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https://en.wikipedia.org/wiki/Steve%20Molnar | Steve Molnar (February 28, 1947 – January 16, 2021) was a Canadian Football League running back.
Molnar played in the 1972 and 1976 Grey Cups for the Saskatchewan Roughriders.
Statistics
When Molnar played for the Saskatchewan Roughriders he scored in the 1975 Western Conference final 3 touchdowns, 1 two point convert, 0 field goals, 0 safeties, for a total of 20 points.
Notes and references
1947 births
2021 deaths
Canadian football running backs
Players of Canadian football from Saskatchewan
Saskatchewan Roughriders players
Utah Utes football players |
https://en.wikipedia.org/wiki/Favard%20operator | In functional analysis, a branch of mathematics, the Favard operators are defined by:
where , . They are named after Jean Favard.
Generalizations
A common generalization is:
where is a positive sequence that converges to 0. This reduces to the classical Favard operators when .
References
This paper also discussed Szász–Mirakyan operators, which is why Favard is sometimes credited with their development (e.g. Favard–Szász operators).
Footnotes
Approximation theory |
https://en.wikipedia.org/wiki/Bowl%20Championship%20Series%20controversies | The Bowl Championship Series (BCS) was a selection system used between 1998 and 2013 that was designed, through polls and computer statistics, to determine a No. 1 and No. 2 ranked team in the NCAA Division I Football Bowl Subdivision (FBS). After the final polls, the two top teams were chosen to play in the BCS National Championship Game which determined the BCS national champion team, but not the champion team for independent voting systems (most notably the AP Poll). This format was intended to be "bowl-centered" rather than a traditional playoff system, since numerous FBS Conferences had expressed their unwillingness to participate in a play-off system. However, due to the unique and often esoteric nature of the BCS format, there had been controversy as to which two teams should play for the national championship and which teams should play in the four other BCS bowl games (Fiesta Bowl, Orange Bowl, Rose Bowl, and Sugar Bowl). In this selection process, the BCS was often criticized for conference favoritism, its inequality of access for teams in non-Automatic Qualifying (non-AQ) Conferences (most likely due to those teams having a lower perceived strength of schedule), and perceived monopolistic, "profit-centered" motives. In terms of this last concern, Congress explored the possibility on more than one occasion of holding hearings to determine the legality of the BCS under the terms of the Sherman Anti-Trust Act, and the United States Justice Department also periodically announced interest in investigating the BCS for similar reasons.
Overview
A survey conducted in 2009 at the Quinnipiac University found that 63% of individuals interested in college football preferred a playoff system to the BCS, while only 26 percent supported the BCS as status quo. Arguments from critics typically centered on the validity of BCS national championship pairings and its designated National Champions. Many critics focused strictly on the BCS methodology itself, which employed subjective voting assessments, while others noted the ability for undefeated teams to finish seasons without an opportunity to play the national championship game. For example, in the last six seasons of Division I FBS football, there have been more undefeated non-BCS champions than undefeated BCS champions. Other criticisms involved discrepancies in the allocation of monetary resources from BCS games, as well as the determination of non-championship BCS game participants, which need not comply with the BCS rankings themselves. Critics note that other sports and divisions of college football complete seasons without disputed national champions which critics attribute to the use of the playoff format.
Critics argued that increasing the number of teams would increase the validity of team comparisons in conferences, which do not compete with one another during the regular season; teams typically only play three or four non-conference games, as the result of pre-determined schedules. BCS pr |
https://en.wikipedia.org/wiki/Optimality%20criterion | In statistics, an optimality criterion provides a measure of the fit of the data to a given hypothesis, to aid in model selection. A model is designated as the "best" of the candidate models if it gives the best value of an objective function measuring the degree of satisfaction of the criterion used to evaluate the alternative hypotheses.
The term has been used to identify the different criteria that are used to evaluate a phylogenetic tree. For example, in order to determine the best topology between two phylogenetic trees using the maximum likelihood optimality criterion, one would calculate the maximum likelihood score of each tree and choose the one that had the better score. However, different optimality criteria can select different hypotheses. In such circumstances caution should be exercised when making strong conclusions.
Many other disciplines use similar criteria or have specific measures geared toward the objectives of the field. Optimality criteria include maximum likelihood, Bayesian, maximum parsimony, sum of squared residuals, least absolute deviations, and many others.
References
Statistical hypothesis testing
Model selection |
https://en.wikipedia.org/wiki/Suslin%27s%20theorem | In mathematics, Suslin's theorem may refer to:
The Quillen–Suslin theorem (formerly the Serre conjecture), due to Andrei Suslin.
Any of several theorems about analytic sets due to Mikhail Yakovlevich Suslin; in particular:
There is an analytic subset of the reals that is not Borel
An analytic set whose complement is also analytic is a Borel set, a special case of the Lusin separation theorem
Any analytic set in Rn is the projection of a Borel set in Rn+1
Analytic sets can be constructed using the Suslin operation |
https://en.wikipedia.org/wiki/Szeg%C5%91%20polynomial | In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product
where dμ is a given positive measure on [−π, π]. Writing for the polynomials, they obey a recurrence relation
where is a parameter, called the reflection coefficient or the Szegő parameter.
See also
Cayley transform
Schur class
Favard's theorem
References
G. Szegő, "Orthogonal polynomials", Colloq. Publ., 33, Amer. Math. Soc. (1967)
Orthogonal polynomials |
https://en.wikipedia.org/wiki/Baskakov%20operator | In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operators, and Lupas operators. They are defined by
where ( can be ), , and is a sequence of functions defined on that have the following properties for all :
. Alternatively, has a Taylor series on .
is completely monotone, i.e. .
There is an integer such that whenever
They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions.
Basic results
The Baskakov operators are linear and positive.
References
Footnotes
Approximation theory |
https://en.wikipedia.org/wiki/Vietnamese%20people%20in%20Japan | form Japan's second-largest community of foreign residents ahead of Koreans in Japan and behind Chinese in Japan, according to the statistics of the Ministry of Justice. By in June 2023, there were 520,154 residents. The majority of the Vietnamese legal residents live in the Kantō region and Keihanshin area.
Migration history
Large numbers of Vietnamese students began to choose Japan as a destination in the early 20th century, spurred by the exiled prince Cường Để and the Đông Du Movement (literally, "Travel East movement" or "Eastern Travel movement") he and Phan Bội Châu pioneered. By 1908, 200 Vietnamese students had gone to study at Japanese universities. However, the community of Vietnamese people in Japan is dominated by Vietnam War refugees and their families, who compose about 70% of the total population. Japan began to accept refugees from Vietnam in the late 1970s. The policy of accepting foreign migrants marked a significant break from Japan's post-World War II orientation towards promoting and maintaining racial homogeneity. Most of these migrants settled in Kanagawa and Hyōgo prefectures, the locations of the initial resettlement centres. As they moved out of the resettlement centres, they often gravitated to Zainichi Korean-dominated neighbourhoods; however, they feel little sense of community with Zainichi Koreans, seeing them not as fellow ethnic minorities but as part of the mainstream.
Guest workers began to follow the refugees to Japan in the so-called "third wave" of Vietnamese migration beginning in the 1990s. As contract workers returned home to Vietnam from the countries of the former Eastern Bloc, which by then had begun their transition away from Communism, they began to look for other foreign destinations in which they could earn good incomes, and Japan proved attractive due to its nearby location and high standard of living. By the end of 1994, the annual number of Vietnamese workers going to Japan totalled 14,305 individuals, mostly under industrial traineeship visas. In contrast to other labour-exporting countries in Southeast Asia, the vast majority of migrants were men, due to the Vietnamese government's restrictions on migration for work in traditionally female-dominated fields such as domestic work or entertainment.
During the COVID-19 pandemic travel between Japan and Vietnam was restricted, temporarily halting migration.
Integration
The refugees have suffered various difficulties adjusting to Japanese society, especially in the areas of education and employment; their attendance rate in senior high school is estimated to be only 40%, as compared to 96.6% for Japanese nationals, a fact attributed both to the refugees' lack of Japanese language proficiency as well as the schools' own inability to adjust to the challenges of educating students with different cultural backgrounds. Tensions have also arisen between migrants admitted to Japan as adults, and 1.5 or 2nd-generation children born or educated in Japa |
https://en.wikipedia.org/wiki/Subcountability | In constructive mathematics, a collection is subcountable if there exists a partial surjection from the natural numbers onto it.
This may be expressed as
where denotes that is a surjective function from a onto . The surjection is a member of and here the subclass of is required to be a set.
In other words, all elements of a subcountable collection are functionally in the image of an indexing set of counting numbers and thus the set can be understood as being dominated by the countable set .
Discussion
Nomenclature
Note that nomenclature of countability and finiteness properties vary substantially - in part because many of them coincide when assuming excluded middle. To reiterate, the discussion here concerns the property defined in terms of surjections onto the set being characterized. The language here is common in constructive set theory texts, but the name subcountable has otherwise also been given to properties in terms of injections out of the set being characterized.
The set in the definition can also be abstracted away, and in terms of the more general notion may be called a subquotient of .
Example
Important cases are where the set in question is some subclass of a bigger class of functions as studied in computability theory.
There cannot be a computable surjection from onto the set of total computable functions , as demonstrated via the function from the diagonal construction, which could never be in such a surjections image. However, via the codes of all possible partial computable functions, which also allows non-terminating programs, such subsets of functions, such as the total functions, are seen to be subcountable sets: The total functions are the range of some strict subset of the natural numbers. Being dominated by an uncomputable, and so constructively uncountable, set of numbers, the name subcountable thus conveys that the constructively uncountable set is no bigger than . Note that no effective map between all counting numbers and the unbounded and non-finite indexing set is asserted here, merely the subset relation . Being total is famously not a decidable property. By Rice's theorem on index sets, most domains of indices are, in fact, not computable sets.
The demonstration that is subcountable also implies that it is classically (non-constructively) formally countable, but this does not reflect any effective countability. In other words, the fact that an algorithm listing all total functions in sequence cannot be coded up is not captured by classical axioms regarding set and function existence. We see that, depending on the axioms of a theory, subcountability may be more likely to be provable than countability.
Relation to excluded middle
In constructive logics and set theories tie the existence of a function between infinite (non-finite) sets to questions of decidability and possibly of effectivity. There, the subcountability property splits from countability and is thus not a redundant notio |
https://en.wikipedia.org/wiki/Championnat%20International%20de%20Jeux%20Math%C3%A9matiques%20et%20Logiques | The Championnat International des Jeux Mathématiques et Logiques () is an international mathematics competition mainly for French-speaking countries, but participation is not limited by language.
History
This competition was created in 1987 and was called Championnat de France des jeux mathématiques et logiques (French competition for mathematical and logical games) at that time. Contestants could come from other countries, but the competition was entirely in French. It became Championnat International des Jeux Mathématiques et Logiques at its fourth edition (1990), but was still in French. Then it became more and more international, with the problems translated in more and more languages.
Since its beginning, this competition is organized by the French federation Fédération Française des Jeux Mathématiques (FFJM). But as it became international, federations were created in other countries and they take part in the organization.
The final of the first edition in 1987 took place in Parthenay. From its second edition in 1988 to the 33rd in 2019, the final took place in Paris or its suburbs. At the final of the 33rd edition, it was announced that the next final would be in Lausanne. However, the 34th edition was eventually canceled due to the COVID-19 pandemic and replaced by an online competition "for the pleasure". The final of the 35th edition is still planned to be in Lausanne.
See also
List of mathematics competitions
Notes
External links
Competition home page
Mathematics competitions |
https://en.wikipedia.org/wiki/Inverse%20curve | In inversive geometry, an inverse curve of a given curve is the result of applying an inverse operation to . Specifically, with respect to a fixed circle with center and radius the inverse of a point is the point for which lies on the ray and . The inverse of the curve is then the locus of as runs over . The point in this construction is called the center of inversion, the circle the circle of inversion, and the radius of inversion.
An inversion applied twice is the identity transformation, so the inverse of an inverse curve with respect to the same circle is the original curve. Points on the circle of inversion are fixed by the inversion, so its inverse is itself.
Equations
The inverse of the point with respect to the unit circle is where
or equivalently
So the inverse of the curve determined by with respect to the unit circle is
It is clear from this that inverting an algebraic curve of degree with respect to a circle produces an algebraic curve of degree at most .
Similarly, the inverse of the curve defined parametrically by the equations
with respect to the unit circle is given parametrically as
This implies that the circular inverse of a rational curve is also rational.
More generally, the inverse of the curve determined by with respect to the circle with center and radius is
The inverse of the curve defined parametrically by
with respect to the same circle is given parametrically as
In polar coordinates, the equations are simpler when the circle of inversion is the unit circle. The inverse of the point with respect to the unit circle is where
So the inverse of the curve is determined by and the inverse of the curve is .
Degrees
As noted above, the inverse with respect to a circle of a curve of degree has degree at most . The degree is exactly unless the original curve passes through the point of inversion or it is circular, meaning that it contains the circular points, , when considered as a curve in the complex projective plane. In general, inversion with respect to an arbitrary curve may produce an algebraic curve with proportionally larger degree.
Specifically, if is -circular of degree , and if the center of inversion is a singularity of order on , then the inverse curve will be an -circular curve of degree and the center of inversion is a singularity of order on the inverse curve. Here if the curve does not contain the center of inversion and if the center of inversion is a nonsingular point on it; similarly the circular points, , are singularities of order on . The value can be eliminated from these relations to show that the set of -circular curves of degree , where may vary but is a fixed positive integer, is invariant under inversion.
Examples
Applying the above transformation to the lemniscate of Bernoulli
gives us
the equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve |
https://en.wikipedia.org/wiki/Magnus%20Myklebust | Magnus Waade Myklebust (born 8 July 1985) is a Norwegian football striker who currently plays for Bergsöy IL.
He is the cousin of Jan Åge Fjørtoft.
Career statistics
References
External links
Guardian's Stats Centre
Magnus Myklebust at NFF
1985 births
Living people
People from Ulstein
Footballers from Møre og Romsdal
Norwegian men's footballers
Norway men's under-21 international footballers
Norway men's youth international footballers
IL Hødd players
Lillestrøm SK players
Odds BK players
Eliteserien players
Kongsvinger IL Toppfotball players
Norwegian First Division players
Men's association football forwards |
https://en.wikipedia.org/wiki/Richard%20R.%20Eakin | Richard Ronald Eakin (born August 6, 1938) was the eighth chancellor of East Carolina University. He was born in New Castle, Pennsylvania and earned his bachelor's degree in mathematics and physics, summa cum laude, from Geneva College. He earned his master's and doctorate degree in mathematics from Washington State University in 1962 and 1964. He started his career on the mathematics faculty at Bowling Green State University. He was named chancellor in 1987.
References
External links
Eakin biography
1938 births
Presidents of East Carolina University
Living people |
https://en.wikipedia.org/wiki/NCSS%20%28statistical%20software%29 | NCSS is a statistics package produced and distributed by NCSS, LLC. Created in 1981 by Jerry L. Hintze, NCSS, LLC specializes in providing statistical analysis software to researchers, businesses, and academic institutions. It also produces PASS Sample Size Software which is used in scientific study planning and evaluation.
The NCSS package includes over 250 documented statistical and plot procedures. NCSS imports and exports all major spreadsheet, database, and statistical file formats.
Major statistical topics in NCSS
Analysis of Variance
Appraisal Methods
Charts and Graphs
Correlation
Cross Tabulation
Curve Fitting
Descriptive Statistics
Design of Experiments
Diagnostic Tests
Forecasting
General Linear Models
Meta-Analysis
Mixed Models
Multivariate
Proportions
Quality Control
Regression Analysis
Reliability Analysis
Repeated Measures
ROC Curves
Survival Analysis
Time Series Analysis
T-Tests
See also
List of statistical packages
Comparison of statistical packages
References
External links
Official NCSS Homepage
NCSS Documentation
NCSS in the cloud
Further reading
Windows-only proprietary software
Regression and curve fitting software
Time series software |
https://en.wikipedia.org/wiki/Semi-s-cobordism | In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the inclusion is a simple homotopy equivalence (as in an s-cobordism), with no further requirement on the inclusion (not even being a homotopy equivalence).
Other notations
The original creator of this topic, Jean-Claude Hausmann, used the notation M− for the right-hand boundary of the cobordism.
Properties
A consequence of (W, M, M−) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups is perfect. A corollary of this is that solves the group extension problem . The solutions to the group extension problem for prescribed quotient group and kernel group K are classified up to congruence by group cohomology (see Mac Lane's Homology pp. 124-129), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-s-cobordism with prescribed left-hand boundary M and superperfect kernel group K.
Relationship with Plus cobordisms
Note that if (W, M, M−) is a semi-s-cobordism, then (W, M−, M) is a plus cobordism. (This justifies the use of M− for the right-hand boundary of a semi-s-cobordism, a play on the traditional use of M+ for the right-hand boundary of a plus cobordism.) Thus, a semi-s-cobordism may be thought of as an inverse to Quillen's Plus construction in the manifold category. Note that (M−)+ must be diffeomorphic (respectively, piecewise-linearly (PL) homeomorphic) to M but there may be a variety of choices for (M+)− for a given closed smooth (respectively, PL) manifold M.
References
.
.
.
Manifolds
Geometric topology
Algebraic topology
Homotopy theory |
https://en.wikipedia.org/wiki/Maths%20Class | Maths Class were a British band from that formed in 2006 in Brighton, England, and achieved publicity through their Myspace. They have a wide range of influences and have played extensively around the UK and in Europe. They have also toured Japan in September 2008.
History
Beginnings
Maths Class first gained publicity when they approached Shitdisco, who were playing at The Great Escape, and asked them if they could support them at their house party.
After supporting some of the best new bands around at the moment they were noticed by the media. The NME have highly recommended them as a breaking band, and they have been featured heavily in Artrocker. Since then they have released their first single, double A-sided "Emporio Laser"/"Cushion Glamour" in November 2007 which received airplay on radio 1 from Steve Lamacq and Huw Stephens.
Maths Class were the only band chosen by VICE Magazine to play both nights at their Vice Spain launch event in Barcelona. They also played at the Great Escape festival in May 2007., and the summer festivals Tales Of The Jackalope and Underage Festival.
Now This Will Take Two Hands, a five-track EP, was released in July 2008 on Gift Music (UK) / 1977 Records (Japan).
In 2008, they played Offset Festival and The Great Escape Festival. After The Great Escape Festival they played in their house with Rolo Tomassi. They also completed a tour with Stephen Malkmus in August 2008. On 23 July 2008, they recorded their first radio session for Marc Riley on BBC 6 music.
A promo single/video 'Peach' was recorded in February, and will be released as a download in April 2009. The video featured Ciaran Griffiths from Shameless in it.
On 10 May 2010, it was announced on Myspace that the band had split.
Musical style
Artrocker recently described them as "Incredible. Best live band in the country right now"."Don't Panic magazine described the band as "the sort of hardcore math-rock that makes the kids bounce off the walls and beat the crap out of each other in spinning mosh-pits". The BBC described their sound as "jagged techno-rock".
Members
Tim Sketchley – Vocals
Piers Cowburn – Synth, Vocals
Andy Davies – Guitar, Vocals
Aleksandar Damms – Bass guitar
Michael Garthside – Drums
Discography
Singles
"Emporio Laser" / "Cushion Glamour" (12 November 2007) (Life Is Easy Records)
"Nerves" (21 July 2008) (Gift Music)
"Peach – Promo Single" (April 2009) (Gift Music)
EPs
Now This Will Take Two Hands (21 July 2008, Gift Music)
References
External links
Official website
Dance-punk musical groups |
https://en.wikipedia.org/wiki/Sampling%20design | In the theory of finite population sampling, a sampling design specifies for every possible sample its probability of being drawn.
Mathematical formulation
Mathematically, a sampling design is denoted by the function which gives the probability of drawing a sample
An example of a sampling design
During Bernoulli sampling, is given by
where for each element is the probability of being included in the sample and is the total number of elements in the sample and is the total number of elements in the population (before sampling commenced).
