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https://en.wikipedia.org/wiki/Patrick%20Flanagan
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Patrick Flanagan (October 11, 1944 - December 19, 2019) was an American New Age author and inventor.
Flanagan wrote books focused on Egyptian sacred geometry and Pyramidology.
In 1958, at the age of 14, while living in Bellaire, Texas, Flanagan invented the neurophone, an electronic device that claims to transmit sound through the body’s nervous system directly to the brain. It was patented in the United States in 1968 (Patent #3,393,279). The invention earned him a profile in Life magazine, which called him a "unique, mature and inquisitive scientist."
Pyramid power
During the 1970s, Flanagan was a proponent of pyramid power. He wrote several books and promoted it with lectures and seminars. According to Flanagan, pyramids with the exact relative dimensions of Egyptian pyramids act as "an effective resonator of randomly polarized microwave signals which can be converted into electrical energy." One of his first books, Pyramid Power, was featured in the lyrics of The Alan Parsons Project album, Pyramid.
Inventions and discoveries
In 1958, at the age of 13, Flanagan invented a device which he called a Neurophone, which he claimed transmitted sound via the nervous system to the brain.
Bibliography
References
External links
PhiSciences Patrick Flanagan Official site
1944 births
Living people
American inventors
Pyramidologists
Sacred geometry
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https://en.wikipedia.org/wiki/Plotkin%20bound
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In the mathematics of coding theory, the Plotkin bound, named after Morris Plotkin, is a limit (or bound) on the maximum possible number of codewords in binary codes of given length n and given minimum distance d.
Statement of the bound
A code is considered "binary" if the codewords use symbols from the binary alphabet . In particular, if all codewords have a fixed length n,
then the binary code has length n. Equivalently, in this case the codewords can be considered elements of vector space over the finite field . Let be the minimum
distance of , i.e.
where is the Hamming distance between and . The expression represents the maximum number of possible codewords in a binary code of length and minimum distance . The Plotkin bound places a limit on this expression.
Theorem (Plotkin bound):
i) If is even and , then
ii) If is odd and , then
iii) If is even, then
iv) If is odd, then
where denotes the floor function.
Proof of case i
Let be the Hamming distance of and , and be the number of elements in (thus, is equal to ). The bound is proved by bounding the quantity in two different ways.
On the one hand, there are choices for and for each such choice, there are choices for . Since by definition for all and (), it follows that
On the other hand, let be an matrix whose rows are the elements of . Let be the number of zeros contained in the 'th column of . This means that the 'th column contains ones. Each choice of a zero and a one in the same column contributes exactly (because ) to the sum and therefore
The quantity on the right is maximized if and only if holds for all (at this point of the proof we ignore the fact, that the are integers), then
Combining the upper and lower bounds for that we have just derived,
which given that is equivalent to
Since is even, it follows that
This completes the proof of the bound.
See also
Singleton bound
Hamming bound
Elias-Bassalygo bound
Gilbert-Varshamov bound
Johnson bound
Griesmer bound
Diamond code
References
Coding theory
Articles containing proofs
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https://en.wikipedia.org/wiki/Antler%2C%20Saskatchewan
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Antler is a special service area in the Rural Municipality of Antler No. 61, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the community had a population of 40 in the Canada 2016 Census.
The community is approximately 120 km east of the city of Estevan and 3 km from the Manitoba border. Antler was dissolved from village status to become part of the Rural Municipality of Antler No. 61 on December 31, 2013.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Antler had a population of 30 living in 14 of its 17 total private dwellings, a change of from its 2016 population of 40. With a land area of , it had a population density of in 2021.
See also
List of communities in Saskatchewan
Special service area
Block settlement
References
External links
Antler No. 61, Saskatchewan
Designated places in Saskatchewan
Former villages in Saskatchewan
Special service areas in Saskatchewan
Populated places disestablished in 2013
Division No. 1, Saskatchewan
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https://en.wikipedia.org/wiki/Ordered%20probit
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In statistics, ordered probit is a generalization of the widely used probit analysis to the case of more than two outcomes of an ordinal dependent variable (a dependent variable for which the potential values have a natural ordering, as in poor, fair, good, excellent). Similarly, the widely used logit method also has a counterpart ordered logit. Ordered probit, like ordered logit, is a particular method of ordinal regression.
For example, in clinical research, the effect a drug may have on a patient may be modeled with ordered probit regression. Independent variables may include the use or non-use of the drug as well as control variables such as age and details from medical history such as whether the patient suffers from high blood pressure, heart disease, etc. The dependent variable would be ranked from the following list: complete cure, relieve symptoms, no effect, deteriorate condition, death.
Another example application are Likert-type items commonly employed in survey research, where respondents rate their agreement on an ordered scale (e.g., "Strongly disagree" to "Strongly agree"). The ordered probit model provides an appropriate fit to these data, preserving the ordering of response options while making no assumptions of the interval distances between options.
Conceptual underpinnings
Suppose the underlying relationship to be characterized is
,
where is the exact but unobserved dependent variable (perhaps the exact level of improvement by the patient); is the vector of independent variables, and is the vector of regression coefficients which we wish to estimate. Further suppose that while we cannot observe , we instead can only observe the categories of response:
Then the ordered probit technique will use the observations on , which are a form of censored data on , to fit the parameter vector .
Estimation
The model cannot be consistently estimated using ordinary least squares; it is usually estimated using maximum likelihood. For details on how the equation is estimated, see the article Ordinal regression.
References
Further reading
Categorical regression models
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https://en.wikipedia.org/wiki/Johnson%20bound
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In applied mathematics, the Johnson bound (named after Selmer Martin Johnson) is a limit on the size of error-correcting codes, as used in coding theory for data transmission or communications.
Definition
Let be a q-ary code of length , i.e. a subset of . Let be the minimum distance of , i.e.
where is the Hamming distance between and .
Let be the set of all q-ary codes with length and minimum distance and let denote the set of codes in such that every element has exactly nonzero entries.
Denote by the number of elements in . Then, we define to be the largest size of a code with length and minimum distance :
Similarly, we define to be the largest size of a code in :
Theorem 1 (Johnson bound for ):
If ,
If ,
Theorem 2 (Johnson bound for ):
(i) If
(ii) If , then define the variable as follows. If is even, then define through the relation ; if is odd, define through the relation . Let . Then,
where is the floor function.
Remark: Plugging the bound of Theorem 2 into the bound of Theorem 1 produces a numerical upper bound on .
See also
Singleton bound
Hamming bound
Plotkin bound
Elias Bassalygo bound
Gilbert–Varshamov bound
Griesmer bound
References
Coding theory
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https://en.wikipedia.org/wiki/International%20Federation%20of%20Football%20History%20%26%20Statistics
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The International Federation of Football History & Statistics (IFFHS) is an organisation that chronicles the history and records of association football. It was founded in 1984 by Alfredo Pöge in Leipzig. The IFFHS was based in Abu Dhabi for some time but, in 2010, relocated to Bonn, Germany, and then in 2014 to Zürich.
From its early stages to 2002, the IFFHS concentrated on publishing the quarterly magazines Fußball-Weltzeitschrift, Libero spezial deutsch and Libero international. When these had to be discontinued for reasons which were not officially told, the organisation published its material in a series of multi-lingual books in co-operation with sponsors. The statistical organisation has now confined its publishing activities to its website. IFFHS has no affiliation with FIFA, but FIFA has cited awards and records conducted by IFFHS on their website.
In 2008, Karl Lennartz, a sports historian and professor at the University of Cologne, Germany, called the organisation "obscure", describing it as a one-man show of its founder Alfredo Pöge. IFFHS rankings and their significance have been a matter of criticism and the largest German news agency, Deutsche Presse-Agentur, refuse to publish them. Furthermore, German die Tageszeitung stated that the IFFHS rankings serves merely for publicity, although Bild, Deutsche Welle, Kicker-Sportmagazin, German Football Association (DFB), and former president of the Association of West German Sports Journalists ( — VWS) Heribert Faßbender have referenced IFFHS.
The World's Best Club
Since 1991, the entity has produced a monthly Club World Ranking.
The ranking takes into consideration the results of twelve months of continental and intercontinental competitions, national league matches (including play-offs) and the most important national cup (excluding points won before the round of 16).
All countries are rated at four levels based upon the national league performance—clubs in the highest level leagues receive 4 points for each match won, 2 for a draw and 0 for a defeat. Level 2 is assigned 3 pts. (win), 1.5 (draw) and 0 (lost), and so on with the next lower levels.
In continental competitions, all clubs receive the same number of points at all stages regardless of the performance level of their leagues. However, the UEFA Champions League and the Copa Libertadores yield more points than UEFA Europa League and Copa Sudamericana, respectively. The point assignment system is still lower for the AFC, CAF, CONCACAF and OFC continental tournaments. Competitions between two continents are evaluated depending upon their importance. Competitions not organised by a continental confederation, or any intercontinental events not recognized by FIFA, are not taken into consideration.
Men's winners
Continental Men's Clubs of the Century (1901–2000)
In 2009, the IFFHS released the results of a statistical study series which determined the best continental clubs of the 20th century. The ranking did not consider
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https://en.wikipedia.org/wiki/Splitting
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Splitting may refer to:
Splitting (psychology)
Lumpers and splitters, in classification or taxonomy
Wood splitting
Tongue splitting
Splitting, railway operation
Mathematics
Heegaard splitting
Splitting field
Splitting principle
Splitting theorem
Splitting lemma
for the numerical method to solve differential equations, see Symplectic integrator
See also
Split (disambiguation)
Splitter (disambiguation)
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https://en.wikipedia.org/wiki/Square-free%20polynomial
|
In mathematics, a square-free polynomial is a polynomial defined over a field (or more generally, an integral domain) that does not have as a divisor any square of a non-constant polynomial. A univariate polynomial is square free if and only if it has no multiple root in an algebraically closed field containing its coefficients. This motivates that, in applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots.
In the case of univariate polynomials, the product rule implies that, if divides , then divides the formal derivative of . The converse is also true and hence, is square-free if and only if is a greatest common divisor of the polynomial and its derivative.
A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free polynomials
where those of the that are non-constant are pairwise coprime square-free polynomials (here, two polynomials are said coprime is their greatest common divisor is a constant; in other words that is the coprimality over the field of fractions of the coefficients that is considered). Every non-zero polynomial admits a square-free factorization, which is unique up to the multiplication and division of the factors by non-zero constants. The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, as for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms that are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra.
Over a field of characteristic 0, the quotient of by its GCD with its derivative is the product of the in the above square-free decomposition. Over a perfect field of non-zero characteristic , this quotient is the product of the such that is not a multiple of . Further GCD computations and exact divisions allow computing the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a better algorithm is known, Yun's algorithm, which is described below. Its computational complexity is, at most, twice that of the GCD computation of the input polynomial and its derivative. More precisely, if is the time needed to compute the GCD of two polynomials of degree and the quotient of these polynomial by the GCD, then is an upper bound for the time needed to compute the square free decomposition.
There are also known algorithms for the computation of the square-free decomposition of multivariate polynomials, that proceed generally by considering a multivariate polynomial as a univariate polynomial with polynomial coefficients, and applying recursively a univariate algorithm.
Yun's algorithm
This section describes Yun'
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https://en.wikipedia.org/wiki/Charles%20Wheelan
|
Charles J. Wheelan (born 1966) is an American professor, journalist, speaker, and is the founder and co-chairman of Unite America. Wheelan is the author of Naked Statistics, Naked Economics, and Naked Money. He was an unsuccessful Democratic candidate in the special election for Illinois's 5th congressional district, the seat vacated by Rahm Emanuel.
Journalist and author
Wheelan graduated from Dartmouth College in 1988; he was a member of Alpha Delta fraternity. From 1997 to 2002, he was the Midwest correspondent for The Economist. He has also written for the Chicago Tribune, The New York Times, The Wall Street Journal and Yahoo! Finance.
Charles Wheelan is a senior lecturer and policy fellow at the Rockefeller Center at Dartmouth College.
Wheelan is a regular contributor to the Motley Fool Radio Show on National Public Radio and to the Eight Forty-Eight program on WBEZ, Chicago Public Radio.
Wheelan's first book, Naked Economics (2002), is an introduction to economics for lay readers; Naked Statistics (2013) is an introduction to statistics. The Centrist Manifesto (2013) attempts to articulate a centrism that is more than a set of compromises between the political extremes, a perspective Wheelan elsewhere characterizes as radical centrist.
Works
Books
Naked Economics: Undressing the Dismal Science, W.W. Norton, 2002.
Revealing Chicago: An Aerial Portrait, Harry N. Abrams, Inc., 2005.
Introduction to Public Policy, W.W. Norton, 2010.
10 1/2 Things No Commencement Speaker Has Ever Said, W. W. Norton, 2012.
The Centrist Manifesto, W.W. Norton, 2013.
Naked Statistics: Stripping the Dread from the Data, W.W. Norton, 2013.
Naked Money: A Revealing Look at What It Is and Why It Matters, W.W. Norton, 2016.
References
External links
Campaign contributions at OpenSecrets.org
Radical centrist writers
Living people
1966 births
Dartmouth College faculty
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https://en.wikipedia.org/wiki/Friendly%20number
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In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; n numbers with the same "abundancy" form a friendly n-tuple.
Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually "friendly numbers".
A number that is not part of any friendly pair is called solitary.
The "abundancy" index of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a "friendly number" if there exists m ≠ n such that σ(m) / m = σ(n) / n. "Abundancy" is not the same as abundance, which is defined as σ(n) − 2n.
"Abundancy" may also be expressed as where denotes a divisor function with equal to the sum of the k-th powers of the divisors of n.
The numbers 1 through 5 are all solitary. The smallest "friendly number" is 6, forming for example, the "friendly" pair 6 and 28 with "abundancy" σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases. Numbers with "abundancy" 2 are also known as perfect numbers. There are several unsolved problems related to the "friendly numbers".
In spite of the similarity in name, there is no specific relationship between the friendly numbers and the amicable numbers or the sociable numbers, although the definitions of the latter two also involve the divisor function.
Examples
As another example, 30 and 140 form a friendly pair, because 30 and 140 have the same "abundancy":
The numbers 2480, 6200 and 40640 are also members of this club, as they each have an "abundancy" equal to 12/5.
For an example of odd numbers being friendly, consider 135 and 819 ("abundancy" 16/9 (deficient)). There are also cases of even being "friendly" to odd, such as 42 and 544635 ("abundancy" 16/7). The odd "friend" may be less than the even one, as in 84729645 and 155315394 ("abundancy" 896/351).
A square number can be friendly, for instance both 693479556 (the square of 26334) and 8640 have "abundancy" 127/36 (this example is accredited to Dean Hickerson).
Status for small n
In the table below, blue numbers are proven friendly , red numbers are proven solitary , numbers n such that n and are coprime are left uncolored, though they are known to be solitary. Other numbers have unknown status and are yellow.
Solitary numbers
A number that belongs to a singleton club, because no other number is "friendly" with it, is a solitary number. All prime numbers are known to be solitary, as are powers of prime numbers. More generally, if the numbers n and σ(n) are coprime – meaning that the greatest common divisor of these numbers is 1, so that σ(n)/n is an irreducible fraction – then the number n is solitary . For a prime number p we have σ(p) = p + 1, which is
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https://en.wikipedia.org/wiki/Influence%20function
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In mathematics, influence function is used to mean either:
a synonym for a Green's function;
Influence function (statistics), the effect on an estimator of changing one point of the sample
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https://en.wikipedia.org/wiki/Richard%20M.%20Dudley
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Richard Mansfield Dudley (July 28, 1938 – January 19, 2020) was Professor of Mathematics at the Massachusetts Institute of Technology.
Education and career
Dudley was born in Cleveland, Ohio. He earned his BA at Harvard College and received his PhD at Princeton University in 1962 under the supervision of Edward Nelson and Gilbert Hunt. He was a Putnam Fellow in 1958. He was an instructor and assistant professor at University of California, Berkeley between 1962 and 1967, before moving to MIT as a professor in mathematics, where he stayed from 1967 until 2015, when he retired.
He died on January 19, 2020, following a long illness.
Research
His work mainly concerned fields of probability, mathematical statistics, and machine learning, with highly influential contributions to the theory of Gaussian processes and empirical processes. He published over a hundred papers in peer-reviewed journals and authored several books. His specialty was probability theory and statistics, especially empirical processes. He is often noted for his results on the so-called Dudley entropy integral. In 2012 he became a fellow of the American Mathematical Society.
Books
References
R. S. Wenocur and R. M. Dudley, "Some special Vapnik–Chervonenkis classes," Discrete Mathematics, vol. 33, pp. 313–318, 1981.
External links
Publications from Google Scholar.
A Conversation with Dick Dudley
1938 births
2020 deaths
20th-century American mathematicians
21st-century American mathematicians
American statisticians
Probability theorists
Princeton University alumni
Massachusetts Institute of Technology School of Science faculty
Fellows of the American Mathematical Society
Fellows of the American Statistical Association
Putnam Fellows
Harvard College alumni
Annals of Probability editors
Mathematical statisticians
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https://en.wikipedia.org/wiki/Cake%20number
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In mathematics, the cake number, denoted by Cn, is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence.
The values of Cn for are given by .
General formula
If n! denotes the factorial, and we denote the binomial coefficients by
and we assume that n planes are available to partition the cube, then the n-th cake number is:
Properties
The only cake number which is prime is 2, since it requires to have prime factorisation where is some prime. This is impossible for as we know must be even, so it must be equal to , , , or , which correspond to the cases: (which has only complex roots), (i.e. ), , and .
The cake numbers are the 3-dimensional analogue of the 2-dimensional lazy caterer's sequence. The difference between successive cake numbers also gives the lazy caterer's sequence.
The fourth column of Bernoulli's triangle (k = 3) gives the cake numbers for n cuts, where n ≥ 3.
The sequence can be alternatively derived from the sum of up to the first 4 terms of each row of Pascal's triangle:
{| class="wikitable" style="text-align:right;"
! !! 0 !! 1 !! 2 !! 3
! rowspan="11" style="padding:0;"| !! Sum
|-
! style="text-align:left;"|1
| 1 || — || — || — || 1
|-
! style="text-align:left;"|2
| 1 || 1 || — || — || 2
|-
! style="text-align:left;"|3
| 1 || 2 || 1 || — || 4
|-
! style="text-align:left;"|4
| 1 || 3 || 3 || 1 || 8
|-
! style="text-align:left;"|5
| 1 || 4 || 6 || 4 || 15
|-
! style="text-align:left;"|6
| 1 || 5 || 10 || 10 || 26
|-
! style="text-align:left;"|7
| 1 || 6 || 15 || 20 || 42
|-
! style="text-align:left;"|8
| 1 || 7 || 21 || 35 || 64
|-
! style="text-align:left;"|9
| 1 || 8 || 28 || 56 || 93
|-
! style="text-align:left;"|10
| 1 || 9 || 36 || 84 || 130
|}
References
External links
Mathematical optimization
Integer sequences
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https://en.wikipedia.org/wiki/Block%20walking
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In combinatorial mathematics, block walking is a method useful in thinking about sums of combinations graphically as "walks" on Pascal's triangle. As the name suggests, block walking problems involve counting the number of ways an individual can walk from one corner A of a city block to another corner B of another city block given restrictions on the number of blocks the person may walk, the directions the person may travel, the distance from A to B, et cetera.
An example block walking problem
Suppose such an individual, say "Fred", must walk exactly k blocks to get to a point B that is exactly k blocks from A. It is convenient to regard Fred's starting point A as the origin, , of a rectangular array of lattice points and B as some lattice point , e units "East" and n units "North" of A, where and both and are nonnegative.
Solution by brute force
A "brute force" solution to this problem may be obtained by systematically counting the number of ways Fred can reach each point where
and
without backtracking (i.e. only traveling North or East from one point to another) until a pattern is observed. For example, the number of ways Fred could go from to or is exactly one; to is two; to or is one; to or is three; and so on. Actually, you could receive the number of ways to get to a particular point by adding up the number of ways you can get to the point south of it and the number of ways you can get to the point west of it.(With the starting point being zero and all the points directly north and south of it one.) In general, one soon discovers that the number of paths from A to any such X corresponds to an entry of Pascal's Triangle.
Combinatorial solution
Since the problem involves counting a finite, discrete number of paths between lattice points, it is reasonable to assume a combinatorial solution exists to the problem. Towards this end, we note that for Fred to still be on a path that will take him from A to B over blocks, at any point X he must either travel along one of the unit vectors and . For the sake of clarity, let and . Given the coordinates of B, regardless of the path Fred travels he must walk along the vectors E and N exactly and times, respectively. As such, the problem reduces to finding the number of distinct rearrangements of the word
,
which is equivalent to finding the number of ways to choose indistinct objects from a group of . Thus the total number of paths Fred could take from A to B traveling only blocks is
Other problems with known block walking combinatorial proofs
Proving that
can be done with a straightforward application of block walking.
