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https://en.wikipedia.org/wiki/Mirko%20Dickhaut
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Mirko Dickhaut (born 11 January 1971) is a German football coach and a former player.
Career statistics
References
External links
1971 births
Living people
2. Bundesliga managers
Footballers from Kassel
Men's association football defenders
Men's association football midfielders
German men's footballers
German football managers
Eintracht Frankfurt players
VfL Bochum players
VfL Bochum II players
Schwarz-Weiß Bregenz players
KSV Hessen Kassel players
Bundesliga players
2. Bundesliga players
Austrian Football Bundesliga players
KSV Hessen Kassel managers
SpVgg Greuther Fürth managers
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https://en.wikipedia.org/wiki/Hypercubic%20honeycomb
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In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group (or ) for .
The tessellation is constructed from 4 -hypercubes per ridge. The vertex figure is a cross-polytope
The hypercubic honeycombs are self-dual.
Coxeter named this family as for an -dimensional honeycomb.
Wythoff construction classes by dimension
A Wythoff construction is a method for constructing a uniform polyhedron or plane tiling.
The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard.
A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements. For example, an expanded cubic honeycomb has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 4 colors of cells around in vertex in 1:3:3:1 counts.
The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are hyperrectangles, also called orthotopes; in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively.
See also
Alternated hypercubic honeycomb
Quarter hypercubic honeycomb
Simplectic honeycomb
Truncated simplectic honeycomb
Omnitruncated simplectic honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
pp. 122–123. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}
p. 296, Table II: Regular honeycombs, δn+1
Honeycombs (geometry)
Polytopes
Regular tessellations
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https://en.wikipedia.org/wiki/Gilberto%20Calvillo%20Vives
|
Gilberto Calvillo Vives (born 3 November 1945 in Mexico City) is the president of the National Institute of Statistics, Geography and Informatics (INEGI).
He obtained a BSc in physics and mathematics at the Instituto Politécnico Nacional (IPN), a MSc in science, and a PhD in Operations Research at the University of Waterloo in Ontario, Canada.
As president of the National Institute of Statistics, Geography and Informatics (INEGI), he is currently president of the executive committee of the Statistical Conference of the Americas and president of the United Nations Statistics Commission. He is also a member of the Food and Agriculture Organization of the United Nations.
Before being appointed president of INEGI, he worked in the Mexican Olympic Committee, PEMEX and the World Bank.
See also
List of University of Waterloo people
External links
National Institute of Statistics, Geography and Informatics (INEGI) Website
National Institute of Statistics and Geography
Living people
1945 births
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https://en.wikipedia.org/wiki/8-simplex
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In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°.
It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.
As a configuration
This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:
More simply, the vertices of the 8-simplex can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.
Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.
Images
Related polytopes and honeycombs
This polytope is a facet in the uniform tessellations: 251, and 521 with respective Coxeter-Dynkin diagrams:
,
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
References
Coxeter, H.S.M.:
(Paper 22)
(Paper 23)
(Paper 24)
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
8-polytopes
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https://en.wikipedia.org/wiki/9-orthoplex
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In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.
It has two constructed forms, the first being regular with Schläfli symbol {37,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {36,31,1} or Coxeter symbol 611.
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.
Alternate names
Enneacross, derived from combining the family name cross polytope with ennea for nine (dimensions) in Greek
Pentacosidodecayotton as a 512-facetted 9-polytope (polyyotton)
Construction
There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or [4,37] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D9 or [36,1,1] symmetry group.
Cartesian coordinates
Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are
(±1,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
Images
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
9-polytopes
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https://en.wikipedia.org/wiki/9-simplex
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In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or approximately 83.62°.
It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions.. The name decayotton is derived from deca for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and -on.
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular decayotton having edge length 2 are:
More simply, the vertices of the 9-simplex can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). These are the vertices of one Facet of the 10-orthoplex.
Images
References
Coxeter, H.S.M.:
(Paper 22)
(Paper 23)
(Paper 24)
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
9-polytopes
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https://en.wikipedia.org/wiki/Probability%20Moon
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Probability Moon is a 2000 science fiction novel by the American writer Nancy Kress. The novel concerns a xenological expedition to the planet World, where aliens live who have developed a strange form of telepathy or collective unconscious, "shared reality", which causes piercing "head-pain" whenever "Worlders" attempt to hold strongly differing opinions. Simultaneously, an artificial satellite is found in orbit of the planet which has uncharted powers, and may be the key to winning a war against a xenocidal alien race, the "Fallers".
Setting
The "Probability" trilogy takes place in a galaxy that has been colonized by humans. This was made possible by the space tunnels, a network of FTL warp gates that were created by a now-lost progenitor race. Humanity is not united under a common government and political system. The Terrans have also discovered a number of alien races, most of them vastly similar in body format, living conditions and even DNA, leading to the hypothesis that the aforementioned progenitor race seeded the galaxy with sentient life, which then evolved according to the conditions on each planet. Of the known alien races, humanity is the only one that has reached space.
Humanity's understanding of the space tunnels is very limited, but several peculiar traits have been discovered. Firstly, if Ship A enters Tunnel 1, exits Tunnel 2 and then turns around and enters Tunnel 2 again, it will emerge from Tunnel 1 again. Unless Ship B emerges from Tunnel 2 in the interim, at which point Ship A will instead emerge from wherever Ship B entered. (The single tunnel leading to World is #438.) Secondly, objects can only enter the Tunnel if they are below a certain mass, about 100,000 tons; anything larger will actually fit into the aperture, but will collapse and explode. The threshold of what the tunnel can handle is determined by the object's Schwarzschild radius. Finally, nobody knows how the tunnels work. Macro-level quantum entanglement has been proposed, but it is so far out of the realm of current physics that nobody believes it.
The space tunnels also lead to the discovery of the Fallers, an alien race who refused to establish communications and immediately launched a war, which they are winning. No Faller has been captured alive—they prefer to suicide or kamikaze—but forensic examination of corpses indicate they evolved separately from humans, instead of being seeded by the progenitors. Like humanity, they were not an interstellar race until the discovery of a space tunnel in their system, though they have been closing the gap quickly. Unlike humans, they did not discover the tunnel independently; it was, in fact, a Terran craft emerging into their home system that catapulted them onto the interstellar stage.
The Probability novels shares two technological quirks with another of Nancy Kress' trilogies, the Beggars trilogy. In both stories, use of both genetic modification and behavior-regulating neuropharmacological drugs is commo
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https://en.wikipedia.org/wiki/Stefan%20Donchev
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Stefan Donchev (; born 28 August 1975 in Varna) is a former Bulgarian footballer who played as a defender.
Career statistics
As of 23 December 2010
References
1975 births
Living people
Bulgarian men's footballers
First Professional Football League (Bulgaria) players
PFC Ludogorets Razgrad players
FC Spartak Varna players
PFC Levski Sofia players
FC Atyrau players
FC Lokomotiv 1929 Sofia players
Bulgarian expatriate sportspeople in Kazakhstan
Expatriate men's footballers in Kazakhstan
Men's association football defenders
Footballers from Varna, Bulgaria
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https://en.wikipedia.org/wiki/Volume%20operator
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A quantum field theory of general relativity provides operators that measure the geometry of spacetime. The volume operator of a region is defined as the operator that yields the expectation value of a volume measurement of the region , given a state of quantum General Relativity. I.e. is the expectation value for the volume of . Loop Quantum Gravity, for example, provides volume operators, area operators and length operators for regions, surfaces and path respectively.
Sources
Carlo Rovelli and Lee Smolin, "Discreteness of Area and Volume in Quantum Gravity", Nuclear Physics B 442, 593 (1995).
Abhay Ashtekar and Jerzy Lewandowski, Quantum Theory of Geometry II: Volume operators
Quantum field theory
General relativity
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https://en.wikipedia.org/wiki/Sidney%20Harris%20%28cartoonist%29
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Sidney Harris, a.k.a. S. Harris, is an American cartoonist who draws cartoons about science, mathematics, and technology.
About
He was born in Brooklyn, New York on May 8, 1933, and has been drawing science-related cartoons at least since 1955. His cartoons have appeared in the scientific journals, science fiction magazines and textbooks. He was elected as the 19th honorary member of Sigma Xi, a scientific honor society, in 1997. Harris obtained his degree from Brooklyn College and then attended the Art Students League in New York. He began his career as a science cartoonist in 1955. Over 600 of Harris' drawings have been published by American Scientist since the 1970s. Harris also has had more than 20 cartoon collections published.
Appearances
Harris's cartoons have appeared the publications Science, Current Contents, Discover, Physics Today, The New Yorker, The Wall Street Journal, Harvard Business Review, The Chronicle of Higher Education, Chicago, Playboy and National Lampoon. Harris has a traveling exhibit, created by Sigma Xi and the New York Hall of Science, to appear in local museums.
Bibliography
What's So Funny About Science? (1977)
Chicken Soup and Other Medical Matters (1979)
All Ends Up: Cartoons from American Scientist (1980)
What's So Funny About Computers (1983)
Science Goes to the Dogs (1985)
You Want Proof? I'll Give You Proof! More Cartoons from Sidney Harris (1990)
Can't You Guys Read? Cartoons on Academia (1991)
Chalk Up Another One: The Best of Sidney Harris (1992)
From Personal Ads to Cloning Labs: More Science Cartoons from Sidney Harris (1992)
So Sue Me! Cartoons on the Law (1993)
Stress Test: Cartoons on Medicine (1994)
What's So Funny About Business? Yuppies, Bosses, and other Capitalists (1995)
There Goes the Neighborhood: Cartoons on the Environment (1996)
Einstein Atomized: More Science Cartoons (1996)
The Interactive Toaster: Cartoons on Business (1996)
Freudian Slips: Cartoons on Psychology (1997)
49 Dogs, 36 Cats and a Platypus: Animal Cartoons (2000)
Einstein Simplified: Cartoons on Science (2004)
101 Funny Things About Global Warming (2007)
Aside From The Cockroach, How Was Everything? Cartoons on the Dangers of Eating (2013)
References
External links
American cartoonists
Living people
1933 births
Brooklyn College alumni
The Magazine of Fantasy & Science Fiction people
The New Yorker cartoonists
The New Yorker people
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https://en.wikipedia.org/wiki/Pleckgate%20High%20School
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Pleckgate High School is a mixed, Ofsted rated Outstanding secondary school located in Blackburn, Lancashire, England.
Previously a community school and Mathematics and Computing College administered by Blackburn with Darwen Borough Council, in February 2016 Pleckgate High School converted to academy status. The school is now sponsored by The Education Partnership Trust, but continues to coordinate with Blackburn with Darwen Borough Council for admissions.
The current Headteacher of the Academy is Aishling McGinty, who took up post in September 2022.
Upon being inspected by Ofsted, in January 2019, the school received a judgement of 'Outstanding' in all categories.
References
External links
Secondary schools in Blackburn with Darwen
Schools in Blackburn
Academies in Blackburn with Darwen
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https://en.wikipedia.org/wiki/Daniel%20Z.%20Freedman
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Daniel Zissel Freedman (born 1939 in Hartford, Connecticut) is an American theoretical physicist. He is an Emeritus Professor of Physics and Applied Mathematics at the Massachusetts Institute of Technology (MIT), and is currently a visiting professor at Stanford University. He is mainly known for his work in supergravity. He is a member of the U. S. National Academy of Sciences.
Education
Daniel Freedman completed his undergraduate degree from Wesleyan University and completed his Ph.D. from the University of Wisconsin–Madison in 1964. In 1967–68, Freedman was a member of the School of Natural Sciences at the Institute for Advanced Study, and returned subsequently in 1973–74 and 1986–87. He was appointed Professor of Applied Mathematics at the Massachusetts Institute of Technology (MIT) in 1980 and joint Professor of Physics in 2001. Before joining MIT, he was a professor at Stony Brook University.
Supergravity
In 1976, Daniel Z. Freedman codiscovered (with Sergio Ferrara and Peter van Nieuwenhuizen) supergravity. Freedman and van Nieuwenhuizen were on the faculty of the Stony Brook University. Supergravity generalizes Einstein's theory of general relativity by incorporating the then-new idea of supersymmetry. In the following decades it had implications for physics beyond the Standard Model, for superstring theory and for mathematics. For his work on supergravity, Freedman, a former Sloan and twice Guggenheim fellow, received in 1993 the Dirac Medal and Prize, in 2006 the Dannie Heineman Prize for Mathematical Physics, in 2016 the Majorana Medal and in 2019 the Breakthrough Prize in Fundamental Physics, in each case together with his codiscoverers Sergio Ferrara and Peter van Nieuwenhuizen. Freedman also gave the 2002 Andrejewski Lectures in Mathematical Physics at the Max Planck Institute for Mathematics in the Sciences in Leipzig.
Research interests
Daniel Freedman is a professor at MIT. His research is in quantum field theory, quantum gravity, and superstring theory with an emphasis on the role of supersymmetry. His most recent area of concentration is the AdS/CFT correspondence in which results on the strong coupling limit of certain 4-dimensional gauge theories can be obtained from calculations in classical 5-dimensional supergravity.
In the academic year 1993/94 Freedman was a visiting scientist in the CERN Theory Division, Geneva, Switzerland.
References
External links
Oral history interview transcript with Daniel Freedman on 26 May 2021, American Institute of Physics, Niels Bohr Library & Archives
1939 births
Living people
21st-century American physicists
Massachusetts Institute of Technology School of Science faculty
Stony Brook University faculty
University of Wisconsin–Madison alumni
Wesleyan University alumni
Theoretical physicists
Institute for Advanced Study visiting scholars
People associated with CERN
MIT Center for Theoretical Physics faculty
Fellows of the American Physical Society
Members of the United States Natio
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https://en.wikipedia.org/wiki/Probability%20Sun
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Probability Sun is a 2001 science fiction novel by American writer Nancy Kress, a sequel to her 2000 publication Probability Moon. It was followed in 2002 by Probability Space, which won the John W. Campbell Memorial Award.
The novel concerns a military expedition to the planet World, where aliens live who have developed a strange form of telepathy or collective unconscious, "shared reality", which causes piercing "head-pain" whenever "Worlders" attempt to hold strongly differing opinions. However, the expedition concerns a crash-landed alien artifact in the planet's crust which has uncharted powers, and may be the key to humanity winning a war against the "Fallers", a genocidal alien race.
Setting
The Probability trilogy takes place in a galaxy that has been colonized by humans. This was made possible by the space tunnels, a network of FTL warp gates that were created by a now-lost progenitor race. Humanity is not united under a common government and political system; instead, the various governments in the Solar System and beyond have united as the "Solar Alliance Defense Network" in light of the war against the Fallers. The Terrans have also discovered a number of alien races, most of them vastly similar in body format, living conditions and even DNA, leading to the hypothesis that the aforementioned progenitor race seeded the galaxy with sentient life, which then evolved according to the conditions on each planet. Of the known alien races, humanity is the only one that has reached space.
Humanity's understanding of the space tunnels is very limited, but several peculiar traits have been discovered. For one: if Ship A enters Tunnel 1, exits Tunnel 2 and then turns around and enters Tunnel 2 again, it will emerge from Tunnel 1 again... Unless Ship B emerges from Tunnel 2 in the interim, at which point Ship A will instead emerge from wherever Ship B entered. (The single tunnel leading to World is #438, which gives an idea of how carefully passage through heavily used tunnels must be coordinated.) For two, objects can only enter the Tunnel if they are below a certain mass, about 100,000 tons; anything larger will actually fit into the aperture, but will collapse and explode. The threshold of what the tunnel can handle is determined by the object's Schwarzschild radius. Finally, nobody knows how the tunnels work. At all. Macro-level quantum entanglement has been proposed, but it is so far out of the realm of current physics that nobody believes it.
The space tunnels also lead to the discovery of the Fallers, an alien race who refused to establish communications and immediately launched a war, which they are winning. No Faller has been captured alive—they prefer to suicide or kamikaze—but forensic examination of corpses indicate they evolved separately from humans, instead of being seeded by the progenitors. Like humanity, they were not an interstellar race until the discovery of a space tunnel in their system, though they have
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https://en.wikipedia.org/wiki/Order%20dual%20%28functional%20analysis%29
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In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space is the set where denotes the set of all positive linear functionals on , where a linear function on is called positive if for all implies
The order dual of is denoted by .
Along with the related concept of the order bound dual, this space plays an important role in the theory of ordered topological vector spaces.
Canonical ordering
An element of the order dual of is called positive if implies
The positive elements of the order dual form a cone that induces an ordering on called the canonical ordering.
If is an ordered vector space whose positive cone is generating (that is, ) then the order dual with the canonical ordering is an ordered vector space.
The order dual is the span of the set of positive linear functionals on .
Properties
The order dual is contained in the order bound dual.
If the positive cone of an ordered vector space is generating and if holds for all positive and , then the order dual is equal to the order bound dual, which is an order complete vector lattice under its canonical ordering.
The order dual of a vector lattice is an order complete vector lattice.
The order dual of a vector lattice can be finite dimension (possibly even ) even if is infinite-dimensional.
Order bidual
Suppose that is an ordered vector space such that the canonical order on makes into an ordered vector space.
Then the order bidual is defined to be the order dual of and is denoted by .
If the positive cone of an ordered vector space is generating and if holds for all positive and , then is an order complete vector lattice and the evaluation map is order preserving.
In particular, if is a vector lattice then is an order complete vector lattice.
Minimal vector lattice
If is a vector lattice and if is a solid subspace of that separates points in , then the evaluation map defined by sending to the map given by , is a lattice isomorphism of onto a vector sublattice of .
However, the image of this map is in general not order complete even if is order complete.
Indeed, a regularly ordered, order complete vector lattice need not be mapped by the evaluation map onto a band in the order bidual.
An order complete, regularly ordered vector lattice whose canonical image in its order bidual is order complete is called minimal and is said to be of minimal type.
Examples
For any , the Banach lattice is order complete and of minimal type;
in particular, the norm topology on this space is the finest locally convex topology for which every order convergent filter converges.
Properties
Let be an order complete vector lattice of minimal type.
For any such that the following are equivalent:
is a weak order unit.
For every non-0 positive linear functional on ,
For each topology on such that is a locally convex vector lattice, is a quasi-interior point of its positive cone.
Related
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https://en.wikipedia.org/wiki/Dialling%20%28mathematics%29
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In somewhat archaic applied mathematics, dialling is the mathematics required to create a sundial face to determine solar time based on the position of the sun. Those skilled in the art were referred to as dialists or gnomonists, the latter derived from the word gnomon, which was a device that used a shadow as an indicator.
The mathematician William Oughtred published a book, Easy Method of Mathematical Dialling, around 1600. Samuel Walker (1716–1782) was a Yorkshire mathematician and diallist. In his later years, Thomas Jefferson was known to practice dialling as a mental exercise. Professor of astronomy at Gresham College (London, UK), Samuel Foster (d. 1652), developed reflex dialling, which describes a device of his own invention: a sundial capable of reflecting a spot of light onto the ceiling of a room.
