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https://en.wikipedia.org/wiki/1923%E2%80%9324%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1923–24.
Scottish League Division One
Scottish League Division Two
Scottish League Division Three
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1924%E2%80%9325%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1924–25.
Scottish League Division One
Scottish League Division Two
Scottish League Division Three
See also
1924–25 in Scottish football
References
Scottish Football League seasons
Scotland |
https://en.wikipedia.org/wiki/1925%E2%80%9326%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1925–26.
Scottish League Division One
Scottish League Division Two
Scottish League Division Three
See also
1925–26 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1926%E2%80%9327%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1926–27.
Scottish League Division One
Scottish League Division Two
See also
1926–27 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1927%E2%80%9328%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1927–28.
Scottish League Division One
Scottish League Division Two
See also
1927–28 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1928%E2%80%9329%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1928–29.
Scottish League Division One
Scottish League Division Two
See also
1928–29 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1929%E2%80%9330%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1929–30.
Scottish League Division One
Scottish League Division Two
See also
1929–30 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1930%E2%80%9331%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1930–31.
Scottish League Division One
Scottish League Division Two
See also
1930–31 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1931%E2%80%9332%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1931–32.
Scottish League Division One
Scottish League Division Two
See also
1931–32 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1932%E2%80%9333%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1932–33.
Scottish League Division One
Scottish League Division Two
See also
1932–33 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1933%E2%80%9334%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1933–34.
Scottish League Division One
Scottish League Division Two
See also
1933–34 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1934%E2%80%9335%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1934–35.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1935%E2%80%9336%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1935–36.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1936%E2%80%9337%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1936–37.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1937%E2%80%9338%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1937–38.
Scottish League Division One
Scottish League Division Two
See also
1937–38 in Scottish football
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1938%E2%80%9339%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1938–39.
Scottish League Division One
Scottish League Division Two
See also
1938–39 in Scottish football
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1946%E2%80%9347%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1946–47.
Scottish League Division A
Scottish League Division B
Scottish League Division C
See also
1946–47 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1947%E2%80%9348%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1947–48.
Scottish League Division A
Scottish League Division B
Scottish League Division C
See also
1947–48 in Scottish football
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1948%E2%80%9349%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1948–49.
Scottish League Division A
Scottish League Division B
Scottish League Division C
See also
1948–49 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1949%E2%80%9350%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1949–50.
Scottish League Division A
Scottish League Division B
See also
1949–50 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1950%E2%80%9351%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1950–51.
Scottish League Division A
Scottish League Division B
See also
1950–51 in Scottish football
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1951%E2%80%9352%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1951–52.
Scottish League Division A
Scottish League Division B
See also
1951–52 in Scottish football
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1952%E2%80%9353%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1952–53.
Scottish League Division A
Scottish League Division B
See also
1952–53 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1953%E2%80%9354%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1953–54.
Scottish League Division A
Scottish League Division B
See also
1953–54 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1954%E2%80%9355%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1954–55.
Scottish League Division A
Scottish League Division B
See also
1954–55 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1955%E2%80%9356%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1955–56.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1956%E2%80%9357%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1956–57.
Scottish League Division One
Scottish League Division Two
See also
1956–57 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1957%E2%80%9358%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1957–58.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1958%E2%80%9359%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1958–59.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1959%E2%80%9360%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1959–60.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1960%E2%80%9361%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1960–61.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1961%E2%80%9362%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1961–62.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1962%E2%80%9363%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1962–63.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1963%E2%80%9364%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1963–64.
Scottish League Division One
Scottish League Division Two
See also
1963–64 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1964%E2%80%9365%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1964–65.
Scottish League Division One
Scottish League Division Two
See also
1964–65 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1965%E2%80%9366%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1965–66.
Scottish League Division One
Scottish League Division Two
See also
1965–66 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1966%E2%80%9367%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1966–67.
Scottish League Division One
Scottish League Division Two
See also
1966–67 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1967%E2%80%9368%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1967–68.