Sample design for managerial research
In business research, companies must often generate samples of customers, clients, employees, and so forth to gather their opinions. Sample design is also a critical component of marketing research and employee research for many organizations. During sample design, firms must answer questions such as:
What is the relevant population, sampling frame, and sampling unit?
What is the appropriate margin of error that should be achieved?
How should sampling error and non-sampling error be assessed and balanced?
These issues require very careful consideration, and good commentaries are provided in several sources.
See also
Bernoulli sampling
Sampling probability
Sampling (statistics)
References
Further reading
Sarndal, Swenson, and Wretman (1992), Model Assisted Survey Sampling, Springer-Verlag,
Sampling (statistics) |
https://en.wikipedia.org/wiki/Schr%C3%B6der%20number | In mathematics, the Schröder number also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner of an grid to the northeast corner using only single steps north, northeast, or east, that do not rise above the SW–NE diagonal.
The first few Schröder numbers are
1, 2, 6, 22, 90, 394, 1806, 8558, ... .
where and They were named after the German mathematician Ernst Schröder.
Examples
The following figure shows the 6 such paths through a grid:
Related constructions
A Schröder path of length is a lattice path from to with steps northeast, east, and southeast, that do not go below the -axis. The th Schröder number is the number of Schröder paths of length . The following figure shows the 6 Schröder paths of length 2.
Similarly, the Schröder numbers count the number of ways to divide a rectangle into smaller rectangles using cuts through points given inside the rectangle in general position, each cut intersecting one of the points and dividing only a single rectangle in two (i.e., the number of structurally-different guillotine partitions). This is similar to the process of triangulation, in which a shape is divided into nonoverlapping triangles instead of rectangles. The following figure shows the 6 such dissections of a rectangle into 3 rectangles using two cuts:
Pictured below are the 22 dissections of a rectangle into 4 rectangles using three cuts:
The Schröder number also counts the separable permutations of length
Related sequences
Schröder numbers are sometimes called large or big Schröder numbers because there is another Schröder sequence: the little Schröder numbers, also known as the Schröder-Hipparchus numbers or the super-Catalan numbers. The connections between these paths can be seen in a few ways:
Consider the paths from to with steps and that do not rise above the main diagonal. There are two types of paths: those that have movements along the main diagonal and those that do not. The (large) Schröder numbers count both types of paths, and the little Schröder numbers count only the paths that only touch the diagonal but have no movements along it.
Just as there are (large) Schröder paths, a little Schröder path is a Schröder path that has no horizontal steps on the -axis.
If is the th Schröder number and is the th little Schröder number, then for
Schröder paths are similar to Dyck paths but allow the horizontal step instead of just diagonal steps. Another similar path is the type of path that the Motzkin numbers count; the Motzkin paths allow the same diagonal paths but allow only a single horizontal step, (1,0), and count such paths from to .
There is also a triangular array associated with the Schröder numbers that provides a recurrence relation (though not just with the Schröder numbers). The first few terms are
1, 1, 2, 1, 4, 6, 1, 6, 16, 22, .... .
It is easier to see the connection with the Schröder numbers when the sequence |
https://en.wikipedia.org/wiki/Direct%20limit%20of%20groups | In mathematics, a direct limit of groups is the direct limit of a of groups. These are central objects of study in algebraic topology, especially stable homotopy theory and homological algebra. They are sometimes called stable groups, though this term normally means something quite different in model theory.
Certain examples of stable groups are easier to study than "unstable" groups, the groups occurring in the limit. This is surprising, given that they are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations.
Examples
Each family of classical groups forms a direct system, via inclusion of matrices in the upper left corner, such as . The stable groups are denoted or .
Bott periodicity computes the homotopy of the stable unitary group and stable orthogonal group.
The Whitehead group of a ring (the first K-group) can be defined in terms of .
Stable homotopy groups of spheres are the stable groups associated with the suspension functor.
See also
References
Homotopy theory
Homological algebra
Algebraic topology |
https://en.wikipedia.org/wiki/Antoine%27s%20necklace | In mathematics Antoine's necklace is a topological embedding of the Cantor set in 3-dimensional Euclidean space, whose complement is not simply connected. It also serves as a counterexample to the claim that all Cantor spaces are ambiently homeomorphic to each other. It was discovered by .
Construction
Antoine's necklace is constructed iteratively like so: Begin with a solid torus A0 (iteration 0). Next, construct a "necklace" of smaller, linked tori that lie inside A0. This necklace is A1 (iteration 1). Each torus composing A1 can be replaced with another smaller necklace as was done for A0. Doing this yields A2 (iteration 2).
This process can be repeated a countably infinite number of times to create an An for all n. Antoine's necklace A is defined as the intersection of all the iterations.
Properties
Since the solid tori are chosen to become arbitrarily small as the iteration number increases, the connected components of A must be single points. It is then easy to verify that A is closed, dense-in-itself, and totally disconnected, having the cardinality of the continuum. This is sufficient to conclude that as an abstract metric space A is homeomorphic to the Cantor set.
However, as a subset of Euclidean space A is not ambiently homeomorphic to the standard Cantor set C, embedded in R3 on a line segment. That is, there is no bi-continuous map from R3 → R3 that carries C onto A. To show this, suppose there was such a map h : R3 → R3, and consider a loop k that is interlocked with the necklace. k cannot be continuously shrunk to a point without touching A because two loops cannot be continuously unlinked. Now consider any loop j disjoint from C. j can be shrunk to a point without touching C because we can simply move it through the gap intervals. However, the loop g = h−1(k) is a loop that cannot be shrunk to a point without touching C, which contradicts the previous statement. Therefore, h cannot exist.
In fact, there is no homeomorphism of R3 sending A to a set of Hausdorff dimension < 1, since the complement of such a set must be simply-connected.
Antoine's necklace was used by to construct Antoine's horned sphere (similar to but not the same as Alexander's horned sphere).
See also
Wild knot
Superhelix
Hawaiian earring
References
Further reading
Topology |
https://en.wikipedia.org/wiki/Anne%20Sjerp%20Troelstra | Anne Sjerp Troelstra (10 August 1939 – 7 March 2019) was a professor of pure mathematics and foundations of mathematics at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam.
He was a constructivist logician, who was influential in the development of intuitionistic logic With Georg Kreisel, he was a developer of the theory of choice sequences. He wrote one of the first texts on linear logic, and, with Helmut Schwichtenberg, he co-wrote an important book on proof theory.
He became a member of the Royal Netherlands Academy of Arts and Sciences in 1976. Troelstra died on 7 March 2019.
Notes
External links
Homepage of A. S. Troelstra : Dead Link - Archived : Homepage of A. S. Troelstra : Retrieved on 27 June 2018
1939 births
2019 deaths
Dutch mathematicians
Members of the Royal Netherlands Academy of Arts and Sciences
People from De Bilt
University of Amsterdam alumni
Academic staff of the University of Amsterdam
20th-century Dutch people |
https://en.wikipedia.org/wiki/Mathukumalli%20V.%20Subbarao | Mathukumalli (Matukumalli) Venkata Subbarao (May 4, 1921 – February 15, 2006) was an Indo-Canadian mathematician, specialising in number theory. He was a long-time resident of Edmonton, Alberta, Canada.
Subbarao was born in the small village of Yazali, Guntur, Andhra Pradesh, India. He received his master's degree from Presidency College, Madras in 1941. He went on to complete a doctorate in functional analysis, advised by Ramaswamy S. Vaidyanathaswamy. He worked at Presidency College, Madras, Sri Venkateswara University, and the University of Missouri, before moving in 1963 to the University of Alberta, where he spent the rest of his professional career.
In the 1960s Subbarao began to study the congruence properties of the partition function, p(n), which became one of his favourite problems. For example, he conjectured that if A and B are integers with 0 ≤ B < A, there are infinitely many n for which p(An+B) is even and infinitely many n for which p(An+B) is odd. Ken Ono showed that the even case is always true and that if there is one number n such that p(An+B) is odd, then there are infinitely many such numbers n. The odd case was finally settled by Silviu Radu. A more general variant of the conjecture was formulated by Morris Newman predicting that for any given r and m, there are infinitely many n such that p(n)= r(mod m). At the end of his life, Subbarao co-authored a book on partition theory with A.K. Agarwal and Padmavathamma. Partition theory is ubiquitous in mathematics with connections to the representation theory of the symmetric group and the general linear group, modular forms, and physics. Thus, Subbarao's conjectures, though seemingly simple, will generate fundamental research activity for years to come. He also researched special classes of divisors and the corresponding analogues of divisor functions and perfect numbers, such as those arising from the exponential divisors ("e-divisors") which he defined. Many other mathematicians have published papers building on his work in these subjects.
A prolific collaborator, Subbarao had more than 40 joint authors (including Paul Erdős, giving him Erdős number 1). He continued producing mathematics papers into the final years of his life. He died in Edmonton at the age of 84.
Subbarao was the father of Prof. Mathukumalli Vidyasagar.
Selected publications
References
External links
1921 births
2006 deaths
Telugu people
20th-century Indian mathematicians
21st-century Indian mathematicians
Indian emigrants to Canada
Indian number theorists
Academic staff of the University of Alberta
University of Missouri faculty
University of Missouri mathematicians
Scientists from Andhra Pradesh
Scientists from Edmonton
People from Guntur district
Presidency College, Chennai alumni
Canadian mathematicians
Canadian people of Indian descent |
https://en.wikipedia.org/wiki/Charles%20S.%20Venable | Charles Scott Venable (March 19, 1827 – August 11, 1900) was a mathematician, astronomer, and military officer. In mathematics, he is noted for authoring a series of publications as a University of Virginia professor.
Early life
He was born at Longwood House in Farmville, Virginia and graduated from Hampden-Sydney College, founded by his Patriot ancestor Nathaniel Venable, at the age of 15. He was a member of Beta Theta Pi fraternity while at Hampden-Sydney College.
Career
For several years following his graduation, he served as a mathematics tutor at the college. He received further education at the University of Virginia as well as in Berlin and Bonn, Germany. He became a professor in mathematics and astronomy in Virginia and South Carolina.
Following the war, Venable resumed his career as an educator.
Civil War
Venable was present at the firing on Fort Sumter in April 1861, serving as a lieutenant in the South Carolina state militia. He then fought as a private in Company A, 2nd South Carolina Infantry. In the spring of 1862, Venable joined the staff of presidential military advisor General Robert E. Lee as an aide-de-camp with the rank of major. He continued serving on Lee's staff when the general took command of the Army of Northern Virginia on June 1, 1862. He served on Lee's staff from the Peninsula Campaign to Appomattox Court House and was promoted to lieutenant colonel.
According to family legend (and some history books) Lee called Venable "Faithful old Venable."
Personal life
Following the war, Venable resumed his career as an educator. During a visit to Prussia, he was invited to the castle of Heros von Borcke, the former aide-de-camp to General J.E.B. Stuart.
Venable died in Charlottesville in 1900. He was buried at the University of Virginia Cemetery in Charlottesville.
Legacy
Venable Elementary School opened in 1925 near the University of Virginia. In December 2022, the Charlottesville City Public Schools Naming of Facilities Committee recommended changing the school name to Trailblazers Elementary.
Notes
References
Inventory of the Charles S. Venable Papers, 1862-1894, in the Southern Historical Collection, UNC-Chapel Hill.
"Charles Scott Venable" , Leander McCormick Observatory Museum, Department of Astronomy, University of Virginia.
Johnson, Francis W., A History of Texas and Texans: Volume III, The American Historical Society, Chicago and New York, 1914. p 1235, 1236.
Sifakis, Stewart. Who Was Who in the Confederacy, Facts on File (November 1989). .
1827 births
1900 deaths
19th-century American mathematicians
American astronomers
Confederate States Army officers
People from Prince Edward County, Virginia
People of Virginia in the American Civil War
University of Virginia faculty
University of Virginia alumni
Burials at the University of Virginia Cemetery
Mathematicians from Virginia
Southern Historical Society |
https://en.wikipedia.org/wiki/William%20Karush | William Karush (1 March 1917 – 22 February 1997) was an American professor of mathematics at California State University at Northridge and was a mathematician best known for his contribution to Karush–Kuhn–Tucker conditions. In his master's thesis he was the first to publish these necessary conditions for the inequality-constrained problem, although he became renowned after a seminal conference paper by Harold W. Kuhn and Albert W. Tucker. He also worked as a physicist for the Manhattan Project, although he signed the Szilárd petition and became a peace activist afterwards.
Selected works
Webster's New World Dictionary of Mathematics, MacMillan Reference Books, Revised edition (April 1989),
On the Maximum Transform and Semigroups of Transformations (1962), Richard Bellman, William Karush,
The crescent dictionary of mathematics, general editor (1962) William Karush, Oscar Tarcov
Isoperimetric problems & index theorems. (1942), William Karush, Thesis (Ph.D.) University of Chicago, department of mathematics.
Minima of functions of several variables with inequalities as side conditions, William Karush. (1939), Thesis (M.S.) – University of Chicago, 1939.
See also
Karush–Kuhn–Tucker conditions
Szilárd petition
References
External links
1917 births
1997 deaths
20th-century American mathematicians
California State University, Northridge faculty
University of Chicago alumni
Manhattan Project people
20th-century American physicists
Mathematical physicists
Activists from Chicago
American anti-war activists |
https://en.wikipedia.org/wiki/Daniel%20Pitbull | Daniel Carlos Silva Anjos (born 23 November 1979), sometimes known as Daniel Pitbull or just Daniel, is a Brazilian football player.
Club statistics
External links
Kawasaki Frontale
1979 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Expatriate men's footballers in Portugal
Primeira Liga players
Men's association football midfielders
J2 League players
Kawasaki Frontale players
Sportspeople from Feira de Santana |
https://en.wikipedia.org/wiki/Roland%20Benz | Roland Benz (born 1943 in Singen, Baden-Württemberg) is a German biophysicist.
Early life and education
Benz studied mathematics, chemistry, and physics at the University of Würzburg. In 1972, he obtained his Ph.D. in biology, with Peter Läuger at University of Konstanz as his supervisor; and, in 1977, he obtained his Habilitation in Biophysics.
Career
A Heisenberg Fellow of the Deutsche Forschungsgemeinschaft (DFG) (German Science Foundation), Benz was a visiting professor at State University of New York at Stony Brook (SUNYSB) in 1980 and 1982. In 1984, he was a visiting professor at University of British Columbia in Vancouver.
In 1986, Benz became a full professor of biotechnology at the University of Würzburg, his alma mater.
Since 2003, Benz has been a Member of the European Graduate College; and, since 2005, a Member in the French–German Graduate College, both sponsored by the DFG.
Since 2009, Benz has held the Wisdom Professorship at the Jacobs University Bremen and has been a research fellow at the Rudolf Virchow Center and the DFG Research Center for Experimental Biomedicine. He remains a professor at the University of Würzburg.
Research interests
Benz's research interests include the periplasmic structure and organization of cell membranes and other biological membranes; biophysical processes and the molecular basis of membrane proteins in microorganisms and higher organisms; and, pore-forming peptides and proteins.
Benz is the leader of several research projects, including:
the molecular basis of signal transduction and membrane transport (SFB 176; 1987–1999);
ecology, physiology, and biochemistry of plants under stress (SFB 251; 1989–1992);
nuclear magnetic resonance in vivo and in vitro for the study of biomedical basic elements (Member in the DFG-Graduate College; 1992–1999); and,
the regulatory membrane proteins: from the mechanism of recognition to the pharmacological structure (SFB 487; seit 2000).
Awards
In 2002, Benz was recognised with the Gay-Lussac/Humboldt Award de la Ministère de recherche français for his role in the development of a Franco–German collaboration.
In 2007, he was awarded an honorary doctorate by the University of Barcelona.
In 2011, he was honoured, with another honorary doctorate, by the Umeå University's Faculty of Medicine
Publications
Benz R. (1980.) Künstliche Lipidmembranen. Modelle für biologische Membranen, Universitätsverlag Konstanz,
Benz R. (2004.) Bacterial and Eukaryotic Porins. Structure, Function, Mechanism, Wiley-VCH,
Notes
1943 births
Living people
German biophysicists
People from Singen
University of Würzburg alumni
Academic staff of the University of Würzburg
University of Konstanz alumni |
https://en.wikipedia.org/wiki/1903%E2%80%9304%20Manchester%20United%20F.C.%20season | The 1903–04 season was Manchester United's 12th season in the Football League.
Second Division
FA Cup
Squad statistics
References
Manchester United F.C. seasons
Manchester United |
https://en.wikipedia.org/wiki/Gregory%20Wheeler | Gregory Wheeler (born 1968) is an American logician, philosopher, and computer scientist, who specializes in formal epistemology. Much of his work has focused on imprecise probability. He is currently Professor of Philosophy and Computer Science at the Frankfurt School of Finance and Management, and has held positions at LMU Munich, Carnegie Mellon University, the Max Planck Institute for Human Development in Berlin, and the New University of Lisbon. He is a member of the PROGIC steering committee, the editorial boards of Synthese, and Minds and Machines, and was the editor-in-chief of Minds and Machines from 2011 to 2016. In 2019 he co-founded Exaloan AG, a financial technology company based in Frankfurt. He obtained a Ph.D. in philosophy and computer science from the University of Rochester under Henry Kyburg.
Select bibliography
Books
Reflections on the Foundations of Probability and Statistics: Essays in Honor of Teddy Seidenfeld, Thomas Augustin, Fabio Cozman, and Gregory Wheeler (eds.) Springer, 2022
New Challenges to Philosophy of Science, Hanne Andersen, Dennis Dieks, Wenceslao Gonzalez, Marcel Weber and Gregory Wheeler (eds.) Springer, 2013.
Probabilistic Logics and Probabilistic Networks, Rolf Haenni, Jan-Willem Romeijn, Gregory Wheeler, and Jon Williamson. The Synthese Library, Springer, 2011.
Probability and Inference: Essays in Honour of Henry E. Kyburg, Jr., William Harper and Gregory Wheeler (eds.), College Publications, 2007.
Articles
"Discounting Desirable Gambles", Proceedings of Machine Learning Research 147: 336-346, 2021.
"Moving Beyond Sets of Probabilities", Statistical Science 36(2): 201-204, 2021.
"Less is More for Bayesians, Too", in Riccardo Viale (Ed.) Routledge Handbook on Bounded Rationality, pp. 471–483, 2020.
"Dilation and Asymmetric Relevance" (w/ Arthur Paul Pedersen), Proceedings of Machine Learning Research 103: 324-26, 2019.
"Bounded Rationality", The Stanford Encyclopedia of Philosophy, Winter 2018 Edition.
"Resolving Peer Disagreements Through Imprecise Probabilities (w/ Lee Elkin), Nous 52(2): 260-94, 2018.
"Scoring Imprecise Credences: A Mildly Immodest Proposal" (w/ Conor Mayo-Wilson), Philosophy and Phenomenological Research 93(1): 55–78, 2016.
"Dilation, Disintegrations, and Delayed Decisions" (with Arthur Paul Pedersen), Proceedings of the 9th International Symposium on Imprecise Probability: Theories and Applications (ISIPTA 2015), Pescara, Italy: 227–236, 2015.
"Is there a Logic of Information?" Journal of Experimental and Theoretical Artificial Intelligence 27(1): 95–98, 2015.
"Demystifying Dilation" (with Arthur Paul Pedersen), Erkenntnis, 79(6): 1305–1342, 2014.
"Defeat Reconsidered and Repaired", The Reasoner, 8(2): 15, 2014.
"Character Matching and the Locke Pocket of Belief", Epistemology, Context, and Formalism, Franck Lihoreau and Manuel Rebuschi (ed.), Dordrecht: The Synthese Library, Springer, pp. 185–94, 2014.
"Coherence and Confirmation Through Causation" (with Richa |
https://en.wikipedia.org/wiki/Stochastic%20ordering | In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders, so that one random variable may be neither stochastically greater than, less than nor equal to another random variable . Many different orders exist, which have different applications.
Usual stochastic order
A real random variable is less than a random variable in the "usual stochastic order" if
where denotes the probability of an event. This is sometimes denoted or . If additionally for some , then is stochastically strictly less than , sometimes denoted . In decision theory, under this circumstance B is said to be first-order stochastically dominant over A.