Etymology
The word dial derives from the Latin term dialis (daily), and comes from the fact that a sundial throws a shadow related to the time of day. It was also used to describe the gear in a medieval clock which turned once per day.
Notes
References
History of mathematics
Timekeeping
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https://en.wikipedia.org/wiki/Pavel%20Har%C3%A7ik
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Pavel Harçik (born 5 April 1979) is a former Turkmenistani professional football goalkeeper. He is ethnically Russian.
Career statistics
Club
International
Statistics accurate as of match played 18 April 2009
References
External links
1979 births
Living people
FC Rubin Kazan players
Turkmenistan men's footballers
Turkmenistan expatriate men's footballers
Turkmenistan men's international footballers
Nisa Aşgabat players
FC Anzhi Makhachkala players
Turkmenistan expatriate sportspeople in Russia
Expatriate men's footballers in Russia
Russian Premier League players
Turkmenistan expatriate sportspeople in Azerbaijan
Expatriate men's footballers in Azerbaijan
FC AGMK players
Sportspeople from Dushanbe
Turkmenistan people of Russian descent
Footballers at the 2002 Asian Games
FC Kristall Smolensk players
Men's association football goalkeepers
FC Neftekhimik Nizhnekamsk players
Karvan FK players
Asian Games competitors for Turkmenistan
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https://en.wikipedia.org/wiki/Doob%27s%20martingale%20convergence%20theorems
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In mathematicsspecifically, in the theory of stochastic processesDoob's martingale convergence theorems are a collection of results on the limits of supermartingales, named after the American mathematician Joseph L. Doob. Informally, the martingale convergence theorem typically refers to the result that any supermartingale satisfying a certain boundedness condition must converge. One may think of supermartingales as the random variable analogues of non-increasing sequences; from this perspective, the martingale convergence theorem is a random variable analogue of the monotone convergence theorem, which states that any bounded monotone sequence converges. There are symmetric results for submartingales, which are analogous to non-decreasing sequences.
Statement for discrete-time martingales
A common formulation of the martingale convergence theorem for discrete-time martingales is the following. Let be a supermartingale. Suppose that the supermartingale is bounded in the sense that
where is the negative part of , defined by . Then the sequence converges almost surely to a random variable with finite expectation.
There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone convergence theorem that the sequence be bounded from below. The condition that the martingale is bounded is essential; for example, an unbiased random walk is a martingale but does not converge.
As intuition, there are two reasons why a sequence may fail to converge. It may go off to infinity, or it may oscillate. The boundedness condition prevents the former from happening. The latter is impossible by a "gambling" argument. Specifically, consider a stock market game in which at time , the stock has price . There is no strategy for buying and selling the stock over time, always holding a non-negative amount of stock, which has positive expected profit in this game. The reason is that at each time the expected change in stock price, given all past information, is at most zero (by definition of a supermartingale). But if the prices were to oscillate without converging, then there would be a strategy with positive expected profit: loosely, buy low and sell high. This argument can be made rigorous to prove the result.
Proof sketch
The proof is simplified by making the (stronger) assumption that the supermartingale is uniformly bounded; that is, there is a constant such that always holds. In the event that the sequence does not converge, then and differ. If also the sequence is bounded, then there are some real numbers and such that and the sequence crosses the interval infinitely often. That is, the sequence is eventually less than , and at a later time exceeds , and at an even later time is less than , and so forth ad infinitum. These periods where the sequence starts below and
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https://en.wikipedia.org/wiki/Point%20distribution%20model
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The point distribution model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes.
Background
The point distribution model concept has been developed by Cootes, Taylor et al. and became a standard in computer vision for the statistical study of shape and for segmentation of medical images where shape priors really help interpretation of noisy and low-contrasted pixels/voxels. The latter point leads to active shape models (ASM) and active appearance models (AAM).
Point distribution models rely on landmark points. A landmark is an annotating point posed by an anatomist onto a given locus for every shape instance across the training set population. For instance, the same landmark will designate the tip of the index finger in a training set of 2D hands outlines. Principal component analysis (PCA), for instance, is a relevant tool for studying correlations of movement between groups of landmarks among the training set population. Typically, it might detect that all the landmarks located along the same finger move exactly together across the training set examples showing different finger spacing for a flat-posed hands collection.
Details
First, a set of training images are manually landmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes. These landmarks are aligned using the generalized procrustes analysis, which minimizes the least squared error between the points.
aligned landmarks in two dimensions are given as
.
It's important to note that each landmark should represent the same anatomical location. For example, landmark #3, might represent the tip of the ring finger across all training images.
Now the shape outlines are reduced to sequences of landmarks, so that a given training shape is defined as the vector . Assuming the scattering is gaussian in this space, PCA is used to compute normalized eigenvectors and eigenvalues of the covariance matrix across all training shapes. The matrix of the top eigenvectors is given as , and each eigenvector describes a principal mode of variation along the set.
Finally, a linear combination of the eigenvectors is used to define a new shape , mathematically defined as:
where is defined as the mean shape across all training images, and is a vector of scaling values for each principal component. Therefore, by modifying the variable an infinite number of shapes can be defined. To ensure that the new shapes are all within the variation seen in the training set, it is common to only allow each element of to be within 3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue.
PDM's can be extended to any arbitrary number of dimensions, but are typically used in 2D image and 3D volume applications (where each landmark point is or ).
Discussion
An eigenvector,
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https://en.wikipedia.org/wiki/2gether%20%282gether%20album%29
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2gether is the debut album by 2gether, released in 2000. It includes the singles, "U + Me = Us (Calculus)", "Say It (Don't Spray It)", and "Before We Say Goodbye". It also contains "2Gether", which later became the theme song for the band's TV show. The name is a Wordplay of the word together. Most of the lyrics were written by the band's creators, Brian Gunn and Mark Gunn.
Track listing
Singles
"U + Me = Us (Calculus)" – released January 2000; did not chart.
"Say It (Don't Spray It)" – released April 2000; did not chart.
"Before We Say Goodbye" – released June 2000; did not chart.
Personnel
Evan Farmer – vocals, background vocals, co-writing on "You're My Baby Girl"
Noah Bastian – vocals, background vocals
Michael Cuccione – vocals, background vocals, co-writing on "Visualize"
Kevin Farley – vocals, background vocals
Alex Solowitz – vocals, background vocals
Nigel Dick – writing
Andrew Fromm – writing
Brian Gunn – writing
Mark Gunn – writing
Brian Kierulf – writing
Veit Renn – writing
Joshua M. Schwartz – writing
"Rub One Out" and "Breaking All The Rules" were performed by uncredited vocalists.
Charts
Album
References
2000 debut albums
2gether (band) albums
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https://en.wikipedia.org/wiki/Alexander%20matrix
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In mathematics, an Alexander matrix is a presentation matrix for the Alexander invariant of a knot. The determinant of an Alexander matrix is the Alexander polynomial for the knot.
References
External links
Knot theory
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https://en.wikipedia.org/wiki/Thomson%20cubic
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In geometry, the Thomson cubic of a triangle is the locus of centers of circumconics whose normals at the vertices are concurrent.
See also
Cubic plane curve – Thomson cubic
References
Viktor Vasilʹevich Prasolov: Essays on numbers and figures. AMS, 2000, ISBN 9780821819449, p. 73
External links
K002 (Thomson cubic) at Cubics in the Triangle Plane
Curves defined for a triangle
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https://en.wikipedia.org/wiki/6-sphere%20coordinates
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In mathematics, 6-sphere coordinates are a coordinate system for three-dimensional space obtained by inverting the 3D Cartesian coordinates across the unit 2-sphere . They are so named because the loci where one coordinate is constant form spheres tangent to the origin from one of six sides (depending on which coordinate is held constant and whether its value is positive or negative). They have nothing whatsoever to do with the 6-sphere, which is an object of considerable interest in its own right.
The three coordinates are
Since inversion is its own inverse, the equations for x, y, and z in terms of u, v, and w are similar:
This coordinate system is -separable for the 3-variable Laplace equation.
See also
Multiplicative inverse (for 1-dimensional version)
References
Moon, P. and Spencer, D. E. 6-sphere Coordinates. Fig. 4.07 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 122–123, 1988.
Six-Sphere Coordinates by Michael Schreiber, the Wolfram Demonstrations Project.
Three-dimensional coordinate systems
Inversive geometry
Orthogonal coordinate systems
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https://en.wikipedia.org/wiki/Osmar%20%28footballer%2C%20born%201980%29
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Osmar Aparecido de Azevedo or simply Osmar (born March 27, 1980 in Marília), is a Brazilian striker. He is also known in his native Brazil by the nickname Cambalhota ("Backflip").
Club statistics
Honours
Santo André
Brazilian Cup: 2004
Grêmio
Brazilian Série B: 2005
Palmeiras
Campeonato Paulista: 2008
External links
sambafoot
Guardian Stats Centre
zerozero.pt
palmeiras.globo.com
CBF
globoesporte
sopalmeiras
1980 births
Living people
Sportspeople from Marília
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Mexico
Campeonato Brasileiro Série A players
J1 League players
Liga MX players
São Paulo state football team players
Rio Branco Esporte Clube players
União São João Esporte Clube players
Esporte Clube Santo André players
Sociedade Esportiva Palmeiras players
Grêmio Foot-Ball Porto Alegrense players
Atlético Morelia players
Oita Trinita players
Fortaleza Esporte Clube players
Esporte Clube Vitória players
Guaratinguetá Futebol players
Men's association football forwards
Footballers from São Paulo (state)
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https://en.wikipedia.org/wiki/New%20Zealand%20men%27s%20national%20football%20team%20results%20%282000%E2%80%932019%29
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This page details the match results and statistics of the New Zealand men's national football team from 2000 until 2019.
Key
Key to matches
Att. = Match attendance
(H) = Home ground
(A) = Away ground
(N) = Neutral ground
Key to record by opponent
Pld = Games played
W = Games won
D = Games drawn
L = Games lost
GF = Goals for
GA = Goals against
A-International results
New Zealand's score is shown first in each case.
Notes
Streaks
Most wins in a row
5, 5 July 2002 – 14 July 2002
5, 17 October 2007 – 10 September 2008
5, 10 June 2012 – 16 October 2012
5, 12 November 2015 – 8 June 2016
Most matches without a loss
9, 10 June 2012 – 5 September 2013
Most draws in a row
4, 15 June 2010 – 9 October 2010
Most losses in a row
7, 15 July 1999 – 21 January 2000
6, 8 June 2003 – 29 May 2004
Most matches without a win
11, 5 June 2010 – 23 May 2012
11, 9 September 2013 – 7 September 2015
Results by opposition
Results by year
Cumulative table includes all results prior to 2000.
See also
New Zealand national football team
New Zealand at the FIFA World Cup
New Zealand at the FIFA Confederations Cup
New Zealand at the OFC Nations Cup
References
2000-19
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https://en.wikipedia.org/wiki/Solomon%20Islands%20national%20football%20team%20results%20%282000%E2%80%93present%29
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This page details the match results and statistics of the Solomon Islands national football team from 2000 to present.
Key
Key to matches
Att.=Match attendance
(H)=Home ground
(A)=Away ground
(N)=Neutral ground
Key to record by opponent
Pld=Games played
W=Games won
D=Games drawn
L=Games lost
GF=Goals for
GA=Goals against
Results
Solomon Islands' score is shown first in each case.
Notes
Record by opponent
References
Solomon Islands national football team results
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https://en.wikipedia.org/wiki/Papua%20New%20Guinea%20national%20football%20team%20results
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This page details the match results and statistics of the Papua New Guinea national football team.
Key
Key to matches
Att. = Match attendance
(H) = Home ground
(A) = Away ground
(N) = Neutral ground
Key to record by opponent
Pld = Games played
W = Games won
D = Games drawn
L = Games lost
GF = Goals for
GA = Goals against
Results
Papua New Guinea's score is shown first in each case.
Notes
Record by opponent
References
Papua New Guinea national football team results
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https://en.wikipedia.org/wiki/Lumer%E2%80%93Phillips%20theorem
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In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.
Statement of the theorem
Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a contraction semigroup if and only if
D(A) is dense in X,
A is dissipative, and
A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.
An operator satisfying the last two conditions is called maximally dissipative.
Variants of the theorem
Reflexive spaces
Let A be a linear operator defined on a linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if
A is dissipative, and
A − λ0I is surjective for some λ0> 0, where I denotes the identity operator.
Note that the conditions that D(A) is dense and that A is closed are dropped in comparison to the non-reflexive case. This is because in the reflexive case they follow from the other two conditions.
Dissipativity of the adjoint
Let A be a linear operator defined on a dense linear subspace D(A) of the reflexive Banach space X. Then A generates a contraction semigroup if and only if
A is closed and both A and its adjoint operator A∗ are dissipative.
In case that X is not reflexive, then this condition for A to generate a contraction semigroup is still sufficient, but not necessary.
Quasicontraction semigroups
Let A be a linear operator defined on a linear subspace D(A) of the Banach space X. Then A generates a quasi contraction semigroup if and only if
D(A) is dense in X,
A is closed,
A is quasidissipative, i.e. there exists a ω ≥ 0 such that A − ωI is dissipative, and
A − λ0I is surjective for some λ0 > ω, where I denotes the identity operator.
Examples
Consider H = L2([0, 1]; R) with its usual inner product, and let Au = u′ with domain D(A) equal to those functions u in the Sobolev space H1([0, 1]; R) with u(1) = 0. D(A) is dense. Moreover, for every u in D(A),
so that A is dissipative. The ordinary differential equation u' − λu = f, u(1) = 0 has a unique solution u in H1([0, 1]; R) for any f in L2([0, 1]; R), namely
so that the surjectivity condition is satisfied. Hence, by the reflexive version of the Lumer–Phillips theorem A generates a contraction semigroup.
There are many more examples where a direct application of the Lumer–Phillips theorem gives the desired result.
In conjunction with translation, scaling and perturbation theory the Lumer–Phillips theorem is the main tool for showing that certain operators generate strongly continuous semigroups. The following is an example in point.
A normal operator (an operator that commutes with its adjoint) on a Hilbert space generates a strongly continuous semigroup if and only if its spectrum is bounded from above.
Notes
References
Semigroup theory
Theor
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https://en.wikipedia.org/wiki/Dissipative%20operator
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In mathematics, a dissipative operator is a linear operator A defined on a linear subspace D(A) of Banach space X, taking values in X such that for all λ > 0 and all x ∈ D(A)
A couple of equivalent definitions are given below. A dissipative operator is called maximally dissipative if it is dissipative and for all λ > 0 the operator λI − A is surjective, meaning that the range when applied to the domain D is the whole of the space X.
An operator that obeys a similar condition but with a plus sign instead of a minus sign (that is, the negation of a dissipative operator) is called an accretive operator.
The main importance of dissipative operators is their appearance in the Lumer–Phillips theorem which characterizes maximally dissipative operators as the generators of contraction semigroups.
Properties
A dissipative operator has the following properties:
From the inequality given above, we see that for any x in the domain of A, if ‖x‖ ≠ 0 then so the kernel of λI − A is just the zero vector and λI − A is therefore injective and has an inverse for all λ > 0. (If we have the strict inequality for all non-null x in the domain, then, by the triangle inequality, which implies that A itself has an inverse.) We may then state that
for all z in the range of λI − A. This is the same inequality as that given at the beginning of this article, with (We could equally well write these as which must hold for any positive κ.)
λI − A is surjective for some λ > 0 if and only if it is surjective for all λ > 0. (This is the aforementioned maximally dissipative case.) In that case one has (0, ∞) ⊂ ρ(A) (the resolvent set of A).
A is a closed operator if and only if the range of λI - A is closed for some (equivalently: for all) λ > 0.
Equivalent characterizations
Define the duality set of x ∈ X, a subset of the dual space X of X, by
By the Hahn–Banach theorem this set is nonempty. In the Hilbert space case (using the canonical duality between a Hilbert space and its dual) it consists of the single element x. More generally, if X is a Banach space with a strictly convex dual, then J(x) consists of a single element.
Using this notation, A is dissipative if and only if for all x ∈ D(A) there exists a x' ∈ J(x) such that
In the case of Hilbert spaces, this becomes for all x in D(A). Since this is non-positive, we have
Since I−A has an inverse, this implies that is a contraction, and more generally, is a contraction for any positive λ. The utility of this formulation is that if this operator is a contraction for some positive λ then A is dissipative. It is not necessary to show that it is a contraction for all positive λ (though this is true), in contrast to (λI−A)−1 which must be proved to be a contraction for all positive values of λ.
Examples
For a simple finite-dimensional example, consider n-dimensional Euclidean space Rn with its usual dot product. If A denotes the negative of the identity operator, defined on all of Rn, then
so A is a dissipa
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https://en.wikipedia.org/wiki/Graph%20of%20groups
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In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups.
There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabilizer subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre.
Definition
A graph of groups over a graph is an assignment to each vertex of of a group and to each edge of of a group as well as monomorphisms and mapping into the groups assigned to the vertices at its ends.
Fundamental group
Let be a spanning tree for and define the fundamental group to be the group generated by the vertex groups and elements for each edge of with the following relations:
if is the edge with the reverse orientation.
for all in .
if is an edge in .
This definition is independent of the choice of .
The benefit in defining the fundamental groupoid of a graph of groups, as shown by , is that it is defined independently of base point or tree. Also there is proved there a nice normal form for the elements of the fundamental groupoid. This includes normal form theorems for a free product with amalgamation and for an HNN extension .
Structure theorem
Let be the fundamental group corresponding to the spanning tree . For every vertex and edge , and can be identified with their images in . It is possible to define a graph with vertices and edges the disjoint union of all coset spaces and respectively. This graph is a tree, called the universal covering tree, on which acts. It admits the graph as fundamental domain. The graph of groups given by the stabilizer subgroups on the fundamental domain corresponds to the original graph of groups.
Examples
A graph of groups on a graph with one edge and two vertices corresponds to a free product with amalgamation.
A graph of groups on a single vertex with a loop corresponds to an HNN extension.
Generalisations
The simplest possible generalisation of a graph of groups is a 2-dimensional complex of groups. These are modeled on orbifolds arising from cocompact properly discontinuous actions of discrete groups on simplicial complexes that have the structure of CAT(0) spaces. The quotient of the simplicial complex has finite stabilizer groups attached to vertices, edges and triangles together with monomorphisms for every inclusion of simplices. A complex of groups is said to be developable if it arises as the quotient of a CAT(0) simplicial complex. Developability is a non-positive curvature condition on the complex of groups: it can be verified locally by checking that all circuits occurring in the links of vertices have length at least six. Such complexes of group
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https://en.wikipedia.org/wiki/Base%20runs
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Base runs (BsR) is a baseball statistic invented by sabermetrician David Smyth to estimate the number of runs a team "should have" scored given their component offensive statistics, as well as the number of runs a hitter or pitcher creates or allows. It measures essentially the same thing as Bill James' runs created, but as sabermetrician Tom M. Tango points out, base runs models the reality of the run-scoring process "significantly better than any other run estimator".