Scottish League Division One
Scottish League Division Two
See also
1967–68 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1977%E2%80%9378%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1977–78.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1977–78 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1978%E2%80%9379%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1978–79.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1978–79 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1980%E2%80%9381%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1980–81.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1980–81 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1981%E2%80%9382%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1981–82.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1981–82 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1982%E2%80%9383%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1982–83.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1982–83 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1983–84.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1983–84 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1984–85.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1984–85 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1985–86.
Scottish Premier Division
Celtic won the League and became champions in one of the closest finishes in League history. On the final day of the season Hearts were leading Celtic by two points - a draw against Dundee would have been sufficient to see them win their first League title since the 1959–60 season. Hearts lost 2–0 to Dundee at Dens Park thanks to two goals by substitute Albert Kidd; the first in the 83rd minute and the second in the final minute while Celtic beat St Mirren 5–0 at Love Street. As a result, Celtic won the league on goal difference.
Relegation was suspended due to league reconstruction, therefore Motherwell and Clydebank retained their Premier Division status.
Scottish First Division
Scottish Second Division
See also
1985–86 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1986–87.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1986–87 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1987–88.
Scottish Premier Division
Scottish First Division
Scottish Second Division
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1988–89.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1988–89 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1989%E2%80%9390%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1989–90.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1989–90 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1990%E2%80%9391%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1990–91.
Scottish Premier Division
Scottish League Division One
Scottish League Division Two
See also
1990–91 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1991%E2%80%9392%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1991–92.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1991–92 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1992%E2%80%9393%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1992–93.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1992–93 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1993%E2%80%9394%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1993–94.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1993–94 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1994%E2%80%9395%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1994–95.
Scottish Premier Division
Scottish League Division One
Scottish League Division Two
Scottish League Division Three
See also
1994–95 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1995%E2%80%9396%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1995–96.
Scottish Premier Division
Scottish League Division One
Scottish League Division Two
Scottish League Division Three
See also
1995–96 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1996%E2%80%9397%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1996–97.
Scottish Premier Division
Scottish League Division One
Scottish League Division Two
Scottish League Division Three
See also
1996–97 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1997%E2%80%9398%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1997–98.
Scottish Premier Division
Table
Top scorers
Scottish League Division One
Table
Top scorers
Scottish League Division Two
Table
Top scorers
Scottish League Division Three
See also
1997–98 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/Mike%20Paterson | Michael Stewart Paterson, is a British computer scientist, who was the director of the Centre for Discrete Mathematics and its Applications (DIMAP) at the University of Warwick until 2007, and chair of the department of computer science in 2005.
He received his Doctor of Philosophy (Ph.D.) from the University of Cambridge in 1967, under the supervision of David Park. He spent three years at the Massachusetts Institute of Technology (MIT) and moved to the University of Warwick in 1971, where he remains Professor Emeritus.
Paterson is an expert on theoretical computer science with more than 100 publications, especially in the design and analysis of algorithms and computational complexity. Paterson's distinguished career was recognised with the EATCS Award in 2006, and a workshop in honour of his 66th birthday in 2008, including contributions of several Turing Award and Gödel Prize laureates. A further workshop was held in 2017 in honour of his 75th birthday, co-located with the workshop for the 10th anniversary of the DIMAP centre. For his work on distributed computing with Fischer and Lynch, he received the Dijkstra Prize in 2001, and his work with Dyer and Goldberg on counting graph homomorphisms received the best paper award at the ICALP conference in 2006. Mike Paterson received a Lester R. Ford Award in 2010. He is a Fellow of the Royal Society since 2001 and been president of the European Association for Theoretical Computer Science (EATCS). According to EATCS president Maurice Nivat, Paterson played a great role in the late 1960s in the recognition of computer science as a science, "and that theoretical computer science, which is very close to mathematics but distinct in its motivation and inspiration, is indeed a challenging and fruitful field of research."
Paterson is also an enthusiastic mountaineer.
Selected publications
M. Dyer, L.A. Goldberg and M. Paterson, On counting homomorphisms to directed acyclic graphs, Electronic Colloquium on Computational Complexity, Report TR05-121, Oct 2005.