See also
Lattice path
References
Combinatorics
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https://en.wikipedia.org/wiki/Glivenko%E2%80%93Cantelli%20theorem
|
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows.
The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.
Statement
Assume that are independent and identically distributed random variables in with common cumulative distribution function . The empirical distribution function for is defined by
where is the indicator function of the set For every (fixed) is a sequence of random variables which converge to almost surely by the strong law of large numbers. Glivenko and Cantelli strengthened this result by proving uniform convergence of to
Theorem
almost surely.
This theorem originates with Valery Glivenko and Francesco Cantelli, in 1933.
Remarks
If is a stationary ergodic process, then converges almost surely to The Glivenko–Cantelli theorem gives a stronger mode of convergence than this in the iid case.
An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm. See asymptotic properties of the empirical distribution function for this and related results.
Proof
For simplicity, consider a case of continuous random variable . Fix such that for . Now for all there exists such that . Note that
Therefore,
Since by strong law of large numbers, we can guarantee that for any positive and any integer such that , we can find such that for all , we have . Combined with the above result, this further implies that , which is the definition of almost sure convergence.
Empirical measures
One can generalize the empirical distribution function by replacing the set by an arbitrary set C from a class of sets to obtain an empirical measure indexed by sets
Where is the indicator function of each set .
Further generalization is the map induced by on measurable real-valued functions f, which is given by
Then it becomes an important property of these classes whether the strong law of large numbers holds uniformly on or .
Glivenko–Cantelli class
Consider a set with a sigma algebra of Borel subsets and a probability measure For a class of subsets,
and a class of functions
define random variables
where is the empirical measure, is the corresponding map, and
assuming that it exists.
Definitions
A class is called a Glivenko–Cantelli class (or GC class) with respect to a probability measure if any of the following equivalent statements is true.
1. almost sur
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https://en.wikipedia.org/wiki/Donsker%27s%20theorem
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In probability theory, Donsker's theorem (also known as Donsker's invariance principle, or the functional central limit theorem), named after Monroe D. Donsker, is a functional extension of the central limit theorem.
Let be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Let . The stochastic process is known as a random walk. Define the diffusively rescaled random walk (partial-sum process) by
The central limit theorem asserts that converges in distribution to a standard Gaussian random variable as . Donsker's invariance principle extends this convergence to the whole function . More precisely, in its modern form, Donsker's invariance principle states that: As random variables taking values in the Skorokhod space , the random function converges in distribution to a standard Brownian motion as
Formal statement
Let Fn be the empirical distribution function of the sequence of i.i.d. random variables with distribution function F. Define the centered and scaled version of Fn by
indexed by x ∈ R. By the classical central limit theorem, for fixed x, the random variable Gn(x) converges in distribution to a Gaussian (normal) random variable G(x) with zero mean and variance F(x)(1 − F(x)) as the sample size n grows.
Theorem (Donsker, Skorokhod, Kolmogorov) The sequence of Gn(x), as random elements of the Skorokhod space , converges in distribution to a Gaussian process G with zero mean and covariance given by
The process G(x) can be written as B(F(x)) where B is a standard Brownian bridge on the unit interval.
History and related results
Kolmogorov (1933) showed that when F is continuous, the supremum and supremum of absolute value, converges in distribution to the laws of the same functionals of the Brownian bridge B(t), see the Kolmogorov–Smirnov test. In 1949 Doob asked whether the convergence in distribution held for more general functionals, thus formulating a problem of weak convergence of random functions in a suitable function space.
In 1952 Donsker stated and proved (not quite correctly) a general extension for the Doob–Kolmogorov heuristic approach. In the original paper, Donsker proved that the convergence in law of Gn to the Brownian bridge holds for Uniform[0,1] distributions with respect to uniform convergence in t over the interval [0,1].
However Donsker's formulation was not quite correct because of the problem of measurability of the functionals of discontinuous processes. In 1956 Skorokhod and Kolmogorov defined a separable metric d, called the Skorokhod metric, on the space of càdlàg functions on [0,1], such that convergence for d to a continuous function is equivalent to convergence for the sup norm, and showed that Gn converges in law in to the Brownian bridge.
Later Dudley reformulated Donsker's result to avoid the problem of measurability and the need of the Skorokhod metric. One can prove that there exist Xi, iid uniform in [0,1] and a sequ
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https://en.wikipedia.org/wiki/Genstat
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Genstat (General Statistics) is a statistical software package with data analysis capabilities, particularly in the field of agriculture.
It was developed in 1968 by the Rothamsted Research in the United Kingdom and was designed to provide modular design, linear mixed models and graphical functions. It was developed and distributed by VSN International (VSNi), which was owned by The Numerical Algorithms Group and Rothamsted Research.
Genstat is used in a number of research areas, including plant science, forestry, animal science, and medicine.
See also
ASReml: a statistical package which fits linear mixed models to large data sets with complex variance models, using Residual Maximum Likelihood (REML)
References
Further reading
External links
Genstat homepage. VSN International (VSNi).
Fortran software
Statistical software
Windows-only proprietary software
Biostatistics
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https://en.wikipedia.org/wiki/Claudia%20Zaslavsky
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Claudia Zaslavsky (January 12, 1917 – January 13, 2006) was an American mathematics teacher and ethnomathematician.
Life
She was born Claudia Natoma Cohen (later changed to Cogan) on January 12, 1917, in Upper Manhattan in New York City and grew up in Allentown, Pennsylvania. She attributed her first interest in mathematics to her early childhood experiences when she helped her parents in their dry goods store.
She studied mathematics at Hunter College and then earned a master's degree in statistics at the University of Michigan.
In the 1950's while raising her children she was the bookkeeper at Chelsea Publishing Co. and taught pre-instrument classes to small children.
Math teacher
She became a mathematics teacher at Woodlands High School in Hartsdale, New York.
She pursued postgraduate study in mathematics education at Teachers College, Columbia University, in 1974–1978.
During that time she sought to learn about mathematics in Africa to better capture the interest of the African-American students in her classes.
She discovered "that little of what was known about this topic [African cultural mathematics] was available in accessible sources." Thus began a years-long project of assembling, organizing and interpreting a vast amount of little-known material on expressions of mathematics in diverse African cultures, including number words and signs, reckoning of time, games, and architectural and decorative patterns. Her field work on a trip to East Africa in 1970 was assisted by the photography of her husband Sam and travel guidance from her son Alan, then teaching in Kenya.
Zaslavsky wrote the book Africa Counts about mathematics in African culture to sum up her discoveries up to that time.
Her work was welcomed into the burgeoning field of ethnomathematics, which studies the ways in which mathematical concepts are expressed and used by people in diverse cultures in the course of everyday life. As she wrote, "scholars of ethnomathematics examine the practice of mathematics from an anthropological point of view."
Zaslavsky was a lifelong activist for civil rights, peace and social justice. She also mentored many new scholars and activists in the field of ethnomathematics, always remembering the importance of discovering and recognizing the mathematical accomplishments of groups currently underrepresented, including women. As a Jew, Zaslavsky had experienced her own struggles with discrimination against women and Jewish people during her formative years in the 1930's and 1940's.
Personal life
One of her children, Alan Zaslavsky, became a teacher in Kenya, a progressive activist, and later a statistician. The other, Thomas Zaslavsky, became a mathematician.
Zaslavsky died of pancreatic cancer in Harlem, New York, on January 13, 2006, survived by her husband Sam and their two sons.
Books
Zaslavsky's books include:
Africa Counts: Number and Pattern in African Cultures (Prindle, Weber, and Schmidt, 1973; 3rd ed., Chicago Review Pres
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https://en.wikipedia.org/wiki/Abhyankar%27s%20conjecture
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In abstract algebra, Abhyankar's conjecture is a conjecture of Shreeram Abhyankar posed in 1957, on the Galois groups of algebraic function fields of characteristic p. The soluble case was solved by Serre in 1990 and the full conjecture was proved in 1994 by work of Michel Raynaud and David Harbater.
Statement
The problem involves a finite group G, a prime number p, and the function field K(C) of a nonsingular integral algebraic curve C defined over an algebraically closed field K of characteristic p.
The question addresses the existence of a Galois extension L of K(C), with G as Galois group, and with specified ramification. From a geometric point of view, L corresponds to another curve , together with
a morphism
π : → C.
Geometrically, the assertion that π is ramified at a finite set S of points on C
means that π restricted to the complement of S in C is an étale morphism.
This is in analogy with the case of Riemann surfaces.
In Abhyankar's conjecture, S is fixed, and the question is what G can be. This is therefore a special type of inverse Galois problem.
Results
The subgroup p(G) is defined to be the subgroup generated by all the Sylow subgroups of G for the prime number p. This is a normal subgroup, and the parameter n is defined as the minimum number of generators of
G/p(G).
Raynaud proved the case where C is the projective line over K, the conjecture states that G can be realised as a Galois group of L, unramified outside S containing s + 1 points, if and only if
n ≤ s.
The general case was proved by Harbater, in which g isthe genus of C and G can be realised if and only if
n ≤ s + 2 g.
References
External links
A layman's perspective of Abhyankar's conjecture from Purdue University
Algebraic curves
Galois theory
Theorems in abstract algebra
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Formal%20derivative
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In mathematics, the formal derivative is an operation on elements of a polynomial ring or a ring of formal power series that mimics the form of the derivative from calculus. Though they appear similar, the algebraic advantage of a formal derivative is that it does not rely on the notion of a limit, which is in general impossible to define for a ring. Many of the properties of the derivative are true of the formal derivative, but some, especially those that make numerical statements, are not.
Formal differentiation is used in algebra to test for multiple roots of a polynomial.
Definition
Fix a ring (not necessarily commutative) and let be the ring of polynomials over .
(If is not commutative, this is the Free algebra over a single indeterminate variable.)
Then the formal derivative is an operation on elements of , where if
then its formal derivative is
In the above definition, for any nonnegative integer and , is defined as usual in a Ring: (with if ).
This definition also works even if does not have a multiplicative identity.
Alternative axiomatic definition
One may also define the formal derivative axiomatically as the map satisfying the following properties.
1) for all
2) The normalization axiom,
3) The map commutes with the addition operation in the polynomial ring,
4) The map satisfies Leibniz's law with respect to the polynomial ring's multiplication operation,
One may prove that this axiomatic definition yields a well-defined map respecting all of the usual ring axioms.
The formula above (i.e. the definition of the formal derivative when the coefficient ring is commutative) is a direct consequence of the aforementioned axioms:
Properties
It can be verified that:
Formal differentiation is linear: for any two polynomials f(x),g(x) in R[x] and elements r,s of R we have
The formal derivative satisfies the Product rule:
Note the order of the factors; when R is not commutative this is important.
These two properties make D a derivation on A (see module of relative differential forms for a discussion of a generalization).
Note that the formal derivative is not a Ring homomorphism, because the product rule is different from saying (and it is not the case) that . However, it is a homomorphism (linear map) of R-modules, by the above rules.
Application to finding repeated factors
As in calculus, the derivative detects multiple roots. If R is a field then R[x] is a Euclidean domain, and in this situation we can define multiplicity of roots; for every polynomial f(x) in R[x] and every element r of R, there exists a nonnegative integer mr and a polynomial g(x) such that
where g(r)≠0. mr is the multiplicity of r as a root of f. It follows from the Leibniz rule that in this situation, mr is also the number of differentiations that must be performed on f(x) before r is no longer a root of the resulting polynomial. The utility of this observation is that although in general not every polynomial of degree n in R[x
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https://en.wikipedia.org/wiki/Comparability
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In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true. They are called incomparable if they are not comparable.
Rigorous definition
A binary relation on a set is by definition any subset of Given is written if and only if in which case is said to be to by
An element is said to be , or (), to an element if or
Often, a symbol indicating comparison, such as (or and many others) is used instead of in which case is written in place of which is why the term "comparable" is used.
Comparability with respect to induces a canonical binary relation on ; specifically, the induced by is defined to be the set of all pairs such that is comparable to ; that is, such that at least one of and is true.
Similarly, the on induced by is defined to be the set of all pairs such that is incomparable to that is, such that neither nor is true.
If the symbol is used in place of then comparability with respect to is sometimes denoted by the symbol , and incomparability by the symbol .
Thus, for any two elements and of a partially ordered set, exactly one of and is true.
Example
A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes comparable.
Properties
Both of the relations and are symmetric, that is is comparable to if and only if is comparable to and likewise for incomparability.
Comparability graphs
The comparability graph of a partially ordered set has as vertices the elements of and has as edges precisely those pairs of elements for which .
Classification
When classifying mathematical objects (e.g., topological spaces), two are said to be comparable when the objects that obey one criterion constitute a subset of the objects that obey the other, which is to say when they are comparable under the partial order ⊂. For example, the T1 and T2 criteria are comparable, while the T1 and sobriety criteria are not.
See also
, a partial ordering in which incomparability is a transitive relation
References
External links
Binary relations
Order theory
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https://en.wikipedia.org/wiki/Trilinear%20coordinates
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In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and .
In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible for all three trilinear coordinates to be non-positive.
Notation
The ratio notation for trilinear coordinates is often used in preference to the ordered triple notation with the latter reserved for triples of directed distances relative to a specific triangle. The trilinear coordinates can be rescaled by any arbitrary value without affecting their ratio. The bracketed, comma-separated triple notation can cause confusion because conventionally this represents a different triple than e.g. but these equivalent ratios represent the same point.
Examples
The trilinear coordinates of the incenter of a triangle are ; that is, the (directed) distances from the incenter to the sidelines are proportional to the actual distances denoted by , where is the inradius of . Given side lengths we have:
Note that, in general, the incenter is not the same as the centroid; the centroid has barycentric coordinates (these being proportional to actual signed areas of the triangles , where = centroid.)
The midpoint of, for example, side has trilinear coordinates in actual sideline distances for triangle area , which in arbitrarily specified relative distances simplifies to . The coordinates in actual sideline distances of the foot of the altitude from to are which in purely relative distances simplifies to .
Formulas
Collinearities and concurrencies
Trilinear coordinates enable many algebraic methods in triangle geometry. For example, three points
are collinear if and only if the determinant
equals zero. Thus if is a variable point, the equation of a line through the points and is . From this, every straight line has a linear equation homogeneous in . Every equation of the form in real coefficients is a real straight line of finite points unless is proportional to , the side lengths, in which case we have the locus of points at infinity.
The dual of this proposition is that the lines
concur in a point if and only if .
Also, if the actual directed distances
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https://en.wikipedia.org/wiki/Strong%20prime
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In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory.
Definition in number theory
In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, writing the sequence of prime numbers as (p, p, p, ...) = (2, 3, 5, ...), p is a strong prime if . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, so 17 is a strong prime.
The first few strong primes are
11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 .
In a twin prime pair (p, p + 2) with p > 5, p is always a strong prime, since 3 must divide p − 2, which cannot be prime.
Definition in cryptography
In cryptography, a prime number p is said to be "strong" if the following conditions are satisfied.
p is sufficiently large to be useful in cryptography; typically this requires p to be too large for plausible computational resources to enable a cryptanalyst to factorise products of p with other strong primes.
p − 1 has large prime factors. That is, p = aq + 1 for some integer a and large prime q.
q − 1 has large prime factors. That is, q = aq + 1 for some integer a and large prime q.
p + 1 has large prime factors. That is, p = aq − 1 for some integer a and large prime q.
It is possible for a prime to be a strong prime both in the cryptographic sense and the number theoretic sense. For the sake of illustration, 439351292910452432574786963588089477522344331 is a strong prime in the number theoretic sense because the arithmetic mean of its two neighboring primes is 62 less. Without the aid of a computer, this number would be a strong prime in the cryptographic sense because 439351292910452432574786963588089477522344330 has the large prime factor 1747822896920092227343 (and in turn the number one less than that has the large prime factor 1683837087591611009), 439351292910452432574786963588089477522344332 has the large prime factor 864608136454559457049 (and in turn the number one less than that has the large prime factor 105646155480762397). Even using algorithms more advanced than trial division, these numbers would be difficult to factor by hand. For a modern computer algebra system, these numbers can be factored almost instantaneously. A cryptographically strong prime has to be much larger than this example.
Application of strong primes in cryptography
Factoring-based cryptosystems
Some people suggest that in the key generation process in RSA cryptosystems, the modulus n should be chosen as the product of two strong primes. This makes the factorization of n = pq using Pollard's p − 1 al
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https://en.wikipedia.org/wiki/Peter%20B.%20Kronheimer
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Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and former chair of the mathematics department.
Education
Kronheimer attended the City of London School. He completed his DPhil at Oxford University under the direction of Michael Atiyah. He has had a long association with Merton College, the oldest of the constituent colleges of Oxford University, being an undergraduate, graduate, and full fellow of the college.
Career
Kronheimer's early work was on gravitational instantons, in particular the classification of hyperkähler 4-manifolds with asymptotical locally Euclidean geometry (ALE spaces), leading to the papers "The construction of ALE spaces as hyper-Kähler quotients" and "A Torelli-type theorem for gravitational instantons." He and Hiraku Nakajima
gave a construction of instantons on ALE spaces generalizing the Atiyah–Hitchin–Drinfeld–Manin construction. This constructions identified these moduli spaces as moduli spaces for certain quivers (see "Yang-Mills instantons on ALE gravitational instantons.") He was the initial recipient of the Oberwolfach prize in 1998 on the basis of some of this work.
Kronheimer has frequently collaborated with Tomasz Mrowka from the Massachusetts Institute of Technology. Their collaboration began at the Mathematical Research Institute of Oberwolfach, and their first work developed analogues of Simon Donaldson's invariants for 4-manifolds with a distinguished surface. They used the tools developed to prove a conjecture of John Milnor, that the four-ball genus of a -torus knot is . They then went on to develop these tools further and established a structure theorem for Donaldson's polynomial invariants using Kronheimer–Mrowka basic classes. After the arrival of Seiberg–Witten theory their work on embedded surfaces culminated in a proof of the Thom conjecture—which had been outstanding for several decades. Another of Kronheimer and Mrowka's results was a proof of the Property P conjecture for knots. They developed an instanton Floer invariant for knots which was used in their proof that Khovanov homology detects the unknot.
Besides his research articles, his writings include a book, with Simon Donaldson, on 4-manifolds, and a book with Mrowka on Seiberg–Witten–Floer homology, entitled "Monopoles and Three-Manifolds". This book won the 2011 Doob Prize of the AMS.
In 1990 he was an invited speaker at the International Congress of Mathematicians (ICM) in Kyoto. In 2018 he gave a plenary lecture at the ICM in Rio de Janeiro, together with Tomasz Mrowka. In 2023 he was awarded the Leroy P. Steele Prize for Seminal Contribution to Research.
Kronheimer's PhD students have included Ian Dowker, Jacob Rasmussen, Ciprian Manolescu, and Olga Plamenevskaya.
References
External links
Peter Kronheimer's home page at Harvard Univ
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https://en.wikipedia.org/wiki/Thom%20conjecture
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In mathematics, a smooth algebraic curve in the complex projective plane, of degree , has genus given by the genus–degree formula
.
The Thom conjecture, named after French mathematician René Thom, states that if is any smoothly embedded connected curve representing the same class in homology as , then the genus of satisfies the inequality
.
In particular, C is known as a genus minimizing representative of its homology class. It was first proved by Peter Kronheimer and Tomasz Mrowka in October 1994, using the then-new Seiberg–Witten invariants.
Assuming that has nonnegative self intersection number this was generalized to Kähler manifolds (an example being the complex projective plane) by John Morgan, Zoltán Szabó, and Clifford Taubes, also using the Seiberg–Witten invariants.
There is at least one generalization of this conjecture, known as the symplectic Thom conjecture (which is now a theorem, as proved for example by Peter Ozsváth and Szabó in 2000). It states that a symplectic surface of a symplectic 4-manifold is genus minimizing within its homology class. This would imply the previous result because algebraic curves (complex dimension 1, real dimension 2) are symplectic surfaces within the complex projective plane, which is a symplectic 4-manifold.
See also
Adjunction formula
Milnor conjecture (topology)
References
Four-dimensional geometry
4-manifolds
Algebraic surfaces
Conjectures that have been proved
Theorems in geometry
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https://en.wikipedia.org/wiki/OGLE-2005-BLG-390L
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OGLE-2005-BLG-390L is a star thought to be a spectral type M (a red dwarf; 95% probability, 4% probability it is a white dwarf, <1% probability it is a neutron star or black hole). This dim magnitude 16 galactic bulge star is located in the Scorpius constellation at a far distance of about 21,500 light years.