Purpose and formula
Base runs has multiple variations, but all take the form
Smyth detailed the following forms of the statistic:
The simplest, uses only the most common batting statistics
A = H + BB - HR
B = (1.4 * TB - .6 * H - 3 * HR + .1 * BB) * 1.02
C = AB - H
D = HR
An offshoot includes significantly more batting statistics
A = H + BB + HBP - HR - .5 * IBB
B = (1.4 * TB - .6 * H - 3 * HR + .1 * (BB + HBP - IBB) + .9 * (SB - CS - GIDP)) * 1.1
C = AB - H + CS + GIDP
D = HR
A third formula uses pitching statistics
A = H + BB - HR
B = (1.4 * (1.12 * H + 4 * HR) - .6 * H - 3 * HR + .1 * BB) * 1.1
C = 3 * IP
D = HR
Other sabermetricians have developed their own formulas using Smyth's general form, mainly by tinkering with the B factor.
Because the base runs statistic attempts to model the team run scoring process, a formula cannot be applied directly to an individual player's statistics. Doing this would result in a run estimate for an entire team that puts out the individual's statistics. A workaround for this issue is to find the team's base runs with the player in the lineup and the team's base runs with a replacement level player in the lineup. The difference between these values approximates the individual's base runs statistic.
Advantages of base runs
Base runs was primarily designed to provide an accurate model of the run scoring process at the Major League Baseball level, and it accomplishes that goal: in recent seasons, base runs has the lowest RMSE of any of the major run estimation methods. In addition, its accuracy holds up in even the most extreme of circumstances and leagues. For instance, when a solo home run is hit, base runs will correctly predict one run having been scored by the batting team. By contrast, when runs created assesses a solo HR, it predicts four runs to be scored; likewise, most linear weights-based formulas will predict a number close to 1.4 runs having been scored on a solo HR. This is because each of these models were developed to fit the sample of a 162-game MLB season; they work well when applied to that sample, of course, but are inaccurate when taken out of the environment for which they were designed. Base runs, on the other hand, can be applied to any sample at any level of baseball (provided it is possible to calculate the B multiplier), because it models the way the game of baseball operates, and not just for a 162-game season at the highest professional level. This means that base runs can be applied to high school o
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https://en.wikipedia.org/wiki/Calder%C3%B3n%E2%80%93Zygmund%20lemma
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In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund.
Given an integrable function , where denotes Euclidean space and denotes the complex numbers, the lemma gives a precise way of partitioning into two sets: one where is essentially small; the other a countable collection of cubes where is essentially large, but where some control of the function is retained.
This leads to the associated Calderón–Zygmund decomposition of , wherein is written as the sum of "good" and "bad" functions, using the above sets.
Covering lemma
Let be integrable and be a positive constant. Then there exists an open set such that:
(1) is a disjoint union of open cubes, , such that for each ,
(2) almost everywhere in the complement of .
Here, denotes the measure of the set .
Calderón–Zygmund decomposition
Given as above, we may write as the sum of a "good" function and a "bad" function , . To do this, we define
and let . Consequently we have that
for each cube .
The function is thus supported on a collection of cubes where is allowed to be "large", but has the beneficial property that its average value is zero on each of these cubes. Meanwhile, for almost every in , and on each cube in , is equal to the average value of over that cube, which by the covering chosen is not more than .
See also
Singular integral operators of convolution type, for a proof and application of the lemma in one dimension.
Rising sun lemma
References
Theorems in Fourier analysis
Lemmas in analysis
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https://en.wikipedia.org/wiki/Michael%20Barr%20%28mathematician%29
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Michael Barr (born January 22, 1937) is an American mathematician who is the Peter Redpath Emeritus Professor of Pure Mathematics at McGill University.
Early life and education
He was born in Philadelphia, Pennsylvania, and graduated from the 202nd class of Central High School in June 1954. He graduated from the University of Pennsylvania in February 1959 and received a PhD from the same school in June 1962.
Career
Barr taught at Columbia University and the University of Illinois before coming to McGill in 1968.
His earlier work was in homological algebra, but his principal research area for a number of years has been category theory. He is well known to theoretical computer scientists for his book Category Theory for Computing Science with Charles Wells, as well as for the development of *-autonomous categories and Chu spaces which have found various applications in computer science. His monograph *-autonomous categories, and his books Toposes, Triples, and Theories, also coauthored with Wells, and Acyclic Models, are aimed at more specialized audiences.
He is on the editorial boards of Mathematical Structures in Computer Science and the electronic journal Homology, Homotopy and Applications, and is editor of the electronic journal Theory and Applications of Categories.
References
External links
Toposes, Triples and Theories, updated edition of text published in 1983.
Category Theory for Computing Science updated edition of text published in 1999.
http://www.tac.mta.ca/tac (Theory and Applications of Categories)
https://web.archive.org/web/20080704125156/http://www.math.rutgers.edu/hha/geninfo.html (Homology, Homotopy and Applications)
1937 births
Living people
Mathematicians from Philadelphia
Central High School (Philadelphia) alumni
University of Pennsylvania School of Arts and Sciences alumni
Academic staff of McGill University
Canadian mathematicians
Canadian computer scientists
Anglophone Quebec people
Category theorists
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https://en.wikipedia.org/wiki/Roads%20and%20motorways%20in%20Cyprus
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Since the arrival of the first motor cars on the island in 1907, Cyprus has developed a modern road network. According to 2002 statistics, the road network in the Republic of Cyprus-administered areas of Cyprus consists of about 7,206 km of paved and 4,387 km of unpaved roads. Although the first motorway in Cyprus, A1, was completed as recently as October 1985, the country already has the most motorway km per capita (36.8 km /100,000 inhabitants) among all European Union members.
There are no toll paying roads in Cyprus to date.
Maintenance
The Public Works Department of the Ministry of Communications and Works is responsible for the maintenance, improvement and construction of motorways, the majority of rural and interurban road network and the main urban roads. The Municipalities are responsible for the secondary and local urban roads; the District Administration Authorities are responsible for the paved and unpaved district (tertiary) roads and village roads. The Forestry Department is responsible for most unpaved roads in forest areas, this is in order to accommodate the administration and protection of forests.
The Turkish invasion of 1974 radically changed the program of road development and created new priorities in order to cover the augmented needs in the government controlled areas, where 80% of the Cyprus population and the greatest portion of development had concentrated.
Under these circumstances New Road Development Schemes were promoted, which were partially financed by foreign Financing Organizations. Under these development projects new 4 lane motorways were constructed and more are on their way as follows:
Cyprus motorways list
The highway network is continuously developed. The first section of the A9 Nicosia - Astromeritis Motorway between Kokkinotrimithia and Akaki has been completed, whereas the rest is under study. Also the upgrading of the Limassol Junctions and the A1 Nicosia - Limassol Motorway to a 6 lane road between the Strovolos Junction and Alampra Interchange are completed.
The following are under design:
A7 Paphos - Polis Motorway is promoted through the D.B.F.O. method (Design, Build, Finance, Operate).
Preliminary and feasibility studies are conducted for the:
Nicosia Ring road
A8 Limassol - Saittas Motorway
Astromeritis - Evrychou Motorway
Nicosia - Klirou Motorway
Road network categories and numbering
Roads and Motorways in Cyprus can be classified into 5 main categories:
Motorways, 2 lanes per direction, free of any at-grade intersections. They are the most important road network on the island, and the letter "A" is used on their official numbering system. Motorways usually run parallel to the same-number "B class" intercity roads that they replaced and sometimes these roads are even transformed to Motorways (e.g. A3 Motorway and B3 road). While there is no formal announcement about the numbering of new motorways under construction and under planning, it's anticipated that they will have
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https://en.wikipedia.org/wiki/Office%20landscape
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Office landscape () was an early (1950s) movement in open plan office space planning that typically used irregular geometry and organic circulation patterns.
History
The general European mentality towards innovative forms of office design in the 1940 and 1950s was that of caution and hesitation following the devastating effects left from WWI and WWII. Before the 1950s, the majority of offices in Europe operated with closed-door offices and scaled-down versions of the massive bullpen offices that were found in skyscrapers across the U.S. But the postwar environment proved fertile to economic growth from massive amounts of reconstruction, and one country that had a particularly fast rate of growth was Germany. The quick emergence in manufacturing, paired with a new mentality of wanting to pave over their brutal past, left Germany open to new thinking. This openness led to the interjection of a concept that would quickly populate areas all over Europe and North America.
In 1958, the Quickborner consulting group was established by two brothers, Wolfgang and Eberhard Schnelle, who had previously been working as assistants in their father’s furniture studio. Upon founding Quickborner outside of Hamburg as a space planning firm, the two brothers soon developed an interest in office space. They saw the current status quo, which used versions of scientific management and consisted of uninspired rows of desks and a strict office hierarchy as an opportunity for change. They wanted to create a system where the individual is the focus, and rebel against the strict grid of corridors and desks with something organic and natural. Their approach was called Bürolandschaft, a German term that translates to “office landscape”.
Social theory
The post-World War II social-democratic environment in many Northern European countries engendered an egalitarian management approach. Office landscape encouraged all levels of staff to sit together in one open floor to create a non-hierarchical environment that increased communication and collaboration.
Typical designs and components
Typical designs used contemporary but conventional furniture which was available at the time. Standard desks and chairs were used, with lateral file cabinets, curved screens, and large potted plants used as visual barriers and space definers. Floor plans frequently used irregular geometry and organic circulation patterns to enhance the egalitarian nature of the plan. Many designs used slightly lower than normal occupancy density to mitigate the acoustical problems inherent in open designs.
The seemingly random cluster of desks was done very intentionally and based on work paths and roles within the company. This flexibly made Bürolandschaft a very exciting option for companies, since it wasn’t “one size fits all”. Their approach to office design allowed for different spaces to be dealt with differently. The consistencies within this approach included an open plan, with no door to be closed
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https://en.wikipedia.org/wiki/Demographics%20of%20Lima
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The demographics of Lima can be reconstructed through the population censuses carried out throughout its history. The government statistics department estimates that a third of Peru's population lives in Lima.
Population by year
Ethnic groups
Mestizos: 47%
European: 40%
Asian: 8%
Amerindian: 2%
Afro-Peruvian: 3%
Evolution of the Lima Metropolitan Area
References
Lima
Lima
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https://en.wikipedia.org/wiki/Samuel%20McLaren
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Professor Samuel Bruce McLaren (16 August 1876 – 13 August 1916) was an Australian mathematician and mathematical physicist. Joint winner of the Adams Prize in 1913 and Professor of Mathematics, University College, Reading from 1913 until his death during the Battle of the Somme.
Early life
McLaren was born in Yedo, near Tokyo, Japan, elder son of Rev. Samuel Gilfillan McLaren M.A. a Scottish missionary and later professor of sacred history and biblical literature at the Presbyterian Union Theological Seminary, and Marjory Millar McLaren née Bruce. The eldest child his siblings included Mary, Charles McLaren (psychiatrist) (later a missionary to Korea), and Marjory. In 1886, the family moved to Australia, where in 1889 his father became principal of Presbyterian Ladies' College, Melbourne. Samuel McLaren was educated at Brighton Grammar School and Scotch College, Melbourne, where he was dux in mathematics in 1893. He gained a scholarship at Ormond College, University of Melbourne, and qualified for the Bachelor of Arts degree at the end of 1896 with first class final honours, and the final honours and Wyselaskie scholarships in mathematics. He also shared the Dixon scholarship in natural philosophy. One of his teachers at the University stated in 1903 that McLaren was by far the ablest student he had met during his twelve years' tenure of office, and one whose ability should be sufficient to place him in a very conspicuous position as an original thinker.
Study in England
Moving to England in 1897, McLaren attended Trinity College, Cambridge and was elected into a major scholarship in 1899, and was third wrangler in the same year. Taking part 2 of the mathematical tripos in his third year, he was placed in the second division of the first class. He was awarded an Isaac Newton studentship in astronomy and physical optics in 1901, and graduated M.A. in 1905. Not absorbed by mathematics alone he was interested in philosophy, literature and art, and played football tennis and boxed.
Mathematical career
McLaren was lecturer in mathematics at University College, Bristol 1904–06. Then from 1906 until 1913 obtained a similar position at the University of Birmingham. Between 1911 and 1913 he wrote some important papers on radiation which were published in the Philosophical Magazine, and he presented some of the more fundamental parts of his work to the mathematical congress at Cambridge in 1912. John William Nicholson, professor of mathematics in the University of London, writing in 1918 said McLaren "undoubtedly anticipated Einstein and Abraham in their suggestion of a variable velocity of light, with the consequent expressions for the energy and momentum of the gravitational field". In 1913 he was made professor of mathematics at University College, Reading where he took much interest in the development of the young university. In 1913 he shared the, at the time, biennial Adams Prize of the University of Cambridge with Nicholson.
Late life
In 1914
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https://en.wikipedia.org/wiki/UEFA%20club%20competition%20records%20and%20statistics
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Clubs
UEFA club competition winners
Real Madrid hold the record for the most overall titles (24) while Milan has the most UEFA Super Cup wins (5), a record shared with Barcelona and Real Madrid. The Madrid club have a record 14 titles achieved in the UEFA Champions League and its predecessor. Barcelona have a record four titles in the Cup Winners' Cup while Sevilla have a record of seven UEFA Cup and Europa League titles. Roma and West Ham United have each won one UEFA Europa Conference League title. Finally, German clubs Hamburger SV, Schalke 04, and VfB Stuttgart, as well as Spanish club Villarreal, are the record holders by titles won in the UEFA Intertoto Cup (two each).
Ranking main European club competitions' winning club sides by winning percentage
This is a ranking of all club sides which have won one of the three main European competitions, past or present.
Bayern Munich are the only team to finish a continental competition with a 100% winning record, achieving that milestone in 2020 as part of a modified tournament structure with a final eight in a neutral venue held in a single elimination match due to the COVID-19 pandemic in Europe.
Top 15 club sides
Qualifying and preliminary round matches are not included, neither are play-off matches; results of penalty shoot-outs are considered the score which preceded them (including extra time).
Table key
List of teams to have won the three main European club competitions
To date, five clubs have won all three main pre-1999 UEFA club competitions, the "European Treble" of European Cup/UEFA Champions League, European/UEFA Cup Winners' Cup, and UEFA Cup/UEFA Europa League.
Although the Cup Winners' Cup no longer exists, 27 of its former winners could still add wins in the other two competitions to achieve this UEFA treble. Ten of those teams are just one trophy away from the feat, including Barcelona and Milan who have both won multiple Champions League and Cup Winners' Cup titles and are one Europa League trophy away from achieving the UEFA treble. Other clubs needing the Europa League title to achieve the treble are Hamburg, Borussia Dortmund and Manchester City, having previously won the European Cup and the Cup Winners' Cup once each. The remaining five clubs that need to win the Champions League: Atlético Madrid, Tottenham Hotspur, Anderlecht, Valencia and Parma.
Upon the commencement of the UEFA Europa Conference League in the 2021–22 season, there is a chance for the 32 former winners of the Cup Winners' Cup to win that competition. Any other existing clubs can also win a modern UEFA treble (counting only the Champions, Europa and Europa Conference League titles) in the future.
Only the first win is shown for any club with multiple wins of the same competition.
Juventus received The UEFA Plaque from the confederation in 1988, in recognition of being the first side in European football history to win all three major UEFA club competitions, and the only one to reach it with in
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https://en.wikipedia.org/wiki/Isotopy%20of%20loops
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In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop.
Isotopy for loops and quasigroups was introduced by , based on his slightly earlier definition of isotopy for algebras, which was in turn inspired by work of Steenrod.
Isotopy of quasigroups
Each quasigroup is isotopic to a loop.
Let and be quasigroups. A quasigroup homotopy from Q to P is a triple of maps from Q to P such that
for all x, y in Q. A quasigroup homomorphism is just a homotopy for which the three maps are equal.
An isotopy is a homotopy for which each of the three maps is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy is given by a permutation of rows α, a permutation of columns β, and a permutation on the underlying element set γ.
An autotopy is an isotopy from a quasigroup to itself. The set of all autotopies of a quasigroup form a group with the automorphism group as a subgroup.
A principal isotopy is an isotopy for which γ is the identity map on Q. In this case the underlying sets of the quasigroups must be the same but the multiplications may differ.
Isotopy of loops
Let and be loops and let be an isotopy. Then it is the product of the principal isotopy from and and the isomorphism between and . Indeed, put , and define the operation by .
Let and be loops and let e be the neutral element of . Let a principal isotopy from to . Then and where and .
A loop L is a G-loop if it is isomorphic to all its loop isotopes.
Pseudo-automorphisms of loops
Let L be a loop and c an element of L. A bijection α of L is called a right pseudo-automorphism of L with companion element c if for all x, y the identity
holds. One defines left pseudo-automorphisms analogously.
Universal properties
We say that a loop property P is universal if it is isotopy invariant, that is, P holds for a loop L if and only if P holds for all loop isotopes of L. Clearly, it is enough to check if P holds for all principal isotopes of L.
For example, since the isotopes of a commutative loop need not be commutative, commutativity is not universal. However, associativity and being an abelian group are universal properties. In fact, every group is a G-loop.
The geometric interpretation of isotopy
Given a loop L, one can define an incidence geometric structure called a 3-net. Conversely, after fixing an origin and an order of the line classes, a 3-net gives rise to a loop. Choosing a different origin or exchanging the line classes may result in nonisomorphic coordinate loops. However, the coordinate loops are always isotopic. In other words, two loops are isotopic if and only if they are equivalent from geometric point of view.
The dictionary between algebraic and geometric concepts is as follows
The group of autotopism of the loop corresponds to the group direction preserving collineations of the 3-net.
Pseudo-automorphisms
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https://en.wikipedia.org/wiki/F%C3%A1tima%20Choi
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Fátima Choi Mei Lei (born 1958) MSc, BSc, was a Commissioner of Audit in Macau.
Born in Macau, Choi obtained a Master of Science degree in statistics and Bachelor of Science degree in mathematics from the University of Essex.
She was an assistant researcher in Hong Kong Polytechnic University and Chinese University of Hong Kong from 1985 to 1986. She joined the statistics and census department of the Macau government. Choi held other positions in the local government:
Senior Technician
Department Chief of Social Affairs and Accountants 1991–1995
Deputy Director 1995–1997
Director 1997–1999
Choi was the first Chinese official at the director's level after localization in Macau prior to the handover.
References
1958 births
Living people
Macau women in politics
Macau people
Alumni of the University of Essex
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https://en.wikipedia.org/wiki/Quantum%20stirring%2C%20ratchets%2C%20and%20pumping
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A pump is an alternating current-driven device that generates a direct current (DC). In the simplest configuration a pump has two leads connected to two reservoirs. In such open geometry, the pump takes particles from one reservoir and emits them into the other. Accordingly, a current is produced even if the reservoirs have the same temperature and chemical potential.
