle, Sigmund was invited to the Illinois Institute of Technology to speak at the inaugural Remembering Menger event on April 9, 2007.
Publications
Sigmund's publications include 133
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https://en.wikipedia.org/wiki/K-set%20%28geometry%29
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In discrete geometry, a -set of a finite point set in the Euclidean plane is a subset of elements of that can be strictly separated from the remaining points by a line. More generally, in Euclidean space of higher dimensions, a -set of a finite point set is a subset of elements that can be separated from the remaining points by a hyperplane. In particular, when (where is the size of ), the line or hyperplane that separates a -set from the rest of is a halving line or halving plane.
The -sets of a set of points in the plane are related by projective duality to the -levels in an arrangement of lines. The -level in an arrangement of lines in the plane is the curve consisting of the points that lie on one of the lines and have exactly lines below them. Discrete and computational geometers have also studied levels in arrangements of more general kinds of curves and surfaces.
Combinatorial bounds
It is of importance in the analysis of geometric algorithms to bound the number of -sets of a planar point set, or equivalently the number of -levels of a planar line arrangement, a problem first studied by Lovász and Erdős et al. The best known upper bound for this problem is , as was shown by Tamal Dey using the crossing number inequality of Ajtai, Chvátal, Newborn, and Szemerédi. However, the best known lower bound is far from Dey's upper bound: it is for some constant , as shown by Tóth.
In three dimensions, the best upper bound known is , and the best lower bound known is .
For points in three dimensions that are in convex position, that is, are the vertices of some convex polytope, the number of -sets is
, which follows from arguments used for bounding the complexity of th order Voronoi diagrams.
For the case when (halving lines), the maximum number of combinatorially distinct lines through two points of that bisect the remaining points when is
Bounds have also been proven on the number of -sets, where a -set is a -set for some . In two dimensions, the maximum number of -sets is exactly , while in dimensions the bound is .
Construction algorithms
Edelsbrunner and Welzl first studied the problem of constructing all -sets of an input point set, or dually of constructing the -level of an arrangement. The -level version of their algorithm can be viewed as a plane sweep algorithm that constructs the level in left-to-right order. Viewed in terms of -sets of point sets, their algorithm maintains a dynamic convex hull for the points on each side of a separating line, repeatedly finds a bitangent of these two hulls, and moves each of the two points of tangency to the opposite hull. Chan surveys subsequent results on this problem, and shows that it can be solved in time proportional to Dey's bound on the complexity of the -level.
Agarwal and Matoušek describe algorithms for efficiently constructing an approximate level; that is, a curve that passes between the -level and the -level for some small approximation parameter . They show that s
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https://en.wikipedia.org/wiki/Riemann%E2%80%93von%20Mangoldt%20formula
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In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function.
The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies
The formula was stated by Riemann in his notable paper "On the Number of Primes Less Than a Given Magnitude" (1859) and was finally proved by Mangoldt in 1905.
Backlund gives an explicit form of the error for all T > 2:
Under the Lindelöf and Riemann hypotheses the error term can be improved to and respectively.
Similarly, for any primitive Dirichlet character χ modulo q, we have
where N(T,χ) denotes the number of zeros of L(s,χ) with imaginary part between -T and T.
Notes
References
Theorems in analytic number theory
Bernhard Riemann
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https://en.wikipedia.org/wiki/Va%E1%B9%ADe%C5%9Bvara-siddh%C4%81nta
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Vaṭeśvara-siddhānta is a mathematical and astronomical treatise by Vaṭeśvara in India in 904. This treatise contains fifteen chapters on astronomy and applied mathematics.
Mathematical exercises are included for students to show their comprehension of the text.
References
K. S. Shukla, "Ancient Indian Mathematical Astronomy Eleven Centuries ago (Vateswara Siddanta of Vateshwaracharya 880 AD)", Indian Institute of Scientific Heritage (IISH)
Indian mathematics
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https://en.wikipedia.org/wiki/Transition%20layer
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Transition layer may refer to:
In mathematics, a mathematical approach to finding an accurate approximation to a problem's solution.
In aviation, a region of airspace between the transition altitude and the transition level.
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https://en.wikipedia.org/wiki/65%2C537
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65537 is the integer after 65536 and before 65538.
In mathematics
65537 is the largest known prime number of the form (). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann Gustav Hermes gave the first explicit construction of this polygon. In number
theory, primes of this form are known as Fermat primes, named after the mathematician
Pierre de Fermat. The only known prime Fermat numbers are
In 1732, Leonhard Euler found that the next Fermat number is composite:
In 1880, showed that
65537 is also the 17th Jacobsthal–Lucas number, and currently the largest known integer n for which the number is a probable prime.
Applications
65537 is commonly used as a public exponent in the RSA cryptosystem. Because it is the Fermat number with , the common shorthand is "F" or "F4". This value was used in RSA mainly for historical reasons; early raw RSA implementations (without proper padding) were vulnerable to very small exponents, while use of high exponents was computationally expensive with no advantage to security (assuming proper padding).
65537 is also used as the modulus in some Lehmer random number generators, such as the one used by ZX Spectrum, which ensures that any seed value will be coprime to it (vital to ensure the maximum period) while also allowing efficient reduction by the modulus using a bit shift and subtract.
References
Integers
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https://en.wikipedia.org/wiki/Circular%20points%20at%20infinity
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In projective geometry, the circular points at infinity (also called cyclic points or isotropic points) are two special points at infinity in the complex projective plane that are contained in the complexification of every real circle.
Coordinates
A point of the complex projective plane may be described in terms of homogeneous coordinates, being a triple of complex numbers , where two triples describe the same point of the plane when the coordinates of one triple are the same as those of the other aside from being multiplied by the same nonzero factor. In this system, the points at infinity may be chosen as those whose z-coordinate is zero. The two circular points at infinity are two of these, usually taken to be those with homogeneous coordinates
and .
Trilinear coordinates
Let A. B. C be the measures of the vertex angles of the reference triangle ABC. Then the trilinear coordinates of the circular points at infinity in the plane of the reference triangle are as given below:
or, equivalently,
or, again equivalently,
where .
Complexified circles
A real circle, defined by its center point (x0,y0) and radius r (all three of which are real numbers) may be described as the set of real solutions to the equation
Converting this into a homogeneous equation and taking the set of all complex-number solutions gives the complexification of the circle. The two circular points have their name because they lie on the complexification of every real circle. More generally, both points satisfy the homogeneous equations of the type
The case where the coefficients are all real gives the equation of a general circle (of the real projective plane). In general, an algebraic curve that passes through these two points is called circular.
Additional properties
The circular points at infinity are the points at infinity of the isotropic lines.
They are invariant under translations and rotations of the plane.
The concept of angle can be defined using the circular points, natural logarithm and cross-ratio:
The angle between two lines is a certain multiple of the logarithm of the cross-ratio of the pencil formed by the two lines and the lines joining their intersection to the circular points.
Sommerville configures two lines on the origin as Denoting the circular points as ω and ω′, he obtains the cross ratio
so that
References
Pierre Samuel (1988) Projective Geometry, Springer, section 1.6;
Semple and Kneebone (1952) Algebraic projective geometry'', Oxford, section II-8.
Projective geometry
Complex manifolds
Infinity
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https://en.wikipedia.org/wiki/Isotropic%20line
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In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form.
Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point that depend on the imaginary unit :
First system:
Second system:
Laguerre then interpreted these lines as geodesics:
An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation . On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines.
In the complex projective plane, points are represented by homogeneous coordinates and lines by homogeneous coordinates . An isotropic line in the complex projective plane satisfies the equation:
In terms of the affine subspace , an isotropic line through the origin is
In projective geometry, the isotropic lines are the ones passing through the circular points at infinity.
In the real orthogonal geometry of Emil Artin, isotropic lines occur in pairs:
A non-singular plane which contains an isotropic vector shall be called a hyperbolic plane. It can always be spanned by a pair N, M of vectors which satisfy
We shall call any such ordered pair N, M a hyperbolic pair. If V is a non-singular plane with orthogonal geometry and N ≠ 0 is an isotropic vector of V, then there exists precisely one M in V such that N, M is a hyperbolic pair. The vectors x N and y M are then the only isotropic vectors of V.
Relativity
Isotropic lines have been used in cosmological writing to carry light. For example, in a mathematical encyclopedia, light consists of photons: "The worldline of a zero rest mass (such as a non-quantum model of a photon and other elementary particles of mass zero) is an isotropic line."
For isotropic lines through the origin, a particular point is a null vector, and the collection of all such isotropic lines forms the light cone at the origin.
Élie Cartan expanded the concept of isotropic lines to multivectors in his book on spinors in three dimensions.
References
Pete L. Clark, Quadratic forms chapter I: Witts theory from University of Miami in Coral Gables, Florida.
O. Timothy O'Meara (1963,2000) Introduction to Quadratic Forms, page 94
Quadratic forms
Theory of relativity
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https://en.wikipedia.org/wiki/Imaginary%20line%20%28mathematics%29
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In complex geometry, an imaginary line is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line.
It is a special case of an imaginary curve.
An imaginary line is found in the complex projective plane P2(C) where points are represented by three homogeneous coordinates
Boyd Patterson described the lines in this plane:
The locus of points whose coordinates satisfy a homogeneous linear equation with complex coefficients
is a straight line and the line is real or imaginary according as the coefficients of its equation are or are not proportional to three real numbers.
Felix Klein described imaginary geometrical structures: "We will characterize a geometric structure as imaginary if its coordinates are not all real.:
According to Hatton:
The locus of the double points (imaginary) of the overlapping involutions in which an overlapping involution pencil (real) is cut by real transversals is a pair of imaginary straight lines.
Hatton continues,
Hence it follows that an imaginary straight line is determined by an imaginary point, which is a double point of an involution, and a real point, the vertex of the involution pencil.
See also
Conic section
Imaginary number
Imaginary point
Real curve
References
Citations
J.L.S. Hatton (1920) The Theory of the Imaginary in Geometry together with the Trigonometry of the Imaginary, Cambridge University Press via Internet Archive
Felix Klein (1928) Vorlesungen über nicht-euklischen Geometrie, Julius Springer.
Algebraic geometry
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https://en.wikipedia.org/wiki/Real%20point
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In geometry, a real point is a point in the complex projective plane with homogeneous coordinates for which there exists a nonzero complex number such that , , and are all real numbers.
This definition can be widened to a complex projective space of arbitrary finite dimension as follows:
are the homogeneous coordinates of a real point if there exists a nonzero complex number such that the coordinates of
are all real.
A point which is not real is called an imaginary point.
Context
Geometries that are specializations of real projective geometry, such as Euclidean geometry, elliptic geometry or conformal geometry may be complexified, thus embedding the points of the geometry in a complex projective space, but retaining the identity of the original real space as special. Lines, planes etc. are expanded to the lines, etc. of the complex projective space. As with the inclusion of points at infinity and complexification of real polynomials, this allows some theorems to be stated more simply without exceptions and for a more regular algebraic analysis of the geometry.
Viewed in terms of homogeneous coordinates, a real vector space of homogeneous coordinates of the original geometry is complexified. A point of the original geometric space is defined by an equivalence class of homogeneous vectors of the form , where is an nonzero complex value and is a real vector. A point of this form (and hence belongs to the original real space) is called a real point, whereas a point that has been added through the complexification and thus does not have this form is called an imaginary point.
Real subspace
A subspace of a projective space is real if it is spanned by real points.
Every imaginary point belongs to exactly one real line, the line through the point and its complex conjugate.
References
Projective geometry
Point (geometry)
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https://en.wikipedia.org/wiki/Complex%20conjugate%20line
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In complex geometry, the complex conjugate line of a straight line is the line that it becomes by taking the complex conjugate of each point on this line.
This is the same as taking the complex conjugates of the coefficients of the line. So if the equation of is , then the equation of its conjugate is .
The conjugate of a real line is the line itself.
The intersection point of two conjugated lines is always real.
References
Complex numbers
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https://en.wikipedia.org/wiki/Heavy-tailed%20distribution
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In probability theory, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded: that is, they have heavier tails than the exponential distribution. In many applications it is the right tail of the distribution that is of interest, but a distribution may have a heavy left tail, or both tails may be heavy.
There are three important subclasses of heavy-tailed distributions: the fat-tailed distributions, the long-tailed distributions, and the subexponential distributions. In practice, all commonly used heavy-tailed distributions belong to the subexponential class, introduced by Jozef Teugels.
There is still some discrepancy over the use of the term heavy-tailed. There are two other definitions in use. Some authors use the term to refer to those distributions which do not have all their power moments finite; and some others to those distributions that do not have a finite variance. The definition given in this article is the most general in use, and includes all distributions encompassed by the alternative definitions, as well as those distributions such as log-normal that possess all their power moments, yet which are generally considered to be heavy-tailed. (Occasionally, heavy-tailed is used for any distribution that has heavier tails than the normal distribution.)
Definitions
Definition of heavy-tailed distribution
The distribution of a random variable X with distribution function F is said to have a heavy (right) tail if the moment generating function of X, MX(t), is infinite for all t > 0.
That means
This is also written in terms of the tail distribution function
as
Definition of long-tailed distribution
The distribution of a random variable X with distribution function F is said to have a long right tail if for all t > 0,
or equivalently
This has the intuitive interpretation for a right-tailed long-tailed distributed quantity that if the long-tailed quantity exceeds some high level, the probability approaches 1 that it will exceed any other higher level.
All long-tailed distributions are heavy-tailed, but the converse is false, and it is possible to construct heavy-tailed distributions that are not long-tailed.
Subexponential distributions
Subexponentiality is defined in terms of convolutions of probability distributions. For two independent, identically distributed random variables with a common distribution function , the convolution of with itself, written and called the convolution square, is defined using Lebesgue–Stieltjes integration by:
and the n-fold convolution is defined inductively by the rule:
The tail distribution function is defined as .
A distribution on the positive half-line is subexponential if
This implies that, for any ,
The probabilistic interpretation of this is that, for a sum of independent random variables with common distribution ,
This is often known as the principle of the single big jump or catastrophe principle.
A distribution on
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https://en.wikipedia.org/wiki/Euclidean%20distance%20matrix
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In mathematics, a Euclidean distance matrix is an matrix representing the spacing of a set of points in Euclidean space.
For points in -dimensional space , the elements of their Euclidean distance matrix are given by squares of distances between them.
That is
where denotes the Euclidean norm on .
In the context of (not necessarily Euclidean) distance matrices, the entries are usually defined directly as distances, not their squares.
However, in the Euclidean case, squares of distances are used to avoid computing square roots and to simplify relevant theorems and algorithms.
Euclidean distance matrices are closely related to Gram matrices (matrices of dot products, describing norms of vectors and angles between them).
The latter are easily analyzed using methods of linear algebra.
