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https://en.wikipedia.org/wiki/1970%E2%80%9371%201.Lig | The following are the statistics of the Turkish First Football League for the 1970–71 season.
Overview
It was contested by 16 teams, and Galatasaray S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1970–71 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1971%E2%80%9372%201.Lig | Statistics of the Turkish First Football League for the 1971–72 season.
Overview
It was contested by 16 teams, and Galatasaray S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1971–72 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1972%E2%80%9373%201.Lig | Statistics of the Turkish First Football League for the 1972–73 season.
Overview
It was contested by 16 teams, and Galatasaray S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1972–73 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1973%E2%80%9374%201.Lig | Statistics of the Turkish First Football League for the 1973–74 season.
Overview
It was contested by 16 teams, and Fenerbahçe S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1973–74 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1974%E2%80%9375%201.Lig | Statistics of the Turkish First Football League for the 1974–75 season.
Overview
It was contested by 16 teams, and Fenerbahçe S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1974–75 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1975%E2%80%9376%201.Lig | Statistics of the Turkish First Football League for the 1975–76 season.
Overview
Sixteen teams participated, and Trabzonspor won the championship, becoming the first team outside of Istanbul to win the league title.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1975–76 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1976%E2%80%9377%201.Lig | Statistics of the Turkish First Football League for the 1976–77 season.
Overview
It was contested by 16 teams, and Trabzonspor won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1976–77 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1977%E2%80%9378%201.Lig | Statistics of the Turkish First Football League for the 1977–78 season.
Overview
It was contested by 16 teams, and Fenerbahçe S.K. won the championship. Ankaragücü and Mersin İdman Yurdu relegated to Second League. Turkish Cup winners Trabzonspor could not play in 1978–79 European Cup Winners' Cup because they were suspended.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1977–78 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1979%E2%80%9380%201.Lig | Statistics of the Turkish First Football League in season 1979/1980.
Overview
It was contested by 16 teams, and Trabzonspor won the championship. The top goal scorer was known as "The Matador" for the '79 series.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1979–80 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1980%E2%80%9381%201.Lig | Statistics of Turkish First Football League in season 1980/1981.
Overview
It was contested by 16 teams, and Trabzonspor won the championship. 1981–82 European Cup Winners' Cup spot goes to Second League team Ankaragücü, who was also promoted and went back to 1. Lig at the end of the 1980/81 season.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1980–81 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1982%E2%80%9383%201.Lig | The following are the statistics of the Turkish First Football League in season 1982/1983.
Overview
It was contested by 18 teams, and Fenerbahçe S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1982–83 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1983%E2%80%9384%201.Lig | The following are the statistics of the Turkish First Football League in season 1983/1984.
Overview
Eighteen teams participated, and Trabzonspor won the championship. Trabzonspor have not won a league title until the 2021–22 season. This season was the first season in which no İzmir team took part in the league.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1983–84 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1984%E2%80%9385%201.Lig | The following are the statistics of the Turkish First Football League in season 1984/1985.
Overview
It was contested by 18 teams, and Fenerbahçe S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1984–85 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1986%E2%80%9387%201.Lig | Statistics of Turkish First Football League in season 1986/1987.
Overview
Nineteen clubs participated, and Galatasaray S.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1986–87 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1987%E2%80%9388%201.Lig | Statistics of Turkish First Football League in season 1987–88.
Overview
Twenty clubs participated, and Galatasaray S.K. won the championship. Denizlispor, Kocaelispor, Gençlerbirliği and Zonguldakspor were relegated to Second League.
This was the first season where clubs were awarded 3 points for victories, in contrast to previous years where a victory had earned the winning club only 2 points.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1987–88 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1988%E2%80%9389%201.Lig | Statistics of Turkish First Football League in season 1988/1989.
Overview
It was contested by 19 teams, and Fenerbahçe S.K. won the championship.
On 20 January 1989, while traveling to Malatya to face Malatyaspor, Samsunspor were involved in a bus accident that killed three of their players and left seven others seriously injured. In addition, two coaches, manager Nuri Asan, and the team's bus driver were also killed in the accident.
Due to the tragedy, Samsunpor were left unable to field a team for their remaining 18 matches, which were scratched and awarded 3-0 to Samsunspor's opponents, while Samsunspor were reprieved from relegation at the end of the season.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1988–89 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1989%E2%80%9390%201.Lig | Statistics of the Turkish First Football League in season 1989/1990.
Overview
It was contested by 18 teams, and Beşiktaş J.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1989–90 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1990%E2%80%9391%201.Lig | Statistics of the Turkish First Football League in season 1990/1991.
Overview
It was contested by 16 teams, and Beşiktaş J.K. won the championship.
League table
Results
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1990–91 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1991%E2%80%9392%201.Lig | The following are the statistics of the Turkish First Football League in season 1991/1992.
Overview
It was contested by 16 teams, and Beşiktaş J.K. won the championship. This is the only season in the Turkish League, when a team has won the championship without a loss.
League table
Galatasaray qualified to UEFA following the Albanian renounce.
Results
Top scorers
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1991–92 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1993%E2%80%9394%201.Lig | The following are the statistics of the Turkish First Football League in season 1993-1994.
Overview
Sixteen teams took part and Galatasaray S.K. won the championship. The teams Karabükspor, Karsiyaka and Sariyer got relegated.
League table
Results
Top scorers
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1993–94 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/1994%E2%80%9395%201.Lig | The following are the statistics of the Turkish First Football League in season 1994/1995.
Overview
It was contested by 18 teams, and Beşiktaş J.K. won the championship.
League table
Results
Top scorers
References
Turkey - List of final tables (RSSSF)
Süper Lig seasons
1994–95 in Turkish football
Turkey |
https://en.wikipedia.org/wiki/Ping-pong%20lemma | In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group.
History
The ping-pong argument goes back to the late 19th century and is commonly attributed to Felix Klein who used it to study subgroups of Kleinian groups, that is, of discrete groups of isometries of the hyperbolic 3-space or, equivalently Möbius transformations of the Riemann sphere. The ping-pong lemma was a key tool used by Jacques Tits in his 1972 paper containing the proof of a famous result now known as the Tits alternative. The result states that a finitely generated linear group is either virtually solvable or contains a free subgroup of rank two. The ping-pong lemma and its variations are widely used in geometric topology and geometric group theory.
Modern versions of the ping-pong lemma can be found in many books such as Lyndon & Schupp, de la Harpe, Bridson & Haefliger and others.
Formal statements
Ping-pong lemma for several subgroups
This version of the ping-pong lemma ensures that several subgroups of a group acting on a set generate a free product. The following statement appears in Olijnyk and Suchchansky (2004), and the proof is from de la Harpe (2000).
Let G be a group acting on a set X and let H1, H2, ..., Hk be subgroups of G where k ≥ 2, such that at least one of these subgroups has order greater than 2.
Suppose there exist pairwise disjoint nonempty subsets of such that the following holds:
For any and for any in , we have .
Then
Proof
By the definition of free product, it suffices to check that a given (nonempty) reduced word represents a nontrivial element of . Let be such a word of length , and let where for some . Since is reduced, we have for any and each is distinct from the identity element of . We then let act on an element of one of the sets . As we assume that at least one subgroup has order at least 3, without loss of generality we may assume that has order at least 3. We first make the assumption that and are both 1 (which implies ). From here we consider acting on . We get the following chain of containments:
By the assumption that different 's are disjoint, we conclude that acts nontrivially on some element of , thus represents a nontrivial element of .
To finish the proof we must consider the three cases:
if , then let (such an exists since by assumption has order at least 3);
if , then let ;
and if , then let .
In each case, after reduction becomes a reduced word with its first and last letter in . Finally, represents a nontrivial element of , and so does . This proves the claim.
The Ping-pong lemma for cyclic subgroups
Let G be a group acting on a set X. Let a1, ...,ak be elements of G of infinite order, where k ≥ 2. Suppose there exist disjoint nonempty subsets
of with the following properties:
for ;
for .
Then the subgroup generated by a1, ..., ak |
https://en.wikipedia.org/wiki/Quasideterminant | In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
In general, there are n2 quasideterminants defined for an n × n matrix (one for each position in the matrix), but the presence of the inverted terms above should give the reader pause: they are not always defined, and even when they are defined, they do not reduce to determinants when the entries commute. Rather,
where means delete the ith row and jth column from A.
The examples above were introduced between 1926 and 1928 by Richardson and Heyting, but they were marginalized at the time because they were not polynomials in the entries of . These examples were rediscovered and given new life in 1991 by Israel Gelfand and Vladimir Retakh. There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if is built from by rescaling its -th row (on the left) by , then .