Characterizations
The following rules describe situations when one random variable is stochastically less than or equal to another. Strict version of some of these rules also exist.
if and only if for all non-decreasing functions , .
If is non-decreasing and then
If is increasing in each variable and and are independent sets of random variables with for each , then and in particular Moreover, the th order statistics satisfy .
If two sequences of random variables and , with for all each converge in distribution, then their limits satisfy .
If , and are random variables such that and for all and such that , then .
Other properties
If and then (the random variables are equal in distribution).
Stochastic dominance
Stochastic dominance relations are a family of stochastic orderings used in decision theory:
Zeroth-order stochastic dominance: if and only if for all realizations of these random variables and for at least one realization.
First-order stochastic dominance: if and only if for all and there exists such that .
Second-order stochastic dominance: if and only if for all , with strict inequality at some .
There also exist higher-order notions of stochastic dominance. With the definitions above, we have .
Multivariate stochastic order
An -valued random variable is less than an -valued random variable in the "usual stochastic order" if
Other types of multivariate stochastic orders exist. For instance the upper and lower orthant order which are similar to the usual one-dimensional stochastic order. is said to be smaller than in upper orthant order if
and is smaller than in lower orthant order if
All three order types also have integral representations, that is for a particular order is smaller than if and only if for all in a class of functions . is then called generator of the respective order.
Other dominance orders
The following stochastic orders are useful in the theory of random social choice. They are used to compare the outcomes of random social choice functions, in order to check them for efficiency or other desirable criteria. The dominance orders below are ordered from the most conservative to the least conservative. They are exemplified on random variab |
https://en.wikipedia.org/wiki/Category%20of%20topological%20vector%20spaces | In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted TVect or TVS.
Fixing a topological field K, one can also consider the subcategory TVectK of topological vector spaces over K with continuous K-linear maps as the morphisms.
TVect is a concrete category
Like many categories, the category TVect is a concrete category, meaning its objects are sets with additional structure (i.e. a vector space structure and a topology) and its morphisms are functions preserving this structure. There are obvious forgetful functors into the category of topological spaces, the category of vector spaces and the category of sets.
TVect is a topological category
The category is topological, which means loosely speaking that it relates to its "underlying category", the category of vector spaces, in the same way that Top relates to Set. Formally, for every K-vector space and every family of topological K-vector spaces and K-linear maps there exists a vector space topology on so that the following property is fulfilled:
Whenever is a K-linear map from a topological K-vector space it holds that
is continuous is continuous.
The topological vector space is called "initial object" or "initial structure" with respect to the given data.
If one replaces "vector space" by "set" and "linear map" by "map", one gets a characterisation of the usual initial topologies in Top. This is the reason why categories with this property are called "topological".
There are numerous consequences of this property. For example:
"Discrete" and "indiscrete" objects exist. A topological vector space is indiscrete iff it is the initial structure with respect to the empty family. A topological vector space is discrete iff it is the initial structure with respect to the family of all possible linear maps into all topological vector spaces. (This family is a proper class, but that does not matter: Initial structures with respect to all classes exists iff they exists with respect to all sets)
Final structures (the similar defined analogue to final topologies) exist. But there is a catch: While the initial structure of the above property is in fact the usual initial topology on with respect to , the final structures do not need to be final with respect to given maps in the sense of Top. For example: The discrete objects (= final with respect to the empty family) in do not carry the discrete topology.
Since the following diagram of forgetful functors commutes
and the forgetful functor from to Set is right adjoint, the forgetful functor from to Top is right adjoint too (and the corresponding left adjoints fit in an analogue commutative diagram). This left adjoint defines "free topological vector spaces". Explicitly these |
https://en.wikipedia.org/wiki/Journal%20of%20Graph%20Theory | The Journal of Graph Theory is a peer-reviewed mathematics journal specializing in graph theory and related areas, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs.
The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. It is published by John Wiley & Sons. The journal was established in 1977 by Frank Harary. The editors-in-chief are Paul Seymour (Princeton University) and Carsten Thomassen (Technical University of Denmark).
Abstracting and indexing
The journal is abstracted and indexed in the Science Citation Index Expanded, Scopus, and Zentralblatt MATH. According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.857.
References
External links
Combinatorics journals
Academic journals established in 1977
Monthly journals
Wiley (publisher) academic journals
English-language journals |
https://en.wikipedia.org/wiki/Range%20space | The term range space has multiple meanings in mathematics:
In linear algebra, it refers to the column space of a matrix, the set of all possible linear combinations of its column vectors.
In computational geometry, it refers to a hypergraph, a pair (X, R) where each r in R is a subset of X. |
https://en.wikipedia.org/wiki/Cone%20of%20curves | In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety is a combinatorial invariant of importance to the birational geometry of .
Definition
Let be a proper variety. By definition, a (real) 1-cycle on is a formal linear combination of irreducible, reduced and proper curves , with coefficients . Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles and are numerically equivalent if for every Cartier divisor on . Denote the real vector space of 1-cycles modulo numerical equivalence by .
We define the cone of curves of to be
where the are irreducible, reduced, proper curves on , and their classes in . It is not difficult to see that is indeed a convex cone in the sense of convex geometry.
Applications
One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor on a complete variety is ample if and only if for any nonzero element in , the closure of the cone of curves in the usual real topology. (In general, need not be closed, so taking the closure here is important.)
A more involved example is the role played by the cone of curves in the theory of minimal models of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety , find a (mildly singular) variety which is birational to , and whose canonical divisor is nef. The great breakthrough of the early 1980s (due to Mori and others) was to construct (at least morally) the necessary birational map from to as a sequence of steps, each of which can be thought of as contraction of a -negative extremal ray of . This process encounters difficulties, however, whose resolution necessitates the introduction of the flip.
A structure theorem
The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the Cone Theorem. The first version of this theorem, for smooth varieties, is due to Mori; it was later generalised to a larger class of varieties by Kawamata, Kollár, Reid, Shokurov, and others. Mori's version of the theorem is as follows:
Cone Theorem. Let be a smooth projective variety. Then
1. There are countably many rational curves on , satisfying , and
2. For any positive real number and any ample divisor ,
where the sum in the last term is finite.
The first assertion says that, in the closed half-space of where intersection with is nonnegative, we know nothing, but in the complementary half-space, the cone is spanned by some countable collection of curves which are quite special: they are rational, and their 'degree' is bounded very tightly by the dimension of . The second assertion then tells us more: it says that, away from the hyperplane , extremal rays of the cone cannot accumulate. When is a Fano variety, because is ample. So the cone theorem shows that the cone of curves of a Fano variety is generated by ra |
https://en.wikipedia.org/wiki/Kazuki%20Hara | is a Japanese footballer for Ococias Kyoto AC.
Career
He joined S-Pulse after graduating at Komazawa University and plays in the forward position.
Career statistics
Updated to 23 February 2020.
Club
International
Appearances in major competitions
References
External links
Profile at Giravanz Kitakyushu
Profile at Kamatamare Sanuki
Yahoo! Sports Profile
1985 births
Living people
Komazawa University alumni
Association football people from Chiba Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Shimizu S-Pulse players
Urawa Red Diamonds players
Kyoto Sanga FC players
Giravanz Kitakyushu players
Kamatamare Sanuki players
Roasso Kumamoto players
Ococias Kyoto AC players
Men's association football forwards
Universiade medalists in football
FISU World University Games gold medalists for Japan |
https://en.wikipedia.org/wiki/Transitive%20model | In mathematical set theory, a transitive model is a model of set theory that is standard and transitive. Standard means that the membership relation is the usual one, and transitive means that the model is a transitive set or class.
Examples
An inner model is a transitive model containing all ordinals.
A countable transitive model (CTM) is, as the name suggests, a transitive model with a countable number of elements.
Properties
If M is a transitive model, then ωM is the standard ω. This implies that the natural numbers, integers, and rational numbers of the model are also the same as their standard counterparts. Each real number in a transitive model is a standard real number, although not all standard reals need be included in a particular transitive model.
References
Set theory |
https://en.wikipedia.org/wiki/Song%20Jung-hyun | Song Jung-Hyun (, born 28 May 1976) is a South Korean footballer.
He was arrested on the charge connected with the match fixing allegations on 7 July 2011.
Club career statistics
References
External links
1976 births
Living people
Men's association football midfielders
South Korean men's footballers
South Korea men's international footballers
Jeonnam Dragons players
Daegu FC players
Ulsan Hyundai FC players
K League 1 players
Footballers from Daegu
Ajou University alumni |
https://en.wikipedia.org/wiki/Roy%20C.%20Geary | Robert (Roy) Charles Geary (April 11, 1896 – February 8, 1983) was an Irish statistician and founder of both the Central Statistics Office and the Economic and Social Research Institute. He held degrees from University College Dublin and the Sorbonne. He lectured in mathematics at University College Southampton (1922–23) and in applied economics at Cambridge University (1946–47). He was a statistician in the Department of Industry and Commerce between 1923 and 1957. The National University of Ireland conferred a Doctorate of Science on him in 1938. He was the founding director of the Central Statistics Office (Ireland) (in 1949). He was head of the National Accounts Branch of the United Nations in New York from 1957 to 1960. He was the founding director of the Economic and Social Research Institute (in 1960) where he stayed till his retirement in 1966. He was an honorary fellow of the American Statistical Association and the Institute of Mathematical Statistics. In 1981, he won the Boyle Medal. To honour his contributions to social sciences, the UCD Geary Institute for Public Policy was named after him in 2005.
Roy Geary is known for his contributions to the estimation of errors-in-variables models, Geary's C, the Geary–Khamis dollar, the Stone–Geary utility function, and Geary's theorem, which has that if the sample mean is distributed independently of the sample variance, then the population is distributed normally.
Works
Europe's Future in Figures, North Holland (1962), ASIN: B002RB858E
Elements of Linear Programming with Economic Applications (with J. E. Spencer), Lubrecht & Cramer Ltd (June 1973),
Exercises in Mathematical Economics and Econometrics, with Outlines of Theory (with J. E. Spencer), London: Charles Griffin & Co (1987),
References
External sources
1896 births
1983 deaths
20th-century Irish mathematicians
University of Paris alumni
Irish statisticians
Fellows of the Institute of Mathematical Statistics
20th-century Irish economists
Economic and Social Research Institute
Fellows of the American Statistical Association
Statistical and Social Inquiry Society of Ireland
Fellows of the Econometric Society
Irish expatriates in France
Alumni of University College Dublin |
https://en.wikipedia.org/wiki/Six%20exponentials%20theorem | In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials.
Statement
If x1, x2, ..., xd are d complex numbers that are linearly independent over the rational numbers, and y1, y2, ..., yl are l complex numbers that are also linearly independent over the rational numbers, and if dl > d + l, then at least one of the following dl numbers is transcendental:
The most interesting case is when d = 3 and l = 2, in which case there are six exponentials, hence the name of the result. The theorem is weaker than the related but thus far unproved four exponentials conjecture, whereby the strict inequality dl > d + l is replaced with dl ≥ d + l, thus allowing d = l = 2.
The theorem can be stated in terms of logarithms by introducing the set L of logarithms of algebraic numbers:
The theorem then says that if λij are elements of L for i = 1, 2 and j = 1, 2, 3, such that λ11, λ12, and λ13 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then the matrix
has rank 2.
History
A special case of the result where x1, x2, and x3 are logarithms of positive integers, y1 = 1, and y2 is real, was first mentioned in a paper by Leonidas Alaoglu and Paul Erdős from 1944 in which they try to prove that the ratio of consecutive colossally abundant numbers is always prime. They claimed that Carl Ludwig Siegel knew of a proof of this special case, but it is not recorded. Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime.
The theorem was first explicitly stated and proved in its complete form independently by Serge Lang and Kanakanahalli Ramachandra in the 1960s.
Five exponentials theorem
A stronger, related result is the five exponentials theorem, which is as follows. Let x1, x2 and y1, y2 be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then at least one of the following five numbers is transcendental:
This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.
Sharp six exponentials theorem
Another related result that implies both the six exponentials theorem and the five exponentials theorem is the sharp six exponentials theorem. This theorem is as follows. Let x1, x2, and x3 be complex numbers that are linearly independent over the rational numbers, and let y1 and y2 be a pair of complex numbers that are linearly independent over the rational numbers, and suppose that βij are six algebraic numbers for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 such that the following six numbers are algebraic:
Then xi yj |
https://en.wikipedia.org/wiki/Four%20exponentials%20conjecture | In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of a certain number of values of the exponential function.
Statement
If x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental:
An alternative way of stating the conjecture in terms of logarithms is the following. For 1 ≤ i, j ≤ 2 let λij be complex numbers such that exp(λij) are all algebraic. Suppose λ11 and λ12 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then
An equivalent formulation in terms of linear algebra is the following. Let M be the 2×2 matrix
where exp(λij) is algebraic for 1 ≤ i, j ≤ 2. Suppose the two rows of M are linearly independent over the rational numbers, and the two columns of M are linearly independent over the rational numbers. Then the rank of M is 2.
While a 2×2 matrix having linearly independent rows and columns usually means it has rank 2, in this case we require linear independence over a smaller field so the rank isn't forced to be 2. For example, the matrix
has rows and columns that are linearly independent over the rational numbers, since π is irrational. But the rank of the matrix is 1. So in this case the conjecture would imply that at least one of e, eπ, and eπ2 is transcendental (which in this case is already known since e is transcendental).
History
The conjecture was considered in the early 1940s by Atle Selberg who never formally stated the conjecture. A special case of the conjecture is mentioned in a 1944 paper of Leonidas Alaoglu and Paul Erdős who suggest that it had been considered by Carl Ludwig Siegel. An equivalent statement was first mentioned in print by Theodor Schneider who set it as the first of eight important, open problems in transcendental number theory in 1957.
The related six exponentials theorem was first explicitly mentioned in the 1960s by Serge Lang and Kanakanahalli Ramachandra, and both also explicitly conjecture the above result. Indeed, after proving the six exponentials theorem Lang mentions the difficulty in dropping the number of exponents from six to four — the proof used for six exponentials "just misses" when one tries to apply it to four.
Corollaries
Using Euler's identity this conjecture implies the transcendence of many numbers involving e and π. For example, taking x1 = 1, x2 = , y1 = iπ, and y2 = iπ, the conjecture—if true—implies that one of the following four numbers is transcendental:
The first of these is just −1, and the fourth is 1, so the |
https://en.wikipedia.org/wiki/Blackhead%20%28New%20York%29 | {
"type": "FeatureCollection",
"features": [
{
"type": "Feature",
"properties": {},
"geometry": {
"type": "Point",
"coordinates": [
-74.10449981689455,
42.26790919743789
]
}
}
]
}Blackhead is a mountain located in Greene County, New York.
The mountain is part of the Blackhead range of the Catskill Mountains.
Blackhead is flanked to the northeast by Black Dome and Acra Point is located north.
Blackhead stands within the watershed of the Hudson River, which drains into New York Bay.
The southwest side of Blackhead drains into East Kill, thence into Schoharie Creek, the Mohawk River, and the Hudson River.
The north side of Blackhead drains into the headwaters of Batavia Kill, and thence into Schoharie Creek.
The east side of Blackhead drains into Trout Brook, thence into Shingle Kill, Catskill Creek, and the Hudson River.
Blackhead is within New York's Catskill Park.
The Long Path, a long-distance hiking trail from New York City to Albany, is contiguous with the Escarpment Trail.
See also
List of mountains in New York
Catskill High Peaks
Catskill Mountain 3500 Club
References
External links
Mountains of Greene County, New York
Catskill High Peaks
Mountains of New York (state) |
https://en.wikipedia.org/wiki/Mehran%20Ghasemi | Mehran Ghasemi (, born November 11, 1991 in Tehran, Iran) is an Iranian football midfielder, who currently plays for Rah Ahan FC in Persian Gulf Pro League.
Club career statistics
Last updated 10 October 2014
Honours
Esteghlal
Iran Pro League (1): 2012–13
Zob Ahan
Iran Pro League (1): 2013–14
Giti Pasand
Azadegan League (1): 2014–15
Rah Ahan FC
Iran Pro League (1): 2015–16
References
External links
Mehran Ghasemi at Persian League
Iranian men's footballers
Living people
Steel Azin F.C. players
Esteghlal F.C. players
1991 births
Men's association football midfielders
Footballers from Tehran |
https://en.wikipedia.org/wiki/Five-term%20exact%20sequence | In mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence.
More precisely, let
be a first quadrant spectral sequence, meaning that vanishes except when p and q are both non-negative. Then there is an exact sequence
0 → E21,0 → H 1(A) → E20,1 → E22,0 → H 2(A).
Here, the map is the differential of the -term of the spectral sequence.
Example
The inflation-restriction exact sequence
0 → H 1(G/N, AN) → H 1(G, A) → H 1(N, A)G/N → H 2(G/N, AN) →H 2(G, A)
in group cohomology arises as the five-term exact sequence associated to the Lyndon–Hochschild–Serre spectral sequence
H p(G/N, H q(N, A)) ⇒ H p+q(G, A)
where G is a profinite group, N is a closed normal subgroup, and A is a discrete G-module.
Construction
The sequence is a consequence of the definition of convergence of a spectral sequence. The second page differential with codomain E21,0 originates from E2−1,1, which is zero by assumption. The differential with domain E21,0 has codomain E23,−1, which is also zero by assumption. Similarly, the incoming and outgoing differentials of Er1,0 are zero for all . Therefore the (1,0) term of the spectral sequence has converged, meaning that it is isomorphic to the degree one graded piece of the abutment H 1(A). Because the spectral sequence lies in the first quadrant, the degree one graded piece is equal to the first subgroup in the filtration defining the graded pieces. The inclusion of this subgroup yields the injection E21,0 → H 1(A) which begins the five-term exact sequence. This injection is called an edge map.
The E20,1 term of the spectral sequence has not converged. It has a potentially non-trivial differential leading to E22,0. However, the differential landing at E20,1 begins at E2−2,2, which is zero, and therefore E30,1 is the kernel of the differential E20,1 → E22,0. At the third page, the (0, 1) term of the spectral sequence has converged, because all the differentials into and out of Er0,1 either begin or end outside the first quadrant when . Consequently E30,1 is the degree zero graded piece of H 1(A). This graded piece is the quotient of H 1(A) by the first subgroup in the filtration, and hence it is the cokernel of the edge map from E21,0. This yields a short exact sequence
0 → E21,0 → H 1(A) → E30,1 → 0.
Because E30,1 is the kernel of the differential E20,1 → E22,0, the last term in the short exact sequence can be replaced with the differential. This produces a four-term exact sequence. The map H 1(A) → E20,1 is also called an edge map.
The outgoing differential of E22,0 is zero, so E32,0 is the cokernel of the differential E20,1 → E22,0. The incoming and outgoing differentials of Er2,0 are zero if , again because the spectral sequence lies in the first quadrant, and hence the spectral sequence has converged. Consequently E32,0 is isomorphic to the degree two graded piece of H 2(A). In particular, it is a subgroup of H 2(A). |
https://en.wikipedia.org/wiki/Iran%20national%20football%20team%20records%20and%20statistics | This is a list of the Iran national football team's competitive records.
Individual records
Appearances
Most capped players
As of 17 October 2023, the players with the most caps for Iran are:
The records are collected based on data from FIFA and RSSSF. bold names denotes a player still playing or available for selection.
Most capped goalkeepers
World Cup appearances
Ehsan Hajsafi, 9 Games, 2014, 2018, 2022
Alireza Jahanbakhsh, 8 Games, 2014, 2018, 2022
Mehdi Mahdavikia, 6 Games, 1998, 2006
Andranik Teymourian, 6 Games, 2006, 2014
World Cup event appearances
Masoud Shojaei, 3 World Cups, 2006, 2014, 2018
Oldest player to feature at the World Cup Ali Daei, 37 years, 2006
Youngest player to feature at the World Cup Hossein Kaebi, 20 years and 9 months, 2006.