Stirring is the operation of inducing a circulating current with a non-vanishing DC component in a closed system. The simplest geometry is obtained by integrating a pump in a closed circuit. More generally one can consider any type of stirring mechanism such as moving a spoon in a cup of coffee.
Main observations
Pumping and stirring effects in quantum physics have counterparts in purely classical stochastic and dissipative processes. The studies of quantum pumping and of quantum stirring emphasize the role of quantum interference in the analysis of the induced current. A major objective is to calculate the amount of transported particles per a driving cycle. There are circumstances in which is an integer number due to the topology of parameter space. More generally is affected by inter-particle interactions, disorder, chaos, noise and dissipation.
Electric stirring explicitly breaks time-reversal symmetry. This property can be used to induce spin polarization in conventional semiconductors by purely electric means. Strictly speaking, stirring is a non-linear effect, because in linear response theory (LRT) an AC driving induces an AC current with the same frequency. Still an adaptation of the LRT Kubo formalism allows the analysis of stirring. The quantum pumping problem (where we have an open geometry) can be regarded as a special limit of the quantum stirring problem (where we have a closed geometry). Optionally the latter can be analyzed within the framework of scattering theory. Pumping and Stirring devices are close relatives of ratchet systems. The latter are defined in this context as AC driven spatially periodic arrays, where DC current is induced.
It is possible to induce a DC current by applying a bias, or if the particles are charged then by applying an electro-motive-force. In contrast to that a quantum pumping mechanism produces a DC current in response to a cyclic deformation of the confining potential. In order to have a DC current from an AC driving, time reversal symmetry (TRS) should be broken. In the absence of magnetic field and dissipation it is the driving itself that can break TRS. Accordingly, an adiabatic pump operation is based on varying more than one parameter, while for non-adiabatic pumps
modulation of a single parameter may suffice for DC current generation. The best known example is the peristaltic mechanism that combines a cyclic squeezing operation with on/off switching of entrance/exit valves.
Adiabatic quantum pumping is closely related to a class of current-driven nanomotors named Adiabatic quantum motor. While in a quantum pump, the
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https://en.wikipedia.org/wiki/Analytic%20Fredholm%20theorem
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In mathematics, the analytic Fredholm theorem is a result concerning the existence of bounded inverses for a family of bounded linear operators on a Hilbert space. It is the basis of two classical and important theorems, the Fredholm alternative and the Hilbert–Schmidt theorem. The result is named after the Swedish mathematician Erik Ivar Fredholm.
Statement of the theorem
Let be a domain (an open and connected set). Let be a real or complex Hilbert space and let Lin(H) denote the space of bounded linear operators from H into itself; let I denote the identity operator. Let be a mapping such that
B is analytic on G in the sense that the limit exists for all ; and
the operator B(λ) is a compact operator for each .
Then either
does not exist for any ; or
exists for every , where S is a discrete subset of G (i.e., S has no limit points in G). In this case, the function taking λ to is analytic on and, if , then the equation has a finite-dimensional family of solutions.
References
(Theorem 8.92)
Fredholm theory
Theorems in functional analysis
Theorems in complex analysis
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https://en.wikipedia.org/wiki/Limits%20of%20integration
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In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral
of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit. The region that is bounded can be seen as the area inside and .
For example, the function is defined on the interval
with the limits of integration being and .
Integration by Substitution (U-Substitution)
In Integration by substitution, the limits of integration will change due to the new function being integrated. With the function that is being derived, and are solved for . In general,
where and . Thus, and will be solved in terms of ; the lower bound is and the upper bound is .
For example,
where and . Thus, and . Hence, the new limits of integration are and .
The same applies for other substitutions.
Improper integrals
Limits of integration can also be defined for improper integrals, with the limits of integration of both
and
again being a and b. For an improper integral
or
the limits of integration are a and ∞, or −∞ and b, respectively.
Definite Integrals
If , then
See also
Integral
Riemann integration
Definite integral
References
Integral calculus
Real analysis
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https://en.wikipedia.org/wiki/Nemytskii%20operator
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In mathematics, Nemytskii operators are a class of nonlinear operators on Lp spaces with good continuity and boundedness properties. They take their name from the mathematician Viktor Vladimirovich Nemytskii.
Definition
Let Ω be a domain (an open and connected set) in n-dimensional Euclidean space. A function f : Ω × Rm → R is said to satisfy the Carathéodory conditions if
f(x, u) is a continuous function of u for almost all x ∈ Ω;
f(x, u) is a measurable function of x for all u ∈ Rm.
Given a function f satisfying the Carathéodory conditions and a function u : Ω → Rm, define a new function F(u) : Ω → R by
The function F is called a Nemytskii operator.
Boundedness theorem
Let Ω be a domain, let 1 < p < +∞ and let g ∈ Lq(Ω; R), with
Suppose that f satisfies the Carathéodory conditions and that, for some constant C and all x and u,
Then the Nemytskii operator F as defined above is a bounded and continuous map from Lp(Ω; Rm) into Lq(Ω; R).
References
(Section 10.3.4)
Operator theory
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https://en.wikipedia.org/wiki/10-simplex
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In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos−1(1/10), or approximately 84.26°.
It can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:
More simply, the vertices of the 10-simplex can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 11-orthoplex.
Images
Related polytopes
The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).
References
Coxeter, H.S.M.:
(Paper 22)
(Paper 23)
(Paper 24)
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
10-polytopes
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https://en.wikipedia.org/wiki/List%20of%20FC%20Steaua%20Bucure%C8%99ti%20records%20and%20statistics
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The FC Steaua București football club has played 74 seasons in Liga I, which it has won 26 times. It has also won Cupa României 22 times, Supercupa României 6 times and Cupa Ligii twice – all competition records. In UEFA competitions the club has won the European Cup and European Super Cup, both in 1986. It has also reached the European Cup final in 1989, the final of the Intercontinental Cup, quarter-finals of the European Cup Winners' Cup, and the semi-finals of the UEFA Cup. Its players have won numerous awards and many of them have represented Romania in international competitions.
Overall seasons table in Liga I
Steaua in European and International competitions
Honours
Steaua won their first trophy in 1948 when they lifted the Cupa Romaniei. The club won the Romanian Championship a record 14 times during the 40-year span of the tournament.
They are the most successful football club in Romanian, having won a total of 77 domestic titles: 26 Divizia A, a record 24 Cupa Romaniei, a record 14 Supercupa Romaniei, a record two Cupa Ligii
The club is also one of the most successful clubs in international club football, having won 2 official trophies in total, 1 of which are UEFA competitions . Ghencea club has won one UEFA Champions League title, a romanian record of one UEFA Super Cup title.
Domestic
Leagues
Liga I / Divizia A
Winners (26) – Record: 1951, 1952, 1953, 1956, 1959–60, 1960–61, 1967–68, 1975–76, 1977–78, 1984–85, 1985–86, 1986–87, 1987–88, 1988–89, 1992–93, 1993–94, 1994–95, 1995–96, 1996–97, 1997–98, 2000–01, 2004–05, 2005–06, 2012–13, 2013–14, 2014–15
Runners-up (20): 1954, 1957–58, 1962–63, 1976–77, 1979–80, 1983–84, 1989–90, 1990–91, 1991–92, 2002–03, 2003–04, 2006–07, 2007–08, 2015–16, 2016–17, 2017–18, 2018–19, 2020–21, 2021–22, 2022–23
Cups
Cupa României
Winners (24) – Record: 1948–49, 1950, 1951, 1952, 1955, 1961–62, 1965–66, 1966–67, 1968–69, 1969–70, 1970–71, 1975–76, 1978–79, 1984–85, 1986–87, 1987–88, 1988–89, 1991–92, 1995–96, 1996–97, 1998–99, 2010–11, 2014–15, 2019–20
Runners-up (8): 1953, 1963–64, 1976–77, 1979–80, 1983–84, 1985–86, 1989–90, 2013–14
Cupa Ligii
Winners (2) – Record: 2014–15, 2015–16
Supercupa României
Winners (6) – Record: 1994, 1995, 1998, 2001, 2006, 2013
Runners-up (6): 1999, 2005, 2011, 2014, 2015, 2020
European
European Cup / UEFA Champions League:
Winners (1): 1985–86
Runners-up (1): 1988–89
Semi-finalists (1): 1987–88
European Super Cup / UEFA Super Cup:
Winners (1): 1986
UEFA Cup / UEFA Europa League:
Semi-finalists (1): 2005–06
European Cup Winners' Cup / UEFA Cup Winners' Cup:
Quarter-finalists (2): 1971–72, 1992–93
International
Intercontinental Cup:
Runners-up (1): 1986
Minor honours
The Autumn Cup:
Winners (1): 1949
Dordrecht Tournament (Dordrecht-Holland):
Winners (1): 1984
Bruges Matins: Link
Winners (1): 1987
Third Place (1): 1988
Norcia Winter Cup: Link
Winners (1): 1999
Runners-up (1): 2001
Third Place (1): 2000
Torneo di Viareggio: Link
Thi
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https://en.wikipedia.org/wiki/Cinderella%20%28software%29
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Cinderella is a proprietary interactive geometry software, written in Java.
History
Cinderella was initially developed by Jürgen Richter-Gebert and Henry Crapo and was used to input incidence theorems and conjectures for automatic theorem proving using the binomial proving method by Richter-Gebert. The initial software was created in Objective-C on the NeXT platform.
In 1996, the software was rewritten in Java from scratch by Jürgen Richter-Gebert and Ulrich Kortenkamp. It still included the binomial prover, but was not suitable for classroom teaching as it still was prototypical. This version won the Multimedia Innovation Award at Learntec '97 in Karlsruhe, Germany. Due to this attention the German educational software publisher Heureka-Klett and the scientific publisher Springer-Verlag, Heidelberg, agreed to produce a commercial version of the software. The school version of Cinderella 1.0 was published in 1998, including about 150 examples, animations and exercises created with Cinderella, the university version was released in 1999.
In 2006, a new version of Cinderella, Cinderella.2, was published in an online-only version. The printed manual for the now current version 2.6 has been published by Springer-Verlag in 2012.
In 2013, the pro version of Cinderella has been made freely available.
Features
Interactive geometry and analysis takes place in the realm of euclidean geometry, spherical geometry or hyperbolic geometry. It includes a physics simulation engine (with real gravity on Apple computers) and a scripting language. An export to blog feature allows for a 1-click publication on the web of a figure. It is currently mainly used in universities in Germany but its ease of use makes it suitable for usage at primary and secondary level as well.
External links
Cinderella official website
Public Beta version
Online Documentation
CindyJS, a reimplementation in JavaScript
Interactive geometry software
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https://en.wikipedia.org/wiki/George%20A.%20Milliken
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George Albert Milliken is emeritus professor of statistics at Kansas State University. He is a Fellow of the American Statistical Association and has published many papers in various statistical journals. Milliken is a co-author of the three volume Analysis of Messy Data series (Volume 1: Designed Experiments; Volume 2: Nonreplicated Experiments; Volume 3: Analysis of Covariance) and the co-author of the book SAS System for Mixed Models.
Milliken's books are widely referenced in the statistical research community. He has placed a significant emphasis of his professional research on the following areas:
Nonlinear mixed models
Linear and nonlinear models
Design of experiments, appropriate experimental units
Mixed models, repeated measures, non-replicated experiments
Complex designs from designed experiments and observational studies
References
External links
Milliken's KSU faculty web page
Milliken's consulting firm
Year of birth missing (living people)
Living people
People from Manhattan, Kansas
Kansas State University alumni
Fellows of the American Statistical Association
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https://en.wikipedia.org/wiki/Deir%20Ballut
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Deir Ballut () is a Palestinian town located in the Salfit Governorate in the northern West Bank, south west of Nablus. According to the Palestinian Central Bureau of Statistics, it had a population of 3,873 in 2017.
Location
Deir Ballut is located west of Salfit. It is bordered by Kafr ad Dik to its east, Al Lubban al Gharbi to the south, Kafr Qasem to the west, and Rafat to the north.
History
Sherds from the Iron Age, Roman, Byzantine, Umayyad/Abbasid and Crusader/Ayyubid eras have been found here.
The "great valley" of Wadi Deir Ballut was identified by Charles William Wilson (1836–1905) as the boundary between Judaea and Samaria, as defined by first-century historian Josephus.
Arab geographer Yaqut al-Hamawi records in 1226 that "Deir al-Ballut was a village of district around ar-Ramla."
Ottoman era
In the 18th and 19th centuries, Deir Ballut belonged to the highland region known as Jūrat ‘Amra or Bilād Jammā‘īn. Situated between Dayr Ghassāna in the south and the present Route 5 in the north, and between Majdal Yābā in the west and Jammā‘īn, Mardā and Kifl Ḥāris in the east, this area served, according to historian Roy Marom, "as a buffer zone between the political-economic-social units of the Jerusalem and the Nablus regions. On the political level, it suffered from instability due to the migration of the Bedouin tribes and the constant competition among local clans for the right to collect taxes on behalf of the Ottoman authorities.”
In 1838, it was noted as a Muslim village, Deir Balut, in Jurat Merda, south of Nablus.
In 1870 Victor Guérin found it to be a village of one hundred and fifty people. However, judging by the extent of the ruins that covered the hill where it stood, Guérin thought it had once been a large city. Most houses were built with large stones.
In 1882 the PEF's Survey of Western Palestine (SWP) described it as "a small village, partly ruinous, but evidently once a place of greater importance, with rock-cut tombs. The huts are principally of stone. The water supply is from wells." To the west of the village are rock-tombs, from a Christian age.
WWI and British Mandate era
During World War I, Deir Ballut was the site of a minor engagement between Turkish and British troops on March 12, 1918.
In the 1922 census of Palestine Deir Ballut had a population of 384 inhabitants, all Muslim, rising to 532 in the 1931 census, still all Muslim, in a total of 91 houses.
In the 1945 statistics the population was 720, all Muslim while the total land area was 14,789 dunams, according to an official land and population survey. Of this, 508 dunams were for plantations and irrigable land, 3,488 for cereals, while 63 dunams were classified as built-up (urban) areas.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Deir Ballut came under Jordanian rule.
In 1961, the population was 1,087.
Post-1967
Since the Six-Day War in 1967, Deir Ballut has been under Israeli occupation.
A
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https://en.wikipedia.org/wiki/Alan%20Nolan
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Alan Nolan (born 4 June 1985) is a hurler for Dublin and St Brigid's. He was named on the Dublin Blue Stars team for 2006.
Career statistics
Honours
Dublin
Leinster Senior Hurling Championship: 2013
National Hurling League Division 1: 2011
National Hurling League Division 1B: 2013
National Hurling League Division 2: 2006
Walsh Cup: 2011, 2013
References
1985 births
Living people
Dublin inter-county hurlers
Hurling goalkeepers
Irish plumbers
St Brigid's (Dublin) hurlers
People educated at St. Declan's College, Dublin
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https://en.wikipedia.org/wiki/Energy%20minimization
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In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a collection of atoms where, according to some computational model of chemical bonding, the net inter-atomic force on each atom is acceptably close to zero and the position on the potential energy surface (PES) is a stationary point (described later). The collection of atoms might be a single molecule, an ion, a condensed phase, a transition state or even a collection of any of these. The computational model of chemical bonding might, for example, be quantum mechanics.
As an example, when optimizing the geometry of a water molecule, one aims to obtain the hydrogen-oxygen bond lengths and the hydrogen-oxygen-hydrogen bond angle which minimize the forces that would otherwise be pulling atoms together or pushing them apart.
The motivation for performing a geometry optimization is the physical significance of the obtained structure: optimized structures often correspond to a substance as it is found in nature and the geometry of such a structure can be used in a variety of experimental and theoretical investigations in the fields of chemical structure, thermodynamics, chemical kinetics, spectroscopy and others.
Typically, but not always, the process seeks to find the geometry of a particular arrangement of the atoms that represents a local or global energy minimum. Instead of searching for global energy minimum, it might be desirable to optimize to a transition state, that is, a saddle point on the potential energy surface. Additionally, certain coordinates (such as a chemical bond length) might be fixed during the optimization.
Molecular geometry and mathematical interpretation
The geometry of a set of atoms can be described by a vector of the atoms' positions. This could be the set of the Cartesian coordinates of the atoms or, when considering molecules, might be so called internal coordinates formed from a set of bond lengths, bond angles and dihedral angles.
Given a set of atoms and a vector, , describing the atoms' positions, one can introduce the concept of the energy as a function of the positions, . Geometry optimization is then a mathematical optimization problem, in which it is desired to find the value of for which is at a local minimum, that is, the derivative of the energy with respect to the position of the atoms, , is the zero vector and the second derivative matrix of the system, , also known as the Hessian matrix, which describes the curvature of the PES at , has all positive eigenvalues (is positive definite).
A special case of a geometry optimization is a search for the geometry of a transition state; this is discussed below.
The computational model that provides an approximate could be based on quantum mechanics (using either density functional theory or semi-empirical methods), force fields, or a combination of those in case
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https://en.wikipedia.org/wiki/John%20B.%20Cosgrave
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Dr. John B. Cosgrave (born 5 January 1946) is an Irish mathematician specialising in number theory. Born in Bailieborough, County Cavan, he was educated at Royal Holloway College, London, he lectured in Carysfort College (Blackrock, Dublin) and St Patrick's College of Education (Drumcondra).
Other
In January 1999, while preparing some work for his students, he identified a highly structured prime number with exactly two thousand digits. Dubbing this prime a millennium prime, he wrote an email about it to a niece and nephew, which was subsequently published by Folding Landscapes, the publishing house of the cartographer Tim Robinson. He donated his author royalties to the Irish Cancer Society, and subsequently wrote an Irishman's Diary column about it for the Irish Times newspaper.
In July 1999 – while a participant in the Proth Search Group – he became the discoverer of the then-largest known composite Fermat number, a record which his St. Patrick's College (Drumcondra) based Proth-Gallot Group twice broke in 2003, the 1999 record having stood until then. The third of those records continued to stand until it was broken in June 2011.
Selected publications
Cosgrave, John B. and Dilcher, Karl. The Multiplicative Orders of Certain Gauss Factorials, International Journal of Number Theory, Volume 7, Number 1, February 2011.
Cosgrave, John B. and Dilcher, Karl. Mod p^3 analogues of theorems of Gauss and Jacobi on binomial coefficients, Acta Arithmetica, Vol. 142, No. 2, 103–118, 2010.
Cosgrave, John B. and Dilcher, Karl. Extensions of the Gauss-Wilson theorem, Integers: Electronic Journal of Combinatorial Number Theory, Vol. 8, #A39, 2008.
Cosgrave, John B. Number Theory and Cryptography (using Maple), in David Joyner USNA (Ed.), Coding Theory and Cryptography: From Enigma to Geheimschreiber to Quantum Theory (United States Naval Academy Conference), Springer-Verlag, 2000, pp 124–143.
Cosgrave, John B. A Prime for the Millennium, published by Folding Landscapes (2000).