L.A. Goldberg, M. Jalsenius, R. Martin and M. Paterson, Improved mixing bounds for the anti-ferromagnetic Potts Model on Z2, LMS J. Comput. Math. 9 (2006) 1–20.
L.A. Goldberg, R. Martin and M. Paterson, Strong spatial mixing for lattice graphs with fewer colours, SICOMP, 35(2) 486–517 (2005).
M. Albert and M. Paterson, Bounds for the growth rate of meander numbers, Proceedings of the 16th Annual International Conference on Formal Power Series and Algebraic Combinatorics, 2004, University of British Columbia (Vancouver B.C., Canada).
L.A. Goldberg, M. Jerrum, S. Kannan and M. Paterson, A bound on the capacity of backoff and acknowledgement-based protocols, SICOMP, 88 (2004) 313–331.
M. Adler, P. Berenbrink, T. Friedetzky, L.A. Goldberg, P. Goldberg and M. Paterson, A proportionate fair scheduling rule with good worst-case performance, Proc. of the 15th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA 2003), 101–108 (2003).
|
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1998–99.
Scottish First Division
Scottish Second Division
Scottish Third Division
See also
1998–99 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1999%E2%80%932000%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 1999–2000.
Scottish First Division
League table
Top scorers
Scottish Second Division
League table
Top scorers
Scottish Third Division
See also
1999–00 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2000–01.
Scottish First Division
Scottish Second Division
Scottish Third Division
See also
2000–01 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2001–02.
Scottish First Division
Scottish Second Division
Scottish Third Division
See also
2001–02 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2002–03. The season saw two new teams, as Airdrie United and Gretna replaced Clydebank and Airdrieonians.
Scottish First Division
Scottish Second Division
Scottish Third Division
References
See also
2002–03 in Scottish football
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2003–04.
Scottish First Division
League standings
Top scorers
Scottish Second Division
League standings
Top scorers
Scottish Third Division
League standings
Top scorers
See also
2003–04 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2004%E2%80%9305%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2004–05.
Scottish First Division
League standings
Top scorers
Scottish Second Division
League standings
Top scorers
Scottish Third Division
League standings
Top scorers
See also
2004–05 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2005–06.
Scottish First Division
League standings
Top scorers
Scottish Second Division
League standings
Top scorers
Scottish Third Division
League standings
Top scorers
See also
2005–06 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2006–07.
Scottish First Division
League standings
Top scorers
Scottish Second Division
League standings
Top scorers
Scottish Third Division
League standings
Top scorers
See also
2006–07 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2007–08.
Scottish First Division
Scottish Second Division
Scottish Third Division
See also
2007–08 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/2008%E2%80%9309%20Scottish%20Football%20League | Statistics of the Scottish Football League in season 2008–09.
Scottish First Division
Scottish Second Division
Scottish Third Division
See also
2008–09 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/List%20of%20Thor%20and%20Delta%20launches%20%281960%E2%80%931969%29 | Between 1960 and 1969, there were 314 Thor missiles launched, of which 272 were successful, giving an 86.6% success rate.
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
1960
There were 29 Thor missiles launched in 1960. 22 of the 29 launches were successful, giving a 75.8% success rate.
1961
There were 27 Thor missiles launched in 1961. 21 of the 27 launches were successful, giving a 77.7% success rate.
1962
There were 50 Thor missiles launched in 1962. 42 of the 50 launches were successful, giving an 84% success rate.
1963
There were 30 Thor missiles launched in 1963. 26 of the 30 launches were successful, giving an 86.6% success rate.
1964
There were 41 Thor missiles launched in 1964. 34 of the 41 launches were successful, giving an 82.9% success rate.
1965
There were 36 Thor missiles launched in 1965. 33 of the 36 launches were successful, giving a 91.6% success rate.
1966
There were 27 Thor missiles launched in 1966. 25 of the 27 launches were successful, giving a 92.6% success rate.
1967
There were 29 Thor missiles launched in 1967. 28 of the 29 launches were successful, giving a 96.6% success rate.
1968
There were 23 Thor missiles launched in 1968. 21 of the 23 launches were successful, giving a 91.3% success rate.
1969
There were 22 Thor missiles launched in 1969. 20 of the 22 launches were successful, giving a 90.9% success rate.
