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https://en.wikipedia.org/wiki/Homotopy%20category%20of%20chain%20complexes
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In homological algebra in mathematics, the homotopy category K(A) of chain complexes in an additive category A is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes Kom(A) of A and the derived category D(A) of A when A is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that A is abelian. Philosophically, while D(A) turns into isomorphisms any maps of complexes that are quasi-isomorphisms in Kom(A), K(A) does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, K(A) is more understandable than D(A).
Definitions
Let A be an additive category. The homotopy category K(A) is based on the following definition: if we have complexes A, B and maps f, g from A to B, a chain homotopy from f to g is a collection of maps (not a map of complexes) such that
or simply
This can be depicted as:
We also say that f and g are chain homotopic, or that is null-homotopic or homotopic to 0. It is clear from the definition that the maps of complexes which are null-homotopic form a group under addition.
The homotopy category of chain complexes K(A) is then defined as follows: its objects are the same as the objects of Kom(A), namely chain complexes. Its morphisms are "maps of complexes modulo homotopy": that is, we define an equivalence relation
if f is homotopic to g
and define
to be the quotient by this relation. It is clear that this results in an additive category if one notes that this is the same as taking the quotient by the subgroup of null-homotopic maps.
The following variants of the definition are also widely used: if one takes only bounded-below (An=0 for n<<0), bounded-above (An=0 for n>>0), or bounded (An=0 for |n|>>0) complexes instead of unbounded ones, one speaks of the bounded-below homotopy category etc. They are denoted by K+(A), K−(A) and Kb(A), respectively.
A morphism which is an isomorphism in K(A) is called a homotopy equivalence. In detail, this means there is another map , such that the two compositions are homotopic to the identities: and
.
The name "homotopy" comes from the fact that homotopic maps of topological spaces induce homotopic (in the above sense) maps of singular chains.
Remarks
Two chain homotopic maps f and g induce the same maps on homology because (f − g) sends cycles to boundaries, which are zero in homology. In particular a homotopy equivalence is a quasi-isomorphism. (The converse is false in general.) This shows that there is a canonical functor to the derived category (if A is abelian).
The triangulated structure
The shift A[1] of a complex A is the following complex
(note that ),
where the differential is .
For the cone of a morphism f we take the mapping cone. There are natural maps
This diagram is called a triangle. The homotopy category K(A) is a triangulated ca
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https://en.wikipedia.org/wiki/2007%20Cricket%20World%20Cup%20statistics
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The following is a list of all the major statistics and records for the 2007 Cricket World Cup held in the West Indies from 13 March to 28 April 2007. Though India were eliminated early, they set the ODI record for the highest victory margin in their 257 run win over Bermuda. In their match against Netherlands, Herschelle Gibbs (South Africa) created ODI and International cricket record when he hit sixes off all six deliveries in Daan van Bunge's over. In the Super 8 stage games, Lasith Malinga (Sri Lanka) created ODI record when he took four wickets in four consecutive deliveries in a losing effort against South Africa. By the end of the tournament, new World Cup records for the fastest fifty (20 balls – Brendon McCullum of New Zealand) and fastest hundred (66 balls – Matthew Hayden of Australia) were established. Glenn McGrath established a new Cricket World Cup record for the most wickets (26) and also finished his ODI career with the most wickets in World Cup history (71). The number of sixes in the overall tournament (373) was 40% higher than the previous record holder, the 2003 Cricket World Cup (266). The tournament also saw 32 century partnerships (previous record of 28 during the 1996 Cricket World Cup) and 10 batsmen over 400 runs (previous record of 4 during the 2003 Cricket World Cup).
Records
Team totals
Highest team total
India's total of 413 runs against Bermuda was, at the time, the record for the highest score in an innings in a World cup match, since bettered by South Africa's 428 runs against Sri Lanka in the 2023 Cricket World Cup.
Lowest team total
Bowling
Most wickets in the tournament
McGrath surpassed Akram's record (55 wickets) for the highest number of wickets in World Cup matches, in the game against Bangladesh. His total of 26 wickets was the highest in any single World Cup tournament, and he finished the tournament with 71 wickets in all World Cup matches.
Note: Only top 10 players shown. Sorted by wickets then bowling average.
Best bowling
Note: Only top ten performances listed.
Batting
Most runs in the tournament
Hayden's 659 runs in the series stands second to only Tendulkar's 673 runs in the 2003 Cricket World Cup. The tournament also saw 10 players exceeding 400 runs for the first time, the previous best being 4 players over 400 runs in world cup tournament (2003 edition).
Note : Only top 10 players shown.
Highest individual scores
Imran Nazir's 160 is the highest score by any individual in West Indies in ODI and List A matches. Matthew Hayden scored the 100th century in World cup history during his innings of 103 against New Zealand.
Note: Only top ten scores listed.
Highest partnerships of the tournament
The 4th wicket partnership between Brad Hodge and Michael Clarke is the world cup record for that wicket.
Note: Top ten would be listed – eleventh place listed due to equal scores.
Highest partnerships for each wicket
Note:
* denotes unfinished partnerships.
Most sixes
In an innings
Note:
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https://en.wikipedia.org/wiki/G-network
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In queueing theory, a discipline within the mathematical theory of probability, a G-network (generalized queueing network, often called a Gelenbe network) is an open network of G-queues first introduced by Erol Gelenbe as a model for queueing systems with specific control functions, such as traffic re-routing or traffic destruction, as well as a model for neural networks. A G-queue is a network of queues with several types of novel and useful customers:
positive customers, which arrive from other queues or arrive externally as Poisson arrivals, and obey standard service and routing disciplines as in conventional network models,
negative customers, which arrive from another queue, or which arrive externally as Poisson arrivals, and remove (or 'kill') customers in a non-empty queue, representing the need to remove traffic when the network is congested, including the removal of "batches" of customers</ref>
"triggers", which arrive from other queues or from outside the network, and which displace customers and move them to other queues
A product-form solution superficially similar in form to Jackson's theorem, but which requires the solution of a system of non-linear equations for the traffic flows, exists for the stationary distribution of G-networks while the traffic equations of a G-network are in fact surprisingly non-linear, and the model does not obey partial balance. This broke previous assumptions that partial balance was a necessary condition for a product-form solution. A powerful property of G-networks is that they are universal approximators for continuous and bounded functions, so that they can be used to approximate quite general input-output behaviours.
Definition
A network of m interconnected queues is a G-network if
each queue has one server, who serves at rate μi,
external arrivals of positive customers or of triggers or resets form Poisson processes of rate for positive customers, while triggers and resets, including negative customers, form a Poisson process of rate ,
on completing service a customer moves from queue i to queue j as a positive customer with probability , as a trigger or reset with probability and departs the network with probability ,
on arrival to a queue, a positive customer acts as usual and increases the queue length by 1,
on arrival to a queue, the negative customer reduces the length of the queue by some random number (if there is at least one positive customer present at the queue), while a trigger moves a customer probabilistically to another queue and a reset sets the state of the queue to its steady-state if the queue is empty when the reset arrives. All triggers, negative customers and resets disappear after they have taken their action, so that they are in fact "control" signals in the network,
note that normal customers leaving a queue can become triggers or resets and negative customers when they visit the next queue.
A queue in such a network is known as a G-queue.
Stationary dist
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https://en.wikipedia.org/wiki/Maximal%20ergodic%20theorem
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The maximal ergodic theorem is a theorem in ergodic theory, a discipline within mathematics.
Suppose that is a probability space, that is a (possibly noninvertible) measure-preserving transformation, and that . Define by
Then the maximal ergodic theorem states that
for any λ ∈ R.
This theorem is used to prove the point-wise ergodic theorem.
References
.
Probability theorems
Ergodic theory
Theorems in dynamical systems
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https://en.wikipedia.org/wiki/Median%20absolute%20deviation
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(MAD) is an acronym for both median absolute deviation and mean absolute deviation; there is no universal agreement on which is correct. It is used here to connote the former.
In statistics, the median absolute deviation (MAD) is a robust measure of the variability of a univariate sample of quantitative data. It can also refer to the population parameter that is estimated by the MAD calculated from a sample.
For a univariate data set X1, X2, ..., Xn, the MAD is defined as the median of the absolute deviations from the data's median :
that is, starting with the residuals (deviations) from the data's median, the MAD is the median of their absolute values.
Example
Consider the data (1, 1, 2, 2, 4, 6, 9). It has a median value of 2. The absolute deviations about 2 are (1, 1, 0, 0, 2, 4, 7) which in turn have a median value of 1 (because the sorted absolute deviations are (0, 0, 1, 1, 2, 4, 7)). So the median absolute deviation for this data is 1.
Uses
The median absolute deviation is a measure of statistical dispersion. Moreover, the MAD is a robust statistic, being more resilient to outliers in a data set than the standard deviation. In the standard deviation, the distances from the mean are squared, so large deviations are weighted more heavily, and thus outliers can heavily influence it. In the MAD, the deviations of a small number of outliers are irrelevant.
Because the MAD is a more robust estimator of scale than the sample variance or standard deviation, it works better with distributions without a mean or variance, such as the Cauchy distribution.
Relation to standard deviation
The MAD may be used similarly to how one would use the deviation for the average.
In order to use the MAD as a consistent estimator for the estimation of the standard deviation , one takes
where is a constant scale factor, which depends on the distribution.
For normally distributed data is taken to be
i.e., the reciprocal of the quantile function (also known as the inverse of the cumulative distribution function) for the standard normal distribution .
Derivation
The argument 3/4 is such that covers 50% (between 1/4 and 3/4) of the standard normal cumulative distribution function, i.e.
Therefore, we must have that
Noticing that
we have that , from which we obtain the scale factor .
Another way of establishing the relationship is noting that MAD equals the half-normal distribution median:
This form is used in, e.g., the probable error.
In the case of complex values (X+iY), the relation of MAD to the standard deviation is unchanged for normally distributed data.
MAD using geometric median
Analogously to how the median generalizes to the geometric median (gm) in multivariate data, MAD can be generalized to MADGM (median of distances to gm) in n dimensions. This is done by replacing the absolute differences in one dimension by euclidian distances of the data points to the geometric median in n dimensions. This gives the identical result a
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https://en.wikipedia.org/wiki/Oleg%20Lupanov
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Oleg Borisovich Lupanov (; 2 June 1932 – 3 May 2006) was a Soviet and Russian mathematician, dean of the Moscow State University's Faculty of Mechanics and Mathematics (1980–2006), head of the Chair of Discrete Mathematics of the Faculty of Mechanics and Mathematics (1981–2006).
Together with his graduate school advisor, Sergey Yablonsky, he is considered one of the founders of the Soviet school of Mathematical Cybernetics. In particular he authored pioneering works on synthesis and complexity of Boolean circuits, and of control systems in general (), the term used in the USSR and Russia for a generalization of finite state automata, Boolean circuits and multi-valued logic circuits.
Ingo Wegener, in his book The Complexity of Boolean Functions, credits O. B. Lupanov for coining the term Shannon effect in his 1970 paper, to refer to the fact that almost all Boolean functions have nearly the same circuit complexity as the hardest function.
O. B. Lupanov is best known for his (k, s)-Lupanov representation of Boolean functions that he used to devise an asymptotically optimal method of Boolean circuit synthesis, thus proving the asymptotically tight upper bound on Boolean circuit complexity:
Biography
O. B. Lupanov graduated from Moscow State University's Faculty of Mechanics and Mathematics in 1955. He received his PhD in 1958 from the Academy of Sciences of the Soviet Union and his Doctorate degree in 1963. He began teaching at Moscow State University in 1959 and became professor there in 1967. From 1955 he had appointment at the Institute of Applied Mathematics and he was a professor at Faculty of Computational Mathematics and Cybernetics (1970–1980). He had served as the Dean of the Moscow State University's Faculty of Mechanics and Mathematics (1980–2006), and as the founding head of the Chair of Discrete Mathematics of the Faculty of Mechanics and Mathematics (1981–2006).
Lupanov became a corresponding member of the Academy of Sciences of the Soviet Union in 1972 and a full member of Russian Academy of Sciences in 2003. He was the lead scientist of the Keldysh Institute of Applied Mathematics since 1993 and was awarded the title of a distinguished professor of Moscow State University in 2002. He was a recipient of the prestigious Lenin Prize (1966) and of the Moscow State University's Lomonosov Award (1993).
His students count more than 30 PhD degree holders and 6 holders of the Soviet/Russian Doctorate degree. As a dean of the Faculty of Mechanics and Mathematics he had a reputation of a democratic and accessible person.
Personal life
Lupanov died at around 7pm on 3 May 2006 in his office at the Faculty of Mechanics and Mathematics of Moscow State University.
References
External links
Oleg Lupanov — scientific works on the website Math-Net.Ru
1932 births
2006 deaths
20th-century Russian mathematicians
21st-century Russian mathematicians
Mathematicians from Saint Petersburg
Academic staff of Moscow State University
Corresponding Me
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https://en.wikipedia.org/wiki/Curve%20radius
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Radius of curvature, the reciprocal of the curvature in differential geometry
Minimum railway curve radius, the shortest allowable design radius for the centerline of railway tracks
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https://en.wikipedia.org/wiki/Data%20transformation%20%28statistics%29
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In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point zi is replaced with the transformed value yi = f(zi), where f is a function. Transforms are usually applied so that the data appear to more closely meet the assumptions of a statistical inference procedure that is to be applied, or to improve the interpretability or appearance of graphs.
Nearly always, the function that is used to transform the data is invertible, and generally is continuous. The transformation is usually applied to a collection of comparable measurements. For example, if we are working with data on peoples' incomes in some currency unit, it would be common to transform each person's income value by the logarithm function.
Motivation
Guidance for how data should be transformed, or whether a transformation should be applied at all, should come from the particular statistical analysis to be performed. For example, a simple way to construct an approximate 95% confidence interval for the population mean is to take the sample mean plus or minus two standard error units. However, the constant factor 2 used here is particular to the normal distribution, and is only applicable if the sample mean varies approximately normally. The central limit theorem states that in many situations, the sample mean does vary normally if the sample size is reasonably large. However, if the population is substantially skewed and the sample size is at most moderate, the approximation provided by the central limit theorem can be poor, and the resulting confidence interval will likely have the wrong coverage probability. Thus, when there is evidence of substantial skew in the data, it is common to transform the data to a symmetric distribution before constructing a confidence interval. If desired, the confidence interval can then be transformed back to the original scale using the inverse of the transformation that was applied to the data.
Data can also be transformed to make them easier to visualize. For example, suppose we have a scatterplot in which the points are the countries of the world, and the data values being plotted are the land area and population of each country. If the plot is made using untransformed data (e.g. square kilometers for area and the number of people for population), most of the countries would be plotted in tight cluster of points in the lower left corner of the graph. The few countries with very large areas and/or populations would be spread thinly around most of the graph's area. Simply rescaling units (e.g., to thousand square kilometers, or to millions of people) will not change this. However, following logarithmic transformations of both area and population, the points will be spread more uniformly in the graph.
Another reason for applying data transformation is to improve interpretability, even if no formal statistical analysis or visualization is to be performed. For example
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https://en.wikipedia.org/wiki/Category%20of%20manifolds
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In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp.
One is often interested only in Cp-manifolds modeled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modeled on a fixed space E is denoted Manp(E).
One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Manω.
Manp is a concrete category
Like many categories, the category Manp is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor
U : Manp → Top
to the category of topological spaces which assigns to each manifold the underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor
U′ : Manp → Set
to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.
Pointed manifolds and the tangent space functor
It is often convenient or necessary to work with the category of manifolds along with a distinguished point: Man•p analogous to Top• - the category of pointed spaces. The objects of Man•p are pairs where is a manifold along with a basepoint and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g. such that The category of pointed manifolds is an example of a comma category - Man•p is exactly where represents an arbitrary singleton set, and the represents a map from that singleton to an element of Manp, picking out a basepoint.
The tangent space construction can be viewed as a functor from Man•p to VectR as follows: given pointed manifolds and with a map between them, we can assign the vector spaces and with a linear map between them given by the pushforward (differential): This construction is a genuine functor because the pushforward of the identity map is the vector space isomorphism and the chain rule ensures that
References
Manifolds
Manifolds
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https://en.wikipedia.org/wiki/R%C3%B4ni
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Roniéliton Pereira Santos or simply Rôni (born 28 April 1977) is a Brazilian former footballer who played as a striker.
Career statistics
Club
International
International goals
Scores and results list Brazil's goal tally first.
Honours
Club
Vila Nova
Goiás State League: 1995
Brazilian League (3rd division): 1996
Fluminense
Brazilian League (3rd division): 1999
Rio de Janeiro State League: 2002
Atlético Mineiro
Brazilian League (2nd division): 2006
Flamengo
Rio de Janeiro State League: 2007
Taça Guanabara: 2007
Goiás
Goiás State League: 2006
Gamba Osaka
AFC Champions League: 2008
Emperor's Cup: 2008
References
External links
CBF
placar
mercadofutebol
1977 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Brazil men's under-20 international footballers
Brazil men's international footballers
1999 FIFA Confederations Cup players
Vila Nova Futebol Clube players
São Paulo FC players
Fluminense FC players
Al Hilal SFC players
FC Rubin Kazan players
PFC Krylia Sovetov Samara players
Russian Premier League players
Expatriate men's footballers in Russia
Goiás Esporte Clube players
CR Flamengo footballers
Clube Atlético Mineiro players
Cruzeiro Esporte Clube players
Expatriate men's footballers in Japan
J1 League players
Yokohama F. Marinos players
Gamba Osaka players
Santos FC players
Campeonato Brasileiro Série A players
Men's association football forwards
Saudi Pro League players
Sportspeople from Tocantins
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https://en.wikipedia.org/wiki/Constant-Q%20transform
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In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transform and very closely related to the complex Morlet wavelet transform. Its design is suited for musical representation.
The transform can be thought of as a series of filters fk, logarithmically spaced in frequency, with the k-th filter having a spectral width δfk equal to a multiple of the previous filter's width:
where δfk is the bandwidth of the k-th filter, fmin is the central frequency of the lowest filter, and n is the number of filters per octave.
Calculation
The short-time Fourier transform of x[n] for a frame shifted to sample m is calculated as follows:
Given a data series at sampling frequency fs = 1/T, T being the sampling period of our data, for each frequency bin we can define the following:
Filter width, δfk.
Q, the "quality factor":
This is shown below to be the integer number of cycles processed at a center frequency fk. As such, this somewhat defines the time complexity of the transform.
Window length for the k-th bin:
Since fs/fk is the number of samples processed per cycle at frequency fk, Q is the number of integer cycles processed at this central frequency.
The equivalent transform kernel can be found by using the following substitutions:
The window length of each bin is now a function of the bin number:
The relative power of each bin will decrease at higher frequencies, as these sum over fewer terms. To compensate for this, we normalize by N[k].
Any windowing function will be a function of window length, and likewise a function of window number. For example, the equivalent Hamming window would be
Our digital frequency, , becomes .
After these modifications, we are left with
Variable-Q bandwidth calculation
The variable-Q transform is the same as constant-Q transform, but the only difference is the filter Q is variable, hence the name variable-Q transform. The variable-Q transform is useful . There are ways to calculate the bandwidth of the VQT, one of them using equivalent rectangular bandwidth as a value for VQT bin's bandwidth.
The simplest way to implement a variable-Q transform is add a bandwidth offset called γ like this one:
This formula can be modified to have extra parameters to adjust sharpness of the transition between constant-Q and constant-bandwidth like this:
with α as a parameter for transition sharpness and where α of 2 is equals to hyperbolic sine frequency scale, in terms of frequency resolution.
Fast calculation
The direct calculation of the constant-Q transform (either using naive DFT or slightly faster Goertzel algorithm) is slow when compared against the fast Fourier transform (FFT). However, the FFT can itself be employed, in conjunction with the use of a kernel, to perform the equivalent calculation but much faster. An approximate inverse to such an implementation was pr
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https://en.wikipedia.org/wiki/Dual%20cone%20and%20polar%20cone
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Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.
Dual cone
In a vector space
The dual cone C of a subset C in a linear space X over the reals, e.g. Euclidean space Rn, with dual space X is the set
where is the duality pairing between X and X, i.e. .
C is always a convex cone, even if C is neither convex nor a cone.
In a topological vector space
If X is a topological vector space over the real or complex numbers, then the dual cone of a subset C ⊆ X is the following set of continuous linear functionals on X:
,
which is the polar of the set -C.
No matter what C is, will be a convex cone.