Planetary system
OGLE-2005-BLG-390L has one known planet, which was discovered using the technique of gravitational microlensing. Indications are that the planet is about five times Earth mass, orbiting at about 2.6 astronomical units from the parent star. The discovery was announced on January 25, 2006. OGLE-2005-BLG-390Lb was once considered one of the smallest known extrasolar planets around a main sequence star, possibly rocky, with a mass around 5.5 times that of the Earth. The orbital
radius (assuming a circular orbit) of the planet is 2.6 AU, however the orbital elements are unknown. Based on its low mass and estimated temperature of around 50 K, the planet is thought to consist mainly of ices, like Pluto or Uranus, rather than being a Jupiter-like gas giant.
See also
List of stars with extrasolar planets
OGLE-2005-BLG-169L
Optical Gravitational Lensing Experiment (OGLE)
References
External links
OGLE: 2005-BLG-390 Event
Small Rocky Planet Found Orbiting Normal Star
OGLE-2005-BLG-390L b
Scorpius
M-type main-sequence stars
Gravitational lensing
Planetary systems with one confirmed planet
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https://en.wikipedia.org/wiki/Ars%20Magna
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Ars Magna may refer to:
Ars Magna (Cardano book), a 16th-century book on algebra
Ars Magna (Llull book), a 14th-century philosophical work
Ars Magna Lucis et Umbrae, a 17th-century work on optics
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https://en.wikipedia.org/wiki/Word%20problem%20%28mathematics%29
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In computational mathematics, a word problem is the problem of deciding whether two given expressions are equivalent with respect to a set of rewriting identities. A prototypical example is the word problem for groups, but there are many other instances as well. A deep result of computational theory is that answering this question is in many important cases undecidable.
Background and motivation
In computer algebra one often wishes to encode mathematical expressions using an expression tree. But there are often multiple equivalent expression trees. The question naturally arises of whether there is an algorithm which, given as input two expressions, decides whether they represent the same element. Such an algorithm is called a solution to the word problem. For example, imagine that are symbols representing real numbers - then a relevant solution to the word problem would, given the input , produce the output EQUAL, and similarly produce NOT_EQUAL from .
The most direct solution to a word problem takes the form of a normal form theorem and algorithm which maps every element in an equivalence class of expressions to a single encoding known as the normal form - the word problem is then solved by comparing these normal forms via syntactic equality. For example one might decide that is the normal form of , , and , and devise a transformation system to rewrite those expressions to that form, in the process proving that all equivalent expressions will be rewritten to the same normal form. But not all solutions to the word problem use a normal form theorem - there are algebraic properties which indirectly imply the existence of an algorithm.
While the word problem asks whether two terms containing constants are equal, a proper extension of the word problem known as the unification problem asks whether two terms containing variables have instances that are equal, or in other words whether the equation has any solutions. As a common example, is a word problem in the integer group ℤ,
while is a unification problem in the same group; since the former terms happen to be equal in ℤ, the latter problem has the substitution as a solution.
History
One of the most deeply studied cases of the word problem is in the theory of semigroups and groups. A timeline of papers relevant to the Novikov-Boone theorem is as follows:
The word problem for semi-Thue systems
The accessibility problem for string rewriting systems (semi-Thue systems or semigroups) can be stated as follows: Given a semi-Thue system and two words (strings) , can be transformed into by applying rules from ? Note that the rewriting here is one-way. The word problem is the accessibility problem for symmetric rewrite relations, i.e. Thue systems.
The accessibility and word problems are undecidable, i.e. there is no general algorithm for solving this problem. This even holds if we limit the systems to have finite presentations, i.e. a finite set of symbols and a finite set of relations
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https://en.wikipedia.org/wiki/Word%20problem
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Word problem may refer to:
Word problem (mathematics education), a type of textbook exercise or exam question to have students apply abstract mathematical concepts to real-world situations
Word problem (mathematics), a decision problem for algebraic identities in mathematics and computer science
Word problem for groups, the problem of recognizing the identity element in a finitely presented group
Word problem (computability), a decision problem concerning formal languages
See also
Word-finding problem; problem using words; language problem: aphasia
Word game
Wordle
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https://en.wikipedia.org/wiki/Compact%20quantum%20group
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In mathematics, a compact quantum group is an abstract structure on a unital separable C*-algebra axiomatized from those that exist on the commutative C*-algebra of "continuous complex-valued functions" on a compact quantum group.
The basic motivation for this theory comes from the following analogy. The space of complex-valued functions on a compact Hausdorff topological space forms a commutative C*-algebra. On the other hand, by the Gelfand Theorem, a commutative C*-algebra is isomorphic to the C*-algebra of continuous complex-valued functions on a compact Hausdorff topological space, and the topological space is uniquely determined by the C*-algebra up to homeomorphism.
S. L. Woronowicz introduced the important concept of compact matrix quantum groups, which he initially called compact pseudogroups. Compact matrix quantum groups are abstract structures on which the "continuous functions" on the structure are given by elements of a C*-algebra. The geometry of a compact matrix quantum group is a special case of a noncommutative geometry.
Formulation
For a compact topological group, , there exists a C*-algebra homomorphism
where is the minimal C*-algebra tensor product — the completion of the algebraic tensor product of and ) — such that
for all , and for all , where
for all and all . There also exists a linear multiplicative mapping
,
such that
for all and all . Strictly speaking, this does not make into a Hopf algebra, unless is finite.
On the other hand, a finite-dimensional representation of can be used to generate a *-subalgebra of which is also a Hopf *-algebra. Specifically, if
is an -dimensional representation of , then
for all , and
for all . It follows that the *-algebra generated by for all and for all is a Hopf *-algebra: the counit is determined by
for all (where is the Kronecker delta), the antipode is , and the unit is given by
Compact matrix quantum groups
As a generalization, a compact matrix quantum group is defined as a pair , where is a C*-algebra and
is a matrix with entries in such that
The *-subalgebra, , of , which is generated by the matrix elements of , is dense in ;
There exists a C*-algebra homomorphism, called the comultiplication, (here is the C*-algebra tensor product - the completion of the algebraic tensor product of and ) such that
There exists a linear antimultiplicative map, called the coinverse, such that for all and where is the identity element of . Since is antimultiplicative, for all .
As a consequence of continuity, the comultiplication on is coassociative.
In general, is a bialgebra, and is a Hopf *-algebra.
Informally, can be regarded as the *-algebra of continuous complex-valued functions over the compact matrix quantum group, and can be regarded as a finite-dimensional representation of the compact matrix quantum group.
Compact quantum groups
For C*-algebras and acting on the Hilbert spaces and respectively, their minimal tensor product
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https://en.wikipedia.org/wiki/Dungeons%20%26%20Dragons%20gameplay
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In the Dungeons & Dragons role-playing game, game mechanics and dice rolls determine much of what happens. These mechanics include:
Ability scores, the most basic statistics of a character, which influence all other statistics
Armor class, how well-protected a character is against physical attack
Hit points, how much punishment a character can take before falling unconscious or dying
Saving throws, a character's defenses against nonphysical or area attacks (like poisons, fireballs, and enchantments)
Attack rolls and damage rolls, how effectively a character can score hits against, and inflict damage on, another character
Skills, how competent a character is in various areas of expertise
Feats, what special advantages a character has through natural aptitude or training
Ability scores
All player characters have six basic statistics:
Strength (STR): Strength is a measure of muscle, endurance and stamina combined; a high strength score indicates superiority in all these attributes. Strength affects the ability of characters to lift and carry weights, melee attack rolls, damage rolls (for both melee and ranged weapons), certain physical skills, several combat actions, and general checks involving moving or breaking objects.
Dexterity (DEX): Dexterity encompasses a number of physical attributes including hand-eye coordination, agility, reflexes, fine motor skills, balance and speed of movement; a high dexterity score indicates superiority in all these attributes. Dexterity affects characters with regard to initiative in combat, ranged attack rolls, armor class, saving throws, and other physical skills. Dexterity is the ability most influenced by outside influences (such as armor).
Constitution (CON): Constitution is a term which encompasses the character's physique, toughness, health and resistance to disease and poison; a high constitution score indicates superiority in all these attributes. The higher a character's constitution, the more hit points that character will have. Constitution also is important for saving throws, and fatigue-based general checks. Unlike the other ability scores, which render the character unconscious or immobile when they hit 0, having 0 Constitution is fatal.
Intelligence (INT): Intelligence is similar to IQ, but also includes mnemonic ability, reasoning and learning ability outside those measured by the written word; a high intelligence score indicates superiority in all these attributes. Intelligence dictates the number of languages a character can learn, and it influences the number of spells a preparation-based arcane spell-caster (like a Wizard) may cast per day, and the effectiveness of said spells. It also affects certain mental skills.
Wisdom (WIS): Wisdom is a composite term for the character's enlightenment, judgment, wile, willpower and intuitiveness; a high wisdom score indicates superiority in all these attributes. Wisdom influences the number of spells a divine spell-caster (such as clerics,
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https://en.wikipedia.org/wiki/Kit%20Fine
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Kit Fine (born 26 March 1946) is a British philosopher, currently university professor and Silver Professor of Philosophy and Mathematics at New York University. Prior to joining the philosophy department of NYU in 1997, he taught at the University of Edinburgh, University of California, Irvine, University of Michigan and UCLA. The author of several books and dozens of articles in international academic journals, he has made notable contributions to the fields of philosophical logic, metaphysics, and the philosophy of language and also has written on ancient philosophy, in particular on Aristotle's account of logic and modality.
He is also a distinguished research professor in the Department of Philosophy, University of Birmingham, UK. Since 2018, Fine is visiting professor at the University of Italian Switzerland.
Education, family and career
After graduating from Balliol College, Oxford (B.A., 1967), Fine received his Ph.D. from the University of Warwick in 1969, under the supervision of A. N. Prior. He then taught at the University of Edinburgh, University of California, Irvine, University of Michigan, and UCLA, before moving to New York University.
He was elected a Corresponding Fellow of the British Academy in 2005 and a Fellow of the American Academy of Arts & Sciences in 2006. He has held fellowships from the John Simon Guggenheim Memorial Foundation and the American Council of Learned Societies and is a former editor of the Journal of Symbolic Logic.
Fine has two daughters from his former marriage to Anne Fine. Anne Fine is an author of children's books; Cordelia Fine is a professor of philosophy at the University of Melbourne; Ione Fine is a professor at the University of Washington.
Philosophical work
In addition to his primary areas of research, he has written papers in ancient philosophy, linguistics, computer science, and economic theory.
Fine has described his general approach to philosophy as follows: "I’m firmly of the opinion that real progress in philosophy can only come from taking common sense seriously. A departure from common sense is usually an indication that a mistake has been made."
Awards
In 2013, Fine held the Gödel Lecture, titled Truthmaker semantics.
Bibliography
Worlds, Times, and Selves (with A. N. Prior). University of Massachusetts Press, 1977.
Reasoning With Arbitrary Objects. Blackwell, 1986.
The Limits of Abstraction. Oxford University Press, 2002.
Modality and Tense: Philosophical Papers. Oxford University Press, 2005.
Semantic Relationism. Blackwell, 2007.
Vagueness: A Global Approach, Oxford University Press, 2020.
Notes
References
Sources
Silver Dialogues: Kit Fine
Kit Fine's CV
External links
Kit Fine's web page at New York University
KIt Fine (1946-) Internet Encyclopedia of Philosophy
Annotated Bibliography of Kit Fine's Writings
Kit Fine. Annotated Bibliography of the Studies on His Philosophy
Interview with 3AM Magazine
Living people
1946 births
British philosophers
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https://en.wikipedia.org/wiki/Geir%20Ellingsrud
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Geir Ellingsrud (born 29 November 1948) is professor of mathematics at the University of Oslo, where he specialises in algebra and algebraic geometry.
He took the cand.real. degree at the University of Oslo in 1973, and the doctorate at Stockholm University in 1982. He was a lecturer at Stockholm University from 1982 to 1984, associate professor at the University of Oslo from 1984 to 1989, professor at the University of Bergen from 1989 to 1993 and at the University of Oslo since 1993. He has been a visiting scholar in Nice, Paris, Bonn and Chicago. He has edited the journals Acta Mathematica and Normat.
In 2005 Ellingsrud was elected to be rector of the University of Oslo for the period 2006-2009. His team also consisted of Inga Bostad and Haakon Breien Benestad. He did not seek reelection to a second term, and was succeeded by Ole Petter Ottersen.
References
1948 births
Living people
Algebraic geometers
University of Oslo alumni
Stockholm University alumni
Academic staff of Stockholm University
Academic staff of the University of Bergen
Academic staff of the University of Oslo
Rectors of the University of Oslo
Members of the Norwegian Academy of Science and Letters
20th-century Norwegian mathematicians
21st-century Norwegian mathematicians
Royal Norwegian Society of Sciences and Letters
Presidents of the Norwegian Mathematical Society
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https://en.wikipedia.org/wiki/Atan2
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In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, is the angle measure (in radians, with ) between the positive -axis and the ray from the origin to the point in the Cartesian plane. Equivalently, is the argument (also called phase or angle) of the complex number
The function first appeared in the programming language Fortran in 1961. It was originally intended to return a correct and unambiguous value for the angle in converting from Cartesian coordinates to polar coordinates . If and , then and
If , the desired angle measure is However, when , the angle is diametrically opposite the desired angle, and ± (a half turn) must be added to place the point in the correct quadrant. Using the function does away with this correction, simplifying code and mathematical formulas.
Motivation
The ordinary single-argument arctangent function only returns angle measures in the interval and when invoking it to find the angle measure between the -axis and an arbitrary vector in the Cartesian plane, there is no simple way to indicate a direction in the left half-plane (that is, a point with ). Diametrically opposite angle measures have the same tangent because so the tangent is not in itself sufficient to uniquely specify an angle.
To determine an angle measure using the arctangent function given a point or vector mathematical formulas or computer code must handle multiple cases; at least one for positive values of and one for negative values of and sometimes additional cases when is negative or one coordinate is zero. Finding angle measures and converting Cartesian to polar coordinates are common in scientific computing, and this code is redundant and error-prone.
To remedy this, computer programming languages introduced the function, at least as early as the Fortran IV language of the 1960s. The quantity is the angle measure between the -axis and a ray from the origin to a point anywhere in the Cartesian plane. The signs of and are used to determine the quadrant of the result and select the correct branch of the multivalued function .
The function is useful in many applications involving Euclidean vectors such as finding the direction from one point to another or converting a rotation matrix to Euler angles.
The function is now included in many other programming languages, and is also commonly found in mathematical formulas throughout science and engineering.
Argument order
In 1961, Fortran introduced the function with argument order so that the argument (phase angle) of a complex number is This follows the left-to-right order of a fraction written so that for positive values of However, this is the opposite of the conventional component order for complex numbers, or as coordinates See section Definition and computation.
Some other programming languages (see § Realizations of the function in common computer languages) picked the opposite order instead. For example Micros
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https://en.wikipedia.org/wiki/N%C3%B8rlund%E2%80%93Rice%20integral
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In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory of finite differences and has also been applied in computer science and graph theory to estimate binary tree lengths. It is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation.
Definition
The nth forward difference of a function f(x) is given by
where is the binomial coefficient.
The Nörlund–Rice integral is given by
where f is understood to be meromorphic, α is an integer, , and the contour of integration is understood to circle the poles located at the integers α, ..., n, but encircles neither integers 0, ..., nor any of the poles of f. The integral may also be written as
where B(a,b) is the Euler beta function. If the function is polynomially bounded on the right hand side of the complex plane, then the contour may be extended to infinity on the right hand side, allowing the transform to be written as
where the constant c is to the left of α.
Poisson–Mellin–Newton cycle
The Poisson–Mellin–Newton cycle, noted by Flajolet et al. in 1985, is the observation that the resemblance of the Nørlund–Rice integral to the Mellin transform is not accidental, but is related by means of the binomial transform and the Newton series. In this cycle, let be a sequence, and let g(t) be the corresponding Poisson generating function, that is, let
Taking its Mellin transform
one can then regain the original sequence by means of the Nörlund–Rice integral:
where Γ is the gamma function.
Riesz mean
A closely related integral frequently occurs in the discussion of Riesz means. Very roughly, it can be said to be related to the Nörlund–Rice integral in the same way that Perron's formula is related to the Mellin transform: rather than dealing with infinite series, it deals with finite series.
Utility
The integral representation for these types of series is interesting because the integral can often be evaluated using asymptotic expansion or saddle-point techniques; by contrast, the forward difference series can be extremely hard to evaluate numerically, because the binomial coefficients grow rapidly for large n.
See also
Table of Newtonian series
List of factorial and binomial topics
References
Niels Erik Nørlund, Vorlesungen uber Differenzenrechnung, (1954) Chelsea Publishing Company, New York.
Donald E. Knuth, The Art of Computer Programming, (1973), Vol. 3 Addison-Wesley.
Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101–124.
Peter Kirschenhofer, "", The Electronic Journal of Combinatorics, Volume 3 (1996) Issue 2 Article 7.
Factorial and binomial topics
Complex analysis
In
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https://en.wikipedia.org/wiki/Pursuit%E2%80%93evasion
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Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit–evasion, and the graph formulation discrete pursuit–evasion (also called graph searching). Current research is typically limited to one of these two formulations.
Discrete formulation
In the discrete formulation of the pursuit–evasion problem, the environment is modeled as a graph.
Problem definition
There are innumerable possible variants of pursuit–evasion, though they tend to share many elements. A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a graph. The two sides take alternate turns, which consist of each member either staying put or moving along an edge to an adjacent node. If a pursuer occupies the same node as an evader the evader is captured and removed from the graph. The question usually posed is how many pursuers are necessary to ensure the eventual capture of all the evaders. If one pursuer suffices, the graph is called a cop-win graph. In this case, a single evader can always be captured in time linear to the number of n nodes of the graph. Capturing r evaders with k pursuers can take in the order of r n time as well, but the exact bounds for more than one pursuer are still unknown.
Often the movement rules are altered by changing the velocity of the evaders. This velocity is the maximum number of edges that an evader can move along in a single turn. In the example above, the evaders have a velocity of one. At the other extreme is the concept of infinite velocity, which allows an evader to move to any node in the graph so long as there is a path between its original and final positions that contains no nodes occupied by a pursuer. Similarly some variants arm the pursuers with "helicopters" which allow them to move to any vertex on their turn.
Other variants ignore the restriction that pursuers and evaders must always occupy a node and allow for the possibility that they are positioned somewhere along an edge. These variants are often referred to as sweeping problems, whilst the previous variants would fall under the category of searching problems.
Variants
Several variants are equivalent to important graph parameters. Specifically, finding the number of pursuers necessary to capture a single evader with infinite velocity in a graph G (when pursuers and evader are not constrained to move turn by turn, but move simultaneously) is equivalent to finding the treewidth of G, and a winning strategy for the evader may be described in terms of a haven in G. If this evader is invisible to the pursuers then the p
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https://en.wikipedia.org/wiki/Ecological%20correlation
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In statistics, an ecological correlation (also spatial correlation) is a correlation between two variables that are group means, in contrast to a correlation between two variables that describe individuals. For example, one might study the correlation between physical activity and weight among sixth-grade children. A study at the individual level might make use of 100 children, then measure both physical activity and weight; the correlation between the two variables would be at the individual level. By contrast, another study might make use of 100 classes of sixth-grade students, then measure the mean physical activity and the mean weight of each of the 100 classes. A correlation between these group means would be an example of an ecological correlation.
Because a correlation describes the measured strength of a relationship, correlations at the group level can be much higher than those at the individual level. Thinking both are equal is an example of ecological fallacy.
See also
General topics
Ecological regression
Geographic information science
Spatial autocorrelation
Complete spatial randomness
Modifiable areal unit problem
Specific applications
Spatial epidemiology
Spatial econometrics
References
Covariance and correlation
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https://en.wikipedia.org/wiki/Babylonian%20mathematics
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Babylonian mathematics (also known as Assyro-Babylonian mathematics) are the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. With respect to time they fall in two distinct groups: one from the Old Babylonian period (1830–1531 BC), the other mainly Seleucid from the last three or four centuries BC. With respect to content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for over a millennium.
In contrast to the scarcity of sources in Egyptian mathematics, knowledge of Babylonian mathematics is derived from hundreds of clay tablets unearthed since the 1850s. Written in cuneiform, tablets were inscribed while the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics that include fractions, algebra, quadratic and cubic equations and the Pythagorean theorem. The Babylonian tablet YBC 7289 gives an approximation to accurate to three significant sexagesimal digits (about six significant decimal digits).