Oldest player to feature at the Asian Cup Ali Daei, 35 years, 2004
First appearance by a player who had never played for an Iranian club Ferydoon Zandi, vs Bahrain on 9 February 2005
First player to debut as a substitute Mohammad Mohajer, on 25 August 1941 vs Afghanistan
Goals
Top goalscorers
First goal Masoud Boroumand vs , Friendly, 26 October 1947
Most goals in a match
Karim Bagheri, 7 goals vs , 2 June 1997 (1998 World Cup qualification)
Karim Bagheri, 6 goals vs , 24 November 2000 (2002 World Cup Qualification)
Ali Daei, 5 goals vs , 12 June 1996 (1996 AFC Asian Cup qualification)
Four goals in a match
Behtash Fariba vs , 22 September 1980 (1980 AFC Asian Cup)
Nasser Mohammadkhani vs , 7 August 1984 (1984 AFC Asian Cup qualification);
Ali Asghar Modir Roosta vs , 25 June 1993 (1994 World Cup qualification)
Ali Daei vs , 10 June 1996 (1996 AFC Asian Cup qualification)
Ali Daei vs , 16 December 1996 (1996 AFC Asian Cup)
Ali Karimi vs , 24 November 2000 (2002 World Cup qualification)
Ali Daei vs , 17 November 2004 (2006 World Cup qualification)
Karim Ansarifard vs , 11 October 2019 (2022 World Cup qualification)
Three goals in a match
Masoud Boroumand vs , 27 October 50
Abbas Hajari vs , 14 December 1959 (1960 AFC Asian Cup qualification)
Ali Jabbari vs , 12 March 1o69
Hossein Kalani vs , 9 May 1972 (1972 AFC Asian Cup)
Ali Jabbari vs , 13 May 1972 (1972 AFC Asian Cup)
Gholam Hossein Mazloumi vs , 3 September 1974 (1974 Asian Games)
Gholam Hossein Mazloumi vs , 8 June 1976 (1976 AFC Asian Cup)
Karim Bavi vs , 4 June 1988 (1988 AFC Asian Cup qualification)
Farshad Pious vs , 24 September 1990 (1990 Asian Games)
Karim Bagheri vs , 17 June 1996 (1996 AFC Asian Cup qualification)
Hamid Estili vs , 2 June 1997 (1998 World Cup qualification)
Ali Daei vs , 14 December 1998 (1998 Asian Games)
Ali Daei vs , 31 March 2000 (2000 AFC Asian Cup qualification)
Ali Daei vs , 24 November 2000 (2002 World Cup qualification)
Farhad Majidi vs , 24 November 2000 (2002 World Cup qualification)
Sirous Dinmohammadi vs , 8 August 2001
Ali Karimi vs , 15 August 2001
Ali Daei vs , 17 June 2004 (2004 WAFF West Asian Championship)
Ali Karimi vs , 31 July 2004 (2004 AFC Asian |
https://en.wikipedia.org/wiki/Division%20by%20zero%20%28disambiguation%29 | Division by zero is a term used in mathematics if the divisor (denominator) is zero.
Division by Zero or Dividing by Zero or Divide by Zero may also refer to:
Division by Zero (album), by Hux Flux, 2003
Dividing by Zero, a 2002 album by Seven Storey Mountain
"Dividing by Zero", a song by the Offspring from the 2012 album Days Go By
Divide by Zero (album), by Killing Floor, 1997
Division by Zero (story), by Ted Chiang, 1991
See also
Zero Divide, video game
American wire gauge, including 1/0, 2/0 etc
"Two Divided by Zero", a song on the 1986 Pet Shop Boys album Please |
https://en.wikipedia.org/wiki/Method%20of%20support | In statistics, the method of support is a technique that is used to make inferences from datasets.
According to A. W. F. Edwards, the method of support aims to make inferences about unknown parameters in terms of the relative support, or log likelihood, induced by a set of data for a particular parameter value. The technique may be used whether or not prior information is available.
The method of maximum likelihood is part of the method of support, but note that the method of support also provides confidence regions that are defined in terms of their support.
Notable proponents of the method of support include A. W. F. Edwards.
Bibliography
Edwards, A.W.F. 1972. Likelihood. Cambridge University Press, Cambridge (expanded edition, 1992, Johns Hopkins University Press, Baltimore).
Likelihood
Maximum likelihood estimation |
https://en.wikipedia.org/wiki/The%20Saudi%20Repatriates%20Report | The Saudi Repatriates Report is a statistics analysis of the cases of 24 repatriated Saudi prisoners released from the Guantanamo Bay naval station since the first planeload of detainees arrived on 11 January 2003. They represent nearly half of the 53 Saudi nationals released from Guantanamo Bay in the period until the publication of the report (19 March 2007). The authors are Anant Raut and Jill M. Friedman, members of the Washington, D.C. office of Weil, Gotshal & Manges law firm. This firm has provided pro-bono representation for five Saudis detained at Guantanamo Bay and provides copies of the supporting documents to the Washington Post.
The report examines the reasons for release of this particular detainees and the timing of their release.
External links
Anant Raut, Jill M. Friedman (March 19, 2007). The Saudi Repatriates Report. Retrieved on April 21, 2007.
2007 documents |
https://en.wikipedia.org/wiki/Minimal%20model%20program | In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry.
Outline
The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety for which any birational morphism with a smooth surface is an isomorphism.
In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety , which for simplicity is assumed non-singular. There are two cases based on its Kodaira dimension, :
We want to find a variety birational to , and a morphism to a projective variety such that with the anticanonical class of a general fibre being ample. Such a morphism is called a Fano fibre space.
We want to find birational to , with the canonical class nef. In this case, is a minimal model for .
The question of whether the varieties and appearing above are non-singular is an important one. It seems natural to hope that if we start with smooth , then we can always find a minimal model or Fano fibre space inside the category of smooth varieties. However, this is not true, and so it becomes necessary to consider singular varieties also. The singularities that appear are called terminal singularities.
Minimal models of surfaces
Every irreducible complex algebraic curve is birational to a unique smooth projective curve, so the theory for curves is trivial. The case of surfaces was first investigated by the geometers of the Italian school around 1900; the contraction theorem of Guido Castelnuovo essentially describes the process of constructing a minimal model of any surface. The theorem states that any nontrivial birational morphism must contract a −1-curve to a smooth point, and conversely any such curve can be smoothly contracted. Here a −1-curve is a smooth rational curve C with self-intersection Any such curve must have which shows that if the canonical class is nef then the surface has no −1-curves.
Castelnuovo's theorem implies that to construct a minimal model for a smooth surface, we simply contract all the −1-curves on the surface, and the resulting variety Y is either a (unique) minimal model with K nef, or a ruled surface (which is the same as a 2-dimensional Fano fiber space, and is either a projective plane or a ruled surface over a curve). In the second case, the ruled surface birational to X is not unique, though there is a unique one isomorphic to the product of the projective line and a curve. A somewhat subt |
https://en.wikipedia.org/wiki/Sou%C4%8Dek%20space | In mathematics, Souček spaces are generalizations of Sobolev spaces, named after the Czech mathematician Jiří Souček. One of their main advantages is that they offer a way to deal with the fact that the Sobolev space W1,1 is not a reflexive space; since W1,1 is not reflexive, it is not always true that a bounded sequence has a weakly convergent subsequence, which is a desideratum in many applications.
Definition
Let Ω be a bounded domain in n-dimensional Euclidean space with smooth boundary. The Souček space W1,μ(Ω; Rm) is defined to be the space of all ordered pairs (u, v), where
u lies in the Lebesgue space L1(Ω; Rm);
v (thought of as the gradient of u) is a regular Borel measure on the closure of Ω;
there exists a sequence of functions uk in the Sobolev space W1,1(Ω; Rm) such that
and
weakly-∗ in the space of all Rm×n-valued regular Borel measures on the closure of Ω.
Properties
The Souček space W1,μ(Ω; Rm) is a Banach space when equipped with the norm given by
i.e. the sum of the L1 and total variation norms of the two components.
References
Banach spaces
Sobolev spaces |
https://en.wikipedia.org/wiki/Pinsky%20phenomenon | In mathematics, the Pinsky phenomenon is a result in Fourier analysis. This phenomenon was discovered by Mark Pinsky of Northwestern University. It involves the spherical inversion of the Fourier transform.
The phenomenon involves a lack of convergence at a point due to a discontinuity at boundary.
This lack of convergence in the Pinsky phenomenon happens far away from the boundary of the discontinuity, rather than at the discontinuity itself seen in the Gibbs phenomenon. This non-local phenomenon is caused by a lensing effect.
Prototypical example
Let a function g(x) = 1 for |x| < c in 3 dimensions, with g(x) = 0 elsewhere. The jump at |x| = c will cause an oscillatory behavior of the spherical partial sums, which prevents convergence at the center of the ball as well as the possibility of Fourier inversion at x = 0. Stated differently, spherical partial sums of a Fourier integral of the indicator function of a ball are divergent at the center of the ball but convergent elsewhere to the desired indicator function. This prototype example was coined the ”Pinsky phenomenon” by Jean-Pierre Kahane, CRAS, 1995.
Generalizations
This prototype example can be suitably generalized to Fourier integral expansions in higher dimensions, both in Euclidean space and other non-compact rank-one symmetric spaces.
Also related are eigenfunction expansions on a geodesic ball in a rank-one symmetric space, but one must consider boundary conditions. Pinsky and others also represent some results on the asymptotic behavior of the Fejer approximation in one dimension, inspired by work
of Bump, Persi Diaconis, and J. B. Keller.
References
Mathematics that describe the Pinsky phenomenon are available on pages 142 to 143, and generalizations on pages 143+, in the book Introduction to Fourier Analysis and Wavelets, by Mark A. Pinsky, 2002, Publisher: Thomson Brooks/Cole.
Real analysis
Fourier series |
https://en.wikipedia.org/wiki/Friedrich%20Christoph%20M%C3%BCller | Christoph Friedrich Müller (8 October 1751, Allendorf (Lumda) – 10 April 1808, Schwelm) was a theologian and cartographer in Schwelm.
Mueller studied theology, mathematics, astronomy and the sciences. In addition, he learned four languages. He was pastor from 1776 in Bad Sassendorf, from 1782 in Unna, and from 1785 in Schwelm.
1751 births
1808 deaths
People from Giessen (district) |
https://en.wikipedia.org/wiki/Involution | Involution may refer to:
Involute, a construction in the differential geometry of curves
Agricultural Involution: The Processes of Ecological Change in Indonesia, a 1963 study of intensification of production through increased labour inputs
Involution (mathematics), a function that is its own inverse
Involution (medicine), the shrinking of an organ (such as the uterus after pregnancy)
Involution (esoterism), several notions of a counterpart to evolution
Involution (Meher Baba), the inner path of the human soul to the self
Involution algebra, a *-algebra: an algebra equipped with an involution
Involution (album), a 1998 album by multi-instrumentalist Michael Marcus, with the Jaki Byard trio |
https://en.wikipedia.org/wiki/Lists%20of%20the%20Arab%20League | All lists and statistics of the Arab League
Economic Lists
Financing
GDP
External Debt
Industrial Growth Rate
Oil
Production
Demographic Lists
Population
Cities
Languages
Religions
Ethnicities
Arab League-related lists |
https://en.wikipedia.org/wiki/Projection%20matrix | In statistics, the projection matrix , sometimes also called the influence matrix or hat matrix , maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes the influence each response value has on each fitted value. The diagonal elements of the projection matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation.
Definition
If the vector of response values is denoted by and the vector of fitted values by ,
As is usually pronounced "y-hat", the projection matrix is also named hat matrix as it "puts a hat on ".
The element in the ith row and jth column of is equal to the covariance between the jth response value and the ith fitted value, divided by the variance of the former:
Application for residuals
The formula for the vector of residuals can also be expressed compactly using the projection matrix:
where is the identity matrix. The matrix is sometimes referred to as the residual maker matrix or the annihilator matrix.
The covariance matrix of the residuals , by error propagation, equals
,
where is the covariance matrix of the error vector (and by extension, the response vector as well). For the case of linear models with independent and identically distributed errors in which , this reduces to:
.
Intuition
From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of . A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
From there, one rearranges, so
Therefore, since is on the column space of , the projection matrix, which maps onto is just , or
Linear model
Suppose that we wish to estimate a linear model using linear least squares. The model can be written as
where is a matrix of explanatory variables (the design matrix), β is a vector of unknown parameters to be estimated, and ε is the error vector.
Many types of models and techniques are subject to this formulation. A few examples are linear least squares, smoothing splines, regression splines, local regression, kernel regression, and linear filtering.
Ordinary least squares
When the weights for each observation are identical and the errors are uncorrelated, the estimated parameters are
so the fitted values are
Therefore, the projection matrix (and hat matrix) is given by
Weighted and generalized least squares
The above may be generalized to the cases where the weights are not identical and/or the errors are correlated. Suppose that the covariance matrix of the errors is Σ. Then since
.
the hat matrix is thus
and again it may be seen that , though now it is no longer symmetric.
Properties
The projection matrix has a number of useful algebraic properties. In the language of linear algebra, the projection matrix is the orthogonal projection onto t |
https://en.wikipedia.org/wiki/Trigram%20%28disambiguation%29 | In the fields of computational linguistics and probability, Trigrams, are a special case of the n-gram, where n is 3.
Trigram may also refer to:
Bagua (called Eight Trigrams in English), a set of eight symbols in Taoist cosmology
A three-letter acronym
Trigram (FIFA), three letter codes used by the football association FIFA
See also
Trigram search
Digram (disambiguation)
Trigraph (disambiguation) |
https://en.wikipedia.org/wiki/Mathieu%20group%20M11 | {{DISPLAYTITLE:Mathieu group M11}}
In the area of modern algebra known as group theory, the Mathieu group M11 is a sporadic simple group of order
2432511 = 111098 = 7920.
History and properties
M11 is one of the 26 sporadic groups and was introduced by . It is the smallest sporadic group and, along with the other four Mathieu groups, the first to be discovered. The Schur multiplier and the outer automorphism group are both trivial.
M11 is a sharply 4-transitive permutation group on 11 objects. It admits many generating sets of permutations, such as the pair (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) of permutations used by the GAP computer algebra system.
Representations
M11 has a sharply 4-transitive permutation representation on 11 points. The point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points.
M11 has a 3-transitive permutation representation on 12 points with point stabilizer PSL2(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M11 in M12, exchanged by an outer automorphism.
The permutation representation on 11 points gives a complex irreducible representation in 10 dimensions. This is the smallest possible dimension of a faithful complex representation, though there are also two other such representations in 10 dimensions forming a complex conjugate pair.
M11 has two 5-dimensional irreducible representations over the field with 3 elements, related to the restrictions of 6-dimensional representations of the double cover of M12. These have the smallest dimension of any faithful linear representations of M11 over any field.
Maximal subgroups
There are 5 conjugacy classes of maximal subgroups of M11 as follows:
M10, order 720, one-point stabilizer in representation of degree 11
PSL(2,11), order 660, one-point stabilizer in representation of degree 12
M9:2, order 144, stabilizer of a 9 and 2 partition.
S5, order 120, orbits of 5 and 6
Stabilizer of block in the S(4,5,11) Steiner system
Q:S3, order 48, orbits of 8 and 3
Centralizer of a quadruple transposition
Isomorphic to GL(2,3).
Conjugacy classes
The maximum order of any element in M11 is 11. Cycle structures are shown for the representations both of degree 11 and 12.
References
Reprinted in
External links
MathWorld: Mathieu Groups
Atlas of Finite Group Representations: M11
Sporadic groups |
https://en.wikipedia.org/wiki/Mathieu%20group%20M12 | {{DISPLAYTITLE:Mathieu group M12}}
In the area of modern algebra known as group theory, the Mathieu group M12 is a sporadic simple group of order
12111098 = 2633511 = 95040.
History and properties
M12 is one of the 26 sporadic groups and was introduced by . It is a sharply 5-transitive permutation group on 12 objects. showed that the Schur multiplier of M12 has order 2 (correcting a mistake in where they incorrectly claimed it has order 1).
The double cover had been implicitly found earlier by , who showed that M12 is a subgroup of the projective linear group of dimension 6 over the finite field with 3 elements.
The outer automorphism group has order 2, and the full automorphism group M12.2 is contained in M24 as the stabilizer of a pair of complementary dodecads of 24 points, with outer automorphisms of M12 swapping the two dodecads.
Representations
calculated the complex character table of M12.
M12 has a strictly 5-transitive permutation representation on 12 points, whose point stabilizer is the Mathieu group M11. Identifying the 12 points with the projective line over the field of 11 elements, M12 is generated by the permutations of PSL2(11) together with the permutation (2,10)(3,4)(5,9)(6,7). This permutation representation preserves a Steiner system S(5,6,12) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code. In fact M12 has two inequivalent actions on 12 points, exchanged by an outer automorphism; these are analogous to the two inequivalent actions of the symmetric group S6 on 6 points.
The double cover 2.M12 is the automorphism group of the extended ternary Golay code, a dimension 6 length 12 code over the field of order 3 of minimum weight 6. In particular the double cover has an irreducible 6-dimensional representation over the field of 3 elements.
The double cover 2.M12 is the automorphism group of any 12×12 Hadamard matrix.
M12 centralizes an element of order 11 in the monster group, as a result of which it acts naturally on a vertex algebra over the field with 11 elements, given as the Tate cohomology of the monster vertex algebra.
Maximal subgroups
There are 11 conjugacy classes of maximal subgroups of M12, 6 occurring in automorphic pairs, as follows:
M11, order 7920, index 12. There are two classes of maximal subgroups, exchanged by an outer automorphism. One is the subgroup fixing a point with orbits of size 1 and 11, while the other acts transitively on 12 points.
S6:2 = M10.2, the outer automorphism group of the symmetric group S6 of order 1440, index 66. There are two classes of maximal subgroups, exchanged by an outer automorphism. One is imprimitive and transitive, acting with 2 blocks of 6, while the other is the subgroup fixing a pair of points and has orbits of size 2 and 10.
PSL(2,11), order 660, index 144, doubly transitive on the 12 points
32:(2.S4), order 432. Th |
https://en.wikipedia.org/wiki/Mathieu%20group%20M22 | {{DISPLAYTITLE:Mathieu group M22}}
In the area of modern algebra known as group theory, the Mathieu group M22 is a sporadic simple group of order
27325711 = 443520
≈ 4.
History and properties
M22 is one of the 26 sporadic groups and was introduced by . It is a 3-fold transitive permutation group on 22 objects. The Schur multiplier of M22 is cyclic of order 12, and the outer automorphism group has order 2.
There are several incorrect statements about the 2-part of the Schur multiplier in the mathematical literature.
incorrectly claimed that the Schur multiplier of M22 has order 3, and in a correction incorrectly claimed that it has order 6. This caused an error in the title of the paper announcing the discovery of the Janko group J4. showed that the Schur multiplier is in fact cyclic of order 12.
calculated the 2-part of all the cohomology of M22.
Representations
M22 has a 3-transitive permutation representation on 22 points, with point stabilizer the group PSL3(4), sometimes called M21. This action fixes a Steiner system S(3,6,22) with 77 hexads, whose full automorphism group is the automorphism group M22.2 of M22.
M22 has three rank 3 permutation representations: one on the 77 hexads with point stabilizer 24:A6, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A7.
M22 is the point stabilizer of the action of M23 on 23 points, and also the point stabilizer of the rank 3 action of the Higman–Sims group on 100 = 1+22+77 points.
The triple cover 3.M22 has a 6-dimensional faithful representation over the field with 4 elements.
The 6-fold cover of M22 appears in the centralizer 21+12.3.(M22:2) of an involution of the Janko group J4.
Maximal subgroups
There are no proper subgroups transitive on all 22 points. There are 8 conjugacy classes of maximal subgroups of M22 as follows:
PSL(3,4) or M21, order 20160: one-point stabilizer
24:A6, order 5760, orbits of 6 and 16
Stabilizer of W22 block
A7, order 2520, orbits of 7 and 15
There are 2 sets, of 15 each, of simple subgroups of order 168. Those of one type have orbits of 1, 7 and 14; the others have orbits of 7, 8, and 7.
A7, orbits of 7 and 15
Conjugate to preceding type in M22:2.