If C ⊆ {0} then .
In a Hilbert space (internal dual cone)
Alternatively, many authors define the dual cone in the context of a real Hilbert space (such as Rn equipped with the Euclidean inner product) to be what is sometimes called the internal dual cone.
Using this latter definition for C, we have that when C is a cone, the following properties hold:
A non-zero vector y is in C if and only if both of the following conditions hold:
y is a normal at the origin of a hyperplane that supports C.
y and C lie on the same side of that supporting hyperplane.
C is closed and convex.
implies .
If C has nonempty interior, then C is pointed, i.e. C* contains no line in its entirety.
If C is a cone and the closure of C is pointed, then C has nonempty interior.
C is the closure of the smallest convex cone containing C (a consequence of the hyperplane separation theorem)
Self-dual cones
A cone C in a vector space X is said to be self-dual if X can be equipped with an inner product ⟨⋅,⋅⟩ such that the internal dual cone relative to this inner product is equal to C.
Those authors who define the dual cone as the internal dual cone in a real Hilbert space usually say that a cone is self-dual if it is equal to its internal dual.
This is slightly different from the above definition, which permits a change of inner product.
For instance, the above definition makes a cone in Rn with ellipsoidal base self-dual, because the inner product can be changed to make the base spherical, and a cone with spherical base in Rn is equal to its internal dual.
The nonnegative orthant of Rn and the space of all positive semidefinite matrices are self-dual, as are the cones with ellipsoidal base (often called "spherical cones", "Lorentz cones", or sometimes "ice-cream cones").
So are all cones in R3 whose base is the convex hull of a regular polygon with an odd number of vertices.
A less regular example is the cone in R3 whose base is the "house": the convex hull of a square and a point outside the square forming an equilateral triangle (of the appropriate height) with one of the sides of the square.
Polar cone
For a set C in X, the polar cone of C is the set
It can be seen that the polar cone is equal to the negative of the dual cone, i.e. Co = −C.
For a closed convex cone C in X, the polar cone is equivalent to the p
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https://en.wikipedia.org/wiki/RobotFest
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Also called the "Day of Playful Invention", Robot fest "is an annual event for anyone interested in the creative use of technology" to promote science, technology, engineering and mathematics (STEM). It takes place at the National Electronics Museum in Linthicum, Maryland and entry is donation based.
This year's Robot Fest will occur in April 2021 and include exhibitors such as The Art Institute of Washington, MakerBot Industries and Lego. 2020 saw no event.
References
External links
FIRST Robotics Wiki
Supporting teams for competition
National Electronics Museum
Festivals in Maryland
Maryland culture
Robotics events
Tourist attractions in Anne Arundel County, Maryland
Recurring events established in 2000
2000 establishments in Maryland
Annual events in Maryland
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https://en.wikipedia.org/wiki/Minimal%20prime%20%28recreational%20mathematics%29
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In recreational number theory, a minimal prime is a prime number for which there is no shorter subsequence of its digits in a given base that form a prime. In base 10 there are exactly 26 minimal primes:
2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 .
For example, 409 is a minimal prime because there is no prime among the shorter subsequences of the digits: 4, 0, 9, 40, 49, 09. The subsequence does not have to consist of consecutive digits, so 109 is not a minimal prime (because 19 is prime). But it does have to be in the same order; so, for example, 991 is still a minimal prime even though a subset of the digits can form the shorter prime 19 by changing the order.
Similarly, there are exactly 32 composite numbers which have no shorter composite subsequence:
4, 6, 8, 9, 10, 12, 15, 20, 21, 22, 25, 27, 30, 32, 33, 35, 50, 51, 52, 55, 57, 70, 72, 75, 77, 111, 117, 171, 371, 711, 713, 731 .
There are 146 primes congruent to 1 mod 4 which have no shorter prime congruent to 1 mod 4 subsequence:
5, 13, 17, 29, 37, 41, 61, 73, 89, 97, 101, 109, 149, 181, 233, 277, 281, 349, 409, 433, 449, 677, 701, 709, 769, 821, 877, 881, 1669, 2221, 3001, 3121, 3169, 3221, 3301, 3833, 4969, 4993, 6469, 6833, 6949, 7121, 7477, 7949, 9001, 9049, 9221, 9649, 9833, 9901, 9949, ...
There are 113 primes congruent to 3 mod 4 which have no shorter prime congruent to 3 mod 4 subsequence:
3, 7, 11, 19, 59, 251, 491, 499, 691, 991, 2099, 2699, 2999, 4051, 4451, 4651, 5051, 5651, 5851, 6299, 6451, 6551, 6899, 8291, 8699, 8951, 8999, 9551, 9851, ...
Other bases
Minimal primes can be generalized to other bases. It can be shown that there are only a finite number of minimal primes in every base. Equivalently, every sufficiently large prime contains a shorter subsequence that forms a prime.
The base 12 minimal primes written in base 10 are listed in .
Number of minimal (probable) primes in base n are
1, 2, 3, 3, 8, 7, 9, 15, 12, 26, 152, 17, 228, 240, 100, 483, 1280, 50, 3463, 651, 2601, 1242, 6021, 306, (17608 or 17609), 5664, 17215, 5784, (57296 or 57297), 220, ...
The length of the largest minimal (probable) prime in base n are
2, 2, 3, 2, 5, 5, 5, 9, 4, 8, 45, 8, 32021, 86, 107, 3545, (≥111334), 33, (≥110986), 449, (≥479150), 764, 800874, 100, (≥136967), (≥8773), (≥109006), (≥94538), (≥174240), 1024, ...
Largest minimal (probable) prime in base n (written in base 10) are
2, 3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921, ... (next term has 35670 digits)
Number of minimal composites in base n are
1, 3, 4, 9, 10, 19, 18, 26, 28, 32, 32, 46, 43, 52, 54, 60, 60, 95, 77, 87, 90, 94, 97, 137, 117, 111, 115, 131, 123, 207, ...
The length of the largest minimal composite in base n are
4, 4, 3, 3, 3, 4, 3, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 4, 3, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, ...
Notes
References
Chris Caldwell,
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https://en.wikipedia.org/wiki/Categorical%20distribution
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In probability theory and statistics, a categorical distribution (also called a generalized Bernoulli distribution, multinoulli distribution) is a discrete probability distribution that describes the possible results of a random variable that can take on one of K possible categories, with the probability of each category separately specified. There is no innate underlying ordering of these outcomes, but numerical labels are often attached for convenience in describing the distribution, (e.g. 1 to K). The K-dimensional categorical distribution is the most general distribution over a K-way event; any other discrete distribution over a size-K sample space is a special case. The parameters specifying the probabilities of each possible outcome are constrained only by the fact that each must be in the range 0 to 1, and all must sum to 1.
The categorical distribution is the generalization of the Bernoulli distribution for a categorical random variable, i.e. for a discrete variable with more than two possible outcomes, such as the roll of a dice. On the other hand, the categorical distribution is a special case of the multinomial distribution, in that it gives the probabilities of potential outcomes of a single drawing rather than multiple drawings.
Terminology
Occasionally, the categorical distribution is termed the "discrete distribution". However, this properly refers not to one particular family of distributions but to a general class of distributions.
In some fields, such as machine learning and natural language processing, the categorical and multinomial distributions are conflated, and it is common to speak of a "multinomial distribution" when a "categorical distribution" would be more precise. This imprecise usage stems from the fact that it is sometimes convenient to express the outcome of a categorical distribution as a "1-of-K" vector (a vector with one element containing a 1 and all other elements containing a 0) rather than as an integer in the range 1 to K; in this form, a categorical distribution is equivalent to a multinomial distribution for a single observation (see below).
However, conflating the categorical and multinomial distributions can lead to problems. For example, in a Dirichlet-multinomial distribution, which arises commonly in natural language processing models (although not usually with this name) as a result of collapsed Gibbs sampling where Dirichlet distributions are collapsed out of a hierarchical Bayesian model, it is very important to distinguish categorical from multinomial. The joint distribution of the same variables with the same Dirichlet-multinomial distribution has two different forms depending on whether it is characterized as a distribution whose domain is over individual categorical nodes or over multinomial-style counts of nodes in each particular category (similar to the distinction between a set of Bernoulli-distributed nodes and a single binomial-distributed node). Both forms have very similar-lookin
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https://en.wikipedia.org/wiki/1/2%20%2B%201/4%20%2B%201/8%20%2B%201/16%20%2B%20%E2%8B%AF
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In mathematics, the infinite series is an elementary example of a geometric series that converges absolutely. The sum of the series is 1.
In summation notation, this may be expressed as
The series is related to philosophical questions considered in antiquity, particularly to Zeno's paradoxes.
Proof
As with any infinite series, the sum
is defined to mean the limit of the partial sum of the first terms
as approaches infinity.
By various arguments, one can show that this finite sum is equal to
As approaches infinity, the term approaches 0 and so tends to 1.
History
Zeno's paradox
This series was used as a representation of many of Zeno's paradoxes. For example, in the paradox of Achilles and the Tortoise, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but the tortoise would already have moved another five meters by then. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on. Zeno argued that the tortoise would always remain ahead of Achilles.
The Dichotomy paradox also states that to move a certain distance, you have to move half of it, then half of the remaining distance, and so on, therefore having infinitely many time intervals. This can be easily resolved by noting that each time interval is a term of the infinite geometric series, and will sum to a finite number.
The Eye of Horus
The parts of the Eye of Horus were once thought to represent the first six summands of the series.
In a myriad ages it will not be exhausted
A version of the series appears in the ancient Taoist book Zhuangzi. The miscellaneous chapters "All Under Heaven" include the following sentence: "Take a chi long stick and remove half every day, in a myriad ages it will not be exhausted."
See also
0.999...
1/2 − 1/4 + 1/8 − 1/16 + ⋯
Actual infinity
Notes
References
Geometric series
1 (number)
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https://en.wikipedia.org/wiki/1/4%20%2B%201/16%20%2B%201/64%20%2B%201/256%20%2B%20%E2%8B%AF
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In mathematics, the infinite series is an example of one of the first infinite series to be summed in the history of mathematics; it was used by Archimedes circa 250–200 BC. As it is a geometric series with first term and common ratio , its sum is
Visual demonstrations
The series lends itself to some particularly simple visual demonstrations because a square and a triangle both divide into four similar pieces, each of which contains the area of the original.
In the figure on the left, if the large square is taken to have area 1, then the largest black square has area × = . Likewise, the second largest black square has area , and the third largest black square has area . The area taken up by all of the black squares together is therefore , and this is also the area taken up by the gray squares and the white squares. Since these three areas cover the unit square, the figure demonstrates that
Archimedes' own illustration, adapted at top, was slightly different, being closer to the equation
See below for details on Archimedes' interpretation.
The same geometric strategy also works for triangles, as in the figure on the right: if the large triangle has area 1, then the largest black triangle has area , and so on. The figure as a whole has a self-similarity between the large triangle and its upper sub-triangle. A related construction making the figure similar to all three of its corner pieces produces the Sierpiński triangle.
Proof by Archimedes
Archimedes encounters the series in his work Quadrature of the Parabola. He finds the area inside a parabola by the method of exhaustion, and he gets a series of triangles; each stage of the construction adds an area times the area of the previous stage. His desired result is that the total area is times the area of the first stage. To get there, he takes a break from parabolas to introduce an algebraic lemma:
Proposition 23. Given a series of areas , of which A is the greatest, and each is equal to four times the next in order, then
Archimedes proves the proposition by first calculating
On the other hand,
Subtracting this equation from the previous equation yields
and adding A to both sides gives the desired result.
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series are:
This form can be proved by multiplying both sides by 1 − and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs. The same strategy works for any finite geometric series.
The limit
Archimedes' Proposition 24 applies the finite (but indeterminate) sum in Proposition 23 to the area inside a parabola by a double reductio ad absurdum. He does not quite take the limit of the above partial sums, but in modern calculus this step is easy enough:
Since the sum of an infinite series is defined as the limit of its partial sums,
Notes
References
Page images at HTML with figures and commentary at
Geomet
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https://en.wikipedia.org/wiki/Eventually%20%28mathematics%29
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In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set (such as sequences and subsets of ).
Notation
The general form where the phrase eventually (or sufficiently large) is found appears as follows:
is eventually true for ( is true for sufficiently large ),
where and are the universal and existential quantifiers, which is actually a shorthand for:
such that is true
or somewhat more formally:
This does not necessarily mean that any particular value for is known, but only that such an exists. The phrase "sufficiently large" should not be confused with the phrases "arbitrarily large" or "infinitely large". For more, see Arbitrarily large#Arbitrarily large vs. sufficiently large vs. infinitely large.
Motivation and definition
For an infinite sequence, one is often more interested in the long-term behaviors of the sequence than the behaviors it exhibits early on. In which case, one way to formally capture this concept is to say that the sequence possesses a certain property eventually, or equivalently, that the property is satisfied by one of its subsequences , for some .
For example, the definition of a sequence of real numbers converging to some limit is:
For each positive number , there exists a natural number such that for all , .
When the term "eventually" is used as a shorthand for "there exists a natural number such that for all ", the convergence definition can be restated more simply as:
For each positive number , eventually .
Here, notice that the set of natural numbers that do not satisfy this property is a finite set; that is, the set is empty or has a maximum element. As a result, the use of "eventually" in this case is synonymous with the expression "for all but a finite number of terms" – a special case of the expression "for almost all terms" (although "almost all" can also be used to allow for infinitely many exceptions as well).
At the basic level, a sequence can be thought of as a function with natural numbers as its domain, and the notion of "eventually" applies to functions on more general sets as well—in particular to those that have an ordering with no greatest element.
More specifically, if is such a set and there is an element in such that the function is defined for all elements greater than , then is said to have some property eventually if there is an element such that whenever , has the said property. This notion is used, for example, in the study of Hardy fields, which are fields made up of real functions, each of which have certain properties eventually.
Examples
"All primes greater than 2 are odd"
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https://en.wikipedia.org/wiki/5-cubic%20honeycomb
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In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(5).
Related polytopes and honeycombs
The [4,33,4], , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.
It is also related to the regular 6-cube which exists in 6-space with 3 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.
Tritruncated 5-cubic honeycomb
A tritruncated 5-cubic honeycomb, , contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled ×2, [[4,33,4]] symmetry, alternately colored from , [4,33,4] symmetry, three colors from , [4,3,3,31,1] symmetry, and 4 colors from , [31,1,3,31,1] symmetry.
See also
List of regular polytopes
Regular and uniform honeycombs in 5-space:
5-demicubic honeycomb
5-simplex honeycomb
Truncated 5-simplex honeycomb
Omnitruncated 5-simplex honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
6-polytopes
Regular tessellations
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https://en.wikipedia.org/wiki/6-cube
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In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.
Related polytopes
It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes.
Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.
As a configuration
This configuration matrix represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Cartesian coordinates
Cartesian coordinates for the vertices of a 6-cube centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with −1 < xi < 1.
Construction
There are three Coxeter groups associated with the 6-cube, one regular, with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry (D6) or [33,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.
Projections
Related polytopes
The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.
The 6-cube is 6th in a series of hypercube:
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
External links
Multi-dimensional Glossary: hypercube Garrett Jones
6-polytopes
Articles containing video clips
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https://en.wikipedia.org/wiki/6-orthoplex
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In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.
It has two constructed forms, the first being regular with Schläfli symbol {34,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,31,1} or Coxeter symbol 311.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.
Alternate names
Hexacross, derived from combining the family name cross polytope with hex for six (dimensions) in Greek.
Hexacontitetrapeton as a 64-facetted 6-polytope.
As a configuration
This configuration matrix represents the 6-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
Construction
There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.
Cartesian coordinates
Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are
(±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
Images
Related polytopes
The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron.
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. 1966
Specific
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
6-polytopes
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https://en.wikipedia.org/wiki/List%20of%20Ipswich%20Town%20F.C.%20records%20and%20statistics
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Ipswich Town Football Club are an English professional association football club based in Ipswich, Suffolk. The club was founded in 1878 and turned professional in 1936. Ipswich have played at all professional levels of English football and have participated in European football since the 1960s. The team currently plays in the second tier of English football.
This list encompasses the major honours won by Ipswich Town, records set by the club, their managers and their players, and details the club's European performances. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Ipswich players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at Portman Road, the club's home ground since 1884, are also included in the list.
Honours
Ipswich Town have won honours both domestically and in European Cup competitions. The club has won the English League Championship (1961–62) and the FA Cup (1978) and, in European competition, won the UEFA Cup in 1980–81. Their last senior league honour was the Football League Second Division title in 1992.
Domestic
League titles
First Division / Premier League (Tier 1)
Winners: 1961–62
Runners-up: 1980–81, 1981–82
Second Division / Championship (Tier 2)
Winners: 1960–61, 1967–68, 1991–92
Play-off winners: 1999–2000
Third Division / League One (Tier 3)
Winners: 1953–54, 1956–57
Runners-up: 2022–23
Southern League
Winners: 1936–37
Cups
FA Cup
Winners: 1977–78
Texaco Cup
Winners: 1972–73
European
UEFA Cup / UEFA Europa League
Winners: 1980–81
Minor titles
Friendly
Amsterdam Tournament
Winners: 1981
Innsbruck Cup
Winners: 2023
Ipswich Charity Cup
Winners: 1896–97, 1897–98, 1898–99, 1899–1900
Ipswich Hospital Cup
Winners: 1929, 1930, 1932, 1936, 1959, 1988, 1990
Paisley Charity Cup
Winners: 1966 (shared)
Uhren Cup
Winners: 1963
Willhire Cup
Winners: 1978, 1979
Other
South East Anglian League
Champions: 1903–04
Runners-up: 1904–05
Southern Amateur League
Champions: 1921–22, 1929–30, 1932–33, 1933–34
Runners-up: 1922–23, 1930–31
Suffolk Challenge Cup
Winners: 1886–87, 1888–89, 1889–90
Suffolk Senior Cup
Winners: 1886–87, 1888–89, 1889–90, 1895–96, 1899–1900, 1903–04, 1904–05, 1905–06, 1906–07, 1907–08, 1911–12, 1912–13, 1913–14, 1927–28, 1928–29, 1929–30
Player records
Appearances
Youngest first-team player: Connor Wickham, 16 years 11 days (against Doncaster Rovers, 11 April 2009).
Oldest first-team player: Mick Burns, 43 years 219 days (against Gateshead, 12 January 1952).
Most appearances
Competitive, professional matches only, appearances as substitutes in brackets.
Goalscorers
Most goals in a season: Ted Phillips, 46 goals (including 41 league goals) in the 1956–57 season.
Most league goals in a season: Ted Phillips, 41 goals in the 1956–57 season.
Youngest goalscorer: Jason Doz
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https://en.wikipedia.org/wiki/Symmetry%20operation
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In group theory, geometry, representation theory and molecular geometry, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, as transformations of an object in space, rotations, reflections and inversions are all symmetry operations. Such symmetry operations are performed with respect to symmetry elements (for example, a point, line or plane). In the context of molecular symmetry, a symmetry operation is a permutation of atoms such that the molecule or crystal is transformed into a state indistinguishable from the starting state.
Two basic facts follow from this definition, which emphasizes its usefulness.
Physical properties must be invariant with respect to symmetry operations.
Symmetry operations can be collected together in groups which are isomorphic to permutation groups.
In the context of molecular symmetry, quantum wavefunctions need not be invariant, because the operation can multiply them by a phase or mix states within a degenerate representation, without affecting any physical property.
Molecules
Identity Operation
The identity operation corresponds to doing nothing to the object. Because every molecule is indistinguishable from itself if nothing is done to it, every object possesses at least the identity operation. The identity operation is denoted by or . In the identity operation, no change can be observed for the molecule. Even the most asymmetric molecule possesses the identity operation. The need for such an identity operation arises from the mathematical requirements of group theory.
Reflection through mirror planes
The reflection operation is carried out with respect to symmetry elements known as planes of symmetry or mirror planes. Each such plane is denoted as (sigma). Its orientation relative to the principal axis of the molecule is indicated by a subscript. The plane must pass through the molecule and cannot be completely outside it.
If the plane of symmetry contains the principal axis of the molecule (i.e., the molecular -axis), it is designated as a vertical mirror plane, which is indicated by a subscript ().
If the plane of symmetry is perpendicular to the principal axis, it is designated as a horizontal mirror plane, which is indicated by a subscript ().
If the plane of symmetry bisects the angle between two 2-fold axes perpendicular to the principal axis, it is designated as a dihedral mirror plane, which is indicated by a subscript ().