Cosgrave, John B. An Introduction to Number Theory with Talented Youth, USA School Science and Mathematics, Vol 99, No 6, October 1999 (Special issue devoted to gifted and talented Mathematics and Science students).
Cosgrave, John B. From divisibility by 6 to the Euclidean Algorithm and the RSA cryptographic method, The American Mathematical Association of Two-Year Colleges Review, Vol 19, No 1, Fall 1997, 38–45.
Cosgrave, John B. Teaching Mathematics by Questioning – The Socratic Method, Newsletter of Irish Mathematics Teachers Association, Nos 81–82, 1993, 32–47.
Cosgrave, John B. A Halmos Problem and a Related Problem, American Mathematical Monthly, Vol. 101, No. 10, 993–996, December 1994.
Cosgrave, John B. A Remark on Euclid's Proof of the Infinitude of Primes, American Mathematical Monthly, Vol. 96, No. 4, 339–341, April 1989.
Cosgrave, John B. Transcendental numbers in the p-adic domain (unpublished PhD thesis, 1972).
Cosgrave, John B. An application of Wilson's t
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https://en.wikipedia.org/wiki/Law%20enforcement%20in%20British%20Columbia%2C%202005
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This is a list of statistics in law enforcement in British Columbia in 2005, including crime rates, police strength, and police costs. In total there were 508,271 reported (non-traffic) incidents of Criminal Code offences, giving the province a crime rate of 120 offences per 1,000 people, the second highest in Canada. This was down 5% from 2004's rate of 125, and was the first decrease since 1999-2000. Of these crimes, only 22% were solved in the same year, including 52% of all violent crimes and 13% of all property crimes. This resulted in 57,817 persons being recommended for charges to the Crown counsel, of which 81% were male and 10% were young offenders (between 12 and 17 years old).
Law enforcement was supplied mainly by municipal forces, either an independent police department or the Royal Canadian Mounted Police (RCMP). Municipalities with populations over 5,000 people using the RCMP paid either 90% or 70% of their costs, depending on their population size, with the federal government paying the remainder. Those municipalities under 5,000 people shared a detachment with the general rural area but did not pay any of the policing costs while the unincorporated rural areas paid a small, varied, amount through a general rural property tax. Other police forces operating within BC include 2 First Nations forces, a RCMP federal force, the Greater Vancouver Transportation Authority Police Service (now South Coast British Columbia Transportation Authority Police Service), the Organized Crime Agency (Combined Forces Special Enforcement Unit), Conservation Officer Service, the Canadian National and Canadian Pacific railway police forces, and various municipal bylaw enforcement officers.
In the following tables crime rates refer to the number of incidents of Criminal Code offences, excluding traffic offences, per 1,000 people. If an incident involved more than one offense, only the most serious was recorded in these statistics. The population figures were based on the Canada 2001 Census and estimated for 2005. These populations only include permanent residents, so municipalities with high numbers of visitors (from the rural areas, commuters, tourists, seasonal workers, etc.) are not counted and will result in a higher crime rate. The total costs are the actual costs of police services at year end, not the budgeted costs. The case burden reflects the workload of each office: the number of offences, excluding traffic offenes, per officer.
Municipal detachments
The 11 municipalities that operated their own police department paid 100% of the total costs. Their number of officers, policies and priorities were set by a municipally-appointed police board. The 59 municipalities which contracted their police duties to the RCMP "E" Division had their officers operate under provincial or federal policies and priorities. Of the municipalities which have population over 15,000 people paid 90% and municipalities between 5,000 and 15,000 people paid 70% with the
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https://en.wikipedia.org/wiki/Gustav%20Jaumann
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Gustav Andreas Johannes Jaumann (1863–1924) was an Austrian physicist.
An assistant to the physicist Ernst Mach, he had a talent for mathematics, but disbelieved the existence of small particles like electrons and atoms. Between 1901 and 1924 he taught physics
at the German Technical University in Brno. He won the Haitinger Prize of the Austrian Academy of Sciences in 1911.
Remembered for
"Corotational derivative" expresses the stress tensor in a rotating body.
Jaumann was offered a professorship at Prague University in 1911, but refused the position. The candidate who was the faculty's first choice, Albert Einstein, would accept the offer after it was turned down by Jaumann, who is alleged to have said in an unsubstantiated quotation from Philipp Frank, "If Einstein has been proposed as the first choice because of the belief that he has greater achievements to his credit, then I will have nothing to do with a university that chases after modernity and does not appreciate merit." The actual reason as alleged by the Austrian Minister of Education, in his official report to Emperor Franz Josef, seems to have been about money. The beleaguered situation of Germans in Prague at the time, with which Jaumann was familiar, may also have been a factor in his declining the post. Jaumann was apparently the candidate preferred by the Austro-Hungarian ministry, presumably because he was Austrian and not a foreigner. Einstein, under the impression he would not receive the job, however, blamed his "Semitic origin [that] the ministry did not approve [of]".
Notes
References
Blackmore J. T., Itagaki R., Tanaka S. (2001), Ernst Mach's Vienna 1895-1930: Or Phenomenalism as Philosophy of Science (Boston Studies in the Philosophy of Science) Springer Verlag, . Available in Google Books.
Isaacson W. (2007) Einstein, Simon and Schuster, .
Müller, I., (2007) A History of Thermodynamics: The Doctrine of Energy and Entropy, Springer Verlag, . footnote on page 75. Available in Google Books.
Teachers of physics and chemistry at the German Technical University in Brno.
, available on Google Books.
External links
Gustav Jaumann in German Wikipedia
Einstein's Job Search
19th-century Austrian physicists
People from Caransebeș
1863 births
1924 deaths
20th-century Austrian physicists
Physicists from Austria-Hungary
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https://en.wikipedia.org/wiki/Lu%C3%ADs%20Augusto
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Luís Augusto Osório Romão do Nascimento(born 20 November 1983), known as Luís Augusto, is a Brazilian retired footballer who played as an attacking midfielder.
Club statistics
References
External links
CBF
1983 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Santos FC players
Paysandu Sport Club players
Expatriate men's footballers in Japan
J1 League players
J2 League players
Yokohama FC players
Oita Trinita players
Albirex Niigata players
Clube Atlético Bragantino players
Comercial Futebol Clube (Ribeirão Preto) players
Guarany Sporting Club players
Brasiliense FC players
River Atlético Clube players
Men's association football midfielders
People from Oeiras, Piauí
Footballers from Piauí
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https://en.wikipedia.org/wiki/Gabriel%20%28footballer%2C%20born%201987%29
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Gabriel Donizette de Santana, or simply Gabriel (born September 8, 1987) is a Brazilianiel played for V in the J1 League during 2006 and 2007.
Club statistics
References
External links
1987 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Mogi Mirim Esporte Clube players
Sanat Naft Abadan F.C. players
Aluminium Hormozgan F.C. players
Expatriate men's footballers in Iran
Expatriate men's footballers in Japan
J1 League players
J2 League players
Vissel Kobe players
Men's association football midfielders
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https://en.wikipedia.org/wiki/Tic%C3%A3o
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Carlos Augusto Bertoldi (; born February 7, 1985, in Curitiba), known simply as Ticão, is a Brazilian professional footballer who is currently a free agent.
Statistics
Club career
Sim.
Honours
Club Athlético Paranaense
Paraná State League: 2005
Sport Club do Recife
Pernambuco State League: 2006, 2007.
Fortaleza Esporte Clube
Ceará State League: 2010.
Yuen Long
Hong Kong Senior Shield: 2017–18
Contract
Náutico (Loan) 1 January 2008 to 31 December 2008
Atlético-PR 1 January 2008 to 1 January 2010
References
External links
1985 births
Living people
Brazilian men's footballers
Men's association football midfielders
Club Athletico Paranaense players
Sport Club do Recife players
Clube Náutico Capibaribe players
Ituano FC players
Olympiacos Volos F.C. players
Brazilian expatriate men's footballers
Expatriate men's footballers in Greece
Expatriate men's footballers in Hong Kong
Footballers from Curitiba
South China AA players
Yuen Long FC players
Southern District FC players
Hong Kong Premier League players
Hong Kong League XI representative players
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https://en.wikipedia.org/wiki/Kesar%20Ordin
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Kesar Filippovich Ordin (1835—1892) was a 19th-century Russian mathematician and historian.
He was a graduate in mathematics of St. Petersburg University and author of a number of articles on Finland, opposing Finnish separatism.
Ordin is perhaps most known of his work in which he tried to disprove the claims published by Finnish (also part of Russia at that time) Leo Mechelin about the 1809 Diet of Porvoo. Whereas Mechelin thought that Finland and Russia had made a treaty which resulted the two countries to form a so-called "permanent union", Ordin's version was that Finland had simply been merged to the motherland.
References
Heikkonen, Ojakoski, Väisänen, "Muutosten maailma 4: Suomen historian käännekohtia" (WSOY, 2003), . Pages 59 and 202.
1892 deaths
Saint Petersburg State University alumni
1835 births
19th-century historians from the Russian Empire
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https://en.wikipedia.org/wiki/Hanan%20grid
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In geometry, the Hanan grid of a finite set of points in the plane is obtained by constructing vertical and horizontal lines through each point in .
The main motivation for studying the Hanan grid stems from the fact that it is known to contain a minimum length rectilinear Steiner tree for . It is named after Maurice Hanan, who was first to investigate the rectilinear Steiner minimum tree and introduced this graph.
References
Graph families
Geometric graphs
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https://en.wikipedia.org/wiki/Boolean%20algebra%20%28disambiguation%29
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Boolean algebra is the algebra of truth values and operations on them.
Boolean algebra may also refer to:
Boolean algebra (structure), any member of a certain class of mathematical structures that can be described in terms of an ordering or in terms of operations on a set
Two-element Boolean algebra, Boolean algebra whose underlying set has two elements
Boolean ring
See also
Boolean (disambiguation)
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https://en.wikipedia.org/wiki/Saint-S%C3%A9v%C3%A8re%2C%20Quebec
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Saint-Sévère is a parish municipality in the Mauricie region of the province of Quebec in Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Saint-Sévère had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
References
External links
Parish municipalities in Quebec
Incorporated places in Mauricie
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https://en.wikipedia.org/wiki/Photoionisation%20cross%20section
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Photoionisation cross section in the context of condensed matter physics refers to the probability of a particle (usually an electron) being emitted from its electronic state.
Cross section in photoemission
The photoemission is a useful experimental method for the determination and the study of the electronic states. Sometimes the small amount of deposited material over a surface has a weak contribution to the photoemission spectra, which makes its identification very difficult.
The knowledge of the cross section of a material can help to detect thin layers or 1D nanowires over a substrate. A right choice of the photon energy can enhance a small amount of material deposited over a surface, otherwise the display of the different spectra won't be possible.
See also
Gamma ray cross section
ARPES
Synchrotron radiation
Cross section (physics)
Absorption cross section
Nuclear cross section
References
External links
Elettra's photoemission cross sections calculations
Electromagnetism
Condensed matter physics
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https://en.wikipedia.org/wiki/Josimar%20%28footballer%2C%20born%201987%29
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Josimar Rodrigues Souza Roberto, or simply Josimar (born August 16, 1987), is a Brazilian striker .
Club statistics
References
External links
1987 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
J2 League players
Ipatinga Futebol Clube players
Ventforet Kofu players
Ehime FC players
Tokyo Verdy players
Clube Náutico Capibaribe players
Clube Esportivo Lajeadense players
Al Fateh SC players
Army United F.C. players
Port F.C. players
PTT Rayong F.C. players
Thai League 1 players
Thai League 2 players
Expatriate men's footballers in Saudi Arabia
Expatriate men's footballers in Thailand
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in Saudi Arabia
Brazilian expatriate sportspeople in Thailand
Saudi Pro League players
Men's association football forwards
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https://en.wikipedia.org/wiki/Fernandinho%20%28footballer%2C%20born%20January%201981%29
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Éldis Fernando Damasio, better known as Fernandinho (フェルナンジーニョ, born January 13, 1981), is a Brazilian footballer who plays as an attacking midfielder for Gainare Tottori.
Club career statistics
Updated to 23 February 2020.
References
External links
Profile at Gainare Tottori
Profile at Oita Trinita
1981 births
Living people
Men's association football midfielders
Brazilian men's footballers
Brazilian expatriate men's footballers
Figueirense FC players
Associação Desportiva São Caetano players
CR Vasco da Gama players
Expatriate men's footballers in Japan
J1 League players
J2 League players
J3 League players
Gamba Osaka players
Shimizu S-Pulse players
Kyoto Sanga FC players
Oita Trinita players
Vegalta Sendai players
Ventforet Kofu players
Gainare Tottori players
Mogi Mirim Esporte Clube players
Footballers from Florianópolis
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https://en.wikipedia.org/wiki/Bounded%20inverse%20theorem
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In mathematics, the bounded inverse theorem ( also called inverse mapping theorem or Banach isomorphism theorem) is a result in the theory of bounded linear operators on Banach spaces.
It states that a bijective bounded linear operator T from one Banach space to another has bounded inverse T−1. It is equivalent to both the open mapping theorem and the closed graph theorem.
Generalization
Counterexample
This theorem may not hold for normed spaces that are not complete.
For example, consider the space X of sequences x : N → R with only finitely many non-zero terms equipped with the supremum norm. The map T : X → X defined by
is bounded, linear and invertible, but T−1 is unbounded.
This does not contradict the bounded inverse theorem since X is not complete, and thus is not a Banach space.
To see that it's not complete, consider the sequence of sequences x(n) ∈ X given by
converges as n → ∞ to the sequence x(∞) given by
which has all its terms non-zero, and so does not lie in X.
The completion of X is the space of all sequences that converge to zero, which is a (closed) subspace of the ℓp space ℓ∞(N), which is the space of all bounded sequences.
However, in this case, the map T is not onto, and thus not a bijection. To see this, one need simply note that the sequence
is an element of , but is not in the range of .
See also
References
Bibliography
(Section 8.2)
Operator theory
Theorems in functional analysis
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https://en.wikipedia.org/wiki/Inverse%20mapping%20theorem
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In mathematics, inverse mapping theorem may refer to:
the inverse function theorem on the existence of local inverses for functions with non-singular derivatives
the bounded inverse theorem on the boundedness of the inverse for invertible bounded linear operators on Banach spaces
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https://en.wikipedia.org/wiki/Hilbert%E2%80%93Schmidt%20integral%20operator
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In mathematics, a Hilbert–Schmidt integral operator is a type of integral transform. Specifically, given a domain (an open and connected set) Ω in n-dimensional Euclidean space Rn, a Hilbert–Schmidt kernel is a function k : Ω × Ω → C with
(that is, the L2(Ω×Ω; C) norm of k is finite), and the associated Hilbert–Schmidt integral operator is the operator K : L2(Ω; C) → L2(Ω; C) given by
Then K is a Hilbert–Schmidt operator with Hilbert–Schmidt norm
Hilbert–Schmidt integral operators are both continuous (and hence bounded) and compact (as with all Hilbert–Schmidt operators).
The concept of a Hilbert–Schmidt operator may be extended to any locally compact Hausdorff spaces. Specifically, let X be a locally compact Hausdorff space equipped with a positive Borel measure. Suppose further that L2(X) is a separable Hilbert space. The above condition on the kernel k on Rn can be interpreted as demanding k belong to L2(X × X). Then the operator
is compact. If
then K is also self-adjoint and so the spectral theorem applies. This is one of the fundamental constructions of such operators, which often reduces problems about infinite-dimensional vector spaces to questions about well-understood finite-dimensional eigenspaces. See Chapter 2 of the book by Bump in the references for examples.
See also
Hilbert–Schmidt operator
References
(Sections 8.1 and 8.5)
Linear operators
Operator theory
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https://en.wikipedia.org/wiki/Distributed%20active%20transformer
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Distributed active transformer is a circuit topology that allows low-voltage transistors to be used to generate large amounts of RF (radio frequency) power. Its main use has been in making integrated CMOS power amplifier for wireless applications, such as GSM/GPRS cellular phones.
At the time it was introduced, the distributed active transformer performance improved more than an order of magnitude relative to the previous state of the art.
Output power of up to 2.2 Watt in S-band was demonstrated back in 2002, utilizing Distributed active transformer which combine the power of four differential power amplifiers.
References
External links
Thesis - Distributed Active Transformer for Integrated Power Amplification - Ichiro Aoki (2002) - California Institute of Technology
Aoki, I.; Kee, S.; Magoon, R.; Aparicio, R.; Bohn, F.; Zachan, J.; Hatcher, G.; McClymont, D.; Hajimiri, A.; "A Integrated Quad-Band GSM/GPRS CMOS Power Amplifier"; Solid-State Circuits, IEEE Journal of; Dec. 2008
Aoki, I.; Kee, S.D.; Rutledge, D.B.; Hajimiri, A.; "Fully Integrated CMOS Power Amplifier Design Using the Distributed Active-Transformer Architecture"; Solid-State Circuits, IEEE Journal of; Mar. 2002
Aoki, I.; Kee, S.D.; Rutledge, D.B.; Hajimiri, A.; "Distributed Active Transformer - A New Power-Combining and Impedance-Transformation Technique"; Microwave Theory and Techniques, IEEE Transactions in January 2002
Electronic circuits
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https://en.wikipedia.org/wiki/Pr%C3%BCfer%20domain
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In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
Examples
The ring of entire functions on the open complex plane form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring of integer polynomials is not . While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Prüfer domain. Just as a Dedekind domain is locally a discrete valuation ring, a Prüfer domain is locally a valuation ring, so that Prüfer domains act as non-noetherian analogues of Dedekind domains. Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain .
Many Prüfer domains are also Bézout domains, that is, not only are finitely generated ideals projective, they are even free (that is, principal). For instance the ring of analytic functions on any non-compact Riemann surface is a Bézout domain , and the ring of algebraic integers is Bézout.
Definitions
A Prüfer domain is a semihereditary integral domain. Equivalently, a Prüfer domain may be defined as a commutative ring without zero divisors in which every non-zero finitely generated ideal is invertible. Many different characterizations of Prüfer domains are known. Bourbaki lists fourteen of them, has around forty, and open with nine.
As a sample, the following conditions on an integral domain R are equivalent to R being a Prüfer domain, i.e. every finitely generated ideal of R is projective:
Ideal arithmetic
Every non-zero finitely generated ideal I of R is invertible: i.e. , where and is the field of fractions of R. Equivalently, every non-zero ideal generated by two elements is invertible.
For any (finitely generated) non-zero ideals I, J, K of R, the following distributivity property holds:
For any (finitely generated) ideals I, J, K of R, the following distributivity property holds:
For any (finitely generated) non-zero ideals I, J of R, the following property holds:
For any finitely generated ideals I, J, K of R, if IJ = IK then J = K or I = 0.
Localizations
For every prime ideal P of R, the localization RP of R at P is a valuation domain.
For every maximal ideal m in R, the localization Rm of R at m is a valuation domain.