Similarly, if is built from by adding a (left) multiple of the -th row to another row, then . They even develop a quasideterminantal
version of Cramer's rule.
Definition
Let be an matrix over a (not necessarily commutative)
ring and fix . Let
denote the ()-entry of , let denote the -th row of with column deleted, and let denote the -th column of with row deleted. The ()-quasideterminant of is defined if the submatrix is invertible over . In this case,
Recall the formula (for commutative rings) relating to the determinant, namely . The above definition is a generalization in that (even for noncommutative rings) one has
whenever the two sides makes sense.
Identities
One of the most important properties of the quasideterminant is what Gelfand and Retakh
call the "heredity principle". It allows one to take a quasideterminant in
stages (and has no commutative counterpart). To illustrate, suppose
is a block matrix decomposition of an matrix with
a matrix. If the ()-entry of lies within , it says that
That is, the quasideterminant of a quasideterminant is a quasideterminant. To put it less succinctly: UNLIKE determinants, quasideterminants treat matrices with block-matrix entries no differently than ordinary matrices (something determinants cannot do since block-matrices generally don't commute with one another). That is, while the precise form of the above identity is quite surprising, the existence of some such identity is less so.
Other identities from the papers are (i) the so-called "homological relations", stating that two quasideterminants in a common row or column are closely related to one another, and (ii) the Sylvester formula.
(i) Two quasideterminants sharing a common row or column satisfy
or
respectively, for all choices , so that the
quasideterminants involved are defined.
(ii) Like the heredity principle, the Sylvester identity is a way to recursively compute a quasideterminant. To ease notation, we display a special cas |
https://en.wikipedia.org/wiki/Srinivasacharya%20Raghavan | Srinivasacharya Raghavan was an Indian mathematician who worked in number theory. He was born on 11 April 1934 in Thillaisthanam, Thanjavur, Tamil Nadu. After
completing B.A. (Hons) from St. Joseph's College, Tiruchirapalli, he joined TIFR in 1954 as research student, and completed his Ph.D. in 1960 under the supervision of Professors K. Chandrasekharan and K.G. Ramanathan. He was affiliated with TIFR from 1956 until retirement in 1994, and served as Dean of Mathematics Faculty during 1986-89. He played an important role
in the development of the TIFR Centre for Applicable Mathematics (now TIFR CAM) at Bangalore in its initial years. He also held visiting appointments at the Institute for Advanced Study, Princeton, USA, Sonderforschungsberiech at University of Goettingen, Germany, SPIC Mathematical Institute (now Chennai Mathematical Institute) and taught at the Centre for Advanced Studies in Mathematics at the Unviersity of Mumbai for many years.
Raghavan estimated the Fourier coefficients of Siegel modular forms yielding a generalization of Hardy-Ramanujan-Hecke asymptotic formula for representation by positive definite quadratic forms. His other notable findings include the determination of the structure of singular Siegel modular forms, application of Hecke's Grenzprozess to analytic continuation of non-holomorphic Eisenstein series of degree 3 as forerunner of Weissauer's deep generalisation, Ramanathan-Raghavan's analogue over algebraic number fields of Oppenheim's result on density of values of irrational indefinite quadratic (zero) forms, and Dani-Raghavan's result on density of irrational euclidean frames under familiar discrete groups following Kronecker, Rangachari-Raghavan's investigation of Ramanujan's integral identities. He also published about 40 research articles and guided four students for their PhD.
He contributed research papers to many international journals of renown and received many honours. He was awarded the Shanti Swarup Bhatnagar Prize for Science and Technology in Mathematical Science in 1979 He was also a Fellow of the Indian Academy of Sciences. He also served as member of the INSA Council and as Chairman of the Editorial Board of the Proceedings (Math.Sci.) of IASc.
Prof. Raghavan served as Academic Secretary and Council Member of the Indian Mathematical Society during 1970-75 and was a member of the Editorial Board of the Journal of the Indian Mathematical Society for many years. He was a coauthor of Homological Methods in Commutative Algebra. He retired as Senior Professor from the Tata Institute of Fundamental Research (TIFR), Mumbai in 1994, died in Chennai on 7 October 2014, peacefully, due to cancer. He was married, and had a son, daughter-in-law and two grandsons.
References
Indian number theorists
20th-century Indian mathematicians
Fellows of the Indian Academy of Sciences
1934 births
2014 deaths
Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science |
https://en.wikipedia.org/wiki/Experimental%20Mathematics%20%28journal%29 | Experimental Mathematics is a quarterly scientific journal of mathematics published by A K Peters, Ltd. until 2010, now by Taylor & Francis. The journal publishes papers in experimental mathematics, broadly construed. The journal's mission statement describes its scope as follows: "Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses." the editor-in-chief is Alexander Kasprzyk (University of Nottingham).
History
Experimental Mathematics was established in 1992 by David Epstein, Silvio Levy, and Klaus Peters. Experimental Mathematics was the first mathematical research journal to concentrate on experimental mathematics and to explicitly acknowledge its importance for mathematics as a general research field. The journal's launching was described as "something of a watershed". Indeed, the launching of the journal in 1992 was surrounded by some controversy in the mathematical community about the value and validity of experimentation in mathematical research. Some critics of the new journal suggested that it be renamed as the "Journal of Unproved Theorems". In a 1995 article in the Notices of the American Mathematical Society, in part responding to such criticism, Epstein and Levy described the journal's aims as follows:
In recent years a number of other research journals in pure mathematics have substantially expanded their coverage of experimental mathematics and new journals devoted in large part to experimental mathematics have been launched. Thus, in 1998 the London Mathematical Society launched LMS Journal of Computation and Mathematics and in 2004 the Journal of Algebra started a new section called "Computational Algebra". LMS Journal of Computation and Mathematics was closed to new submissions in October 2015.
Despite the initial controversy, Experimental Mathematics quickly established a solid reputation and is now a highly respected mathematical publication. The journal is reviewed cover-to-cover in Mathematical Reviews and Zentralblatt MATH and is indexed in the Web of Science.
References
External links
Mathematics journals
Academic journals established in 1992
Quarterly journals
English-language journals
Experimental mathematics |
https://en.wikipedia.org/wiki/Persepolis%20F.C.%20in%20Asia | These are the records of Persepolis F.C. and their statistics in Asian football competitions. They have won the Asian Cup Winners' Cup once (in 1990–91) and were also runners-up once in 1992–93. Persepolis have finished third place on three occasions and finished in fourth place once in the Asian Club Championship. They were also emerged as runners-up of the AFC Champions League in 2018 and 2020.
The first international game of that Persepolis played was on 26 July 1968 in Tehran, against the South Korean Army team.
Overall
Persepolis was the first Iranian team to participate in the Asian Champion League. Persepolis earned three victories against Shahrbanai, Oghab and Pas, and a draw with Taj, to qualify for the 1969 Asian Cup. Persepolis was eliminated after two wins, a draw and a defeat in the group stage, and failed to reach the next two rounds.
AFC did not hold any club competitions on the continent from 1971 to 1985 due to political and security problems, and the opposition of the Arab countries with the presence of Israel in the competition.
Persepolis' second appearance in the Asian Cup came after winning the Hazfi Cup; Persepolis was eliminated by Mohammed Bangladesh, whose coach was Nasser Hejazi.
Following the introduction of the Asian Cup Winners' Cup, Persepolis qualified for the tournament on three occasions. In their first appearance in the competition, they won the title, and finished as runners-up on the second occasion.
In the 1990s (before the solar decade of the 1370s), and the early years of the new millennium, Persepolis participated in four editions of the AFC Champions League, reaching the semi-finals on each occasion. They subsequently reached the competition's final in 2018 and 2020, losing on both occasions.
Correct as of 24 October 2023
Only official matches included (AFC Champions League and Asian Cup Winners' Cup matches).
Record by country of opposition
Correct as of 24 October 2023
P – Played; W – Won; D – Drawn; L – Lost
Year by year performance
Below is a table of the performance of Persepolis in Asian competition.
Records
Asian Cup Winners' Cup
Winners: 1
Asian Cup Winners' Cup 1990–91
(Oct 4 & 18, 1991)
|}
Runners-Up: 1
Asian Cup Winners' Cup 1992–93
(Jan 17 & Apr 16, 1993)
|}
AFC Champions League
Runners-Up: 2
AFC Champions League 2018
(Nov 3 & 10, 2018)
|}
AFC Champions League 2020
(Dec 19, 2020)
Top goalscorers in Asia
Below is the list of most goalscorer of Persepolis in Asia.
Bold names denote a player still playing for club.