Through the reflection of each mirror plane, the molecule must be able to produce an identical image of itself.
Inversion operation
In an inversion through a centre of symmetry, (the element), we imagine taking each point in a molecule and then moving it out the same distance on the other side. In summary, the inversion operation projects each atom through the centre of inversion and out to the same distance on the opposite side. The inversion center is a point in space
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https://en.wikipedia.org/wiki/1984%20Alpine%20Skiing%20World%20Cup%20%E2%80%93%20Men%27s%20slalom
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This is a list of statistics for the Men's slalom in the World Cup 1983/1984.
Calendar
Final point standings
In men's slalom World Cup 1983/84 the best 5 results count. Deduction are given in ().
External links
FIS-ski.com - World Cup standings - Slalom 1984
World Cup
FIS Alpine Ski World Cup men's slalom discipline titles
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https://en.wikipedia.org/wiki/Order%20of%20integration
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In statistics, the order of integration, denoted I(d), of a time series is a summary statistic, which reports the minimum number of differences required to obtain a covariance-stationary series.
Integration of order d
A time series is integrated of order d if
is a stationary process, where is the lag operator and is the first difference, i.e.
In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process.
In particular, if a series is integrated of order 0, then is stationary.
Constructing an integrated series
An I(d) process can be constructed by summing an I(d − 1) process:
Suppose is I(d − 1)
Now construct a series
Show that Z is I(d) by observing its first-differences are I(d − 1):
where
See also
ARIMA
ARMA
Random walk
Unit root test
References
Hamilton, James D. (1994) Time Series Analysis. Princeton University Press. p. 437. .
Time series
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https://en.wikipedia.org/wiki/Peano%20existence%20theorem
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In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems.
History
Peano first published the theorem in 1886 with an incorrect proof. In 1890 he published a new correct proof using successive approximations.
Theorem
Let be an open subset of with
a continuous function and
a continuous, explicit first-order differential equation defined on D, then every initial value problem
for f with
has a local solution
where is a neighbourhood of in ,
such that for all .
The solution need not be unique: one and the same initial value may give rise to many different solutions .
Proof
By replacing with , with , we may assume . As is open there is a rectangle .
Because is compact and is continuous, we have and by the Stone–Weierstrass theorem there exists a sequence of Lipschitz functions converging uniformly to in . Without loss of generality, we assume for all .
We define Picard iterations as follows, where . , and . They are well-defined by induction: as
is within the domain of .
We have
where is the Lipschitz constant of . Thus for maximal difference , we have a bound , and
By induction, this implies the bound which tends to zero as for all .
The functions are equicontinuous as for we have
so by the Arzelà–Ascoli theorem they are relatively compact. In particular, for each there is a subsequence
converging uniformly to a continuous function . Taking limit
in
we conclude that . The functions are in the closure of a relatively compact set, so they are themselves relatively compact. Thus there is a subsequence converging uniformly to a continuous function . Taking limit in we conclude that , using the fact that are equicontinuous by the Arzelà–Ascoli theorem. By the fundamental theorem of calculus, in .
Related theorems
The Peano theorem can be compared with another existence result in the same context, the Picard–Lindelöf theorem. The Picard–Lindelöf theorem both assumes more and concludes more. It requires Lipschitz continuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness where the Peano theorem proves only the existence of solutions. To illustrate, consider the ordinary differential equation
on the domain
According to the Peano theorem, this equation has solutions, but the Picard–Lindelöf theorem does not apply since the right hand side is not Lipschitz continuous in any neighbourhood containing 0. Thus we can conclude existence but not uniqueness. It turns out that this ordinary differential equation has two kinds of solutions when starting at , either or . The transition between and can happen at any .
The Carathéodory existence theorem is a generalization of the Peano existence theorem with weaker conditions tha
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https://en.wikipedia.org/wiki/Nagata%20ring
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In commutative algebra, an N-1 ring is an integral domain whose integral closure in its quotient field is a finitely generated -module. It is called a Japanese ring (or an N-2 ring) if for every finite extension of its quotient field , the integral closure of in is a finitely generated -module (or equivalently a finite -algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N-2 rings). A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring , but this concept is not used much.
Examples
Fields and rings of polynomials or power series in finitely many indeterminates over fields are examples of Japanese rings. Another important example is a Noetherian integrally closed domain (e.g. a Dedekind domain) having a perfect field of fractions. On the other hand, a principal ideal domain or even a discrete valuation ring is not necessarily Japanese.
Any quasi-excellent ring is a Nagata ring, so in particular almost all Noetherian rings that occur in algebraic geometry are Nagata rings.
The first example of a Noetherian domain that is not a Nagata ring was given by .
Here is an example of a discrete valuation ring that is not a Japanese ring. Choose a prime and an infinite degree field extension of a characteristic field , such that . Let the discrete valuation ring be the ring of formal power series over whose coefficients generate a finite extension of . If is any formal power series not in then the ring is not an N-1 ring (its integral closure is not a finitely generated module) so is not a Japanese ring.
If is the subring of the polynomial ring in infinitely many generators generated by the squares and cubes of all generators, and is obtained from by adjoining inverses to all elements not in any of the ideals generated by some , then is a Noetherian domain that is not an N-1 ring, in other words its integral closure in its quotient field is not a finitely generated -module. Also has a cusp singularity at every closed point, so the set of singular points is not closed.
References
Bosch, Güntzer, Remmert, Non-Archimedean Analysis, Springer 1984,
A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique, Ch. 0IV § 23, Publ. Math. IHÉS 20, (1964).
H. Matsumura, Commutative algebra , chapter 12.
Nagata, Masayoshi Local rings. Interscience Tracts in Pure and Applied Mathematics, No. 13 Interscience Publishers a division of John Wiley & Sons,New York-London 1962, reprinted by R. E. Krieger Pub. Co (1975)
External links
http://stacks.math.columbia.edu/tag/032E
Algebraic geometry
Commutative algebra
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https://en.wikipedia.org/wiki/G-ring
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In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number theory are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after Alexander Grothendieck.
A ring that is both a G-ring and a J-2 ring is called a quasi-excellent ring, and if in addition it is universally catenary it is called an excellent ring.
Definitions
A (Noetherian) ring R containing a field k is called geometrically regular over k if for any finite extension K of k the ring R ⊗k K is a regular ring.
A homomorphism of rings from R to S is called regular if it is flat and for every p ∈ Spec(R) the fiber S ⊗R k(p) is geometrically regular over the residue field k(p) of p. (see also Popescu's theorem.)
A ring is called a local G-ring if it is a Noetherian local ring and the map to its completion (with respect to its maximal ideal) is regular.
A ring is called a G-ring if it is Noetherian and all its localizations at prime ideals are local G-rings. (It is enough to check this just for the maximal ideals, so in particular local G-rings are G-rings.)
Examples
Every field is a G-ring
Every complete Noetherian local ring is a G-ring
Every ring of convergent power series in a finite number of variables over R or C is a G-ring.
Every Dedekind domain in characteristic 0, and in particular the ring of integers, is a G-ring, but in positive characteristic there are Dedekind domains (and even discrete valuation rings) that are not G-rings.
Every localization of a G-ring is a G-ring
Every finitely generated algebra over a G-ring is a G-ring. This is a theorem due to Grothendieck.
Here is an example of a discrete valuation ring A of characteristic p>0 which is not a G-ring. If k is any field of characteristic p with [k : kp] = ∞ and R = k[[x]] and A is the subring of power series Σaixi such that [kp(a0,a1,...) : kp] is finite then the formal fiber of A over the generic point is not geometrically regular so A is not a G-ring. Here kp denotes the image of k under the Frobenius morphism a→ap.
References
A. Grothendieck, J. Dieudonné, Eléments de géométrie algébrique IV Publ. Math. IHÉS 24 (1965), section 7
H. Matsumura, Commutative algebra , chapter 13.
Commutative algebra
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https://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood%20maximal%20function
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In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis.
Definition
The operator takes a locally integrable function f : Rd → C and returns another function Mf.
For any point x ∈ Rd, the function Mf returns the maximum of a set of reals, namely the set of average values of f for all the balls B(x, r) of any radius r at x. Formally,
where |E| denotes the d-dimensional Lebesgue measure of a subset E ⊂ Rd.
The averages are jointly continuous in x and r, therefore the maximal function Mf, being the supremum over r > 0, is measurable. It is not obvious that Mf is finite almost everywhere. This is a corollary of the Hardy–Littlewood maximal inequality.
Hardy–Littlewood maximal inequality
This theorem of G. H. Hardy and J. E. Littlewood states that M is bounded as a sublinear operator from Lp(Rd) to itself for p > 1. That is, if f ∈ Lp(Rd) then the maximal function Mf is weak L1-bounded and Mf ∈ Lp(Rd). Before stating the theorem more precisely, for simplicity, let {f > t} denote the set {x | f(x) > t}. Now we have:
Theorem (Weak Type Estimate). For d ≥ 1, there is a constant Cd > 0 such that for all λ > 0 and f ∈ L1(Rd), we have:
With the Hardy–Littlewood maximal inequality in hand, the following strong-type estimate is an immediate consequence of the Marcinkiewicz interpolation theorem:
Theorem (Strong Type Estimate). For d ≥ 1, 1 < p ≤ ∞, and f ∈ Lp(Rd),
there is a constant Cp,d > 0 such that
In the strong type estimate the best bounds for Cp,d are unknown. However subsequently Elias M. Stein used the Calderón-Zygmund method of rotations to prove the following:
Theorem (Dimension Independence). For 1 < p ≤ ∞ one can pick Cp,d = Cp independent of d.
Proof
While there are several proofs of this theorem, a common one is given below: For p = ∞, the inequality is trivial (since the average of a function is no larger than its essential supremum). For 1 < p < ∞, first we shall use the following version of the Vitali covering lemma to prove the weak-type estimate. (See the article for the proof of the lemma.)
Lemma. Let X be a separable metric space and a family of open balls with bounded diameter. Then has a countable subfamily consisting of disjoint balls such that
where 5B is B with 5 times radius.
If Mf(x) > t, then, by definition, we can find a ball Bx centered at x such that
By the lemma, we can find, among such balls, a sequence of disjoint balls Bj such that the union of 5Bj covers {Mf > t}.
It follows:
This completes the proof of the weak-type estimate. We next deduce from this the Lp bounds. Define b by b(x) = f(x) if |f(x)| > t/2 and 0 otherwise. By the weak-type estimate applied to b, we have:
with C = 5d. Then
By the estimate above we have:
where the constant Cp depends only on p and d. This completes the proof of the theorem.
Note that the constant in the proof can be improved to by using the inner regularity of the Lebesgue measure, and th
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https://en.wikipedia.org/wiki/Kenneth%20M%C3%B8ller%20Pedersen
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Kenneth Møller Pedersen (born 18 April 1973) is a former Danish professional football midfielder.
External links
Official Danish Superliga player statistics at danskfodbold.com
1973 births
Living people
Danish men's footballers
Danish Superliga players
Ikast FC players
Odense Boldklub players
Esbjerg fB players
FC Midtjylland players
Randers FC players
Footballers from Odense
Men's association football midfielders
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https://en.wikipedia.org/wiki/Newton%27s%20inequalities
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In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are real numbers and let denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by
satisfy the inequality
If all the numbers ai are non-zero, then equality holds if and only if all the numbers ai are equal.
It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.
See also
Maclaurin's inequality
References
D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55
Isaac Newton
Inequalities
Symmetric functions
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https://en.wikipedia.org/wiki/Vitali%20covering%20lemma
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In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
Vitali covering lemma
There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space . In both theorems we will use the following notation: if is a ball and , we will write for the ball .
Finite version
Theorem (Finite Covering Lemma). Let be any finite collection of balls contained in an arbitrary metric space. Then there exists a subcollection of these balls which are disjoint and satisfy Proof: Without loss of generality, we assume that the collection of balls is not empty; that is, n > 0. Let be the ball of largest radius. Inductively, assume that have been chosen. If there is some ball in that is disjoint from , let be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set m := k and terminate the inductive definition.
Now set . It remains to show that for every . This is clear if . Otherwise, there necessarily is some such that intersects and the radius of is at least as large as that of . The triangle inequality then easily implies that , as needed. This completes the proof of the finite version.
Infinite version
Theorem (Infinite Covering Lemma). Let be an arbitrary collection of balls in a separable metric space such that where denotes the radius of the ball B. Then there exists a countable sub-collection such that the balls of are pairwise disjoint, and satisfyAnd moreover, each intersects some with .
Proof: Consider the partition of F into subcollections Fn, n ≥ 0, defined by
That is, consists of the balls B whose radius is in (2−n−1R, 2−nR]. A sequence Gn, with Gn ⊂ Fn, is defined inductively as follows. First, set H0 = F0 and let G0 be a maximal disjoint subcollection of H0 (such a subcollection exists by Zorn's lemma). Assuming that G0,...,Gn have been selected, let
and let Gn+1 be a maximal disjoint subcollection of Hn+1. The subcollection
of F satisfies the requirements of the theorem: G is a disjoint collection, and is thus countable since the given metric space is separable. Moreover, every ball B ∈ F intersects a ball C ∈ G such that B ⊂ 5 C.
Indeed, if we are given some , there must be some n be such that B belongs to Fn. Either B does not belong to Hn, which implies n > 0 and means that B intersects a ball from the union of G0, ..., Gn−1, or B ∈ Hn and by maximality of Gn, B intersects a ball in Gn. In any case, B intersects
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https://en.wikipedia.org/wiki/Kuratowski%27s%20closure-complement%20problem
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In point-set topology, Kuratowski's closure-complement problem asks for the largest number of distinct sets obtainable by repeatedly applying the set operations of closure and complement to a given starting subset of a topological space. The answer is 14. This result was first published by Kazimierz Kuratowski in 1922. It gained additional exposure in Kuratowski's fundamental monograph Topologie (first published in French in 1933; the first English translation appeared in 1966) before achieving fame as a textbook exercise in John L. Kelley's 1955 classic, General Topology.
Proof
Letting denote an arbitrary subset of a topological space, write for the closure of , and for the complement of . The following three identities imply that no more than 14 distinct sets are obtainable:
. (The closure operation is idempotent.)
. (The complement operation is an involution.)
. (Or equivalently , using identity (2)).
The first two are trivial. The third follows from the identity where is the interior of which is equal to the complement of the closure of the complement of , . (The operation is idempotent.)
A subset realizing the maximum of 14 is called a 14-set. The space of real numbers under the usual topology contains 14-sets. Here is one example:
where denotes an open interval and denotes a closed interval. Let denote this set. Then the following 14 sets are accessible:
, the set shown above.
Further results
Despite its origin within the context of a topological space, Kuratowski's closure-complement problem is actually more algebraic than topological. A surprising abundance of closely related problems and results have appeared since 1960, many of which have little or nothing to do with point-set topology.
The closure-complement operations yield a monoid that can be used to classify topological spaces.
References
External links
The Kuratowski Closure-Complement Theorem by B. J. Gardner and Marcel Jackson
The Kuratowski Closure-Complement Problem by Mark Bowron
Topology
Mathematical problems
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https://en.wikipedia.org/wiki/Sylvester%20equation
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In mathematics, in the field of control theory, a Sylvester equation is a matrix equation of the form:
It is named after English mathematician James Joseph Sylvester. Then given matrices A, B, and C, the problem is to find the possible matrices X that obey this equation. All matrices are assumed to have coefficients in the complex numbers. For the equation to make sense, the matrices must have appropriate sizes, for example they could all be square matrices of the same size. But more generally, A and B must be square matrices of sizes n and m respectively, and then X and C both have n rows and m columns.
A Sylvester equation has a unique solution for X exactly when there are no common eigenvalues of A and −B.
More generally, the equation AX + XB = C has been considered as an equation of bounded operators on a (possibly infinite-dimensional) Banach space. In this case, the condition for the uniqueness of a solution X is almost the same: There exists a unique solution X exactly when the spectra of A and −B are disjoint.
Existence and uniqueness of the solutions
Using the Kronecker product notation and the vectorization operator , we can rewrite Sylvester's equation in the form
where is of dimension , is of dimension , of dimension and is the identity matrix. In this form, the equation can be seen as a linear system of dimension .
Theorem.
Given matrices and , the Sylvester equation has a unique solution for any if and only if and do not share any eigenvalue.
Proof.
The equation is a linear system with unknowns and the same number of equations. Hence it is uniquely solvable for any given if and only if the homogeneous equation
admits only the trivial solution .
(i) Assume that and do not share any eigenvalue. Let be a solution to the abovementioned homogeneous equation. Then , which can be lifted to
for each
by mathematical induction. Consequently,
for any polynomial . In particular, let be the characteristic polynomial of . Then
due to the Cayley-Hamilton theorem; meanwhile, the spectral mapping theorem tells us
where denotes the spectrum of a matrix. Since and do not share any eigenvalue, does not contain zero, and hence is nonsingular. Thus as desired. This proves the "if" part of the theorem.
(ii) Now assume that and share an eigenvalue . Let be a corresponding right eigenvector for , be a corresponding left eigenvector for , and . Then , and
Hence is a nontrivial solution to the aforesaid homogeneous equation, justifying the "only if" part of the theorem. Q.E.D.
As an alternative to the spectral mapping theorem, the nonsingularity of in part (i) of the proof can also be demonstrated by the Bézout's identity for coprime polynomials.
Let be the characteristic polynomial of . Since and do not share any eigenvalue, and are coprime. Hence there exist polynomials and such that . By the Cayley–Hamilton theorem, . Thus , implying that is nonsingular.
The theorem remains true for real matr
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https://en.wikipedia.org/wiki/Fort%20space
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In mathematics, there are a few topological spaces named after M. K. Fort, Jr.
Fort space
Fort space is defined by taking an infinite set X, with a particular point p in X, and declaring open the subsets A of X such that:
A does not contain p, or
A contains all but a finite number of points of X.
Note that the subspace has the discrete topology and is open and dense in X.
X is homeomorphic to the one-point compactification of an infinite discrete space.
Modified Fort space
Modified Fort space is similar but has two particular points. So take an infinite set X with two distinct points p and q, and declare open the subsets A of X such that:
A contains neither p nor q, or
A contains all but a finite number of points of X.
The space X is compact and T1, but not Hausdorff.
Fortissimo space
Fortissimo space is defined by taking an uncountable set X, with a particular point p in X, and declaring open the subsets A of X such that:
A does not contain p, or
A contains all but a countable number of points of X.
Note that the subspace has the discrete topology and is open and dense in X. The space X is not compact, but it is a Lindelöf space. It is obtained by taking an uncountable discrete space, adding one point and defining a topology such that the resulting space is Lindelöf and contains the original space as a dense subspace. Similarly to Fort space being the one-point compactification of an infinite discrete space, one can describe Fortissimo space as the one-point Lindelöfication of an uncountable discrete space.
See also
Notes
References
M. K. Fort, Jr. "Nested neighborhoods in Hausdorff spaces." American Mathematical Monthly vol.62 (1955) 372.
Topological spaces
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https://en.wikipedia.org/wiki/Centre%20for%20Statistics%20in%20Medicine
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The Centre for Statistics in Medicine (CSM) at the University of Oxford, United Kingdom was founded by Professor Douglas G. Altman until 2018. He was succeeded by Professor Sallie Lamb until 2019, then by Professor Gary Collins. In 1995 it was based at the Institute of Health Sciences in Headington, Oxford, it relocated to the annexe of Wolfson College, Oxford in 2005, and in 2013 moved to the Botnar Research Centre in Headington.
The CSM incorporates the Cancer Research UK Medical Statistics Group (MSG), Oxford Clinical Trial Research Unit statisticians and the UK EQUATOR Centre. It is based in the Nuffield Department of Orthopaedics, Rheumatology & Musculoskeletal Sciences in the University of Oxford.
CSM collaborates in health care research, conducts applied statistical research and runs training courses/workshops for both health care workers and statisticians.
Statisticians within the CSM are involved in many collaborative projects with clinicians in Oxford and further afield, some working across the medical spectrum and others focusing on cancer. Other statisticians within the CSM work primarily on a programme of methodological research, in particular relating to studies of diagnosis and prognosis, and to systematic reviews and meta-analysis.
References
External links
Centre for Statistics in Medicine
Medical associations based in the United Kingdom
Statistical organisations in the United Kingdom
1988 establishments in the United Kingdom
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https://en.wikipedia.org/wiki/Courant%20algebroid
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In a field of mathematics known as differential geometry, a Courant geometry was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier in 1990 the standard prototype of Courant algebroid through his discovery of a skew symmetric bracket on , called Courant bracket today, which fails to satisfy the Jacobi identity. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids.