This allows to characterize Euclidean distance matrices and recover the points that realize it.
A realization, if it exists, is unique up to rigid transformations, i.e. distance-preserving transformations of Euclidean space (rotations, reflections, translations).
In practical applications, distances are noisy measurements or come from arbitrary dissimilarity estimates (not necessarily metric).
The goal may be to visualize such data by points in Euclidean space whose distance matrix approximates a given dissimilarity matrix as well as possible — this is known as multidimensional scaling.
Alternatively, given two sets of data already represented by points in Euclidean space, one may ask how similar they are in shape, that is, how closely can they be related by a distance-preserving transformation — this is Procrustes analysis.
Some of the distances may also be missing or come unlabelled (as an unordered set or multiset instead of a matrix), leading to more complex algorithmic tasks, such as the graph realization problem or the turnpike problem (for points on a line).
Properties
By the fact that Euclidean distance is a metric, the matrix has the following properties.
All elements on the diagonal of are zero (i.e. it is a hollow matrix); hence the trace of is zero.
is symmetric (i.e. ).
(by the triangle inequality)
In dimension , a Euclidean distance matrix has rank less than or equal to . If the points are in general position, the rank is exactly
Distances can be shrunk by any power to obtain another Euclidean distance matrix. That is, if is a Euclidean distance matrix, then is a Euclidean distance matrix for every .
Relation to Gram matrix
The Gram matrix of a sequence of points in -dimensional space
is the matrix of their dot products (here a point is thought of as a vector from 0 to that point):
, where is the angle between the vector and .
In particular
is the square of the distance of from 0.
Thus the Gram matrix describes norms and angles of vectors (from 0 to) .
Let be the matrix containing as columns.
Then
, because (seeing as a column vector).
Matrices that can be decomposed as , that is, Gram matrices of some sequenc
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https://en.wikipedia.org/wiki/Hollow%20matrix
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In mathematics, a hollow matrix may refer to one of several related classes of matrix: a sparse matrix; a matrix with a large block of zeroes; or a matrix with diagonal entries all zero.
Definitions
Sparse
A hollow matrix may be one with "few" non-zero entries: that is, a sparse matrix.
Block of zeroes
A hollow matrix may be a square n × n matrix with an r × s block of zeroes where r + s > n.
Diagonal entries all zero
A hollow matrix may be a square matrix whose diagonal elements are all equal to zero. That is, an n × n matrix A = (aij) is hollow if aij = 0 whenever i = j (i.e. aii = 0 for all i). The most obvious example is the real skew-symmetric matrix. Other examples are the adjacency matrix of a finite simple graph, and a distance matrix or Euclidean distance matrix.
In other words, any square matrix that takes the form
is a hollow matrix, where the symbol denotes an arbitrary entry.
For example,
is a hollow matrix.
Properties
The trace of a hollow matrix is zero.
If A represents a linear map with respect to a fixed basis, then it maps each basis vector e into the complement of the span of e. That is, where .
The Gershgorin circle theorem shows that the moduli of the eigenvalues of a hollow matrix are less or equal to the sum of the moduli of the non-diagonal row entries.
References
Matrices
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https://en.wikipedia.org/wiki/Institute%20of%20Mathematical%20Statistics
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The Institute of Mathematical Statistics is an international professional and scholarly society devoted to the development, dissemination, and application of statistics and probability. The Institute currently has about 4,000 members in all parts of the world. Beginning in 2005, the institute started offering joint membership with the Bernoulli Society for Mathematical Statistics and Probability as well as with the International Statistical Institute. The Institute was founded in 1935 with Harry C. Carver and Henry L. Rietz as its two most important supporters. The institute publishes a variety of journals, and holds several international conference every year.
Publications
The Institute publishes five journals:
Annals of Statistics
Annals of Applied Statistics
Annals of Probability
Annals of Applied Probability
Statistical Science
In addition, it co-sponsors:
The Current Index to Statistics
Electronic Communications in Probability
Electronic Journal of Probability
Electronic Journal of Statistics
Journal of Computational and Graphical Statistics (A joint publication with the American Statistical Association and the Interface Foundation of North America)
Probability Surveys (A joint publication with the International Statistical Institute and the Bernoulli Society for Mathematical Statistics and Probability)
Statistics Surveys (A joint publication with the American Statistical Association, the Bernoulli Society for Mathematical Statistics and Probability, and the Statistical Society of Canada)
There are also some affiliated journals:
Probability and Mathematical Statistics (Wrocław University of Technology)
Latin American Journal of Probability and Mathematical Statistics
Furthermore, five journals are supported by the IMS:
Annales de l'Institut Henri Poincaré
Bayesian Analysis (published by the International Society for Bayesian Analysis)
Bernoulli (published by the Bernoulli Society for Mathematical Statistics and Probability)
Brazilian Journal of Probability and Statistics (published by the Brazilian Statistical Association)
Stochastic Systems''
Fellows
The IMS selects an annual class of Fellows who have demonstrated distinction in research or leadership in statistics or probability.
Meetings
Meetings gives scholars and practitioners a platform to present research results, disseminate job opportunities and exchange ideas with each other. The IMS holds an annual meeting called Joint Statistical Meetings (JSM), and sponsors multiple international meetings, for example, Spring Research Conference (SRC).
See also
List of presidents of the Institute of Mathematical Statistics
References
External links
Statistical societies
1935 establishments in the United States
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https://en.wikipedia.org/wiki/Stieltjes%20matrix
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In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse of this statement is not true in general for n > 2.
From the above definition, a Stieltjes matrix is a symmetric invertible Z-matrix whose eigenvalues have positive real parts. As it is a Z-matrix, its off-diagonal entries are less than or equal to zero.
See also
Hurwitz matrix
Metzler matrix
References
Matrices
Numerical linear algebra
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https://en.wikipedia.org/wiki/Households%20Below%20Average%20Income
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Households below average income is an annual publication on poverty statistics in the United Kingdom. The data is based on the Family Resources Survey.
Poverty is defined as having an equivalised household income below the 60% median line.
References
External links
HBAI
Family economics
Government publications
Household income
Measurements and definitions of poverty
Poverty in the United Kingdom
Publications with year of establishment missing
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https://en.wikipedia.org/wiki/Riesz%20space
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In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.
Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.
Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.
Definition
Preliminaries
If is an ordered vector space (which by definition is a vector space over the reals) and if is a subset of then an element is an upper bound (resp. lower bound) of if (resp. ) for all
An element in is the least upper bound or supremum (resp. greater lower bound or infimum) of if it is an upper bound (resp. a lower bound) of and if for any upper bound (resp. any lower bound) of (resp. ).
Definitions
Preordered vector lattice
A preordered vector lattice is a preordered vector space in which every pair of elements has a supremum.
More explicitly, a preordered vector lattice is vector space endowed with a preorder, such that for any :
Translation Invariance: implies
Positive Homogeneity: For any scalar implies
For any pair of vectors there exists a supremum (denoted ) in with respect to the order
The preorder, together with items 1 and 2, which make it "compatible with the vector space structure", make a preordered vector space.
Item 3 says that the preorder is a join semilattice.
Because the preorder is compatible with the vector space structure, one can show that any pair also have an infimum, making also a meet semilattice, hence a lattice.
A preordered vector space is a preordered vector lattice if and only if it satisfies any of the following equivalent properties:
For any their supremum exists in
For any their infimum exists in
For any their infimum and their supremum exist in
For any exists in
Riesz space and vector lattices
A Riesz space or a vector lattice is a preordered vector lattice whose preorder is a partial order.
Equivalently, it is an ordered vector space
for which the ordering is a lattice.
Note that many authors required that a vector lattice be a partially ordered vector space (rather than merely a preordered vector space) while others only require that it be a preordered vector space.
We will henceforth assume that every Riesz space and every vector lattice is an ordered vector space but that a preordered vector lattice is not necessarily partially ordered.
If is an ordered vector space over whose positive cone (the elements ) is generating (that is, such that ), and if for every either or exists, then is a vector lattice.
Intervals
A
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https://en.wikipedia.org/wiki/Ordered%20vector%20space
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In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space over the real numbers and a preorder on the set the pair is called a preordered vector space and we say that the preorder is compatible with the vector space structure of and call a vector preorder on if for all and with the following two axioms are satisfied
implies
implies
If is a partial order compatible with the vector space structure of then is called an ordered vector space and is called a vector partial order on
The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their addition operation.
Note that if and only if
Positive cones and their equivalence to orderings
A subset of a vector space is called a cone if for all real A cone is called pointed if it contains the origin. A cone is convex if and only if The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone);
the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones). A cone in a vector space is said to be generating if
Given a preordered vector space the subset of all elements in satisfying is a pointed convex cone with vertex (that is, it contains ) called the positive cone of and denoted by
The elements of the positive cone are called positive.
If and are elements of a preordered vector space then if and only if The positive cone is generating if and only if is a directed set under
Given any pointed convex cone with vertex one may define a preorder on that is compatible with the vector space structure of by declaring for all that if and only if
the positive cone of this resulting preordered vector space is
There is thus a one-to-one correspondence between pointed convex cones with vertex and vector preorders on
If is preordered then we may form an equivalence relation on by defining is equivalent to if and only if and
if is the equivalence class containing the origin then is a vector subspace of and is an ordered vector space under the relation: if and only there exist and such that
A subset of of a vector space is called a proper cone if it is a convex cone of vertex satisfying
Explicitly, is a proper cone if (1) (2) for all and (3)
The intersection of any non-empty family of proper cones is again a proper cone. Each proper cone in a real vector space induces an order on the vector space by defining if and only if and furthermore, the positive cone of this ordered vector space will be Therefore, there exists a one-to-one correspondence between the proper convex cones of and the vector partial orders on
By a total v
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https://en.wikipedia.org/wiki/Necktie%20paradox
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The necktie paradox is a puzzle and paradox with a subjective interpretation of probability theory describing a paradoxical bet advantageous to both involved parties. The two-envelope paradox is a variation of the necktie paradox.
Statement of paradox
Two persons, each given a necktie, start arguing over who has the cheaper one. The person with the more expensive necktie must give it to the other person.
The first person reasons as follows: winning and losing are equally likely. If I lose, then I will lose the value of my necktie. But if I win, then I will win more than the value of my necktie. Therefore, the wager is to my advantage. The second person can consider the wager in exactly the same way; thus, paradoxically, it seems both persons have the advantage in the bet.
Resolution
The paradox can be resolved by giving more careful consideration to what is lost in one scenario ("the value of my necktie") and what is won in the other ("more than the value of my necktie"). If one assumes for simplicity that the only possible necktie prices are $20 and $40, and that a person has equal chances of having a $20 or $40 necktie, then four outcomes (all equally likely) are possible:
The first person has a 50% chance of a neutral outcome, a 25% chance of gaining a necktie worth $40, and a 25% chance of losing a necktie worth $40. Turning to the losing and winning scenarios: if the person loses $40, then it is true that they have lost the value of their necktie; and if they gain $40, then it is true that they have gained more than the value of their necktie. The win and the loss are equally likely, but what we call "the value of the necktie" in the losing scenario is the same amount as what we call "more than the value of the necktie" in the winning scenario. Accordingly, neither person has the advantage in the wager.
This paradox is a rephrasing of the simplest case of the two envelopes problem, and the explanation of the resolution is essentially the same.
See also
Bayesian probability
Bertrand paradox
Decision theory
Dutch book
Monty Hall problem
Two envelopes problem
Newcomb's paradox
St. Petersburg paradox
References
Probability theory paradoxes
Decision-making paradoxes
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https://en.wikipedia.org/wiki/Ultraconnected%20space
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In mathematics, a topological space is said to be ultraconnected if no two nonempty closed sets are disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T1 space with more than one point is ultraconnected.
Properties
Every ultraconnected space is path-connected (but not necessarily arc connected). If and are two points of and is a point in the intersection , the function defined by if , and if , is a continuous path between and .
Every ultraconnected space is normal, limit point compact, and pseudocompact.
Examples
The following are examples of ultraconnected topological spaces.
A set with the indiscrete topology.
The Sierpiński space.
A set with the excluded point topology.
The right order topology on the real line.
See also
Hyperconnected space
Notes
References
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).
Properties of topological spaces
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https://en.wikipedia.org/wiki/Robb%2C%20Alberta
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Robb is a hamlet in west-central Alberta, Canada within Yellowhead County that is recognized as a designated place by Statistics Canada. It is located on Highway 47, approximately southwest of Edson. It has an elevation of .
It was named after Peter (Baldy) Addison Robb (1887–1954), a freighter and prospector. Robb was born in Gamrie, Banffshire, Scotland on 24 November 1887 to master blacksmith George Robb, and his wife Jane Addison.
The hamlet is located in Census Division No. 14 and in the federal riding of Yellowhead.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Robb had a population of 144 living in 76 of its 125 total private dwellings, a change of from its 2016 population of 170. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Robb had a population of 170 living in 82 of its 111 total private dwellings, a change of from its 2011 population of 171. With a land area of , it had a population density of in 2016.
Climate
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Designated places in Alberta
Hamlets in Alberta
Yellowhead County
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https://en.wikipedia.org/wiki/Wiener%20deconvolution
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In mathematics, Wiener deconvolution is an application of the Wiener filter to the noise problems inherent in deconvolution. It works in the frequency domain, attempting to minimize the impact of deconvolved noise at frequencies which have a poor signal-to-noise ratio.
The Wiener deconvolution method has widespread use in image deconvolution applications, as the frequency spectrum of most visual images is fairly well behaved and may be estimated easily.
Wiener deconvolution is named after Norbert Wiener.
Definition
Given a system:
where denotes convolution and:
is some original signal (unknown) at time .
is the known impulse response of a linear time-invariant system
is some unknown additive noise, independent of
is our observed signal
Our goal is to find some so that we can estimate as follows:
where is an estimate of that minimizes the mean square error
,
with denoting the expectation.
The Wiener deconvolution filter provides such a . The filter is most easily described in the frequency domain:
where:
and are the Fourier transforms of and ,
is the mean power spectral density of the original signal ,
is the mean power spectral density of the noise ,
, , and are the Fourier transforms of , and , and , respectively,
the superscript denotes complex conjugation.
The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain:
and then performing an inverse Fourier transform on to obtain .
Note that in the case of images, the arguments and above become two-dimensional; however the result is the same.
Interpretation
The operation of the Wiener filter becomes apparent when the filter equation above is rewritten:
Here, is the inverse of the original system, is the signal-to-noise ratio, and is the ratio of the pure filtered signal to noise spectral density. When there is zero noise (i.e. infinite signal-to-noise), the term inside the square brackets equals 1, which means that the Wiener filter is simply the inverse of the system, as we might expect. However, as the noise at certain frequencies increases, the signal-to-noise ratio drops, so the term inside the square brackets also drops. This means that the Wiener filter attenuates frequencies according to their filtered signal-to-noise ratio.