Hat-tricks
Most successful coaches
Correct as of 24 October 2023
Honours
AFC Champions League
Runners-up (2): 2018, 2020
Asian Club Championship
Third place (3): 1996–97, 1999–00, 2000–01
Fourth place (1): 1997–98
Asian Cup Winners' Cup
Champions (1): 1990–91
Runners-up (1): 1992–93
See also
Iranian clubs in the AFC Champions League
AFC Champions League
Asian Cup Winners' Cup
References
Asia
Iranian football club statistics |
https://en.wikipedia.org/wiki/Regular%204-polytope | In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
There are six convex and ten star regular 4-polytopes, giving a total of sixteen.
History
The convex regular 4-polytopes were first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. He discovered that there are precisely six such figures.
Schläfli also found four of the regular star 4-polytopes: the grand 120-cell, great stellated 120-cell, grand 600-cell, and great grand stellated 120-cell. He skipped the remaining six because he would not allow forms that failed the Euler characteristic on cells or vertex figures (for zero-hole tori: F − E + V 2). That excludes cells and vertex figures such as the great dodecahedron {5,} and small stellated dodecahedron {,5}.
Edmund Hess (1843–1903) published the complete list in his 1883 German book Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder.
Construction
The existence of a regular 4-polytope is constrained by the existence of the regular polyhedra which form its cells and a dihedral angle constraint
to ensure that the cells meet to form a closed 3-surface.
The six convex and ten star polytopes described are the only solutions to these constraints.
There are four nonconvex Schläfli symbols {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,,3}, {4,3,}, {,3,4}, {,3,}.
Regular convex 4-polytopes
The regular convex 4-polytopes are the four-dimensional analogues of the Platonic solids in three dimensions and the convex regular polygons in two dimensions.
Five of the six are clearly analogues of the five corresponding Platonic solids. The sixth, the 24-cell, has no regular analogue in three dimensions. However, there exists a pair of irregular solids, the cuboctahedron and its dual the rhombic dodecahedron, which are partial analogues to the 24-cell (in complementary ways). Together they can be seen as the three-dimensional analogue of the 24-cell.
Each convex regular 4-polytope is bounded by a set of 3-dimensional cells which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the surface of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).
Properties
Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content within the same radius. The 4-simplex (5-cell |
https://en.wikipedia.org/wiki/Nonstandard%20integer | In mathematics, a nonstandard integer may refer to
Hyperinteger, the integer part of a hyperreal number
an integer in a non-standard model of arithmetic |
https://en.wikipedia.org/wiki/Jacobi%27s%20four-square%20theorem | In number theory, Jacobi's four-square theorem gives a formula for the number of ways that a given positive integer can be represented as the sum of four squares (of integers).
History
The theorem was proved in 1834 by Carl Gustav Jakob Jacobi.
Theorem
Two representations are considered different if their terms are in different order or if the integer being squared (not just the square) is different; to illustrate, these are three of the eight different ways to represent 1:
The number of ways to represent as the sum of four squares is eight times the sum of the divisors of if is odd and 24 times the sum of the odd divisors of if is even (see divisor function), i.e.
Equivalently, it is eight times the sum of all its divisors which are not divisible by 4, i.e.
We may also write this as
where the second term is to be taken as zero if is not divisible by 4. In particular, for a prime number we have the explicit formula .
Some values of occur infinitely often as whenever is even. The values of can be arbitrarily large: indeed, is infinitely often larger than
Proof
The theorem can be proved by elementary means starting with the Jacobi triple product.
The proof shows that the Theta series for the lattice Z4 is a modular form of a certain level, and hence equals a linear combination of Eisenstein series.
See also
Lagrange's four-square theorem
Lambert series
Sum of squares function
Notes
References
External links
Squares in number theory
Theorems in number theory |
https://en.wikipedia.org/wiki/2006%20Kabul%20Premier%20League | Statistics of Kabul Premier League in season 2006.
Overview
Ordu Kabul F.C. won the championship.
Group A
Top three of group A were:
Group B
Top three of group B were:
References
Afghanistan 2006
Kabul Premier League seasons
1
Afghan
Afghan |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 1983–84 season.
Overview
It was contested by 9 teams, and Muharraq Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1983–84 in Bahraini football |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League in season 1984–85.
Overview
It was contested by 9 teams, and Bahrain won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1984–85 in Bahraini football |
https://en.wikipedia.org/wiki/1994%E2%80%9395%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 1994–95 season.
Overview
It was contested by 10 teams, and Muharraq Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1994–95 in Bahraini football |
https://en.wikipedia.org/wiki/1995%E2%80%9396%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 1995–96 season.
Overview
Al-Ahli won the championship.
Championship playoff
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1995–96 in Bahraini football |
https://en.wikipedia.org/wiki/1997%E2%80%9398%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 1997–98 season.
Overview
It was contested by 10 teams, and Bahrain Riffa Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1997–98 in Bahraini football |
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 1998–99 season.
Overview
It was contested by 10 teams, and Muharraq Club won the championship.
League standings
References
Bahrain – List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1998–99 in Bahraini football |
https://en.wikipedia.org/wiki/1999%E2%80%932000%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 1999–2000 season.
Overview
It was contested by 12 teams, and Bahrain Riffa Club won the championship.
Regular season
Group 1
Group 2
Championship playoff
Group A
Group B
Championship playoff
Semifinals
Muharraq Club 0-3 : 1-0 Bahrain Riffa Club
Al-Ahli 2-3 : 0-1 East Riffa Club
Third-place match
Muharraq Club 2-3 Al-Ahli
Final
Bahrain Riffa Club 4-0 East Riffa Club
Relegation playoff
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
Bah
1999–2000 in Bahraini football |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 2000–01 season.
Overview
It was contested by 12 teams, and Muharraq Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bah
1
Bahraini Premier League seasons |
https://en.wikipedia.org/wiki/2002%20Bahraini%20Classification%20League | Statistics of Bahraini Premier League for the 2002 season.
Overview
It was contested by 18 teams, playing in a single round-robin format. Muharraq Club won the championship. The top 10 teams would go to the next Premier League season, while the bottom 8 teams would form the 2nd Division of the next season.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
1
1
Bah
Bah |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 2002–03 season.
Overview
It was contested by 10 teams, and Bahrain Riffa Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
1
Bah |
https://en.wikipedia.org/wiki/2004%E2%80%9305%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 2004–05 season.
Overview
It was contested by 10 teams, and Bahrain Riffa Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
1
Bah |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 2005–06 season.
Overview
It was contested by 10 teams, and Muharraq Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
1
Bah |
https://en.wikipedia.org/wiki/2006%E2%80%9307%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 2006–07 season.
Overview
It was contested by 12 teams, and Muharraq Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
1
Bah |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Bahraini%20Premier%20League | Statistics of Bahraini Premier League for the 2007–08 season.
Overview
It was contested by 12 teams, and Muharraq Club won the championship.
League standings
References
Bahrain - List of final tables (RSSSF)
Bahraini Premier League seasons
1
Bah |
https://en.wikipedia.org/wiki/Herma%20%28Xenakis%29 | Herma (from Greek ἕρμα "a stringing together, a foundation") is a piece for solo piano composed by Iannis Xenakis in 1961. About ten minutes long, it is based on a formulation of the algebraic equations of Boolean algebra, and is also an example of what Xenakis called symbolic music.
Composition
Herma was the composer's first major work for piano. It was composed after a visit to Japan in 1961, where Xenakis befriended pianist and composer Yūji Takahashi. Xenakis completed the piece upon his return to Paris and dedicated it to Takahashi, who premièred the piece on February 2, 1962. The pianist's impression of that concert was that the piece "made some excited and wonder, others feel painful".
Boolean algebra is the main mathematical principle behind Herma. Xenakis defines several pitch sets and proceeds to apply various logical operations to them. The results are incorporated into music by using successions and combinations of various sets. Stochastic procedures are used to select the order and place of notes within each set.
The piece has been described by the pianist and critic Susan Bradshaw as "[deserving] the label of the most difficult piano piece ever written", because of its extreme tempo.
References
Sources
Further reading
Hill, Peter. 1975. "Xenakis and the Performer". Tempo 112:17–22.
Montague, Eugene. 1995. "The Limits of Logic: Structure and Aesthetics in Xenakis's Herma". M.A. thesis. Amherst: University of Massachusetts Amherst. Study based on the thesis available online.
Sevrette, Daniel. 1973. "Étude statistique sur Herma". Dissertation, Schola Cantorum.
Solomos, Makis. "À propos des premières œuvres (1953–69) de I. Xenakis". Thesis, University of Paris.
Squibbs, Ron. 1996. "An Analytical Approach to the Music of Iannis Xenakis". Dissertation. New Haven: Yale University.
Sward, Rosalie. 1981. "An Examination of the Mathematical Systems used in Selected Compositions of Milton Babbitt and Iannis Xenakis". Dissertation. Evanston: Northwestern University.