Definition
A Courant algebroid consists of the data a vector bundle with a bracket , a non degenerate fiber-wise inner product , and a bundle map subject to the following axioms,
where are sections of E and f is a smooth function on the base manifold M. D is the combination with d the de Rham differential, the dual map of , and κ the map from E to induced by the inner product.
Skew-Symmetric Definition
An alternative definition can be given to make the bracket skew-symmetric as
This no longer satisfies the Jacobi-identity axiom above. It instead fulfills a homotopic Jacobi-identity.
where T is
The Leibniz rule and the invariance of the scalar product become modified by the relation and the violation of skew-symmetry gets replaced by the axiom
The skew-symmetric bracket together with the derivation D and the Jacobiator T form a strongly homotopic Lie algebra.
Properties
The bracket is not skew-symmetric as one can see from the third axiom. Instead it fulfills a certain Jacobi-identity (first axiom) and a Leibniz rule (second axiom). From these two axioms one can derive that the anchor map ρ is a morphism of brackets:
The fourth rule is an invariance of the inner product under the bracket. Polarization leads to
Examples
An example of the Courant algebroid is the Dorfman bracket on the direct sum with a twist introduced by Ševera, (1998) defined as:
where X,Y are vector fields, ξ,η are 1-forms and H is a closed 3-form twisting the bracket. This bracket is used to describe the integrability of generalized complex structures.
A more general example arises from a Lie algebroid A whose induced differential on will be written as d again. Then use the same formula as for the Dorfman bracket with H an A-3-form closed under d.
Another example of a Courant algebroid is a quadratic Lie algebra, i.e. a Lie algebra with an invariant scalar product. Here the base manifold is just a point and thus the anchor map (and D) are trivial.
The example described in the paper by Weinstein et al. comes from a Lie bialgebroid, i.e. A a Lie algebroid (with anchor and bracket ), also its dual a Lie algebroid (inducing the differential on ) and (where on the RHS you extend the A-bracket to using graded Leibniz rule). This notion is symmetric in A and (see Roytenberg). Here with anchor and the bracket is the skew-symmetrization of the above in X and α (equivalen
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https://en.wikipedia.org/wiki/Chilton%20and%20Colburn%20J-factor%20analogy
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Chilton–Colburn J-factor analogy (also known as the modified Reynolds analogy) is a successful and widely used analogy between heat, momentum, and mass transfer. The basic mechanisms and mathematics of heat, mass, and momentum transport are essentially the same. Among many analogies (like Reynolds analogy, Prandtl–Taylor analogy) developed to directly relate heat transfer coefficients, mass transfer coefficients, and friction factors Chilton and Colburn J-factor analogy proved to be the most accurate.
It is written as follows,
This equation permits the prediction of an unknown transfer coefficient when one of the other coefficients is known. The analogy is valid for fully developed turbulent flow in conduits with Re > 10000, 0.7 < Pr < 160, and tubes where L/d > 60 (the same constraints as the Sieder–Tate correlation). The wider range of data can be correlated by Friend–Metzner analogy.
Relationship between Heat and Mass;
See also
Reynolds analogy
Thomas H. Chilton
References
Geankoplis, C.J. Transport processes and separation process principles (2003). Fourth Edition, p. 475.
External links
Lecture notes on mass transfer coefficients: http://facstaff.cbu.edu/rprice/lectures/mtcoeff.html
Transport phenomena
Analogy
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https://en.wikipedia.org/wiki/Cophenetic%20correlation
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In statistics, and especially in biostatistics, cophenetic correlation (more precisely, the cophenetic correlation coefficient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. Although it has been most widely applied in the field of biostatistics (typically to assess cluster-based models of DNA sequences, or other taxonomic models), it can also be used in other fields of inquiry where raw data tend to occur in clumps, or clusters. This coefficient has also been proposed for use as a test for nested clusters.
Calculating the cophenetic correlation coefficient
Suppose that the original data {Xi} have been modeled using a cluster method to produce a dendrogram {Ti}; that is, a simplified model in which data that are "close" have been grouped into a hierarchical tree. Define the following distance measures.
, the Euclidean distance between the ith and jth observations.
, the dendrogrammatic distance between the model points and . This distance is the height of the node at which these two points are first joined together.
Then, letting be the average of the x(i, j), and letting be the average of the t(i, j), the cophenetic correlation coefficient c is given by
Software implementation
It is possible to calculate the cophenetic correlation in R using the dendextend R package.
In Python, the SciPy package also has an implementation.
In MATLAB, the Statistic and Machine Learning toolbox contains an implementation.
See also
Cophenetic
References
External links
Numerical example of cophenetic correlation
Computing and displaying Cophenetic distances
Covariance and correlation
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https://en.wikipedia.org/wiki/Essential%20extension
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In mathematics, specifically module theory, given a ring R and an R-module M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or large submodule of M) if for every submodule H of M,
implies that
As a special case, an essential left ideal of R is a left ideal that is essential as a submodule of the left module RR. The left ideal has non-zero intersection with any non-zero left ideal of R. Analogously, an essential right ideal is exactly an essential submodule of the right R module RR.
The usual notations for essential extensions include the following two expressions:
, and
The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule N is superfluous if for any other submodule H,
implies that .
The usual notations for superfluous submodules include:
, and
Properties
Here are some of the elementary properties of essential extensions, given in the notation introduced above. Let M be a module, and K, N and H be submodules of M with K N
Clearly M is an essential submodule of M, and the zero submodule of a nonzero module is never essential.
if and only if and
if and only if and
Using Zorn's Lemma it is possible to prove another useful fact:
For any submodule N of M, there exists a submodule C such that
.
Furthermore, a module with no proper essential extension (that is, if the module is essential in another module, then it is equal to that module) is an injective module. It is then possible to prove that every module M has a maximal essential extension E(M), called the injective hull of M. The injective hull is necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the sense that any other injective module containing M contains a copy of E(M).
Many properties dualize to superfluous submodules, but not everything. Again let M be a module, and K, N and H be submodules of M with K N.
The zero submodule is always superfluous, and a nonzero module M is never superfluous in itself.
if and only if and
if and only if and .
Since every module can be mapped via a monomorphism whose image is essential in an injective module (its injective hull), one might ask if the dual statement is true, i.e. for every module M, is there a projective module P and an epimorphism from P onto M whose kernel is superfluous? (Such a P is called a projective cover). The answer is "No" in general, and the special class of rings whose right modules all have projective covers is the class of right perfect rings.
One form of Nakayama's lemma is that J(R)M is a superfluous submodule of M when M is a finitely-generated module over R.
Generalization
This definition can be generalized to an arbitrary abelian category C. An essential extension is a monomorphism u : M → E such that for every non-zero subobject s : N → E, the fibre product N ×E M ≠ 0.
In a general category, a morphism f : X → Y
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https://en.wikipedia.org/wiki/Rusty%20Kruger
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Rusty Kruger (born March 26, 1975) is a Canadian retired lacrosse player in the National Lacrosse League and a current assistant coach with the Buffalo Bandits.
Statistics
NLL
Reference:
References
1975 births
Buffalo Bandits players
Canadian lacrosse players
Chicago Shamrox players
Lacrosse people from Ontario
Living people
People from Orangeville, Ontario
Rochester Knighthawks players
San Jose Stealth players
Toronto Rock players
New York Saints players
Albany Attack players
National Lacrosse League coaches
National Lacrosse League players
Buffalo Bandits coaches
Lacrosse forwards
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https://en.wikipedia.org/wiki/Trivial%20measure
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In mathematics, specifically in measure theory, the trivial measure on any measurable space (X, Σ) is the measure μ which assigns zero measure to every measurable set: μ(A) = 0 for all A in Σ.
Properties of the trivial measure
Let μ denote the trivial measure on some measurable space (X, Σ).
A measure ν is the trivial measure μ if and only if ν(X) = 0.
μ is an invariant measure (and hence a quasi-invariant measure) for any measurable function f : X → X.
Suppose that X is a topological space and that Σ is the Borel σ-algebra on X.
μ trivially satisfies the condition to be a regular measure.
μ is never a strictly positive measure, regardless of (X, Σ), since every measurable set has zero measure.
Since μ(X) = 0, μ is always a finite measure, and hence a locally finite measure.
If X is a Hausdorff topological space with its Borel σ-algebra, then μ trivially satisfies the condition to be a tight measure. Hence, μ is also a Radon measure. In fact, it is the vertex of the pointed cone of all non-negative Radon measures on X.
If X is an infinite-dimensional Banach space with its Borel σ-algebra, then μ is the only measure on (X, Σ) that is locally finite and invariant under all translations of X. See the article There is no infinite-dimensional Lebesgue measure.
If X is n-dimensional Euclidean space Rn with its usual σ-algebra and n-dimensional Lebesgue measure λn, μ is a singular measure with respect to λn: simply decompose Rn as A = Rn \ {0} and B = {0} and observe that μ(A) = λn(B) = 0.
References
Measures (measure theory)
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https://en.wikipedia.org/wiki/Mapping%20cone%20%28homological%20algebra%29
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In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi-isomorphism; if we pass to the derived category of complexes, this means that f is an isomorphism there, which recalls the familiar property of maps of groups, modules over a ring, or elements of an arbitrary abelian category that if the kernel and cokernel both vanish, then the map is an isomorphism. If we are working in a t-category, then in fact the cone furnishes both the kernel and cokernel of maps between objects of its core.
Definition
The cone may be defined in the category of cochain complexes over any additive category (i.e., a category whose morphisms form abelian groups and in which we may construct a direct sum of any two objects). Let be two complexes, with differentials i.e.,
and likewise for
For a map of complexes we define the cone, often denoted by or to be the following complex:
on terms,
with differential
(acting as though on column vectors).
Here is the complex with and .
Note that the differential on is different from the natural differential on , and that some authors use a different sign convention.
Thus, if for example our complexes are of abelian groups, the differential would act as
Properties
Suppose now that we are working over an abelian category, so that the homology of a complex is defined. The main use of the cone is to identify quasi-isomorphisms: if the cone is acyclic, then the map is a quasi-isomorphism. To see this, we use the existence of a triangle
where the maps are given by the direct summands (see Homotopy category of chain complexes). Since this is a triangle, it gives rise to a long exact sequence on homology groups:
and if is acyclic then by definition, the outer terms above are zero. Since the sequence is exact, this means that induces an isomorphism on all homology groups, and hence (again by definition) is a quasi-isomorphism.
This fact recalls the usual alternative characterization of isomorphisms in an abelian category as those maps whose kernel and cokernel both vanish. This appearance of a cone as a combined kernel and cokernel is not accidental; in fact, under certain circumstances the cone literally embodies both. Say for example that we are working over an abelian category and have only one nonzero term in degree 0:
and therefore is just (as a map of objects of the underlying abelian category). Then the cone is just
(Underset text indicates the degree of each term.) The homology of this complex is then
This is not an accident and in fact occurs in every t-category.
Mapping cylinder
A related notion is the mapping cylinder: let be a morphism of chain complexes, let
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https://en.wikipedia.org/wiki/Fundamental%20matrix%20%28linear%20differential%20equation%29
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In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations
is a matrix-valued function whose columns are linearly independent solutions of the system.
Then every solution to the system can be written as , for some constant vector (written as a column vector of height ).
One can show that a matrix-valued function is a fundamental matrix of if and only if and is a non-singular matrix for all
Control theory
The fundamental matrix is used to express the state-transition matrix, an essential component in the solution of a system of linear ordinary differential equations.
See also
Linear differential equation
Liouville's formula
Systems of ordinary differential equations
References
Matrices
Differential calculus
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https://en.wikipedia.org/wiki/Rupture%20field
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In abstract algebra, a rupture field of a polynomial over a given field is a field extension of generated by a root of .
For instance, if and then is a rupture field for .
The notion is interesting mainly if is irreducible over . In that case, all rupture fields of over are isomorphic, non-canonically, to : if where is a root of , then the ring homomorphism defined by for all and is an isomorphism. Also, in this case the degree of the extension equals the degree of .
A rupture field of a polynomial does not necessarily contain all the roots of that polynomial: in the above example the field does not contain the other two (complex) roots of (namely and where is a primitive cube root of unity). For a field containing all the roots of a polynomial, see Splitting field.
Examples
A rupture field of over is . It is also a splitting field.
The rupture field of over is since there is no element of which squares to (and all quadratic extensions of are isomorphic to ).
See also
Splitting field
References
Field (mathematics)
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https://en.wikipedia.org/wiki/Irenaean%20theodicy
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The Irenaean theodicy is a Christian theodicy (a response to the problem of evil). It defends the probability of an omnipotent and omnibenevolent (all-powerful and perfectly loving) God in the face of evidence of evil in the world. Numerous variations of theodicy have been proposed which all maintain that, while evil exists, God is either not responsible for creating evil, or he is not guilty for creating evil. Typically, the Irenaean theodicy asserts that the world is the best of all possible worlds because it allows humans to fully develop. Most versions of the Irenaean theodicy propose that creation is incomplete, as humans are not yet fully developed, and experiencing evil and suffering is necessary for such development.
Second-century theologian and philosopher Irenaeus, after whom the theodicy is named, proposed a two-stage creation process in which humans require free will and the experience of evil to develop. Another early Christian theologian, Origen, presented a response to the problem of evil which cast the world as a schoolroom or hospital for the soul; theologian Mark Scott has argued that Origen, rather than Irenaeus, ought to be considered the father of this kind of theodicy. Friedrich Schleiermacher argued in the nineteenth century that God must necessarily create flawlessly, so this world must be the best possible world because it allows God's purposes to be naturally fulfilled. In 1966, philosopher John Hick discussed the similarities of the preceding theodicies, calling them all "Irenaean". He supported the view that creation is incomplete and argued that the world is best placed for the full moral development of humans, as it presents genuine moral choices. British philosopher Richard Swinburne proposed that, to make a free moral choice, humans must have experience of the consequences of their own actions and that natural evil must exist to provide such choices.
The development of process theology has challenged the Irenaean tradition by teaching that God using suffering for his own ends would be immoral. Twentieth-century philosopher Alvin Plantinga's freewill defense argues that, while this may be the best world God could have created, God's options were limited by the need to allow freewill. Alvin Plantinga's ultimate response to the problem of evil is that it is not a problem that can be solved. Christians simply cannot claim to know the answer to the "Why?” of suffering and evil. Plantinga stresses that this is why he does not proffer a theodicy but only a defense of theistic belief as rational in the face of unanswered questions. D. Z. Phillips and Fyodor Dostoyevsky challenged the instrumental use of suffering, suggesting that love cannot be expressed through suffering. However, Dostoyevsky also states that the beauty of love is evident, in that love can continue to grow, withstand and overcome even the most evil acts. Michael Tooley argued that the magnitude of suffering is excessive and that, in some cases, cannot
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https://en.wikipedia.org/wiki/Tsen%27s%20theorem
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In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve.
The theorem was published by Chiungtze C. Tsen in 1933.
See also
Tsen rank
References
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/PDIFF
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In geometric topology, PDIFF, for piecewise differentiable, is the category of piecewise-smooth manifolds and piecewise-smooth maps between them. It properly contains DIFF (the category of smooth manifolds and smooth functions between them) and PL (the category of piecewise linear manifolds and piecewise linear maps between them), and the reason it is defined is to allow one to relate these two categories. Further, piecewise functions such as splines and polygonal chains are common in mathematics, and PDIFF provides a category for discussing them.
Motivation
PDIFF is mostly a technical point: smooth maps are not piecewise linear (unless linear), and piecewise linear maps are not smooth (unless globally linear) – the intersection is linear maps, or more precisely affine maps (because not based) – so they cannot directly be related: they are separate generalizations of the notion of an affine map.
However, while a smooth manifold is not a PL manifold, it carries a canonical PL structure – it is uniquely triangularizable; conversely, not every PL manifold is smoothable. For a particular smooth manifold or smooth map between smooth manifolds, this can be shown by breaking up the manifold into small enough pieces, and then linearizing the manifold or map on each piece: for example, a circle in the plane can be approximated by a triangle, but not by a 2-gon, since this latter cannot be linearly embedded.
This relation between Diff and PL requires choices, however, and is more naturally shown and understood by including both categories in a larger category, and then showing that the inclusion of PL is an equivalence: every smooth manifold and every PL manifold is a PDiff manifold. Thus, going from Diff to PDiff and PL to PDiff are natural – they are just inclusion. The map PL to PDiff, while not an equality – not every piecewise smooth function is piecewise linear – is an equivalence: one can go backwards by linearize pieces. Thus it can for some purposes be inverted, or considered an isomorphism, which gives a map These categories all sit inside TOP, the category of topological manifold and continuous maps between them.
In summary, PDiff is more general than Diff because it allows pieces (corners), and one cannot in general smooth corners, while PL is no less general that PDiff because one can linearize pieces (more precisely, one may need to break them up into smaller pieces and then linearize, which is allowed in PDiff).
History
That every smooth (indeed, C1) manifold has a unique PL structure was originally proven in . A detailed expositionary proof is given in . The result is elementary and rather technical to prove in detail, so it is generally only sketched in modern texts, as in the brief proof outline given in . A very brief outline is given in , while a short but detailed proof is given in .
References
Geometric topology
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https://en.wikipedia.org/wiki/Olive%20Hazlett
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Olive Clio Hazlett (October 27, 1890 – March 8, 1974) was an American mathematician who spent most of her career working for the University of Illinois. She mainly researched algebra, and wrote seventeen research papers on subjects such as nilpotent algebras, division algebras, modular invariants, and the arithmetic of algebras.
Background
Hazlett was born in Cincinnati, Ohio, but grew up in Boston, Massachusetts, where she attended public school. In 1912 she received her bachelor's degree from Radcliffe College. She then attended the University of Chicago for her master's degree (1913) and Ph.D. (1915), for which she wrote a thesis titled On the Classification and Invariantive Characterization of Nilpotent Algebras with L. E. Dickson as thesis advisor. After receiving her doctoral degree Hazlett was awarded an Alice Freeman Palmer Fellowship by Harvard, which allowed her to research invariants of nilpotent algebras at Wellesley College for the next year.
Career
In 1916 she was appointed to Bryn Mawr College, where she worked for two years before accepting an appointment as assistant professor at Mount Holyoke College. She was promoted to associate professor in 1924, the same year she gave a talk on The Arithmetic of a General Associative Algebra at the International Congress of Mathematicians in Toronto, but in 1925 she left Mount Holyoke because she felt she was not given enough time or resources to pursue her research in algebra. It was then that she took a job as assistant professor at the University of Illinois, where she would spend the rest of her career.
In 1928 Hazlett received a Guggenheim Fellowship that allowed her to spend a year visiting Italy, Germany, and Switzerland. While in Italy she presented a paper called Integers as Matrices to the International Congress of Mathematicians in Bologna. Near the end of her visits she requested an extension of her Guggenheim Fellowship, which was granted and allowed her to spend another year in Europe. When she finally returned to the University of Illinois in 1930, she was promoted to associate professor and received a pay raise. However, her teaching schedule was rigorous and required her to teach service courses to large classes of non-math-majors, and after 1930 she did not publish any more research papers. In 1935 she wrote to the chair of the mathematics department complaining that the service courses left her no time for research, but her teaching schedule was not changed and by December 1936 she took a sick leave after having a mental breakdown from the stress of her job. The sick leave was supposed to end in August 1937, but her health had not improved enough by this time and she took another year off. She was, however, able to return to teaching by the end of 1938.
In 1940, she was appointed a member of the American Mathematical Society's Cryptanalysis Committee, for which she worked until the end of World War II. She maintained her teaching job for most of this period, though, (e
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https://en.wikipedia.org/wiki/Tensor%20product%20of%20quadratic%20forms
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In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible (that is, R has characteristic ), and if and are two quadratic spaces over R, then their tensor product is the quadratic space whose underlying R-module is the tensor product of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to and .
In particular, the form satisfies
(which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in R), i.e.,
then the tensor product has diagonalization
Quadratic forms
Tensors
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https://en.wikipedia.org/wiki/Lola%20J.%20May
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Lola J. May (October 29, 1923 – March 13, 2007) was a mathematics educator, consultant, author, producer of audio-visual materials, an early proponent of the new math educational process, and a household name among mathematics.