Origins of Babylonian mathematics
Babylonian mathematics is a range of numeric and more advanced mathematical practices in the ancient Near East, written in cuneiform script. Study has historically focused on the Old Babylonian period in the early second millennium BC due to the wealth of data available. There has been debate over the earliest appearance of Babylonian mathematics, with historians suggesting a range of dates between the 5th and 3rd millennia BC. Babylonian mathematics was primarily written on clay tablets in cuneiform script in the Akkadian or Sumerian languages.
"Babylonian mathematics" is perhaps an unhelpful term since the earliest suggested origins date to the use of accounting devices, such as bullae and tokens, in the 5th millennium BC.
Babylonian numerals
The Babylonian system of mathematics was a sexagesimal (base 60) numeral system. From this we derive the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a superior highly composite number, having factors of 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 (including those that are themselves composite), facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values (much as, in our base ten system, 734 = 7×100 + 3×10 + 4×1).
Sumerian mathematics
The ancient Sumerians of Mesopotamia developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exe
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https://en.wikipedia.org/wiki/Composition%20algebra
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In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies
for all and in .
A composition algebra includes an involution called a conjugation: The quadratic form is called the norm of the algebra.
A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a null vector. When x is not a null vector, the multiplicative inverse of x is When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".
Structure theorem
Every unital composition algebra over a field can be obtained by repeated application of the Cayley–Dickson construction starting from (if the characteristic of is different from ) or a 2-dimensional composition subalgebra (if ). The possible dimensions of a composition algebra are , , , and .
1-dimensional composition algebras only exist when .
Composition algebras of dimension 1 and 2 are commutative and associative.
Composition algebras of dimension 2 are either quadratic field extensions of or isomorphic to .
Composition algebras of dimension 4 are called quaternion algebras. They are associative but not commutative.
Composition algebras of dimension 8 are called octonion algebras. They are neither associative nor commutative.
For consistent terminology, algebras of dimension 1 have been called unarion, and those of dimension 2 binarion.
Every composition algebra is an alternative algebra.
Using the doubled form ( _ : _ ): A × A → K by then the trace of a is given by (a:1) and the conjugate by a* = (a:1)e – a where e is the basis element for 1. A series of exercises prove that a composition algebra is always an alternative algebra.
Instances and usage
When the field is taken to be complex numbers and the quadratic form , then four composition algebras over are , the bicomplex numbers, the biquaternions (isomorphic to the complex matrix ring ), and the bioctonions , which are also called complex octonions.
The matrix ring has long been an object of interest, first as biquaternions by
Hamilton (1853), later in the isomorphic matrix form, and especially as Pauli algebra.
The squaring function on the real number field forms the primordial composition algebra.
When the field is taken to be real numbers , then there are just six other real composition algebras.
In two, four, and eight dimensions there are both a division algebra and a split algebra:
binarions: complex numbers with quadratic form and split-complex numbers with quadratic form ,
quaternions and split-quaternions,
octonions and split-octonions.
Every composition algebra has an associated bilinear form B(x,y) constructed with the norm N and a polarization identity:
History
The composition of sums of squares was noted by several early authors. Diophantus was aware of the identity involving the sum of two squares
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https://en.wikipedia.org/wiki/Split-octonion
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In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0).
Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers. They are also the only two octonion algebras over the real numbers. Split-octonion algebras analogous to the split-octonions can be defined over any field.
Definition
Cayley–Dickson construction
The octonions and the split-octonions can be obtained from the Cayley–Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit ℓ and write a pair of quaternions (a, b) in the form a + ℓb. The product is defined by the rule:
where
If λ is chosen to be −1, we get the octonions. If, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley–Dickson doubling of the split-quaternions. Here either choice of λ (±1) gives the split-octonions.
Multiplication table
A basis for the split-octonions is given by the set .
Every split-octonion can be written as a linear combination of the basis elements,
with real coefficients .
By linearity, multiplication of split-octonions is completely determined by the following multiplication table:
A convenient mnemonic is given by the diagram at the right, which represents the multiplication table for the split-octonions. This one is derived from its parent octonion (one of 480 possible), which is defined by:
where is the Kronecker delta and is the Levi-Civita symbol with value when and:
with the scalar element, and
The red arrows indicate possible direction reversals imposed by negating the lower right quadrant of the parent creating a split octonion with this multiplication table.
Conjugate, norm and inverse
The conjugate of a split-octonion x is given by
just as for the octonions.
The quadratic form on x is given by
This quadratic form N(x) is an isotropic quadratic form since there are non-zero split-octonions x with N(x) = 0. With N, the split-octonions form a pseudo-Euclidean space of eight dimensions over R, sometimes written R4,4 to denote the signature of the quadratic form.
If N(x) ≠ 0, then x has a (two-sided) multiplicative inverse x−1 given by
Properties
The split-octonions, like the octonions, are noncommutative and nonassociative. Also like the octonions, they form a composition algebra since the quadratic form N is multiplicative. That is,
The split-octonions satisfy the Moufang identities and so form an alternative algebra. Therefore, by Artin's theorem, the subalgebra generated by any two elements is associative. The set of all invertible elements (i.e. those elements for which N(x) ≠ 0) form a
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https://en.wikipedia.org/wiki/Symmetric%20monoidal%20category
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In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" is defined) such that the tensor product is symmetric (i.e. is, in a certain strict sense, naturally isomorphic to for all objects and of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field k, using the ordinary tensor product of vector spaces.
Definition
A symmetric monoidal category is a monoidal category (C, ⊗, I) such that, for every pair A, B of objects in C, there is an isomorphism called the swap map that is natural in both A and B and such that the following diagrams commute:
The unit coherence:
The associativity coherence:
The inverse law:
In the diagrams above, a, l, and r are the associativity isomorphism, the left unit isomorphism, and the right unit isomorphism respectively.
Examples
Some examples and non-examples of symmetric monoidal categories:
The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be fixed as the unit object.
The category of groups. Like before, the tensor product is just the cartesian product of groups, and the trivial group is the unit object.
More generally, any category with finite products, that is, a cartesian monoidal category, is symmetric monoidal. The tensor product is the direct product of objects, and any terminal object (empty product) is the unit object.
The category of bimodules over a ring R is monoidal (using the ordinary tensor product of modules), but not necessarily symmetric. If R is commutative, the category of left R-modules is symmetric monoidal. The latter example class includes the category of all vector spaces over a given field.
Given a field k and a group (or a Lie algebra over k), the category of all k-linear representations of the group (or of the Lie algebra) is a symmetric monoidal category. Here the standard tensor product of representations is used.
The categories (Ste,) and (Ste,) of stereotype spaces over are symmetric monoidal, and moreover, (Ste,) is a closed symmetric monoidal category with the internal hom-functor .
Properties
The classifying space (geometric realization of the nerve) of a symmetric monoidal category is an space, so its group completion is an infinite loop space.
Specializations
A dagger symmetric monoidal category is a symmetric monoidal category with a compatible dagger structure.
A cosmos is a complete cocomplete closed symmetric monoidal category.
Generalizations
In a symmetric monoidal category, the natural isomorphisms are their own inverses in the sense that . If we abandon this requirement (but still require that be naturally isomorphic to ), we obtain the more general notion of a braided monoidal category.
References
Monoidal categories
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https://en.wikipedia.org/wiki/%2A-autonomous%20category
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In mathematics, a *-autonomous (read "star-autonomous") category C is a symmetric monoidal closed category equipped with a dualizing object . The concept is also referred to as Grothendieck—Verdier category in view of its relation to the notion of Verdier duality.
Definition
Let C be a symmetric monoidal closed category. For any object A and , there exists a morphism
defined as the image by the bijection defining the monoidal closure
of the morphism
where is the symmetry of the tensor product. An object of the category C is called dualizing when the associated morphism is an isomorphism for every object A of the category C.
Equivalently, a *-autonomous category is a symmetric monoidal category C together with a functor such that for every object A there is a natural isomorphism , and for every three objects A, B and C there is a natural bijection
.
The dualizing object of C is then defined by . The equivalence of the two definitions is shown by identifying .
Properties
Compact closed categories are *-autonomous, with the monoidal unit as the dualizing object. Conversely, if the unit of a *-autonomous category is a dualizing object then there is a canonical family of maps
.
These are all isomorphisms if and only if the *-autonomous category is compact closed.
Examples
A familiar example is the category of finite-dimensional vector spaces over any field k made monoidal with the usual tensor product of vector spaces. The dualizing object is k, the one-dimensional vector space, and dualization corresponds to transposition. Although the category of all vector spaces over k is not *-autonomous, suitable extensions to categories of topological vector spaces can be made *-autonomous.
On the other hand, the category of topological vector spaces contains an extremely wide full subcategory, the category Ste of stereotype spaces, which is a *-autonomous category with the dualizing object and the tensor product .
Various models of linear logic form *-autonomous categories, the earliest of which was Jean-Yves Girard's category of coherence spaces.
The category of complete semilattices with morphisms preserving all joins but not necessarily meets is *-autonomous with dualizer the chain of two elements. A degenerate example (all homsets of cardinality at most one) is given by any Boolean algebra (as a partially ordered set) made monoidal using conjunction for the tensor product and taking 0 as the dualizing object.
The formalism of Verdier duality gives further examples of *-autonomous categories. For example, mention that the bounded derived category of constructible l-adic sheaves on an algebraic variety has this property. Further examples include derived categories of constructible sheaves on various kinds of topological spaces.
An example of a self-dual category that is not *-autonomous is finite linear orders and continuous functions, which has * but is not autonomous: its dualizing object is the two-element chain but there is
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https://en.wikipedia.org/wiki/Dinatural%20transformation
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In category theory, a branch of mathematics, a dinatural transformation between two functors
written
is a function that to every object of associates an arrow
of
and satisfies the following coherence property: for every morphism of the diagram
commutes.
The composition of two dinatural transformations need not be dinatural.
See also
Extranatural transformation
Natural transformation
References
External links
Functors
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https://en.wikipedia.org/wiki/Transversal
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Transversal may refer to:
Transversal (combinatorics), a set containing exactly one member of each of several other sets
Transversal (geometry), a line that intersects two or more lines at different points
Transversal (instrument making), a technique for subdividing graduations
Transversal Corporation, a software company
Transversal plane, a geometric concept
Transversal, relating to the transverse plane in anatomy
See also
Transverse (disambiguation)
Transversality (disambiguation)
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https://en.wikipedia.org/wiki/Table%20of%20Newtonian%20series
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In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence written in the form
where
is the binomial coefficient and is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus.
List
The generalized binomial theorem gives
A proof for this identity can be obtained by showing that it satisfies the differential equation
The digamma function:
The Stirling numbers of the second kind are given by the finite sum
This formula is a special case of the kth forward difference of the monomial xn evaluated at x = 0:
A related identity forms the basis of the Nörlund–Rice integral:
where is the Gamma function and is the Beta function.
The trigonometric functions have umbral identities:
and
The umbral nature of these identities is a bit more clear by writing them in terms of the falling factorial . The first few terms of the sin series are
which can be recognized as resembling the Taylor series for sin x, with (s)n standing in the place of xn.
In analytic number theory it is of interest to sum
where B are the Bernoulli numbers. Employing the generating function its Borel sum can be evaluated as
The general relation gives the Newton series
where is the Hurwitz zeta function and the Bernoulli polynomial. The series does not converge, the identity holds formally.
Another identity is
which converges for . This follows from the general form of a Newton series for equidistant nodes (when it exists, i.e. is convergent)
See also
Binomial transform
List of factorial and binomial topics
Nörlund–Rice integral
Carlson's theorem
References
Philippe Flajolet and Robert Sedgewick, "Mellin transforms and asymptotics: Finite differences and Rice's integrals", Theoretical Computer Science 144 (1995) pp 101–124.
Finite differences
Factorial and binomial topics
Newton series
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https://en.wikipedia.org/wiki/Smooth%20infinitesimal%20analysis
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Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. Lawvere and employing the methods of category theory, it views all functions as being continuous and incapable of being expressed in terms of discrete entities. As a theory, it is a subset of synthetic differential geometry.
The nilsquare or nilpotent infinitesimals are numbers ε where ε² = 0 is true, but ε = 0 need not be true at the same time.
Overview
This approach departs from the classical logic used in conventional mathematics by denying the law of the excluded middle, e.g., NOT (a ≠ b) does not imply a = b. In particular, in a theory of smooth infinitesimal analysis one can prove for all infinitesimals ε, NOT (ε ≠ 0); yet it is provably false that all infinitesimals are equal to zero. One can see that the law of excluded middle cannot hold from the following basic theorem (again, understood in the context of a theory of smooth infinitesimal analysis):
Every function whose domain is R, the real numbers, is continuous and infinitely differentiable.
Despite this fact, one could attempt to define a discontinuous function f(x) by specifying that f(x) = 1 for x = 0, and f(x) = 0 for x ≠ 0. If the law of the excluded middle held, then this would be a fully defined, discontinuous function. However, there are plenty of x, namely the infinitesimals, such that neither x = 0 nor x ≠ 0 holds, so the function is not defined on the real numbers.
In typical models of smooth infinitesimal analysis, the infinitesimals are not invertible, and therefore the theory does not contain infinite numbers. However, there are also models that include invertible infinitesimals.
Other mathematical systems exist which include infinitesimals, including nonstandard analysis and the surreal numbers. Smooth infinitesimal analysis is like nonstandard analysis in that (1) it is meant to serve as a foundation for analysis, and (2) the infinitesimal quantities do not have concrete sizes (as opposed to the surreals, in which a typical infinitesimal is , where ω is a von Neumann ordinal). However, smooth infinitesimal analysis differs from nonstandard analysis in its use of nonclassical logic, and in lacking the transfer principle. Some theorems of standard and nonstandard analysis are false in smooth infinitesimal analysis, including the intermediate value theorem and the Banach–Tarski paradox. Statements in nonstandard analysis can be translated into statements about limits, but the same is not always true in smooth infinitesimal analysis.
Intuitively, smooth infinitesimal analysis can be interpreted as describing a world in which lines are made out of infinitesimally small segments, not out of points. These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved. The construction of discontinuous functions fails because a function is identified with a curve, and the curve cannot be construc
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https://en.wikipedia.org/wiki/Synthetic%20differential%20geometry
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In mathematics, synthetic differential geometry is a formalization of the theory of differential geometry in the language of topos theory. There are several insights that allow for such a reformulation. The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle). The second insight is that the operation of assigning a bundle of jets to a smooth manifold is functorial in nature. The third insight is that over a certain category, these are representable functors. Furthermore, their representatives are related to the algebras of dual numbers, so that smooth infinitesimal analysis may be used.
Synthetic differential geometry can serve as a platform for formulating certain otherwise obscure or confusing notions from differential geometry. For example, the meaning of what it means to be natural (or invariant) has a particularly simple expression, even though the formulation in classical differential geometry may be quite difficult.
Further reading
John Lane Bell, Two Approaches to Modelling the Universe: Synthetic Differential Geometry and Frame-Valued Sets (PDF file)
F.W. Lawvere, Outline of synthetic differential geometry (PDF file)
Anders Kock, Synthetic Differential Geometry (PDF file), Cambridge University Press, 2nd Edition, 2006.
R. Lavendhomme, Basic Concepts of Synthetic Differential Geometry, Springer-Verlag, 1996.
Michael Shulman, Synthetic Differential Geometry
Ryszard Paweł Kostecki, Differential Geometry in Toposes
Differential geometry
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https://en.wikipedia.org/wiki/Order-5%20dodecahedral%20honeycomb
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In hyperbolic geometry, the order-5 dodecahedral honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five dodecahedral cells around each edge, and each vertex is surrounded by twenty dodecahedra. Its vertex figure is an icosahedron.
Description
The dihedral angle of a Euclidean regular dodecahedron is ~116.6°, so no more than three of them can fit around an edge in Euclidean 3-space. In hyperbolic space, however, the dihedral angle is smaller than it is in Euclidean space, and depends on the size of the figure; the smallest possible dihedral angle is 60°, for an ideal hyperbolic regular dodecahedron with infinitely long edges. The dodecahedra in this dodecahedral honeycomb are sized so that all of their dihedral angles are exactly 72°.
Images
Related polytopes and honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
There is another honeycomb in hyperbolic 3-space called the order-4 dodecahedral honeycomb, {5,3,4}, which has only four dodecahedra per edge. These honeycombs are also related to the 120-cell which can be considered as a honeycomb in positively curved space (the surface of a 4-dimensional sphere), with three dodecahedra on each edge, {5,3,3}. Lastly the dodecahedral ditope, {5,3,2} exists on a 3-sphere, with 2 hemispherical cells.
There are nine uniform honeycombs in the [5,3,5] Coxeter group family, including this regular form. Also the bitruncated form, t1,2{5,3,5}, , of this honeycomb has all truncated icosahedron cells.
The Seifert–Weber space is a compact manifold that can be formed as a quotient space of the order-5 dodecahedral honeycomb.
This honeycomb is a part of a sequence of polychora and honeycombs with icosahedron vertex figures:
This honeycomb is a part of a sequence of regular polytopes and honeycombs with dodecahedral cells:
Rectified order-5 dodecahedral honeycomb
The rectified order-5 dodecahedral honeycomb, , has alternating icosahedron and icosidodecahedron cells, with a pentagonal prism vertex figure.
Related tilings and honeycomb
There are four rectified compact regular honeycombs:
Truncated order-5 dodecahedral honeycomb
The truncated order-5 dodecahedral honeycomb, , has icosahedron and truncated dodecahedron cells, with a pentagonal pyramid vertex figure.
Related honeycombs
Bitruncated order-5 dodecahedral honeycomb
The bitruncated order-5 dodecahedral honeycomb, , has truncated icosahedron cells, with a tetragonal disphenoid vertex figure.
Related honeycombs
Cantellated order-5 dodecahedral honeycomb
The cantellated order-5 dodecahedral honeycomb, , has rhombicosidodecahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.
Related honeycombs
Cantitruncated order-5 dodecahedral honeycomb
The cantitruncated order-5 dodecahedral honeycomb, , has truncated icosidodecahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphen
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https://en.wikipedia.org/wiki/Order-5%20cubic%20honeycomb
|
In hyperbolic geometry, the order-5 cubic honeycomb is one of four compact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol it has five cubes around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.
Description
Symmetry
It has a radial subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)*], index 120.
Related polytopes and honeycombs
The order-5 cubic honeycomb has a related alternated honeycomb, ↔ , with icosahedron and tetrahedron cells.
The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:
There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:
The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.
It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.
Rectified order-5 cubic honeycomb
The rectified order-5 cubic honeycomb, , has alternating icosahedron and cuboctahedron cells, with a pentagonal prism vertex figure.
Related honeycomb
There are four rectified compact regular honeycombs:
Truncated order-5 cubic honeycomb
The truncated order-5 cubic honeycomb, , has truncated cube and icosahedron cells, with a pentagonal pyramid vertex figure.
It can be seen as analogous to the 2D hyperbolic truncated order-5 square tiling, t{4,5}, with truncated square and pentagonal faces:
It is similar to the Euclidean (order-4) truncated cubic honeycomb, t{4,3,4}, which has octahedral cells at the truncated vertices.
Related honeycombs
Bitruncated order-5 cubic honeycomb
The bitruncated order-5 cubic honeycomb is the same as the bitruncated order-4 dodecahedral honeycomb.
Cantellated order-5 cubic honeycomb
The cantellated order-5 cubic honeycomb, , has rhombicuboctahedron, icosidodecahedron, and pentagonal prism cells, with a wedge vertex figure.
Related honeycombs
It is similar to the Euclidean (order-4) cantellated cubic honeycomb, rr{4,3,4}:
Cantitruncated order-5 cubic honeycomb
The cantitruncated order-5 cubic honeycomb, , has truncated cuboctahedron, truncated icosahedron, and pentagonal prism cells, with a mirrored sphenoid vertex figure.
Related honeycombs
It is similar to the Euclidean (order-4) cantitruncated cubic honeycomb, tr{4,3,4}:
Runcinated order-5 cubic honeycomb
The runcinated order-5 cubic honeycomb or runcinated order-4 dodecahedral honeycomb , has cube, dodecahedron, and pentagonal prism cells, with an irregular triangular antiprism vertex figure.
It is analogous to the 2D hyperbolic rhombitetrapentagonal tiling, rr{4,5}, with square and pentagonal faces:
Related honeycombs
It is similar to the Euclidean (order-4) runcinated cubic honeycomb, t0,3{4,3,4}:
Runcitruncated or
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https://en.wikipedia.org/wiki/Icosahedral%20honeycomb
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In geometry, the icosahedral honeycomb is one of four compact, regular, space-filling tessellations (or honeycombs) in hyperbolic 3-space. With Schläfli symbol there are three icosahedra around each edge, and 12 icosahedra around each vertex, in a regular dodecahedral vertex figure.