24:S5, order 1920, orbits of 2 and 20 (5 blocks of 4)
A 2-point stabilizer in the sextet group
23:PSL(3,2), order 1344, orbits of 8 and 14
M10, order 720, orbits of 10 and 12 (2 blocks of 6)
A one-point stabilizer of M11 (point in orbit of 11)
A non-split group extension of form A6.2
PSL(2,11), order 660, orbits of 11 and 11
Another one-point stabilizer of M11 (point in orbit of 12)
Conjugacy classes
There are 12 conjugacy classes, though the two classes of elements of order 11 are fused under an outer automorphism.
See also
M22 graph
References
Reprinted in
(The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.)
External links
MathWor |
https://en.wikipedia.org/wiki/Mathieu%20group%20M23 | {{DISPLAYTITLE:Mathieu group M23}}
In the area of modern algebra known as group theory, the Mathieu group M23 is a sporadic simple group of order
2732571123 = 10200960
≈ 1 × 107.
History and properties
M23 is one of the 26 sporadic groups and was introduced by . It is a 4-fold transitive permutation group on 23 objects. The Schur multiplier and the outer automorphism group are both trivial.
calculated the integral cohomology, and showed in particular that M23 has the unusual property that the first 4 integral homology groups all vanish.
The inverse Galois problem seems to be unsolved for M23. In other words, no polynomial in Z[x] seems to be known to have M23 as its Galois group. The inverse Galois problem is solved for all other sporadic simple groups.
Construction using finite fields
Let be the finite field with 211 elements. Its group of units has order − 1 = 2047 = 23 · 89, so it has a cyclic subgroup of order 23.
The Mathieu group M23 can be identified with the group of -linear automorphisms of that stabilize . More precisely, the action of this automorphism group on can be identified with the 4-fold transitive action of M23 on 23 objects.
Representations
M23 is the point stabilizer of the action of the Mathieu group M24 on 24 points, giving it a 4-transitive permutation representation on 23 points with point stabilizer the Mathieu group M22.
M23 has 2 different rank 3 actions on 253 points. One is the action on unordered pairs with orbit sizes 1+42+210 and point stabilizer M21.2, and the other is the action on heptads with orbit sizes 1+112+140 and point stabilizer 24.A7.
The integral representation corresponding to the permutation action on 23 points decomposes into the trivial representation and a 22-dimensional representation. The 22-dimensional representation is irreducible over any field of characteristic not 2 or 23.
Over the field of order 2, it has two 11-dimensional representations, the restrictions of the corresponding representations of the Mathieu group M24.
Maximal subgroups
There are 7 conjugacy classes of maximal subgroups of M23 as follows:
M22, order 443520
PSL(3,4):2, order 40320, orbits of 21 and 2
24:A7, order 40320, orbits of 7 and 16
Stabilizer of W23 block
A8, order 20160, orbits of 8 and 15
M11, order 7920, orbits of 11 and 12
(24:A5):S3 or M20:S3, order 5760, orbits of 3 and 20 (5 blocks of 4)
One-point stabilizer of the sextet group
23:11, order 253, simply transitive
Conjugacy classes
References
Reprinted in
External links
MathWorld: Mathieu Groups
Atlas of Finite Group Representations: M23
Sporadic groups |
https://en.wikipedia.org/wiki/Mathieu%20group%20M24 | {{DISPLAYTITLE:Mathieu group M24}}
In the area of modern algebra known as group theory, the Mathieu group M24 is a sporadic simple group of order
21033571123 = 244823040
≈ 2.
History and properties
M24 is one of the 26 sporadic groups and was introduced by . It is a 5-transitive permutation group on 24 objects. The Schur multiplier and the outer automorphism group are both trivial.
The Mathieu groups can be constructed in various ways. Initially, Mathieu and others constructed them as permutation groups. It was difficult to see that M24 actually existed, that its generators did not just generate the alternating group A24. The matter was clarified when Ernst Witt constructed M24 as the automorphism (symmetry) group of an S(5,8,24) Steiner system W24 (the Witt design). M24 is the group of permutations that map every block in this design to some other block. The subgroups M23 and M22 then are easily defined to be the stabilizers of a single point and a pair of points respectively.
Construction as a permutation group
M24 is the subgroup of S24 that is generated by the three permutations:
and
.
M24 can also be generated by two permutations:
and
M24 from PSL(3,4)
M24 can be built starting from PSL(3,4), the projective special linear group of 3-dimensional space over the finite field with 4 elements . This group, sometimes called M21, acts on the projective plane over the field F4, an S(2,5,21) system called W21. Its 21 blocks are called lines. Any 2 lines intersect at one point.
M21 has 168 simple subgroups of order 360 and 360 simple subgroups of order 168. In the larger projective general linear group PGL(3,4) both sets of subgroups form single conjugacy classes, but in M21 both sets split into 3 conjugacy classes. The subgroups respectively have orbits of 6, called hyperovals, and orbits of 7, called Fano subplanes. These sets allow creation of new blocks for larger Steiner systems. M21 is normal in PGL(3,4), of index 3. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F4 (the field automorphism). PGL(3,4) can therefore be extended to the group PΓL(3,4) of projective semilinear transformations, which is a split extension of M21 by the symmetric group S3. PΓL(3,4) has an embedding as a maximal subgroup of M24.
A hyperoval has no 3 points that are collinear. A Fano subplane likewise satisfies suitable uniqueness conditions.
To W21 append 3 new points and let the automorphisms in PΓL(3,4) but not in M21 permute these new points. An S(3,6,22) system W22 is formed by appending just one new point to each of the 21 lines and new blocks are 56 hyperovals conjugate under M21.
An S(5,8,24) system would have 759 blocks, or octads. Append all 3 new points to each line of W21, a different new point to the Fano subplanes in each of the sets of 120, and append appropriate pairs of new points to all the hyperovals. That accounts for all but 210 of the octads. Those remaining octads are subsets of W21 and a |
https://en.wikipedia.org/wiki/Central%20series | In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central series means it is a nilpotent group; for matrix rings (considered as Lie algebras), it means that in some basis the ring consists entirely of upper triangular matrices with constant diagonal.
This article uses the language of group theory; analogous terms are used for Lie algebras.
A general group possesses a lower central series and upper central series (also called the descending central series and ascending central series, respectively), but these are central series in the strict sense (terminating in the trivial subgroup) if and only if the group is nilpotent. A related but distinct construction is the derived series, which terminates in the trivial subgroup whenever the group is solvable.
Definition
A central series is a sequence of subgroups
such that the successive quotients are central; that is, , where denotes the commutator subgroup generated by all elements of the form , with g in G and h in H. Since , the subgroup is normal in G for each i. Thus, we can rephrase the 'central' condition above as: is normal in G and is central in for each i. As a consequence, is abelian for each i.
A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action (more prosaically, a basis in which each element is represented by a strictly upper triangular matrix); compare Engel's theorem.
A group need not have a central series. In fact, a group has a central series if and only if it is a nilpotent group. If a group has a central series, then there are two central series whose terms are extremal in certain senses. Since A0 = {1}, the center Z(G) satisfies A1 ≤ Z(G). Therefore, the maximal choice for A1 is A1 = Z(G). Continuing in this way to choose the largest possible Ai + 1 given Ai produces what is called the upper central series. Dually, since An = G, the commutator subgroup [G, G] satisfies [G, G] = [G, An] ≤ An − 1. Therefore, the minimal choice for An − 1 is [G, G]. Continuing to choose Ai minimally given Ai + 1 such that [G, Ai + 1] ≤ Ai produces what is called the lower central series. These series can be constructed for any group, and if a group has a central series (is a nilpotent group), these procedures will yield central series.
Lower central series
The lower central series (or descending central series) of a group G is the descending series of subgroups
G = G1 ⊵ G2 ⊵ ⋯ ⊵ Gn ⊵ ⋯,
where, for each n,
,
the subgroup of G generated by all commutators with and . Thus, , the derived subgroup of G, while , etc. The lower central series is often denoted . We say the series terminates or stablizes when , and the smallest such n is the length of the series.
This should not be confused with the derived series, whose terms are
,
not . The two s |
https://en.wikipedia.org/wiki/Tweedie%20distribution | In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poisson–gamma distributions which have positive mass at zero, but are otherwise continuous.
Tweedie distributions are a special case of exponential dispersion models and are often used as distributions for generalized linear models.
The Tweedie distributions were named by Bent Jørgensen after Maurice Tweedie, a statistician and medical physicist at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984.
Definitions
The (reproductive) Tweedie distributions are defined as subfamily of (reproductive) exponential dispersion models (ED), with a special mean-variance relationship.
A random variable Y is Tweedie distributed Twp(μ, σ2), if with mean , positive dispersion parameter and
where is called Tweedie power parameter.
The probability distribution Pθ,σ2 on the measurable sets A, is given by
for some σ-finite measure νλ.
This representation uses the canonical parameter θ of an exponential dispersion model and cumulant function
where we used , or equivalently .
Properties
Additive exponential dispersion models
The models just described are in the reproductive form. An exponential dispersion model has always a dual: the additive form. If Y is reproductive, then with is in the additive form ED*(θ,λ), for Tweedie Tw*p(μ, λ). Additive models have the property that the distribution of the sum of independent random variables,
for which Zi ~ ED*(θ,λi) with fixed θ and various λ are members of the family of distributions with the same θ,
Reproductive exponential dispersion models
A second class of exponential dispersion models exists designated by the random variable
where σ2 = 1/λ, known as reproductive exponential dispersion models. They have the property that for n independent random variables Yi ~ ED(μ,σ2/wi), with weighting factors wi and
a weighted average of the variables gives,
For reproductive models the weighted average of independent random variables with fixed μ and σ2 and various values for wi is a member of the family of distributions with same μ and σ2.
The Tweedie exponential dispersion models are both additive and reproductive; we thus have the duality transformation
Scale invariance
A third property of the Tweedie models is that they are scale invariant: For a reproductive exponential dispersion model Twp(μ, σ2) and any positive constant c we have the property of closure under scale transformation,
The Tweedie power variance function
To define the variance function for exponential dispersion models we make use of the mean value mapping, the relationship between the canonical parameter θ and the mean μ. It is defined by the function
with cumulative function .
The variance function V(μ) is c |
https://en.wikipedia.org/wiki/Khuza%CA%BDa%2C%20Khan%20Yunis | Khuza'a () is a Palestinian town in the Khan Yunis Governorate in the southern Gaza Strip. According to the Palestinian Central Bureau of Statistics, Khuza'a had a population of 11,388 inhabitants in 2017.
The town of Khuza'a is around 500 metres from the Green Line.
The society is strongly influenced by tribal structure so, there are many extended families such as Qudayh in () (the Ashraaf of the Holy Land), Alshawaf, Al-Daghmah, M'ssabih, Abu Yousef, Abu Mustafa, Abu Tair, Abu Dagga, Abu Tabash, Abu Draz, Abu Mutlaq, Abu Hamed, Abu subha and Abu Amer. The families generally turn to custom to solve disputes amongst themselves.
History
In the 1945 statistics, Khuza'a (named Khirbat Ikhzaa), had a population of 990, all Muslims, with 8,179 dunams of land, according to an official land and population survey. Of this, 7,987 dunams were used for cereals, while 8 dunams were built-up land.
Allegations of war crimes in the 2008–09 war
The Observer collected allegations from residents that during the 2008–09 Gaza War, the Israeli military bulldozed houses in Khuza'a with civilians still inside and that civilians were shot despite carrying white flags. B'Tselem collected accounts from residents consistent with what The Observer reported.
Bruno Stevens, a Western journalist who was among the first to get access to Gaza, reported that white phosphorus was used in the shelling of houses. Stevens reported "What I can tell you is that many, many houses were shelled and that they used white phosphorus" and that "it appears to have been indiscriminate".
Killings and destruction of most homes in the 2014 war
During the 2014 Israel–Gaza conflict, most of the over 500 houses were destroyed when the Israeli military went in with their tanks.
Dozens of civilians were fired on and killed by the Israeli army during the ground offensive, according to human right groups, which some called "apparent violations of the laws of war". Israel dropped leaflets warning civilians to flee, and most did, with only a few hundred remaining. Witnesses said they were used as human shields by Israeli soldiers, though these reports come from biased sources and are uncorroborated.
Helsingborgs Dagblad reported that the 5,000 residents fled after warning leaflets were dropped and most took refuge in UNRWA schools. Many residents were trapped because of Israeli shelling. Several Israeli soldiers said they were told Hamas had threatened to kill civilians who left their homes but this was "strongly denied" by more than a dozen residents of the town, who said Israel did not let them leave the fighting. Israeli soldiers said they, per instructions, fired warning shots to anyone who came close to them and then killed if they came closer. They also blamed Hamas' tactics, which they thought "made it impossible to determine who was or was not a threat". these tactics included Hamas fighters waving white flags as though they were civilians. However, more than a dozen of Khuza'a residents, a |
https://en.wikipedia.org/wiki/Restricted%20maximum%20likelihood | In statistics, the restricted (or residual, or reduced) maximum likelihood (REML) approach is a particular form of maximum likelihood estimation that does not base estimates on a maximum likelihood fit of all the information, but instead uses a likelihood function calculated from a transformed set of data, so that nuisance parameters have no effect.
In the case of variance component estimation, the original data set is replaced by a set of contrasts calculated from the data, and the likelihood function is calculated from the probability distribution of these contrasts, according to the model for the complete data set. In particular, REML is used as a method for fitting linear mixed models. In contrast to the earlier maximum likelihood estimation, REML can produce unbiased estimates of variance and covariance parameters.
The idea underlying REML estimation was put forward by M. S. Bartlett in 1937. The first description of the approach applied to estimating components of variance in unbalanced data was by Desmond Patterson and Robin Thompson of the University of Edinburgh in 1971, although they did not use the term REML.
A review of the early literature was given by Harville.
REML estimation is available in a number of general-purpose statistical software packages, including Genstat (the REML directive), SAS (the MIXED procedure), SPSS (the MIXED command), Stata (the mixed command), JMP (statistical software), and R (especially the lme4 and older nlme packages),
as well as in more specialist packages such as MLwiN, HLM, ASReml, BLUPF90, wombat, Statistical Parametric Mapping and CropStat.
REML estimation is implemented in Surfstat, a Matlab toolbox for the statistical analysis of univariate and multivariate surface and volumetric neuroimaging data using linear mixed effects models and random field theory, but more generally in the fitlme package for modeling linear mixed effects models in a domain-general way.
References
Maximum likelihood estimation |
https://en.wikipedia.org/wiki/Quasi-likelihood | In statistics, quasi-likelihood methods are used to estimate parameters in a statistical model when exact likelihood methods, for example maximum likelihood estimation, are computationally infeasible. Due to the wrong likelihood being used, quasi-likelihood estimators lose asymptotic efficiency compared to, e.g., maximum likelihood estimators. Under broadly applicable conditions, quasi-likelihood estimators are consistent and asymptotically normal. The asymptotic covariance matrix can be obtained using the so-called sandwich estimator. Examples of quasi-likelihood methods include the generalized estimating equations and pairwise likelihood approaches.
History
The term quasi-likelihood function was introduced by Robert Wedderburn in 1974 to describe a function that has similar properties to the log-likelihood function but is not the log-likelihood corresponding to any actual probability distribution. He proposed to fit certain quasi-likelihood models using a straightforward extension of the algorithms used to fit generalized linear models.
Application to overdispersion modelling
Quasi-likelihood estimation is one way of allowing for overdispersion, that is, greater variability in the data than would be expected from the statistical model used. It is most often used with models for count data or grouped binary data, i.e. data that would otherwise be modelled using the Poisson or binomial distribution.
Instead of specifying a probability distribution for the data, only a relationship between the mean and the variance is specified in the form of a variance function giving the variance as a function of the mean. Generally, this function is allowed to include a multiplicative factor known as the overdispersion parameter or scale parameter that is estimated from the data. Most commonly, the variance function is of a form such that fixing the overdispersion parameter at unity results in the variance-mean relationship of an actual probability distribution such as the binomial or Poisson. (For formulae, see the binomial data example and count data example under generalized linear models.)
Comparison to alternatives
Random-effects models, and more generally mixed models (hierarchical models) provide an alternative method of fitting data exhibiting overdispersion using fully specified probability models. However, these methods often become complex and computationally intensive to fit to binary or count data. Quasi-likelihood methods have the advantage of relative computational simplicity, speed and robustness, as they can make use of the more straightforward algorithms developed to fit generalized linear models.
See also
Quasi-maximum likelihood estimate
Extremum estimator
Notes
References
Likelihood
Maximum likelihood estimation |
https://en.wikipedia.org/wiki/Vedic%20square | In Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table where the entry in each cell is the digital root of the product of the column and row headings i.e. the remainder when the product of the row and column headings is divided by 9 (with remainder 0 represented by 9). Numerous geometric patterns and symmetries can be observed in a Vedic square, some of which can be found in traditional Islamic art.
Algebraic properties
The Vedic Square can be viewed as the multiplication table of the monoid where is the set of positive integers partitioned by the residue classes modulo nine. (the operator refers to the abstract "multiplication" between the elements of this monoid).
If are elements of then can be defined as , where the element 9 is representative of the residue class of 0 rather than the traditional choice of 0.
This does not form a group because not every non-zero element has a corresponding inverse element; for example but there is no such that .
Properties of subsets
The subset forms a cyclic group with 2 as one choice of generator - this is the group of multiplicative units in the ring . Every column and row includes all six numbers - so this subset forms a Latin square.
From two dimensions to three dimensions
A Vedic cube is defined as the layout of each digital root in a three-dimensional multiplication table.
Vedic squares in a higher radix
Vedic squares with a higher radix (or number base) can be calculated to analyse the symmetric patterns that arise. Using the calculation above, . The images in this section are color-coded so that the digital root of 1 is dark and the digital root of (base-1) is light.
See also
Latin square
Modular arithmetic
Monoid
References
Indian mathematics
Modular arithmetic |
https://en.wikipedia.org/wiki/F-divergence | In probability theory, an -divergence is a function that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of -divergence.
History
These divergences were introduced by Alfréd Rényi in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease in Markov processes. f-divergences were studied further independently by , and and are sometimes known as Csiszár -divergences, Csiszár–Morimoto divergences, or Ali–Silvey distances.
Definition
Non-singular case
Let and be two probability distributions over a space , such that , that is, is absolutely continuous with respect to . Then, for a convex function such that is finite for all , , and (which could be infinite), the -divergence of from is defined as
We call the generator of .
In concrete applications, there is usually a reference distribution on (for example, when , the reference distribution is the Lebesgue measure), such that , then we can use Radon–Nikodym theorem to take their probability densities and , giving
When there is no such reference distribution ready at hand, we can simply define , and proceed as above. This is a useful technique in more abstract proofs.
Extension to singular measures
The above definition can be extended to cases where is no longer satisfied (Definition 7.1 of ).
Since is convex, and , the function must nondecrease, so there exists , taking value in .
Since for any , we have , we can extend f-divergence to the .
Properties
Basic properties
Linearity: given a finite sequence of nonnegative real numbers and generators .
iff for some .
In particular, the monotonicity implies that if a Markov process has a positive equilibrium probability distribution then is a monotonic (non-increasing) function of time, where the probability distribution is a solution of the Kolmogorov forward equations (or Master equation), used to describe the time evolution of the probability distribution in the Markov process. This means that all f-divergences are the Lyapunov functions of the Kolmogorov forward equations. The converse statement is also true: If is a Lyapunov function for all Markov chains with positive equilibrium and is of the trace-form
() then , for some convex function f. For example, Bregman divergences in general do not have such property and can increase in Markov processes.
Analytic properties
The f-divergences can be expressed using Taylor series and rewritten using a weighted sum of chi-type distances ().
Naive variational representation
Let be the convex conjugate of . Let be the effective domain of
, that is, . Then we have two variational representations of , which we describe below.
Basic variational representation
Under the above setup,
This is Theorem 7.24 in.
Example applications
Using this theorem on total variation di |
https://en.wikipedia.org/wiki/SYT13 | Synaptotagmin-13 is a protein that in humans is encoded by the SYT13 gene.