The Wiener filter equation above requires us to know the spectral content of a typical image, and also that of the noise. Often, we do not have access to these exact quantities, but we may be in a situation where good estimates can be made. For instance, in the case of photographic images, the signal (the original image) typically has strong low frequencies and weak high frequencies, while in many cases the noise content will be relatively flat with frequency.
Derivation
As mentioned above, we want to produce an estimate of the original signal that minimizes the mean square error, which may be expressed:
.
The equivalence to the previous definition of , can be
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https://en.wikipedia.org/wiki/Bernard%20Beckett
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Bernard Beckett (born 13 October 1967) is a New Zealand writer of fiction for young adults. His work includes novels and plays. Beckett has taught Drama, Mathematics and English at several high schools in the Wellington Region, and is currently teaching at Hutt Valley High School in Lower Hutt.
Selected works
Lester (novel, 1999)
Red Cliff (novel, 2000)
Jolt (novel, 2001)
No Alarms (novel, 2002)
3 Plays: Puck, Plan 10 From Outer Space, The End Of The World As We Know It 2003
Home Boys (novel, 2003)
Malcolm and Juliet (novel, 2004)
Deep Fried - with Clare Knighton (novel, 2005)
Genesis (novel, 2006)
Falling for Science (non-fiction, 2007)
Limbo (film, 2008)
Loaded (film, 2009)
Last Dance (film, 2011)
Lament (film, 2012)
Awards
2005: Esther Glen Award at the LIANZA Children's Book Awards, for Malcolm and Juliet.
2005: Winner Young Adult Fiction Category of the New Zealand Post Book Awards for Children and Young Adults, for Malcolm and Juliet.
2007: Winner Young Adult Fiction Category of the New Zealand Post Book Awards for Children and Young Adults, for Genesis.
2010: Winner of Prix Sorcières in the Adolescent novels category, for Genesis
References
External links
Longacre press pages on Beckett
NZ Book Council biography
Audio: In conversation on BBC World Service discussion programme The Forum
Bernard Beckett website
Living people
1967 births
21st-century New Zealand male writers
21st-century New Zealand novelists
New Zealand children's writers
New Zealand schoolteachers
People from Featherston, New Zealand
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https://en.wikipedia.org/wiki/Sarah%20Darby
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Sarah C. Darby is Professor of Medical Statistics at the University of Oxford. Her research has focused the beneficial effects of smoking cessation, the risk of lung cancer from residential radon, and treatments for early breast cancer. She is also a Principal Scientist with the Cancer Research UK in the Clinical Trial Service Unit (CTSU) and Epidemiological Studies Unit at the Nuffield Department of Clinical Medicine, at the Radcliffe Infirmary, Oxford.
Education
Darby studied Mathematics at Imperial College London (BSc) and Mathematical Statistics at the University of Birmingham (MSc). She completed her PhD at the London School of Hygiene and Tropical Medicine in 1977 where her research investigated Bayesian approaches to analysing bioassays.
Career and research
After her PhD, she worked at St Thomas's Hospital Medical School, the National Radiological Protection Board, and the Radiation Effects Research Foundation in Hiroshima, before moving to the University of Oxford in 1984. Her major funder since then has been Cancer Research UK.
Darby and her team have demonstrated that there is a linear relationship between the dose of radiation delivered incidentally to the heart during breast cancer radiotherapy and the subsequent risk of ischaemic heart disease, and that the absolute size of the radiation-related risk is bigger for women already at increased risk of heart disease.
She and her team have also estimated the absolute size of the benefit of radiotherapy to breast cancer patients and their work is enabling comparison of the likely absolute benefit of radiotherapy with its likely absolute risk for individual patients. Therefore, it is now becoming possible to assess which patients can receive standard radiotherapy, which should be considered for advanced techniques, and which should avoid radiotherapy altogether.
Other topics that Darby has worked on include estimating the risk of lung cancer from residential radon, the risk of invasive breast cancer after a diagnosis of ductal carcinoma in situ, and the risk of cancer after computerised tomography (CT) scans in young people.
Awards and honours
Darby was awarded the Guy Medal in Bronze in 1988 by the Royal Statistical Society. She was elected a Fellow of the Royal Society (FRS) in 2019.
References
British women epidemiologists
Living people
Fellows of Green Templeton College, Oxford
Women statisticians
Year of birth missing (living people)
Fellows of the Royal Society
British statisticians
Female Fellows of the Royal Society
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https://en.wikipedia.org/wiki/Preclosure%20operator
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In topology, a preclosure operator or Čech closure operator is a map between subsets of a set, similar to a topological closure operator, except that it is not required to be idempotent. That is, a preclosure operator obeys only three of the four Kuratowski closure axioms.
Definition
A preclosure operator on a set is a map
where is the power set of
The preclosure operator has to satisfy the following properties:
(Preservation of nullary unions);
(Extensivity);
(Preservation of binary unions).
The last axiom implies the following:
4. implies .
Topology
A set is closed (with respect to the preclosure) if . A set is open (with respect to the preclosure) if its complement is closed. The collection of all open sets generated by the preclosure operator is a topology; however, the above topology does not capture the notion of convergence associated to the operator, one should consider a pretopology, instead.
Examples
Premetrics
Given a premetric on , then
is a preclosure on
Sequential spaces
The sequential closure operator is a preclosure operator. Given a topology with respect to which the sequential closure operator is defined, the topological space is a sequential space if and only if the topology generated by is equal to that is, if
See also
Eduard Čech
References
A.V. Arkhangelskii, L.S.Pontryagin, General Topology I, (1990) Springer-Verlag, Berlin. .
B. Banascheski, Bourbaki's Fixpoint Lemma reconsidered, Comment. Math. Univ. Carolinae 33 (1992), 303-309.
Closure operators
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https://en.wikipedia.org/wiki/Dan%20Walls
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Daniel Frank Walls FRS (13 September 1942 – 12 May 1999) was a New Zealand theoretical physicist specialising in quantum optics.
Education
Walls gained a BSc in physics and mathematics and a first class honours MSc in physics at the University of Auckland. He then went to Harvard University as a Fulbright Scholar, obtaining his PhD in 1969. He was supervised by Roy J. Glauber who was later awarded a Nobel prize in 2005.
Career and research
After holding postdoctoral research positions in Auckland and Stuttgart, Walls became a senior lecturer in physics at the University of Waikato in 1972, where he became professor in 1980. Together with his colleague Crispin Gardiner, during the next 25 years he established a major research centre for theoretical quantum optics in New Zealand and built active and productive collaborations with groups throughout the world.
In 1987 he moved to the University of Auckland as professor of theoretical physics.
His major research interests centred on the interaction and similarities between light and atoms. He was notable for his wide-ranging expertise in relating theory to experiment, and was involved in all major efforts to understand non-classical light. A seminal paper by Walls with his first graduate student Howard Carmichael, showed how to create antibunched light, in which photons arrive at regular intervals, rather than randomly.
Walls was a pioneer in the study of ways that the particle-like nature of light (photons) could be controlled to make optical systems less susceptible to unwanted fluctuations, in particular by the use of squeezed light, a concept formulated by Carlton Caves. In squeezed light, some fluctuations can be made very small provided other fluctuations are correspondingly large.
He made major contributions to the theory of quantum measurement such as those involving Albert Einstein's"which-path" experiment, and the quantum nondemolition measurement. Walls also used a simple field theoretical approach to explain and corroborate Dirac's description of photon interference and in particular Dirac's statement "that a photon interferes only with itself."
In the later stages of his career he focused his research efforts on the theoretical aspects of the newly created state of matter, the Bose–Einstein condensate (BECs). Some of his contributions in the field include the prediction of the interference signature of quantized vortices, and the collapses and revivals of the Josephson coupled BECs.
Awards and honours
Walls was elected a Fellow of the Royal Society (FRS) in 1992. Walls was also elected Fellow of the American Physical Society (1981) and the Royal Society of New Zealand (FRSNZ). In 1995 he was awarded the Dirac Medal by the Institute of Physics for theoretical physics.
The Dodd-Walls Centre for Photonic and Quantum Technologies, a New Zealand Centre of Research excellence based in the University of Otago, was named after Jack Dodd and Dan Walls in recognition of their pione
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https://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus%20theorem
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The Schur–Zassenhaus theorem is a theorem in group theory which states that if is a finite group, and is a normal subgroup whose order is coprime to the order of the quotient group , then is a semidirect product (or split extension) of and . An alternative statement of the theorem is that any normal Hall subgroup of a finite group has a complement in . Moreover if either or is solvable then the Schur–Zassenhaus theorem also states that all complements of in are conjugate. The assumption that either or is solvable can be dropped as it is always satisfied, but all known proofs of this require the use of the much harder Feit–Thompson theorem.
The Schur–Zassenhaus theorem at least partially answers the question: "In a composition series, how can we classify groups with a certain set of composition factors?" The other part, which is where the composition factors do not have coprime orders, is tackled in extension theory.
History
The Schur–Zassenhaus theorem was introduced by . Theorem 25, which he credits to Issai Schur, proves the existence of a complement, and theorem 27 proves that all complements are conjugate under the assumption that or is solvable. It is not easy to find an explicit statement of the existence of a complement in Schur's published works, though the results of on the Schur multiplier imply the existence of a complement in the special case when the normal subgroup is in the center. Zassenhaus pointed out that the Schur–Zassenhaus theorem for non-solvable groups would follow if all groups of odd order are solvable, which was later proved by Feit and Thompson. Ernst Witt showed that it would also follow from the Schreier conjecture (see for Witt's unpublished 1937 note about this), but the Schreier conjecture has only been proved using the classification of finite simple groups, which is far harder than the Feit–Thompson theorem.
Examples
If we do not impose the coprime condition, the theorem is not true: consider for example the cyclic group and its normal subgroup . Then if were a semidirect product of and then would have to contain two elements of order 2, but it only contains one. Another way to explain this impossibility of splitting (i.e. expressing it as a semidirect product) is to observe that the automorphisms of are the trivial group, so the only possible [semi]direct product of with itself is a direct product (which gives rise to the Klein four-group, a group that is non-isomorphic with ).
An example where the Schur–Zassenhaus theorem does apply is the symmetric group on 3 symbols, , which has a normal subgroup of order 3 (isomorphic with ) which in turn has index 2 in (in agreement with the theorem of Lagrange), so . Since 2 and 3 are relatively prime, the Schur–Zassenhaus theorem applies and . Note that the automorphism group of is and the automorphism of used in the semidirect product that gives rise to is the non-trivial automorphism that permutes the two non-identity elements of .
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https://en.wikipedia.org/wiki/David%20Henderson%20%28economist%29
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Patrick David Henderson (10 April 1927 – 30 September 2018) was a British economist. He was the chief economist at the Economics and Statistics Department at the OECD during 1984–1992. Before that he worked as an academic economist in Britain, first at Oxford (Fellow of Lincoln College) and later at University College London (Professor of Economics, 1975–1983); as a British civil servant (first as an Economic Advisor in HM Treasury, and later as Chief Economist in the Ministry of Aviation); and as a staff member of the World Bank (1969–1975). In 1985 he gave the BBC Reith Lectures, which were published in the book Innocence and Design: The Influence of Economic Ideas on Policy (Blackwell, 1986).
After leaving the OECD, Henderson was an independent author and consultant, and acted as Visiting Fellow or Professor at the OECD Development Centre (Paris), the Centre for European Policy Studies (Brussels), Monash University, the Fondation Nationale des Sciences Politiques, the University of Melbourne, the Royal Institute of International Affairs, the New Zealand Business Roundtable, the Melbourne Business School, and Westminster Business School. Subsequently he was a Fellow of the Institute of Economic Affairs.
In 1992, Henderson was appointed to the Order of St Michael and St George as a Companion (CMG).
Henderson has published books that strongly criticize "corporate social responsibility" (see §Books, below).
Henderson and Nigel Lawson appealed to then-Prime Minister Tony Blair to investigate the economic implications of the potential implementation of policies put forth by the UN Intergovernmental Panel on Climate Change (IPCC) findings.
Henderson and Ian Castles, a former head of the Australian Bureau of Statistics argued that the IPCC's projections of future emissions of greenhouse gases was flawed. The IPCC's forecasts of global output were based on national GDP converted to dollars using market exchange rates. Henderson and Ian Castles were critical of the Special Report on Emissions Scenarios (SRES) report by the Intergovernmental Panel on Climate Change (IPCC) that was published in 2000. The core of their critique was the use of market exchange rates (MER) for international comparison, in lieu of the theoretically favoured PPP exchange rate which corrects for differences in purchasing power. The IPCC rebutted this criticism. Castles and Henderson later acknowledged that they were mistaken that future greenhouse gas emissions had been significantly overestimated.
Henderson has suggested about climate change that the science is not settled, and he specifically criticized the Stern Review regarding the economics of global warming. Relatedly, in 2008, Nigel Lawson published the book An Appeal to Reason: A Cool Look at Global Warming; the dedication for the book says "To David Henderson, who first aroused my interest in all this". The following year, Lawson went on to found The Global Warming Policy Foundation, a climate change denialist
|
https://en.wikipedia.org/wiki/UEFA%20Cup%20Winners%27%20Cup%20records%20and%20statistics
|
Below are tables of the clubs that have won the Cup Winners' Cup.
Performances
By club
By nation
By manager
Four managers hold the record of winning the competition on two occasions:
Nereo Rocco: 1968 and 1973 (Milan)
Valeriy Lobanovskyi in 1975 and 1986 (Dynamo Kyiv)
Johan Cruyff: 1987 (Ajax) and 1989 (Barcelona)
Alex Ferguson: 1983 (Aberdeen) and 1991 (Manchester United)
By player
Most UEFA Cup Winners' Cup titles: Lobo Carrasco (3)
FC Barcelona (3): (1978–79, 1981–82, 1988–89)
Clubs
By number of appearances
(Years marked in bold denote Cups won by the respective club)
By semi-final appearances
{|class="wikitable sortable"
|-
!Club
!No.