Images
See also
Lists of Thor and Delta launches
Lists of Thor launches
Lists of Delta launches |
https://en.wikipedia.org/wiki/Approximately%20finite-dimensional | In operator algebras, an algebra is said to be approximately finite-dimensional if it contains an increasing sequence of finite-dimensional subalgebras that is dense. One can consider
Approximately finite dimensional C*-algebras, or
Approximately finite-dimensional von Neumann algebras.
C*-algebras
Von Neumann algebras |
https://en.wikipedia.org/wiki/Minimum%20polynomial%20extrapolation | In mathematics, minimum polynomial extrapolation is a sequence transformation used for convergence acceleration of vector sequences, due to Cabay and Jackson.
While Aitken's method is the most famous, it often fails for vector sequences. An effective method for vector sequences is the minimum polynomial extrapolation. It is usually phrased in terms of the fixed point iteration:
Given iterates in , one constructs the matrix whose columns are the differences. Then, one computes the vector where denotes the Moore–Penrose pseudoinverse of . The number 1 is then appended to the end of , and the extrapolated limit is
where is the matrix whose columns are the iterates starting at 2.
The following 4 line MATLAB code segment implements the MPE algorithm:
U = x(:, 2:end - 1) - x(:, 1:end - 2);
c = - pinv(U) * (x(:, end) - x(:, end - 1));
c(end + 1, 1) = 1;
s = (x(:, 2:end) * c) / sum(c);
References
Numerical analysis
Articles with example MATLAB/Octave code |
https://en.wikipedia.org/wiki/Universal%20algebraic%20geometry | In algebraic geometry, universal algebraic geometry generalizes the geometry of rings to geometries of arbitrary varieties of algebras, so that every variety of algebras has its own algebraic geometry. The two terms algebraic variety and variety of algebras should not be confused.
See also
Algebraic geometry
Universal algebra
References
Seven Lectures on the Universal Algebraic Geometry
Algebraic geometry
Universal algebra |
https://en.wikipedia.org/wiki/Krawtchouk%20matrices | In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K(N) is an matrix. The first few Krawtchouk matrices are:
Definition
In general, for positive integer , the entries are given by the generating function:
where the row and column indices and run from to . Explicitly:
or in terms of the Krawtchouk polynomials:
The values of a Krawchouk matrix can also be calculated using a recurrence relation. Filling the top row with ones and the rightmost column with alternating binomial coefficients, the other entries are each given by the sum of the neighbouring entries to the top, topright and right.
Properties
The Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, .
As a transformation, a Krawtchouk matrix is an involution up to scaling:
Krawchouk matrices have an LDU decomposition involving triangular Pascal matrices and a diagonal matrix of the powers of 2.
The eigenvalues are , and the determinant is .
See also
Krawtchouk polynomial
Pascal matrix
References
External links
Krawtchouk encyclopedia
Matrices |
https://en.wikipedia.org/wiki/Topological%20data%20analysis | In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA provides a general framework to analyze such data in a manner that is insensitive to the particular metric chosen and provides dimensionality reduction and robustness to noise. Beyond this, it inherits functoriality, a fundamental concept of modern mathematics, from its topological nature, which allows it to adapt to new mathematical tools.
The initial motivation is to study the shape of data. TDA has combined algebraic topology and other tools from pure mathematics to allow mathematically rigorous study of "shape". The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types of data across many fields. Moreover, its mathematical foundation is also of theoretical importance. The unique features of TDA make it a promising bridge between topology and geometry.
Basic theory
Intuition
TDA is premised on the idea that the shape of data sets contains relevant information. Real high-dimensional data is typically sparse, and tends to have relevant low dimensional features. One task of TDA is to provide a precise characterization of this fact. For example, the trajectory of a simple predator-prey system governed by the Lotka–Volterra equations forms a closed circle in state space. TDA provides tools to detect and quantify such recurrent motion.