Function
SYT13 belongs to the large synaptotagmin protein family. All synaptotagmins show type I membrane topology, with an extracellular N terminus, a single transmembrane region, and a cytoplasmic C terminus containing tandem C2 domains. Major functions of synaptotagmins include vesicular traffic, exocytosis, and secretion.[supplied by OMIM]
References
Further reading
External links |
https://en.wikipedia.org/wiki/Petter%20Lennartsson | Petter Oskar Lennartsson (born 13 March 1988) is a Swedish footballer, currently playing for Elverum. He has made two appearances as a sub in Allsvenskan.
Career statistics
External links
Living people
1988 births
Swedish men's footballers
Kalmar FF players
Nybergsund IL-Trysil players
Allsvenskan players
Norwegian First Division players
Swedish expatriate men's footballers
Expatriate men's footballers in Norway
Swedish expatriate sportspeople in Norway
Men's association football defenders |
https://en.wikipedia.org/wiki/Gerhard%20Streminger | Gerhard Streminger is an Austrian Philosopher and author, born in Graz in 1952 . From 1970, he studied philosophy and mathematics in Graz, Goettingen, Edinburgh with G.E.Davie and Oxford with J. L. Mackie. He gained his PhD in 1978 at the University of Graz, where he held posts from 1975 until 1997. In 1981 he was Visiting Professor at the University of Minnesota, Minneapolis.
Streminger was appointed Assistant Professor at the University of Graz in 1988 and received the title of University Professor in 1995.
He received several awards and prizes: 1974 a scholarship of the Deutsche Akademische Auslandsdienst; 1978 one of the British Council; and he was awarded the 1991/92 Humboldt Scholarship. In 2006 he gained the David Hume Award of the Kellmann Society for Humanism and Enlightenment.
Streminger is generally considered as an engaged agnostic/agnosticism. He is a member of the Giordano Bruno Stiftung, a society to promote evolutionary humanism.
Streminger is widely known as editor and translator of works of David Hume. His biographies and commentaries on Hume and Adam Smith are seen as the standard of research on the Scottish Enlightenment in the German-speaking world. Besides, he published many articles on this subject and the Philosophy of Religion. His philosophically most important work Gottes Guete und die Uebel der Welt deals comprehensively with Theodicy (the Problem of evil).
Publications
His most important publications are:
Adam Smith. Reinbek: Rowohlt 1989 (2. ed.: 1999, German) (New edition: Rowohlt e-book 2022).
Gottes Guete und die Uebel der Welt. Das Theodizeeproblem. Tuebingen: Mohr 1992, 2nd revised edition 2016 (German and Italian).
David Hume. Sein Leben und sein Werk. Paderborn: Schoeningh 1994 (2. ed.: 1994; paperback: 1995, German).
Der natuerliche Lauf der Dinge. Marburg: Metropolis 1995 (German).
David Hume. Der Philosoph und sein Zeitalter. München: C.H.Beck, 2011. Revised edition: Munich 2017.
ECCE TERRA. Norderstedt b. Hamburg (books on demand) 2008 (German). New Impression: Weitra - Bibliothek der Provinz 2013
DALRIADA. Ein schottisches Märchen. Graz: Leykam 2015
DIE FREMDE. Novel (in German). Vienna: Baumueller 2016 (also as e-book)
Adam Smith. Wohlstand und Moral. Eine Biographie. München: C.H.Beck, 2017, .
Die Welt gerät ins Wanken. Das Erdbeben von Lissabon im Jahre 1755 und seine Nachwirkungen auf das europäische Geistesleben. Ein literarischer Essay.. Alibri Verlag 2021, .
References
External links
Gerhard Stremingers Website, partly English
20th-century Austrian philosophers
Continental philosophers
Hume scholars
Academic staff of the University of Graz
1952 births
Living people |
https://en.wikipedia.org/wiki/Square%20pyramidal%20molecular%20geometry | Square pyramidal geometry describes the shape of certain chemical compounds with the formula where L is a ligand. If the ligand atoms were connected, the resulting shape would be that of a pyramid with a square base. The point group symmetry involved is of type C4v. The geometry is common for certain main group compounds that have a stereochemically-active lone pair, as described by VSEPR theory. Certain compounds crystallize in both the trigonal bipyramidal and the square pyramidal structures, notably .
As a transition state in Berry pseudorotation
As a trigonal bipyramidal molecule undergoes Berry pseudorotation, it proceeds via an intermediary stage with the square pyramidal geometry. Thus even though the geometry is rarely seen as the ground state, it is accessed by a low energy distortion from a trigonal bipyramid.
Pseudorotation also occurs in square pyramidal molecules. Molecules with this geometry, as opposed to trigonal bipyramidal, exhibit heavier vibration. The mechanism used is similar to the Berry mechanism.
Examples
Some molecular compounds that adopt square pyramidal geometry are XeOF4, and various halogen pentafluorides (XF5, where X = Cl, Br, I). Complexes of vanadium(IV), such as vanadyl acetylacetonate, [VO(acac)2], are square pyramidal (acac = acetylacetonate, the deprotonated anion of acetylacetone (2,4-pentanedione)).
See also
AXE method
Square pyramid
Hypervalent molecule
Molecular geometry
References
External links
Chem| Chemistry, Structures, and 3D Molecules
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
Molecular geometry |
https://en.wikipedia.org/wiki/Generalized%20linear%20mixed%20model | In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random effects in addition to the usual fixed effects. They also inherit from GLMs the idea of extending linear mixed models to non-normal data.
GLMMs provide a broad range of models for the analysis of grouped data, since the differences between groups can be modelled as a random effect. These models are useful in the analysis of many kinds of data, including longitudinal data.
Model
GLMMs are generally defined such that, conditioned on the random effects , the dependent variable is distributed according to the exponential family
with its expectation
related to the linear predictor via a link function
:
.
Here and are the fixed effects design matrix, and fixed effects respectively; and are the random effects design matrix and random effects respectively. To understand this very brief definition you will first need to understand the definition of a generalized linear model and of a mixed model.
Generalized linear mixed models are a special cases of hierarchical generalized linear models in which the random effects are normally distributed.
The complete likelihood
has no general closed form, and integrating over the random effects is usually extremely computationally intensive. In addition to numerically approximating this integral(e.g. via Gauss–Hermite quadrature), methods motivated by Laplace approximation have been proposed. For example, the penalized quasi-likelihood method, which essentially involves repeatedly fitting (i.e. doubly iterative) a weighted normal mixed model with a working variate, is implemented by various commercial and open source statistical programs.
Fitting a model
Fitting GLMMs via maximum likelihood (as via AIC) involves integrating over the random effects. In general, those integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data sets (e.g. ungrouped binary data are particularly problematic). For this reason, methods involving numerical quadrature or Markov chain Monte Carlo have increased in use, as increasing computing power and advances in methods have made them more practical.
The Akaike information criterion (AIC) is a common criterion for model selection. Estimates of AIC for GLMMs based on certain exponential family distributions have recently been obtained.
Software
Several contributed packages in R provide GLMM functionality, including lme4 and glmm.
GLMM can be fitted using SAS and SPSS
MATLAB also provides a function called "fitglme" to fit GLMM models.
The Python package Statsmodels supports binomial and poisson implementation
The Julia package MixedModels.jl provides a function called GeneralizedLinearMixedModel that fits a GLMM to provided data.
DHARMa: residual diagnostics for hierarchical (multi-level/mixed) regression models (utk. |
https://en.wikipedia.org/wiki/Wallman%20compactification | In mathematics, the Wallman compactification, generally called Wallman–Shanin compactification is a compactification of T1 topological spaces that was constructed by .
Definition
The points of the Wallman compactification ωX of a space X are the maximal proper filters in the poset of closed subsets of X. Explicitly, a point of ωX is a family of closed nonempty subsets of X such that is closed under finite intersections, and is maximal among those families that have these properties. For every closed subset F of X, the class ΦF of points of ωX containing F is closed in ωX. The topology of ωX is generated by these closed classes.
Special cases
For normal spaces, the Wallman compactification is essentially the same as the Stone–Čech compactification.
See also
Lattice (order)
Pointless topology
References
General topology
Compactification (mathematics) |
https://en.wikipedia.org/wiki/Arithmetic%20hyperbolic%203-manifold | In mathematics, more precisely in group theory and hyperbolic geometry, Arithmetic Kleinian groups are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic hyperbolic three-manifold is the quotient of hyperbolic space by an arithmetic Kleinian group.
Definition and examples
Quaternion algebras
A quaternion algebra over a field is a four-dimensional central simple -algebra. A quaternion algebra has a basis where and .
A quaternion algebra is said to be split over if it is isomorphic as an -algebra to the algebra of matrices ; a quaternion algebra over an algebraically closed field is always split.
If is an embedding of into a field we shall denote by the algebra obtained by extending scalars from to where we view as a subfield of via .
Arithmetic Kleinian groups
A subgroup of is said to be derived from a quaternion algebra if it can be obtained through the following construction. Let be a number field which has exactly two embeddings into whose image is not contained in (one conjugate to the other). Let be a quaternion algebra over such that for any embedding the algebra is isomorphic to the Hamilton quaternions. Next we need an order in . Let be the group of elements in of reduced norm 1 and let be its image in via . We then consider the Kleinian group obtained as the image in of .
The main fact about these groups is that they are discrete subgroups and they have finite covolume for the Haar measure on . Moreover, the construction above yields a cocompact subgroup if and only if the algebra is not split over . The discreteness is a rather immediate consequence of the fact that is only split at its complex embeddings. The finiteness of covolume is harder to prove.
An arithmetic Kleinian group is any subgroup of which is commensurable to a group derived from a quaternion algebra. It follows immediately from this definition that arithmetic Kleinian groups are discrete and of finite covolume (this means that they are lattices in ).
Examples
Examples are provided by taking to be an imaginary quadratic field, and where is the ring of integers of (for example and ). The groups thus obtained are the Bianchi groups. They are not cocompact, and any arithmetic Kleinian group which is not commensurable to a conjugate of a Bianchi group is cocompact.
If is any quaternion algebra over an imaginary quadratic number field which is not isomorphic to a matrix algebra then the unit groups of orders in are cocompact.
Trace field of arithmetic manifolds
The invariant trace field of a Kleinian group (or, through the monodromy image of the fundamental group, of an hyperbolic manifold) is the field generated by the traces of the squares of its elements. In the case of an arithmetic manifold whose fundamental groups is commensurable with that of a manifold derived from a quaternion algebra over a number field the invar |
https://en.wikipedia.org/wiki/Residual%20property | Residual property may refer to:
Residual property (mathematics), a property of groups
Residual property (physics), a thermodynamic term
it:Grandezze residue |
https://en.wikipedia.org/wiki/Cross%20product%20%28disambiguation%29 | The cross product is a product in vector algebra.
Cross product may also refer to:
Seven-dimensional cross product, a related product in seven dimensions
A product in a Künneth theorem
A crossed product in von Neumann algebras
A Cartesian product in set theory
See also
Cross-multiplication |
https://en.wikipedia.org/wiki/Jean-Louis%20Colliot-Th%C3%A9l%C3%A8ne | Jean-Louis Colliot-Thélène (born 2 December 1947) is a French mathematician. He is a Directeur de Recherches at CNRS at the Université Paris-Saclay in Orsay.
He studies mainly number theory and arithmetic geometry.
Awards
Prize of the French Academy of Sciences "Charles Louis de Saulces de Freycine" (1985)
Invited Speaker to the International Congress of Mathematicians (Berkeley 1986)
Fermat Prize for mathematical research (1991)
Grand prize of the French Academy of Sciences "Léonid Frank" (2009)
Fellow of the American Mathematical Society (2012)
References
External links
The personal web page of Jean-Louis Colliot-Thélène
Léonid Frank prize
1947 births
Number theorists
Paris-Sud University alumni
École Normale Supérieure alumni
20th-century French mathematicians
21st-century French mathematicians
Fellows of the American Mathematical Society
Living people
Academic staff of Paris-Saclay University |
https://en.wikipedia.org/wiki/Kafr%20ad-Dik | Kafr ad-Dik () is a Palestinian town located 9.5 kilometers west of Salfit in the Salfit Governorate of Palestine, in the northern West Bank. According to the Palestinian Central Bureau of Statistics (PCBS), the town had a population of 5,551 in 2017.
Approximately 70% of the families in Kafr ad-Dik are dependent on agriculture as the main source of income, while the remaining 30% work in the private and public sectors. The unemployment rate in the town is 60%.
The town's total land area consists of 15,228 dunams of which 578 dunams are built-up. In the town's area, the archaeological ruins of Deir Samaan are located.
The Israeli settlements of Peduel and Alei Zahav, as well as nearby Israeli outposts, are built over 1,448 dunams of Kafr ad-Dik's land. As a result of the Interim Agreement on the West Bank and the Gaza Strip, the Palestinian National Authority controls the civil affairs of 1,953 dunums of Kafr ad-Dik's land (Area B), while 13,275 dunams are classified as Area C, which is under full Israeli control.
Location
Kafr ad Dik located west of Salfit. It is bordered by Bruqin to the east, Bani Zaid to the south, Rafat and Deir Ballut to the west, and Biddya and Sarta to the north.
History
It has been suggested that this is the place mentioned in Crusader sources under the name of Caphaer; a village connected with the Casale Santa Maria. In 1175, Crusader sources mentions a former cistern-keeper of the village. In 1176, the revenues from Caphaer (=Kafr ad-Dik) and caslia S. Maria (=Aboud) were given for the provision of white bread for the sick in the Hospital in Jerusalem.
The coat of arms the Mamluk Sultan Qaitbay (1468-1496 C.E.) have been found in a mosque in the village.
Ottoman era
It has been suggested that this village is the Kafr Bani Hamid of the 1596 Ottoman tax records, with 83 Muslim families.
In the 18th and 19th centuries, the village formed part of 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 village el-Kufr, part of the Jurat Merda district, south of Nablus.
In 1870 Victor Guérin found here very considerable remains. They included two birkets cut in the rock, one 15 paces long by 12 broad, the other not quite so large; about 30 cisterns and 20 tombs cut in the rock, some with sepulchral chambers, their walls pierced with loculi, others simple graves, either intended for a single body or having right and left vaulted tombs with arcosolia. These gra |
https://en.wikipedia.org/wiki/Coarse%20space | In mathematics, coarse space may refer to
Coarse structure, a family of sets in geometry and topology to measure large-scale properties of a space
Coarse space (numerical analysis), a reduced representation of a numerical problem |
https://en.wikipedia.org/wiki/Yangian | In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970s and early 1980s concerning the quantum inverse scattering method. The name Yangian was introduced by Vladimir Drinfeld in 1985 in honor of C.N. Yang.
Initially, they were considered a convenient tool to generate the solutions of the quantum Yang–Baxter equation.
The center of the Yangian can be described by the quantum determinant.
The Yangian is a degeneration of the quantum loop algebra (i.e. the quantum affine algebra at vanishing central charge).
Description
For any finite-dimensional semisimple Lie algebra a, Drinfeld defined an infinite-dimensional Hopf algebra Y(a), called the Yangian of a. This Hopf algebra is a deformation of the universal enveloping algebra U(a[z]) of the Lie algebra of polynomial loops of a given by explicit generators and relations. The relations can be encoded by identities involving a rational R-matrix. Replacing it with a trigonometric R-matrix, one arrives at affine quantum groups, defined in the same paper of Drinfeld.
In the case of the general linear Lie algebra glN, the Yangian admits a simpler description in terms of a single ternary (or RTT) relation on the matrix generators due to Faddeev and coauthors.
The Yangian Y(glN) is defined to be the algebra generated by elements with 1 ≤ i, j ≤ N and p ≥ 0, subject to the relations
Defining , setting
and introducing the R-matrix R(z) = I + z−1 P on CNCN,
where P is the operator permuting the tensor factors, the above relations can be written more simply as the ternary relation:
The Yangian becomes a Hopf algebra with comultiplication Δ, counit ε and antipode s given by
At special values of the spectral parameter , the R-matrix degenerates to a rank one projection. This can be used to define the quantum determinant of , which generates the center of the Yangian.
The twisted Yangian Y−(gl2N), introduced by G. I. Olshansky, is the co-ideal generated by the coefficients of
where σ is the involution of gl2N given by
Applications
Classical representation theory
G.I. Olshansky and I.Cherednik discovered that the Yangian of glN is closely related with the branching properties of irreducible finite-dimensional representations of general linear algebras. In particular, the classical Gelfand–Tsetlin construction of a basis in the space of such a representation has a natural interpretation in the language of Yangians, studied by M.Nazarov and V.Tarasov. Olshansky, Nazarov and Molev later discovered a generalization of this theory to other classical Lie algebras, based on the twisted Yangian.
Physics
The Yangian appears as a symmetry group in different models in physics.
Yangian appears as a symmetry group of one-dimensional exactly solvable models such as spin chains, Hubbard model and in models of one-dimensional relativistic quantum field theory.
The m |
https://en.wikipedia.org/wiki/List%20of%20Bradford%20City%20A.F.C.%20records%20and%20statistics | These are a list of player and club records for Bradford City Association Football Club.
Honours
League
Division One
Runners-up (1): 1998–99
Division Two
Winners (1): 1907–08
Play-off winners (1): 1995–96
Division Three
Winners (1): 1984–85
Division Three (North)
Winners (1): 1928–29
Division Four
Runners-up (1): 1981–82
Cup
FA Cup
Winners (1): 1911
Football League Cup
Runners-up(1): 2013
Third Division North Challenge Cup
Winners (1): 1939
Runners-up (1): 1938
Player records
Youngest and oldest
Youngest player: 15 years 332 days – Reece Staunton v Rotherham United, 7 November 2017.
Oldest player: 41 years 178 days – Neville Southall v Leeds United, 12 March 2000.
Most appearances
The following players have played more than 300 league appearances for Bradford City.
Most appearances : 574 – Ces Podd.
Goalscorers
Most goals in a season: 36 – David Layne, 1961–62.
Most league goals in a season: 34 – David Layne, 1961–62.
Most goals scored in a match: 7 – Albert Whitehurst v Tranmere Rovers, Division Three (North), 6 March 1929.
Most goals scored : 143 – Bobby Campbell.
Top goalscorers
The following players have scored more than 60 league goals for Bradford City.
Transfers
Record transfer fees paid
The following players are all the players for whom Bradford City have paid at least £1 million.
Record transfer fees received
Highest transfer fee received: £2 million – Des Hamilton, to Newcastle United, March 1997.
Highest transfer fee received: £2 million – Andy O'Brien, to Newcastle United, March 2001.
Managerial records
First manager: Robert Campbell (managed the club for 79 matches from June 1903 to October 1905).
Longest serving manager: Peter O'Rourke (managed the club for 497 matches from November 1905 to June 1921).
Club records
Goals
Most league goals scored in a season: 128 in 42 matches – Division Three (North), 1928–29.
Fewest league goals scored in a season: 30 in 38 matches – Premiership, 2000–01.
Most league goals conceded in a season: 94 in 42 matches – Division Two, 1936–37.
Most league goals conceded in a season: 94 in 46 matches – Division Four, 1965–66.
Fewest league goals conceded in a season: 40 in 46 matches – League One, 2015 - 16.
Points
Most points in a season
Two points for a win: 63 in 42 matches – Division Three (North), 1928–29.
Three points for a win: 94 in 46 matches – Division Three, 1984–85.
Fewest points in a season
Two points for a win: 23 in 42 matches – Division Two, 1926–27.
Three points for a win: 26 in 38 matches – Premiership, 2000–01.
Matches
Firsts
First league match: Grimsby Town 2–0 Bradford City, Division Two at Blundell Park, 1 September 1903.
First FA Cup match: Bradford City 6–1 Rockingham Colliery, first round qualifying at Valley Parade, 3 October 1903.
First League Cup match: Bradford City 2–1 Manchester United, second round at Valley Parade, 2 November 1960.
First European match: FK Atlantas 1–3 Bradford City, Intertoto Cup second round at Žalgiris Stadium, 2 July 2000.
Record v |
https://en.wikipedia.org/wiki/Daoxing%20Xia | Daoxing Xia () is a Chinese American mathematician. He is currently a professor at the Department of Mathematics, Vanderbilt University in the United States. He was elected an academician of the Chinese Academy of Science in 1980.