Life
Her father was a salesman and her mother was a homemaker. Her father taught her mathematics every night using a movable blackboard and a collection of coins. She found her early schooling boring and too strict, and she did not initially consider becoming a teacher.
A native of Kenosha, Wisconsin and a summa cum laude graduate of the University of Wisconsin–Madison in 1945, where she received her B.S. in mathematics and science. After teaching high school for three years, she studied and achieved her master's degree in mathematics at Northwestern University in 1950 and her doctorate in mathematics education from there in 1964. She taught mathematics at New Trier Township High School in the Chicago area until 1960, and was a mathematics consultant at the Winnetka, Illinois public schools until 1998. Her summers were often spent teaching at the university level, but she taught mathematics to all grades over the course of her career.
She promised herself to make her students laugh and ask questions. She did not want her students to be bored by or scared of mathematics. She succeeded; her students cheered when they figured out the answers to math problems and lamented when class time with Dr. May was over.
May explained, "The big thing I have going for me is my enthusiasm. There are people who are brighter than I am. There are people who may be better teachers-although I'm pretty good at teaching-and there are certainly people who are better writers. But I have enthusiasm." This enthusiasm was not unnoticed; teachers in the same hallway as her described how loud she was.
May died on March 13, 2007, in Evanston, Illinois, at the age of 83.
Contributions
Her authored works include her autobiography "Lola May Who?", the book "Teaching Mathematics in the Elementary School", a number of Harcourt Brace textbooks, monthly articles for a regular column in the Teaching K-8 magazine, and a series of articles for the Chicago Tribune Magazine. May created videotapes, film strips, audiocassettes, and students' audiovisual programs for teaching mathematics. She led 20 shows about "new math" for parents and teachers on NCB TV from 1962 to 1964. She also designed a cartoon series about new math for an adult audience, called Space Age Math for Stone Age Parents.
She frequently was a speaker at the annual California Math Conference and Northwest Math Conference during the 1970s, 1980s, and 1990s. May also spoke at National Council of Teachers of Mathematics (NCTM) and National Council of Supervisors of Mathematics (NCSM) conferences. She gave talks in all 50 states and around the world.
Recognition
Her awards include the Northwestern Alumni Merit Award in 1999, the Lifetime Achievement Award of the National Council of Teachers of Ma
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https://en.wikipedia.org/wiki/Centre%20for%20Research%20and%20Development%20on%20Information%20Technology%20and%20Telecommunication%20%28Albania%29
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The Centre for Research and Development on Information Technology and Telecommunication (), formerly known as INIMA or Institute of Informatics and Applied Mathematics is a research institute on technology in Tirana, Albania, affiliated since 2007 with the Polytechnic University of Tirana. It was founded in 1986 on the basis of the Center of Computational Mathematics (QMLL). The latter former was founded in 1971, depending from Tirana University (UT) and, in 1973, when the Academy of Sciences of Albania was founded, became one of the scientific institutions the Academy was composed of.
Having some of the most prominent experts in informatics and applied mathematics, INIMA has played a prime hand role in all informatics developments in Albania, in introduction of modern methods and technologies in different domains of Albanian reality such as: economics, engineering, geology&mining, medicine and health care, farming, animal breeding, etc.; in the preparation of new specialists as well as in offering different scientific services, installation and maintenance of computer systems, etc.
In 2007, with a Council of Ministers decision (#146 dated 28 March 2007), INIMA was dissolved and restructured as part of the Polytechnic University of Tirana. INIMA was renamed to "Centre for Research and Development on Information Technology and Telecommunication" (after Council of Ministers' decision #824 dated 05.12.2007).
See also
List of universities in Albania
References
Educational organizations based in Albania
Science and technology in Albania
Scientific organizations based in Albania
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https://en.wikipedia.org/wiki/Kneser%27s%20theorem%20%28differential%20equations%29
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In mathematics, the Kneser theorem can refer to two distinct theorems in the field of ordinary differential equations:
the first one, named after Adolf Kneser, provides criteria to decide whether a differential equation is oscillating or not;
the other one, named after Hellmuth Kneser, is about the topology of the set of all solutions of an initial value problem with continuous right hand side.
Statement of the theorem due to A. Kneser
Consider an ordinary linear homogeneous differential equation of the form
with
continuous.
We say this equation is oscillating if it has a solution y with infinitely many zeros, and non-oscillating otherwise.
The theorem states that the equation is non-oscillating if
and oscillating if
Example
To illustrate the theorem consider
where is real and non-zero. According to the theorem, solutions will be oscillating or not depending on whether is positive (non-oscillating) or negative (oscillating) because
To find the solutions for this choice of , and verify the theorem for this example, substitute the 'Ansatz'
which gives
This means that (for non-zero ) the general solution is
where and are arbitrary constants.
It is not hard to see that for positive the solutions do not oscillate while for negative the identity
shows that they do.
The general result follows from this example by the Sturm–Picone comparison theorem.
Extensions
There are many extensions to this result, such as the Gesztesy–Ünal criterion.
Statement of the theorem due to H. Kneser
While Peano's existence theorem guarantees the existence of solutions of certain initial values problems with continuous right hand side, H. Kneser's theorem deals with the topology of the set of those solutions. Precisely, H. Kneser's theorem states the following:
Let be a continuous function on the region , and such that for all .
Given a real number satisfying , define the set as the set of points for which there is a solution of such that and . The set is a closed and connected set.
References
Ordinary differential equations
Theorems in analysis
Oscillation
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https://en.wikipedia.org/wiki/Oscillation%20theory
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In mathematics, in the field of ordinary differential equations, a nontrivial solution to an ordinary differential equation
is called oscillating if it has an infinite number of roots; otherwise it is called non-oscillating. The differential equation is called oscillating if it has an oscillating solution.
The number of roots carries also information on the spectrum of associated boundary value problems.
Examples
The differential equation
is oscillating as sin(x) is a solution.
Connection with spectral theory
Oscillation theory was initiated by Jacques Charles François Sturm in his investigations of Sturm–Liouville problems from 1836. There he showed that the n'th eigenfunction of a Sturm–Liouville problem has precisely n-1 roots. For the one-dimensional Schrödinger equation the question about oscillation/non-oscillation answers the question whether the eigenvalues accumulate at the bottom of the continuous spectrum.
Relative oscillation theory
In 1996 Gesztesy–Simon–Teschl showed that the number of roots of the Wronski determinant of two eigenfunctions of a Sturm–Liouville problem gives the number of eigenvalues between the corresponding eigenvalues. It was later on generalized by Krüger–Teschl to the case of two eigenfunctions of two different Sturm–Liouville problems. The investigation of the number of roots of the Wronski determinant of two solutions is known as relative oscillation theory.
See also
Classical results in oscillation theory are:
Kneser's theorem (differential equations)
Sturm–Picone comparison theorem
Sturm separation theorem
References
Ordinary differential equations
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https://en.wikipedia.org/wiki/Wallgau
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Wallgau is a municipality in the district of Garmisch-Partenkirchen, in Bavaria, Germany.
Population
Growth
*Statistics according to the Bavarian government, as of 2007.
Demographics
*Statistics according to the Bavarian government, as of 2007.
Notable people
Magdalena Neuner, (born 1987), twelve-time biathlon world champion, Olympic champion, Biathlon World Cup winner. Neuner has lived in the Bavarian village of Wallgau since birth.
Gallery
References
External links
Wallgau.de, official web site
Woiga.de, unofficial web site for citizens Information in Wallgau
Garmisch-Partenkirchen (district)
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https://en.wikipedia.org/wiki/Random%20regular%20graph
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A random r-regular graph is a graph selected from , which denotes the probability space of all r-regular graphs on vertices, where and is even. It is therefore a particular kind of random graph, but the regularity restriction significantly alters the properties that will hold, since most graphs are not regular.
Properties of random regular graphs
As with more general random graphs, it is possible to prove that certain properties of random –regular graphs hold asymptotically almost surely. In particular, for , a random r-regular graph of large size is asymptotically almost surely r-connected. In other words, although –regular graphs with connectivity less than exist, the probability of selecting such a graph tends to 0 as increases.
If is a positive constant, and is the least integer satisfying
then, asymptotically almost surely, a random r-regular graph has diameter at most d. There is also a (more complex) lower bound on the diameter of r-regular graphs, so that almost all r-regular graphs (of the same size) have almost the same diameter.
The distribution of the number of short cycles is also known: for fixed , let be the number of cycles of lengths up to . Then the are asymptotically independent Poisson random variables with means
Algorithms for random regular graphs
It is non-trivial to implement the random selection of r-regular graphs efficiently and in an unbiased way, since most graphs are not regular. The pairing model (also configuration model) is a method which takes nr points, and partitions them into n buckets with r points in each of them. Taking a random matching of the nr points, and then contracting the r points in each bucket into a single vertex, yields an r-regular graph or multigraph. If this object has no multiple edges or loops (i.e. it is a graph), then it is the required result. If not, a restart is required.
A refinement of this method was developed by Brendan McKay and Nicholas Wormald.
References
Random graphs
Regular graphs
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https://en.wikipedia.org/wiki/Ranked%20poset
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In mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is
a graded poset, or
a poset with the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or
a poset in which all maximal chains have the same finite length.
The second definition differs from the first in that it requires all minimal elements to have the same rank; for posets with a least element, however, the two requirements are equivalent. The third definition is even more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank. Richard P. Stanley defines a graded poset of length n as one in which all maximal chains have length n.
References
Order theory
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https://en.wikipedia.org/wiki/Pseudoreplication
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Pseudoreplication (sometimes unit of analysis error) has many definitions. Pseudoreplication was originally defined in 1984 by Stuart H. Hurlbert as the use of inferential statistics to test for treatment effects with data from experiments where either treatments are not replicated (though samples may be) or
replicates are not statistically independent. Subsequently, Millar and Anderson identified it as a special case of inadequate specification of random factors where both random and fixed factors are present. It is sometimes narrowly interpreted as an inflation of the number of samples or replicates which are not statistically independent. This definition omits the confounding of unit and treatment effects in a misspecified F-ratio. In practice, incorrect F-ratios for statistical tests of fixed effects often arise from a default F-ratio that is formed over the error rather the mixed term.
Lazic defined pseudoreplication as a problem of correlated samples (e.g. from longitudinal studies) where correlation is not taken into account when computing the confidence interval for the sample mean. For the effect of serial or temporal correlation also see Markov chain central limit theorem.
The problem of inadequate specification arises when treatments are assigned to units that are subsampled and the treatment F-ratio in an analysis of variance (ANOVA) table is formed with respect to the residual mean square rather than with respect to the among unit mean square. The F-ratio relative to the within unit mean square is vulnerable to the confounding of treatment and unit effects, especially when experimental unit number is small (e.g. four tank units, two tanks treated, two not treated, several subsamples per tank). The problem is eliminated by forming the F-ratio relative to the correct mean square in the ANOVA table (tank by treatment MS in the example above), where this is possible. The problem is addressed by the use of mixed models.
Hurlbert reported "pseudoreplication" in 48% of the studies he examined, that used inferential statistics. Several studies examining scientific papers published up to 2016 similarly found about half of the papers were suspected of pseudoreplication. When time and resources limit the number of experimental units, and unit effects cannot be eliminated statistically by testing over the unit variance, it is important to use other sources of information to evaluate the degree to which an F-ratio is confounded by unit effects.
Replication
Replication increases the precision of an estimate, while randomization addresses the broader applicability of a sample to a population. Replication must be appropriate: replication at the experimental unit level must be considered, in addition to replication within units.
Hypothesis testing
Statistical tests (e.g. t-test and the related ANOVA family of tests) rely on appropriate replication to estimate statistical significance. Tests based on the t and F distributions assume homogeneo
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https://en.wikipedia.org/wiki/Spinors%20in%20three%20dimensions
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In mathematics, the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product. This is part of the detailed algebraic discussion of the rotation group SO(3).
Formulation
The association of a spinor with a 2×2 complex Hermitian matrix was formulated by Élie Cartan.
In detail, given a vector x = (x1, x2, x3) of real (or complex) numbers, one can associate the complex matrix
In physics, this is often written as a dot product , where is the vector form of Pauli matrices. Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
, where denotes the determinant.
, where I is the identity matrix.
where Z is the matrix associated to the cross product .
If is a unit vector, then is the matrix associated with the vector that results from reflecting in the plane orthogonal to .
The last property can be used to simplify rotational operations. It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (More generally, any orientation-reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation which decomposes as the reflection in the plane perpendicular to a unit vector followed by the reflection in the plane perpendicular to , then the matrix represents the rotation of the vector through R.
Having effectively encoded all the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors) play. Provisionally, a spinor is a column vector
with complex entries ξ1 and ξ2.
The space of spinors is evidently acted upon by complex 2×2 matrices. As shown above, the product of two reflections in a pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation. So there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique. Clearly, if is a representation of a rotation, then replacing R by −R will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always double-valued.
History
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σi, so that the Hermitian matrix is written as a Pauli vector In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as biquaternions.
Michael Stone and Paul Goldbar, in Mathematics for Physics, contest this, saying, "The spin representations were discovered by ´Elie Cartan in 1913, some years before they were needed in physics."
Formulation using isotropic vectors
Spinors
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https://en.wikipedia.org/wiki/Spin%20representation
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In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equivalent representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.
Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron.
The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures.
The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
Set-up
Let be a finite-dimensional real or complex vector space with a nondegenerate quadratic form . The (real or complex) linear maps preserving form the orthogonal group . The identity component of the group is called the special orthogonal group . (For real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, has a unique connected double cover, the spin group . There is thus a group homomorphism whose kernel has two elements denoted , where is the identity element. Thus, the group elements and of are equivalent after the homomorphism to ; that is, for any in .
The groups and are all Lie groups, and for fixed they have the same Lie algebra, . If is real, then is a real vector subspace of its complexification , and the quadratic form extends naturally to a quadratic form on . This embeds as a subgroup of , and hence we may realise as a subgroup of . Furthermore, is the complexification of .
In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimension of . Concretely, we may assume and
The corresponding Lie groups are denoted and their Lie algebra as .
In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers where is the dimension of , and is the signature. Concretely, we may a
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https://en.wikipedia.org/wiki/Overdetermined%20system
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In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system will have solutions in some cases, for example if some equation occurs several times in the system, or if some equations are linear combinations of the others.
The terminology can be described in terms of the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom.
Therefore, the critical case occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint. The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns. Such systems usually have an infinite number of solutions.
Overdetermined linear systems of equations
An example in two dimensions
Consider the system of 3 equations and 2 unknowns ( and ), which is overdetermined because 3 > 2, and which corresponds to Diagram #1:
There is one solution for each pair of linear equations: for the first and second equations (0.2, −1.4), for the first and third (−2/3, 1/3), and for the second and third (1.5, 2.5). However, there is no solution that satisfies all three simultaneously. Diagrams #2 and 3 show other configurations that are inconsistent because no point is on all of the lines. Systems of this variety are deemed inconsistent.
The only cases where the overdetermined system does in fact have a solution are demonstrated in Diagrams #4, 5, and 6. These exceptions can occur only when the overdetermined system contains enough linearly dependent equations that the number of independent equations does not exceed the number of unknowns. Linear dependence means that some equations can be obtained from linearly combining other equations. For example, Y = X + 1 and 2Y = 2X + 2 are linearly dependent equations because the second one can be obtained by taking twice the first one.
Matrix form
Any system of linear equations can be written as a matrix equation.
The previous system of equations (in Diagram #1) can be written as follows:
Notice that the rows of the coefficient matrix (corresponding to equations) outnumber the columns (corresponding to unknowns), meaning that the system is overdetermined. The rank of this matrix is 2, which corresponds to the number of dependent variables in the system. A linear system is consistent if and only if the coefficient matrix has the same rank as its augmented matrix (the coefficient matrix with an extra column added, that column be
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https://en.wikipedia.org/wiki/University%20of%20Arkansas%20Office%20of%20Distance%20Education
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The Office of Distance Education (ODE) was founded in July 1998 on the campus of the Arkansas School for Mathematics, Sciences, and the Arts in Hot Springs, Arkansas and is now a part of the University of Arkansas System. Originally established in order to expand educational opportunities in Arkansas’ rural schools, the Office of Distance Education uses H.323-based video conferencing to provide highly qualified, fully certified teachers to school districts nationwide unable to hire qualified faculty locally.
Operations
During its first year of operation, ODE enrolled 228 high school students from 23 school districts across Arkansas. For school year 2010 - 2011, ODE initial enrollment exceeded 3,600 students from approximately one hundred school districts in eight states.
ODE offers complete elementary, middle grades and high school curricula as well as College Board-approved Advanced Placement courses and Concurrent Enrollment courses that allow students to earn college credit.
Of particular note, ODE has been recognized repeatedly by the United States Distance Learning Association for the excellence of its instructors and programming in years 2007, 2008, 2009, 2010 and 2011 including awards for Excellence in Distance Learning Teaching, Excellence in Distance Learning Programing and Leadership in the Field of Distance Education. In addition, ODE was recognized by ComputerWorld as a Laureate in 2010. ODE’s programs are accredited by the North Central Association Commission on Accreditation and School Improvement.
ODE also offers training to students and teachers nationwide, including ACT preparation, using Wiki’s in the classroom, teaching language with TPRS, and other programs.
ODE operates through a pay for service model which allows instruction to be delivered with real teachers in real time at any location. The Office has partnered with St. George's Church of England Primary School in Birmingham, England in order to share Spanish classes between British and Arkansas elementary students, as well as the East Central Board of Cooperative Education and Swink High School in Eastern Colorado providing Spanish, French, and German instruction.
External links
Office of Distance Education
ECBOCES
Arkansas School for Mathematics, Sciences and the Arts
United States Distance Learning Association
Distance education institutions based in the United States
Educational technology non-profits
Public education in Arkansas
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https://en.wikipedia.org/wiki/Sturm%E2%80%93Picone%20comparison%20theorem
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In mathematics, in the field of ordinary differential equations, the Sturm–Picone comparison theorem, named after Jacques Charles François Sturm and Mauro Picone, is a classical theorem which provides criteria for the oscillation and non-oscillation of solutions of certain linear differential equations in the real domain.
Let , for be real-valued continuous functions on the interval and let
be two homogeneous linear second order differential equations in self-adjoint form with
and
Let be a non-trivial solution of (1) with successive roots at and and let be a non-trivial solution of (2). Then one of the following properties holds.
There exists an in such that or
there exists a in R such that .
The first part of the conclusion is due to Sturm (1836), while the second (alternative) part of the theorem is due to Picone (1910) whose simple proof was given using his now famous Picone identity. In the special case where both equations are identical one obtains the Sturm separation theorem.
Notes
References
Diaz, J. B.; McLaughlin, Joyce R. Sturm comparison theorems for ordinary and partial differential equations. Bull. Amer. Math. Soc. 75 1969 335–339
Heinrich Guggenheimer (1977) Applicable Geometry, page 79, Krieger, Huntington .
Ordinary differential equations
Theorems in analysis
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https://en.wikipedia.org/wiki/Super%2030
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Super 30 is an Indian educational program started in Patna, India under the banner of Ramanujan School of Mathematics. It was founded by Anand Kumar, a mathematics teacher, and Abhayanand, the former D.G.P of Bihar. The program selects 30 talented candidates each year from economically underprivileged sections of Indian society and trains them for the JEE. The program is portrayed in the 2019 film, Super 30, starring Hrithik Roshan as Anand
Kumar, and his school, have been the subject of several smear campaigns, some of which have been carried in Indian media sources.
History
In 2002, Anand Kumar and Abhayanand started Super 30 with the plan to select 30 talented students from economically impoverished sections who could not afford IIT coaching. These 30 students were then prepared to pass IIT-JEE examinations. Anand Kumar's mother, Jayanti Devi, volunteered to cook for the students while Anand Kumar, Abhayanand, and other teachers tutored them. The students were also provided study materials and lodging for a year free of cost.
In the first year of the coaching, 18 out of 30 students made it to IIT. The following year, application numbers soared due to the popularity of the program and written examination was conducted to select 30 students. In 2004, 22 out of 30 students qualified for IIT-JEE, increasing the popularity of the program which attracted even more applications.
In 2005, 26 out of 30 students cleared the IIT-JEE exam, while 28 in 2006 - this despite the fact that IIT changed the examination structure. In appreciation of their efforts, Bihar Chief Minister at the time Nitish Kumar congratulated the students with a cash prize of ₹50,000 each.