Wannamaker, Robert. 2001. "Structure and Perception in Herma by Iannis Xenakis". Music Theory Online 7/3.
External links
, Anton Gerzenberg, 2019 Festival of the Accademia Musicale Chigiana, Siena
, [Martin von der Heydt, 2009]
Compositions by Iannis Xenakis
Compositions for solo piano
1961 compositions
Music dedicated to ensembles or performers |
https://en.wikipedia.org/wiki/Szpiro%27s%20conjecture | In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem.
Original statement
The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have
Modified Szpiro conjecture
The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f (using notation from Tate's algorithm), we have
abc conjecture
The abc conjecture originated as the outcome of attempts by Joseph Oesterlé and David Masser to understand Szpiro's conjecture, and was then shown to be equivalent to the modified Szpiro's conjecture.
Claimed proofs
In August 2012, Shinichi Mochizuki claimed a proof of Szpiro's conjecture by developing a new theory called inter-universal Teichmüller theory (IUTT). However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture, with Peter Scholze and Jakob Stix concluding in March 2018 that the gap was "so severe that … small modifications will not rescue the proof strategy".
See also
Arakelov theory
References
Bibliography
Conjectures
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Regular%20chain | In mathematics, and more specifically in computer algebra and elimination theory, a regular chain is a particular kind of triangular set of multivariate polynomials over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more indeterminate than the preceding one. The condition that a triangular set must satisfy to be a regular chain is that, for every , every common zero (in an algebraically closed field) of the first polynomials may be prolongated to a common zero of the th polynomial. In other words, regular chains allow solving systems of polynomial equations by solving successive univariate equations without considering different cases.
Regular chains enhance the notion of Wu's characteristic sets in the sense that they provide a better result with a similar method of computation.
Introduction
Given a linear system, one can convert it to a triangular system via Gaussian elimination. For the non-linear case, given a polynomial system F over a field, one can convert (decompose or triangularize) it to a finite set of triangular sets, in the sense that the algebraic variety V(F) is described by these triangular sets.
A triangular set may merely describe the empty set. To fix this degenerated case, the notion of regular chain was introduced, independently by Kalkbrener (1993), Yang and Zhang (1994). Regular chains also appear in Chou and Gao (1992). Regular chains are special triangular sets which are used in different algorithms for computing unmixed-dimensional decompositions of algebraic varieties. Without using factorization, these decompositions have better properties that the ones produced by Wu's algorithm. Kalkbrener's original definition was based on the following observation: every irreducible variety is uniquely determined by one of its generic points and varieties can be represented by describing the generic points of their irreducible components. These generic points are given by regular chains.
Examples
Denote Q the rational number field. In Q[x1, x2, x3] with variable ordering ,
is a triangular set and also a regular chain. Two generic points given by T are (a, a, a) and (a, −a, a) where a is transcendental over Q.
Thus there are two irreducible components, given by and , respectively.
Note that: (1) the content of the second polynomial is x2, which does not contribute to the generic points represented and thus can be removed; (2) the dimension of each component is 1, the number of free variables in the regular chain.
Formal definitions
The variables in the polynomial ring
are always sorted as .
A non-constant polynomial f in can be seen as a univariate polynomial in its greatest variable.
The greatest variable in f is called its main variable, denoted by mvar(f). Let u be the main variable of f and write it as
where e is the degree of f with respect to u and is the leading coefficient of f with respect to u. Then the initial of f is and e is its mai |
https://en.wikipedia.org/wiki/Shafarevich%20conjecture | In mathematics, the Shafarevich conjecture, named for Igor Shafarevich, may refer to:
The Tate–Shafarevich conjecture that the Tate–Shafarevich group is finite
The Shafarevich conjecture that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a given finite set of places, now proved as Faltings's theorem
The conjecture that the absolute Galois group of the maximal abelian extension of the rational numbers is a free profinite group of countable rank |
https://en.wikipedia.org/wiki/Parity%20problem%20%28sieve%20theory%29 | In number theory, the parity problem refers to a limitation in sieve theory that prevents sieves from giving good estimates in many kinds of prime-counting problems. The problem was identified and named by Atle Selberg in 1949. Beginning around 1996, John Friedlander and Henryk Iwaniec developed some parity-sensitive sieves that make the parity problem less of an obstacle.
Statement
Terence Tao gave this "rough" statement of the problem:
This problem is significant because it may explain why it is difficult for sieves to "detect primes," in other words to give a non-trivial lower bound for the number of primes with some property. For example, in a sense Chen's theorem is very close to a solution of the twin prime conjecture, since it says that there are infinitely many primes p such that p + 2 is either prime or the product of two primes. The parity problem suggests that, because the case of interest has an odd number of prime factors (namely 1), it won't be possible to separate out the two cases using sieves.
Example
This example is due to Selberg and is given as an exercise with hints by Cojocaru & Murty.
The problem is to estimate separately the number of numbers ≤ x with no prime divisors ≤ x1/2, that have an even (or an odd) number of prime factors. It can be shown that, no matter what the choice of weights in a Brun- or Selberg-type sieve, the upper bound obtained will be at least (2 + o(1)) x / ln x for both problems. But in fact the set with an even number of factors is empty and so has size 0. The set with an odd number of factors is just the primes between x1/2 and x, so by the prime number theorem its size is (1 + o(1)) x / ln x. Thus these sieve methods are unable to give a useful upper bound for the first set, and overestimate the upper bound on the second set by a factor of 2.
Parity-sensitive sieves
Beginning around 1996 John Friedlander and Henryk Iwaniec developed some new sieve techniques to "break" the parity problem.
One of the triumphs of these new methods is the Friedlander–Iwaniec theorem, which states that there are infinitely many primes of the form a2 + b4.
Glyn Harman relates the parity problem to the distinction between Type I and Type II information in a sieve.
Karatsuba phenomenon
In 2007 Anatolii Alexeevitch Karatsuba discovered an imbalance between the numbers in an arithmetic progression with given parities of the number of prime factors. His papers were published after his death.
Let be a set of natural numbers (positive integers) that is, the numbers . The set of primes, that is, such integers , , that have just two distinct divisors (namely, and ), is denoted by , . Every natural number , , can be represented as a product of primes (not necessarily distinct), that is where ,
and such representation is unique up to the order of factors.
If we form two sets, the first consisting of positive integers having even number of prime factors, the second consisting of positive integers having an odd num |
https://en.wikipedia.org/wiki/Research%20Institute%20for%20Symbolic%20Computation | The Research Institute for Symbolic Computation (RISC Linz) is a research institute in the area of symbolic computation, including automated theorem proving and computer algebra. It is located in Schloß Hagenberg in Hagenberg near Linz in Austria. RISC was founded in 1987 under Bruno Buchberger and moved to Hagenberg in 1989. The present chairman of RISC is Peter Paule.
External links
RISC Linz
Softwarepark Hagenberg
Computer science organizations |
https://en.wikipedia.org/wiki/Ronald%20%C5%A0ikli%C4%87 | Ronald Šiklić (born 24 November 1980 in Zagreb) is a Croatian retired football defender.
External links
Šiklić profile, detailed club statistics
1980 births
Living people
Footballers from Zagreb
Men's association football defenders
Croatian men's footballers
GNK Dinamo Zagreb players
HNK Šibenik players
NK Inter Zaprešić players
Odra Wodzisław Śląski players
Dyskobolia Grodzisk Wielkopolski players
GKS Górnik Łęczna players
Lechia Gdańsk players
FC Kryvbas Kryvyi Rih players
NK Slaven Belupo players
SK Slavia Prague players
SK Dynamo České Budějovice players
FC Hlučín players
Croatian Football League players
Czech First League players
Croatian expatriate men's footballers
Expatriate men's footballers in Poland
Croatian expatriate sportspeople in Poland
Expatriate men's footballers in Ukraine
Croatian expatriate sportspeople in Ukraine
Expatriate men's footballers in the Czech Republic
Croatian expatriate sportspeople in the Czech Republic |
https://en.wikipedia.org/wiki/Orthogonality%20principle | In statistics and signal processing, the orthogonality principle is a necessary and sufficient condition for the optimality of a Bayesian estimator. Loosely stated, the orthogonality principle says that the error vector of the optimal estimator (in a mean square error sense) is orthogonal to any possible estimator. The orthogonality principle is most commonly stated for linear estimators, but more general formulations are possible. Since the principle is a necessary and sufficient condition for optimality, it can be used to find the minimum mean square error estimator.