!Years
|-
| Barcelona||align="center"|6||1969, 1979, 1982, 1989, 1991, 1997
|-
| Atlético Madrid||align="center"|5||1962, 1963, 1977, 1986, 1993
|-
| Chelsea||align="center"|4||1971, 1995, 1998, 1999
|-
| Anderlecht||align="center"|4||1976, 1977, 1978, 1990
|-
| Bayern Munich||align="center"|4||1967, 1968, 1972, 1985
|-
| Paris Saint-Germain||align="center"|3||1994, 1996, 1997
|-
| Fiorentina||align="center"|3||1961, 1962, 1997
|-
| Feyenoord||align="center"|3||1981, 1992, 1996
|-
| Arsenal||align="center"|3||1980, 1994, 1995
|-
| Sampdoria||align="center"|3||1989, 1990, 1995
|-
| Zaragoza||align="center"|3||1965, 1987, 1995|-
| Juventus||align="center"|3||1980, 1984, 1991
|-
| Dynamo Moscow||align="center"|3||1972, 1978, 1985
|-
| West Ham United||align="center"|3||1965, 1966, 1976|-
| Milan||align="center"|3||1968, 1973, 1974|-
| Rangers||align="center"|3||1961, 1967, 1972|-
| Lokomotiv Moscow||align="center"|2||1998, 1999
|-
| Liverpool||align="center"|2||1966, 1997
|-
| Rapid Wien||align="center"|2||1985, 1996|-
| Parma||align="center"|2||1993, 1994|-
| Benfica||align="center"|2||1981, 1994
|-
| Monaco||align="center"|2||1990, 1992|-
| Manchester United||align="center"|2||1984, 1991|-
| Mechelen||align="center"|2||1988, 1989
|-
| Ajax||align="center"|2||1987, 1988|-
| Dynamo Kyiv||align="center"|2||1975, 1986|-
| Aberdeen||align="center"|2||1983, 1984
|-
| Austria Wien||align="center"|2||1978, 1983
|-
| Real Madrid||align="center"|2||1971, 1983|-
| Dinamo Tbilisi||align="center"|2||1981, 1982
|-
| Standard Liège||align="center"|2||1967, 1982|-
| Tottenham Hotspur||align="center"|2||1963, 1982
|-
| Carl Zeiss Jena||align="center"|2||1962, 1981|-
| Hamburger SV||align="center"|2||1968, 1977|-
| PSV Eindhoven||align="center"|2||1971, 1975
|-
| Sporting CP||align="center"|2||1964, 1974
|-
| Manchester City||align="center"|2||1970, 1971
|-
| Celtic||align="center"|2||1964, 1966
|-
| Lazio||align="center"|1||1999|-
| Mallorca||align="center"|1||1999|-
| VfB Stuttgart||align="center"|1||1998|-
| Vicenza||align="center"|1||1998
|-
| Deportivo La Coruña||align="center"|1||1996
|-
| Antwerp||align="center"|1||1993|-
| Spartak Moscow||align="center"|1||1993
|-
| Club Brugge||align="center"|1||1992
|-
| Werder Bremen||align="center"|1||1992|-
| Legia Warsaw||align="center"|1||1991
|-
| Dinamo București||
|
https://en.wikipedia.org/wiki/Quasi-set%20theory
|
Quasi-set theory is a formal mathematical theory for dealing with collections of objects, some of which may be indistinguishable from one another. Quasi-set theory is mainly motivated by the assumption that certain objects treated in quantum physics are indistinguishable and don't have individuality.
Motivation
The American Mathematical Society sponsored a 1974 meeting to evaluate the resolution and consequences of the 23 problems Hilbert proposed in 1900. An outcome of that meeting was a new list of mathematical problems, the first of which, due to Manin (1976, p. 36), questioned whether classical set theory was an adequate paradigm for treating collections of indistinguishable elementary particles in quantum mechanics. He suggested that such collections cannot be sets in the usual sense, and that the study of such collections required a "new language".
The use of the term quasi-set follows a suggestion in da Costa's 1980 monograph Ensaio sobre os Fundamentos da Lógica (see da Costa and Krause 1994), in which he explored possible semantics for what he called "Schrödinger Logics". In these logics, the concept of identity is restricted to some objects of the domain, and has motivation in Schrödinger's claim that the concept of identity does not make sense for elementary particles (Schrödinger 1952). Thus in order to provide a semantics that fits the logic, da Costa submitted that "a theory of quasi-sets should be developed", encompassing "standard sets" as particular cases, yet da Costa did not develop this theory in any concrete way. To the same end and independently of da Costa, Dalla Chiara and di Francia (1993) proposed a theory of quasets to enable a semantic treatment of the language of microphysics. The first quasi-set theory was proposed by D. Krause in his PhD thesis, in 1990 (see Krause 1992). A related physics theory, based on the logic of adding fundamental indistinguishability to equality and inequality, was developed and elaborated independently in the book The Theory of Indistinguishables by A. F. Parker-Rhodes.
Summary of the theory
We now expound Krause's (1992) axiomatic theory , the first quasi-set theory; other formulations and improvements have since appeared. For an updated paper on the subject, see French and Krause (2010). Krause builds on the set theory ZFU, consisting of Zermelo-Fraenkel set theory with an ontology extended to include two kinds of urelements:
m-atoms, whose intended interpretation is elementary quantum particles;
M-atoms, macroscopic objects to which classical logic is assumed to apply.
Quasi-sets (q-sets) are collections resulting from applying axioms, very similar to those for ZFU, to a basic domain composed of m-atoms, M-atoms, and aggregates of these. The axioms of include equivalents of extensionality, but in a weaker form, termed "weak extensionality axiom"; axioms asserting the existence of the empty set, unordered pair, union set, and power set; the axiom of separation; an axiom stating the i
|
https://en.wikipedia.org/wiki/Introduction%20to%20Commutative%20Algebra
|
Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It deals with elementary concepts of commutative algebra including localization, primary decomposition, integral dependence, Noetherian and Artinian rings and modules, Dedekind rings, completions and a moderate amount of dimension theory. It is notable for being among the shorter English-language introductory textbooks in the subject, relegating a good deal of material to the exercises.
(Hardcover 1969, ) (Paperback 1994, )
1969 non-fiction books
Mathematics textbooks
Commutative algebra
|
https://en.wikipedia.org/wiki/Cipher%20%28disambiguation%29
|
A cipher is a method of encryption or decryption.
Cipher may also refer to:
Science and mathematics
CIPHER (DOS command), an external filter command in some versions of MS-DOS 2.xx
One of the names for the number 0 in English
Entertainment and culture
Cipher (manga), a manga series by Minako Narita
Cipher (comics), a Marvel Comics X-Men character
Cipher (newuniversal), a Marvel Comics character in the newuniversal imprint
Bill Cipher, a dream demon in Gravity Falls
Cipher, the player character in Ace Combat Zero: The Belkan War
Team Cipher, the villainous team from Pokémon Colosseum and the sequel Pokémon XD: Gale of Darkness
Cipher, a criminal mastermind and cyber terrorist in The Fate of the Furious
A codename for The Patriots in the video game series Metal Gear Solid
A word used by the Five-Percent Nation to refer to zero, letter "O" or a circle
A playable character class in the role-playing video game Pillars of Eternity
Music
Cipher (album), by The Alpha Conspiracy
Cipher (band), a hardcore punk band
Ciphers (album), a 1996 album by SETI
Cipher notation, a type of musical notation
A note that continues to sound in a pipe organ when the organist does not intend for it to sound
A freestyle rap session
People
A stage name of Ichiro Takigawa, the guitarist of the Japanese rock band D'erlanger
Cipha Sounds, the alias of Luis Diaz, an American radio and television personality
See also
Cyphers
|
https://en.wikipedia.org/wiki/Control%20point%20%28mathematics%29
|
In computer-aided geometric design a control point is a member of a set of points used to determine the shape of a spline curve or, more generally, a surface or higher-dimensional object.
For Bézier curves, it has become customary to refer to the -vectors in a parametric representation of a curve or surface in -space as control points, while the scalar-valued functions , defined over the relevant parameter domain, are the corresponding weight or blending functions.
Some would reasonably insist, in order to give intuitive geometric meaning to the word "control", that the blending functions form a partition of unity, i.e., that the are nonnegative and sum to one. This property implies that the curve lies within the convex hull of its control points. This is the case for Bézier's representation of a polynomial curve as well as for the B-spline representation of a spline curve or tensor-product spline surface.
References
Splines (mathematics)
|
https://en.wikipedia.org/wiki/Igusa%20zeta%20function
|
In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on.
Definition
For a prime number p let K be a p-adic field, i.e. , R the valuation ring and P the maximal ideal. For we denote by the valuation of z, , and for a uniformizing parameter π of R.
Furthermore let be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let be a character of .
In this situation one associates to a non-constant polynomial the Igusa zeta function
where and dx is Haar measure so normalized that has measure 1.
Igusa's theorem
showed that is a rational function in . The proof uses Heisuke Hironaka's theorem about the resolution of singularities. Later, an entirely different proof was given by Jan Denef using p-adic cell decomposition. Little is known, however, about explicit formulas. (There are some results about Igusa zeta functions of Fermat varieties.)
Congruences modulo powers of
Henceforth we take to be the characteristic function of and to be the trivial character. Let denote the number of solutions of the congruence
.
Then the Igusa zeta function
is closely related to the Poincaré series
by
References
Information for this article was taken from J. Denef, Report on Igusa's Local Zeta Function, Séminaire Bourbaki 43 (1990-1991), exp. 741; Astérisque 201-202-203 (1991), 359-386
Zeta and L-functions
Diophantine geometry
|
https://en.wikipedia.org/wiki/Roller%20hockey%20at%20the%201992%20Summer%20Olympics%20%E2%80%93%20Squads%20and%20statistics
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Below are listed the squad constitutions for every nation which presented a team to the roller hockey competition at the 1992 Summer Olympics. Along with the rosters are also displayed total statistics per player and some general competition statistics.
Angola
Argentina
Australia
Brazil
Germany
Italy
Japan
Netherlands
21
*
*
*
*
*
*
*
Portugal
Spain
Switzerland
United States
General statistics
Players
Top scorers:
– 32
– 24
– 15
Shots:
– 229
– 153
– 146
Efficiency (% goals):
– 30
– 22 (5 goals)
– 22 (5 goals)
Teams
Top scorers:
– 83
– 78
– 70
Shots:
– 729
– 684
– 582
Efficiency (% goals):
– 13
– 11
– 10
Assists:
– 30
– 28
– 21
Steals:
– 250
– 226
– 207
Turnovers:
– 261
– 250
– 239
Fouls:
– 364
– 357
– 334
Others
Most goals in a match – 38 (Japan 0–38 Portugal)
Most goals by a player in a match – 16 → (Japan 0–38 Portugal)
Most goals by a team in a match – 38 (Japan 0–38 Portugal)
Biggest goal difference in a match – 38 (Japan 0–38 Portugal)
See also
Roller hockey at the 1992 Summer Olympics - Preliminary round
Roller hockey at the 1992 Summer Olympics - Semi-finals
References
Roller hockey at the 1992 Summer Olympics
|
https://en.wikipedia.org/wiki/Samuel%20S.%20Wilks
|
Samuel Stanley Wilks (June 17, 1906 – March 7, 1964) was an American mathematician and academic who played an important role in the development of mathematical statistics, especially in regard to practical applications.
Early life and education
Wilks was born in Little Elm, Texas and raised on a farm. He studied Industrial Arts at the North Texas State Teachers College in Denton, Texas, obtaining his bachelor's degree in 1926. He received his master's degree in mathematics in 1928 from the University of Texas. He obtained his Ph.D. at the University of Iowa under Everett F. Lindquist; his thesis dealt with a problem of statistical measurement in education, and was published in the Journal of Educational Psychology.
Career
Wilks became an instructor in mathematics at Princeton University in 1933; in 1938 he assumed the editorship of the journal Annals of Mathematical Statistics in place of Harry C. Carver. Wilks assembled an advisory board for the journal that included major figures in statistics and probability, among them Ronald Fisher, Jerzy Neyman, and Egon Pearson.
During World War II he was a consultant with the Office of Naval Research. Both during and after the War he had a profound impact on the application of statistical methods to all aspects of military planning.
Wilks was named professor of mathematics and director of the Section of Mathematical Statistics at Princeton in 1944, and became chairman of the Division of Mathematics at the university in 1958.
Wilks died in 1964 in Princeton.
Work in mathematical statistics
He was noted for his work on multivariate statistics. He also conducted work on unit-weighted regression, proving the idea that under a wide variety of common conditions, almost all sets of weights will yield composites that are very highly correlated (Wilks, 1938), a result that has been dubbed Wilks's theorem (Ree, Carretta, & Earles, 1998).
Another result, also called “Wilks' theorem” occurs in the theory of likelihood ratio tests, where Wilks showed the distribution of log likelihood ratios is asymptotically .
From the start of his career, Wilks favored a strong focus on practical applications for the increasingly abstract field of mathematical statistics; he also influenced other researchers, notably John Tukey, in a similar direction. Drawing upon the background of his thesis, Wilks worked with the Educational Testing Service in developing the standardized tests like the SAT that have had a profound effect on American education. He also worked with Walter Shewhart on statistical applications in quality control in manufacturing.
Wilks's lambda distribution is a probability distribution related to two independent Wishart distributed variables. It is important in multivariate statistics and likelihood-ratio tests.
Honors
The American Statistical Association named its Wilks Memorial Award in his honor. Wilks was elected to the American Philosophical Society in 1948 and the American Academy of Arts and Scie
|
https://en.wikipedia.org/wiki/List%20of%20AC%20Milan%20records%20and%20statistics
|
Associazione Calcio Milan are an Italian professional football club based in Milan, Lombardy. The club was founded as Milan Foot-Ball and Cricket Club in 1899 and has competed in the Italian football league since the following year. Milan currently play in Serie A, the top tier of Italian football. They have been out of the top tier in only two seasons since the establishment of the Serie A as the single division top tier. They have also been involved in European football ever since they became the first Italian club to enter the European Cup in 1955.
This list encompasses the major honours won by Milan, records set by the club, its managers and its players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Milan players on the international stage.
The club currently has the record for the second most Italian top-flight titles (Scudetti) with 19, tied with cross-city rivals Inter Milan and behind Juventus' 36. They also hold the record for the most European Cup victories by an Italian team, winning the competition seven times. Furthermore, in the 1991–92 season Milan became the first team to win the Serie A title without losing a single game. The club's record appearance maker is Paolo Maldini, who has made 902 official appearances between 1985 and 2009. Gunnar Nordahl is the club's record goalscorer, scoring 221 goals during his Milan career.
All figures are correct as of 23 May 2022.
Honours
Milan have won honours both domestically and in European cup competitions. They have won the Scudetto nineteen times, the Coppa Italia five times and the Supercoppa Italiana seven times. They won their first league title in their second season, winning the 1901 Italian Football Championship, while their most recent success came in 2022, when they won their 19th Scudetto. Internationally, they are the most successful Italian club, with 18 trophies which include seven UEFA Champions League titles, five UEFA Super Cups, two European Cup Winners' Cups, three Intercontinental Cups and one FIFA Club World Cup.