The following year 28 more students cleared the IIT-JEE, and in 2008, all of the Super 30 students cleared the IIT-JEE, after which Abhayanand quit Super 30 saying "the experiment is over." Some of Kumar's former students joined as Super 30 teachers and in 2009 and 2010 all 30 students again qualified the IIT JEE exams. In subsequent years the success rates from the 30 students were: 2011 (24 passed), 2012 (27 passed), 2013 (28 passed), 2014 (27 passed), 2015 (25 passed), and in 2016 (28 passed). In 2017, all Super 30 candidates made it to the IIT-JEE. In 2018, 26 of the 30 students cleared the exam.
Awards and recognition
Time Magazine included Super 30 in its list of The Best of Asia 2010. The organization also received praise from US President Barack Obama's special envoy Rashad Hussain, who termed it the "best" institute in the country. Newsweek Magazine included Super 30 in its list of Four Most Innovative Schools in the World. Anand Kumar was awarded the Maulana Abul Kalam Azad Shiksha Puraskar in November 2010, the highest award given by the Bihar state government in the field of education.
Smear campaigns
On July 23, 2018, an article in Dainik Jagran cited former Super 30 students who said that only three students from the program had passed the IIT JEE exam that year, contrary to
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https://en.wikipedia.org/wiki/Bruhat%20order
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In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties.
History
The Bruhat order on the Schubert varieties of a flag manifold or a Grassmannian was first studied by , and the analogue for more general semisimple algebraic groups was studied by . started the combinatorial study of the Bruhat order on the Weyl group, and introduced the name "Bruhat order" because of the relation to the Bruhat decomposition introduced by François Bruhat.
The left and right weak Bruhat orderings were studied by .
Definition
If (W, S) is a Coxeter system with generators S, then the Bruhat order is a partial order on the group W. Recall that a reduced word for an element w of W is a minimal length expression of w as a product of elements of S, and the length ℓ(w) of w is the length of a reduced word.
The (strong) Bruhat order is defined by u ≤ v if some substring of some (or every) reduced word for v is a reduced word for u. (Note that here a substring is not necessarily a consecutive substring.)
The weak left (Bruhat) order is defined by u ≤L v if some final substring of some reduced word for v is a reduced word for u.
The weak right (Bruhat) order is defined by u ≤R v if some initial substring of some reduced word for v is a reduced word for u.
For more on the weak orders, see the article weak order of permutations.
Bruhat graph
The Bruhat graph is a directed graph related to the (strong) Bruhat order. The vertex set is the set of elements of the Coxeter group and the edge set consists of directed edges (u, v) whenever u = tv for some reflection t and ℓ(u) < ℓ(v). One may view the graph as an edge-labeled directed graph with edge labels coming from the set of reflections. (One could also define the Bruhat graph using multiplication on the right; as graphs, the resulting objects are isomorphic, but the edge labelings are different.)
The strong Bruhat order on the symmetric group (permutations) has Möbius function given by ,
and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset.
See also
Kazhdan–Lusztig polynomial
References
Coxeter groups
Order theory
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https://en.wikipedia.org/wiki/%C3%89tale
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In mathematics, more specifically in algebra, the adjective étale refers to several closely related concepts:
Étale morphism
Formally étale morphism
Étale cohomology
Étale topology
Étale fundamental group
Étale group scheme
Étale algebra
Other
Étale (mountain) in Savoie and Haute-Savoie, France
See also
Étalé space
Etail, or online commerce
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https://en.wikipedia.org/wiki/Crinkill
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Crinkill (), sometimes spelt Crinkle, is a village in County Offaly, Ireland, close to Birr. Crinkill was designated as a census town by the Central Statistics Office for the first time in the 2016 census, at which time it had a population of 682 people.
History
The village originally grew up around a British Army military barracks, Birr Barracks, which was constructed around 1805. However, the barracks was abandoned by the British army around the time of Irish independence, and was burnt down in July 1922 as a result of the civil war that followed. Today only the ruins of the outer wall remain. In 2013, the Regimental Association of the Prince of Wales's Leinster Regiment (Royal Canadians) erected a memorial to commemorate the regiment's strong linkages with the area.
Features
The Thatch, a 200-year-old thatched restaurant and bar in the center of Crinkill, has been in the same family ownership for nearly 200 years. The Thatch has won several awards, including the Offaly Pub of the Year and the All Ireland Pub Of The Year in 1999 and 2001.
Education
Crinkill National School is the local national (primary) school.
Sport
The village is home to Crinkill GAA and Handball Club. Together with neighbouring clubs Carrig and Riverstown the club fields teams at all levels in the name of CRC Gaels. Kevin Breen, a member of the East Tennessee State University cross country team known as the Irish Brigade, is a native of Crinkle. Breen was member of the ETSU team that placed second in the 1972 NCAA Division One cross country championships in Houston, Texas. He competed for Tullamore Harriers and has won Irish National Track championships medals.
References
Towns and villages in County Offaly
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https://en.wikipedia.org/wiki/Sturm%20separation%20theorem
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In mathematics, in the field of ordinary differential equations, Sturm separation theorem, named after Jacques Charles François Sturm, describes the location of roots of solutions of homogeneous second order linear differential equations. Basically the theorem states that given two linear independent solutions of such an equation the zeros of the two solutions are alternating.
Sturm separation theorem
If u(x) and v(x) are two non-trivial continuous linearly independent solutions to a homogeneous second order linear differential equation with x0 and x1 being successive roots of u(x), then v(x) has exactly one root in the open interval (x0, x1). It is a special case of the Sturm-Picone comparison theorem.
Proof
Since and are linearly independent it follows that the Wronskian must satisfy for all where the differential equation is defined, say . Without loss of generality, suppose that . Then
So at
and either and are both positive or both negative. Without loss of generality, suppose that they are both positive. Now, at
and since and are successive zeros of it causes . Thus, to keep we must have . We see this by observing that if then would be increasing (away from the -axis), which would never lead to a zero at . So for a zero to occur at at most (i.e., and it turns out, by our result from the Wronskian that ). So somewhere in the interval the sign of changed. By the Intermediate Value Theorem there exists such that .
On the other hand, there can be only one zero in , because otherwise would have two zeros and there would be no zeros of in between, and it was just proved that this is impossible.
References
Ordinary differential equations
Theorems in analysis
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https://en.wikipedia.org/wiki/Quantile%20function
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In probability and statistics, the quantile function outputs the value of a random variable such that its probability is less than or equal to an input probability value. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function (after the percentile), percent-point function or inverse cumulative distribution function (after the cumulative distribution function).
Definition
Strictly monotonic distribution function
With reference to a continuous and strictly monotonic cumulative distribution function of a random variable X, the quantile function maps its input p to a threshold value x so that the probability of X being less or equal than x is p. In terms of the distribution function F, the quantile function Q returns the value x such that
which can be written as inverse of the c.d.f.
General distribution function
In the general case of distribution functions that are not strictly monotonic and therefore do not permit an inverse c.d.f., the quantile is a (potentially) set valued functional of a distribution function F, given by the interval
It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of F)
Here we capture the fact that the quantile function returns the minimum value of x from amongst all those values whose c.d.f value exceeds p, which is equivalent to the previous probability statement in the special case that the distribution is continuous. Note that the infimum function can be replaced by the minimum function, since the distribution function is right-continuous and weakly monotonically increasing.
The quantile is the unique function satisfying the Galois inequalities
if and only if
If the function F is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have:
In general, even though the distribution function F may fail to possess a left or right inverse, the quantile function Q behaves as an "almost sure left inverse" for the distribution function, in the sense that
almost surely.
Simple example
For example, the cumulative distribution function of Exponential(λ) (i.e. intensity λ and expected value (mean) 1/λ) is
The quantile function for Exponential(λ) is derived by finding the value of Q for which :
for 0 ≤ p < 1. The quartiles are therefore:
first quartile (p = 1/4)
median (p = 2/4)
third quartile (p = 3/4)
Applications
Quantile functions are used in both statistical applications and Monte Carlo methods.
The quantile function is one way of prescribing a probability distribution, and it is an alternative to the probability density function (pdf) or probability mass function, the cumulative distribution function (cdf) and the characteristic function. The quantile function, Q, of a probability distribution is the inverse o
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https://en.wikipedia.org/wiki/Classical%20group
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In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.
The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group is a symmetry group of spacetime of special relativity. The special unitary group is the symmetry group of quantum chromodynamics and the symplectic group finds application in Hamiltonian mechanics and quantum mechanical versions of it.
The classical groups
The classical groups are exactly the general linear groups over and together with the automorphism groups of non-degenerate forms discussed below. These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is not used consistently in the interest of greater generality.
The complex classical groups are , and . A group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups. These are, in turn, , and . One characterization of the compact real form is in terms of the Lie algebra . If , the complexification of , and if the connected group generated by } is compact, then is a compact real form.
The classical groups can uniformly be characterized in a different way using real forms. The classical groups (here with the determinant 1 condition, but this is not necessary) are the following:
The complex linear algebraic groups , and together with their real forms.
For instance, is a real form of , is a real form of , and is a real form of . Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization. The algebraic groups in question are Lie groups, but the "algebraic" qualifier is needed to get the ri
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https://en.wikipedia.org/wiki/Li%20Zitong
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Li Zitong (died 622 CE) was an agrarian leader who claimed the title of emperor in the aftermaths of the death of Emperor Yang of Sui at the hands of the general Yuwen Huaji in 618. After Yuwen vacated the city of Jiangdu (, in modern Yangzhou, Jiangsu), the region was in a state of confusion and, in 619, Li captured Jiangdu and declared a new state of Wu. In 620, he was defeated by the Tang dynasty general Li Fuwei and he headed south, defeating another rebel leader, Shen Faxing King of Liang and seizing Shen's territory (roughly, modern Zhejiang). In 621, however, Li Fuwei attacked him again and forced his surrender. He was taken to the Tang capital at Chang'an but was spared by Emperor Gaozu. In 622, believing that he could try to re-establish his state, he fled from Chang'an. He was captured and executed.
Initial uprising
Li Zitong was from Donghai Commandery (, roughly modern Lianyungang, Jiangsu). He was said to be poor in his youth and supported himself by fishing and hunting. While living in the country, whenever he saw youngster bearing heavy burdens, he would bear the burdens for them. He said also said to be generous with the little he had, but vindictive, repaying every single slight. In or sometime before 615, with agrarian rebels rising against Sui dynasty rule, Li joined the rebel leader Zuo Caixiang (), then at Changbai Mountain (, in modern Binzhou, Shandong). At that time, the agrarian rebel leaders tended to be cruel, but Li was considered kind and tolerant, and therefore many people joined him. Within half a year, he gathered 10,000 men. Zuo began to be jealous and suspicious of him, and Li took his men and left in 615, heading south and crossing the Huai River to join another rebel leader, Du Fuwei. Soon thereafter, for reasons unknown, Li wanted to kill Du and set an ambush for him, but while Du was wounded, he was not killed. Li was then defeated by the Sui general Lai Zheng () and fled to Hailing (, in modern Taizhou, Jiangsu), gathering 20,000 men and claiming the title of general.
Struggle for modern Jiangsu and Zhejiang
Li Zitong's activities for the next several years were not clear, other than that he continued to occupy Hailing. After Emperor Yang was killed in the spring of 618 at Jiangdu (, in modern Yangzhou, Jiangsu) in a coup led by the general Yuwen Huaji, Yuwen then left Jiangdu and Li contended for control of the lower Yangtze River region against Du Fuwei, Shen Faxing the King of Liang, and the former Sui general Chen Leng (). In the fall of 619, Li put Jiangdu, then controlled by Chen, under siege. Chen sought help from Shen and Du; Du led forces to relieve Chen personally while Shen sent his son Shen Guan (). Accepting the suggestion of his official Mao Wenshen (), Li tricked Du and Shen Guan into battling each other by launching an attack on Du with soldiers that pretended to be Shen's troops. Neither was then able to assist Chen, and Li soon captured Jiangdu. He claimed the title of
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https://en.wikipedia.org/wiki/Zero%20dagger
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In set theory, 0† (zero dagger) is a particular subset of the natural numbers, first defined by Robert M. Solovay in unpublished work in the 1960s. (The superscript † should be a dagger, but it appears as a plus sign on some browsers.) The definition is a bit awkward, because there might be no set of natural numbers satisfying the conditions. Specifically, if ZFC is consistent, then ZFC + "0† does not exist" is consistent. ZFC + "0† exists" is not known to be inconsistent (and most set theorists believe that it is consistent). In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows:
0† exists if and only if there exists a non-trivial elementary embedding j : L[U] → L[U] for the relativized Gödel constructible universe L[U], where U is an ultrafilter witnessing that some cardinal κ is measurable.
If 0† exists, then a careful analysis of the embeddings of L[U] into itself reveals that there is a closed unbounded subset of κ, and a closed unbounded proper class of ordinals greater than κ, which together are indiscernible for the structure , and 0† is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in L[U].
Solovay showed that the existence of 0† follows from the existence of two measurable cardinals. It is traditionally considered a large cardinal axiom, although it is not a large cardinal, nor indeed a cardinal at all.
See also
0#: a set of formulas (or subset of the integers) defined in a similar fashion, but simpler.
References
External links
Definition by "Zentralblatt math database" (PDF)
Large cardinals
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https://en.wikipedia.org/wiki/Complete%20homogeneous%20symmetric%20polynomial
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In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials.
Definition
The complete homogeneous symmetric polynomial of degree in variables , written for , is the sum of all monomials of total degree in the variables. Formally,
The formula can also be written as:
Indeed, is just the multiplicity of in the sequence .
The first few of these polynomials are
Thus, for each nonnegative integer , there exists exactly one complete homogeneous symmetric polynomial of degree in variables.
Another way of rewriting the definition is to take summation over all sequences , without condition of ordering :
here is the multiplicity of number in the sequence .
For example
The polynomial ring formed by taking all integral linear combinations of products of the complete homogeneous symmetric polynomials is a commutative ring.
Examples
The following lists the basic (as explained below) complete homogeneous symmetric polynomials for the first three positive values of .
For :
For :
For :
Properties
Generating function
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in :
(this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials). Here each fraction in the final expression is the usual way to represent the formal geometric series that is a factor in the middle expression. The identity can be justified by considering how the product of those geometric series is formed: each factor in the product is obtained by multiplying together one term chosen from each geometric series, and every monomial in the variables is obtained for exactly one such choice of terms, and comes multiplied by a power of equal to the degree of the monomial.
The formula above can be seen as a special case of the MacMahon master theorem. The right hand side can be interpreted as where and . On the left hand side, one can identify the complete homogeneous symmetric polynomials as special cases of the multinomial coefficient that appears in the MacMahon expression.
Performing some standard computations, we can also write the generating function as which is the power series expansion of the plethystic exponential of (and note that is precisely the j-th power sum symmetric polynomial).
Relation with the elementary symmetric polynomials
There is a fundamental relation between the elementary symmetric polynomials and the complete homogeneous ones:
which is valid for all , and any number of variables . The easiest way to see that it holds is from an identity of formal power series in for the elementary symmetric polynomials, analogous to the one given above for the complete homogeneous ones, which can also be written in te
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https://en.wikipedia.org/wiki/Larry%20Cedar
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Larry Frank Cedar (born March 6, 1955) is an American voice, film and television actor, best known as one of the players of the Children's Television Workshop mathematics show Square One TV on PBS from 1987 to 1994. He played Max, Alex the Butcher's assistant, in a series of commercials for Kroger in 1989. He is also known for playing Leon, the opium-addicted thief and faro dealer, in the internationally acclaimed HBO series Deadwood.
Life and career
Cedar's professional acting career did not begin until shortly after his admission to Hastings Law School when, on an impulse, he decided to audition for, and was accepted into the MFA Theater program at UCLA, from which he graduated in 1978. While there, he won the Hugh O'Brian Acting Competition award for Best Actor, resulting in a one-year artist development contract with Universal Studios. He went on to star in various television films, and numerous episodics and feature films, including a starring role opposite Rebecca De Mornay in the Ivan Reitman-produced Feds, and an appearance as The Creature on the Wing, opposite John Lithgow, in the Steven Spielberg remake, Twilight Zone: The Movie, directed by George Miller. He has also won an L.A. Theater Alliance Ovation Award for Best Featured Actor in a Musical. Other actors in Cedar's family include Jon Cedar and George Cedar.
Cedar spent six seasons in New York starring in the award-winning PBS series Square One TV, and later starred in 40 episodes of the Fox television series A.J.'s Time Travelers. A veteran stage performer, he appeared in the one-man play Billy Bishop Goes to War at the Colony Theatre. He has been nominated for two Los Angeles Theater Alliance Ovation awards for his performances in Anything Goes (as Lord Oakley) opposite Rachel York, and in She Loves Me (as Sipos, for which he won Best Featured Actor in a Musical). His other stage work includes portraying Hoagy Carmichael in Hoagy, Bix, and Wolfgang Beethoven Bunkhaus at L.A.'s Mark Taper Forum; as Vernon opposite Lea Thompson in They're Playing Our Song; and as Secretary Thompson in 1776 opposite Roger Rees.
In August 2008, Cedar appeared in Towelhead, the directorial debut of Alan Ball (creator of Six Feet Under). He co-starred opposite Adrien Brody as the demented Chester Sinclair in the Ben Affleck/Diane Lane noir feature film Hollywoodland, directed by Allen Coulter, and recurred for three seasons as Leon, the opium-addicted card dealer and thief, in the David Milch helmed HBO series Deadwood opposite Powers Boothe and Ian McShane. His independent film work includes the award winning short Tel Aviv, the science fiction thriller Forecast, and the full-length horror film Midnight Son. He has also done voice-over work for hundreds of commercials, cartoon series, and video games.
In 2010, Cedar had a role in The Crazies, playing Principal Ben Sandborn. From 2011-2012 he portrayed Cornelius Hawthorne, father of Chevy Chase's character Pierce Hawthorne, on 2 episodes of Communi
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https://en.wikipedia.org/wiki/Quadrature%20domains
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In the branch of mathematics called potential theory, a quadrature domain in two dimensional real Euclidean space is a domain D (an open connected set) together with
a finite subset {z1, …, zk} of D such that, for every function u harmonic and integrable over D with respect to area measure, the integral of u with respect to this measure is given by a "quadrature formula"; that is,
where the cj are nonzero complex constants independent of u.
The most obvious example is when D is a circular disk: here k = 1, z1 is the center of the circle, and c1 equals the area of D. That quadrature formula expresses the mean value property of harmonic functions with respect to disks.
It is known that quadrature domains exist for all values of k. There is an analogous definition of quadrature domains in Euclidean space of dimension d larger than 2. There is also an alternative, electrostatic interpretation of quadrature domains: a domain D is a quadrature domain if a uniform distribution of electric charge on D creates the same electrostatic field outside D as does a k-tuple of point charges at the points z1, …, zk.
Quadrature domains and numerous generalizations thereof (e.g., replace area measure by length measure on the boundary of D) have in recent years been encountered in various connections such as inverse problems of Newtonian gravitation, Hele-Shaw flows of viscous fluids, and purely mathematical isoperimetric problems, and interest in them seems to be steadily growing. They were the subject of an international conference at the University of California at Santa Barbara in 2003 and the state of the art as of that date can be seen in the proceedings of that conference, published by Birkhäuser Verlag.
References
Potential theory
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https://en.wikipedia.org/wiki/Adequate%20pointclass
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In the mathematical field of descriptive set theory, a pointclass can be called adequate if it contains all recursive pointsets and is closed under recursive substitution, bounded universal and existential quantification and preimages by recursive functions.
References
Descriptive set theory
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https://en.wikipedia.org/wiki/Engineering%20and%20Science%20Education%20Program
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The Science, Technology, Engineering and Mathematics Education Program (STEM, formerly Engineering and Science Education Program or ESEP) is a science and mathematics-oriented curriculum devised for high schools in the Philippines. The STEM program is offered by specialized high schools, whether public or private, supervised by the Department of Education. Currently, there are 110 high schools offering the STEM program, the majority being public. It was piloted in 1994 by the Department of Science & Technology (DOST).
Comparison between the STEM, the RSHS Union and the PSHS System
All three types of science high schools in the Philippines (STEM high schools, high schools in the Regional Science High School Union and the Philippine Science High School System) offer a curriculum placing importance in mathematics and the sciences, as well as research. It is noted though that the RSHS Union and the PSHS System have much higher standards of science and mathematics education than STEM high schools. Likewise, STEM high schools and the RSHS Union are operated by Department of Education, while the PSHS system is operated by Department of Science and Technology.