Orthogonality principle for linear estimators
The orthogonality principle is most commonly used in the setting of linear estimation. In this context, let x be an unknown random vector which is to be estimated based on the observation vector y. One wishes to construct a linear estimator for some matrix H and vector c. Then, the orthogonality principle states that an estimator achieves minimum mean square error if and only if
and
If x and y have zero mean, then it suffices to require the first condition.
Example
Suppose x is a Gaussian random variable with mean m and variance Also suppose we observe a value where w is Gaussian noise which is independent of x and has mean 0 and variance We wish to find a linear estimator minimizing the MSE. Substituting the expression into the two requirements of the orthogonality principle, we obtain
and
Solving these two linear equations for h and c results in
so that the linear minimum mean square error estimator is given by
This estimator can be interpreted as a weighted average between the noisy measurements y and the prior expected value m. If the noise variance is low compared with the variance of the prior (corresponding to a high SNR), then most of the weight is given to the measurements y, which are deemed more reliable than the prior information. Conversely, if the noise variance is relatively higher, then the estimate will be close to m, as the measurements are not reliable enough to outweigh the prior information.
Finally, note that because the variables x and y are jointly Gaussian, the minimum MSE estimator is linear. Therefore, in this case, the estimator above minimizes the MSE among all estimators, not only linear estimators.
General formulation
Let be a Hilbert space of random variables with an inner product defined by . Suppose is a closed subspace of , representing the space of all possible estimators. One wishes to find a vector which will approximate a vector . More accurately, one would like to minimize the mean squared error (MSE) between and .
In the special case of linear estimators described above, the space is the set of all functions of and , while is the set of linear estimators, i.e., linear functions of only. Other settings which can be formulated in this way include the subspace of causal linear filters and the subspace of all (possibly nonlinear) estimators.
Geometric |
https://en.wikipedia.org/wiki/2002%20Brunei%20Premier%20League | Statistics of the Brunei Premier League football for the 2002 season.
Overview
It was contested by 16 teams, and DPMM FC won the championship.
First stage
Group A
Group B
Second stage
References
External links
Brunei 2002 (RSSSF)
Brunei Premier League seasons
Brunei
Brunei
1 |
https://en.wikipedia.org/wiki/2003%20Brunei%20Premier%20League | Statistics of the Brunei Premier League for the 2003 season.
Overview
It was contested by 20 teams, and Wijaya FC won the championship.
On 5 October, Wijaya beat Indera in their final match of the season courtesy of a solitary Norsillmy Taha goal. DPMM FC would win the championship if they produced the same result against ABDB the following day, as they had a superior goal difference to Wijaya's in their identical league win–loss record.
On 6 October, ABDB beat DPMM 3-1 through a penalty by future DPMM captain Rosmin Kamis followed by goals scored by Sardillah Abdullah and Samdani Judin to hand the title to Wijaya.
First stage
Group A
Group B
Second stage
References
Brunei 2003 (RSSSF)
Brunei Premier League seasons
Brunei
Brunei
1 |
https://en.wikipedia.org/wiki/2004%20Brunei%20Premier%20League | Statistics of the Brunei Premier League for the 2004 season.
Overview
It was contested by 10 teams, and DPMM FC won the championship.
League standings
References
Brunei 2004 (RSSSF)
Brunei Premier League seasons
Brunei
Brunei
1 |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Brunei%20Premier%20League | Statistics of Brunei Premier League for the 2005–06 season.
Overview
It was contested by 10 teams, and QAF FC won the championship.
League standings
References
Brunei 2005/06 (RSSSF)
Brunei Premier League seasons
Brunei
1
1 |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Brunei%20Premier%20League | Statistics of the Brunei Premier League for the 2007–08 season.
Overview
It was contested by 12 teams, and QAF FC won the championship.
League standings
Promotion/relegation playoff
to be held before start 2009 season
March United n/p LLRC FT
NB: cancelled as top level is reduced to 10 clubs
References
Brunei 2007/08 (RSSSF)
Brunei Premier League seasons
Brunei
1
1 |
https://en.wikipedia.org/wiki/2000%20Cambodian%20League | Statistics of the Cambodian League for the 2000 season.
Overview
It was contested by 10 teams. The top four teams qualified to the Championship play-off and Nokorbal Cheat won the championship.
League standings
Championship play-off
Semi-finals
02 Dec 2000 Nokorbal Cheat 3-2 Sala Vekvoeun Yothes
09 Dec 2000 Kang Yothipoi KP 1-2 Keila Rith
Third place
16 Dec 2000 Sala Vekvoeun Yothes 3-1 Kang Yothipoi KP
Final
16 Dec 2000 Nokorbal Cheat 2-0 Kelia Rith
References
Cambodia - List of final tables (RSSSF)
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/2006%20Cambodian%20League | Statistics of the Cambodian League for the 2006 season.
Overview
It was contested by 10 teams, and Khemara Keila FC won the championship.
League standings
Note: Resumed Sep 2 after a break of 4 months due to financial problems in the wake of a power struggle within the CFA which led to a temporary suspension by the AFC; league eventually played over 9 rounds instead of 18 as originally planned.
Top final table
Semifinals
07 Oct 2006 Phnomh Penh United 5-1 Keila Rith
07 Oct 2006 Khemara Keila FC 2-1 Nagacorp
Final
14 Oct 2006 Phnomh Penh United 4-5 Khemara Keila FC
References
Cambodia - List of final tables (RSSSF)
C-League seasons
Cambodia
Cambodia
football |
https://en.wikipedia.org/wiki/2007%20Cambodian%20League | Statistics of Cambodian League for the 2007 season.
Overview
It was contested by 8 teams, and Naga Corp FC won the championship.
League standings
References
Cambodia - List of final tables (RSSSF)
C-League seasons
Cambodia
Cambodia
1 |
https://en.wikipedia.org/wiki/2008%20Cambodian%20League | Statistics of the Cambodian League for the 2008 season.
Clubs
Khemara Keila
National Defense
Phu Chung Neak
Nagacorp FC
Phnom Penh Empire (it was called Hello United)
Post Tel Club
Preah Khan Reach
Build Bright United
Moha Garuda
Kirivong Sok Sen Chey
League standings
References
Cambodia - List of final tables (RSSSF)
C-League seasons
Cambodia
Cambodia
1 |
https://en.wikipedia.org/wiki/1994%20Chinese%20Taipei%20National%20Football%20League | Statistics of the Chinese Taipei National Football League for the 1994 season.
Overview
It was contested by 8 teams, and Tatung won the championship.
League standings
References
Chinese Taipei - List of final tables (RSSSF)
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/1997%20Chinese%20Taipei%20National%20Football%20League | Statistics of the Chinese Taipei National Football League for the 1997 season.
Overview
It was contested by 8 teams, and Taipower won the championship.
League standings
References
Chinese Taipei - List of final tables (RSSSF)
Chinese Taipei National Football League seasons
1
Taipei
Taipei |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Chinese%20Taipei%20National%20Football%20League | Statistics of the Chinese Taipei National Football League for the 2000–01 season.
Overview
It was contested by 7 teams, and Taipower won the championship.
League standings
Notes
References
Chinese Taipei - List of final tables (RSSSF)
Chinese Taipei National Football League seasons
Chinese Taipei
1
1 |
https://en.wikipedia.org/wiki/Fundamental%20lemma%20of%20sieve%20theory | In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richert
write:
Diamond & Halberstam
attribute the terminology Fundamental Lemma to Jonas Kubilius.
Common notation
We use these notations:
is a set of positive integers, and is its subset of integers divisible by
and are functions of and of that estimate the number of elements of that are divisible by , according to the formula
Thus represents an approximate density of members divisible by , and represents an error or remainder term.
is a set of primes, and is the product of those primes
is the number of elements of not divisible by any prime in that is
is a constant, called the sifting density, that appears in the assumptions below. It is a weighted average of the number of residue classes sieved out by each prime.
Fundamental lemma of the combinatorial sieve
This formulation is from Tenenbaum. Other formulations are in Halberstam & Richert, in Greaves,
and in Friedlander & Iwaniec.
We make the assumptions:
is a multiplicative function.
The sifting density satisfies, for some constant and any real numbers and with :
There is a parameter that is at our disposal. We have uniformly in , , , and that
In applications we pick to get the best error term. In the sieve it is related to the number of levels of the inclusion–exclusion principle.
Fundamental lemma of the Selberg sieve
This formulation is from Halberstam & Richert. Another formulation is in Diamond & Halberstam.
We make the assumptions:
is a multiplicative function.
The sifting density satisfies, for some constant and any real numbers and with :
for some small fixed and all .
for all squarefree whose prime factors are in .
The fundamental lemma has almost the same form as for the combinatorial sieve. Write . The conclusion is:
Note that is no longer an independent parameter at our disposal, but is controlled by the choice of .