Domestic
League
Italian Football Championship / Serie A (first division):
Winners (19): 1901, 1906, 1907, 1950–51, 1954–55, 1956–57, 1958–59, 1961–62, 1967–68, 1978–79, 1987–88, 1991–92, 1992–93, 1993–94, 1995–96, 1998–99, 2003–04, 2010–11, 2021–22
Runners-up (16): 1902, 1947–48, 1949–50, 1951–52, 1955–56, 1960–61, 1964–65, 1968–69, 1970–71, 1971–72, 1972–73, 1989–90, 1990–91, 2004–05, 2011–12, 2020–21
Serie B (second division):
Winners (2): 1980–81, 1982–83
Cups
Coppa Italia:
Winners (5): 1966–67, 1971–72, 1972–73, 1976–77, 2002–03
Runners-up (9): 1941–42, 1967–68, 1970–71, 1974–75, 1984–85, 1989–90, 1997–98, 2015–16, 2017–18
Supercoppa Italiana:
Winners (7): 1988, 1992, 1993, 1994, 2004, 2011, 2016
Runners-up (5): 1996, 1999, 2003, 2018, 2022
International
European Cup/UEFA Champions League:
Winners (7): 19
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https://en.wikipedia.org/wiki/International%20Journal%20of%20Mathematics%20and%20Mathematical%20Sciences
|
The International Journal of Mathematics and Mathematical Sciences is a biweekly peer-reviewed mathematics journal. It was established in 1978 by Lokenath Debnath and is published by the Hindawi Publishing Corporation.
The journal publishes articles in all areas of mathematics such as pure and applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology.
Indexing and abstracting
The journal is or has been indexed and abstracted in the following bibliographic databases:
EBSCO Information Services
Emerging Sources Citation Index
Mathematical Reviews
ProQuest databases
Scopus
Zentralblatt MATH
References
External links
Website prior to 3 March 2001
Academic journals established in 1978
Mathematics journals
Hindawi Publishing Corporation academic journals
Biweekly journals
English-language journals
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https://en.wikipedia.org/wiki/Adherent%20point
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In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset of a topological space is a point in such that every neighbourhood of (or equivalently, every open neighborhood of ) contains at least one point of A point is an adherent point for if and only if is in the closure of thus
if and only if for all open subsets if
This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of contains at least one point of Thus every limit point is an adherent point, but the converse is not true. An adherent point of is either a limit point of or an element of (or both). An adherent point which is not a limit point is an isolated point.
Intuitively, having an open set defined as the area within (but not including) some boundary, the adherent points of are those of including the boundary.
Examples and sufficient conditions
If is a non-empty subset of which is bounded above, then the supremum is adherent to In the interval is an adherent point that is not in the interval, with usual topology of
A subset of a metric space contains all of its adherent points if and only if is (sequentially) closed in
Adherent points and subspaces
Suppose and where is a topological subspace of (that is, is endowed with the subspace topology induced on it by ). Then is an adherent point of in if and only if is an adherent point of in
By assumption, and Assuming that let be a neighborhood of in so that will follow once it is shown that The set is a neighborhood of in (by definition of the subspace topology) so that implies that Thus as desired. For the converse, assume that and let be a neighborhood of in so that will follow once it is shown that By definition of the subspace topology, there exists a neighborhood of in such that Now implies that From it follows that and so as desired.
Consequently, is an adherent point of in if and only if this is true of in every (or alternatively, in some) topological superspace of
Adherent points and sequences
If is a subset of a topological space then the limit of a convergent sequence in does not necessarily belong to however it is always an adherent point of Let be such a sequence and let be its limit. Then by definition of limit, for all neighbourhoods of there exists such that for all In particular, and also so is an adherent point of
In contrast to the previous example, the limit of a convergent sequence in is not necessarily a limit point of ; for example consider as a subset of Then the only sequence in is the constant sequence whose limit is but is not a limit point of it is only an adherent point of
See also
Notes
Citations
References
Adamson, Iain T., A General Topology Workbook, Birkhäuser Boston; 1st edition (November 29, 1995). .
Apostol, Tom M., Mathematical Analysis, Addison Wesley Longman; second editi
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https://en.wikipedia.org/wiki/Dvoretzky%E2%80%93Kiefer%E2%80%93Wolfowitz%20inequality
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In the theory of probability and statistics, the Dvoretzky–Kiefer–Wolfowitz–Massart inequality (DKW inequality) bounds how close an empirically determined distribution function will be to the distribution function from which the empirical samples are drawn. It is named after Aryeh Dvoretzky, Jack Kiefer, and Jacob Wolfowitz, who in 1956 proved the inequality
with an unspecified multiplicative constant C in front of the exponent on the right-hand side.
In 1990, Pascal Massart proved the inequality with the sharp constant C = 2, confirming a conjecture due to Birnbaum and McCarty. In 2021, Michael Naaman proved the multivariate version of the DKW inequality and generalized Massart's tightness result to the multivariate case, which results in a sharp constant of twice the dimension k of the space in which the observations are found: C = 2k.
The DKW inequality
Given a natural number n, let X1, X2, …, Xn be real-valued independent and identically distributed random variables with cumulative distribution function F(·). Let Fn denote the associated empirical distribution function defined by
so is the probability that a single random variable is smaller than , and is the fraction of random variables that are smaller than .
The Dvoretzky–Kiefer–Wolfowitz inequality bounds the probability that the random function Fn differs from F by more than a given constant ε > 0 anywhere on the real line. More precisely, there is the one-sided estimate
which also implies a two-sided estimate
This strengthens the Glivenko–Cantelli theorem by quantifying the rate of convergence as n tends to infinity. It also estimates the tail probability of the Kolmogorov–Smirnov statistic. The inequalities above follow from the case where F corresponds to be the uniform distribution on [0,1]
as Fn has the same distributions as Gn(F) where Gn is the empirical distribution of
U1, U2, …, Un where these are independent and Uniform(0,1), and noting that
with equality if and only if F is continuous.
Multivariate case
In the multivariate case, X1, X2, …, Xn is an i.i.d. sequence of k-dimensional vectors. If Fn is the multivariate empirical cdf, then
for every ε, n, k > 0. The (n + 1) term can be replaced with a 2 for any sufficiently large n.
Kaplan–Meier estimator
The Dvoretzky–Kiefer–Wolfowitz inequality is obtained for the Kaplan–Meier estimator which is a right-censored data analog of the empirical distribution function
for every and for some constant , where is the Kaplan–Meier estimator, and is the censoring distribution function.
Building CDF bands
The Dvoretzky–Kiefer–Wolfowitz inequality is one method for generating CDF-based confidence bounds and producing a confidence band, which is sometimes called the Kolmogorov–Smirnov confidence band. The purpose of this confidence interval is to contain the entire CDF at the specified confidence level, while alternative approaches attempt to only achieve the confidence level on each individual point, whi
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https://en.wikipedia.org/wiki/Viggo%20Stoltenberg-Hansen
|
Viggo Stoltenberg-Hansen, born 1942, professor at Uppsala University, Department of Mathematics, is a Swedish mathematician/logician and expert on domain theory and recursion theory (also known as computability theory). Viggo received his PhD in Mathematics (titled "On Priority Arguments In Friedberg Theories") from University of Toronto in 1973.
Work on domain theory
Viggo Stoltenberg-Hansen and John Tucker developed in the early 1980s a general method of domain representations of topological algebras.
Viggo is the main author of the textbook "Mathematical Theory of Domains", Cambridge University Press, 1994 (coauthored by I. Lindström and E. Griffor), and also of a set of Marktoberdorf summer school lecture notes on domain theory.
Work on effective domains
Viggo Stoltenberg-Hansen and John Tucker made a thorough analysis of the computability associated to effective algebras and continuity of homomorphisms between such.
Some References
V Stoltenberg-Hansen and J V Tucker, Effective algebras, in S Abramsky, D Gabbay and T Maibaum (eds.), Handbook of Logic in Computer Science, Volume IV: Semantic Modelling, Oxford University Press (1995), pp357–526.
V Stoltenberg-Hansen and J V Tucker, Computable rings and fields, in E Griffor (ed.), Handbook of Computability Theory, Elsevier (1999), pp363–447.
External links
Living people
Swedish mathematicians
Swedish logicians
Logicians
Academic staff of Uppsala University
1942 births
Swedish philosophers
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https://en.wikipedia.org/wiki/Inverse-Wishart%20distribution
|
In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a
multivariate normal distribution.
We say follows an inverse Wishart distribution, denoted as , if its inverse has a Wishart distribution . Important identities have been derived for the inverse-Wishart distribution.
Density
The probability density function of the inverse Wishart is:
where and are positive definite matrices, is the determinant, and Γp(·) is the multivariate gamma function.
Theorems
Distribution of the inverse of a Wishart-distributed matrix
If and is of size , then has an inverse Wishart distribution .
Marginal and conditional distributions from an inverse Wishart-distributed matrix
Suppose has an inverse Wishart distribution. Partition the matrices and conformably with each other
where and are matrices, then we have
is independent of and , where is the Schur complement of in ;
;
, where is a matrix normal distribution;
, where ;
Conjugate distribution
Suppose we wish to make inference about a covariance matrix whose prior has a distribution. If the observations are independent p-variate Gaussian variables drawn from a distribution, then the conditional distribution has a distribution, where .
Because the prior and posterior distributions are the same family, we say the inverse Wishart distribution is conjugate to the multivariate Gaussian.
Due to its conjugacy to the multivariate Gaussian, it is possible to marginalize out (integrate out) the Gaussian's parameter , using the formula and the linear algebra identity :
(this is useful because the variance matrix is not known in practice, but because is known a priori, and can be obtained from the data, the right hand side can be evaluated directly). The inverse-Wishart distribution as a prior can be constructed via existing transferred prior knowledge.
Moments
The following is based on Press, S. J. (1982) "Applied Multivariate Analysis", 2nd ed. (Dover Publications, New York), after reparameterizing the degree of freedom to be consistent with the p.d.f. definition above.
Let with and , so that .
The mean:
The variance of each element of :
The variance of the diagonal uses the same formula as above with , which simplifies to:
The covariance of elements of are given by:
The same results are expressed in Kronecker product form by von Rosen as follows:
where
commutation matrix
There appears to be a typo in the paper whereby the coefficient of is given as rather than , and that the expression for the mean square inverse Wishart, corollary 3.1, should read
To show how the interacting terms become sparse when the covariance is diagonal, let and introduce some arbitrary parameters :
where denotes the matrix vectorization operator. Then the second mome
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https://en.wikipedia.org/wiki/Neighbourhood%20Statistics%20Service
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The Neighbourhood Statistics Service (NeSS) was established in 2001 by the UK's Office for National Statistics (ONS) and the Neighbourhood Renewal Unit (NRU) - then part of the Office of the Deputy Prime Minister (ODPM), now Communities and Local Government (CLG) - to provide good quality small area data to support the Government's Neighbourhood Renewal agenda. This cross-Government initiative also involved the co-operation and partnership of data suppliers across departments, agencies and other organisations. The ONS closed the Neighbourhood Statistics website for England and Wales on the 12 May 2017. To offset this, the ONS is aiming to meet the needs of users via the ONS website, although direct postcode searches are no longer available to users.
The Scottish Government continues to provide local statistics via Statistics.Gov.Scot and Census area profiles.
Neighbourhood statistics for Northern Ireland continue to be made available from the Northern Ireland Neighbourhood Information Service (NINIS).
Purpose
The need for Neighbourhood Statistics can be traced back to the Social Exclusion Unit's 1998 report on deprived neighbourhoods. The absence of information about neighbourhoods produced a series of failings at all levels, with policy makers unaware of the scale and location of problems but when small area information is collected and made easily available, it can radically improve strategies and service delivery.
Resource
NeSS provided a powerful platform through which high quality small area data for England and Wales was disseminated to an expanding audience. It allowed users to paint a statistical pictures of communities at a local level. Neighbourhood Statistics contained datasets covering Health, Housing, Education, Deprivation, Age, Ethnicity and 2011 Census data.
References
External links
Advice on obtaining local statistics for England and Wales
statistics.gov.scot Data previously available on Scottish Neighbourhood Statistics and more.
www.nisra.gov.uk/ninis Northern Ireland Neighbourhood Information Service (NINIS)
Department for Levelling Up, Housing and Communities
National statistical services
Office for National Statistics
Public bodies and task forces of the United Kingdom government
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https://en.wikipedia.org/wiki/Marino%20Ba%C5%BEdari%C4%87
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Marino Baždarić (born 25 November 1978) is a Croatian former professional basketball player, who is currently the sports director for Cedevita Junior.
Career statistics
External links
ACB Profile
1978 births
Living people
ABA League players
Croatian expatriate basketball people in Spain
Croatian men's basketball players
KK Cedevita players
KK Cibona players
KK Olimpija players
Liga ACB players
Menorca Bàsquet players
Small forwards
Basketball players from Rijeka
KK Kvarner players
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https://en.wikipedia.org/wiki/Alpha%20Sigma%20Kappa
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Alpha Sigma Kappa – Women in Technical Studies ( – WiTS) is a social sorority for women in the fields of mathematics, architecture, engineering, technology and the sciences.
The sorority was founded at the University of Minnesota in 1989 by a group of women who had formerly been affiliated with the Sisters of Triangle Fraternity program. Alpha Sigma Kappa became a national organization in 1996.
History
Alpha Sigma Kappa originally grew from a Little Sisters of Triangle organization at the University of Minnesota. In the late 1980s, Little Sister programs were being phased out by fraternal organizations across the country; Triangle Fraternity's National Council resolved to do so with their local Little Sisters organizations. To maintain a formal relationship, the University of Minnesota's Little Sisters group chose to found Alpha Sigma Kappa. The sorority was created on May 1, 1989, by eighteen Founding Sisters. The founding sisters include:
When Alpha Sigma Kappa was founded, scientific careers were filled primarily by men. In 1989, only seventeen percent of the students enrolled in the Institute of Technology at the University of Minnesota were female. The Founding Sisters of the organization wished to create a sorority dedicated to supporting women who entered these fields. Alpha Sigma Kappa was intended to bring women pursuing technical studies together in a social setting: working to develop, encourage, and support the academic and social needs of these women. At the time of its founding, Alpha Sigma Kappa was the only social sorority for technical women whose scope included those in architecture and non-engineering sciences.
The stated purpose of Alpha Sigma Kappa is to promote friendship, academic achievement, unity within the organization, and philanthropy throughout the community. The sorority supports women in their academic goals and promotes women in technical fields through leadership, friendship, and support.
On March 4, 1996, an Interest Group of women was formed at the University of Oklahoma, which would ultimately become the Beta chapter on September 13, 1997. The National Organization of Alpha Sigma Kappa was officially formed on April 29, 1996, at the first national meeting in Minneapolis, Minnesota. Between 1999 and 2005, four more chapters were installed. In 2010, Alpha Sigma Kappa became the first sorority established on the campus of New Mexico Tech. Since 2015, the organization's growth has accelerated rapidly, establishing a presence on an additional ten universities across the United States as a social sorority for women in STEM.