Description
The dihedral angle of a regular icosahedron is around 138.2°, so it is impossible to fit three icosahedra around an edge in Euclidean 3-space. However, in hyperbolic space, properly scaled icosahedra can have dihedral angles of exactly 120 degrees, so three of those can fit around an edge.
Related regular honeycombs
There are four regular compact honeycombs in 3D hyperbolic space:
Related regular polytopes and honeycombs
It is a member of a sequence of regular polychora and honeycombs {3,p,3} with deltrahedral cells:
It is also a member of a sequence of regular polychora and honeycombs {p,5,p}, with vertex figures composed of pentagons:
Uniform honeycombs
There are nine uniform honeycombs in the [3,5,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2{3,5,3}, , also called truncated dodecahedral honeycomb, each of whose cells are truncated dodecahedra.
Rectified icosahedral honeycomb
The rectified icosahedral honeycomb, t1{3,5,3}, , has alternating dodecahedron and icosidodecahedron cells, with a triangular prism vertex figure:
Perspective projections from center of Poincaré disk model
Related honeycomb
There are four rectified compact regular honeycombs:
Truncated icosahedral honeycomb
The truncated icosahedral honeycomb, t0,1{3,5,3}, , has alternating dodecahedron and truncated icosahedron cells, with a triangular pyramid vertex figure.
Related honeycombs
Bitruncated icosahedral honeycomb
The bitruncated icosahedral honeycomb, t1,2{3,5,3}, , has truncated dodecahedron cells with a tetragonal disphenoid vertex figure.
Related honeycombs
Cantellated icosahedral honeycomb
The cantellated icosahedral honeycomb, t0,2{3,5,3}, , has rhombicosidodecahedron, icosidodecahedron, and triangular prism cells, with a wedge vertex figure.
Related honeycombs
Cantitruncated icosahedral honeycomb
The cantitruncated icosahedral honeycomb, t0,1,2{3,5,3}, , has truncated icosidodecahedron, truncated dodecahedron, and triangular prism cells, with a mirrored sphenoid vertex figure.
Related honeycombs
Runcinated icosahedral honeycomb
The runcinated icosahedral honeycomb, t0,3{3,5,3}, , has icosahedron and triangular prism cells, with a pentagonal antiprism vertex figure.
Viewed from center of triangular prism
Related honeycombs
Runcitruncated icosahedral honeycomb
The runcitruncated icosahedral honeycomb, t0,1,3{3,5,3}, , has truncated icosahedron, rhombicosidodecahedron, hexagonal prism, and triangular prism cells, with an isosceles-trapezoidal pyramid vertex figure.
The runcicantellated icosahedral honeycomb is equivalent to the runcitruncated icosahedral honeycomb.
Viewed from center of triangular pris
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https://en.wikipedia.org/wiki/Fr%C3%A9chet%20manifold
|
In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.
More precisely, a Fréchet manifold consists of a Hausdorff space with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus has an open cover and a collection of homeomorphisms onto their images, where are Fréchet spaces, such that
is smooth for all pairs of indices
Classification up to homeomorphism
It is by no means true that a finite-dimensional manifold of dimension is homeomorphic to or even an open subset of However, in an infinite-dimensional setting, it is possible to classify "well-behaved" Fréchet manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Fréchet manifold can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, (up to linear isomorphism, there is only one such space).
The embedding homeomorphism can be used as a global chart for Thus, in the infinite-dimensional, separable, metric case, up to homeomorphism, the "only" topological Fréchet manifolds are the open subsets of the separable infinite-dimensional Hilbert space. But in the case of or Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.
See also
, of which a Fréchet manifold is a generalization
References
Generalized manifolds
Manifolds
Nonlinear functional analysis
Structures on manifolds
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https://en.wikipedia.org/wiki/JoCaml
|
JoCaml is an experimental functional programming language derived from OCaml. It integrates the primitives of the join-calculus to enable flexible, type-checked concurrent and distributed programming. The current version of JoCaml is a re-implementation of the now unmaintained JoCaml made by Fabrice Le Fessant, featuring a modified syntax and improved OCaml compatibility compared to the original.
JoCaml was used by team Camls 'R Us to implement a distributed ray tracer, earning 2nd place on the ICFP 2000 programming contest.
The name is a reference to Joe Camel, a cartoon camel used in advertisements for Camel-brand cigarettes.
Example
type coins = Nickel | Dime
and drinks = Coffee | Tea
and buttons = BCoffee | BTea | BCancel;;
(* def defines a Join-pattern alternatives set clause
* '&' in the left side of '=' means join (channel synchronism)
* '&' in the right hand side is parallel processing
* synchronous_reply :== "reply" [x] "to" channel_name
* synchronous channels have function-like types (`a -> `b)
* while asynchronous ones have type `a Join.chan
* only the last statement in a pattern rhs expression can be an asynchronous message
* 0 in an asynchronous message position means STOP ("no sent message" in CSP terminology).
*)
def put(s) = print_endline s ; 0 (* STOP *)
;; (* put: string Join.chan *)
def give(d) = match d with
Coffee -> put("Coffee")
| Tea -> put("Tea")
;; (* give: drink Join.chan *)
def refund(v) = let s = Printf.sprintf "Refund %d" v in put(s)
;; (* refund: int Join.chan *)
let new_vending give refund =
let vend (cost:int) (credit:int) = if credit >= cost
then (true, credit - cost)
else (false, credit)
in
def coin(Nickel) & value(v) = value(v+5) & reply to coin
or coin(Dime) & value(v) = value(v+10) & reply to coin
or button(BCoffee) & value(v) =
let should_give, remainder = vend 10 v in
(if should_give then give(Coffee) else 0 (* STOP *))
& value(remainder) & reply to button
or button(BTea) & value(v) =
let should_give, remainder = vend 5 v in
(if should_give then give(Tea) else 0 (* STOP *))
& value(remainder) & reply to button
or button(BCancel) & value(v) = refund( v) & value(0) & reply to button
in spawn value(0) ;
coin, button (* coin, button: int -> unit *)
;; (* new_vending: drink Join.chan -> int Join.chan -> (int->unit)*(int->unit) *)
let ccoin, cbutton = new_vending give refund in
ccoin(Nickel); ccoin(Nickel); ccoin(Dime);
Unix.sleep(1); cbutton(BCoffee);
Unix.sleep(1); cbutton(BTea);
Unix.sleep(1); cbutton(BCancel);
Unix.sleep(1) (* let the last message show up *)
;;
execution
$ jocamlc example.ml -o test
$ ./test
Coffee
Tea
Refund 5
See also
Join-calculus
References
External links
The join-calculus language
The JoCaml system
Concurrent programming languages
OCaml programming lan
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https://en.wikipedia.org/wiki/Simple%20%28abstract%20algebra%29
|
In mathematics, the term simple is used to describe an algebraic structure which in some sense cannot be divided by a smaller structure of the same type. Put another way, an algebraic structure is simple if the kernel of every homomorphism is either the whole structure or a single element. Some examples are:
A group is called a simple group if it does not contain a nontrivial proper normal subgroup.
A ring is called a simple ring if it does not contain a nontrivial two sided ideal.
A module is called a simple module if it does not contain a nontrivial submodule.
An algebra is called a simple algebra if it does not contain a nontrivial two sided ideal.
The general pattern is that the structure admits no non-trivial congruence relations.
The term is used differently in semigroup theory. A semigroup is said to be simple if it has no nontrivial
ideals, or equivalently, if Green's relation J is
the universal relation. Not every congruence on a semigroup is associated with an ideal, so a simple semigroup may
have nontrivial congruences. A semigroup with no nontrivial congruences is called congruence simple.
See also
semisimple
simple universal algebra
Abstract algebra
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https://en.wikipedia.org/wiki/Brauer%20algebra
|
In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality.
Structure
The Brauer algebra is a -algebra depending on the choice of a positive integer . Here is an indeterminate, but in practice is often specialised to the dimension of the fundamental representation of an orthogonal group . The Brauer algebra has the dimension
Diagrammatic definition
A basis of consists of all pairings on a set of elements (that is, all perfect matchings of a complete graph : any two of the elements may be matched to each other, regardless of their symbols). The elements are usually written in a row, with the elements beneath them.
The product of two basis elements and is obtained by concatenation: first identifying the endpoints in the bottom row of and the top row of (Figure AB in the diagram), then deleting the endpoints in the middle row and joining endpoints in the remaining two rows if they are joined, directly or by a path, in AB (Figure AB=nn in the diagram). Thereby all closed loops in the middle of AB are removed. The product of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by where is the number of deleted loops. In the example .
Generators and relations
can also be defined as the -algebra with generators satisfying the following relations:
Relations of the symmetric group:
whenever
Almost-idempotent relation:
Commutation:
whenever
Tangle relations
Untwisting:
:
In this presentation represents the diagram in which is always connected to directly beneath it except for and which are connected to and respectively. Similarly represents the diagram in which is always connected to directly beneath it except for being connected to and to .
Basic properties
The Brauer algebra is a subalgebra of the partition algebra.
The Brauer algebra is semisimple if .
The subalgebra of generated by the generators is the group algebra of the symmetric group .
The subalgebra of generated by the generators is the Temperley-Lieb algebra .
The Brauer algebra is a cellular algebra.
For a pairing let be the number of closed loops formed by identifying with for any : then the Jones trace obeys i.e. it is indeed a trace.
Representations
Brauer-Specht modules
Brauer-Specht modules are finite-dimensional modules of the Brauer algebra.
If is such that is semisimple,
they form a complete set of simple modules of . These modules are parametrized by partitions, because they are built from the Specht modules of the symmetric group, which are themselves parametrized by partitions.
For with , let be the set of perfect matchings of elements , such that is matched with one of the elements . For any ring , the space is
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https://en.wikipedia.org/wiki/Semisimple%20algebra
|
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras.
Definition
The Jacobson radical of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite-dimensional algebra is then said to be semisimple if its radical contains only the zero element.
An algebra A is called simple if it has no proper ideals and A2 = {ab | a, b ∈ A} ≠ {0}. As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra A are A and {0}. Thus if A is simple, then A is not nilpotent. Because A2 is an ideal of A and A is simple, A2 = A. By induction, An = A for every positive integer n, i.e. A is not nilpotent.
Any self-adjoint subalgebra A of n × n matrices with complex entries is semisimple. Let Rad(A) be the radical of A. Suppose a matrix M is in Rad(A). Then M*M lies in some nilpotent ideals of A, therefore (M*M)k = 0 for some positive integer k. By positive-semidefiniteness of M*M, this implies M*M = 0. So M x is the zero vector for all x, i.e. M = 0.
If {Ai} is a finite collection of simple algebras, then their Cartesian product A=Π Ai is semisimple. If (ai) is an element of Rad(A) and e1 is the multiplicative identity in A1 (all simple algebras possess a multiplicative identity), then (a1, a2, ...) · (e1, 0, ...) = (a1, 0..., 0) lies in some nilpotent ideal of Π Ai. This implies, for all b in A1, a1b is nilpotent in A1, i.e. a1 ∈ Rad(A1). So a1 = 0. Similarly, ai = 0 for all other i.
It is less apparent from the definition that the converse of the above is also true, that is, any finite-dimensional semisimple algebra is isomorphic to a Cartesian product of a finite number of simple algebras.
Characterization
Let A be a finite-dimensional semisimple algebra, and
be a composition series of A, then A is isomorphic to the following Cartesian product:
where each
is a simple algebra.
The proof can be sketched as follows. First, invoking the assumption that A is semisimple, one can show that the J1 is a simple algebra (therefore unital). So J1 is a unital subalgebra and an ideal of J2. Therefore, one can decompose
By maximality of J1 as an ideal in J2 and also the semisimplicity of A, the algebra
is simple. Proceed by induction in similar fashion proves the claim. For example, J3 is the Cartesian product of simple algebras
The above result can be restated in a different way. For a semisimple algebra A = A1 ×...× An expressed in terms of its simple factors, consider the units ei ∈ Ai. The elements Ei = (0,...,ei,...,0) are idempotent elements in A an
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https://en.wikipedia.org/wiki/Hereditary%20set
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In set theory, a hereditary set (or pure set) is a set whose elements are all hereditary sets. That is, all elements of the set are themselves sets, as are all elements of the elements, and so on.
Examples
For example, it is vacuously true that the empty set is a hereditary set, and thus the set containing only the empty set is a hereditary set. Similarly, a set that contains two elements: the empty set and the set that contains only the empty set, is a hereditary set.
In formulations of set theory
In formulations of set theory that are intended to be interpreted in the von Neumann universe or to express the content of Zermelo–Fraenkel set theory, all sets are hereditary, because the only sort of object that is even a candidate to be an element of a set is another set. Thus the notion of hereditary set is interesting only in a context in which there may be urelements.
Assumptions
The inductive definition of hereditary sets presupposes that set membership is well-founded (i.e., the axiom of regularity), otherwise the recurrence may not have a unique solution. However, it can be restated non-inductively as follows: a set is hereditary if and only if its transitive closure contains only sets.
In this way the concept of hereditary sets can also be extended to non-well-founded set theories in which sets can be members of themselves. For example, a set that contains only itself is a hereditary set.
See also
Hereditarily countable set
Hereditarily finite set
Well-founded set
References
Set theory
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https://en.wikipedia.org/wiki/Innisfree%2C%20Alberta
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Innisfree is a village in central Alberta, Canada. It is located 52 km west of Vermilion along the Yellowhead Highway.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, the Village of Innisfree had a population of 187 living in 94 of its 124 total private dwellings, a change of from its 2016 population of 193. With a land area of , it had a population density of in 2021.
The population of the Village of Innisfree according to its 2017 municipal census is 223.
In the 2016 Census of Population conducted by Statistics Canada, the Village of Innisfree recorded a population of 193 living in 96 of its 126 total private dwellings, a change from its 2011 population of 220. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of villages in Alberta
References
External links
1911 establishments in Alberta
County of Minburn No. 27
Villages in Alberta
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https://en.wikipedia.org/wiki/Coefficient%20%28disambiguation%29
|
Coefficient could have one of the following meanings:
Mathematics
A coefficient is a constant multiplication of a function.
The term differential coefficient has been mostly displaced by the modern term derivative.
Computing
In computer arithmetics, the term coefficient (floating point number) is also sometimes used as a synonym for mantissa or significand.
Probability theory and Statistics
The coefficient of determination, denoted R2 and pronounced R squared, is the proportion of total variation of outcomes explained by a statistical model.
The coefficient of variation (CV) is a normalized measure of dispersion of a probability distribution or frequency distribution.
The correlation coefficient (Pearson's r) is a measure of the linear correlation (dependence) between two variables.
Science
In physics, a physical coefficient is an important number that characterizes some physical property of an object.
In chemistry, a stoichiometric coefficient is a number placed in front of a term in a chemical equation to indicate how many molecules (or atoms) take part in the reaction.
Other
UEFA coefficient, used by the governing body for association football in Europe to calculate ranking points for its member clubs and national federations
The Coefficients were an Edwardian London dining club.
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https://en.wikipedia.org/wiki/Second%20partial%20derivative%20test
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In mathematics, the second partial derivative test is a method in multivariable calculus used to determine if a critical point of a function is a local minimum, maximum or saddle point.
Functions of two variables
Suppose that is a differentiable real function of two variables whose second partial derivatives exist and are continuous. The Hessian matrix of is the 2 × 2 matrix of partial derivatives of :
Define to be the determinant
of . Finally, suppose that is a critical point of , that is, that . Then the second partial derivative test asserts the following:
If and then is a local minimum of .
If and then is a local maximum of .
If then is a saddle point of .
If then the point could be any of a minimum, maximum, or saddle point (that is, the test is inconclusive).
Sometimes other equivalent versions of the test are used. In cases 1 and 2, the requirement that is positive at implies that and have the same sign there. Therefore, the second condition, that be greater (or less) than zero, could equivalently be that or be greater (or less) than zero at that point.
A condition implicit in the statement of the test is that if or , it must be the case that and therefore only cases 3 or 4 are possible.
Functions of many variables
For a function f of three or more variables, there is a generalization of the rule above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point. The following test can be applied at any critical point a for which the Hessian matrix is invertible:
If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a.
If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate).
In those cases not listed above, the test is inconclusive.
For functions of three or more variables, the determinant of the Hessian does not provide enough information to classify the critical point, because the number of jointly sufficient second-order conditions is equal to the number of variables, and the sign condition on the determinant of the Hessian is only one of the conditions. Note that in the one-variable case, the Hessian condition simply gives the usual second derivative test.
In the two variable case, and are the principal minors of the Hessian. The first two conditions listed above on the signs of these minors are the conditions for the positive or negative definiteness of the Hessian. For the general case of an arbitrary number n of variables, there are n sign conditions on the n principal minors of the Hessian matrix that together are equivalent to positive or negative definiteness of the Hessian (Sylvester's criterion): for a local minimum,
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https://en.wikipedia.org/wiki/Polynomial%20transformation
|
In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.
Simple examples
Translating the roots
Let
be a polynomial, and
be its complex roots (not necessarily distinct).
For any constant , the polynomial whose roots are
is
If the coefficients of are integers and the constant is a rational number, the coefficients of may be not integers, but the polynomial has integer coefficients and has the same roots as .
A special case is when The resulting polynomial does not have any term in .
Reciprocals of the roots
Let
be a polynomial. The polynomial whose roots are the reciprocals of the roots of as roots is its reciprocal polynomial
Scaling the roots
Let
be a polynomial, and be a non-zero constant. A polynomial whose roots are the product by of the roots of is
The factor appears here because, if and the coefficients of are integers or belong to some integral domain, the same is true for the coefficients of .
In the special case where , all coefficients of are multiple of , and is a monic polynomial, whose coefficients belong to any integral domain containing and the coefficients of . This polynomial transformation is often used to reduce questions on algebraic numbers to questions on algebraic integers.
Combining this with a translation of the roots by , allows to reduce any question on the roots of a polynomial, such as root-finding, to a similar question on a simpler polynomial, which is monic and does not have a term of degree . For examples of this, see Cubic function § Reduction to a depressed cubic or Quartic function § Converting to a depressed quartic.
Transformation by a rational function
All preceding examples are polynomial transformations by a rational function, also called Tschirnhaus transformations. Let
be a rational function, where and are coprime polynomials. The polynomial transformation of a polynomial by is the polynomial (defined up to the product by a non-zero constant) whose roots are the images by of the roots of .
Such a polynomial transformation may be computed as a resultant. In fact, the roots of the desired polynomial are exactly the complex numbers such that there is a complex number such that one has simultaneously (if the coefficients of and are not real or complex numbers, "complex number" has to be replaced by "element of an algebraically closed field containing the coefficients of the input polynomials")
This is exactly the defining property of the resultant
This is generally difficult to compute by hand. However, as most computer algebra systems have a built-in function to compute resultants, it is straightforward to compute it with a computer.
Properties
If the polynomial is irreducible, then either the resulting polynomial is irreducible, or it is
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https://en.wikipedia.org/wiki/J.%20Fuller
|
J. Fuller was a publisher in 18th-century England.
Publications
"A Lover of the Mathematics". A Mathematical Miscellany in Four Parts. 2nd ed., S. Fuller, Dublin, 1735. The First Part is: An Essay towards the Probable Solution of the Forty five Surprising PARADOXES, in GORDON's Geography.
Gentleman's Diary or The Mathematical Repository (1741-1745)
English publishers (people)
Businesspeople from Dublin (city)
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https://en.wikipedia.org/wiki/Near-miss%20Johnson%20solid
|
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whose faces are all regular, though it "can often be physically constructed without noticing the discrepancy" between its regular and irregular faces. The precise number of near-misses depends on how closely the faces of such a polyhedron are required to approximate regular polygons.
Some near-misses with high symmetry are also symmetrohedra with some truly regular polygon faces.
Some near-misses are also zonohedra.
Examples
Coplanar misses
Some failed Johnson solid candidates have coplanar faces. These polyhedra can be perturbed to become convex with faces that are arbitrarily close to regular polygons. These cases use 4.4.4.4 vertex figures of the square tiling, 3.3.3.3.3.3 vertex figure of the triangular tiling, as well as 60 degree rhombi divided double equilateral triangle faces, or a 60 degree trapezoid as three equilateral triangles. It is possible to take an infinite amount of distinct coplanar misses from sections of the cubic honeycomb (alternatively convex polycubes) or alternated cubic honeycomb, ignoring any obscured faces.
Examples:
3.3.3.3.3.3
4.4.4.4
3.4.6.4:
See also
Geodesic polyhedron
Goldberg polyhedron
Johnson solid
Platonic solid
Semiregular polyhedron
Archimedean solid
Prism
Antiprism
References
External links
Near-miss Johnson solid, Polytope Wiki (74)
Johnson Solid Near Misses, Polyhedra by Jim McNeil (31)
Near Misses, Craig S. Kaplan (5)
Polyhedra
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https://en.wikipedia.org/wiki/Bosonic%20field
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In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose–Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeyed by fermionic fields.