Many algorithms for data analysis, including those used in TDA, require setting various parameters. Without prior domain knowledge, the correct collection of parameters for a data set is difficult to choose. The main insight of persistent homology is to use the information obtained from all parameter values by encoding this huge amount of information into an understandable and easy-to-represent form. With TDA, there is a mathematical interpretation when the information is a homology group. In general, the assumption is that features that persist for a wide range of parameters are "true" features. Features persisting for only a narrow range of parameters are presumed to be noise, although the theoretical justification for this is unclear.
Early history
Precursors to the full concept of persistent homology appeared gradually over time. In 1990, Patrizio Frosini introduced a pseudo-distance between submanifolds, and later the size function, which on 1-dim curves is equivalent to the 0th persistent homology. Nearly a decade later, Vanessa Robins studied the images of homomorphisms induced by inclusion. Finally, shortly thereafter, Edelsbrunner et al. introduced the concept of persistent homology together with an efficient algorithm and its visualization as a persistence diagram. Carlsson et al. reformulated the initial definition and gave an equivalent visualization method called persistence b |
https://en.wikipedia.org/wiki/Perceptrons%20%28book%29 | Perceptrons: an introduction to computational geometry is a book written by Marvin Minsky and Seymour Papert and published in 1969. An edition with handwritten corrections and additions was released in the early 1970s. An expanded edition was further published in 1988 after the revival of neural networks, containing a chapter dedicated to counter the criticisms made of it in the 1980s.
The main subject of the book is the perceptron, a type of artificial neural network developed in the late 1950s and early 1960s. The book was dedicated to psychologist Frank Rosenblatt, who in 1957 had published the first model of a "Perceptron". Rosenblatt and Minsky knew each other since adolescence, having studied with a one-year difference at the Bronx High School of Science. They became at one point central figures of a debate inside the AI research community, and are known to have promoted loud discussions in conferences, yet remained friendly.
This book is the center of a long-standing controversy in the study of artificial intelligence. It is claimed that pessimistic predictions made by the authors were responsible for a change in the direction of research in AI, concentrating efforts on so-called "symbolic" systems, a line of research that petered out and contributed to the so-called AI winter of the 1980s, when AI's promise was not realized.
The crux of Perceptrons is a number of mathematical proofs which acknowledge some of the perceptrons' strengths while also showing major limitations. The most important one is related to the computation of some predicates, such as the XOR function, and also the important connectedness predicate. The problem of connectedness is illustrated at the awkwardly colored cover of the book, intended to show how humans themselves have difficulties in computing this predicate. One reviewer, Earl Hunt, noted that the XOR function is difficult for humans to acquire as well during concept learning experiments.
Publication history
The first edition was printed in 1969. Handwritten alterations were made by the authors for the second printing in 1972. The handwritten notes include some references to the review for the first edition.
An "expanded edition" was published in 1988, which adds a prologue and an epilogue to discuss the revival of neural networks in the 1980s, but no new scientific results. In 2017, the expanded edition was re-printed, with a foreword by Léon Bottou that discusses the book from the perspective of someone working in deep learning.
Background
The perceptron is a neural net developed by psychologist Frank Rosenblatt in 1958 and is one of the most famous machines of its period. In 1960, Rosenblatt and colleagues were able to show that the perceptron could in finitely many training cycles learn any task that its parameters could embody. The perceptron convergence theorem was proved for single-layer neural nets.
During this period, neural net research was a major approach to the brain-machine issue that h |
https://en.wikipedia.org/wiki/1929%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1929 season.
Overview
In 1929 six teams entered with ÍBA playing for the first time and ÍBV returning. The rules of the competition stated that, if more than five teams entered, the championship should be decided in a knock-out competition. Because two teams had travelled a far distance to get to the championship (all matches were still played in Reykjavík), it was decided that the competition would be a two-loss elimination.
.
KR won the championship.
League standings
||Fram is eliminated
||Víkingur is eliminated
||ÍBA is eliminated
||ÍBV is eliminated
||Final
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1930%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1930 season.
Overview
ÍBA did not enter in 1930 so the number of teams dropped to five again. Valur won the club's first championship.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1931%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1931 season.
Overview
Four teams participated in the 20th season of Icelandic league football. KR won the championship for the 7th time.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1933%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1933 season.