R is integrally closed and every overring of R (that is, a ring contained between R and its field of fractions) is the intersection of localizations of R
Flatness
Every torsion-free R-module is flat.
Every torsionless R-module is flat.
Every ideal of R is flat
Every overring of R is R-flat
Every submodule of a flat R-module is flat.
If M and N are torsion-free R-modules then their tensor product M ⊗R N is torsion-free.
If I and J are two idea
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https://en.wikipedia.org/wiki/Gerrit%20Krol
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Gerrit Krol (1 August 1934 − 24 November 2013) was a Dutch author, essayist and writer.
Krol was born in Groningen. He studied mathematics and worked with Royal Dutch Shell and some of its operating units as computer programmer and system designer. Krol's debut consisted of poems published in 1961 in various Dutch literary magazines. In 1962 his first book De rokken van Joy Scheepmaker was published. Thereafter, he developed a typical writing style consisting of text mingled with abstract thoughts expressed in drawings and mathematical equations. His novel Het gemillimeterde hoofd is typical for this Krollesque style.
In 1986 Krol received the Constantijn Huygens Prize, and in 2001 the P. C. Hooft Award - the highest Dutch Governmental award for literature - for his complete oeuvre. On 20 October 2005, the 125th anniversary of the Amsterdam Free University, Krol received a Doctorate Honoris causa from this university.
Published works
De rokken van Joy Scheepmaker, 1962
Kwartslag, 1964
De zoon van de levende stad, 1966
Het gemillimeterde hoofd (English translation: The Cropped Head), 1967
De ziekte van Middleton, 1969
De laatste winter, 1970
APPI, (essay) 1971
De man van het lateraal denken, (essay) 1971
De chauffeur verveelt zich, 1973
In dienst van de 'Koninklijke', 1974
Hoe ziet ons gevoel er uit?, essay 1974
De gewone man en het geluk of Waarom het niet goed is lid van een vakbond te zijn, (essays) 1975
Halte opgeheven, (short stories) 1976
Polaroid, (poetry) 1976
De weg naar Sacramento, 1977
Over het huiselijk geluk en andere gedachten, (columns) 1978
De t.v.-b.h., (columns) 1979
Een Fries huilt niet, 1980
De schrijver, zijn schaamte en zijn spiegels, (essay) 1981
De man achter het raam, 1982
Het vrije vers, (essays) 1982
Scheve levens, 1983
De schriftelijke natuur, (essays) 1985
Maurits en de feiten, 1986
Bijna voorjaar, (columns) 1986
De weg naar Tuktoyaktuk, 1987
De schoonheid van de witregel, (essays) 1987
Helmholtz' paradijs, (essays) 1987
Een ongenode gast, 1988
De Hagemeijertjes, 1990
Voor wie kwaad wil, (essay) 1990
Wat mooi is moeilijk, (essays) 1991
Oude foto's, (short stories) 1992
Wat mooi is is moeilijk, (essays) 1992
Omhelzingen, 1993
Okoka's Wonderpark, 1994
De mechanica van het liegen, (essays) 1995
Middleton's dood, 1996
De kleur van Groningen, (poetry) 1997
60000 uur, (autobiography) 1998
Missie Novgorod, 1999
De vitalist, 2000
Geen man, want geen vrouw, (poetry) 2001
Minnaar, (poetry) 2001
'n Kleintje Krol, 2001
Een schaaknovelle, 2002
Rondo Veneziano, 2004
Duivelskermis, 2007
De industrie geneest alle leed (verzamelde gedichten, 2009)
Verplaatste personen (verhalen, schilderijen Otto Krol, 2009)
References
External links
Gerrit Krol at Poetry International
Gerrit Krol at Digital Library for Dutch Literature (in Dutch - De schrijver, zijn schaamte en zijn spiegels available for free download)
1934 births
2013 deaths
20th-century Dutch poets
20th-century Dutch male writers
20th-century Dut
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https://en.wikipedia.org/wiki/Aliquot%20sum
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In number theory, the aliquot sum of a positive integer is the sum of all proper divisors of , that is, all divisors of other than itself.
That is,
It can be used to characterize the prime numbers, perfect numbers, sociable numbers, deficient numbers, abundant numbers, and untouchable numbers, and to define the aliquot sequence of a number.
Examples
For example, the proper divisors of 12 (that is, the positive divisors of 12 that are not equal to 12) are , and 6, so the aliquot sum of 12 is 16 i.e. ().
The values of for are:
0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 11, 14, 1, 36, 6, 16, 13, 28, 1, 42, 1, 31, 15, 20, 13, 55, 1, 22, 17, 50, 1, 54, 1, 40, 33, 26, 1, 76, 8, 43, ...
Characterization of classes of numbers
The aliquot sum function can be used to characterize several notable classes of numbers:
1 is the only number whose aliquot sum is 0.
A number is prime if and only if its aliquot sum is 1.
The aliquot sums of perfect, deficient, and abundant numbers are equal to, less than, and greater than the number itself respectively. The quasiperfect numbers (if such numbers exist) are the numbers whose aliquot sums equal . The almost perfect numbers (which include the powers of 2, being the only known such numbers so far) are the numbers whose aliquot sums equal .
The untouchable numbers are the numbers that are not the aliquot sum of any other number. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable. Paul Erdős proved that their number is infinite. The conjecture that 5 is the only odd untouchable number remains unproven, but would follow from a form of Goldbach's conjecture together with the observation that, for a semiprime number , the aliquot sum is .
The mathematicians noted that one of Erdős' "favorite subjects of investigation" was the aliquot sum function.
Iteration
Iterating the aliquot sum function produces the aliquot sequence of a nonnegative integer (in this sequence, we define ).
Sociable numbers are numbers whose aliquot sequence is a periodic sequence. Amicable numbers are sociable numbers whose aliquot sequence has period 2.
It remains unknown whether these sequences always end with a prime number, a perfect number, or a periodic sequence of sociable numbers.
See also
Sum of positive divisors function, the sum of the (th powers of the) positive divisors of a number
William of Auberive, medieval numerologist interested in aliquot sums
References
External links
Arithmetic dynamics
Arithmetic functions
Divisor function
Perfect numbers
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https://en.wikipedia.org/wiki/Unsolved%20Problems%20in%20Number%20Theory
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Unsolved Problems in Number Theory may refer to:
Unsolved problems in mathematics in the field of number theory.
A book with this title by Richard K. Guy published by Springer Verlag:
First edition 1981, 161 pages,
Second edition 1994, 285 pages,
Third edition 2004, 438 pages,
Books with a similar title include:
Solved and Unsolved Problems in Number Theory, by Daniel Shanks
First edition, 1962
Second edition, 1978
Third edition, 1985,
Fourth edition, 1993
Old and New Unsolved Problems in Plane Geometry and Number Theory, by Victor Klee and Stan Wagon, 1991, .
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https://en.wikipedia.org/wiki/Anscombe%20transform
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In statistics, the Anscombe transform, named after Francis Anscombe, is a variance-stabilizing transformation that transforms a random variable with a Poisson distribution into one with an approximately standard Gaussian distribution. The Anscombe transform is widely used in photon-limited imaging (astronomy, X-ray) where images naturally follow the Poisson law. The Anscombe transform is usually used to pre-process the data in order to make the standard deviation approximately constant. Then denoising algorithms designed for the framework of additive white Gaussian noise are used; the final estimate is then obtained by applying an inverse Anscombe transformation to the denoised data.
Definition
For the Poisson distribution the mean and variance are not independent: . The Anscombe transform
aims at transforming the data so that the variance is set approximately 1 for large enough mean; for mean zero, the variance is still zero.
It transforms Poissonian data (with mean ) to approximately Gaussian data of mean
and standard deviation .
This approximation gets more accurate for larger , as can be also seen in the figure.
For a transformed variable of the form , the expression for the variance has an additional term ; it is reduced to zero at , which is exactly the reason why this value was picked.
Inversion
When the Anscombe transform is used in denoising (i.e. when the goal is to obtain from an estimate of ), its inverse transform is also needed
in order to return the variance-stabilized and denoised data to the original range.
Applying the algebraic inverse
usually introduces undesired bias to the estimate of the mean , because the forward square-root
transform is not linear. Sometimes using the asymptotically unbiased inverse
mitigates the issue of bias, but this is not the case in photon-limited imaging, for which
the exact unbiased inverse given by the implicit mapping
should be used. A closed-form approximation of this exact unbiased inverse is
Alternatives
There are many other possible variance-stabilizing transformations for the Poisson distribution. Bar-Lev and Enis report a family of such transformations which includes the Anscombe transform. Another member of the family is the Freeman-Tukey transformation
A simplified transformation, obtained as the primitive of the reciprocal of the standard deviation of the data, is
which, while it is not quite so good at stabilizing the variance, has the advantage of being more easily understood.
Indeed, from the delta method,
.
Generalization
While the Anscombe transform is appropriate for pure Poisson data, in many applications the data presents also an additive Gaussian component. These cases are treated by a Generalized Anscombe transform and its asymptotically unbiased or exact unbiased inverses.
See also
Variance-stabilizing transformation
Box–Cox transformation
References
Further reading
Poisson distribution
Normal distribution
Statistical data transform
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https://en.wikipedia.org/wiki/Hyperarithmetical%20theory
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In recursion theory, hyperarithmetic theory is a generalization of Turing computability. It has close connections with definability in second-order arithmetic and with weak systems of set theory such as Kripke–Platek set theory. It is an important tool in effective descriptive set theory.
The central focus of hyperarithmetic theory is the sets of natural numbers known as hyperarithmetic sets. There are three equivalent ways of defining this class of sets; the study of the relationships between these different definitions is one motivation for the study of hyperarithmetical theory.
Hyperarithmetical sets and definability
The first definition of the hyperarithmetic sets uses the analytical hierarchy.
A set of natural numbers is classified at level of this hierarchy if it is definable by a formula of second-order arithmetic with only existential set quantifiers and no other set quantifiers. A set is classified at level of the analytical hierarchy if it is definable by a formula of second-order arithmetic with only universal set quantifiers and no other set quantifiers. A set is if it is both and . The hyperarithmetical sets are exactly the sets.
Hyperarithmetical sets and iterated Turing jumps: the hyperarithmetical hierarchy
The definition of hyperarithmetical sets as does not directly depend on computability results. A second, equivalent, definition shows that the hyperarithmetical sets can be defined using infinitely iterated Turing jumps. This second definition also shows that the hyperarithmetical sets can be classified into a hierarchy extending the arithmetical hierarchy; the hyperarithmetical sets are exactly the sets that are assigned a rank in this hierarchy.
Each level of the hyperarithmetical hierarchy is indexed by a countable ordinal number (ordinal), but not all countable ordinals correspond to a level of the hierarchy. The ordinals used by the hierarchy are those with an ordinal notation, which is a concrete, effective description of the ordinal.
An ordinal notation is an effective description of a countable ordinal by a natural number. A system of ordinal notations is required in order to define the hyperarithmetic hierarchy. The fundamental property an ordinal notation must have is that it describes the ordinal in terms of smaller ordinals in an effective way. The following inductive definition is typical; it uses a pairing function .
The number 0 is a notation for the ordinal 0.
If n is a notation for an ordinal λ then is a notation for λ + 1;
Suppose that δ is a limit ordinal. A notation for δ is a number of the form , where e is the index of a total computable function such that for each n, is a notation for an ordinal λn less than δ and δ is the sup of the set .
This may also be defined by taking effective joins at all levels instead of only notations for limit ordinals.
There are only countably many ordinal notations, since each notation is a natural number; thus there is a countable ordinal whic
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https://en.wikipedia.org/wiki/Von%20Neumann%27s%20theorem
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In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.
Statement of the theorem
Let and be Hilbert spaces, and let be an unbounded operator from into Suppose that is a closed operator and that is densely defined, that is, is dense in Let denote the adjoint of Then is also densely defined, and it is self-adjoint. That is,
and the operators on the right- and left-hand sides have the same dense domain in
References
Operator theory
Theorems in functional analysis
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https://en.wikipedia.org/wiki/SIPTA
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The Society for Imprecise Probability: Theories and Applications (SIPTA) was created in February 2002, with the aim of promoting the research on Imprecise probability. This is done through a series of activities for bringing together researchers from different groups, creating resources for information dissemination and documentation, and making other people aware of the potential of Imprecise Probability models.
Background
The Society was originally created to manage the series of International Symposia on Imprecise
Probabilities and Their Applications (ISIPTA). The first ISIPTA happened in 1999 in Ghent, Belgium;
due to the success of the event, a second edition took place in Cornell, United States, in 2001.
The Society was then created in Switzerland,
during the year of 2002. The first general meeting of the Society happened during the third ISIPTA,
in Lugano, Switzerland.
The Society is now concerned with many activities around the theme of imprecise probabilities.
Imprecise probability is understood in a very wide sense. It is used as a generic term to cover all mathematical models which measure chance or uncertainty without sharp numerical probabilities. It includes both qualitative (comparative probability, partial preference orderings,...) and quantitative models (interval probabilities, belief functions, upper and lower previsions,...). Imprecise probability models are needed in inference problems where the relevant information is scarce, vague or conflicting, and in decision problems where preferences may also be incomplete.
Bibliography
Walley, Peter: Statistical reasoning with imprecise probabilities. London; New York: Chapman and Hall, 1991. Monographs on statistics and applied probability: 42. .
References
"On the Use of Imprecise Probabilities in Reliability", F. P. A. Coolen, : 15th ARTS Advances in Reliability Technology Symposium abstract
Editorial: Imprecise probability perspectives on artificial intelligence by Marco Zaffalon and Gert de Cooman. — Annals of Mathematics and Artificial Intelligence (subscription required)
External links
The Society for Imprecise Probability: Theories and Applications
Mathematical societies
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https://en.wikipedia.org/wiki/Picture%20%28disambiguation%29
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A picture is an artifact that depicts or records visual perception.
Picture(s) may also refer to:
Mathematics and science
Picture (mathematics), a combinatorial structure
Picture (string theory), a representation of states
PICTURE clause, a COBOL data type
Music
Picture (band), a Dutch heavy metal band
Albums
Picture (album), a 2005 album by Kino
Pictures (Atlanta album), 1984
Pictures (Jack DeJohnette album), 1976
Pictures (John Michael Montgomery album), 2002
Pictures (Katie Melua album), 2007
Pictures (Leon Bolier album), 2008
Pictures (Niels-Henning Ørsted Pedersen and Kenneth Knudsen album), 1977
Pictures (Timo Maas album), 2005
Pictures, a 2006 album by Tony Rich
Songs
"Picture" (song), a 2001 song by Kid Rock and Sheryl Crow
"Pictures" (song), a 2006 song by Sneaky Sound System
"Pictures", a song by Sia from Lady Croissant
"Pictures", a song by AM Conspiracy from AM Conspiracy
"Pictures", a song by Mojave 3 from Ask Me Tomorrow
"Pictures", a song by System of a Down from Steal This Album!
"Pictures", a song by Terry McDermott
"Pictures", a song by Tommy Keene from Based on Happy Times
Other media
Film or motion picture
Movie theater, a building in which films are shown
Pictures (film), a 1981 New Zealand film
"Pictures" (short story), a 1917 short story by Katherine Mansfield
Picture, a 1952 book by Lillian Ross
See also
The Picture (disambiguation)
Image (disambiguation)
Pictura: An Adventure in Art, a documentary film
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https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Maclane%20spectrum
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In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group pg 134. Note, this construction can be generalized to commutative rings as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the derived category of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the Adams spectral sequence.
Definition
For a fixed abelian group let denote the set of Eilenberg–MacLane spaces with the adjunction map coming from the property of loop spaces of Eilenberg–Maclane spaces: namely, because there is a homotopy equivalencewe can construct maps from the adjunction giving the desired structure maps of the set to get a spectrum. This collection is called the Eilenberg–Maclane spectrum of pg 134.
Properties
Using the Eilenberg–Maclane spectrum we can define the notion of cohomology of a spectrum and the homology of a spectrum pg 42. Using the functorwe can define cohomology simply asNote that for a CW complex , the cohomology of the suspension spectrum recovers the cohomology of the original space . Note that we can define the dual notion of homology aswhich can be interpreted as a "dual" to the usual hom-tensor adjunction in spectra. Note that instead of , we take for some Abelian group , we recover the usual (co)homology with coefficients in the abelian group and denote it by .
Mod-p spectra and the Steenrod algebra
For the Eilenberg–Maclane spectrum there is an isomorphismfor the p-Steenrod algebra .
Tools for computing Adams resolutions
One of the quintessential tools for computing stable homotopy groups is the Adams spectral sequence. In order to make this construction, the use of Adams resolutions are employed. These depend on the following properties of Eilenberg–Maclane spectra. We define a generalized Eilenberg–Maclane spectrum as a finite wedge of suspensions of Eilenberg–Maclane spectra , soNote that for and a spectrum so it shifts the degree of cohomology classes. For the rest of the article for some fixed abelian group
Equivalence of maps to K
Note that a homotopy class represents a finite collection of elements in . Conversely, any finite collection of elements in is represented by some homotopy class .
Constructing a surjection
For a locally finite collection of elements in generating it as an abelian group, the associated map induces a surjection on cohomology, meaning if we evaluate these spectra on some topological space , there is always a surjectionof Abelian groups.
Steenrod-module structure on cohomology of spectra
For a spectrum taking the wedge constructs a spectrum which is homotopy equivalent
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https://en.wikipedia.org/wiki/Triangle%20center
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In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.
Each of these classical centers has the property that it is invariant (more precisely equivariant) under similarity transformations. In other words, for any triangle and any similarity transformation (such as a rotation, reflection, dilation, or translation), the center of the transformed triangle is the same point as the transformed center of the original triangle.
This invariance is the defining property of a triangle center. It rules out other well-known points such as the Brocard points which are not invariant under reflection and so fail to qualify as triangle centers.
For an equilateral triangle, all triangle centers coincide at its centroid. However the triangle centers generally take different positions from each other on all other triangles. The definitions and properties of thousands of triangle centers have been collected in the Encyclopedia of Triangle Centers.
History
Even though the ancient Greeks discovered the classic centers of a triangle they had not formulated any definition of a triangle center. After the ancient Greeks, several special points associated with a triangle like the Fermat point, nine-point center, Lemoine point, Gergonne point, and Feuerbach point were discovered.
During the revival of interest in triangle geometry in the 1980s it was noticed that these special points share some general properties that now form the basis for a formal definition of triangle center. , Clark Kimberling's Encyclopedia of Triangle Centers contains an annotated list of 50,730 triangle centers. Every entry in the Encyclopedia of Triangle Centers is denoted by or where is the positional index of the entry. For example, the centroid of a triangle is the second entry and is denoted by or .
Formal definition
A real-valued function of three real variables may have the following properties:
Homogeneity: for some constant and for all .
Bisymmetry in the second and third variables:
If a non-zero has both these properties it is called a triangle center function. If is a triangle center function and are the side-lengths of a reference triangle then the point whose trilinear coordinates are is called a triangle center.