Examples include scalar fields, describing spin-0 particles such as the Higgs boson, and gauge fields, describing spin-1 particles such as the photon.
Basic properties
Free (non-interacting) bosonic fields obey canonical commutation relations. Those relations also hold for interacting bosonic fields in the interaction picture, where the fields evolve in time as if free and the effects of the interaction are encoded in the evolution of the states. It is these commutation relations that imply Bose–Einstein statistics for the field quanta.
Examples
Examples of bosonic fields include scalar fields, gauge fields, and symmetric 2-tensor fields, which are characterized by their covariance under Lorentz transformations and have spins 0, 1 and 2, respectively. Physical examples, in the same order, are the Higgs field, the photon field, and the graviton field. Of the last two, only the photon field can be quantized using the conventional methods of canonical or path integral quantization. This has led to the theory of quantum electrodynamics, one of the most successful theories in physics. Quantization of gravity, on the other hand, is a long-standing problem that has led to development of theories such as string theory and loop quantum gravity.
Spin and statistics
The spin–statistics theorem implies that quantization of local, relativistic field theories in 3+1 dimensions may lead either to bosonic or fermionic quantum fields, i.e., fields obeying commutation or anti-commutation relations, according to whether they have integer or half-integer spin, respectively. Thus bosonic fields are one of the two theoretically possible types of quantum field, namely those corresponding to particles with integer spin.
In a non-relativistic many-body theory, the spin and the statistical properties of the quanta are not directly related. In fact, the commutation or anti-commutation relations are assumed based on whether the theory one intends to study corresponds to particles obeying Bose–Einstein or Fermi–Dirac statistics. In this context the spin remains an internal quantum number that is only phenomenologically related to the statistical properties of the quanta. Examples of non-relativistic bosonic fields include those describing cold bosonic atoms, such as Helium-4.
Such non-relativistic fields are not as fundamental as their relativistic counterparts: they provide a convenient 're-packaging' of the many-body wave function describing the state of the system, whereas the relativistic fields described above are a necessary consequence of the consistent union of relativity and quantum mechanics.
See also
Quantum triviality
Composite field
Auxiliary field
References
Peskin, M and Schroeder,
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https://en.wikipedia.org/wiki/Albert%20A.%20Murphree
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Albert Alexander Murphree (April 29, 1870 – December 20, 1927) was an American college professor and university president. Murphree was a native of Alabama, and became a mathematics instructor after earning his bachelor's degree. He later served as the third president of Florida State College (later renamed Florida State University) from 1897 to 1909, and the second president of the University of Florida from 1909 to 1927. Murphree is the only person to have been the president of both of Florida's original state universities, the University of Florida and Florida State University, and he played an important role in the organization, growth and ultimate success of both institutions.
Early life and education
Murphree was born near Chepultepec, Alabama in 1870. His father was Jesee Ellis Murphree, a Confederate veteran of the Civil War; his mother was Emily Helen Cornelius. His parents raised him in a family of ten children in Walnut Grove, Alabama, where he attended community schools and a local two-year college. He graduated from the University of Nashville with a Bachelor of Arts degree in 1894, and taught mathematics at several high schools and small colleges in Alabama, Tennessee and Texas. In 1895, he became a mathematics instructor at the West Florida Seminary (now known as Florida State University) in Tallahassee, Florida, and two years later, its board of trustees appointed him as the seminary's third president in 1897, at the age of 27. Later, Murphree married Jennie Henderson, the daughter of John A. Henderson, one of the seminary's trustees. He subsequently started and completed the academic work for a Master of Arts degree while serving as president of the seminary, renamed Florida State College in 1901.
Professor and university president
As President of West Florida Seminary, Murphree worked to create Florida's first liberal arts college by 1897, and in 1901 it was reorganized into the Florida State College with four departments: the College, the College Academy, the School for Teachers and the School of Music. Under his leadership, the Florida State College produced the state's first Rhodes Scholar in 1905, Frederic "Fritz" Buchholz (1885-65).
In 1905, several prominent political backers advanced Murphree's name to be the first president of the new University of the State of Florida located in Gainesville, Florida, which was the newly consolidated men's university and land-grant college created by the Florida Legislature's passage of the Buckman Act, which segregated Florida's schools of higher learning by race and gender. Instead, the Florida Board of Control selected Andrew Sledd, then the president of the University of Florida in Lake City, to be the first president of the new men's university. Murphree continued to serve as the president of Florida State College, which became the all-female Florida Female College under the Buckman Act. From 1905 to 1909, Murphree emphasized greater academic expectations for his
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https://en.wikipedia.org/wiki/Probabilistic%20metric%20space
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In mathematics, probabilistic metric spaces are a generalization of metric spaces where the distance no longer takes values in the non-negative real numbers , but in distribution functions.
Let D+ be the set of all probability distribution functions F such that F(0) = 0 (F is a nondecreasing, left continuous mapping from R into [0, 1] such that max(F) = 1).
Then given a non-empty set S and a function F: S × S → D+ where we denote F(p, q) by Fp,q for every (p, q) ∈ S × S, the ordered pair (S, F) is said to be a probabilistic metric space if:
For all u and v in S, if and only if for all x > 0.
For all u and v in S, .
For all u, v and w in S, and for .
Probability metric of random variables
A probability metric D between two random variables X and Y may be defined, for example, as
where F(x, y) denotes the joint probability density function of the random variables X and Y. If X and Y are independent from each other then the equation above transforms into
where f(x) and g(y) are probability density functions of X and Y respectively.
One may easily show that such probability metrics do not satisfy the first metric axiom or satisfies it if, and only if, both of arguments X and Y are certain events described by Dirac delta density probability distribution functions. In this case:
the probability metric simply transforms into the metric between expected values , of the variables X and Y.
For all other random variables X, Y the probability metric does not satisfy the identity of indiscernibles condition required to be satisfied by the metric of the metric space, that is:
Example
For example if both probability distribution functions of random variables X and Y are normal distributions (N) having the same standard deviation , integrating yields:
where
and is the complementary error function.
In this case:
Probability metric of random vectors
The probability metric of random variables may be extended into metric D(X, Y) of random vectors X, Y by substituting with any metric operator d(x, y):
where F(X, Y) is the joint probability density function of random vectors X and Y. For example substituting d(x, y) with Euclidean metric and providing the vectors X and Y are mutually independent would yield to:
References
Probability distributions
Metric geometry
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https://en.wikipedia.org/wiki/Difference%20hierarchy
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In set theory, a branch of mathematics, the difference hierarchy over a pointclass is a hierarchy of larger pointclasses
generated by taking differences of sets. If Γ is a pointclass, then the set of differences in Γ is . In usual notation, this set is denoted by 2-Γ. The next level of the hierarchy is denoted by 3-Γ and consists of differences of three sets:
. This definition can be extended recursively into the transfinite to α-Γ for some ordinal α.
In the Borel hierarchy, Felix Hausdorff and Kazimierz Kuratowski proved that the countable levels of the
difference hierarchy over Π0γ give
Δ0γ+1.
References
Descriptive set theory
Mathematical logic hierarchies
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https://en.wikipedia.org/wiki/Bjarni%20J%C3%B3nsson
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Bjarni Jónsson (February 15, 1920 – September 30, 2016) was an Icelandic mathematician and logician working in universal algebra, lattice theory, model theory and set theory. He was emeritus distinguished professor of mathematics at Vanderbilt University and the honorary editor in chief of Algebra Universalis. He received his PhD in 1946 at UC Berkeley under supervision of Alfred Tarski.
In 2012, he became a fellow of the American Mathematical Society.
Work
Jónsson's lemma as well as several mathematical objects are named after him, among them Jónsson algebras, ω-Jónsson functions, Jónsson cardinals, Jónsson terms, Jónsson–Tarski algebras and Jónsson–Tarski duality.
Publications
References
Further reading
Kirby A. Baker, Bjarni Jónsson's contributions in algebra, Algebra Universalis, September 1994, Volume 31, Issue 3, pp. 306–336.
J. B. Nation, Jónsson's contributions to lattice theory, Algebra Universalis, September 1994, Volume 31, Issue 3, pp. 430–445.
External links
Bjarni Jónsson's homepage
1920 births
Bjarni Jonsson
Algebraists
Lattice theorists
20th-century mathematicians
University of California, Berkeley alumni
Vanderbilt University faculty
Fellows of the American Mathematical Society
Bjarni Jonsson
2016 deaths
Brown University faculty
Icelandic expatriates in the United States
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https://en.wikipedia.org/wiki/Schur-convex%20function
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In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function that for all such that is majorized by , one has that . Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).
Schur-concave function
A function f is 'Schur-concave' if its negative, −f, is Schur-convex.
Schur-Ostrowski criterion
If f is symmetric and all first partial derivatives exist, then
f is Schur-convex if and only if
for all
holds for all .
Examples
is Schur-concave while is Schur-convex. This can be seen directly from the definition.
The Shannon entropy function is Schur-concave.
The Rényi entropy function is also Schur-concave.
is Schur-convex.
The function is Schur-concave, when we assume all . In the same way, all the elementary symmetric functions are Schur-concave, when .
A natural interpretation of majorization is that if then is more spread out than . So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the median absolute deviation is not.
If is a convex function defined on a real interval, then is Schur-convex.
A probability example: If are exchangeable random variables, then the function is Schur-convex as a function of , assuming that the expectations exist.
The Gini coefficient is strictly Schur convex.
References
See also
Quasiconvex function
Convex analysis
Inequalities
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https://en.wikipedia.org/wiki/National%20Bureau%20of%20Statistics%20of%20the%20Republic%20of%20Moldova
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The National Bureau of Statistics of the Republic of Moldova (NBS; , abbr. BNS) is the central administrative authority which, as the central statistical body, manages and coordinates the activity in the field of statistics from the country.
In its activity, NBS acts according to the Constitution of the Republic of Moldova, the Law on official statistics, other legislative acts, Parliament decisions, decrees of the President of the Republic of Moldova, ordinances, decisions and Government orders, international treaties of which the Republic of Moldova is part of.
The NBS elaborates independently or in collaboration with other central administrative bodies and approves the methodologies of statistical and calculation surveys of statistical indicators, in accordance with international standards, especially those of the European Union, and with the advanced practice of other countries, as well as taking into account the peculiarities of the socio-economic conditions of the Republic of Moldova, organizes, following the programme of statistical works, annually approved by the Government, statistical surveys regarding the situation and economic, social, demographic development of the country, performing the works related to the collection, processing, centralizing, storage and dissemination of statistical data.
The content published by National Bureau of Statistics on its website may be reused completely or partly, in original or modified, as well as its storage in a retrieval system, or transmitted, in any form and by any means, unless otherwise stated, under the Creative Commons Attribution 4.0 International License.
References
External links
National Bureau of Statistics of the Republic of Moldova - official page
Statistical databank of Moldova
Statistical Yearbook of the Republic of Moldova
Animated map "Vital statistics rates"
Animated population pyramid
Consumer Price Index Calculator
Government agencies of Moldova
Moldova
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https://en.wikipedia.org/wiki/Quintile
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Quintile may refer to:
In statistics, a quantile for the case where the sample or population is divided into fifths
Quintiles, a biotechnology research company based in the United States
Quintile (astrology), a type of astrological aspect formed by a 72° angle
See also
1/5 (disambiguation)
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https://en.wikipedia.org/wiki/Hermann%E2%80%93Mauguin%20notation
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In geometry, Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann (who introduced it in 1928) and the French mineralogist Charles-Victor Mauguin (who modified it in 1931). This notation is sometimes called international notation, because it was adopted as standard by the International Tables For Crystallography since their first edition in 1935.
The Hermann–Mauguin notation, compared with the Schoenflies notation, is preferred in crystallography because it can easily be used to include translational symmetry elements, and it specifies the directions of the symmetry axes.
Point groups
Rotation axes are denoted by a number n — 1, 2, 3, 4, 5, 6, 7, 8 ... (angle of rotation φ = ). For improper rotations, Hermann–Mauguin symbols show rotoinversion axes, unlike Schoenflies and Shubnikov notations, that shows rotation-reflection axes. The rotoinversion axes are represented by the corresponding number with a macron, — , , , , , , , , ... . is equivalent to a mirror plane and usually notated as m. The direction of the mirror plane is defined as the direction perpendicular to it (the direction of the axis).
Hermann–Mauguin symbols show non-equivalent axes and planes in a symmetrical fashion. The direction of a symmetry element corresponds to its position in the Hermann–Mauguin symbol. If a rotation axis n and a mirror plane m have the same direction (i.e. the plane is perpendicular to axis n), then they are denoted as a fraction or n/m.
If two or more axes have the same direction, the axis with higher symmetry is shown. Higher symmetry means that the axis generates a pattern with more points. For example, rotation axes 3, 4, 5, 6, 7, 8 generate 3-, 4-, 5-, 6-, 7-, 8-point patterns, respectively. Improper rotation axes , , , , , generate 6-, 4-, 10-, 6-, 14-, 8-point patterns, respectively. If a rotation and a rotoinversion axis generate the same number of points, the rotation axis should be chosen. For example, the combination is equivalent to . Since generates 6 points, and 3 generates only 3, should be written instead of (not , because already contains the mirror plane m). Analogously, in the case when both 3 and axes are present, should be written. However we write , not , because both 4 and generate four points. In the case of the combination, where 2, 3, 6, , and axes are present, axes , , and 6 all generate 6-point patterns, as we can see on the figure in the right, but the latter should be used because it is a rotation axis — the symbol will be .
Finally, the Hermann–Mauguin symbol depends on the type of the group.
Groups without higher-order axes (axes of order three or more)
These groups may contain only two-fold axes, mirror planes, and/or an inversion center. These are the crystallographic point groups 1 and (triclinic crystal system), 2, m, and (monoclinic), and 222, , and mm2 (orthorhombic). (The sh
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https://en.wikipedia.org/wiki/Seed%20dormancy
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Seed dormancy is an evolutionary adaptation that prevents seeds from germinating during unsuitable ecological conditions that would typically lead to a low probability of seedling survival. Dormant seeds do not germinate in a specified period of time under a combination of environmental factors that are normally conducive to the germination of non-dormant seeds.
An important function of seed dormancy is delayed germination, which allows dispersal and prevents simultaneous germination of all seeds. The staggering of germination safeguards some seeds and seedlings from suffering damage or death from short periods of bad weather or from transient herbivores; it also allows some seeds to germinate when competition from other plants for light and water might be less intense. Another form of delayed seed germination is seed quiescence, which is different from true seed dormancy and occurs when a seed fails to germinate because the external environmental conditions are too dry or warm or cold for germination.
Many species of plants have seeds that delay germination for many months or years, and some seeds can remain in the soil seed bank for more than 50 years before germination. Seed dormancy is especially adaptive in fire-prone ecosystems. Some seeds have a very long viability period, and the oldest documented germinating seed was nearly 2000 years old based on radiocarbon dating.
Overview
True dormancy or inherent (or innate) dormancy is caused by conditions within the seed that prevent germination even if the conditions are favorable. Imposed dormancy is caused by the external conditions that remain unsuitable for germination Seed dormancy can be divided into two major categories based on what part of the seed produces dormancy: exogenous and endogenous. There are three types of inherent dormancy based on their mode of action: physical, physiological and morphological.
There have been a number of classification schemes developed to group different dormant seeds, but none have gained universal usage. Dormancy occurs because of a wide range of reasons that often overlap, producing conditions in which definitive categorization is not clear. Compounding this problem is that the same seed that is dormant for one reason at a given point may be dormant for another reason at a later point. Some seeds fluctuate from periods of dormancy to non dormancy, and despite the fact that a dormant seed appears to be static or inert, in reality they are still receiving and responding to environmental cues.
Not all seeds undergo a period of dormancy, many species of plants release their seeds late in the year when the soil temperature is too low for germination or when the environment is dry. If these seeds are collected and sown in an environment that is warm enough, and/or moist enough, they will germinate. Under natural conditions non dormant seeds released late in the growing season wait until spring when the soil temperature rises or in the case of seeds dis
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https://en.wikipedia.org/wiki/Freescape
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Freescape is a video game engine, an early 3D game engine used in video games such as 1987's Driller. Graphics were composed mostly of solid geometry rendered without shading.
History
Developed in-house by Incentive Software, Freescape is considered to be one of the first proprietary 3D engines to be used in video games, although the engine was not used commercially outside of Incentive's own titles. The project was originally thought to be so ambitious that according to Incentive designer Ian Andrew, the company struggled to recruit programmers for the project, with many believing that it could not be achieved.
Paul Gregory (graphics artist for Major Developments, Incentive's in-house design team) mentions that Freescape was developed by Chris Andrew starting in September 1986 on an Amstrad CPC, as it was the most suitable development system with 128K memory and had adequate power to run 3D environments. Due to the engine's success, it was later ported to all the dominant systems of the era such as the ZX Spectrum, IBM PC, Commodore 64, Amiga and Atari ST.
Freescape development ended in 1992 with the release of 3D Construction Kit II. Its legacy continued in the latter Superscape VRT virtual reality authoring engine, from the same developer and advertised on the 3D Construction Kit II software.
Technology
Geometry
The Freescape engine allowed the generation of complete 3D environments that consist of a floor and as many primitives as memory and processor speed realistically allowed for. These primitives were cuboids, four-sided frustums (called pyramids by Freescape), triangles, rectangles, quadrilaterals, pentagons, hexagons and line segments. A further primitive, "sensor", was used for gaming purpose to detect the position of the camera relative to the sensor in the game world.
Freescape was designed with limited hardware in mind and as such contains a number of inherent limitations that are necessary to enable the games to run properly on these computers:
Individual regions were restricted to a size of 8192 × 4096 × 8192 units. These units were arbitrary but each region always corresponded to the dimensions.
The engine did not allow for fractional movements. On 16-bit machines each movement—camera or object—must be a multiple of one unit. On 8-bit machines the angles at which the world may be viewed are further restricted to steps of 5 degrees.
The x and z axes were subdivided into only 128 discrete locations, and the y axis is subdivided into only 64 discrete locations. As a result, objects can only be placed at 64 unit intervals, for example, 0,64,128 or 128,64,32.
Objects may not overlap.
All objects possessed a "bounding cube", for which detection rules apply as per a cube, i.e. no overlapping.
Interaction
Games used the Freescape Command Language ('FCL'), an early in-game scripting language, to add interactive elements to Freescape worlds. Scripts may be set to run constantly for the entire world or run constantly for a certa
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https://en.wikipedia.org/wiki/Projection%20%28mathematics%29
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In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost.
An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency). The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are:
The projection from a point onto a plane or central projection: If C is a point, called the center of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the line CP with the plane. The points P such that the line CP is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined.
The projection parallel to a direction D, onto a plane or parallel projection: The image of a point P is the intersection with the plane of the line parallel to D passing through P. See for an accurate definition, generalized to any dimension.
The concept of projection in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground. This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.
In cartography, a map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The 3D projections are also at the basis of the theory of perspective.
The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property not shared with the projections of this article.
Definition
Generally, a mapping where the domain and codomain are the same set (or mathematical structure) is a projection if the mapping is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a right inverse. Both notions are st
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https://en.wikipedia.org/wiki/Multiple%20%28mathematics%29
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In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, it can be said that b is a multiple of a if b = na for some integer n, which is called the multiplier. If a is not zero, this is equivalent to saying that is an integer.
When a and b are both integers, and b is a multiple of a, then a is called a divisor of b. One says also that a divides b. If a and b are not integers, mathematicians prefer generally to use integer multiple instead of multiple, for clarification. In fact, multiple is used for other kinds of product; for example, a polynomial p is a multiple of another polynomial q if there exists third polynomial r such that p = qr.
Examples
14, 49, −21 and 0 are multiples of 7, whereas 3 and −6 are not. This is because there are integers that 7 may be multiplied by to reach the values of 14, 49, 0 and −21, while there are no such integers for 3 and −6. Each of the products listed below, and in particular, the products for 3 and −6, is the only way that the relevant number can be written as a product of 7 and another real number:
is not an integer;
is not an integer.
Properties
0 is a multiple of every number ().
The product of any integer and any integer is a multiple of . In particular, , which is equal to , is a multiple of (every integer is a multiple of itself), since 1 is an integer.
If and are multiples of then and are also multiples of .
Submultiple
In some texts, "a is a submultiple of b" has the meaning of "a being a unit fraction of b" (a1/b) or, equivalently, "b being an integer multiple n of a" (bna). This terminology is also used with units of measurement (for example by the BIPM and NIST), where a unit submultiple is obtained by prefixing the main unit, defined as the quotient of the main unit by an integer, mostly a power of 103. For example, a millimetre is the 1000-fold submultiple of a metre. As another example, one inch may be considered as a 12-fold submultiple of a foot, or a 36-fold submultiple of a yard.