In STEM high schools, transfer students are permitted to enroll provided the student is coming from another STEM high school, from an RSHS or from the PSHS System. In the Regional Science High School Union and the PSHS System, transfers are only allowed within their respective systems for incoming sophomores only. Students who wish to transfer from an STEM high school to the RSHS or PSHS systems will not be admitted, although the reverse is permissible.
All three types of science high school also maintain different grading systems. STEM high schools and the RSHS Union apply the standard grading system for high schools in the Philippines, while the PSHS System maintains a unique grading system using the 1.00-5.00 scale.
Academic Programme
Philippine Science High School (PSHS) Core Curriculum
Electives
Regional Science High School (RSHS) Curriculum
Science, Technology, Engineering and Mathematics Program (STEM) Curriculum
Other programs in other fields
Technical-Vocational Education Program (TVEP)
The Tech-Voc program seeks to provide early training for labor skills, particularly on machine works, trade, agriculture, information technology, among others. The program is offered to graduating high school students and its main purpose is to either prepare them for college or to enable them to work in various industries.
The technical-vocational program has 18 areas of specialization which includes: machine shop, automotive technology, welding, electronics technology, building construction, furniture and cabinet making, plumbing, electricity, computer technology, food processing, animal production, fish processing, fish capture, fish culture, agriculture, PC operations and technical drawing.
Currently, there are 280 Tech-Voc high schools in the Philippines, 140 of which are priority Tech-Voc
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https://en.wikipedia.org/wiki/Power%20sum%20symmetric%20polynomial
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In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the rationals, but not over the integers.
Definition
The power sum symmetric polynomial of degree k in variables x1, ..., xn, written pk for k = 0, 1, 2, ..., is the sum of all kth powers of the variables. Formally,
The first few of these polynomials are
Thus, for each nonnegative integer , there exists exactly one power sum symmetric polynomial of degree in variables.
The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring.
Examples
The following lists the power sum symmetric polynomials of positive degrees up to n for the first three positive values of In every case, is one of the polynomials. The list goes up to degree n because the power sum symmetric polynomials of degrees 1 to n are basic in the sense of the theorem stated below.
For n = 1:
For n = 2:
For n = 3:
Properties
The set of power sum symmetric polynomials of degrees 1, 2, ..., n in n variables generates the ring of symmetric polynomials in n variables. More specifically:
Theorem. The ring of symmetric polynomials with rational coefficients equals the rational polynomial ring The same is true if the coefficients are taken in any field of characteristic 0.
However, this is not true if the coefficients must be integers. For example, for n = 2, the symmetric polynomial
has the expression
which involves fractions. According to the theorem this is the only way to represent in terms of p1 and p2. Therefore, P does not belong to the integral polynomial ring
For another example, the elementary symmetric polynomials ek, expressed as polynomials in the power sum polynomials, do not all have integral coefficients. For instance,
The theorem is also untrue if the field has characteristic different from 0. For example, if the field F has characteristic 2, then , so p1 and p2 cannot generate e2 = x1x2.
Sketch of a partial proof of the theorem: By Newton's identities the power sums are functions of the elementary symmetric polynomials; this is implied by the following recurrence relation, though the explicit function that gives the power sums in terms of the ej is complicated:
Rewriting the same recurrence, one has the elementary symmetric polynomials in terms of the power sums (also implicitly, the explicit formula being complicated):
This implies that the elementary polynomials are rational, though not integral, linear combinations of the power sum polyn
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https://en.wikipedia.org/wiki/Nilpotent%20orbit
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In mathematics, nilpotent orbits are generalizations of nilpotent matrices that play an important role
in representation theory of real and complex semisimple Lie groups and semisimple Lie algebras.
Definition
An element X of a semisimple Lie algebra g is called nilpotent if its adjoint endomorphism
ad X: g → g, ad X(Y) = [X,Y]
is nilpotent, that is, (ad X)n = 0 for large enough n. Equivalently, X is nilpotent if its characteristic polynomial pad X(t) is equal to tdim g.
A semisimple Lie group or algebraic group G acts on its Lie algebra via the adjoint representation, and the property of being nilpotent is invariant under this action. A nilpotent orbit is an orbit of the adjoint action such that any (equivalently, all) of its elements is (are) nilpotent.
Examples
Nilpotent matrices with complex entries form the main motivating case for the general theory, corresponding to the complex general linear group. From the Jordan normal form of matrices we know that each nilpotent matrix is conjugate to a unique matrix with Jordan blocks of sizes where is a partition of n. Thus in the case n=2 there are two nilpotent orbits, the zero orbit consisting of the zero matrix and corresponding to the partition (1,1) and the principal orbit consisting of all non-zero matrices A with zero trace and determinant,
with
corresponding to the partition (2). Geometrically, this orbit is a two-dimensional complex quadratic cone in four-dimensional vector space of matrices minus its apex.
The complex special linear group is a subgroup of the general linear group with the same nilpotent orbits. However, if we replace the complex special linear group with the real special linear group, new nilpotent orbits may arise. In particular, for n=2 there are now 3 nilpotent orbits: the zero orbit and two real half-cones (without the apex), corresponding to positive and negative values of in the parametrization above.
Properties
Nilpotent orbits can be characterized as those orbits of the adjoint action whose Zariski closure contains 0.
Nilpotent orbits are finite in number.
The Zariski closure of a nilpotent orbit is a union of nilpotent orbits.
Jacobson–Morozov theorem: over a field of characteristic zero, any nilpotent element e can be included into an sl2-triple {e,h,f} and all such triples are conjugate by ZG(e), the centralizer of e in G. Together with the representation theory of sl2, this allows one to label nilpotent orbits by finite combinatorial data, giving rise to the Dynkin–Kostant classification of nilpotent orbits.
Poset structure
Nilpotent orbits form a partially ordered set: given two nilpotent orbits, O1 is less than or equal to O2 if O1 is contained in the Zariski closure of O2. This poset has a unique minimal element, zero orbit, and unique
maximal element, the regular nilpotent orbit, but in general, it is not a graded poset.
If the ground field is algebraically closed then the zero orbit is covered by a unique orbit, called the
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https://en.wikipedia.org/wiki/Modular%20Lie%20algebra
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In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic.
The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential map to establish a tight connection between properties of a modular Lie algebra and the corresponding algebraic group.
Although serious study of modular Lie algebras was initiated by Nathan Jacobson in 1950s, their representation theory in the semisimple case was advanced only recently due to the influential Lusztig conjectures, which have been partially proved.
References
Lie algebras
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https://en.wikipedia.org/wiki/Minimal%20prime
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In mathematics, the term minimal prime may refer to
Minimal prime ideal, in commutative algebra
Minimal prime (recreational mathematics), the minimal prime number satisfying some property
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https://en.wikipedia.org/wiki/Anne%20Chu
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"Anne Chu was born in 1959 in New York City. Her parents came from China, and her father was a mathematics professor at Columbia University. When she was in middle school, her family moved to Westchester County, north of the city. She graduated from the Philadelphia College of Art (now the University of the Arts) in 1982 and received an MFA from Columbia University in 1985".
Chu's works, influenced by the combination of eastern and western elements, create a "strong dichotomy between that which is modern and ancient, abstract and figurative, unknown and fantastical". She applies multiple techniques that "unite form, content, and color" in a "seemingly effortless, cohesive manner". Despite being primarily a sculptor, "creating monumental works from wood, ceramic, and papier-mâché", “Chu also makes watercolors and monotypes”. In these mediums she chooses the themes of "landscapes, castles, and knights", creating exotic works that seem abstract but thematically connect her works through figurative elements.
“Ms. Chu was the recipient of the 2001 Penny McCall award and was awarded grants from the Anonymous Was a Woman Foundation and the Joan Mitchell Foundation”. “Her work has been widely exhibited, including at the Dallas Art Museum, the Berkeley Art Museum, and the Indianapolis Museum of Art”.
Artistic style
Despite being a Chinese American, Anne Chu does not identify closely with Chinese culture. Instead, her sculptures "reflect a thorough knowledge of world art, much of it coming from Western sources and years of going to galleries and museums". Chu's focus on art history has resulted in a singular vision, affecting her works which "place a contemporary sensibility in genuine dialogue with the past", giving an "ad hoc, but never excessively informal, sense of the present".
According to Heidi Zuckerman Jacobson, a curator at the Berkeley Museum of Art, Anne Chu explores issues in sculpture and painting by "infusing painting into materials that are themselves used in unexpected ways. By shifting conventional expectations of the appearance of sculpture, the artist allows a reconsideration of the familiar." For example, she uses "traditional Chinese artifacts as a base from which to work". She has remade "T'ang dynasty ceramic funerary figures, sculpted Asian and Western–inspired landscapes, and painted luminous watercolors characterized by a subtle tension between abstraction and figuration".
Chu's work consists largely of bold mixed-media sculptures in "wood, metal, resin, fabric, leather, or porcelain, as well as delicate watercolors and ink-based works on paper". She often combined "figures and animals with elements drawn from folklore; just as often, however, her work could be nearly unrecognizable from one piece to the next. In deliberately remaking herself this way, Chu developed a unique visual language, celebrated in more than 30 solo exhibitions over a 25-year period".
Works and publications
Ballplayer on Horse
In Chu's Ballplayer
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https://en.wikipedia.org/wiki/Gordon%E2%80%93Luecke%20theorem
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In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian.
The theorem is usually stated as "knots are determined by their complements"; however this is slightly ambiguous as it considers two knots to be equivalent if there is a self-homeomorphism taking one knot to the other. Thus mirror images are neglected. Often two knots are considered equivalent if they are isotopic. The correct version in this case is that if two knots have complements which are orientation-preserving homeomorphic, then they are isotopic.
These results follow from the following (also called the Gordon–Luecke theorem): no nontrivial Dehn surgery on a nontrivial knot in the 3-sphere can yield the 3-sphere.
The theorem was proved by Cameron Gordon and John Luecke. Essential ingredients of the proof are their joint work with Marc Culler and Peter Shalen on the cyclic surgery theorem, combinatorial techniques in the style of Litherland, thin position, and Scharlemann cycles.
For link complements, it is not in fact true that links are determined by their complements. For example, JHC Whitehead proved that there are infinitely many links whose complements are all homeomorphic to the Whitehead link. His construction is to twist along a disc spanning an unknotted component (as is the case for either component of the Whitehead link). Another method is to twist along an annulus spanning two components. Gordon proved that for the class of links where these two constructions are not possible there are finitely many links in this class with a given complement.
References
Cameron Gordon and John Luecke, Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.
Cameron Gordon, Links and their complements. Topology and geometry: commemorating SISTAG, 71–82, Contemp. Math., 314, Amer. Math. Soc., Providence, RI, 2002.
Knot theory
3-manifolds
Theorems in topology
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https://en.wikipedia.org/wiki/List%20of%20Nottingham%20Forest%20F.C.%20records%20and%20statistics
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This article contains statistics and records related to Nottingham Forest F.C..
Honours
Football League First Division: 1977–78
FA Cup: 1897–98, 1958–59
Football League Cup: 1977–78, 1978–79, 1988–89, 1989–90
Full Members Cup: 1988–89, 1991–92
FA Charity Shield: 1978
European Cup: 1978–79, 1979–80
European Super Cup: 1979
Source:
Club records
Record win (in all competitions):
14–0, vs. Clapton (away), 1st round FA Cup, 17 January 1891
Record defeat (in all competitions):
1–9, vs. Blackburn Rovers, Division 2, 10 April 1937
Most league points in one season:
94, Division 1, 1997-1998
Most league goals in one season:
101, Division 3, 1950-1951
Player records
Most appearances for the club (in all competitions, as of 2012):
Bob McKinlay: 685
Ian Bowyer: 564
Steve Chettle: 526
Stuart Pearce: 522
John Robertson: 514
Jack Burkitt: 503
Jack Armstrong: 460
Grenville Morris: 460
Geoff Thomas: 431
Viv Anderson: 430
Most goals for the club (in all competitions, as of 2012):
Grenville Morris: 217
Nigel Clough: 131
Wally Ardron: 124
Johnny Dent: 122
Ian Storey-Moore: 118
References
Records and Statistics
Nottingham Forest
Forest
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https://en.wikipedia.org/wiki/The%20Mathematics%20of%20Magic%3A%20The%20Enchanter%20Stories%20of%20L.%20Sprague%20de%20Camp%20and%20Fletcher%20Pratt
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The Mathematics of Magic: The Enchanter Stories of L. Sprague de Camp and Fletcher Pratt is an omnibus collection of seven fantasy stories by American science fiction and fantasy authors L. Sprague de Camp and Fletcher Pratt, gathering material previously published in three volumes as The Incomplete Enchanter (1941), The Castle of Iron (1950), and Wall of Serpents (1960) together with additional material from The Enchanter Reborn (1992) and The Exotic Enchanter (1995). It represents an expansion of the earlier omnibuses The Compleat Enchanter, which contained only the material in the first two volumes, and The Complete Compleat Enchanter, which contained only the material in the first three volumes. The expanded version also differs from the previous omnibuses in its selection of supplementary material. The Mathematics of Magic is the first edition of the authors' Harold Shea series to include every one of their contributions to it in one volume. Contributions to the series of other authors from the collections of the 1990s are omitted.
The collection was edited by Mark L. Olson and first published in hardcover by NESFA Press in February, 2007. The stories in the collection were originally published in magazine form in the May 1940, August 1940 and April 1941 issues of Unknown, the June 1953 issue of Fantasy Fiction, the November 1954 issue of Beyond Fantasy Fiction, the World Fantasy Convention program book for 1990, and the collection The Exotic Enchanter in 1995. De Camp's essay "Fletcher and I" was originally published in The Compleat Enchanter in 1975, and Jerry Pournelle's essay "Arming the Incomplete Enchanter" was originally published in George H. Scithers' fanzine Amra.
Summary
The Harold Shea stories are parallel world tales in which universes where magic works coexist with our own, and in which those based on the mythologies, legends, and literary fantasies of our world and can be reached by aligning one's mind to them by a system of symbolic logic. Psychologist Harold Shea and his colleagues Reed Chalmers, Walter Bayard, and Vaclav Polacek (Votsy), travel to several such worlds, joined in the course of their adventures by Belphebe and Florimel of Faerie, who become the wives of Shea and Chalmers, and Pete Brodsky, a policeman who is accidentally swept up into the chaos. The seven stories collected in The Complete Compleat Enchanter explore the worlds of Norse mythology in "The Roaring Trumpet", Edmund Spenser's The Faerie Queene in "The Mathematics of Magic", Ludovico Ariosto's Orlando Furioso (with a brief stop in Samuel Taylor Coleridge's Kubla Khan) in "The Castle of Iron", the Kalevala in "The Wall of Serpents", Irish mythology in "The Green Magician", L. Frank Baum's land of Oz in "Sir Harold and the Gnome King", and Edgar Rice Burroughs's Barsoom in "Sir Harold of Zodanga".
Contents
"Introduction" (by Christopher Stasheff)
"The Roaring Trumpet" (by L. Sprague de Camp and Fletcher Pratt)
"The Mathematics of Magic" (by L. Spr
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https://en.wikipedia.org/wiki/Amaral%20%28footballer%2C%20born%201983%29
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Carlos Rafael do Amaral or simply Amaral (born 28 November 1983, in Mogi Mirim), is a Brazilian defensive midfielder who last played for Passo Fundo in the Campeonato Gaúcho.
Club statistics
Honours
Brazilian Série C: 2003
Brazilian Cup: 2005
Campeonato Brasileiro Série B: 2009
References
External links
Guardian Stats Centre
globoesporte.globo.com
netvasco.com.br
crvascodagama.com
CBF
ntevasco statistics
1983 births
Living people
Brazilian men's footballers
Paulista Futebol Clube players
Ituano FC players
CR Vasco da Gama players
Grêmio Foot-Ball Porto Alegrense players
Cerezo Osaka players
J1 League players
Expatriate men's footballers in Japan
América Futebol Clube (MG) players
Cruzeiro Esporte Clube players
Botafogo de Futebol e Regatas players
Criciúma Esporte Clube players
Ceará Sporting Club players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Men's association football midfielders
People from Mogi Mirim
Footballers from São Paulo (state)
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https://en.wikipedia.org/wiki/Crystalline%20cohomology
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In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by and developed by .
Crystalline cohomology is partly inspired by the p-adic proof in of part of the Weil conjectures and is closely related to the algebraic version of de Rham cohomology that was introduced by Grothendieck (1963). Roughly speaking, crystalline cohomology of a variety X in characteristic p is the de Rham cohomology of a smooth lift of X to characteristic 0, while de Rham cohomology of X is the crystalline cohomology reduced mod p (after taking into account higher Tors).
The idea of crystalline cohomology, roughly, is to replace the Zariski open sets of a scheme by infinitesimal thickenings of Zariski open sets with divided power structures. The motivation for this is that it can then be calculated by taking a local lifting of a scheme from characteristic p to characteristic 0 and employing an appropriate version of algebraic de Rham cohomology.
Crystalline cohomology only works well for smooth proper schemes. Rigid cohomology extends it to more general schemes.
Applications
For schemes in characteristic p, crystalline cohomology theory can handle questions about p-torsion in cohomology groups better than p-adic étale cohomology. This makes it a natural backdrop for much of the work on p-adic L-functions.
Crystalline cohomology, from the point of view of number theory, fills a gap in the l-adic cohomology information, which occurs exactly where there are 'equal characteristic primes'. Traditionally the preserve of ramification theory, crystalline cohomology converts this situation into Dieudonné module theory, giving an important handle on arithmetic problems. Conjectures with wide scope on making this into formal statements were enunciated by Jean-Marc Fontaine, the resolution of which is called p-adic Hodge theory.
Coefficients
For a variety X over an algebraically closed field of characteristic p > 0, the -adic cohomology groups for any prime number other than p give satisfactory cohomology groups of X, with coefficients in the ring of -adic integers. It is not possible in general to find similar cohomology groups with coefficients in Q (or Z, or Q, or Z) having reasonable properties.
The classic reason (due to Serre) is that if X is a supersingular elliptic curve, then its endomorphism ring is a maximal order in a quaternion algebra B over Q ramified at p and ∞. If X had a cohomology group over Q of the expected dimension 2, then (the opposite algebra of) B would act on this 2-dimensional space over Q, which is impossible since B is ramified at p.
Grothendieck's crystalline cohomology theory gets around this obstruction because it produces modules over the ring of Witt vectors of the ground field. So if the ground field is an algebraic closure of F, its values are modules over the p-adic completion of the maximal unramifi
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https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Zilber%20theorem
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In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space and those of the spaces and . The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.
Statement of the theorem
The theorem can be formulated as follows. Suppose and are topological spaces, Then we have the three chain complexes , , and . (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex , whose differential is, by definition,
for and , the differentials on ,.
Then the theorem says that we have chain maps
such that is the identity and is chain-homotopic to the identity. Moreover, the maps are natural in and . Consequently the two complexes must have the same homology:
Statement in terms of composite maps
The original theorem was proven in terms of acyclic models but more mileage was gotten in a phrasing by Eilenberg and Mac Lane using explicit maps. The standard map they produce is traditionally referred to as the Alexander–Whitney map and the Eilenberg–Zilber map. The maps are natural in both and and inverse up to homotopy: one has
for a homotopy natural in both and such that further, each of , , and is zero. This is what would come to be known as a contraction or a homotopy retract datum.
The coproduct
The diagonal map induces a map of cochain complexes which, followed by the Alexander–Whitney yields a coproduct inducing the standard coproduct on . With respect to these coproducts on
and , the map
,
also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite itself is not a map of coalgebras.
Statement in cohomology
The Alexander–Whitney and Eilenberg–Zilber maps dualize (over any choice of commutative coefficient ring with unity) to a pair of maps
which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy . The coproduct does not dualize straightforwardly, because dualization does not distribute over tensor products of infinitely-generated modules, but there is a natural injection of differential graded algebras given by , the product being taken in the coefficient ring . This induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps
inducing a product in cohomology, known as the cup product, because and are isomorphisms. Replacing with so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by
,
which, since cochain evaluation vanishes unless , reduces to the more familiar expression.
Note that if this direct map of cochain complexes were in fact a map of differential graded algebras, then the cup product would make a commutative graded alge
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https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Moore%20spectral%20sequence
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In mathematics, in the field of algebraic topology, the Eilenberg–Moore spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from knowledge of the homology of the remaining spaces. Samuel Eilenberg and John C. Moore's original paper addresses this for singular homology.