In 2014, the Alpha Sigma Kappa Educational Foundation, a 501(c)(3) organization, was created to support women in technical fields by managing academic scholarships.
Symbols
The Greek letters Alpha Sigma Kappa represent social aspects; "Women in Technical Studies" specifies its membership. The sorority's colors are royal blue and silver. Its flower is the white rose and its jewel is the blu
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https://en.wikipedia.org/wiki/Egbert%20van%20Kampen
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Egbert Rudolf van Kampen (28 May 1908 – 11 February 1942) was a Dutch mathematician. He made important contributions to topology, especially to the study of fundamental groups.
Life
Van Kampen was born to Dutch parents in Belgium, where his father had recently taken a job as an accountant in Antwerp. At the outbreak of World War I the family moved back to the Netherlands, first to Amsterdam and in 1918 to The Hague. At the age of 16 he graduated from high school and entered Leiden University to study mathematics. After his undergraduate studies he continued with a doctorate study at the same university under the guidance of Willem van der Woude. In 1927, Van Kampen traveled to the University of Göttingen to meet with Bartel van der Waerden and Pavel Aleksandrov. In the summer of 1928 he worked with Emil Artin at the University of Hamburg. Around that time, while still only 20 years old, he was offered a position by Johns Hopkins University in the United States. He received his Ph.D. degree with Van der Woude in Leiden in 1929, writing a dissertation entitled Die kombinatorische Topologie und die Dualitaetssaetze.
In 1931, Van Kampen took up the position which he had been offered at Johns Hopkins University in Baltimore, Maryland, and travelled to the United States. There he met Oscar Zariski who had taught at Johns Hopkins as a Johnston Scholar from 1927 until 1929, when he had joined the Faculty. Zariski had been working on the fundamental group of the complement of an algebraic curve, and he had found generators and relations for the fundamental group but was unable to show that he had found sufficient relations to give a presentation for the group. Van Kampen solved the problem, showing that Zariski's relations were sufficient, and the result is now known as the Zariski–Van Kampen theorem. This led Van Kampen to formulate and prove what is nowadays known as the Seifert–Van Kampen theorem.
By the late 1930s, Van Kampen started to suffer from headaches which in 1941 were diagnosed to arise from a tumor originating from a birth mark near his ear. After three surgeries in 1941 and 1942, Van Kampen succumbed to cancer and died on 11 February 1942.
Publications
References
1908 births
1942 deaths
People from Berchem
Leiden University alumni
20th-century Dutch mathematicians
Dutch expatriates in the United States
Johns Hopkins University faculty
Topologists
Deaths from cancer in Maryland
Dutch expatriates in Belgium
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https://en.wikipedia.org/wiki/Amsterdam%20Science%20Park
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Amsterdam Science Park is a science park in the Oost city district of Amsterdam, Netherlands with foci on physics, mathematics, information technology and the life sciences. The 70 hectare (175 acre) park provides accommodations for science, business and housing. Resident groups include institutes of the natural science faculties of the University of Amsterdam, several research institutes, and related companies. Three of the colocations of the Amsterdam Internet Exchange are at the institutes SURFsara, NIKHEF, and Equinix-AM3 at the science park.
In 2009, the Amsterdam Science Park railway station was by opened then-mayor Job Cohen.
Science and business
FOM Institute AMOLF (Physics of Biomolecular systems and Nanophotonics)
Advanced Research Center for Nanolithography (ARCNL)
National Research Institute for Mathematics and Computer Science (CWI)
Faculty of Science (FNWI) of the University of Amsterdam offering education programmes in biology, chemistry, computer science, earth science, physics, mathematics etc. and comprising eight research institutes, including:
Institute for Biodiversity and Ecosystem Dynamics (IBED)
Institute for Logic, Language and Computation (ILLC)
Institute of Physics (IoP)
Korteweg-de Vries Institute for Mathematics (KdVI)
Netherlands eScience Center (NLeSC)
National Institute for Subatomic Physics (Nikhef)
SURFsara (computer centre)
EGI.eu, the coordinating organisation for the European Grid Infrastructure
More than 90 companies in the fields of ICT, life sciences, and related fields (e.g. BioDetection Systems)
Housing
At the science park, 314 residences and 721 student units have been completed. An additional 423 residences and 617 student units are planned.
Leisure
Café-restaurant 'Polder' (temporary location adjacent to the historical Anna Hoeve farm)
University Sports Centre 'Universum' (official opening October 8, 2010)
Sports Café 'Oerknal'
Meet & Eat Restaurant
Café Maslow
Ann's Farm
References
External links
Amsterdam Science Park
Neighbourhoods of Amsterdam
Economy of Amsterdam
Amsterdam, Science Park
University of Amsterdam
Amsterdam-Oost
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https://en.wikipedia.org/wiki/Kumon
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Kumon Institute Education Co. Ltd. is an educational network based in Japan and created by Toru Kumon. It uses his Kumon Method to teach mathematics and reading primarily for young students.
History
Kumon was founded by Toru Kumon, a Japanese educator, in July 1958, when he opened the first Kumon Maths Centre in Moriguchi, Osaka. Prior to creating the Kumon franchise, Kumon taught at Kochi Municipal High School and Tosa Junior/Senior High School. Inspired by teaching his own son, Takeshi, Kumon developed a curriculum focused on rote memorization.
Kumon initially grew slowly, only gaining 63,000 students over its first 16 years. However, in 1974, Kumon published a book titled The Secret of Kumon Math, leading to a doubling of its size in the next two years. Kumon opened their first United States locations in 1983, and by 1985, Kumon reached 1.4 million students.
Kumon soon added more educational subjects, leading them to change their name from Kumon Institute of Mathematics to Kumon Institute of Education. At this point, the first Kumon Logo was created. In 1985, Kumon's success lead to an increase in enrollments.
Kumon attracted national attention in the United States after it was implemented at Sumiton Elementary School, in Sumiton, Alabama. This was the first instance in which an American school integrated the Kumon Math Method into the regular K–4 mathematics curriculum. Sumiton continued to use the Kumon program through 2001, and influenced other schools to also adopt the Kumon method in their curriculum.
As of 2008, Kumon had over 26,000 centers around the globe with over 4 million registered students. As of 2018, there were 410,000 students enrolled in 2,200 centers across the United States.
Kumon introduced "Baby Kumon" in Japan in 2012, a tutoring program targeted for children between 1 and 2 years old. Baby Kumon hasn't been utilized in most Kumon Centres in other countries outside Japan. In North America, Kumon began a "Junior Kumon" program in 2001, targeted at children aged 2–5 years old.
Kumon method of learning
Kumon is an enrichment or remedial program, where instructors and assistants tailor instruction for individual students.
All Kumon programs are pencil-and-worksheet-based, with a digital program starting in 2023. The worksheets increase in difficulty in small increments. The program recommends that students study 30 minutes per subject on their own for five days of the week, and visit their local Kumon Center or attend a virtual class the other two days.
Psychologist Kathy Hirsh-Pasek claims that using such techniques, when used in 2, 3, and 4-year-olds, "does not give your child a leg up on anything". However, studies have shown a high percentage of efficacy.
Programs
Mathematics program
As a high school mathematics teacher, Mr. Kumon understood that an understanding of calculus was essential for Japanese university entrance exams so in writing worksheets for his son, Mr. Kumon focused on all the topics needed fo
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https://en.wikipedia.org/wiki/List%20of%20science%20and%20technology%20awards%20for%20women
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This list of science and technology awards for women is an index to articles about notable awards made to women for work in science and the STEM (Science, technology, engineering, and mathematics) fields generally. It includes awards for astronomy, space and atmospheric science; biology and medicine; chemistry; engineering; mathematics; neuroscience; physics; technology; and general or multiple fields.
Astronomy, space, atmospheric science
Annie Jump Cannon Award in Astronomy – annual award for outstanding contributions to astronomy by a woman within five years of earning a doctorate degree
Peter B. Wagner Memorial Award for Women in Atmospheric Sciences – awarded annually since 1998, based on paper completion, to a woman studying for a Masters or PhD in atmospheric science at a university in the United States
Biology and medicine
Elizabeth Blackwell Medal, given by the American Medical Women's Association to a woman physician "who has made the most outstanding contributions to the cause of women in the field of medicine"
Federation of American Societies for Experimental Biology (FASEB) Excellence in Science Award
Group on Women in Medicine and Science Leadership Awards, Association of American Medical Colleges
Margaret Oakley Dayhoff Award from the Biophysical Society, Rockville, Maryland – given to a woman who "has achieved prominence for 'substantial contributions to science'" and showing high promise in the early part of her career
Pearl Meister Greengard Prize – established 2004
WICB Junior and Senior Awards from Women in Cell Biology (WICB)
Chemistry
ACS Award for Encouraging Women into Careers in the Chemical Sciences, sponsored by the Camille and Henry Dreyfus Foundation
Garvan–Olin Medal – annual award that recognizes distinguished service to chemistry by women chemists
Awards by the Iota Sigma Pi honorary society for women in chemistry:
Agnes Fay Morgan Research Award
Anna Louise Hoffman Award for Outstanding Achievement in Graduate Research
Centennial Award for Excellence in Undergraduate Teaching
Gladys Anderson Emerson Undergraduate Scholarship
Members-at-Large Re-entry Award
National Honorary Member
Outstanding Young Women in Chemistry award.
Undergraduate Excellence in Chemistry
Violet Diller Professional Excellence Award
Engineering
Sharon Keillor Award for Women in Engineering Education
Achievement Award of the Society of Women Engineers
Young Woman Engineer of the Year Award
Mathematics
Awards by the Association for Women in Mathematics:
Alice T. Schafer Prize – established 1991
Biographies of Contemporary Women in Mathematics Essay Contest – established in 2001 for biographical essays
Emmy Noether Lectures – an honorary lecture award
M. Gweneth Humphreys Award
Louise Hay Award for Contributions to Mathematics Education – established 1991
Ruth I. Michler Memorial Prize
Ruth Lyttle Satter Prize in Mathematics – established 1990
additional awards by the AWM
Awards sponsored by the Kovalevskaia
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https://en.wikipedia.org/wiki/Dutra%20%28footballer%2C%20born%201973%29
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Antônio Monteiro Dutra (born August 11, 1973), or simply Dutra, is a Brazilian left back.
Club statistics
Honours
Santos
Torneio Rio-São Paulo: 1997
Coritiba
Campeonato Paranaense: 1999
Yokohama F. Marinos
J.League Cup: 2001
J1 League : 2003, 2004
Emperor's Cup: 2013
Sport
Campeonato Pernambucano: 2007, 2008, 2009, 2010
Copa do Brasil: 2008
Santa Cruz
Campeonato Pernambucano: 2012
Individual
J.League Best XI: 2003, 2004
References
External links
zerozero.pt
1973 births
América Futebol Clube (MG) players
Brazilian men's footballers
Brazilian expatriate men's footballers
Coritiba Foot Ball Club players
Living people
Mogi Mirim Esporte Clube players
Paysandu Sport Club players
Santos FC players
Sport Club do Recife players
Santa Cruz Futebol Clube players
Yokohama F. Marinos players
Expatriate men's footballers in Japan
J1 League players
Men's association football fullbacks
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https://en.wikipedia.org/wiki/An%20Yong-hak
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An Yong-Hak (born 25 October 1978) is a Japanese-born North Korean football midfielder. He is a former member of the North Korea national football team.
Club statistics
Updated to 23 February 2016.
Honours
Albirex Niigata
J2 League (1): 2003
Suwon Bluewings
K League 1 (1): 2008
Korean FA Cup (1): 2009
Korean League Cup (1): 2008
Kashiwa Reysol
J1 League (1): 2011
Japanese Super Cup (1): 2012
International goals
.''Scores and results are list North Korea's goal tally first.
References
External links
Yokohama FC Top Team
1978 births
Living people
Men's association football midfielders
Association football people from Okayama Prefecture
North Korean men's footballers
North Korean expatriate men's footballers
North Korea men's international footballers
2010 FIFA World Cup players
2011 AFC Asian Cup players
Albirex Niigata players
Nagoya Grampus players
Busan IPark players
Suwon Samsung Bluewings players
Omiya Ardija players
Kashiwa Reysol players
Yokohama FC players
J1 League players
J2 League players
K League 1 players
Expatriate men's footballers in Japan
Expatriate men's footballers in South Korea
North Korean expatriate sportspeople in Japan
South Korean people of North Korean origin
People from Kurashiki
Zainichi Korean men's footballers
People's Athletes
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https://en.wikipedia.org/wiki/Lucio%20Russo
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Lucio Russo (born 22 November 1944) is an Italian physicist, mathematician and historian of science. Born in Venice, he teaches at the Mathematics Department of the University of Rome Tor Vergata.
Among his main areas of interest are Gibbs measure of the Ising model, percolation theory, and finite Bernoulli schemes, within which he proved an approximate version of the classical Kolmogorov's zero–one law.
In the history of science, he has reconstructed some contributions of the Hellenistic astronomer Hipparchus, through the analysis of his surviving works, and the proof of heliocentrism attributed by Plutarch to Seleucus of Seleucia and studied the history of theories of tides, from the Hellenistic to modern age.
Books
The Forgotten Revolution
In The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had to Be Reborn (Italian: La rivoluzione dimenticata), Russo promotes the belief that Hellenistic science in the period 320–144 BC reached heights not achieved by Classical age science, and proposes that it went further than ordinarily thought, in multiple fields not normally associated with ancient science.
According to Russo, Hellenistic scientists were not simply forerunners, but actually achieved scientific results of high importance, in the fields of "mathematics, solid and fluid mechanics, optics, astronomy, anatomy, physiology, scientific medicine," even psychoanalysis. They may have even discovered the inverse square law of gravitation (Russo's argument on this point hinges on well-established, but seldom discussed, evidence). Hellenistic scientists, among them Euclid, Archimedes, Eratosthenes, developed an axiomatic and deductive way of argumentation. When this way of argumentation was dropped, the ability to understand the results were lost as well. Thus Russo conjectures that the definitions of elementary geometric objects were introduced in Euclid's Elements by Heron of Alexandria, 400 years after the work was completed. More concretely, Russo shows how the theory of tides must have been well-developed in Antiquity, because several pre-Newtonian sources relay various complementary parts of the theory without grasping their import or justification (getting the empirical facts wrong but the theory right).
Hellenistic science was focused on the city of Alexandria. The emerging scientific revolution in Alexandria was ended when Ptolemy VIII Physcon came to power. He engaged in mass purges and expulsions of all intellectuals. Other centers of Hellenistic science mentioned in Russo's book were Antioch, Pergamon, Cyzicus, Rhodes, Syracuse and Massilia.
He also concludes that the 17th-century scientific revolution in Europe was due in large part to the recovery of Hellenistic science. The Forgotten Revolution has received mixed reviews, praising Russo's enthusiasm but noting that his conclusions outreach his sources.