Career
Xia was born on October 20, 1930, in Taizhou, Jiangsu. He pursued his undergraduate studies at the Department of Mathematics at Shandong University and subsequently obtained his postgraduate degree from the Department of Mathematics at Zhejiang University in 1952. His advisor was Chen Jiangong, a pioneer of modern Chinese mathematics who was then dean of the Department of Mathematics.
In 1952, Xia went to Fudan University in Shanghai as an assistant. In 1954 he became a lecturer and in 1956 he received a position as an associate professor. In September 1957, he was sent to Moscow State University in the USSR where he did one year of research.
In 1978 he obtained his professorship at Fudan University and rose to the position of vice director of the university's Mathematics Research Institute. In 1980 he was elected a member of the Chinese Academy of Science. He was also an adjunct professor in the Chinese Academy of Science Mathematical Physics Research Institute and the Department of Mathematics at Shandong University. Xia was a visiting professor of many universities and gave lectures.
In 1984 he went to the United States to become a professor at the Department of Mathematics, Vanderbilt University.
Bibliography
Spectral Theory of Hyponormal Operators, by Daoxing Xia, Springer Verlag (January 1984)
Spectral Theory of Linear Operators, (with S. Yan), Press Chinese Academy of Science, Beijing (1987).
The Second of Functional Analysis, (with S. Yan, W. Su and Y. Tong), Press Higher Education, Beijing (1986).
An Invitation to the Theory of Linear Topological Spaces, (with Y.L. Yang), Science & Technology Press Shanghai (1986).
Theory of a Real Variable and Functional Analysis, (with S. Yan. Z. Wu and W. Su), Press Chinese Academy of Science, Beijing, (1980).
Measures and Integration on Infinite-dimensional spaces, Science & Technology Press Shanghai (1965), Acad. Press, New York, London (1972).
Theory of Functions of a Real Variable and Essentials of Functional Analysis, (with S. Yan and Z. Wu), Science & Technology Press Shanghai (1956).
References
External links
Daoxing Xia's homepage at the Department of Mathematics, Vanderbilt University, including photo
Daoxing Xia's publications
Xia's profile in the website of Chinese Academy of Science (Branch of Shanghai)
20th-century American mathematicians
21st-century American mathematicians
1930 births
American people of Chinese descent
Zhejiang University alumni
Members of the Chinese Academy of Sciences
Living people
Mathematicians from Jiangsu
Academic staff of Fudan University
Vanderbilt University faculty
Scientists from Taizhou, Jiangsu
Academic staff of Shandong University
20th-century Chinese science writers
Writers from Taizhou, Jiangsu
Edu |
https://en.wikipedia.org/wiki/Emile%20Waxweiler | Emile Waxweiler (1867–1916) was a Belgian engineer and sociologist. He was a member of the Royal Academy of Belgium as well as the International Institute of Statistics (Sarton 1917: 168).
Waxweiler was born in Mechelen, Belgium, 22 May 1867, and died in a street accident in London, where he was attached to the London School of Economics, in late June 1916 (Sarton 1917: 168).
Waxweiler's education included taking the “highest degree” in engineering from the University of Ghent, and then spending a year in the United States, where he studied labor questions and industrial organization (Sarton 1917: 168). In 1895, he was appointed head of the statistics section of the Belgian Office of Labor, and from 1897 on, Waxweiler taught courses in political and financial economics, statistics and demographics, as well as descriptive sociology, at the Université Libre de Bruxelles (Sauveur 1924: 395–396). These teaching obligations did not prevent him, however, from serving, beginning in 1901–1902, as director of the Solvay Institute of Sociology (Sarton 1917: 168; Sauveur 1924: 395).
In addition to his career-long emphasis on the importance of statistics as an analytical tool for all of the life sciences (Sauveur 1924: 397; Waxweiler 1909a), Waxweiler's major scientific contribution was his conception of sociology as a subfield of biology, in particular, ethology (Waxweiler 1906). In his Esquisse d’une sociologie of 1906, Waxweiler defined sociology (along with its alternative names of “social ethology” and “social energetics”), as “the science, one could almost say, the physiology of reactive phenomena caused by the mutual excitations of individuals of the same species, without distinctions of sex” (Waxweiler 1906: 62–63).
Furthermore, Waxweiler early on advocated a system of profit-sharing by which employees become co-partners with their employers (Waxweiler 1898; Gide 1899: 240; Willoughby 1899: 121), and also argued for compulsory education laws and limits on child labor in Belgium (McLean and Waxweiler 1906).
In the final two years of his life, Waxweiler published two popular books dealing with Germany's invasion of Belgium in 1914 (Waxweiler 1915; 1916).
Esquisse d’une sociologie
Waxweiler's Esquisse d’une sociologie [Sketch of a sociology] was published as the second fascicule of the Solvay Institute of Sociology’s Notes et Mémoires series. As George Sarton (1924: 168) explained, “The Esquisse displayed a vast programme of research that Waxweiler had been obliged to outline as a working basis for the Institute of Sociology. This Institute had been founded a few years before, thanks to Ernest Solvay’s munificence, and entrusted to Waxweiler in 1902.”
The Esquisse, along with the other fascicules of the Notes et Mémoires series published by the Solvay Institute of Sociology in 1906, was reviewed by A. F. Chamberlain in the April 1907 issue of the American Journal of Psychology:
In his “Outlines of Sociology,” Emile Waxweiler, who is a professor |
https://en.wikipedia.org/wiki/Chen%20Jiangong | Chen Jiangong (; 1893–1971), or Jian-gong Chen, was a Chinese mathematician. He was a pioneer of modern Chinese mathematics. He was the dean of the Department of Mathematics, National Chekiang University (now Zhejiang University), and a founding academician the Chinese Academy of Sciences (elected 1955).
Life
Chen was born in Shanyin County (now Shaoxing), Zhejiang Province during the late Qing dynasty. He studied at Shanyin School and later Shaoxing Prefecture School. In 1910 he entered the Zhejiang Advanced Normal School, a teacher-training institution which was later merged into National Chekiang University.
Chen later went to Japan to continue his studies. In 1916 he graduated from the Tokyo Institute of Technology, where he majored in textile technology, and the Tokyo Academy of Physics (now known as the Tokyo University of Science).
After graduating from Tohoku Imperial University in 1923, Chen returned to China and became a lecturer at the Zhejiang Industrial School, which was later merged into National Chekiang University. In 1924 he went to Wuhan, Hubei Province, and became a professor at National Wuchang University (now Wuhan University).
In 1926, Chen returned to Tohoku Imperial University to continue his studies in mathematics, completing his Ph.D. in 1929 and becoming the first international student awarded a Ph.D. by a Japanese university. It was during this time that he met his later colleague Su Buqing, a fellow mathematics Ph.D. candidate.
After earning his doctorate, Chen was offered teaching positions at institutions including Peking University and Wuhan University. However, on the invitation of National Chekiang University president Shao Feizhi, Chen returned to Zhejiang University to serve as dean of the Department of Mathematics, a position he held for the next 20 years.
After earning his doctorate in 1931, Su Buqing was invited to join Chen's department and take over his position as department chair, allowing Chen to focus more on research. Their collaboration resulted in the Chen-Su school of mathematics in Hangzhou.
The outbreak of the Second Sino-Japanese War in 1937 forced Chen, and much of Zhejiang University, to relocate from Hangzhou. In February 1940, Chen arrived at Zunyi, and then subsequently Meitan, Guizhou Province, where he helped re-establish the colleges of engineering and sciences.
In 1945, after the end of the Second Sino-Japanese War, Chen was invited by biologist Luo Zongluo (Lo Tsung-lo), who was serving as the 1st president of National Taiwan University (formerly Taihoku Imperial University), as well as the Nationalist government in Nanjing, to travel to Taipei and serve as acting dean of NTU during its reorganization.
In the spring of 1946, Chen returned to mainland China (then still controlled by Nationalist government), where he continued teaching in National Chekiang University and became a researcher in the Mathematics Research Institute at the Academia Sinica. From 1947 to 1948, Chen t |
https://en.wikipedia.org/wiki/Predicate%20functor%20logic | In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors (or predicate modifiers) that operate on terms to yield terms. PFL is mostly the invention of the logician and philosopher Willard Quine.
Motivation
The source for this section, as well as for much of this entry, is Quine (1976). Quine proposed PFL as a way of algebraizing first-order logic in a manner analogous to how Boolean algebra algebraizes propositional logic. He designed PFL to have exactly the expressive power of first-order logic with identity. Hence the metamathematics of PFL are exactly those of first-order logic with no interpreted predicate letters: both logics are sound, complete, and undecidable. Most work Quine published on logic and mathematics in the last 30 years of his life touched on PFL in some way.
Quine took "functor" from the writings of his friend Rudolf Carnap, the first to employ it in philosophy and mathematical logic, and defined it as follows:
"The word functor, grammatical in import but logical in habitat... is a sign that attaches to one or more expressions of given grammatical kind(s) to produce an expression of a given grammatical kind." (Quine 1982: 129)
Ways other than PFL to algebraize first-order logic include:
Cylindric algebra by Alfred Tarski and his American students. The simplified cylindric algebra proposed in Bernays (1959) led Quine to write the paper containing the first use of the phrase "predicate functor";
The polyadic algebra of Paul Halmos. By virtue of its economical primitives and axioms, this algebra most resembles PFL;
Relation algebra algebraizes the fragment of first-order logic consisting of formulas having no atomic formula lying in the scope of more than three quantifiers. That fragment suffices, however, for Peano arithmetic and the axiomatic set theory ZFC; hence relation algebra, unlike PFL, is incompletable. Most work on relation algebra since about 1920 has been by Tarski and his American students. The power of relation algebra did not become manifest until the monograph Tarski and Givant (1987), published after the three important papers bearing on PFL, namely Bacon (1985), Kuhn (1983), and Quine (1976);
Combinatory logic builds on combinators, higher order functions whose domain is another combinator or function, and whose range is yet another combinator. Hence combinatory logic goes beyond first-order logic by having the expressive power of set theory, which makes combinatory logic vulnerable to paradoxes. A predicate functor, on the other hand, simply maps predicates (also called terms) into predicates.
PFL is arguably the simplest of these formalisms, yet also the one about which the least has been written.
Quine had a lifelong fascination with combinatory logic, attested to by his introduction to the |
https://en.wikipedia.org/wiki/Flat%20bar%20road%20bike | A flat bar road bike, also called a fitness bike, is a hybrid bike optimized for road usage or a road bike with a flat handlebar in place of a drop bar.
Frame construction and geometry borrow significantly from conventional road design. The frame is constructed to a light or middleweight standard with a shape that promotes an aggressive, aerodynamic posture suited to riding at higher speeds. There is no conventional suspension for either wheel, though the front fork may use carbon or steel to quell vibration. Wheel size is almost universally 700c with a width of 28 mm to 32 mm, somewhat wider than the 23 mm to 25 mm road bike standard.
The drivetrain of a flat bar bike often borrows features from multiple bike styles, pairing the trigger-shifting approach of mountain bikes with the taller cassette ratios of road bikes. The brakes of purpose-built flat-bar designs tend to be linear-pull, a mechanism nonexistent in road bikes and largely displaced by discs with mountain bikes. Disc brake penetration in road cycling continues to increase, however, with flat bar and cyclocross bikes leading the curve.
Relative to more upright hybrids, flat bar road bikes are often lighter and more efficient to pedal, though less so than a drop bar bike by equal measure. Drop bar bikes will have considerable aerodynamic advantages above about 15 MPH. Conventional hybrids will have a measure of off-road capability lacking in entirety from flat bar bikes. For efficient speed with the familiarity and stability of a flat bar, however, the flat bar road bike is an optimal compromise.
See also
Outline of cycling
References
Cycle types |
https://en.wikipedia.org/wiki/Judith%20Roitman | Judith "Judy" Roitman (born November 12, 1945) is a mathematician, a retired professor at the University of Kansas. She specializes in set theory, topology, Boolean algebras, and mathematics education.
Biography
Roitman was born in 1945 in New York City. She attended Oberlin College, followed by Sarah Lawrence College, graduating in 1966 with a degree in English literature. Next, she became interested in mathematical linguistics. As she had little formal mathematical education, Roitman started taking mathematics classes at the University of California, Berkeley and San Francisco State University. She had enjoyed mathematics as a high school student and found her interest renewed. In 1969 she started graduate studies in mathematics at Berkeley. During graduate school, she spent some time teaching mathematics in elementary schools as a Community Teaching Fellow with Project SEED. Roitman received her Ph.D. in 1974 from UC Berkeley with a thesis in topology; her thesis advisor was Robert M. Solovay. She taught at Wellesley College for three years, then spent a semester at the Institute for Advanced Study. She has been at the University of Kansas since then.
She has been involved in the field of mathematics education for much of her career, running workshops for elementary school teachers and high school teachers and observing them in the classroom. She has encouraged individual mathematicians and the mathematical community at large to get involved and take mathematics education more seriously. She was in the National Council of Teachers of Mathematics writing group that produced Principles and Standards for School Mathematics. Dismayed at the politicization of U.S. mathematics education, Roitman has insisted, "There is no math war."
Roitman has been active in the Association for Women in Mathematics since its early years, and she served as president for the term 1979–1981. She has been a Zen Buddhist since 1976, and is currently the guiding teacher of the Kansas Zen Center, of which she and her husband Stanley Lombardo were founding members.
Roitman is also a poet. Her poetry has appeared in a number of magazines, seven chapbooks, and two books.
Awards and honors
In 1996, she received the Louise Hay Award as recognition for her role as a math educator. In 2012 she became a fellow of the American Mathematical Society. In 2017, she was selected as a fellow of the Association for Women in Mathematics in the inaugural class.
Selected publications
Selected poetry
References
External links
Roitman's Blog
1945 births
20th-century American mathematicians
21st-century American mathematicians
Living people
Educators from New York City
American Buddhists
American former Christians
American women educators
Sarah Lawrence College alumni
Set theorists
Topologists
University of California, Berkeley alumni
University of Kansas faculty
American Zen Buddhist spiritual teachers
Wellesley College faculty
Fellows of the American Mathematical Society
American |
https://en.wikipedia.org/wiki/Seesaw%20molecular%20geometry | Disphenoidal or seesaw (also known as sawhorse) is a type of molecular geometry where there are four bonds to a central atom with overall C2v molecular symmetry. The name "seesaw" comes from the observation that it looks like a playground seesaw. Most commonly, four bonds to a central atom result in tetrahedral or, less commonly, square planar geometry.
The seesaw geometry occurs when a molecule has a steric number of 5, with the central atom being bonded to 4 other atoms and 1 lone pair (AX4E1 in AXE notation). An atom bonded to 5 other atoms (and no lone pairs) forms a trigonal bipyramid with two axial and three equatorial positions, but in the seesaw geometry one of the atoms is replaced by a lone pair of electrons, which is always in an equatorial position. This is true because the lone pair occupies more space near the central atom (A) than does a bonding pair of electrons. An equatorial lone pair is repelled by only two bonding pairs at 90°, whereas a hypothetical axial lone pair would be repelled by three bonding pairs at 90° which would make it stable. Repulsion by bonding pairs at 120° is much smaller and less important.
Structure
Compounds with disphenoidal (see-saw) geometry have two types of ligands: axial and equatorial. The axial pair lie along a common bond axis so that are related by a bond angle of 180°. The equatorial pair of ligands is situated in a plane orthogonal to the axis of the axial pair. Typically the bond distance to the axial ligands is longer than to the equatorial ligands. The ideal angle between the axial ligands and the equatorial ligands is 90°; whereas the ideal angle between the two equatorial ligands themselves is 120°.
Disphenoidal molecules, like trigonal bipyramidal ones, are subject to Berry pseudorotation in which the axial ligands move to equatorial positions and vice versa. This exchange of positions results in similar time-averaged environments for the two types of ligands. Thus, the 19F NMR spectrum of SF4 (like that of PF5) consists of single resonance near room temperature. The four atoms in motion act as a lever about the central atom; for example, the four fluorine atoms of sulfur tetrafluoride rotate around the sulfur atom.
Examples
Sulfur tetrafluoride is the premier example of a molecule with the disphenoidal molecular geometry (see image at upper right). The following compounds and ions have disphenoidal geometry:
SF4
SeF4
IOF3
TeF4
XeO2F2
SCl4
See also
Molecular geometry
AXE method
References
External links
Chem| Chemistry, Structures, and 3D Molecules
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
VSEPR
Molecular geometry |
https://en.wikipedia.org/wiki/Linear%20molecular%20geometry | The linear molecular geometry describes the geometry around a central atom bonded to two other atoms (or ligands) placed at a bond angle of 180°. Linear organic molecules, such as acetylene (), are often described by invoking sp orbital hybridization for their carbon centers.
According to the VSEPR model (Valence Shell Electron Pair Repulsion model), linear geometry occurs at central atoms with two bonded atoms and zero or three lone pairs ( or ) in the AXE notation. Neutral molecules with linear geometry include beryllium fluoride () with two single bonds, carbon dioxide () with two double bonds, hydrogen cyanide () with one single and one triple bond. The most important linear molecule with more than three atoms is acetylene (), in which each of its carbon atoms is considered to be a central atom with a single bond to one hydrogen and a triple bond to the other carbon atom. Linear anions include azide () and thiocyanate (), and a linear cation is the nitronium ion ().
Linear geometry also occurs in molecules, such as xenon difluoride () and the triiodide ion () with one iodide bonded to the two others. As described by the VSEPR model, the five valence electron pairs on the central atom form a trigonal bipyramid in which the three lone pairs occupy the less crowded equatorial positions and the two bonded atoms occupy the two axial positions at the opposite ends of an axis, forming a linear molecule.
See also
AXE method
Molecular geometry
References
External links
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
Molecular geometry |
https://en.wikipedia.org/wiki/Bent%20molecular%20geometry | In chemistry, molecules with a non-collinear arrangement of two adjacent bonds have bent molecular geometry, also known as angular or V-shaped. Certain atoms, such as oxygen, will almost always set their two (or more) covalent bonds in non-collinear directions due to their electron configuration. Water (H2O) is an example of a bent molecule, as well as its analogues. The bond angle between the two hydrogen atoms is approximately 104.45°. Nonlinear geometry is commonly observed for other triatomic molecules and ions containing only main group elements, prominent examples being nitrogen dioxide (NO2), sulfur dichloride (SCl2), and methylene (CH2).
This geometry is almost always consistent with VSEPR theory, which usually explains non-collinearity of atoms with a presence of lone pairs. There are several variants of bending, where the most common is AX2E2 where two covalent bonds and two lone pairs of the central atom (A) form a complete 8-electron shell. They have central angles from 104° to 109.5°, where the latter is consistent with a simplistic theory which predicts the tetrahedral symmetry of four sp3 hybridised orbitals. The most common actual angles are 105°, 107°, and 109°: they vary because of the different properties of the peripheral atoms (X).
Other cases also experience orbital hybridisation, but in different degrees. AX2E1 molecules, such as SnCl2, have only one lone pair and the central angle about 120° (the centre and two vertices of an equilateral triangle). They have three sp2 orbitals. There exist also sd-hybridised AX2 compounds of transition metals without lone pairs: they have the central angle about 90° and are also classified as bent. (See further discussion at VSEPR theory#Complexes with strong d-contribution).
See also
AXE method
References
External links
3D Chem: Chemistry, Structures, and 3D Molecules
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
Stereochemistry
Molecular geometry |
https://en.wikipedia.org/wiki/T-shaped%20molecular%20geometry | In chemistry, T-shaped molecular geometry describes the structures of some molecules where a central atom has three ligands. Ordinarily, three-coordinated compounds adopt trigonal planar or pyramidal geometries. Examples of T-shaped molecules are the halogen trifluorides, such as ClF3.
According to VSEPR theory, T-shaped geometry results when three ligands and two lone pairs of electrons are bonded to the central atom, written in AXE notation as AX3E2. The T-shaped geometry is related to the trigonal bipyramidal molecular geometry for AX5 molecules with three equatorial and two axial ligands. In an AX3E2 molecule, the two lone pairs occupy two equatorial positions, and the three ligand atoms occupy the two axial positions as well as one equatorial position. The three atoms bond at 90° angles on one side of the central atom, producing the T shape.