Note that the error term here is weaker than for the fundamental lemma of the combinatorial sieve. Halberstam & Richert remark: "Thus it is not true to say, as has been asserted from time to time in the literature, that Selberg's sieve is always better than Brun's."
Notes
Sieve theory
Theorems in analytic number theory |
https://en.wikipedia.org/wiki/Circle%20of%20antisimilitude | In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, α and β, is a reference circle for which α and β are inverses of each other. If α and β are non-intersecting or tangent, a single circle of antisimilitude exists; if α and β intersect at two points, there are two circles of antisimilitude. When α and β are congruent, the circle of antisimilitude degenerates to a line of symmetry through which α and β are reflections of each other.
Properties
If the two circles α and β cross each other, another two circles γ and δ are each tangent to both α and β, and in addition γ and δ are tangent to each other, then the point of tangency between γ and δ necessarily lies on one of the two circles of antisimilitude. If α and β are disjoint and non-concentric, then the locus of points of tangency of γ and δ again forms two circles, but only one of these is the (unique) circle of antisimilitude. If α and β are tangent or concentric, then the locus of points of tangency degenerates to a single circle, which again is the circle of antisimilitude.
If the two circles α and β cross each other, then their two circles of antisimilitude each pass through both crossing points, and bisect the angles formed by the arcs of α and β as they cross.
If a circle γ crosses circles α and β at equal angles, then γ is crossed orthogonally by one of the circles of antisimilitude of α and β; if γ crosses α and β in supplementary angles, it is crossed orthogonally by the other circle of antisimilitude, and if γ is orthogonal to both α and β then it is also orthogonal to both circles of antisimilitude.
For three circles
Suppose that, for three circles α, β, and γ, there is a circle of antisimilitude for the pair (α,β) that crosses a second circle of antisimilitude for the pair (β,γ). Then there is a third circle of antisimiltude for the third pair (α,γ) such that the three circles of antisimilitude cross each other in two triple intersection points. Altogether, at most eight triple crossing points may be generated in this way, for there are two ways of choosing each of the first two circles and two points where the two chosen circles cross. These eight or fewer triple crossing points are the centers of inversions that take all three circles α, β, and γ to become equal circles. For three circles that are mutually externally tangent, the (unique) circles of antisimilitude for each pair again cross each other at 120° angles in two triple intersection points that are the isodynamic points of the triangle formed by the three points of tangency.
See also
Inversive geometry
Limiting point (geometry), the center of an inversion that transforms two circles into concentric position
Radical axis
References
External links
Circles
Inversive geometry |
https://en.wikipedia.org/wiki/Primera%20Divisi%C3%B3n%20de%20Paraguay%20topscorers | The following article contains a year-by-year list and statistics of football topscorers in the Primera División de Paraguay (Paraguayan First Division).
Topscorers by year
The following list only comprises the professional era and is missing data from 1906 to 1934 (amateur era).
Also, since 2008 the Paraguayan football association (APF) awards two national champions per year; one for the Torneo Apertura and another for the Torneo Clausura. Therefore, the following legends are used from 2008 and on:
[A] = Apertura
[C] = Clausura
[O] = Overall topscorer for that year
Topscorers by club
The number of topscorers included in the following table only represent the overall topscorers for the entire year (those labeled with a [O] since 2008). Topscorers for specific tournaments (like Apertura [A] and Clausura [C]) are not included.
Records and statistics
Juan Samudio is the all-time goalscoring leader with 111 goals, all of them scored for Libertad and Guaraní from 1997 to the present day. Mauro Caballero is second on the list with 107 goals between 1992 and 2007, playing for Olimpia, Cerro Porteño, Nacional and Libertad.
Flaminio Silva is the player to have scored more goals in a single season: 34 goals in 1936, playing for Olimpia.
Máximo Rolón is the player with the most consecutive goalscoring titles (3 times): he was the topscorer in 1954, 1955 and 1956; playing for Libertad.
Héctor Núñez is the only foreign player to win the goalscoring title back-to-back (1994–1995), playing for Cerro Porteño.
See also
Primera División de Paraguay
Paraguayan football league system
References
External links
Asociacion Paraguaya de Futbol
Teledeportes Digital
Topscorers
Association football player non-biographical articles |
https://en.wikipedia.org/wiki/Formal%20scheme | In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of the scheme. For this reason, formal schemes frequently appear in topics such as deformation theory. But the concept is also used to prove a theorem such as the theorem on formal functions, which is used to deduce theorems of interest for usual schemes.
A locally Noetherian scheme is a locally Noetherian formal scheme in the canonical way: the formal completion along itself. In other words, the category of locally Noetherian formal schemes contains all locally Noetherian schemes.
Formal schemes were motivated by and generalize Zariski's theory of formal holomorphic functions.
Algebraic geometry based on formal schemes is called formal algebraic geometry.
Definition
Formal schemes are usually defined only in the Noetherian case. While there have been several definitions of non-Noetherian formal schemes, these encounter technical problems. Consequently, we will only define locally noetherian formal schemes.
All rings will be assumed to be commutative and with unit. Let A be a (Noetherian) topological ring, that is, a ring A which is a topological space such that the operations of addition and multiplication are continuous. A is linearly topologized if zero has a base consisting of ideals. An ideal of definition for a linearly topologized ring is an open ideal such that for every open neighborhood V of 0, there exists a positive integer n such that . A linearly topologized ring is preadmissible if it admits an ideal of definition, and it is admissible if it is also complete. (In the terminology of Bourbaki, this is "complete and separated".)
Assume that A is admissible, and let be an ideal of definition. A prime ideal is open if and only if it contains . The set of open prime ideals of A, or equivalently the set of prime ideals of , is the underlying topological space of the formal spectrum of A, denoted Spf A. Spf A has a structure sheaf which is defined using the structure sheaf of the spectrum of a ring. Let be a neighborhood basis for zero consisting of ideals of definition. All the spectra of have the same underlying topological space but a different structure sheaf. The structure sheaf of Spf A is the projective limit .
It can be shown that if f ∈ A and Df is the set of all open prime ideals of A not containing f, then , where is the completion of the localization Af.
Finally, a locally noetherian formal scheme is a topologically ringed space (that is, a ringed space whose sheaf of rings is a sheaf of topological rings) such that each point of admits an open neighborhood isomorphic (as topologically ringed spaces) to the formal spectrum of a noetherian ring.
Morphisms between formal schemes
A morphism of locally noetherian formal schemes is a morphism of them as locally ringed space |
https://en.wikipedia.org/wiki/Valuative%20criterion | In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes, is universally closed, separated, or proper.
Statement of the valuative criteria
Recall that a valuation ring A is a domain, so if K is the field of fractions of A, then Spec K is the generic point of Spec A.
Let X and Y be schemes, and let f : X → Y be a morphism of schemes. Then the following are equivalent:
f is separated (resp. universally closed, resp. proper)
f is quasi-separated (resp. quasi-compact, resp. of finite type and quasi-separated) and for every valuation ring A, if Y' = Spec A and X' denotes the generic point of Y' , then for every morphism Y' → Y and every morphism X' → X which lifts the generic point, then there exists at most one (resp. at least one, resp. exactly one) lift Y' → X.
The lifting condition is equivalent to specifying that the natural morphism
is injective (resp. surjective, resp. bijective).
Furthermore, in the special case when Y is (locally) noetherian, it suffices to check the case that A is a discrete valuation ring.
References
Algebraic geometry
Scheme theory |
https://en.wikipedia.org/wiki/Littlewood%20polynomial | In mathematics, a Littlewood polynomial is a polynomial all of whose coefficients are +1 or −1.
Littlewood's problem asks how large the values of such a polynomial must be on the unit circle in the complex plane. The answer to this would yield information about the autocorrelation of binary sequences.
They are named for J. E. Littlewood who studied them in the 1950s.
Definition
A polynomial
is a Littlewood polynomial if all the . Littlewood's problem asks for constants c1 and c2 such that there are infinitely many Littlewood polynomials pn , of increasing degree n satisfying
for all on the unit circle. The Rudin–Shapiro polynomials provide a sequence satisfying the upper bound with . In 2019, an infinite family of Littlewood polynomials satisfying both the upper and lower bound was constructed by Paul Balister, Béla Bollobás, Robert Morris, Julian Sahasrabudhe, and Marius Tiba.
References
Polynomials
Conjectures |
https://en.wikipedia.org/wiki/Hall%E2%80%93Littlewood%20polynomials | In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials.
They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961).
Definition
The Hall–Littlewood polynomial P is defined by
where λ is a partition of at most n with elements λi, and m(i) elements equal to i, and Sn is the symmetric group of order n!.