L'America dimenticata
In L'America dimenticata, Russo suggests that the Americas were known to som
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https://en.wikipedia.org/wiki/Elements%20of%20Algebra
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Elements of Algebra is an elementary mathematics textbook written by mathematician Leonhard Euler around 1765 in German. It was first published in Russian as "Universal Arithmetic" (Универсальная арифметика), two volumes appearing in 1768-9 and in 1770 was printed from the original text. Elements of Algebra is one of the earliest books to set out algebra in the modern form we would recognize today (another early book being Elements of Algebra by Nicholas Saunderson, published in 1740), and is one of Euler's few writings, along with Letters to a German Princess, that are accessible to the general public. Written in numbered paragraphs as was common practice till the 19th century, Elements begins with the definition of mathematics and builds on the fundamental operations of arithmetic and number systems, and gradually moves towards more abstract topics.
In 1771, Joseph-Louis Lagrange published an addendum titled Additions to Euler's Elements of Algebra, which featured a number of important mathematical results.
The original German title of the book was Vollständige Anleitung zur Algebra, which literally translates to Complete Instruction to Algebra. Two English translations are now extant, one by John Hewlett (1822), and the other, which is translated to English from a French translation of the book, by Charles Tayler (1824). On the 300th birth anniversary of Euler in 2007, mathematician Christopher Sangwin working with Tarquin Publications published a digitized copy based on Hewlett's translation of the first four sections (or Part I) of the book.
In 2015, Scott Hecht published both print and Kindle versions of Elements of Algebra () with Euler's Part I (Containing the Analysis of Determinate Quantities), Part II (Containing the Analysis of Indeterminate Quantities), Lagrange's Additions, and footnotes by Johann Bernoulli and others.
See also
Introductio in analysin infinitorum (1748)
Institutiones calculi differentialis (1755)
References
External links
Elements of Algebra, 1822, Full text
Elements of Algebra, Part I, HTML
The origin of the problems in Euler's Algebra
Mathematics textbooks
1760s books
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https://en.wikipedia.org/wiki/JSJ
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JSJ may refer to:
IATA code for Jiansanjiang Airport
JSJ decomposition, a process in mathematics of decomposing a topological space
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https://en.wikipedia.org/wiki/Tudor%20Ganea
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Tudor Ganea (October 17, 1922 –August 1971) was a Romanian-American mathematician, known for his work in algebraic topology, especially homotopy theory. Ganea left Communist Romania to settle in the United States in the early 1960s. He taught at the University of Washington.
Life and work
He studied mathematics at the University of Bucharest, and then started his research as a member of Simion Stoilow's seminar on complex functions. His papers from 1949–1952 were on covering spaces, topological groups, symmetric products, and the Lusternik–Schnirelmann category. During this time, he earned his candidate thesis in topology under the direction of Stoilow.
In 1957, Ganea published in the Annals of Mathematics a short, yet influential paper with Samuel Eilenberg, in which the Eilenberg–Ganea theorem was proved and the celebrated Eilenberg–Ganea conjecture was formulated. The conjecture is still open.
By 1958, Ganea and his mentee, , were the two leading algebraic topologists in Romania. Later that year at an international conference on geometry and topology in Iași, the two met Peter Hilton, starting long mathematical collaborations. Ganea left for France in 1961, where he obtained in 1962 his Ph.D. from the University of Paris under Henri Cartan, with thesis Sur quelques invariants numeriques du type d'homotopie. He then emigrated to the United States. After spending a year at Purdue University in West Lafayette, Indiana, he joined the faculty at the University of Washington in Seattle.
During this time, he tried to get Aurora Cornu (his fiancée at the time) out of Romania, but did not succeed. In 1962, he gave an invited talk at the International Congress of Mathematicians in Stockholm, titled On some numerical homotopy invariants.
Just before he died, Ganea attended the Symposium on Algebraic Topology, held February 22–26, 1971 at the Battelle Seattle Research Center, in Seattle. At the symposium, he was not able to give a talk, but he did distribute a preprint containing a list of unsolved problems. One of these problems, regarding the Lusternik–Schnirelmann category, came to be known as Ganea's conjecture. A version of this conjecture for rational spaces was proved by Kathryn Hess in her 1989 MIT Ph.D. thesis. Many particular cases of Ganea's original conjecture were proved, until Norio Iwase provided a counterexample in 1998. A minimum dimensional counterexample to Ganea’s conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010.
Ganea is buried at Lake View Cemetery in Seattle.
References
Publications
External links
1922 births
1971 deaths
20th-century Romanian mathematicians
20th-century American mathematicians
University of Bucharest alumni
University of Paris alumni
Topologists
Romanian expatriates in France
Romanian emigrants to the United States
Purdue University faculty
University of Washington faculty
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https://en.wikipedia.org/wiki/Ganea%20conjecture
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Ganea's conjecture is a now disproved claim in algebraic topology. It states that
for all , where is the Lusternik–Schnirelmann category of a topological space X, and Sn is the n-dimensional sphere.
The inequality
holds for any pair of spaces, and . Furthermore, , for any sphere , . Thus, the conjecture amounts to .
The conjecture was formulated by Tudor Ganea in 1971. Many particular cases of this conjecture were proved, and Norio Iwase gave a counterexample to the general case in 1998. In a follow-up paper from 2002, Iwase gave an even stronger counterexample, with X a closed smooth manifold. This counterexample also disproved a related conjecture, which stated that
for a closed manifold and a point in .
A minimum dimensional counterexample to the conjecture was constructed by Don Stanley and Hugo Rodríguez Ordóñez in 2010.
This work raises the question: For which spaces X is the Ganea condition, , satisfied? It has been conjectured that these are precisely the spaces X for which equals a related invariant,
References
Disproved conjectures
Algebraic topology
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https://en.wikipedia.org/wiki/GEOS%20circle
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In geometry, the GEOS circle is derived from the intersection of four lines that are associated with a generalized triangle: the Euler line, the Soddy line, the orthic axis and the Gergonne line. Note that the Euler line is orthogonal to the orthic axis and that the Soddy line is orthogonal to the Gergonne line.
These four lines provide six points of intersection of which two points occur at line intersections that are orthogonal. Consequently, the other four points form an orthocentric system.
The GEOS circle is that circle centered at a point equidistant from X650 (the intersection of the orthic axis with the Gergonne line) and X20 (the intersection of the Euler line with the Soddy line and is known as the de Longchamps point) and passes through these points as well as the two points of orthogonal intersection.
The orthogonal intersection points are X468 (the intersection of the orthic axis with the Euler line) and X1323 (the Fletcher point, the intersection of the Gergonne line with the Soddy line).
The orthocentric system comprises X650, X20, X1375 (the intersection of the Euler line with the Gergonne line and is known as the Evans point) and X3012 (the intersection of the Soddy line and the orthic axis).
The X(i) point notation is the Clark Kimberling ETC classification of triangle centers.
References
Circles defined for a triangle
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https://en.wikipedia.org/wiki/B%20%28disambiguation%29
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B is the second letter of the Latin alphabet.
B may also refer to:
Science, technology, and mathematics
Astronomy
Astronomical objects in the Barnard list of dark nebulae (abbreviation B)
Latitude (b) in the galactic coordinate system
Biology and medicine
Haplogroup B (mtDNA), a human mitochondrial DNA (mtDNA) haplogroup
Haplogroup B (Y-DNA), a Y-chromosomal DNA (Y-DNA) haplogroup
Blood type B
ATC code B Blood and blood forming organs, a section of the Anatomical Therapeutic Chemical Classification System
Vitamin B
Hepatitis B
Berlin Botanical Garden and Botanical Museum, assigned the abbreviation B as a repository of herbarium specimens
Computing
B (programming language)
B-Method, for computer software development
B-tree, a data structure
Bit (b)
Byte (B)
, an HTML element denoting bold text
Physical and chemical quantities and units
One of the reciprocal lattice vectors (b*)
Breadth (b); see length
Impact parameter (b)
Molality (b)
Barn (unit) (b), a unit of area
Magnetic field (B)
Napierian absorbance (B)
Rotational constant (B)
Second virial coefficient (B)
Susceptance (B)
Bel (B), a logarithmic unit equal to ten decibels
Other uses in science, technology, and mathematics
B, a modal logic
bottom quark (symbol: b), an elementary particle
B meson, a type of meson
Boolean domain (), in mathematics
Boron, symbol B, a chemical element
Bulb (photography), a shutter-speed setting
B battery, a battery used to provide the plate voltage of a vacuum tube
B horizon, a layer of soil commonly known as subsoil
B, B+ and B−, expressions of buoyancy in convective available potential energy
Linguistics
В, a letter of the Cyrillic alphabet
Voiced bilabial stop (b, in the International Phonetic Alphabet)
Arts and media
Music
Musical notation
B (musical note)
B major, a scale
B minor, a scale
B major chord, Chord names and symbols (popular music)
Flat (music), a note lower in pitch by a semitone
Performers
Beyoncé
B, member of Superorganism (band)
Recordings
B (BamBam EP), an extended play by BamBam
B (Battles EP), an extended play by Battles
B (I Am Kloot album), a compilation album by I Am Kloot
b (Moxy Früvous EP), an extended play by Moxy Früvous
"b", an Iamamiwhoami song
"B", a song by Todrick Hall from his 2018 album Forbidden
Other media
/b/, a board on 4chan
b (newspaper), a free daily, tabloid sized newspaper, published by The Baltimore Sun
Codex Vaticanus Graecus 1209, an ancient bible
Rajawali Televisi, previously named B Channel, a television channel in Indonesia
B, the production code for the 1963–64 Doctor Who serial The Daleks
B, a character in Total Drama: Revenge of the Island
"B" Is for Burglar, the second novel in Sue Grafton's "Alphabet mystery" series, published in 1985
Places
Belgium, on the vehicle registration plates of the European Union
Bucharest, on the vehicle registration plates of Romania
Sport
Australian rules football positions#Full back, in Australian Ru
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https://en.wikipedia.org/wiki/Progress%20in%20Physics
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Progress in Physics is an open-access academic journal, publishing papers in theoretical and experimental physics, including related themes from mathematics. The journal was founded by Dmitri Rabounski, Florentin Smarandache, and Larissa Borissova in 2005, and is published quarterly. Rabounski is the editor-in-chief, while Smarandache and Borissova act as associate editors. It was included on Beall's List of potentially-predatory journals at the time that list was last updated. Since 2008, the Norwegian Scientific Index has rated it a "Level 0" journal, indicating that publication there does not count for official academic career or public funding purposes.
Aims and reviewing process
The journal aims to promote fair and non-commercialized science, as stated in its Declaration of Academic Freedom:
The journal describes itself as peer-reviewed. The review procedure is specified as follows:
The referees of the papers published are not listed, although anonymity of referees is specifically criticized in "Article 8: Freedom to publish scientific results" of the Declaration of Academic Freedom. This document harshly criticizes the current peer-review system using the words "censorship", "alleged expert referees", "blacklisting", and "bribes". The journal has published papers by several authors, who, along with some of the editors, claim to have been blacklisted by the Cornell University arXiv as proponents of fringe scientific theories.
Indexing and abstracting
The journal is or has been indexed and abstracted in the following bibliographic databases:
It was indexed in the (paywalled) aggregator Open J-Gate and in the website Scientific Commons.
References and notes
External links
Progress in Physics website
Physics journals
Open access journals
Academic journals established in 2005
Quarterly journals
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https://en.wikipedia.org/wiki/AGG
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Agg or AGG may refer to:
As an acronym:
Anti-Grain Geometry, computer graphics rendering library
Aesthetic group gymnastics, gymnastics In a group
Abnormal grain growth, materials science phenomenon
Art Gallery of Guelph
AGG (programming language)
Attorney General of the Gambia
Attorney General of Georgia
Attorney General of Ghana
Attorney General of Gibraltar
Attorney General of Grenada
Attorney General of Guam
Attorney General of Guatemala
Attorney General of Guyana
Auditor-General of Ghana
As another abbreviation or symbol:
Angor language (ISO 639-3 code)
Arginine, an amino acid with codon AGG
iShares Core Total US Bond Market ETF, an exchange-traded fund
Tirofiban, trade name Aggrastat, an antiplatelet drug
People:
Alfred John Agg (1830–1886), Australian colonial public servant
Antonio Gandy-Golden (born 1998), American football player
Lily Agg (born 1993), Irish professional footballer
See also
Species aggregate, abbreviated "agg."
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https://en.wikipedia.org/wiki/Anti-Grain%20Geometry
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Anti-Grain Geometry (AGG) is a 2D rendering graphics library written in C++. It features anti-aliasing and sub-pixel resolution. It is not a graphics library, per se, but rather a framework to build a graphics library upon.
The library is operating system independent and renders to an abstract memory object. It comes with examples interfaced to the X Window System, Microsoft Windows, Mac OS X, AmigaOS, BeOS, SDL. The examples also include an SVG viewer.
The design of AGG uses C++ templates only at a very high level, rather than extensively, to achieve the flexibility to plug custom classes into the rendering pipeline, without requiring a rigid class hierarchy, and allows the compiler to inline many of the method calls for high performance. For a library of its complexity, it is remarkably lightweight: it has no dependencies above the standard C++ libraries and it avoids the C++ STL in the implementation of the basic algorithms. The implicit interfaces are not well documented, however, and this can make the learning process quite cumbersome.
While AGG version 2.5 is licensed under the GNU General Public License, version 2 or greater, AGG version 2.4 is still available under the 3-clause BSD license and is virtually the same as version 2.5.
History
Active development of the AGG codebase stalled in 2006, around the time of the v2.5 release, due to shifting priorities of its main developer and maintainer Maxim Shemanarev. M. Shemanarev remained active in the community until his sudden death in 2013. Development has continued on a fork of the more liberally licensed v2.4 on SourceForge.net.
Usage
The Haiku operating system uses AGG in its windowing system.
It is one of the renderers available for use in GNU's Gnash Flash player.
Graphical version of Rebol language interpreter is using AGG for scalable vector graphics DRAW dialect.
Hilti uses it in some of their rebar detection tools, like the PS 1000.
Matplotlib uses AGG as its canonical renderer for interactive user interfaces.
fpGUI Toolkit has an optional AggPas back-end rendering engine. Work is being done to make AggPas the default or sole rendering engine for fpGUI.
Mapnik, the toolkit that renders the maps on the OpenStreetMap website, uses AGG for all its map rendering.
HTTPhotos uses AGG to scale photos.
Pdfium, the PDF rendering engine used by Google Chrome makes use of AGG, although work is progressing to replace this with Skia Graphics Engine.
Graphics Mill, the .NET imaging SDK uses AGG as its drawing engine.