Overview
It was contested by 4 teams, and Valur won the championship.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1934%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1934 season.
Overview
It was contested by 5 teams, and KR won the championship.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1935%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1935 season.
Overview
It was contested by 4 teams, and Valur won the championship. KR's Þorsteinn Einarsson and Bjarni Ólafsson, as well as Valur's Magnús Bergsteinsson, were the joint top scorers with 3 goals each.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1936%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1936 season.
Overview
It was contested by four teams, and Valur won the championship. Valur's Óskar Jónsson was the top scorer with five goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1937%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1937 season.
Overview
It was contested by 3 teams, and Valur won the championship. Valur's Óskar Jónsson was the top scorer with 3 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1938%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1938 season.
Overview
It was contested by 4 teams, and Valur won the championship. Valur's Magnús Bergsteinsson, Fram's Jón Sigurðsson and Víkingur's Björgvin Bjarnason were the joint top scorers with 3 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1939%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1939 season.
Overview
It was contested by 4 teams, and Fram won the championship. KR's Birgir Guðjónsson and Þorsteinn Einarsson were the joint top scorers with 3 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1940%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1940 season.
Overview
It was contested by 4 teams, and Valur won the championship. Valur's Sigurpáll Jónsson and Víkingur's Ingólfur Isebarn were the joint top scorers with 4 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1941%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1941 season.
Overview
It was contested by 5 teams, and KR won the championship. KR's Björgvin Schram was the top scorer with 7 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1942%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1942 season.
Overview
It was contested by 5 teams, and Valur won the championship. Valur's Ellert Sölvason was the top scorer with 6 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1943%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1943 season.
Overview
It was contested by 5 teams, and Valur won the championship. KR's Jón Jónasson was the top scorer with 5 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1944%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1944 season.
Overview
It was contested by 4 teams, and Valur won the championship. Valur's Sveinn Sveinsson, Sveinn Helgason and Jóhann Eyjólfsson, as well as Víkingur's Eiríkur Bergsson, were the joint top scorers with 2 goals. Íþróttafélag Reykjavíkur had initially participated but withdrew from the tournament after losing the first game 0-8 to Fram.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1945%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1945 season.
Overview
It was contested by 4 teams, and Valur won the championship. KR's Hörður Óskarsson was the top scorer with 6 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1946%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1946 season.
Overview
It was contested by 6 teams, and Fram won the championship. Fram's Valtýr Guðmundsson was the top scorer with 6 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1947%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1947 season.
Overview
It was contested by 5 teams, and Fram won the championship. Valur's Einar Halldórsson (Valur)and Albert Guðmundsson, as well as ÍA's Ríkharður Jónsson and KR's Hörður Óskarsson, were the joint top scorers with 3 goals. ·
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1949%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1949 season.
Overview
It was contested by 5 teams, and KR won the championship. KR's Ólafur Hannesson and Hörður Óskarsson, as well as Fram's Guðmundur Jónsson, were the joint top scorers with 4 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1950%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1950 season.
Overview
It was contested by 5 teams, and KR won the championship. Fram's Ríkharður Jónsson and Lárus Hallbjörnsson, as well as ÍA's Þórður Þórðarson, Valur's Halldór Halldórsson and Víkingur's Gunnlaugur Lárusson, were the joint top scorers with 3 goals.
Teams
League standings
Results
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1952%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1952 season.
Overview
It was contested by 5 teams, and KR won the championship. ÍA's Ríkharður Jónsson was the top scorer with 6 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1953%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1953 season.
Overview
It was contested by 6 teams, and ÍA won the championship. ÍA's Ríkharður Jónsson was the top scorer with 5 goals.
Group A
Group B
Final
ÍA 3-2 Valur
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
https://en.wikipedia.org/wiki/1954%20%C3%9Arvalsdeild | Statistics of Úrvalsdeild in the 1954 season.
Overview
It was contested by 6 teams, and ÍA won the championship. ÍA's Ríkharður Jónsson was the top scorer with 6 goals.
League standings
Results
References
Úrvalsdeild karla (football) seasons
Iceland
Iceland
Urvalsdeild |
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