As an example,
Specializations
We have that , and
where the latter is the Schur P polynomials.
Properties
Expanding the Schur polynomials in terms of the Hall–Littlewood polynomials, one has
where are the Kostka–Foulkes polynomials.
Note that as , these reduce to the ordinary Kostka coefficients.
A combinatorial description for the Kostka–Foulkes polynomials was given by Lascoux and Schützenberger,
where "charge" is a certain combinatorial statistic on semistandard Young tableaux,
and the sum is taken over all semi-standard Young tableaux with shape λ and type μ.
See also
Hall polynomial
References
External links
Orthogonal polynomials
Algebraic combinatorics
Symmetric functions |
https://en.wikipedia.org/wiki/Juan%20Ignacio%20Carrera | Juan Ignacio Carrera (born 10 May 1981 in Pergamino, Buenos Aires) is an Argentine football goalkeeper currently playing for San Martín de Tucumán.
External links
Statistics at Futbol XXI
Football-Lineups player profile
1981 births
Living people
People from Pergamino
Argentine men's footballers
Sarmiento de Resistencia footballers
Men's association football goalkeepers
Argentinos Juniors footballers
Argentine Primera División players
Footballers from Buenos Aires Province |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Super%20Liga%20Timorense | Statistics of Super Liga in season 2005/2006.
Club's
List of Participating team in Super Liga 2005/06
Académica
AD Esperança
ADR União
As Lero
Bulgaria
Cacussa
FC Café
FC Irmãos Unidos
FC Porto Taibesi
FC Rusa Fuik
FC Zebra
Fima Sporting
SLB Laulara
First stage
Grup A
Grup B
Grup C
Grup D
Playoff
Grup E
Round 1
Fima Sporting 3-1 SLB Laulara
FC Rusa Fuik 0-0 Académica
Round 2
Académica - Fima Sporting
SLB Laulara - FC Rusa Fuik
Round 3
Académica 3-2 SLB Laulara
Fima Sporting - FC Rusa Fuik
Winners: Fima Sporting
Runners-Up: Académica
Grup F
Round 1
FC Zebra 0-0 FC Café
FC Porto Taibesi 0-2 AD Esperança
Winners: AD Esperança
Runners-Up: FC Zebra
Semifinals
AD Esperança 2-0 Académica
Fima Sporting 3-2 F C Zebra
Third place match
Académica 2-0 FC Zebra
Final
Fima Sporting 0-0 (pen 5-4) AD Esperança
References
East Timor - List of final tables (RSSSF)
2005
2005 in East Timorese sport
2006 in East Timorese sport
East |
https://en.wikipedia.org/wiki/2006%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League for the 2006 season.
Spring League
Group stage
Semifinals
Guam Shipyard 2-1 Orange Crushers
Quality Distributors 4-2 Dodge Rams
Third-place match
Orange Crushers 8-2 Dodge Rams
Final
Guam Shipyard 6-1 Quality Distributors
Fall League
References
Guam 2006 (RSSSF)
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2007%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League for the 2007 season.
League standings
References
Guam 2007 (RSSSF)
Guam Soccer League seasons
Guam
Guam
football |
https://en.wikipedia.org/wiki/2007%E2%80%9308%20Guam%20Men%27s%20Soccer%20League | Statistics of Guam League for the 2007–08 season.
League standings
References
Guam 2007/08 (RSSSF)
Guam Soccer League seasons
Guam
Mens |
https://en.wikipedia.org/wiki/Mike%20Abou-Mechrek | Mike Abou-Mechrek (born October 14, 1975) is a former professional Canadian football offensive linemen. He played CIS Football for the Western Ontario Mustangs.
Season statistics
References
1975 births
Living people
Canadian football offensive linemen
Ottawa Renegades players
Canadian football people from Toronto
Players of Canadian football from Ontario
Saskatchewan Roughriders players
Western Mustangs football players
Winnipeg Blue Bombers players |
https://en.wikipedia.org/wiki/Cyclotomic%20character | In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring , its representation space is generally denoted by (that is, it is a representation ).
p-adic cyclotomic character
Fix a prime, and let denote the absolute Galois group of the rational numbers.
The roots of unity form a cyclic group of order , generated by any choice of a primitive th root of unity .
Since all of the primitive roots in are Galois conjugate, the Galois group acts on by automorphisms. After fixing a primitive root of unity generating , any element of can be written as a power of , where the exponent is a unique element in . One can thus write
where is the unique element as above, depending on both and . This defines a group homomorphism called the mod cyclotomic character:
which is viewed as a character since the action corresponds to a homomorphism .
Fixing and and varying , the form a compatible system in the sense that they give an element of the inverse limit the units in the ring of p-adic integers. Thus the assemble to a group homomorphism called -adic cyclotomic character:
encoding the action of on all -power roots of unity simultaneously. In fact equipping with the Krull topology and with the -adic topology makes this a continuous representation of a topological group.
As a compatible system of -adic representations
By varying over all prime numbers, a compatible system of ℓ-adic representations is obtained from the -adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol to denote a prime instead of ). That is to say, is a "family" of -adic representations
satisfying certain compatibilities between different primes. In fact, the form a strictly compatible system of ℓ-adic representations.
Geometric realizations
The -adic cyclotomic character is the -adic Tate module of the multiplicative group scheme over . As such, its representation space can be viewed as the inverse limit of the groups of th roots of unity in .
In terms of cohomology, the -adic cyclotomic character is the dual of the first -adic étale cohomology group of . It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of .
In terms of motives, the -adic cyclotomic character is the -adic realization of the Tate motive . As a Grothendieck motive, the Tate motive is the dual of .
Properties
The -adic cyclotomic character satisfies several nice properties.
It is unramified at all primes (i.e. the inertia subgroup at acts trivially).
If is a Frobenius element for , then
It is crystalline at .
See also
Tate twist
References
Algebraic number theory |
https://en.wikipedia.org/wiki/Conductor%20of%20an%20abelian%20variety | In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points.
Definition
For an abelian variety A defined over a field F as above, with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over
Spec(R)
(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism
Spec(F) → Spec(R)
gives back A. Let A0 denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a maximal ideal P of R with residue field k, A0k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let uP be the dimension of the unipotent group and tP the dimension of the torus. The order of the conductor at P is
where is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by
Properties
A has good reduction at P if and only if (which implies ).
A has semistable reduction if and only if (then again ).
If A acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic at P, then δP = 0.
If , where d is the dimension of A, then .
If and F is a finite extension of of ramification degree , there is an upper bound expressed in terms of the function , which is defined as follows:
Write with and set . Then
Further, for every with there is a field with and an abelian variety of dimension so that is an equality.
References
Abelian varieties
Diophantine geometry
Algebraic number theory |
https://en.wikipedia.org/wiki/Limiting%20point | Limiting point has the following meanings in mathematics:
Limit (mathematics)
Limit point in mathematics
Limiting point (geometry), one of two points defined from two disjoint circles
See also
Point at infinity
Ideal point |
https://en.wikipedia.org/wiki/Inversive%20distance | In inversive geometry, the inversive distance is a way of measuring the "distance" between two circles, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other.
Properties
The inversive distance remains unchanged if the circles are inverted, or transformed by a Möbius transformation. One pair of circles can be transformed to another pair by a Möbius transformation if and only if both pairs have the same inversive distance.
An analogue of the Beckman–Quarles theorem holds true for the inversive distance: if a bijection of the set of circles in the inversive plane preserves the inversive distance between pairs of circles at some chosen fixed distance , then it must be a Möbius transformation that preserves all inversive distances.
Distance formula
For two circles in the Euclidean plane with radii and , and distance between their centers, the inversive distance can be defined
by the formula
This formula gives:
a value greater than 1 for two disjoint circles,
a value of 1 for two circles that are tangent to each other and both outside each other,
a value between −1 and 1 for two circles that intersect,
a value of 0 for two circles that intersect each other at right angles ,
a value of −1 for two circles that are tangent to each other, one inside of the other,
and a value less than −1 when one circle contains the other.
(Some authors define the absolute inversive distance as the absolute value of the inversive distance.)
Some authors modify this formula by taking the inverse hyperbolic cosine of the value given above, rather than the value itself. That is, rather than using the number as the inversive distance, the distance is instead defined as the number obeying the equation
Although transforming the inversive distance in this way makes the distance formula more complicated, and prevents its application to crossing pairs of circles, it has the advantage that (like the usual distance for points on a line) the distance becomes additive for circles in a pencil of circles. That is, if three circles belong to a common pencil, then (using in place of as the inversive distance) one of their three pairwise distances will be the sum of the other two.
In other geometries
It is also possible to define the inversive distance for circles on a sphere, or for circles in the hyperbolic plane.