Image-Line FL Studio, a digital audio workstation, since version 10.8 released on September 30, 2012, uses AGG for drawing.
Author
Main author of the library was Maxim Shemanarev (). On November 26, 2013 Shemanarev (born June 15, 1966, Nizhny Novgorod, Russia) was reported dead at the age of 47 at his home in Columbia, Maryland (US). He died suddenly, allegedly from an epileptic seizure that he had suffered for a while. He was a graduate from Nizhny Novgorod State
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https://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely%20primality%20test
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In computational number theory, the Adleman–Pomerance–Rumely primality test is an algorithm for determining whether a number is prime. Unlike other, more efficient algorithms for this purpose, it avoids the use of random numbers, so it is a deterministic primality test. It is named after its discoverers, Leonard Adleman, Carl Pomerance, and Robert Rumely. The test involves arithmetic in cyclotomic fields.
It was later improved by Henri Cohen and Hendrik Willem Lenstra, commonly referred to as APR-CL. It can test primality of an integer n in time:
Software implementations
UBASIC provides an implementation under the name APRT-CLE (APR Test CL extended)
A factoring applet that uses APR-CL on certain conditions (source code included)
Pari/GP uses APR-CL conditionally in its implementation of isprime().
mpz_aprcl is an open source implementation using C and GMP.
Jean Penné's LLR uses APR-CL on certain types of prime tests as a fallback option.
References
APR and APR-CL
Primality tests
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https://en.wikipedia.org/wiki/Stationary%20sequence
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In probability theory – specifically in the theory of stochastic processes, a stationary sequence is a random sequence whose joint probability distribution is invariant over time. If a random sequence X j is stationary then the following holds:
where F is the joint cumulative distribution function of the random variables in the subscript.
If a sequence is stationary then it is wide-sense stationary.
If a sequence is stationary then it has a constant mean (which may not be finite):
See also
Stationary process
References
Probability and Random Processes with Application to Signal Processing: Third Edition by Henry Stark and John W. Woods. Prentice-Hall, 2002.
Sequences and series
Time series
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https://en.wikipedia.org/wiki/Riemann%20form
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In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:
A lattice Λ in a complex vector space Cg.
An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:
the real linear extension αR:Cg × Cg→R of α satisfies αR(iv, iw)=αR(v, w) for all (v, w) in Cg × Cg;
the associated hermitian form H(v, w)=αR(iv, w) + iαR(v, w) is positive-definite.
(The hermitian form written here is linear in the first variable.)
Riemann forms are important because of the following:
The alternatization of the Chern class of any factor of automorphy is a Riemann form.
Conversely, given any Riemann form, we can construct a factor of automorphy such that the alternatization of its Chern class is the given Riemann form.
References
Abelian varieties
Bernhard Riemann
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https://en.wikipedia.org/wiki/Alternating%20multilinear%20map
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In mathematics, more specifically in multilinear algebra, an alternating multilinear map is a multilinear map with all arguments belonging to the same vector space (for example, a bilinear form or a multilinear form) that is zero whenever any pair of arguments is equal. More generally, the vector space may be a module over a commutative ring.
The notion of alternatization (or alternatisation) is used to derive an alternating multilinear map from any multilinear map with all arguments belonging to the same space.
Definition
Let be a commutative ring and be modules over . A multilinear map of the form is said to be alternating if it satisfies the following equivalent conditions:
whenever there exists such that then
whenever there exists such that then
Vector spaces
Let be vector spaces over the same field. Then a multilinear map of the form is alternating iff it satisfies the following condition:
if are linearly dependent then .
Example
In a Lie algebra, the Lie bracket is an alternating bilinear map.
The determinant of a matrix is a multilinear alternating map of the rows or columns of the matrix.
Properties
If any component of an alternating multilinear map is replaced by for any and in the base ring then the value of that map is not changed.
Every alternating multilinear map is antisymmetric, meaning that
or equivalently,
where denotes the permutation group of order and is the sign of
If is a unit in the base ring then every antisymmetric -multilinear form is alternating.
Alternatization
Given a multilinear map of the form the alternating multilinear map defined by
is said to be the alternatization of
Properties
The alternatization of an n-multilinear alternating map is n! times itself.
The alternatization of a symmetric map is zero.
The alternatization of a bilinear map is bilinear. Most notably, the alternatization of any cocycle is bilinear. This fact plays a crucial role in identifying the second cohomology group of a lattice with the group of alternating bilinear forms on a lattice.
See also
Alternating algebra
Bilinear map
Map (mathematics)
Multilinear algebra
Multilinear map
Multilinear form
Symmetrization
Notes
References
Functions and mappings
Mathematical relations
Multilinear algebra
fr:Application multilinéaire#Application alternée
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https://en.wikipedia.org/wiki/Mohammad%20Gholamin
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Mohammad Gholamin Noveirsari (, born 11 February 1986 in Bandar-e Anzali, Iran) is an Iranian football player.
He was also a member of Iran national under-23 football team.
Club statistics
Assist Goals
External links
Persian League Profile
Iranian men's footballers
1986 births
Living people
Footballers from Bandar-e Anzali
Persian Gulf Pro League players
Azadegan League players
Malavan F.C. players
Paykan F.C. players
Sanat Naft Abadan F.C. players
Steel Azin F.C. players
Payam Khorasan F.C. players
F.C. Aboomoslem players
Gahar Zagros F.C. players
Asian Games bronze medalists for Iran
Asian Games medalists in football
Footballers at the 2006 Asian Games
Men's association football forwards
Medalists at the 2006 Asian Games
21st-century Iranian people
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https://en.wikipedia.org/wiki/Right%20inverse
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A right inverse in mathematics may refer to:
A right inverse element with respect to a binary operation on a set
A right inverse function for a mapping between sets
See also
Right-cancellative
Loop (algebra), an algebraic structure with identity element where every element has a unique left and right inverse
Section (category theory), a right inverse of some morphism
Left inverse (disambiguation)
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https://en.wikipedia.org/wiki/Cotlar%E2%80%93Stein%20lemma
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In mathematics, in the field of functional analysis, the Cotlar–Stein almost orthogonality lemma is named after mathematicians Mischa Cotlar
and Elias Stein. It may be used to obtain information on the operator norm on an operator, acting from one Hilbert space into another
when the operator can be decomposed into almost orthogonal pieces.
The original version of this lemma
(for self-adjoint and mutually commuting operators)
was proved by Mischa Cotlar in 1955 and allowed him to conclude that the Hilbert transform
is a continuous linear operator in
without using the Fourier transform.
A more general version was proved by Elias Stein.
Cotlar–Stein almost orthogonality lemma
Let be two Hilbert spaces.
Consider a family of operators
, ,
with each
a bounded linear operator from to .
Denote
The family of operators
,
is almost orthogonal if
The Cotlar–Stein lemma states that if
are almost orthogonal,
then the series
converges in the strong operator topology,
and that
Proof
If R1, ..., Rn is a finite collection of bounded operators, then
So under the hypotheses of the lemma,
It follows that
and that
Hence the partial sums
form a Cauchy sequence.
The sum is therefore absolutely convergent with limit satisfying the stated inequality.
To prove the inequality above set
with |aij| ≤ 1 chosen so that
Then
Hence
Taking 2mth roots and letting m tend to ∞,
which immediately implies the inequality.
Generalization
There is a generalization of the Cotlar–Stein lemma with sums replaced by integrals. Let X be a locally compact space and μ a Borel measure on X. Let T(x) be a map from X into bounded operators from E to F which is uniformly bounded and continuous in the strong operator topology. If
are finite, then the function T(x)v is integrable for each v in E with
The result can be proved by replacing sums by integrals in the previous proof or by using Riemann sums to approximate the integrals.
Example
Here is an example of an orthogonal family of operators. Consider the inifite-dimensional matrices
and also
Then
for each ,
hence the series
does not converge in the uniform operator topology.
Yet, since
and
for ,
the Cotlar–Stein almost orthogonality lemma tells us that
converges in the strong operator topology and is bounded by 1.
Notes
References
Hilbert spaces
Harmonic analysis
Operator theory
Inequalities
Theorems in functional analysis
Lemmas in analysis
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https://en.wikipedia.org/wiki/Sunflower%20%28mathematics%29
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In the mathematical fields of set theory and extremal combinatorics, a sunflower or -system is a collection of sets in which all possible distinct pairs of sets share the same intersection. This common intersection is called the kernel of the sunflower.
The naming arises from a visual similarity to the botanical sunflower, arising when a Venn diagram of a sunflower set is arranged in an intuitive way. Suppose the shared elements of a sunflower set are clumped together at the centre of the diagram, and the nonshared elements are distributed in a circular pattern around the shared elements. Then when the Venn diagram is completed, the lobe-shaped subsets, which encircle the common elements and one or more unique elements, take on the appearance of the petals of a flower.
The main research question arising in relation to sunflowers is: under what conditions does there exist a large sunflower (a sunflower with many sets) in a given collection of sets? The -lemma, sunflower lemma, and the Erdős-Rado sunflower conjecture give successively weaker conditions which would imply the existence of a large sunflower in a given collection, with the latter being one of the most famous open problems of extremal combinatorics.
Formal definition
Suppose is a set system over , that is, a collection of subsets of a set . The collection is a sunflower (or -system) if there is a subset of such that for each distinct and in , we have . In other words, a set system or collection of sets is a sunflower if all sets in W share the same common subset of elements. An element in is either found the common subset or else appears in at most one of the elements. No element of is shared by just some of the subset, but not others. Note that this intersection, , may be empty; a collection of pairwise disjoint subsets is also a sunflower. Similarly, a collection of sets each containing the same elements is also trivially a sunflower.
Sunflower lemma and conjecture
The study of sunflowers generally focuses on when set systems contain sunflowers, in particular, when a set system is sufficiently large to necessarily contain a sunflower.
Specifically, researchers analyze the function for nonnegative integers , which is defined to be the smallest nonnegative integer such that, for any set system such that every set has cardinality at most , if has more than sets, then contains a sunflower of sets. Though it is not clear that such an must exist, a basic and simple result of Erdős and Rado, the Delta System Theorem, indicates that it does.
Erdos-Rado Delta System Theorem:
For each , , there is an integer such that if a set system of -sets is of cardinality greater than , then contains a sunflower of size .
In the literature, is often assumed to be a set rather than a collection, so any set can appear in at most once. By adding dummy elements, it suffices to only consider set systems such that every set in has cardinality , so often the sunflower lemm
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https://en.wikipedia.org/wiki/Geoboard
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A geoboard is a mathematical manipulative used to explore basic concepts in plane geometry such as perimeter, area and the characteristics of triangles and other polygons. It consists of a physical board with a certain number of nails half driven in, around which are wrapped geo bands that are made of rubber. Normal rubber bands can also be used.
Geoboards were invented and popularized in the 1950s by Egyptian mathematician Caleb Gattegno (1911-1988).
Structure and use
Geoboard is a board. A variety of boards are used. Originally made out of plywood and brass nails or pegs, geoboards are now usually made out of plastic. They may have an upright square lattice of 9, 16 or 25 nails or more, or a circle of nails around a central nail. Students are asked to place rubber bands around the nails to explore geometric concepts or to solve mathematical puzzles.
Geoboards may be used to learn about:
plane shapes;
translation;
rotation;
reflection;
similarity;
co-ordination;
counting;
right angles;
pattern;
classification;
scaling;
position;
congruence;
area;
perimeter.
Two-dimensional representations of the geoboard may be applied to ordinary paper using rubber stamps or special "geoboard paper" with diagrams of geoboards may be used to help capture a student's explanations of the concept they have discovered or illustrated on the geoboard. There are also a number of online virtual geoboards.
References
External links
Educational and Supplemental Materials for K-12
Virtual geoboards on nrich.maths.org
Free Web-based Geoboard from Math Learning Center
Virtual geoboard on mathplayground.com
Various geoboard activities
Tom Scavo's collection of geoboard activities
Virtual geoboard and activities on nlvm.usu.edu
Authentic guide to using the Geoboards by the inventor Dr. Caleb Gattegno
Mathematical manipulatives
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https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Ganea%20conjecture
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The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group G has cohomological dimension 2, then it has a 2-dimensional Eilenberg–MacLane space . For n different from 2, a group G of cohomological dimension n has an n-dimensional Eilenberg–MacLane space. It is also known that a group of cohomological dimension 2 has a 3-dimensional Eilenberg−MacLane space.
In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, not both conjectures can be true.
References
Conjectures
Theorems in algebraic topology
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/Whitehead%20conjecture
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The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical.
A group presentation is called aspherical if the two-dimensional CW complex associated with this presentation is aspherical or, equivalently, if . The Whitehead conjecture is equivalent to the conjecture that every sub-presentation of an aspherical presentation is aspherical.
In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, it is not possible for both conjectures to be true.
References
Algebraic topology
Conjectures
Unsolved problems in mathematics
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https://en.wikipedia.org/wiki/1993%E2%80%9394%20First%20League%20of%20FR%20Yugoslavia
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Statistics of First League of FR Yugoslavia () for the 1993–94 season.
Overview
The league was divided into 2 groups, A and B, consisting each of 10 clubs. Both groups were played in league system. By winter break all clubs in each group meet each other twice, home and away, with the bottom four classified from A group moving to the group B, and being replaced by the top four from the B group. At the end of the season the same situation happened with four teams being replaced from A and B groups, adding the fact that the bottom three clubs from the B group were relegated into the Second League of FR Yugoslavia for the next season and replaced by the top three from that league.
At the end of the season FK Partizan became champions, with their striker Savo Milošević the league's top-scorer with 21 goals.
The relegated clubs were OFK Kikinda, FK Mogren and FK Jastrebac Niš.
Teams
Autumn
IA league
Table
Bonus point
13: Partizan (7 for 1st place, 6 for obtaining 27-29 points)
11: Red Star (6 for 2nd place, 5 for obtaining 24-26 points)
10: Vojvodina (5 for 3rd place, 5 for obtaining 24-26 points)
8: Zemun (5 for 4th place, 3 for obtaining 18-20 points)
7: Proleter (4 for 5th place, 3 for obtaining 18-20 points)
7: Budućnost (4 for 6th place, 3 for obtaining 18-20 points)
Results
IB league
Table
Bonus point
7: OFK Beograd (3 for 1st place, 4 for obtaining 21-23 points)
6: Spartak Subotica (2 for 2nd place, 4 for obtaining 21-23 points)
4: Radnički Jugopetrol (1 for 3rd place, 3 for obtaining 18-20 points)
4: FK Bečej (1 for 4th place, 3 for obtaining 18-20 points)
Results
Spring
IA league
Table
Results
IB league
Table
Results
Final table
Winning squad
Top goalscorers
External links
Table and results at RSSSF
Yugoslav First League seasons
Yugo
1993–94 in Yugoslav football
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