The trifluoroxenate(II) anion, , has been investigated as a possible first example of an AX3E3 molecule, which might be expected by VSEPR reasoning to have six electron pairs in an octahedral arrangement with both the three lone pairs and the three ligands in a mer or T-shaped orientations. Although this anion has been detected in the gas phase, attempts at synthesis in solution and experimental structure determination were unsuccessful. A computational chemistry study showed a distorted planar Y-shaped geometry with the smallest F–Xe–F bond angle equal to 69°, rather than 90° as in a T-shaped geometry.
See also
AXE method
References
External links
Chem| Chemistry, Structures, and 3D Molecules
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
Molecular geometry |
https://en.wikipedia.org/wiki/Nortel%20Discovery%20Protocol | The Nortel Discovery Protocol (NDP) is a data link layer (OSI Layer 2) network protocol for discovery of Nortel networking devices and certain products from Avaya and Ciena. The device and topology information may be graphically displayed network management software.
The Nortel Discovery Protocol had its origin in the SynOptics Network Management Protocol (SONMP), developed before the SynOptics and Wellfleet Communications merger in 1994. The protocol was rebranded as the Bay Network Management Protocol (BNMP) and some protocol analyzers referenced it as the Bay Discovery Protocol (BDP). Four years later, in 1998, Bay Networks was acquired by Nortel and renamed it to Nortel Discovery Protocol.
The IEEE 802.1AB or Link Layer Discovery Protocol that is supported on most Nortel equipment is a standards based (vendor-neutral) protocol that supports multi-vendor environments.
Historical names
Bay Discovery Protocol (BDP)
Bay Topology Protocol
Bay Network Management Protocol (BNMP)
Nortel Management MIB (NMM)
Nortel Topology Discovery Protocol (NTDP)
SynOptics Network Management Protocol (SONMP)
References
Further reading
External links
Nortel Bay Topology Discovery Packets -Retrieved 29 July 2011
Device discovery protocols
Nortel protocols
Logical link control |
https://en.wikipedia.org/wiki/Category%20of%20rings | In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.
As a concrete category
The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor
U : Ring → Set
for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint
F : Set → Ring
which assigns to each set X the free ring generated by X.
One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are forgetful functors
A : Ring → Ab
M : Ring → Mon
which "forget" multiplication and addition, respectively. Both of these functors have left adjoints. The left adjoint of A is the functor which assigns to every abelian group X (thought of as a Z-module) the tensor ring T(X). The left adjoint of M is the functor which assigns to every monoid X the integral monoid ring Z[X].
Properties
Limits and colimits
The category Ring is both complete and cocomplete, meaning that all small limits and colimits exist in Ring. Like many other algebraic categories, the forgetful functor U : Ring → Set creates (and preserves) limits and filtered colimits, but does not preserve either coproducts or coequalizers. The forgetful functors to Ab and Mon also create and preserve limits.
Examples of limits and colimits in Ring include:
The ring of integers Z is an initial object in Ring.
The zero ring is a terminal object in Ring.
The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups. The coproduct of nonzero rings can be the zero ring; in particular, this happens whenever the factors have relatively prime characteristic (since the characteristic of the coproduct of (Ri)i∈I must divide the characteristics of each of the rings Ri).
The equalizer in Ring is just the set-theoretic equalizer (the equalizer of two ring homomorphisms is always a subring).
The coequalizer of two ring homomorphisms f and g from R to S is the quotient of S by the ideal generated by all elements of the form f(r) − g(r) for r ∈ R.
Given a ring homomorphism f : R → S the kernel pair of f (this is just the pullback of f with itself) is a congruence relation on R. The ideal determined by this congruence relation is precisely the (ring-theoretic) kernel of f. Note that category-theore |
https://en.wikipedia.org/wiki/Parallelogon | In geometry, a parallelogon is a polygon with parallel opposite sides (hence the name) that can tile a plane by translation (rotation is not permitted).
Parallelogons have an even number of sides and opposite sides that are equal in length. A less obvious corollary is that parallelogons can only have either four or six sides; Parallelogons have 180-degree rotational symmetry around the center.
A four-sided parallelogon is called a parallelogram.
The faces of a parallelohedron (the three dimensional analogue) are called parallelogons.
Two polygonal types
Quadrilateral and hexagonal parallelogons each have varied geometric symmetric forms. They all have central inversion symmetry, order 2. Every convex parallelogon is a zonogon, but hexagonal parallelogons enable the possibility of nonconvex polygons.
Geometric variations
A parallelogram can tile the plane as a distorted square tiling while a hexagonal parallelogon can tile the plane as a distorted regular hexagonal tiling.
References
The facts on file: Geometry handbook, Catherine A. Gorini, 2003, , p.117
list of 107 isohedral tilings, p.473-481
External links
Fedorov's Five Parallelohedra
Types of polygons |
https://en.wikipedia.org/wiki/Moran%27s%20I | In statistics, Moran's I is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran. Spatial autocorrelation is characterized by a correlation in a signal among nearby locations in space. Spatial autocorrelation is more complex than one-dimensional autocorrelation because spatial correlation is multi-dimensional (i.e. 2 or 3 dimensions of space) and multi-directional.
Global Moran's I
Global Moran's I is a measure of the overall clustering of the spatial data. It is defined as
where
is the number of spatial units indexed by and ;
is the variable of interest;
is the mean of ;
are the elements of a matrix of spatial weights with zeroes on the diagonal (i.e., );
and is the sum of all (i.e. ).
Defining weights matrices
The value of can depend quite a bit on the assumptions built into the spatial weights matrix . The matrix is required because, in order to address spatial autocorrelation and also model spatial interaction, we need to impose a structure to constrain the number of neighbors to be considered. This is related to Tobler's first law of geography, which states that Everything depends on everything else, but closer things more so—in other words, the law implies a spatial distance decay function, such that even though all observations have an influence on all other observations, after some distance threshold that influence can be neglected.
The idea is to construct a matrix that accurately reflects your assumptions about the particular spatial phenomenon in question. A common approach is to give a weight of 1 if two zones are neighbors, and 0 otherwise, though the definition of 'neighbors' can vary. Another common approach might be to give a weight of 1 to nearest neighbors, 0 otherwise. An alternative is to use a distance decay function for assigning weights. Sometimes the length of a shared edge is used for assigning different weights to neighbors. The selection of spatial weights matrix should be guided by theory about the phenomenon in question. The value of is quite sensitive to the weights and can influence the conclusions you make about a phenomenon, especially when using distances.
Expected value
The expected value of Moran's I under the null hypothesis of no spatial autocorrelation is
The null distribution used for this expectation is that the input is permuted by a permutation picked uniformly at random (and the expectation is over picking the permutation).
At large sample sizes (i.e., as N approaches infinity), the expected value approaches zero.
Its variance equals
where
Values of I usually range from −1 to +1. Values significantly below -1/(N-1) indicate negative spatial autocorrelation and values significantly above -1/(N-1) indicate positive spatial autocorrelation. For statistical hypothesis testing, Moran's I values can be transformed to z-scores.
Moran's I is inversely related to Geary's C, but it is not identical. Moran's I is a measure of global spatial autocorrelation, w |
https://en.wikipedia.org/wiki/Accessible%20category | The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects.
The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic.
A standard text book by Adámek and Rosický appeared in 1994.
Accessible categories also have applications in homotopy theory. Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs.
Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties and Vopěnka's principle.
-directed colimits and -presentable objects
Let be an infinite regular cardinal, i.e. a cardinal number that is not the sum of a smaller number of smaller cardinals; examples are (aleph-0), the first infinite cardinal number, and , the first uncountable cardinal). A partially ordered set is called -directed if every subset of of cardinality less than has an upper bound in . In particular, the ordinary directed sets are precisely the -directed sets.
Now let be a category. A direct limit (also known as a directed colimit) over a -directed set is called a -directed colimit. An object of is called -presentable if the Hom functor preserves all -directed colimits in . It is clear that every -presentable object is also -presentable whenever , since every -directed colimit is also a -directed colimit in that case. A -presentable object is called finitely presentable.
Examples
In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The -presentable objects are the sets of cardinality smaller than .
In the category of all groups, an object is finitely presentable if and only if it is a finitely presented group, i.e. if it has a presentation with finitely many generators and finitely many relations. For uncountable regular , the -presentable objects are precisely the groups with cardinality smaller than .
In the category of left -modules over some (unitary, associative) ring , the finitely presentable objects are precisely the finitely presented modules.
-accessible and locally presentable categories
The category is called -accessible provided that:
has all -directed colimits
contains a set of -presentable objects such that every object of is a -directed colimit of objects of .
An -accessible category is called finitely accessible.
A category is called accessible if it is -accessible for some infinite regular cardinal .
When an accessible category is also cocomplete, it is called locally presentable.
A functor between -accessible categories is called -accessible provided that preserves -directed colimits.
Examples |
https://en.wikipedia.org/wiki/List%20of%20presidents%20of%20the%20Institute%20of%20Mathematical%20Statistics | The president of the Institute of Mathematical Statistics is the highest officer of the Institute of Mathematical Statistics (IMS), and, together with the president-elect and past president, sets the directions for IMS during his or her term of office.
Duties
According to the IMS Handbook for Officers, Editors, Council Members and Committee Chairs (2003), the president makes appointments to IMS committee vacancies, represents the Society to other organizations such as the Committee of Presidents of Statistical Societies (COPSS) and the Conference Board of the Mathematical Sciences (CBMS). In addition, the president is an ex officio member of the corporation of the National Institute for Statistical Sciences (NISS), and is responsible for appointing another member of the corporation and a member of the NISS Board of Trustees.
List of presidents
20th century
1936 Henry L. Rietz
1937 Walter A. Shewhart
1938 Burton H. Camp
1939 Paul R. Rider
1940 Samuel S. Wilks
1941 Harold Hotelling
1942-43 Cecil C. Craig
1944 Walter A. Shewhart
1945 W. Edwards Deming
1946 William Gemmell Cochran
1947 William Feller
1948 Abraham Wald
1949 Jerzy Neyman
1950 Joseph Leo Doob
1951 Paul S. Dwyer
1952 Meyer Abraham Girshick
1953 Morris H. Hansen
1954 Edwin G. Olds
1955 Henry Scheffé
1956 David Blackwell
1957 Alexander Mood
1958 Leonard Jimmie Savage
1959 Jacob Wolfowitz
1960 John Tukey
1961 Erich L. Lehmann
1962 Albert H. Bowker
1963 Theodore W. Anderson
1964 Z. W. Birnbaum
1965 Herbert Solomon
1966 Herbert Robbins
1967 Ted Harris
1968 Herman Chernoff
1969 Wassily Hoeffding
1970 Jack Kiefer
1971 William Kruskal
1972 Raj Chandra Bose
1973 Lucien Le Cam
1974 R. R. Bahadur
1975 Frederick Mosteller
1976 Donald L. Burkholder
1977 C. R. Rao
1978 Elizabeth Scott
1979 Samuel Karlin
1980 George E. P. Box
1981 Peter J. Bickel
1982 Mark Kac
1983 Patrick Billingsley
1984 Ingram Olkin
1985 Oscar Kempthorne
1986 Paul Meier
1987 Ronald Pyke
1988 Bradley Efron
1989 Ramanathan Gnanadesikan
1990 Shanti S. Gupta
1991 David O. Siegmund
1992 Willem van Zwet
1993 Larry Brown
1994 Stephen Stigler
1995 David R. Brillinger
1996 James O. Berger
1997 Nancy Reid
1998 Persi Diaconis
1999 Stephen Fienberg
21st century
2000 Morris Eaton
2001 Bernard Silverman
2002 Iain M. Johnstone
2003 S. R. S. Varadhan
2004 Terry Speed
2005 Louis Chen Hsiao Yun
2006 Thomas G. Kurtz
2007 Jim Pitman
2008 Jianqing Fan
2009 Nanny Wermuth
2010 J. Michael Steele
2011 Peter Gavin Hall
2012 Ruth J. Williams
2013 Hans-Rudolf Künsch
2014 Bin Yu
2015 Erwin Bolthausen
2016 Richard Davis
2017 Jon A. Wellner
2018 Alison Etheridge
2019 Xiao-Li Meng
2020 Susan Murphy
2021 Regina Liu
2022 Krzysztof Burdzy
2023 Michael Kosorok
External links
Past Executive Committee Members
References
IMS Handbook for Officers, Editors, Council Members and Committee Chairs
Institute of Mathematical Statistics
Statistics-related lists
Lists of members of learned societies |
https://en.wikipedia.org/wiki/Algebraic%20reconstruction%20technique | The algebraic reconstruction technique (ART) is an iterative reconstruction technique used in computed tomography. It reconstructs an image from a series of angular projections (a sinogram). Gordon, Bender and Herman first showed its use in image reconstruction; whereas the method is known as Kaczmarz method in numerical linear algebra.
An advantage of ART over other reconstruction methods (such as filtered backprojection) is that it is relatively easy to incorporate prior knowledge into the reconstruction process.
ART can be considered as an iterative solver of a system of linear equations , where:
is a sparse matrix whose values represent the relative contribution of each output pixel to different points in the sinogram ( being the number of individual values in the sinogram, and being the number of output pixels);
represents the pixels in the generated (output) image, arranged as a vector, and:
is a vector representing the sinogram. Each projection (row) in the sinogram is made up of a number of discrete values, arranged along the transverse axis. is made up of all of these values, from each of the individual projections.
Given a real or complex matrix and a real or complex vector , respectively, the method computes an approximation of the solution of the linear systems of equations as in the following formula,
where , is the i-th row of the matrix , is the i-th component of the vector .
is an optional relaxation parameter, of the range . The relaxation parameter is used to slow the convergence of the system. This increases computation time, but can improve the signal-to-noise ratio of the output. In some implementations, the value of is reduced with each successive iteration.
A further development of the ART algorithm is the simultaneous algebraic reconstruction technique (SART) algorithm.
References
Medical imaging
Radiography |
https://en.wikipedia.org/wiki/Designated%20place | A designated place (DPL) is a type of community or settlement identified by Statistics Canada that does not meet the criteria used to define municipalities or population centres. DPLs are delineated every 5 years for the Canadian census as the statistical counterparts of incorporated places such as cities, towns, and villages.
Though lacking separate municipal government, DPLs otherwise physically resemble incorporated places. They are created by provincial or territorial governments for the purpose of providing data for settled concentrated populations that are identifiable by name but are not legally incorporated under the laws of the province/territory in which they are located. The boundaries of a DPL therefore have no legal status, and not all unincorporated communities are necessarily granted DPL status.
Some designated places may have a quasi-governmental status, such as a local services board in Ontario or an organized hamlet in Saskatchewan. Others may be formerly unincorporated settlements or formerly independent municipalities that have been merged into larger governments, and have retained DPL status in order to ensure statistical continuity with past censuses.
DPLs are similar to the function of census-designated places in the United States, but are defined differently. One significant difference is that Statistics Canada applies the designation to much smaller communities than does the United States Census Bureau.
Statistics Canada indexes designated places numerically, with each designated place referred to by a unique six-digit code, the first two digits of which are the Standard Geographical Classification code for the province or territory in which the place is located, an example being 590066 for Shawnigan Lake in British Columbia.
Criteria
As of the 2016 census, Statistics Canada requires small communities or settlements to meet the following criteria in order to become a designated place:
an area less than or equal to
"a boundary that respects the block structure from the previous census, where possible."
In 2006, the criteria required for a community to be defined as a designated place included:
a minimum population of 100 and a maximum population of 1,000. The maximum population limit may be exceeded provided that the population density is less than 400 persons per square kilometre, which is the population density that defines a population centre.
a population density of 150 persons or more per square kilometre
an area less than or equal to 10 square kilometres
a boundary that respects the block structure from the previous census, where possible
a boundary that respects census subdivision (CSD) limits. If a named area with DPL status crosses the boundary of two or more census subdivisions, then it is enumerated as multiple DPLs, each designated "Part A", "Part B", etc., rather than as a single DPL.
The status of designated place was created for the first time in the Canada 1996 Census. Prior to 1996, such a |
https://en.wikipedia.org/wiki/White%20Bear%2C%20Saskatchewan | White Bear is an unincorporated community in the Rural Municipality of Lacadena No. 228, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 15 in the Canada 2006 Census. The community is approximately northwest of Swift Current on the north side of the South Saskatchewan River.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, White Bear had a population of 25 living in 10 of its 13 total private dwellings, a change of from its 2016 population of 10. With a land area of , it had a population density of in 2021.
History
The community's name comes from the sighting of a probable but now extirpated white prairie grizzly bear by an Assiniboine warrior on the shores of a neighbouring lake during the Palliser Expedition of the 1850s. Records from early Metis settlers and the North-West Mounted Police state the last roaming herd of American buffalo being slaughtered in the hills of the Missouri Coteau located northeast around 1879.
During the 1930s, White Bear was a bustling community of approximately 250 residents with two grocery stores, a school, four grain elevators and three garages servicing an area of 200 families, but has since dwindled to a population of 15 in 2006. Part of the decline is attributed to federal policy Canadian National Railway rail line in that area of Saskatchewan. The region rarely suffered poor crops, except during the droughts of the Great Depression and 1988. It is connected to the rest of the province through Highway 342, a now-decrepit road featuring signs with Imperial units in portions. Farmers from the area played prominently in the socialism that later defined Saskatchewan and then Canada through the introduction of Medicare and state-ran insurance. The White Bear Hotel remains the only business in operation, noted for its hot wings and hospitality.
See also
List of communities in Saskatchewan
References
Designated places in Saskatchewan
Lacadena No. 228, Saskatchewan
Populated places established in 1910
Unincorporated communities in Saskatchewan |
https://en.wikipedia.org/wiki/Irwin%E2%80%93Hall%20distribution | In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a uniform distribution. For this reason it is also known as the uniform sum distribution.
The generation of pseudo-random numbers having an approximately normal distribution is sometimes accomplished by computing the sum of a number of pseudo-random numbers having a uniform distribution; usually for the sake of simplicity of programming. Rescaling the Irwin–Hall distribution provides the exact distribution of the random variates being generated.
This distribution is sometimes confused with the Bates distribution, which is the mean (not sum) of n independent random variables uniformly distributed from 0 to 1.
Definition
The Irwin–Hall distribution is the continuous probability distribution for the sum of n independent and identically distributed U(0, 1) random variables:
The probability density function (pdf) for is given by
where denotes the positive part of the expression:
Thus the pdf is a spline (piecewise polynomial function) of degree n − 1 over the knots 0, 1, ..., n. In fact, for x between the knots located at k and k + 1, the pdf is equal to
where the coefficients aj(k,n) may be found from a recurrence relation over k
The coefficients are also A188816 in OEIS. The coefficients for the cumulative distribution is A188668.
The mean and variance are n/2 and n/12, respectively.
Special cases
For n = 1, X follows a uniform distribution:
For n = 2, X follows a triangular distribution:
For n = 3,
For n = 4,
For n = 5,
Approximating a Normal distribution
By the Central Limit Theorem, as n increases, the Irwin–Hall distribution more and more strongly approximates a Normal distribution with mean and variance . To approximate the standard Normal distribution , the Irwin–Hall distribution can be centered by shifting it by its mean of n/2, and scaling the result by the square root of its variance:
This derivation leads to a computationally simple heuristic that removes the square root, whereby a standard Normal distribution can be approximated with the sum of 12 uniform U(0,1) draws like so:
Similar and related distributions
The Irwin–Hall distribution is similar to the Bates distribution, but still featuring only integers as parameter. An extension to real-valued parameters is possible by adding also a random uniform variable with N − trunc(N) as width.
Extensions to the Irwin–Hall distribution
When using the Irwin–Hall for data fitting purposes one problem is that the IH is not very flexible because the parameter n needs to be an integer. However, instead of summing n equal uniform distributions, we could also add e.g. U + 0.5U to address also the case n = 1.5 (giving a trapezoidal distribution).
The Irwin–Hall distribution has an application to beamforming and pattern synthesis in Figure |
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