This definition ensures that triangle centers of similar triangles meet the invariance criteria specified above. By convention only the first of the three trilinear coordinates of a triangle center is quoted since the other two are obtained by cyclic permutation of . This process is known as cyclicity.
Every triangle center function corresponds to a unique triangle center. This correspondence is not bijective. Different functions may define the same triangle center. For example, the functions and both correspond to the centroid
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https://en.wikipedia.org/wiki/Louis-Jacques%20Goussier
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Louis-Jacques Goussier (Paris, 7 March 1722 - Paris, 23 October 1799) was a French illustrator and encyclopedist.
Career
Born poor, he first studied mathematics at Pierre Le Guay de Prémontval's (1716–1764) free school, and then became a teacher himself. The school closed in 1744 and Goussier started an illustrator career. He worked with scientists such as La Condamine, Étienne-Claude de Marivetz and Roland de La Platière. In 1792, he was hired by the Minister of the Interior (arts and craft division) and in 1794 by the Comité de Salut public (weapons division).
Personal life
In 1751, he married Marie-Anne-Françoise Simmonneau. They had two children.
His wife sent him to jail, once, allegating he had no religion and that he didn't have respect for divine and human laws. Ten days later she changed her mind, telling others that he was an honest man with spirit.
As a person, he was a beloved to many, a good husband and a good friend. He liked both pleasure and science.
Denis Diderot made a portrait of Goussier in Jacques le fataliste et son maître, where he stands as La Gousse.
Diderot's encyclopedia
Louis-Jacques Goussier is famous for his work on Diderot's encyclopedia. He was the first drawer to be hired on that project, in 1747 and he did himself more than 900 plates and directed the drawing of the others. Some call Goussier the third encyclopedist, after Diderot and d'Alembert.
Goussier spent ten years visiting people of all arts and techniques (textile, smith, mill, glass, etc.), and twenty-five years drawing. He also wrote sixty-one articles.
References
French illustrators
Contributors to the Encyclopédie (1751–1772)
Artists from Paris
1799 deaths
1722 births
French male non-fiction writers
18th-century French male writers
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https://en.wikipedia.org/wiki/List%20of%20Hannover%2096%20records%20and%20statistics
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This is a list of records set by the football team Hannover 96.
Team records
Biggest home win: 13–1 v Rasen SV Hildesheim 06 (1933–34 Gauliga Niedersachsen)
Biggest home win (Bundesliga): 6–0 v Borussia Neunkirchen (18 September 1965)
Biggest home win (DFB Cup): 8–1 v Borussia Hannover (29 August 1980)
Biggest away win: 7–2 v VfB Oldenburg (2. Bundesliga; 30 May 1981)
Biggest away win (Bundesliga):
5–1 v SC Tasmania 1900 Berlin (2 October 1965)
5–1 v Kickers Offenbach (27 February 1971)
Biggest away win (DFB Cup): 7-0 v Marathon 1902 Berlin (1 August 1991)
Biggest home defeat: 0–10 v FC St. Pauli (1947–48 Oberliga Nord)
Biggest home defeat (Bundesliga):
0–5 v FC Bayern Munich (1 February 1986)
0–5 v KFC Uerdingen 05 (12 November 1988)
0–5 v VfL Wolfsburg (16 May 2009)
Biggest home defeat (DFB Cup): 0–4 v SV Darmstadt 98 (27 August 1982)
Biggest away defeat (Bundesliga):
0–7 v VfB Stuttgart (8 February 1986)
0–7 v FC Bayern Munich (18 April 2010)
Biggest away defeat (DFB Cup): 1–5 v SV Waldhof Mannheim (12 November 1985)
Most league goals (season): 120 (1997–98 Regionalliga Nord)
Most points:
Two points for a win: 56 (1980–81 2. Bundesliga, 1986–87 2. Bundesliga)
Three points for a win: 89 (1997–98 Regionalliga Nord)
Appearances
Most league appearances: Jörg Sievers - 384 (1989–2003)
Most appearances (all games): Peter Anders - 458 (1966–1981)
Most Bundesliga appearances: Steven Cherundolo - 300 (1999–2014)
Most international appearances while at club: Steven Cherundolo - 87 (1999–2014)
Goals
Most league goals: Dieter Schatzschneider - 135 (1978–82; 1988–89)
Most Bundesliga goals: Hans Siemensmeyer - 72 (1965–74)
Most European goals: Hans Siemensmeyer - 7 (1965–74)
Most league goals (season): Dieter Schatzschneider - 34 (1981–82)
Transfers
Highest fee paid: €9 million to Rubin Kazan for Jonathas (2017)
Highest fee received: €8 million from Stoke City for Joselu (2015)
Records
German football club statistics
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https://en.wikipedia.org/wiki/Surface%20of%20constant%20width
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In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes. One defines the width of the surface in a given direction to be the perpendicular distance between the parallels perpendicular to that direction. Thus, a surface of constant width is the three-dimensional analogue of a curve of constant width, a two-dimensional shape with a constant distance between pairs of parallel tangent lines.
Definition
More generally, any compact convex body D has one pair of parallel supporting planes in a given direction. A supporting plane is a plane that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the body itself is referred to as a spheroform.
Examples
A sphere, a surface of constant radius and thus diameter, is a surface of constant width.
Contrary to common belief the Reuleaux tetrahedron is not a surface of constant width. However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width. These shapes were conjectured by to have the minimum volume among all shapes with the same constant width, but this conjecture remains unsolved.
Among all surfaces of revolution with the same constant width, the one with minimum volume is the shape swept out by a Reuleaux triangle rotating about one of its axes of symmetry, while the one with maximum volume is the sphere.
Properties
Every parallel projection of a surface of constant width is a curve of constant width. By Barbier's theorem, the perimeter of this projection is times the width, regardless of the direction of projection. It follows that every surface of constant width is also a surface of constant girth, where the girth of a shape is the perimeter of one of its parallel projections. Conversely, Hermann Minkowski proved that every surface of constant girth is also a surface of constant width.
The shapes whose parallel projections have constant area (rather than constant perimeter) are called bodies of constant brightness.
References
Notes
Sources
.
.
.
Further reading
.
External links
Spheroforms
T. Lachand-Robert & É. Oudet, "Bodies of constant width in arbitrary dimension"
How Round is Your Circle? Solids of constant width
Euclidean solid geometry
Geometric shapes
Constant width
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https://en.wikipedia.org/wiki/Quasi-compact%20morphism
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In algebraic geometry, a morphism between schemes is said to be quasi-compact if Y can be covered by open affine subschemes such that the pre-images are quasi-compact (as topological space). If f is quasi-compact, then the pre-image of a quasi-compact open subscheme (e.g., open affine subscheme) under f is quasi-compact.
It is not enough that Y admits a covering by quasi-compact open subschemes whose pre-images are quasi-compact. To give an example, let A be a ring that does not satisfy the ascending chain conditions on radical ideals, and put . X contains an open subset U that is not quasi-compact. Let Y be the scheme obtained by gluing two X'''s along U. X, Y are both quasi-compact. If is the inclusion of one of the copies of X, then the pre-image of the other X, open affine in Y, is U, not quasi-compact. Hence, f is not quasi-compact.
A morphism from a quasi-compact scheme to an affine scheme is quasi-compact.
Let be a quasi-compact morphism between schemes. Then is closed if and only if it is stable under specialization.
The composition of quasi-compact morphisms is quasi-compact. The base change of a quasi-compact morphism is quasi-compact.
An affine scheme is quasi-compact. In fact, a scheme is quasi-compact if and only if it is a finite union of open affine subschemes. Serre’s criterion gives a necessary and sufficient condition for a quasi-compact scheme to be affine.
A quasi-compact scheme has at least one closed point.
See also
fpqc morphism
References
Hartshorne, Algebraic Geometry''.
Angelo Vistoli, "Notes on Grothendieck topologies, fibered categories and descent theory."
External links
When is an irreducible scheme quasi-compact?
Morphisms of schemes
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https://en.wikipedia.org/wiki/Tom%20Hull%20%28mathematician%29
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Thomas C. Hull is an associate professor of mathematics at Western New England University and is known for his expertise in the mathematics of paper folding.
Career
Hull was an undergraduate at Hampshire College. He earned a master's degree and Ph.D. in mathematics at the University of Rhode Island. His 1997 dissertation, Some Problems in List Coloring Bipartite Graphs, involved graph coloring, and was supervised by Nancy Eaton.
Prior to his appointment at Western New England, Hull taught at Merrimack College. He has also taught at the Hampshire College Summer Studies in Mathematics for many years: as junior staff from 1991 to 1995, and as senior staff in 1998 to 2007. Since 2013, he has taught at MathILy, an intensive residential summer program for mathematically excellent high school students.
Hull was a member of the board of directors of origami association OrigamiUSA from 1995 to 2008.
Author
Hull is the author or co-author of several books on origami, including:
Origametry: Mathematical Methods in Paper Folding (Cambridge University Press, 2021)
Project Origami: Activities for Exploring Mathematics (AK Peters, 2006; 2nd ed., CRC Press, 2013)
Russian Origami: 40 Original Models Designed by the Top Folders in the Former Soviet Union (with Sergei Afonkin, St. Martin's Press, 1998)
Origami, Plain and Simple (with Robert E. Neale, St. Martin's Press, 1994)
He is also featured in the 2010 origami documentary Between the Folds.
Awards and honors
With Tomohiro Tachi of the University of Tokyo, Hull was the recipient of the 2016 A. T. Yang Memorial Award in Theoretical Kinematics of the American Society of Mechanical Engineers, for their joint work on predicting the motion of rigid origami patterns when forces are applied to them in their flat state.
References
External links
Home page
Google scholar profile
Origami videos on Youtube
Innovative Math: from Origami to Calculus. Report from a visit of Hull to Phillips Exeter Academy, April 14, 2009.
MathILy Website
Living people
Origami artists
20th-century American mathematicians
21st-century American mathematicians
Geometers
Hampshire College alumni
University of Rhode Island alumni
Western New England University faculty
Mathematical artists
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Action%20algebra
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In algebraic logic, an action algebra is an algebraic structure which is both a residuated semilattice and a Kleene algebra. It adds the star or reflexive transitive closure operation of the latter to the former, while adding the left and right residuation or implication operations of the former to the latter. Unlike dynamic logic and other modal logics of programs, for which programs and propositions form two distinct sorts, action algebra combines the two into a single sort. It can be thought of as a variant of intuitionistic logic with star and with a noncommutative conjunction whose identity need not be the top element. Unlike Kleene algebras, action algebras form a variety, which furthermore is finitely axiomatizable, the crucial axiom being a•(a → a)* ≤ a. Unlike models of the equational theory of Kleene algebras (the regular expression equations), the star operation of action algebras is reflexive transitive closure in every model of the equations. Action algebras were introduced by Vaughan Pratt in the European Workshop JELIA'90.
Definition
An action algebra (A, ∨, 0, •, 1, ←, →, *) is an algebraic structure such that (A, ∨, •, 1, ←, →) forms a residuated semilattice in the sense of Ward and Dilworth, while (A, ∨, 0, •, 1, *) forms a Kleene algebra in the sense of Dexter Kozen. That is, it is any model of the joint theory of both classes of algebras. Now Kleene algebras are axiomatized with quasiequations, that is, implications between two or more equations, whence so are action algebras when axiomatized directly in this way. However, action algebras have the advantage that they also have an equivalent axiomatization that is purely equational. The language of action algebras extends in a natural way to that of action lattices, namely by the inclusion of a meet operation.
In the following we write the inequality a ≤ b as an abbreviation for the equation a ∨ b = b. This allows us to axiomatize the theory using inequalities yet still have a purely equational axiomatization when the inequalities are expanded to equalities.
The equations axiomatizing action algebra are those for a residuated semilattice, together with the following equations for star.
1 ∨ a*•a* ∨ a ≤ a*
a* ≤ (a ∨ b)*
(a → a)* ≤ a → a
The first equation can be broken out into three equations, 1 ≤ a*, a*•a* ≤ a*, and a ≤ a*. Defining a to be reflexive when 1 ≤ a and transitive when a•a ≤ a by abstraction from binary relations, the first two of those equations force a* to be reflexive and transitive while the third forces a* to be greater or equal to a. The next axiom asserts that star is monotone. The last axiom can be written equivalently as a•(a → a)* ≤ a, a form which makes its role as induction more apparent. These two axioms in conjunction with the axioms for a residuated semilattice force a* to be the least reflexive transitive element of the semilattice of elements greater or equal to a. Taking that as the definition of reflexive transiti
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https://en.wikipedia.org/wiki/Claw-free%20graph
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In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw as an induced subgraph.
A claw is another name for the complete bipartite graph K1,3 (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood of any vertex is the complement of a triangle-free graph.
Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, and the characterization of claw-free perfect graphs. They are the subject of hundreds of mathematical research papers and several surveys.
Examples
The line graph L(G) of any graph G is claw-free; L(G) has a vertex for every edge of G, and vertices are adjacent in L(G) whenever the corresponding edges share an endpoint in G. A line graph L(G) cannot contain a claw, because if three edges e1, e2, and e3 in G all share endpoints with another edge e4 then by the pigeonhole principle at least two of e1, e2, and e3 must share one of those endpoints with each other. Line graphs may be characterized in terms of nine forbidden subgraphs; the claw is the simplest of these nine graphs. This characterization provided the initial motivation for studying claw-free graphs.
The de Bruijn graphs (graphs whose vertices represent n-bit binary strings for some n, and whose edges represent (n − 1)-bit overlaps between two strings) are claw-free. One way to show this is via the construction of the de Bruijn graph for n-bit strings as the line graph of the de Bruijn graph for (n − 1)-bit strings.
The complement of any triangle-free graph is claw-free. These graphs include as a special case any complete graph.
Proper interval graphs, the interval graphs formed as intersection graphs of families of intervals in which no interval contains another interval, are claw-free, because four properly intersecting intervals cannot intersect in the pattern of a claw. The same is true more generally for proper circular-arc graphs.
The Moser spindle, a seven-vertex graph used to provide a lower bound for the chromatic number of the plane, is claw-free.
The graphs of several polyhedra and polytopes are claw-free, including the graph of the tetrahedron and more generally of any simplex (a complete graph), the graph of the octahedron and more generally of any cross polytope (isomorphic to the cocktail party graph formed by removing a perfect matching from a complete graph), the graph of the regular icosahedron, and the graph of the 16-cell.
The Schläfli graph, a strongly regular graph with 27 vertices
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https://en.wikipedia.org/wiki/Arthur%20T.%20Benjamin
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Arthur T. Benjamin (born March 19, 1961) is an American mathematician who specializes in combinatorics. Since 1989 he has been a professor of mathematics at Harvey Mudd College, where he is the Smallwood Family Professor of Mathematics.
He is known for mental math capabilities and "Mathemagics" performances in front of live audiences. His mathematical abilities have been highlighted in newspaper and magazine articles, at TED Talks and on the Colbert Report.
Education
Benjamin earned a Bachelor of Science with highest honors in applied mathematics at Carnegie Mellon University in 1983. He then went on to receive a Master of Science in Engineering in 1985 and a Doctor of Philosophy in 1989 in mathematical sciences at Johns Hopkins University. His PhD dissertation was titled "Turnpike Structures for Optimal Maneuvers", and was supervised by Alan J. Goldman.
During his freshman year at CMU he wrote the lyrics and created the magic effects for the musical comedy, Kije!, in collaboration with author Scott McGregor and composer Arthur Darrell Turner. This musical was the winner of an annual competition and was first performed as the CMU's Spring Musical in 1980.
Career
Academic
Benjamin held several mathematics positions while attending university, including stints with the National Bureau of Standards, the National Security Agency, and the Institute for Defense Analyses. Upon receipt of his PhD he was hired as an assistant professor of mathematics at Harvey Mudd College. He is currently a full professor at Harvey Mudd and was chair of the mathematics department from 2002 to 2004. He has published over 90 academic papers and five books. He has also filmed several sets of lectures on mathematical topics for The Great Courses series from The Teaching Company, including a course on Discrete Mathematics, Mental Math, and The Mathematics of Games and Puzzles: From Cards to Sudoku. He served as co-editor of Math Horizons magazine for five years.
Mathemagics
Benjamin has long had an interest in magic. While in college he honed his skills as a magician and attended magic conferences. At one of these conferences he met well-known magician and skeptic James Randi, who greatly influenced Benjamin's decision to perform Mathemagics shows for live audiences. Randi invited him to perform his mathematical tricks on a television program called Exploring Psychic Powers Live, co-hosted by Uri Geller. Randi also encouraged Benjamin to become involved in the growing skeptical movement. He attended early meetings of the Southern California Skeptics in the 1990s, which later evolved into the Skeptics Society. It was at these meetings that he met Skeptics Society President Michael Shermer, who would later become a co-author on three of Benjamin's books.
Benjamin regularly performs his Mathemagics program for live audiences at schools, colleges, conferences, and even at The Magic Castle in Hollywood, California. These shows feature Benjamin performing mathematical feat
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https://en.wikipedia.org/wiki/Semi-elliptic%20operator
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In mathematics — specifically, in the theory of partial differential equations — a semi-elliptic operator is a partial differential operator satisfying a positivity condition slightly weaker than that of being an elliptic operator. Every elliptic operator is also semi-elliptic, and semi-elliptic operators share many of the nice properties of elliptic operators: for example, much of the same existence and uniqueness theory is applicable, and semi-elliptic Dirichlet problems can be solved using the methods of stochastic analysis.
Definition
A second-order partial differential operator P defined on an open subset Ω of n-dimensional Euclidean space Rn, acting on suitable functions f by
is said to be semi-elliptic if all the eigenvalues λi(x), 1 ≤ i ≤ n, of the matrix a(x) = (aij(x)) are non-negative. (By way of contrast, P is said to be elliptic if λi(x) > 0 for all x ∈ Ω and 1 ≤ i ≤ n, and uniformly elliptic if the eigenvalues are uniformly bounded away from zero, uniformly in i and x.) Equivalently, P is semi-elliptic if the matrix a(x) is positive semi-definite for each x ∈ Ω.
References
(See Section 9)
Differential operators
Partial differential equations
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https://en.wikipedia.org/wiki/Feller-continuous%20process
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In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously on the initial condition of the process. The concept is named after Croatian-American mathematician William Feller.
Definition
Let X : [0, +∞) × Ω → Rn, defined on a probability space (Ω, Σ, P), be a stochastic process. For a point x ∈ Rn, let Px denote the law of X given initial value X0 = x, and let Ex denote expectation with respect to Px. Then X is said to be a Feller-continuous process if, for any fixed t ≥ 0 and any bounded, continuous and Σ-measurable function g : Rn → R, Ex[g(Xt)] depends continuously upon x.
Examples
Every process X whose paths are almost surely constant for all time is a Feller-continuous process, since then Ex[g(Xt)] is simply g(x), which, by hypothesis, depends continuously upon x.
Every Itô diffusion with Lipschitz-continuous drift and diffusion coefficients is a Feller-continuous process.