See also
Unit fraction
Ideal (ring theory)
Decimal and SI prefix
Multiplier (linguistics)
References
Arithmetic
Multiplication
Integers
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https://en.wikipedia.org/wiki/Projection%20%28set%20theory%29
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In set theory, a projection is one of two closely related types of functions or operations, namely:
A set-theoretic operation typified by the th projection map, written that takes an element of the Cartesian product to the value
A function that sends an element to its equivalence class under a specified equivalence relation or, equivalently, a surjection from a set to another set. The function from elements to equivalence classes is a surjection, and every surjection corresponds to an equivalence relation under which two elements are equivalent when they have the same image. The result of the mapping is written as when is understood, or written as when it is necessary to make explicit.
See also
References
Basic concepts in set theory
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https://en.wikipedia.org/wiki/WFO
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WFO may refer to:
Well-founded ordering, in mathematics, see well-founded relation
W.F.O. (album), a 1994 album by the thrash metal band Overkill
Workforce optimization, strategy for managing contact center staffing, processes, and workflows.
Weather Forecast Office, a local forecasting and warning office of the United States National Weather Service: See List of National Weather Service Weather Forecast Offices
Washington Field Office, of the United States Secret Service
Washington Field Office, of the Federal Bureau of Investigation
World Flora Online
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https://en.wikipedia.org/wiki/Lyndon%20word
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In mathematics, in the areas of combinatorics and computer science, a Lyndon word is a nonempty string that is strictly smaller in lexicographic order than all of its rotations. Lyndon words are named after mathematician Roger Lyndon, who investigated them in 1954, calling them standard lexicographic sequences. Anatoly Shirshov introduced Lyndon words in 1953 calling them regular words. Lyndon words are a special case of Hall words; almost all properties of Lyndon words are shared by Hall words.
Definitions
Several equivalent definitions exist.
A -ary Lyndon word of length is an -character string over an alphabet of size , and which is the unique minimum element in the lexicographical ordering in the multiset of all its rotations. Being the singularly smallest rotation implies that a Lyndon word differs from any of its non-trivial rotations, and is therefore aperiodic.
Alternately, a word is a Lyndon word if and only if it is nonempty and lexicographically strictly smaller than any of its proper suffixes, that is for all nonempty words such that and is nonempty.
Another characterisation is the following: A Lyndon word has the property that it is nonempty and, whenever it is split into two nonempty substrings, the left substring is always lexicographically less than the right substring. That is, if is a Lyndon word, and is any factorization into two substrings, with and understood to be non-empty, then . This definition implies that a string of length is a Lyndon word if and only if there exist Lyndon words and such that and . Although there may be more than one choice of and with this property, there is a particular choice, called the standard factorization, in which is as long as possible.
Enumeration
The Lyndon words over the two-symbol binary alphabet {0,1}, sorted by length and then lexicographically within each length class, form an infinite sequence that begins
0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00101, 00111, 01011, 01111, ...
The first string that does not belong to this sequence, "00", is omitted because it is periodic (it consists of two repetitions of the substring "0"); the second omitted string, "10", is aperiodic but is not minimal in its permutation class as it can be cyclically permuted to the smaller string "01".
The empty string also meets the definition of a Lyndon word of length zero. The numbers of binary Lyndon words of each length, starting with length zero, form the integer sequence
1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, ...
Lyndon words correspond to aperiodic necklace class representatives and can thus be counted with Moreau's necklace-counting function.
Generation
provides an efficient algorithm for listing the Lyndon words of length at most with a given alphabet size in lexicographic order. If is one of the words in the sequence, then the next word after can be found by the following steps:
Repeat the symbols from to form a new word of length exactly , where
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https://en.wikipedia.org/wiki/Q-derivative
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In mathematics, in the area of combinatorics and quantum calculus, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inverse of Jackson's q-integration. For other forms of q-derivative, see .
Definition
The q-derivative of a function f(x) is defined as
It is also often written as . The q-derivative is also known as the Jackson derivative.
Formally, in terms of Lagrange's shift operator in logarithmic variables, it amounts to the operator
which goes to the plain derivative, as .
It is manifestly linear,
It has a product rule analogous to the ordinary derivative product rule, with two equivalent forms
Similarly, it satisfies a quotient rule,
There is also a rule similar to the chain rule for ordinary derivatives. Let . Then
The eigenfunction of the q-derivative is the q-exponential eq(x).
Relationship to ordinary derivatives
Q-differentiation resembles ordinary differentiation, with curious differences. For example, the q-derivative of the monomial is:
where is the q-bracket of n. Note that so the ordinary derivative is regained in this limit.
The n-th q-derivative of a function may be given as:
provided that the ordinary n-th derivative of f exists at x = 0. Here, is the q-Pochhammer symbol, and is the q-factorial. If is analytic we can apply the Taylor formula to the definition of to get
A q-analog of the Taylor expansion of a function about zero follows:
Higher order q-derivatives
The following representation for higher order -derivatives is known:
is the -binomial coefficient. By changing the order of summation as , we obtain the next formula:
Higher order -derivatives are used to -Taylor formula and the -Rodrigues' formula (the formula used to construct -orthogonal polynomials).
Generalizations
Post Quantum Calculus
Post quantum calculus is a generalization of the theory of quantum calculus, and it uses the following operator:
Hahn difference
Wolfgang Hahn introduced the following operator (Hahn difference):
When this operator reduces to -derivative, and when it reduces to forward difference. This is a successful tool for constructing families of orthogonal polynomials and investigating some approximation problems.
β-derivative
-derivative is an operator defined as follows:
In the definition, is a given interval, and is any continuous function that strictly monotonically increases (i.e. ). When then this operator is -derivative, and when this operator is Hahn difference.
Applications
The q-calculus has been used in machine learning for designing stochastic activation functions.
See also
Derivative (generalizations)
Jackson integral
Q-exponential
Q-difference polynomials
Quantum calculus
Tsallis entropy
Citations
Bibliography
Differential calculus
Generalizations of the derivative
Linear operators in calculus
Q-analogs
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https://en.wikipedia.org/wiki/Categorical%20set%20theory
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Categorical set theory is any one of several versions of set theory developed from or treated in the context of mathematical category theory.
See also
Categorical logic
References
External links
Category theory
Set theory
Formal methods
Categorical logic
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https://en.wikipedia.org/wiki/2004%20Australian%20Lacrosse%20League%20season
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Results and statistics for the Australian Lacrosse League season of 2004, the inaugural season for the ALL.
Game 1
Saturday, 23 October 2004, Melbourne, Victoria
Goalscorers:
Vic: D Pusvacietis 3–1, D Stiglich 3–1, W Henderson 2-2, D Nicholas 2–1, R Stark 2, R Garnsworthy 1, M Sevior 1.
WA: C Hayes 2, J Stack 2, L Blackie 1, W Curran 1, K Gillespie 1, N Rainey 1, T Roost 1, A Sear 1, D Whiteman 0–3.
Game 2
Sunday, 24 October 2004, Melbourne, Victoria
Goalscorers:
Vic: D Stiglich 3, W Henderson 2–1, D Nicholas 2, D Pusvacietis 1–3, D Arnell 1, J Brammell 1, R Stark 0–1.
WA: D Whiteman 3–1, C Hayes 1-1, D Spreadborough 1-1, N Rainey 1, T Roost 1, J Stack 1, L Blackie 0–1, B Goddard 0–1.
Game 3
Saturday, 30 October 2004, Adelaide, South Australia
Goalscorers:
SA: B Howe 4, M Mangan 3, L Perham 2, A Carter 1, A Feleppa 1, S Gilbert 1, N Wapper 1.
Vic: D Stiglich 3, D Arnell 2, D Pusvacietis 2, J Ardossi 1, M Sevoir 1, knocked-in 2.
Game 4
Sunday, 31 October 2004, Adelaide, South Australia
Goalscorers:
SA: L Perham 3–1, A Feleppa 2-2, B Howe 2, N Wapper 2, M Mangan 1–4, A Carter 0–1, R Stone 0–1.
Vic: D Nicholas 3–1, D Stiglich 3–1, J Joy 3, D Pusvacietis 1–3, A Lawman 1-1, T Fry 1, W Henderson 1, M Sevoir 1, N Le Guen 0–1.
Game 5
Saturday, 6 November 2004, Perth, Western Australia
Goalscorers:
WA: A Sear 3, A Ettridge 2, D Spreadborough 1-1, G Allen 1, A Brown 1, N Rainey 1.
SA: N Wapper 2, S Robb 1–2, L Perham 1-1, A Feleppa 1, S Gilbert 1, M Mangan 1.
Game 6
Sunday, 7 November 2004, Perth, Western Australia
Goalscorers:
WA: D Whiteman 6, R Brown 2, A Sear 2, A Ettridge 1, C Hayes 1, D Spreadborough 1.
SA: B Howe 4, S Gilbert 2, L Perham 2, A Carter 1, M Mangan 1, J Pangrazio 1.
ALL Table 2004
Table after completion of round-robin tournament
FINAL (Game 7)
Saturday, 13 November 2004, Melbourne, Victoria
Goalscorers:
Vic: D Nicholas 5–1, D Pusvacietis 3–2, J Ardossi 2–1, A Lawman 2, D Stiglich 1–2, D Arnall 1, J Joy 1, J Tokarua 1, M Sevoir 0–1.
WA: A Sear 3, D Whiteman 2, D Spreadborough 2, N Rainey 0–2, G Morley 0–1.
All-Stars
ALL 2004 Champions: Victoria
ALL 2004 Most Valuable Player: Russell Brown (WA)
ALL 2004 All-Stars: Darren Nicholas, Daniel Pusvacietis, John Tokarua, Scott Garnsworthy, Cameron Shepherd, Daniel Stiglich (Vic), Russell Brown, David Whiteman, Nathan Rainey, Adam Sear (WA), Leigh Perham, Brett Howe, Mark Mangan (SA). Coach: Duncan McKenzie (Vic). Referee: Rolf Kraus
See also
Lacrosse
Australian Lacrosse League
Lacrosse in Australia
External links
Australian Lacrosse League
Lacrosse Australia
Lacrosse South Australia
Lacrosse Victoria
Western Australian Lacrosse Association
Australian Lacrosse League
lacrosse
Australian
|
https://en.wikipedia.org/wiki/2005%20Australian%20Lacrosse%20League%20season
|
These are the results and statistics for the Australian Lacrosse League season of 2005.
Game 8
Friday, 21 October 2005, Adelaide, South Australia
Goalscorers:
SA: A Lawman 5, L Perham 3–2, M Mangan 2, S Connolly 1, S Robb 1, P Inge 0–2, A Feleppa 0–1, S Gilbert 0–1, R Stone 0–1.
WA: D Whiteman 3, R Brown 2–1, K Delfs 2–1, A Sear 2–1, W Curran 2, L Blackie 1-1, J Stack 1, G Allan 0–1.
Game 9
Saturday, 22 October 2005, Adelaide, South Australia
Goalscorers:
SA: A Lawman 3, M Mangan 3, L Perham 2–1, C Averay 1-1, S Robb 0–2, J Casagrande 0–1.
WA: G Allan 3–1, D Whiteman 3, R Brown 2–1, K Delfs 2–1, A Sear 2, J Stack 1-1, L Blackie 1, W Curran 1, knocked-in 1.
Game 10
Saturday, 29 October 2005, Perth, Western Australia
Goalscorers:
WA: W Curran 2, A Sear 2, K Gillespie 1, D Whiteman 1, B Smith 0–1.
Vic: J Buchanan 3–1, D Nicholas 3, T Fry 1, B Ross 1, R Stark 1, D Stiglich 1, D Pusvacietis 0–1, M Sevior 0–1.
Game 11
Sunday, 30 October 2005, Perth, Western Australia
Goalscorers:
WA: D Whiteman 2–1, J Stack 1-1, G Allan 1, R Brown 1, K Gillespie 1, A Sear 1, L Blackie 0–1.
Vic: B Ross 5–3, D Pusvacietis 3-3, J Buchanan 2-2, D Stiglich 2-2, R Stark 2, N Stiglich 2, D Nicholas 1–2, R Garnsworthy 1, M McInerney 0–1.
Game 12
Friday, 4 November 2005, Melbourne, Victoria
Goalscorers:
Vic: B Ross 2, M Sevior 2, R Stark 2, D Pusvacietis 1–2, D Stiglich 1, N Stiglich 1, J Ardossi 0–1.
SA: A Feleppa 3, S Robb 1-1, A Lawman 1, M Mangan 1, S Gilbert 0–1.
Game 13
Saturday, 5 November 2005, Melbourne, Victoria
Goalscorers:
Vic: J Buchanan 3, J Ardossi 2–1, R Stark 2–1, N Stiglich 2–1, T Fry 2, D Arnell 1, D Pusvacietis 1, M McInerney 0–1.
SA: M Mangan 2–1, S Robb 2, S Connolly 2, L Perham 1-1, A Carter 1, A Feleppa 0–1.
ALL Table 2005
Table after completion of round-robin tournament
FINAL (Game 14)
Friday, 11 November 2005, Adelaide, South Australia
Goalscorers:
Vic: R Stark 3, D Stiglich 3, J Buchanan 2-2, R Garnsworthy 2–1, D Pusvacietis 2–1, J Ardossi 2, D Nicholas 1–3, T Fry 1, J Tokarua 1, N Stiglich 0–2.
WA: W Curran 2, R Brown 1–2, G Allan 1, B Smith 1, D Whiteman 1.
All-Stars
ALL 2005 Champions: Victoria
ALL 2005 Most Valuable Player: Peter Inge (SA)
ALL 2005 All-Stars: Jamie Buchanan, Sam Marquard, Brad Ross, Robbie Stark, Daniel Stiglich, John Tokarua (Vic), Warren Brown, Wayne Curran, Gavin Leavy, Glenn Morley, David Whiteman (WA), Peter Inge (SA). Coach: David Joy (Vic). Referee: ...
See also
Lacrosse
Australian Lacrosse League
Lacrosse in Australia
External links
Australian Lacrosse League
Lacrosse Australia
Lacrosse South Australia
Lacrosse Victoria
Western Australian Lacrosse Association
Australian Lacrosse League
2005 in Australian sport
2005 in lacrosse
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https://en.wikipedia.org/wiki/Rotation%20number
|
In mathematics, the rotation number is an invariant of homeomorphisms of the circle.
History
It was first defined by Henri Poincaré in 1885, in relation to the precession of the perihelion of a planetary orbit. Poincaré later proved a theorem characterizing the existence of periodic orbits in terms of rationality of the rotation number.
Definition
Suppose that is an orientation-preserving homeomorphism of the circle Then may be lifted to a homeomorphism of the real line, satisfying
for every real number and every integer .
The rotation number of is defined in terms of the iterates of :
Henri Poincaré proved that the limit exists and is independent of the choice of the starting point . The lift is unique modulo integers, therefore the rotation number is a well-defined element of Intuitively, it measures the average rotation angle along the orbits of .
Example
If is a rotation by (where ), then
and its rotation number is (cf. irrational rotation).
Properties
The rotation number is invariant under topological conjugacy, and even monotone topological semiconjugacy: if and are two homeomorphisms of the circle and
for a monotone continuous map of the circle into itself (not necessarily homeomorphic) then and have the same rotation numbers. It was used by Poincaré and Arnaud Denjoy for topological classification of homeomorphisms of the circle. There are two distinct possibilities.
The rotation number of is a rational number (in the lowest terms). Then has a periodic orbit, every periodic orbit has period , and the order of the points on each such orbit coincides with the order of the points for a rotation by . Moreover, every forward orbit of converges to a periodic orbit. The same is true for backward orbits, corresponding to iterations of , but the limiting periodic orbits in forward and backward directions may be different.
The rotation number of is an irrational number . Then has no periodic orbits (this follows immediately by considering a periodic point of ). There are two subcases.
There exists a dense orbit. In this case is topologically conjugate to the irrational rotation by the angle and all orbits are dense. Denjoy proved that this possibility is always realized when is twice continuously differentiable.
There exists a Cantor set invariant under . Then is a unique minimal set and the orbits of all points both in forward and backward direction converge to . In this case, is semiconjugate to the irrational rotation by , and the semiconjugating map of degree 1 is constant on components of the complement of .
The rotation number is continuous when viewed as a map from the group of homeomorphisms (with topology) of the circle into the circle.
See also
Circle map
Denjoy diffeomorphism
Poincaré section
Poincaré recurrence
Poincaré–Bendixson theorem
References
, also SciSpace for smaller file size in pdf ver 1.3
Sebastian van Strien, Rotation Numbers and Poincaré's Theorem (2
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https://en.wikipedia.org/wiki/Decagonal%20prism
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In geometry, the decagonal prism is the eighth in the infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra. The decagonal prism has 12 faces, 30 edges, and 20 vertices, so, it is a dodecahedron (while the term is usually applied to regular dodecahedron or rhombic dodecahedron.). If faces are all regular, it is a semiregular or prismatic uniform polyhedron.
Uses
The decagonal prism exists as cells in two four-dimensional uniform 4-polytopes:
Decagonal prisms are often used in combination locks.
Related polyhedra
External links
3-d model of a Decagonal Prism
Prismatoid polyhedra
Zonohedra
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https://en.wikipedia.org/wiki/Dodecagonal%20prism
|
In geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.
If faces are all regular, it is a uniform polyhedron.
Use
It is used in the construction of two prismatic uniform honeycombs:
The new British one pound (£1) coin, which entered circulation in March 2017, is shaped like a dodecagonal prism.
Related polyhedra
References
External links
Prismatoid polyhedra
Zonohedra
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https://en.wikipedia.org/wiki/Decagonal%20antiprism
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In geometry, the decagonal antiprism is the eighth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterals.
In the case of a regular 10-sided base, one usually considers the case where its copy is twisted by an angle 180°/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.
If faces are all regular, it is a semiregular polyhedron.
See also
External links
Decagonal Antiprism: 3-d polyhedron model
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
VRML model
polyhedronisme A10
Prismatoid polyhedra
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https://en.wikipedia.org/wiki/E.%20J.%20G.%20Pitman
|
Edwin James George Pitman (29 October 1897 – 21 July 1993) was an Australian mathematician who made significant contributions to statistics and probability theory. In particular, he is remembered primarily as the originator of the Pitman permutation test, Pitman nearness and Pitman efficiency.
His work the Pitman measure of closeness or Pitman nearness concerning the exponential families of probability distributions has been studied extensively since the 1980s by C. R. Rao, Pranab K. Sen, and others.
The Pitman–Koopman–Darmois theorem states that only exponential families of probability distributions admit a sufficient statistic whose dimension remains bounded as the sample size grows.
Biography
Pitman was born in Melbourne on 29 October 1897, and attended University of Melbourne, residing at Ormond College, where he graduated with First Class Honours. In 1926 he was appointed Professor of Mathematics at the University of Tasmania, which he held until his retirement in 1962.
He was a founding member and second President of the Australian Mathematical Society. He was also active within the Statistical Society of Australia, which in 1978 named the Pitman medal in his honour.
Terminology
For "the sum of squares of deviations from the mean," he coined the term squariance.
For "the logarithm of the likelihood" he coined the term loglihood.
However, neither of these terms caught on.
Pitman's published work (selected)
Sufficient statistics and intrinsic accuracy, Proc. Camb. Phil. Soc. 32, (1936), 567–579.
The "closest" estimates of statistical parameters. Proc. Camb. Phil. Soc. 33 (1937), 212–222.
Significance tests which may be applied to samples from any populations. Suppl.J .R. Statist. Soc. 4, (1937), 119–130.
Significance tests which may be applied to samples from any populations. II. The correlation coefficient test. Suppl. J. R. Statist. Soc. 4, (1937), 225–232.
Significance tests which may be applied to samples from any populations. III. The analysis of variance test. Biometrika 29, (1938), 322–335.
The estimation of the location and scale parameters of a continuous population of any given form, Biometrika 30, (1939) 391–421.
Tests of hypotheses concerning location and scale parameters. Biometrika 31, (1939) 200–215.
Statistics and science. Journal of the American Statistical Association 25, (1957), 322–330.
Some remarks on statistical inference. Proc. Int. Res. Seminar, Berkeley (Bernoulli–Bayes–Laplace Anniversary Volume), (1965), 209–216. New York: Springer-Verlag.
Autobiography
Pitman contributed a chapter, "Reminiscences of a mathematician who strayed into statistics", to the volume
Joseph M. Gani (ed.) (1982) The Making of Statisticians, New York: Springer-Verlag.
Family
He had four children, including Jim Pitman, a Professor of Statistics at UC Berkeley.