Motivation
Let be a field and let
and
denote singular homology and singular cohomology with coefficients in k, respectively.
Consider the following pullback of a continuous map p:
A frequent question is how the homology of the fiber product, , relates to the homology of B, X and E. For example, if B is a point, then the pullback is just the usual product . In this case the Künneth formula says
However this relation is not true in more general situations. The Eilenberg−Moore spectral sequence is a device which allows the computation of the (co)homology of the fiber product in certain situations.
Statement
The Eilenberg−Moore spectral sequences generalizes the above isomorphism to the situation where p is a fibration of topological spaces and the base B is simply connected. Then there is a convergent spectral sequence with
This is a generalization insofar as the zeroeth Tor functor is just the tensor product and in the above special case the cohomology of the point B is just the coefficient field k (in degree 0).
Dually, we have the following homology spectral sequence:
Indications on the proof
The spectral sequence arises from the study of differential graded objects (chain complexes), not spaces. The following discusses the original homological construction of Eilenberg and Moore. The cohomology case is obtained in a similar manner.
Let
be the singular chain functor with coefficients in . By the Eilenberg–Zilber theorem, has a differential graded coalgebra structure over with
structure maps
In down-to-earth terms, the map assigns to a singular chain s: Δn → B the composition of s and the diagonal inclusion B ⊂ B × B. Similarly, the maps and induce maps of differential graded coalgebras
, .
In the language of comodules, they endow and with differential graded comodule structures over , with structure maps
and similarly for E instead of X. It is now possible to construct the so-called cobar resolution for
as a differential graded comodule. The cobar resolution is a standard technique in differential homological algebra:
where the n-th term is given by
The maps are given by
where is the structure map for as a left comodule.
The cobar resolution is a bicomplex, one degree coming from the grading of the chain complexes S∗(−), the other one is the simplicial degree n. The total complex of the bicomplex is denoted .
The link of the above algebraic construction with the topological situation is as follows. Under the above assumptions, there is a map
that induces a quasi-isomorphism (i.e. inducing an isomorphism on homology groups)
where is the cotensor p
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https://en.wikipedia.org/wiki/Municipality%20of%20the%20District%20of%20Digby
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Digby, officially named the Municipality of the District of Digby, is a district municipality in Digby County, Nova Scotia, Canada. Statistics Canada classifies the district municipality as a municipal district.
The district municipality forms the eastern part of Digby County. It is one of three municipal units in the county, the other two being the Town of Digby and the Municipality of the District of Clare.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, the Municipality of the District of Digby had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Communities
Bear River
Gilberts Cove
Little River, Digby, Nova Scotia
Marshalltown
Jordantown
Conway
Acaciaville
Hillgrove
North Range
Brighton
Barton
Weymouth
Sissiboo Falls
Weymouth Falls
Doucetteville
Ashmore
Morganville
Plympton
Smith's Cove
Culloden
Bayview
Rossway
Sandy Cove
Centreville
Mink Cove
Tiddville
Whale Cove
East Ferry
Tiverton
Central Grove
Freeport
Westport
Seabrook
Access routes
Highways and numbered routes that run through the district municipality, including external routes that start or finish at the municipal boundary:
Highways
Trunk Routes
Collector Routes:
External Routes:
None
See also
List of municipalities in Nova Scotia
References
External links
Communities in Digby County, Nova Scotia
District municipalities in Nova Scotia
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https://en.wikipedia.org/wiki/Municipality%20of%20the%20District%20of%20Guysborough
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Guysborough, officially named the Municipality of the District of Guysborough, is a district municipality in Guysborough County, Nova Scotia, Canada. Statistics Canada classifies the district municipality as a municipal district.
It is home to the Boylston and Salsman Provincial Parks. The parks are located between Boylston and Guysborough.
History
The area was originally called Chedabouctou and was the site of one of a fishing post of Nicolas Denys. In 1682, a permanent settlement was started by Clerbaud Bergier. A group cleared land and spent the winter with the first crops being planted in 1683. Louis-Alexandre des Friches de Meneval landed at Chedabouctou in 1687 when arriving to take up his position as governor of Acadia. The community is named after Sir Guy Carleton.
Geography
Occupying the eastern half of Guysborough County, the district municipality's administrative centre is the community of Guysborough. The district completely surrounds the Town of Mulgrave and it borders the Municipality of the District of St. Mary's to the west, the Municipality of the County of Antigonish to the north and the Strait of Canso to the east.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, the Municipality of the District of Guysborough had a population of living in of its total private dwellings, a change of from its 2016 population of . With a land area of , it had a population density of in 2021.
Economy
Various mining and energy (natural gas) projects have been developed in Guysborough, including around the Southwestern town of Goldboro, Nova Scotia where a $10 Billion liquefied natural gas LNG terminal and pipeline project is being pursued by the Canadian firm Pieridae Energy.
See also
List of municipalities in Nova Scotia
References
External links
Guysborough
Communities in Guysborough County, Nova Scotia
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https://en.wikipedia.org/wiki/Alan%20Weiss%20%28mathematician%29
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Alan Weiss (born December 5, 1955) is an American mathematician, a pioneer in the usage of large deviations theory in performance evaluation and related areas.
Weiss received his B.Sc. in mathematics and physics from Case Western Reserve University taking courses from Lajos Takács and being advised by Arthur J. Lohwater (1976). He received his M.Sc. in mathematics from Courant Institute (1979) and Ph.D. from New York University in 1981; his advisor was S. R. S. Varadhan, and his dissertation was entitled Invariant Measures of Diffusion Processes on Domains with Boundaries. He worked at Bell Labs (1981-2007), before joining MathWorks of Natick. Weiss had appointments with University of Maryland, College Park (1986), Columbia University (1993) and Drew University (2005).
Books
Large Deviations for Performance Analysis (Chapman & Hall, 1995). Coauthored with Adam Shwartz.
Publications
Digital Adaptive Filters: Conditions for Convergence, Rates of Convergence, Effects of Noise and Errors Arising from the Implementation, in IEEE Trans. on Information Theory, 25(6):637-652, 1979. With Debasis Mitra
Allocating Independent Substaks on Parallel Processors, IEEE Trans. Softw. Eng., 11(10):1001-1016, 1985. With Clyde Kruskal.
A Lower Bound for Probabilistic Algorithms for Finite State Machines, in Jnr. Comp. and Systems Sci., 33(1):88-105, 1986. With Albert Greenberg
References
External links
apdoo.org, Weiss's personal site
20th-century American mathematicians
21st-century American mathematicians
Scientists at Bell Labs
Case Western Reserve University alumni
Courant Institute of Mathematical Sciences alumni
1955 births
Living people
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https://en.wikipedia.org/wiki/Spacetime%20algebra
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In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of special relativity and relativistic spacetime.
It is a vector space that allows not only vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or blades (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.
Structure
The spacetime algebra may be built up from an orthogonal basis of one time-like vector and three space-like vectors, , with the multiplication rule
where is the Minkowski metric with signature .
Thus, , , otherwise .
The basis vectors share these properties with the Dirac matrices, but no explicit matrix representation need be used in STA.
This generates a basis of one scalar , four vectors , six bivectors , four pseudovectors and one pseudoscalar , where .
The spacetime algebra also contains a non-trivial sub-algebra containing only the even grade elements, i.e. scalars, bivectors, and pseudoscalars. In the even sub-algebra, scalars and pseudoscalars both commute with all elements, and act like complex numbers. However, the pseudoscalar anticommutes with all odd-grade elements of the spacetime algebra, corresponding to the fact that under parity transformations, vectors and pseudovectors become negated.
Reciprocal frame
Associated with the orthogonal basis is the reciprocal basis for
, satisfying the relation
These reciprocal frame vectors differ only by a sign, with , and for .
A vector may be represented in either upper or lower index coordinates with summation over , according to the Einstein notation, where the coordinates may be extracted by taking dot products with the basis vectors or their reciprocals.
Like in tensor calculus, a change of index position can be achieved using the metric and the use of index gymnastics:
Multivector division
The spacetime algebra is not a division algebra, because it contains idempotent elements and nonzero zero divisors: . These can be interpreted as projectors onto the light-cone and orthogonality relations for such projectors, respectively. But in some cases it is possible to divide one multivector quantity by another, and make sense of the result: so, for example, a directed area divided by a vector in the same plane gives another vector, orthogonal to the first.
Spacetime gradient
The spacetime gradient, like the gradient in a Euclidean space, is defined such that the directional derivative relationship is satisfied:
This requires the
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https://en.wikipedia.org/wiki/UBIGEO
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Ubigeo is the coding system for geographical locations (Spanish: Código Ubicacíon Geográfica) in Peru used by the National Statistics and Computing Institute (Spanish: Instituto Nacional de Estadística e Informática INEI) to code the first-level administrative subdivision: regions (Spanish: regiones, singular: región), the second-level administrative subdivision: provinces (Spanish: provincias, singular: provincia) and the third-level administrative subdivision: districts (Spanish: distritos, singular: distrito). There are 1874 different ubigeos in Peru.
Syntax
The coding system uses two-digit numbers for each level of subdivision. The first level starts numbering at 01 for the Amazonas Region and continues in alphabetical order up to 25 for the Ucayali Region. Additional regions will be added to the end of the list, starting with the first available number.
The second level starts with 0101 for the first province in the Amazonas region: Chachapoyas Province and continues up to 2504 for the last province Purús in the Ucayali Region. The provinces are numbered per region with the first province always being the one in which the regions capital is located. The remaining provinces are coded in alphabetical order. Additional provinces will be added per region to the end of the list, starting with the first available province number.
The third level; starts with 010101 for the first district in the first province in the Amazonas region: Chachapoyas District and continues up to 250401 for the last district in the last province of the Ucayali region: Purús District. The districts are numbered per province with the first district always being the one in which the province’ capital is located. The remaining districts are coded in alphabetical order. Additional districts will be added per province to the end of the list, starting with the first available district number.
Examples
Regions
01 Amazonas Region
02 Ancash Region
03 Apurímac Region
Provinces
0101 Chachapoyas Province in the Amazonas region.
0102 Bagua Province in the Amazonas region.
0103 Bongará Province in the Amazonas region.
0104 Condorcanqui Province in the Amazonas region.
0105 Luya Province in the Amazonas region.
0106 Rodríguez de Mendoza Province in the Amazonas region.
0107 Utcubamba Province in the Amazonas region.
0201 Huaraz Province in the Ancash region.
Districts
010101 Chachapoyas District in the Chachapoyas province.
010102 Asunción District in the Chachapoyas province.
010103 Balsas District in the Chachapoyas province.
010104 Cheto District in the Chachapoyas province.
010105 Chiliquín District in the Chachapoyas province.
Recent additions
See also
Census in Peru
References
External links
INEI website
Geocodes
Geography of Peru
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https://en.wikipedia.org/wiki/Alexander%20Beilinson
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Alexander A. Beilinson (born 1957) is the David and Mary Winton Green University professor at the University of Chicago and works on mathematics. His research has spanned representation theory, algebraic geometry and mathematical physics. In 1999, Beilinson was awarded the Ostrowski Prize with Helmut Hofer.
In 2017, he was elected to the National Academy of Sciences. In 2018, he received the Wolf Prize in Mathematics and in 2020 the Shaw Prize in Mathematics.
Work
In 1978, Beilinson published a paper on coherent sheaves and several problems in linear algebra. His two-page note in the journal Functional Analysis and Its Applications was one of the papers on the study of derived categories of coherent sheaves.
In 1981, Beilinson announced a proof of the Kazhdan–Lusztig conjectures and Jantzen conjectures with Joseph Bernstein. Independent of Beilinson and Bernstein, Brylinski and Kashiwara obtained a proof of the Kazhdan–Lusztig conjectures. However, the proof of Beilinson–Bernstein introduced a method of localization. This established a geometric description of the entire category of representations of the Lie algebra, by "spreading out" representations as geometric objects living on the flag variety. These geometric objects naturally have an intrinsic notion of parallel transport: they are D-modules.
In 1982, Beilinson published his own conjectures about the existence of motivic cohomology groups for schemes, provided as hypercohomology groups of a complex of abelian groups and related to algebraic K-theory by a motivic spectral sequence, analogous to the Atiyah–Hirzebruch spectral sequence in algebraic topology. These conjectures have since been dubbed the Beilinson-Soulé conjectures; they are intertwined with Vladimir Voevodsky's program to develop a homotopy theory for schemes.
In 1984, Beilinson published the paper Higher Regulators and values of L-functions, in which he related higher regulators for K-theory and their relationship to L-functions. The paper also provided a generalization to arithmetic varieties of the Lichtenbaum conjecture for K-groups of number rings, the Hodge conjecture, the Tate conjecture about algebraic cycles, the Birch and Swinnerton-Dyer conjecture about elliptic curves, and Bloch's conjecture about K2 of elliptic curves.
Beilinson continued to work on algebraic K-theory throughout the mid-1980s. He collaborated with Pierre Deligne on the developing a motivic interpretation of Don Zagier's polylogarithm conjectures.
From the early 1990s onwards, Beilinson worked with Vladimir Drinfeld to rebuild the theory of vertex algebras. After some informal circulation, this research was published in 2004 in a form of a monograph on chiral algebras. This has led to new advances in conformal field theory, string theory and the geometric Langlands program. He was elected a Fellow of the American Academy of Arts and Sciences in 2008. He was a visiting scholar at the Institute for Advanced Study in the fall of 1994
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https://en.wikipedia.org/wiki/Robert%20K%C3%A1ntor
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Robert Kántor (born February 25, 1977) is a former professional ice hockey defenceman. He last played in Austria with the Graz 99ers during the 2011–12 season.
Career statistics
External links
Bio from Kometa Brno historical website
1977 births
Czech ice hockey defencemen
Czech expatriate ice hockey players in Russia
Färjestad BK players
Ak Bars Kazan players
HC Dynamo Moscow players
Living people
HC Slovan Bratislava players
Ice hockey people from Brno
Czech expatriate ice hockey players in Slovakia
Czech expatriate ice hockey players in Sweden
Czech expatriate ice hockey players in Finland
Czech expatriate ice hockey players in Germany
Czech expatriate sportspeople in Austria
Expatriate ice hockey players in Austria
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https://en.wikipedia.org/wiki/Geometric%20algebra%20%28disambiguation%29
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In mathematics, a geometric algebra is a specific algebraic structure. The term is also used as a blanket term for the theory of geometric algebras.
Geometric algebra may also refer to:
Algebraic geometry
Algebraic geometry and analytic geometry
Analytic geometry
%C3%89l%C3%A9ments de g%C3%A9om%C3%A9trie alg%C3%A9brique, a 1960-7 book by Alexander Grothendieck
Clifford algebra
Geometric Algebra (book), a 1957 book by Emil Artin
Greek geometric algebra
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https://en.wikipedia.org/wiki/CM-field
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In mathematics, a CM-field is a particular type of number field, so named for a close connection to the theory of complex multiplication. Another name used is J-field.
The abbreviation "CM" was introduced by .
Formal definition
A number field K is a CM-field if it is a quadratic extension K/F where the base field F is totally real but K is totally imaginary. I.e., every embedding of F into lies entirely within , but there is no embedding of K into .
In other words, there is a subfield F of K such that K is generated over F by a single square root of an element, say
β = ,
in such a way that the minimal polynomial of β over the rational number field has all its roots non-real complex numbers. For this α should be chosen totally negative, so that for each embedding σ of into the real number field,
σ(α) < 0.
Properties
One feature of a CM-field is that complex conjugation on induces an automorphism on the field which is independent of its embedding into . In the notation given, it must change the sign of β.
A number field K is a CM-field if and only if it has a "units defect", i.e. if it contains a proper subfield F whose unit group has the same -rank as that of K . In fact, F is the totally real subfield of K mentioned above. This follows from Dirichlet's unit theorem.
Examples
The simplest, and motivating, example of a CM-field is an imaginary quadratic field, for which the totally real subfield is just the field of rationals.
One of the most important examples of a CM-field is the cyclotomic field , which is generated by a primitive nth root of unity. It is a totally imaginary quadratic extension of the totally real field The latter is the fixed field of complex conjugation, and is obtained from it by adjoining a square root of
The union QCM of all CM fields is similar to a CM field except that it has infinite degree. It is a quadratic extension of the union of all totally real fields QR. The absolute Galois group Gal(/QR) is generated (as a closed subgroup) by all elements of order 2 in Gal(/Q), and Gal(/QCM) is a subgroup of index 2. The Galois group Gal(QCM/Q) has a center generated by an element of order 2 (complex conjugation) and the quotient by its center is the group Gal(QR/Q).
If V is a complex abelian variety of dimension n, then any abelian algebra F of endomorphisms of V has rank at most 2n over Z. If it has rank 2n and V is simple then F is an order in a CM-field. Conversely any CM field arises like this from some simple complex abelian variety, unique up to isogeny.
One example of a totally imaginary field which is not CM is the number field defined by the polynomial .
References
Field (mathematics)
Algebraic number theory
Complex numbers
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https://en.wikipedia.org/wiki/Unbiased%20estimation%20of%20standard%20deviation
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In statistics and in particular statistical theory, unbiased estimation of a standard deviation is the calculation from a statistical sample of an estimated value of the standard deviation (a measure of statistical dispersion) of a population of values, in such a way that the expected value of the calculation equals the true value. Except in some important situations, outlined later, the task has little relevance to applications of statistics since its need is avoided by standard procedures, such as the use of significance tests and confidence intervals, or by using Bayesian analysis.
However, for statistical theory, it provides an exemplar problem in the context of estimation theory which is both simple to state and for which results cannot be obtained in closed form. It also provides an example where imposing the requirement for unbiased estimation might be seen as just adding inconvenience, with no real benefit.
Motivation
In statistics, the standard deviation of a population of numbers is often estimated from a random sample drawn from the population. This is the sample standard deviation, which is defined by
where is the sample (formally, realizations from a random variable X) and is the sample mean.
One way of seeing that this is a biased estimator of the standard deviation of the population is to start from the result that s2 is an unbiased estimator for the variance σ2 of the underlying population if that variance exists and the sample values are drawn independently with replacement. The square root is a nonlinear function, and only linear functions commute with taking the expectation. Since the square root is a strictly concave function, it follows from Jensen's inequality that the square root of the sample variance is an underestimate.
The use of n − 1 instead of n in the formula for the sample variance is known as Bessel's correction, which corrects the bias in the estimation of the population variance, and some, but not all of the bias in the estimation of the population standard deviation.
It is not possible to find an estimate of the standard deviation which is unbiased for all population distributions, as the bias depends on the particular distribution. Much of the following relates to estimation assuming a normal distribution.
Bias correction
Results for the normal distribution
When the random variable is normally distributed, a minor correction exists to eliminate the bias. To derive the correction, note that for normally distributed X, Cochran's theorem implies that has a chi square distribution with degrees of freedom and thus its square root, has a chi distribution with degrees of freedom. Consequently, calculating the expectation of this last expression and rearranging constants,
where the correction factor is the scale mean of the chi distribution with degrees of freedom, . This depends on the sample size n, and is given as follows:
where Γ(·) is the gamma function. An unbiased estimator of σ can be obta
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https://en.wikipedia.org/wiki/Algebraic%20character
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An algebraic character is a formal expression attached to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous to the Harish-Chandra character of the representations of semisimple Lie groups.
Definition
Let be a semisimple Lie algebra with a fixed Cartan subalgebra and let the abelian group consist of the (possibly infinite) formal integral linear combinations of , where , the (complex) vector space of weights. Suppose that is a locally-finite weight module. Then the algebraic character of is an element of
defined by the formula:
where the sum is taken over all weight spaces of the module
Example
The algebraic character of the Verma module with the highest weight is given by the formula
with the product taken over the set of positive roots.
Properties
Algebraic characters are defined for locally-finite weight modules and are additive, i.e. the character of a direct sum of modules is the sum of their characters. On the other hand, although one can define multiplication of the formal exponents by the formula and extend it to their finite linear combinations by linearity, this does not make into a ring, because of the possibility of formal infinite sums. Thus the product of algebraic characters is well defined only in restricted situations; for example, for the case of a highest weight module, or a finite-dimensional module. In good situations, the algebraic character is multiplicative, i.e., the character of the tensor product of two weight modules is the product of their characters.
Generalization
Characters also can be defined almost verbatim for weight modules over a Kac–Moody or generalized Kac–Moody Lie algebra.
See also
Algebraic representation
Weyl-Kac character formula
References
Lie algebras
Representation theory of Lie algebras
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