See also
Continuous stochastic process
References
(See Lemma 8.1.4)
Stochastic processes
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https://en.wikipedia.org/wiki/Residuated%20Boolean%20algebra
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In mathematics, a residuated Boolean algebra is a residuated lattice whose lattice structure is that of a Boolean algebra. Examples include Boolean algebras with the monoid taken to be conjunction, the set of all formal languages over a given alphabet Σ under concatenation, the set of all binary relations on a given set X under relational composition, and more generally the power set of any equivalence relation, again under relational composition. The original application was to relation algebras as a finitely axiomatized generalization of the binary relation example, but there exist interesting examples of residuated Boolean algebras that are not relation algebras, such as the language example.
Definition
A residuated Boolean algebra is an algebraic structure such that
An equivalent signature better suited to the relation algebra application is where the unary operations x\ and x▷ are intertranslatable in the manner of De Morgan's laws via
x\y = ¬(x▷¬y), x▷y = ¬(x\¬y),
and dually /y and ◁y as
x/y = ¬(¬x◁y), x◁y = ¬(¬x/y),
with the residuation axioms in the residuated lattice article reorganized accordingly (replacing z by ¬z) to read
⇔ ⇔
This De Morgan dual reformulation is motivated and discussed in more detail in the section below on conjugacy.
Since residuated lattices and Boolean algebras are each definable with finitely many equations, so are residuated Boolean algebras, whence they form a finitely axiomatizable variety.
Examples
Any Boolean algebra, with the monoid multiplication • taken to be conjunction and both residuals taken to be material implication x→y. Of the remaining 15 binary Boolean operations that might be considered in place of conjunction for the monoid multiplication, only five meet the monotonicity requirement, namely 0, 1, x, y, and . Setting y = z = 0 in the residuation axiom y ≤ x\z ⇔ x•y ≤ z, we have 0 ≤ x\0 ⇔ x•0 ≤ 0, which is falsified by taking x = 1 when x•y = 1, x, or . The dual argument for z/y rules out x•y = y. This just leaves x•y = 0 (a constant binary operation independent of x and y), which satisfies almost all the axioms when the residuals are both taken to be the constant operation x/y = x\y = 1. The axiom it fails is , for want of a suitable value for . Hence conjunction is the only binary Boolean operation making the monoid multiplication that of a residuated Boolean algebra.
The power set 2X2 made a Boolean algebra as usual with ∩, ∪ and complement relative to X2, and made a monoid with relational composition. The monoid unit is the identity relation {(x,x)|x ∈ X}. The right residual R\S is defined by x(R\S)y if and only if for all z in X, zRx implies zSy. Dually the left residual S/R is defined by y(S/R)x if and only if for all z in X, xRz implies ySz.
The power set 2Σ* made a Boolean algebra as for Example 2, but with language concatenation for the monoid. Here the set Σ is used as an alphabet while Σ* denotes the set of all finite (including empty) words over
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https://en.wikipedia.org/wiki/Problems%20in%20Latin%20squares
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In mathematics, the theory of Latin squares is an active research area with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Problems posed here appeared in, for instance, the Loops (Prague) conferences and the Milehigh (Denver) conferences.
Open problems
Bounds on maximal number of transversals in a Latin square
A transversal in a Latin square of order n is a set S of n cells such that every row and every column contains exactly one cell of S, and such that the symbols in S form {1, ..., n}. Let T(n) be the maximum number of transversals in a Latin square of order n. Estimate T(n).
Proposed: by Ian Wanless at Loops '03, Prague 2003
Comments: Wanless, McKay and McLeod have bounds of the form cn < T(n) < dn n!, where c > 1 and d is about 0.6. A conjecture by Rivin, Vardi and Zimmermann (Rivin et al., 1994) says that you can place at least exp(c n log n) queens in non-attacking positions on a toroidal chessboard (for some constant c). If true this would imply that T(n) > exp(c n log n). A related question is to estimate the number of transversals in the Cayley tables of cyclic groups of odd order. In other words, how many orthomorphisms do these groups have?
The minimum number of transversals of a Latin square is also an open problem. H. J. Ryser conjectured (Oberwolfach, 1967) that every Latin square of odd order has one. Closely related is the conjecture, attributed to Richard Brualdi, that every Latin square of order n has a partial transversal of order at least n − 1.
Characterization of Latin subsquares in multiplication tables of Moufang loops
Describe how all Latin subsquares in multiplication tables of Moufang loops arise.
Proposed: by Aleš Drápal at Loops '03, Prague 2003
Comments: It is well known that every Latin subsquare in a multiplication table of a group G is of the form aH x Hb, where H is a subgroup of G and a, b are elements of G.
Densest partial Latin squares with Blackburn property
A partial Latin square has Blackburn property if whenever the cells (i, j) and (k, l) are occupied by the same symbol, the opposite corners (i, l) and (k, j) are empty. What is the highest achievable density of filled cells in a partial Latin square with the Blackburn property? In particular, is there some constant c > 0 such that we can always fill at least c n2 cells?
Proposed: by Ian Wanless at Loops '03, Prague 2003
Comments: In a paper to appear, Wanless has shown that if c exists then c < 0.463. He also constructed a family of partial Latin squares with the Blackburn property and asymptotic density of at least exp(-d(log n)1/2) for constant d > 0.
Largest power of 2 dividing the number of Latin squares
Let be the number of Latin squares of order n. What is the largest integer such that divides ? Does grow quadratically in n?
Proposed: by Ian Wanless at Loops '03, Prague 2003
Comments: Of course, where is the number of reduced Latin squares of
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https://en.wikipedia.org/wiki/Z%C3%A9%20Ant%C3%B4nio
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José Antônio Pereira (born March 19, 1984 in Monte Azul Paulista), or simply Zé Antônio, is a Brazilian footballer, who plays as a midfielder.
Career statistics
Club
Honours
Brazilian League (2nd division): 2006
Minas Gerais State League: 2007
Campeonato Pernambucano in 2010 with Sport Recife
External links
CBF
footballbusiness
websoccerclub
References
1984 births
Living people
Footballers from São Paulo (state)
Brazilian men's footballers
Brazilian expatriate men's footballers
Botafogo Futebol Clube (SP) players
Clube Atlético Mineiro players
BK Häcken players
Club Athletico Paranaense players
Sport Club do Recife players
Goiás Esporte Clube players
Associação Portuguesa de Desportos players
Paysandu Sport Club players
Guarani FC players
Clube Atlético Linense players
Figueirense FC players
Esporte Clube Santo André players
Joinville Esporte Clube players
Men's association football midfielders
Brazilian expatriate sportspeople in Sweden
Expatriate men's footballers in Sweden
People from Monte Azul Paulista
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https://en.wikipedia.org/wiki/Sainte-C%C3%A9cile-de-L%C3%A9vrard
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Sainte-Cécile-de-Lévrard is a parish municipality in the Centre-du-Québec region of the province of Quebec in Canada.
Demographics
In the 2021 Canadian census conducted by Statistics Canada, Sainte-Cécile-de-Lévrard had a population of 349 living in 153 of its 163 total private dwellings, a change of from its 2016 population of 372. With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec
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https://en.wikipedia.org/wiki/Sainte-Sophie-de-L%C3%A9vrard%2C%20Quebec
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Sainte-Sophie-de-Lévrard is a parish municipality in the Centre-du-Québec region of the province of Quebec in Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Sainte-Sophie-de-Lévrard had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
See also
List of parish municipalities in Quebec
References
External links
Parish municipalities in Quebec
Incorporated places in Centre-du-Québec
Canada geography articles needing translation from French Wikipedia
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https://en.wikipedia.org/wiki/Continuous%20stochastic%20process
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In probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for (the sample paths of) a process to have, since it implies that they are well-behaved in some sense, and, therefore, much easier to analyze. It is implicit here that the index of the stochastic process is a continuous variable. Some authors define a "continuous (stochastic) process" as only requiring that the index variable be continuous, without continuity of sample paths: in another terminology, this would be a continuous-time stochastic process, in parallel to a "discrete-time process". Given the possible confusion, caution is needed.
Definitions
Let (Ω, Σ, P) be a probability space, let T be some interval of time, and let X : T × Ω → S be a stochastic process. For simplicity, the rest of this article will take the state space S to be the real line R, but the definitions go through mutatis mutandis if S is Rn, a normed vector space, or even a general metric space.
Continuity almost surely
Given a time t ∈ T, X is said to be continuous with probability one at t if
Mean-square continuity
Given a time t ∈ T, X is said to be continuous in mean-square at t if E[|Xt|2] < +∞ and
Continuity in probability
Given a time t ∈ T, X is said to be continuous in probability at t if, for all ε > 0,
Equivalently, X is continuous in probability at time t if
Continuity in distribution
Given a time t ∈ T, X is said to be continuous in distribution at t if
for all points x at which Ft is continuous, where Ft denotes the cumulative distribution function of the random variable Xt.
Sample continuity
X is said to be sample continuous if Xt(ω) is continuous in t for P-almost all ω ∈ Ω. Sample continuity is the appropriate notion of continuity for processes such as Itō diffusions.
Feller continuity
X is said to be a Feller-continuous process if, for any fixed t ∈ T and any bounded, continuous and Σ-measurable function g : S → R, Ex[g(Xt)] depends continuously upon x. Here x denotes the initial state of the process X, and Ex denotes expectation conditional upon the event that X starts at x.
Relationships
The relationships between the various types of continuity of stochastic processes are akin to the relationships between the various types of convergence of random variables. In particular:
continuity with probability one implies continuity in probability;
continuity in mean-square implies continuity in probability;
continuity with probability one neither implies, nor is implied by, continuity in mean-square;
continuity in probability implies, but is not implied by, continuity in distribution.
It is tempting to confuse continuity with probability one with sample continuity. Continuity with probability one at time t means that P(At) = 0, where the event At is given by
and it is perfectly feasible to check whether or not this holds for each t ∈ T. Sam
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https://en.wikipedia.org/wiki/Closed%20testing%20procedure
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In statistics, the closed testing procedure is a general method for performing more than one hypothesis test simultaneously.
The closed testing principle
Suppose there are k hypotheses H1,..., Hk to be tested and the overall type I error rate is α. The closed testing principle allows the rejection of any one of these elementary hypotheses, say Hi, if all possible intersection hypotheses involving Hi can be rejected by using valid local level α tests; the adjusted p-value is the largest among those hypotheses. It controls the family-wise error rate for all the k hypotheses at level α in the strong sense.
Example
Suppose there are three hypotheses H1,H2, and H3 to be tested and the overall type I error rate is 0.05. Then H1 can be rejected at level α if H1 ∩ H2 ∩ H3, H1 ∩ H2, H1 ∩ H3 and H1 can all be rejected using valid tests with level α.
Special cases
The Holm–Bonferroni method is a special case of a closed test procedure for which each intersection null hypothesis is tested using the simple Bonferroni test. As such, it controls the family-wise error rate for all the k hypotheses at level α in the strong sense.
Multiple test procedures developed using the graphical approach for constructing and illustrating multiple test procedures are a subclass of closed testing procedures.
See also
Multiple comparisons
Holm–Bonferroni method
Bonferroni correction
References
Statistical hypothesis testing
Statistical tests
Multiple comparisons
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https://en.wikipedia.org/wiki/Christine%20Hamill
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Christine Mary Hamill (24 July 1923 – 24 March 1956) was an English mathematician who specialised in group theory and finite geometry.
Education
Hamill was one of the four children of English physiologist Philip Hamill. She attended St Paul's Girls' School and the Perse School for Girls. In 1942, she won a scholarship to Newnham College, Cambridge, becoming a wrangler in 1945.
She won a Newnham research fellowship in 1948, and received her Ph.D. at the University of Cambridge in 1951. Her dissertation, The Finite Primitive Collineation Groups which contain Homologies of Period Two,
concerned the group-theoretic properties of collineations, geometric transformations preserving straight lines;
she also published this material in three journal papers. J. A. Todd, who supervised her research work, observed that "the detailed results contained in her papers" were "of permanent value".
Career
After completing her doctorate, Hamill was appointed to a lectureship in the University of Sheffield. In 1954, she was appointed lecturer in the University College, Ibadan, Nigeria. She died of polio there in 1956, four months before she was to have married.
Notes
1923 births
1956 deaths
20th-century English mathematicians
People educated at St Paul's Girls' School
Alumni of Newnham College, Cambridge
Fellows of Newnham College, Cambridge
Academics of the University of Sheffield
Academic staff of the University of Ibadan
Group theorists
People educated at the Perse School for Girls
Deaths from polio
British geometers
20th-century women mathematicians
English people with disabilities
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https://en.wikipedia.org/wiki/Arlie%20Petters
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Arlie Oswald Petters, MBE (born February 8, 1964) is a Belizean-American mathematical physicist, who is the Benjamin Powell Professor of mathematics and a professor of physics and economics at Duke University. Petters became the provost at New York University Abu Dhabi effective September 1, 2020. Petters is a founder of mathematical astronomy, focusing on problems connected to the interplay of gravity and light and employing tools from astrophysics, cosmology, general relativity, high energy physics, differential geometry, singularities, and probability theory. His monograph "Singularity Theory and Gravitational Lensing" developed a mathematical theory of gravitational lensing. Petters was also the dean of academic affairs for Trinity College of Arts and Sciences and associate vice provost for undergraduate education at Duke University (2016-2019).
Biography
Petters was raised by his grandparents in the rural community of Dangriga, Belize (formerly Stann Creek Town, British Honduras). His mother immigrated to Brooklyn, New York, and married a U.S. citizen, with Arlie joining them when he was 14 years old.
Petters earned a B.A./M.A. in Mathematics and Physics from Hunter College, CUNY in 1986 with a thesis on "The Mathematical Theory of General Relativity", and began his Ph.D. at the Massachusetts Institute of Technology Department of Mathematics in the same year. After two years of doctoral studies, he became an exchange scholar in the Princeton University Department of Physics in absentia from MIT. Petters earned his Ph.D. in mathematics in 1991 under advisors Bertram Kostant (MIT) and David Spergel (Princeton University). He remained at MIT for two years as an instructor of pure mathematics (1991–1993) and then joined the faculty at Princeton University in the Department of Mathematics. He was an assistant professor at Princeton for five years (1993–1998) before moving to Duke University.
Many media outlets have profiled Arlie Petters and his scholarship, including The New York Times, NOVA, The HistoryMakers (a digital archive of oral histories featuring African-Americans and preserved at the Library of Congress), Big Think, and Duke University's news outlet, The Chronicle.
Research
Petters is known for his work in the mathematical theory of gravitational lensing.
Over the ten-year period from 1991 to 2001, Petters systematically developed a mathematical theory of weak-deflection gravitational lensing, beginning with his 1991 MIT Ph.D. thesis on "Singularities in Gravitational Microlensing". In a series of papers, he and his collaborators resolved an array of theoretical problems in weak-deflection gravitational lensing covering image counting, fixed-point images, image magnification, image time delays, local geometry of caustics, global geometry of caustics, wavefronts, caustic surfaces, and caustic surfing. His work culminated in book, entitled Singularity Theory and Gravitational Lensing (Springer 2012), which he co-authored with
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https://en.wikipedia.org/wiki/Leandro%20Amaral
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Leandro Câmara do Amaral (born 6 August 1977), is a Brazilian former footballer who played as a striker.
Career statistics
Club
Honours
Club
Fiorentina
Coppa Italia (1): 2000–01
Individual
Bola de Prata (1): 2007
References
External links
Zerozero.pt
Guardian Stats Centre
1977 births
Living people
Men's association football forwards
Brazilian men's footballers
Brazil men's international footballers
2001 FIFA Confederations Cup players
Associação Portuguesa de Desportos players
ACF Fiorentina players
Grêmio Foot-Ball Porto Alegrense players
São Paulo FC players
Sociedade Esportiva Palmeiras players
Sport Club Corinthians Paulista players
Ituano FC players
FC Istres players
Fluminense FC players
CR Vasco da Gama players
CR Flamengo footballers
Campeonato Brasileiro Série A players
Serie A players
Ligue 1 players
Expatriate men's footballers in Italy
Expatriate men's footballers in France
Brazilian expatriate men's footballers
Footballers from São Paulo
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https://en.wikipedia.org/wiki/Bergman%20metric
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In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold. It is so called because it is derived from the Bergman kernel, both of which are named after Stefan Bergman.
Definition
Let be a domain and let be the Bergman kernel
on G. We define a Hermitian metric on the tangent bundle by
for . Then the length of a tangent vector is
given by
This metric is called the Bergman metric on G.
The length of a (piecewise) C1 curve is
then computed as
The distance of two points is then defined as
The distance dG is called the Bergman distance.
The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain. More importantly, the distance dG is invariant under
biholomorphic mappings of G to another domain . That is if f
is a biholomorphism of G and , then .
References
Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
Complex manifolds
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https://en.wikipedia.org/wiki/Liouville%27s%20theorem%20%28conformal%20mappings%29
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In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of Rn, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions). This theorem severely limits the variety of possible conformal mappings in R3 and higher-dimensional spaces. By contrast, conformal mappings in R2 can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.
Generalizations of the theorem hold for transformations that are only weakly differentiable . The focus of such a study is the non-linear Cauchy–Riemann system that is a necessary and sufficient condition for a smooth mapping to be conformal:
where Df is the Jacobian derivative, T is the matrix transpose, and I is the identity matrix. A weak solution of this system is defined to be an element f of the Sobolev space with non-negative Jacobian determinant almost everywhere, such that the Cauchy–Riemann system holds at almost every point of Ω. Liouville's theorem is then that every weak solution (in this sense) is a Möbius transformation, meaning that it has the form
where a, b are vectors in Rn, α is a scalar, A is a rotation matrix, , and the matrix in parentheses is I or a Householder matrix (so, orthogonal). Equivalently stated, any quasiconformal map of a domain in Euclidean space that is also conformal is a Möbius transformation. This equivalent statement justifies using the Sobolev space W1,n, since then follows from the geometrical condition of conformality and the ACL characterization of Sobolev space. The result is not optimal however: in even dimensions , the theorem also holds for solutions that are only assumed to be in the space W, and this result is sharp in the sense that there are weak solutions of the Cauchy–Riemann system in W1,p for any that are not Möbius transformations. In odd dimensions, it is known that W1,n is not optimal, but a sharp result is not known.
Similar rigidity results (in the smooth case) hold on any conformal manifold. The group of conformal isometries of an n-dimensional conformal Riemannian manifold always has dimension that cannot exceed that of the full conformal group . Equality of the two dimensions holds exactly when the conformal manifold is isometric with the n-sphere or projective space. Local versions of the result also hold: The Lie algebra of conformal Killing fields in an open set has dimension less than or equal to that of the conformal group, with equality holding if and only if the open set is locally conformally flat.
Notes
References
.
Harley Flanders (1966) "Liouville's theorem on conformal mapping", Journal of Mathematics and Mechanics 15: 157–61,
.
.
Conformal mappings
Theorems in geometry
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