References
External links
1897 births
1993 deaths
Australian statisticians
Mathematicians from Melbourne
Fellows of the Australian Academy of Science
Acad
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https://en.wikipedia.org/wiki/List%20of%20statistical%20software
|
Statistical software are specialized computer programs for analysis in statistics and econometrics.
Open-source
ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management
ADMB – a software suite for non-linear statistical modeling based on C++ which uses automatic differentiation
Chronux – for neurobiological time series data
DAP – free replacement for SAS
Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) a software framework for developing data mining algorithms in Java
Epi Info – statistical software for epidemiology developed by Centers for Disease Control and Prevention (CDC). Apache 2 licensed
Fityk – nonlinear regression software (GUI and command line)
GNU Octave – programming language very similar to MATLAB with statistical features
gretl – gnu regression, econometrics and time-series library
intrinsic Noise Analyzer (iNA) – For analyzing intrinsic fluctuations in biochemical systems
jamovi – A free software alternative to IBM SPSS Statistics
JASP – A free software alternative to IBM SPSS Statistics with additional option for Bayesian methods
JMulTi – For econometric analysis, specialised in univariate and multivariate time series analysis
Just another Gibbs sampler (JAGS) – a program for analyzing Bayesian hierarchical models using Markov chain Monte Carlo developed by Martyn Plummer. It is similar to WinBUGS
KNIME – An open source analytics platform built with Java and Eclipse using modular data pipeline workflows
LIBSVM – C++ support vector machine libraries
mlpack – open-source library for machine learning, exploits C++ language features to provide maximum performance and flexibility while providing a simple and consistent application programming interface (API)
Mondrian – data analysis tool using interactive statistical graphics with a link to R
Neurophysiological Biomarker Toolbox – Matlab toolbox for data-mining of neurophysiological biomarkers
OpenBUGS
OpenEpi – A web-based, open-source, operating-independent series of programs for use in epidemiology and statistics based on JavaScript and HTML
OpenMx – A package for structural equation modeling running in R (programming language)
OpenNN – A software library written in the programming language C++ which implements neural networks, a main area of deep learning research
Orange, a data mining, machine learning, and bioinformatics software
Pandas – High-performance computing (HPC) data structures and data analysis tools for Python in Python and Cython (statsmodels, scikit-learn)
Perl Data Language – Scientific computing with Perl
Ploticus – software for generating a variety of graphs from raw data
PSPP – A free software alternative to IBM SPSS Statistics
R – free implementation of the S (programming language)
Programming with Big Data in R (pbdR) – a series of R packages enhanced by SPMD parallelism for big data analysis
R Commander – GUI interface for R
Rattle GUI – GUI inte
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https://en.wikipedia.org/wiki/Schreier%20vector
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In mathematics, especially the field of computational group theory, a Schreier vector is a tool for reducing the time and space complexity required to calculate orbits of a permutation group.
Overview
Suppose G is a finite group with generating sequence which acts on the finite set . A common task in computational group theory is to compute the orbit of some element under G. At the same time, one can record a Schreier vector for . This vector can then be used to find an element satisfying , for any . Use of Schreier vectors to perform this requires less storage space and time complexity than storing these g explicitly.
Formal definition
All variables used here are defined in the overview.
A Schreier vector for is a vector such that:
For (the manner in which the are chosen will be made clear in the next section)
for
Use in algorithms
Here we illustrate, using pseudocode, the use of Schreier vectors in two algorithms
Algorithm to compute the orbit of ω under G and the corresponding Schreier vector
Input: ω in Ω,
for i in { 0, 1, …, n }:
set v[i] = 0
set orbit = { ω }, v[ω] = −1
for α in orbit and i in { 1, 2, …, r }:
if is not in orbit:
append to orbit
set
return orbit, v
Algorithm to find a g in G such that ωg = α for some α in Ω, using the v from the first algorithm
Input: v, α, X
if v[α] = 0:
return false
set g = e, and k = v[α] (where e is the identity element of G)
while k ≠ −1:
set
return g
References
Computational group theory
Permutation groups
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https://en.wikipedia.org/wiki/Base%20%28group%20theory%29
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Let be a finite permutation group acting on a set . A sequence
of k distinct elements of is a base for G if the only element of which fixes every pointwise is the identity element of .
Bases and strong generating sets are concepts of importance in computational group theory. A base and a strong generating set (together often called a BSGS) for a group can be obtained using the Schreier–Sims algorithm.
It is often beneficial to deal with bases and strong generating sets as these may be easier to work with than the entire group. A group may have a small base compared to the set it acts on. In the "worst case", the symmetric groups and alternating groups have large bases (the symmetric group Sn has base size n − 1), and there are often specialized algorithms that deal with these cases.
References
Permutation groups
Computational group theory
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https://en.wikipedia.org/wiki/AN/APG-77
|
The AN/APG-77 is a multifunction low probability of intercept radar installed on the F-22 Raptor fighter aircraft. The radar was designed and initially built by Westinghouse and Texas Instruments, and production continued with their respective successors Northrop Grumman and Raytheon after acquisition.
It is a solid-state, active electronically scanned array (AESA) radar. Composed of 1956 transmit/receive modules, each about the size of a gum stick, it can perform a near-instantaneous beam steering (in the order of tens of nanoseconds).
The APG-77 provides 120° field of view in azimuth and elevation,. APG-77 has an operating range of while unconfirmed sources suggest an operating range of , against a target. A range of 400 km or more, with the APG-77v1 with newer GaAs modules is believed to be possible while using more narrow beams.
More than 100 APG-77 AESA radars have been produced to date by Northrop Grumman, and much of the technology developed for the APG-77 is being used in the APG-81 radar for the F-35 Lightning II. The AN/APG-77 system itself exhibits a very low radar cross-section, supporting the F-22's stealthy design.
The APG-77v1 was installed on F-22 Raptors from Lot 5 and on. This provided full air-to-ground functionality (high-resolution synthetic aperture radar mapping, ground moving target indication and track (GMTI/GMTT), automatic cueing and recognition, combat identification, and many other advanced features).
See also
Phased array
Active electronically scanned array
References
External links
AN/APG-77 radar technology explained
f22fighter.com: AN/APG-77
Aircraft radars
Military radars of the United States
Military electronics of the United States
Northrop Grumman radars
Radars of the United States Air Force
Synthetic aperture radar
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https://en.wikipedia.org/wiki/Converse%20relation
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In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'. In formal terms, if and are sets and is a relation from to then is the relation defined so that if and only if In set-builder notation,
The notation is analogous with that for an inverse function. Although many functions do not have an inverse, every relation does have a unique converse. The unary operation that maps a relation to the converse relation is an involution, so it induces the structure of a semigroup with involution on the binary relations on a set, or, more generally, induces a dagger category on the category of relations as detailed below. As a unary operation, taking the converse (sometimes called conversion or transposition) commutes with the order-related operations of the calculus of relations, that is it commutes with union, intersection, and complement.
Since a relation may be represented by a logical matrix, and the logical matrix of the converse relation is the transpose of the original, the converse relation is also called the transpose relation. It has also been called the opposite or dual of the original relation, or the inverse of the original relation, or the reciprocal of the relation
Other notations for the converse relation include or
Examples
For the usual (maybe strict or partial) order relations, the converse is the naively expected "opposite" order, for examples,
A relation may be represented by a logical matrix such as
Then the converse relation is represented by its transpose matrix:
The converse of kinship relations are named: " is a child of " has converse " is a parent of ". " is a nephew or niece of " has converse " is an uncle or aunt of ". The relation " is a sibling of " is its own converse, since it is a symmetric relation.
Properties
In the monoid of binary endorelations on a set (with the binary operation on relations being the composition of relations), the converse relation does not satisfy the definition of an inverse from group theory, that is, if is an arbitrary relation on then does equal the identity relation on in general. The converse relation does satisfy the (weaker) axioms of a semigroup with involution: and
Since one may generally consider relations between different sets (which form a category rather than a monoid, namely the category of relations Rel), in this context the converse relation conforms to the axioms of a dagger category (aka category with involution). A relation equal to its converse is a symmetric relation; in the language of dagger categories, it is self-adjoint.
Furthermore, the semigroup of endorelations on a set is also a partially ordered structure (with inclusion of relations as sets), and actually an involutive quantale. Similarly, the category of heterogeneous relations, Rel is also an ordered c
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https://en.wikipedia.org/wiki/Delay%20differential%20equation
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In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times.
DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:
Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, and communication networks that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering.
Delay systems are still resistant to many classical controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation (constant and known delays), it leads to the same degree of complexity in the control design. In worst cases (time-varying delays, for instance), it is potentially disastrous in terms of stability and oscillations.
Voluntary introduction of delays can benefit the control system.
In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs).
A general form of the time-delay differential equation for is
where represents the trajectory of the solution in the past. In this equation, is a functional operator from to
Examples
Continuous delay
Discrete delay for
Linear with discrete delays where .
Pantograph equation where a, b and λ are constants and 0 < λ < 1. This equation and some more general forms are named after the pantographs on trains.
Solving DDEs
DDEs are mostly solved in a stepwise fashion with a principle called the method of steps. For instance, consider the DDE with a single delay
with given initial condition . Then the solution on the interval is given by which is the solution to the inhomogeneous initial value problem
with . This can be continued for the successive intervals by using the solution to the previous interval as inhomogeneous term. In practice, the initial value problem is often solved numerically.
Example
Suppose and . Then the ini
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https://en.wikipedia.org/wiki/Method%20of%20averaging
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In mathematics, more specifically in dynamical systems, the method of averaging (also called averaging theory) exploits systems containing time-scales separation: a fast oscillation versus a slow drift. It suggests that we perform an averaging over a given amount of time in order to iron out the fast oscillations and observe the qualitative behavior from the resulting dynamics. The approximated solution holds under finite time inversely proportional to the parameter denoting the slow time scale. It turns out to be a customary problem where there exists the trade off between how good is the approximated solution balanced by how much time it holds to be close to the original solution.
More precisely, the system has the following form
of a phase space variable The fast oscillation is given by versus a slow drift of . The averaging method yields an autonomous dynamical system
which approximates the solution curves of inside a connected and compact region of the phase space and over time of .
Under the validity of this averaging technique, the asymptotic behavior of the original system is captured by the dynamical equation for . In this way, qualitative methods for autonomous dynamical systems may be employed to analyze the equilibria and more complex structures, such as slow manifold and invariant manifolds, as well as their stability in the phase space of the averaged system.
In addition, in a physical application it might be reasonable or natural to replace a mathematical model, which is given in the form of the differential equation for , with the corresponding averaged system , in order to use the averaged system to make a prediction and then test the prediction against the results of a physical experiment.
The averaging method has a long history, which is deeply rooted in perturbation problems that arose in celestial mechanics (see, for example in ).
First example
Consider a perturbed logistic growth
and the averaged equation
The purpose of the method of averaging is to tell us the qualitative behavior of the vector field when we average it over a period of time. It guarantees that the solution approximates for times Exceptionally: in this example the approximation is even better, it is valid for all times. We present it in a section below.
Definitions
We assume the vector field to be of differentiability class with (or even we will only say smooth), which we will denote . We expand this time-dependent vector field in a Taylor series (in powers of ) with remainder . We introduce the following notation:
where is the -th derivative with . As we are concerned with averaging problems, in general is zero, so it turns out that we will be interested in vector fields given by
Besides, we define the following initial value problem to be in the standard form:
Theorem: averaging in the periodic case
Consider for every connected and bounded and every there exist and such that the original system (a non-autonomous dynamical sys
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https://en.wikipedia.org/wiki/Topology%20table
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A topology table is used by routers that route traffic in a network. It consists of all routing tables inside the Autonomous System where the router is positioned. Each router using the routing protocol EIGRP then maintains a topology table for each configured network protocol — all routes learned, that are leading to a destination are found in the topology table. EIGRP must have a reliable connection. The routing table of all routers of an Autonomous System is same.
Routing
Table
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https://en.wikipedia.org/wiki/Nonnegative%20matrix
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In mathematics, a nonnegative matrix, written
is a matrix in which all the elements are equal to or greater than zero, that is,
A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-negative matrix.
A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via non-negative matrix factorization.
Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem.
Properties
The trace and every row and column sum/product of a nonnegative matrix is nonnegative.
Inversion
The inverse of any non-singular M-matrix is a non-negative matrix. If the non-singular M-matrix is also symmetric then it is called a Stieltjes matrix.
The inverse of a non-negative matrix is usually not non-negative. The exception is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus the inverse of a positive matrix is not positive or even non-negative, as positive matrices are not monomial, for dimension .
Specializations
There are a number of groups of matrices that form specializations of non-negative matrices, e.g. stochastic matrix; doubly stochastic matrix; symmetric non-negative matrix.
See also
Metzler matrix
Bibliography
Abraham Berman, Robert J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, 1994, SIAM. .
A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, 1979 (chapter 2),
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990 (chapter 8).
Henryk Minc, Nonnegative matrices, John Wiley&Sons, New York, 1988,
Seneta, E. Non-negative matrices and Markov chains. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973)
Richard S. Varga 2002 Matrix Iterative Analysis, Second ed. (of 1962 Prentice Hall edition), Springer-Verlag.
Matrices
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https://en.wikipedia.org/wiki/Fundamental%20vector%20field
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In the study of mathematics and especially differential geometry, fundamental vector fields are an instrument that describes the infinitesimal behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry, and the study of Hamiltonian group actions.
Motivation
Important to applications in mathematics and physics is the notion of a flow on a manifold. In particular, if is a smooth manifold and is a smooth vector field, one is interested in finding integral curves to . More precisely, given one is interested in curves such that:
for which local solutions are guaranteed by the Existence and Uniqueness Theorem of Ordinary Differential Equations. If is furthermore a complete vector field, then the flow of , defined as the collection of all integral curves for , is a diffeomorphism of . The flow given by is in fact an action of the additive Lie group on .
Conversely, every smooth action defines a complete vector field via the equation:
It is then a simple result that there is a bijective correspondence between actions on and complete vector fields on .
In the language of flow theory, the vector field is called the infinitesimal generator. Intuitively, the behaviour of the flow at each point corresponds to the "direction" indicated by the vector field. It is a natural question to ask whether one may establish a similar correspondence between vector fields and more arbitrary Lie group actions on .
Definition
Let be a Lie group with corresponding Lie algebra . Furthermore, let be a smooth manifold endowed with a smooth action . Denote the map such that , called the orbit map of corresponding to . For , the fundamental vector field corresponding to is any of the following equivalent definitions:
where is the differential of a smooth map and is the zero vector in the vector space .
The map can then be shown to be a Lie algebra homomorphism.
Applications
Lie groups
The Lie algebra of a Lie group may be identified with either the left- or right-invariant vector fields on . It is a well-known result that such vector fields are isomorphic to , the tangent space at identity. In fact, if we let act on itself via right-multiplication, the corresponding fundamental vector fields are precisely the left-invariant vector fields.
Hamiltonian group actions
In the motivation, it was shown that there is a bijective correspondence between smooth actions and complete vector fields. Similarly, there is a bijective correspondence between symplectic actions (the induced diffeomorphisms are all symplectomorphisms) and complete symplectic vector fields.
A closely related idea is that of Hamiltonian vector fields. Given a symplectic manifold , we say that is a Hamiltonian vector field if there exists a smooth function satisfying:
where the map is the interior product. This motivatives the definition of a Hamiltonian group action as follows: If
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https://en.wikipedia.org/wiki/ISBL
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ISBL (Information Systems Base Language) is the relational algebra notation that was invented for PRTV, one of the earliest database management systems to implement E.F. Codd's relational model of data.
Example
OS = ORDERS * SUPPLIERS
LIST OS: NAME="Brooks" % SNAME, ITEM, PRICE
See also
IBM Business System 12 - An IBM industrial strength relational DBMS influenced by ISBL. It was developed for use by customers of IBM's time-sharing service bureaux in various countries in the early 1980s.
External links
Sample ISBL usage
Query languages
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https://en.wikipedia.org/wiki/Generalized%20quadrangle
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In geometry, a generalized quadrangle is an incidence structure whose main feature is the lack of any triangles (yet containing many quadrangles). A generalized quadrangle is by definition a polar space of rank two. They are the with n = 4 and near 2n-gons with n = 2. They are also precisely the partial geometries pg(s,t,α) with α = 1.
Definition
A generalized quadrangle is an incidence structure (P,B,I), with I ⊆ P × B an incidence relation, satisfying certain axioms. Elements of P are by definition the points of the generalized quadrangle, elements of B the lines. The axioms are the following:
There is an s (s ≥ 1) such that on every line there are exactly s + 1 points. There is at most one point on two distinct lines.
There is a t (t ≥ 1) such that through every point there are exactly t + 1 lines. There is at most one line through two distinct points.
For every point p not on a line L, there is a unique line M and a unique point q, such that p is on M, and q on M and L.
(s,t) are the parameters of the generalized quadrangle. The parameters are allowed to be infinite. If either s or t is one, the generalized quadrangle is called trivial. For example, the 3x3 grid with P = {1,2,3,4,5,6,7,8,9} and B = {123, 456, 789, 147, 258, 369} is a trivial GQ with s = 2 and t = 1. A generalized quadrangle with parameters (s,t) is often denoted by GQ(s,t).
The smallest non-trivial generalized quadrangle is GQ(2,2), whose representation was dubbed "the doily" by Stan Payne in 1973.
Properties
Graphs
There are two interesting graphs that can be obtained from a generalized quadrangle.
The collinearity graph having as vertices the points of a generalized quadrangle, with the collinear points connected. This graph is a strongly regular graph with parameters ((s+1)(st+1), s(t+1), s-1, t+1) where (s,t) is the order of the GQ.
The incidence graph whose vertices are the points and lines of the generalized quadrangle and two vertices are adjacent if one is a point, the other a line and the point lies on the line. The incidence graph of a generalized quadrangle is characterized by being a connected, bipartite graph with diameter four and girth eight. Therefore, it is an example of a Cage. Incidence graphs of configurations are today generally called Levi graphs, but the original Levi graph was the incidence graph of the GQ(2,2).
Duality
If (P,B,I) is a generalized quadrangle with parameters (s,t), then (B,P,I−1), with I−1 the inverse incidence relation, is also a generalized quadrangle. This is the dual generalized quadrangle. Its parameters are (t,s). Even if s = t, the dual structure need not be isomorphic with the original structure.
Generalized quadrangles with lines of size 3
There are precisely five (possible degenerate) generalized quadrangles where each line has three points incident with it, the quadrangle with empty line set, the quadrangle with all lines through a fixed point corresponding to the windmill graph Wd(3,n), grid of si
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https://en.wikipedia.org/wiki/Polar%20space
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In mathematics, in the field of geometry, a polar space of rank n (), or projective index , consists of a set P, conventionally called the set of points, together with certain subsets of P, called subspaces, that satisfy these axioms:
Every subspace is isomorphic to a projective space with and K a division ring. (That is, it is a Desarguesian projective geometry.) For each subspace the corresponding d is called its dimension.
The intersection of two subspaces is always a subspace.
For each subspace A of dimension and each point p not in A, there is a unique subspace B of dimension containing p and such that is -dimensional. The points in are exactly the points of A that are in a common subspace of dimension 1 with p.
There are at least two disjoint subspaces of dimension .
It is possible to define and study a slightly bigger class of objects using only relationship between points and lines: a polar space is a partial linear space (P,L), so that for each point p ∈ P and
each line l ∈ L, the set of points of l collinear to p, is either a singleton or the whole l.
Finite polar spaces (where P is a finite set) are also studied as combinatorial objects.
Generalized quadrangles
A polar space of rank two is a generalized quadrangle; in this case, in the latter definition, the set of points of a line collinear with a point p is the whole of only if p ∈ . One recovers the former definition from the latter under the assumptions that lines have more than 2 points, points lie on more than 2 lines, and there exist a line and a point p not on so that p is collinear to all points of .
Finite classical polar spaces
Let be the projective space of dimension over the finite field and let be a reflexive sesquilinear form or a quadratic form on the underlying vector space. The elements of the finite classical polar space associated with this form are the elements of the totally isotropic subspaces (when is a sesquilinear form) or the totally singular subspaces (when is a quadratic form) of with respect to . The Witt index of the form is equal to the largest vector space dimension of the subspace contained in the polar space, and it is called the rank of the polar space. These finite classical polar spaces can be summarised by the following table, where is the dimension of the underlying projective space and is the rank of the polar space. The number of points in a is denoted by and it is equal to . When is equal to , we get a generalized quadrangle.
Classification
Jacques Tits proved that a finite polar space of rank at least three is always isomorphic with one of the three types of classical polar spaces given above. This lefts open only the problem of classifying the finite generalized quadrangles.
References
.
Families of sets
Projective geometry
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