Applications
Steiner chains
A Steiner chain for two disjoint circles is a finite cyclic sequence of additional circles, each of which is tangent to the two given circles and to its two neighbors in the chain.
Steiner's porism states that if two circles have a Steiner chain, they have infinitely many such chains.
The chain is allowed to wrap more than once around the two circles, and can be characterized by a rational number whose numerator is the number of circles in the chain and whose denominator is the number of times it wraps around. All chains for the same two circles have the same value of . If the |
https://en.wikipedia.org/wiki/Dave%20Gans | David Gans (born June 6, 1964) is a Canadian former professional ice hockey player who played for the Los Angeles Kings during the 1982–83 and the 1985–86 seasons.
Career statistics
External links
1964 births
Living people
Canadian ice hockey centres
Hershey Bears players
HC Ambrì-Piotta players
Los Angeles Kings draft picks
Los Angeles Kings players
New Haven Nighthawks players
Newmarket Saints players
Oshawa Generals players
Ice hockey people from Brantford
Toledo Goaldiggers players |
https://en.wikipedia.org/wiki/Truncated%20octahedral%20prism | In 4-dimensional geometry, a truncated octahedral prism or omnitruncated tetrahedral prism is a convex uniform 4-polytope. This 4-polytope has 16 cells (2 truncated octahedra connected by 6 cubes, 8 hexagonal prisms.) It has 64 faces (48 squares and 16 hexagons), and 96 edges and 48 vertices.
It has two symmetry constructions, one from the truncated octahedron, and one as an omnitruncation of the tetrahedron.
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Images
Alternative names
Truncated octahedral dyadic prism (Norman W. Johnson)
Truncated octahedral hyperprism
Tope (Jonathan Bowers: for truncated octahedral prism)
Related polytopes
The snub tetrahedral prism (also called an icosahedral prism), , sr{3,3}×{ }, is related to this polytope just like a snub tetrahedron (icosahedron), is the alternation of the truncated octahedron in its tetrahedral symmetry . The snub tetrahedral prism has symmetry [(3,3)+,2], order 24, although as an icosahedral prism, its full symmetry is [5,3,2], order 240.
Also related, the full snub tetrahedral antiprism or omnisnub tetrahedral antiprism is defined as an alternation of an omnitruncated tetrahedral prism, represented by = ht0,1,2,3{3,3,2}, or , although it cannot be constructed as a uniform 4-polytope. It can also be seen as an alternated truncated octahedral prism or pyritohedral icosahedral antiprism, . It has 2 icosahedra connected by 6 tetrahedra and 8 octahedra, with 24 irregular tetrahedra in the alternated gaps. In total it has 40 cells, 112 triangular faces, 96 edges, and 24 vertices. It has [4,(3,2)+] symmetry, order 48, and also [3,3,2]+ symmetry, order 24.
A construction exists with two regular icosahedra in snub positions with two edge lengths in a ratio of around 0.831 : 1.
Vertex figure for the omnisnub tetrahedral antiprism
See also
Truncated 16-cell,
External links
4-polytopes |
https://en.wikipedia.org/wiki/Adriano%20Garsia | Adriano Mario Garsia (born 20 August 1928) is a Tunisian-born Italian American mathematician who works in analysis, combinatorics, representation theory, and algebraic geometry. He is a student of Charles Loewner and has published work on representation theory, symmetric functions, and algebraic combinatorics. He and Mark Haiman made the N!_conjecture. He is also the namesake of the Garsia–Wachs algorithm for optimal binary search trees, which he published with his student Michelle L. Wachs in 1977.
Born to Italian Tunisians in Tunis on 20 August 1928, Garsia moved to Rome in 1946.
, he had 36 students and at least 200 descendants, according to the data at the Mathematics Genealogy Project. He was on the faculty of the University of California, San Diego. He retired in 2013 after 57 years at UCSD as a founding member of the Mathematics Department. At his 90 Birthday Conference in 2019, it was notable that he was the oldest principal investigator of a grant from the National Science Foundation in the country.
In 2012, he became a fellow of the American Mathematical Society.
Books by A. Garsia
Adriano M. Garsia, Topics in Almost Everywhere Convergence, Lectures in Advanced Mathematics Volume 4, Markham Publishing Co., Chicago, Ill., 1970.
Adriano M. Garsia, Martingale inequalities: Seminar Notes on Recent Progress, Mathematics Lecture Notes Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973.
Adriano M. Garsia and Mark Haiman, Orbit Harmonics and Graded Representations, Research Monograph, to appear as part of the collection published by the Laboratoire de Combinatoire et d'Informatique Mathématique, edited by S. Brlek, Université du Québec à Montréal.
Adriano M. Garsia and Ömer Eğecioğlu, Lessons in Enumerative Combinatorics, Graduate Texts in Mathematics 290, Springer Nature, Switzerland AG, 2021. ISBN 978-3-030-71249-5.
References
External links
1928 births
Living people
Italian emigrants to the United States
20th-century American mathematicians
21st-century American mathematicians
Stanford University alumni
University of California, San Diego faculty
Fellows of the American Mathematical Society
People from Tunis
Combinatorialists
Scientists from California
Italian expatriates in Tunisia |
https://en.wikipedia.org/wiki/Pir%C9%99qan%C4%B1m | Pirəqanım (also, Pir-Agany and Piraganym) is a village in the Ismailli Rayon of Azerbaijan. The village forms part of the municipality of Müdrəsə. According to Azerbaijan's State Statistics Committee, only two people lived in the village as of 2014.
References
Populated places in Ismayilli District |
https://en.wikipedia.org/wiki/Dahar%2C%20%C4%B0smay%C4%B1ll%C4%B1 | Dahar (also, Daxar and Dakhar) is a village in the Ismailli Rayon of Azerbaijan. The village forms part of the municipality of Cülyan. According to Azerbaijan's State Statistics Committee, only five people lived in the village as of 2014.
References
Populated places in Ismayilli District |
https://en.wikipedia.org/wiki/K%C9%99nz%C9%99%2C%20Ismailli | Kənzə (also, Ganza and Genzya) is a village in the Ismailli Rayon of Azerbaijan. The village forms part of the municipality of Qoşakənd. According to Azerbaijan's State Statistics Committee, only three people lived in the village as of 2014.
Notes
References
Populated places in Ismayilli District |
https://en.wikipedia.org/wiki/JASON%20Project | The JASON Project is a US K-12 science curriculum program that is designed to motivate and inspire students to pursue interests and careers in science, technology, engineering and mathematics.
The JASON Project's approach to science education immerses students in real-world situations where they are mentored by scientists from organizations like NASA, NOAA, the U.S. Department of Energy, and parent company National Geographic Society. JASON creates these connections using educational games, videos, live interactivity and social networking to embed its partners' research in the curriculum.
History
The JASON Project was started in 1989 by Dr. Robert Ballard, the oceanographer who discovered the wreck of the RMS Titanic. The JASON Foundation for Education was founded in 1990 as a 501(c)(3) non-profit organization to administer the project. The Foundation became a subsidiary of the National Geographic Society in 2005.
The project won a scientific public engagement award from the American Association for the Advancement of Science, and Computerworld's Smithsonian Award for its use of technology. The JASON curricula are available in print and free online, aligned to national and state standards. The JASON Mission Center contains all student and teacher content, communications systems, digital experiences, and tools to manage, assess and track student performance and online usage.
JASON curricula are available free online, free print-on-demand and in print editions for purchase.
References
External links
Distance education institutions based in the United States
Science education in the United States
NASA programs |
https://en.wikipedia.org/wiki/Supersymmetry%20algebra | In theoretical physics, a supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains not only the Poincaré algebra and a compact subalgebra of internal symmetries, but also contains some fermionic supercharges, transforming as a sum of N real spinor representations of the Poincaré group. Such symmetries are allowed by the Haag–Łopuszański–Sohnius theorem. When N>1 the algebra is said to have extended supersymmetry. The supersymmetry algebra is a semidirect sum of a central extension of the super-Poincaré algebra by a compact Lie algebra B of internal symmetries.
Bosonic fields commute while fermionic fields anticommute. In order to have a transformation that relates the two kinds of fields, the introduction of a Z2-grading under which the even elements are bosonic and the odd elements are fermionic is required. Such an algebra is called a Lie superalgebra.
Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra, called supermultiplets. For each Lie algebra, there exists an associated Lie group which is connected and simply connected, unique up to isomorphism, and the representations of the algebra can be extended to create group representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.
Structure of a supersymmetry algebra
The general supersymmetry algebra for spacetime dimension d, and with the fermionic piece consisting of a sum of N irreducible real spinor representations, has a structure of the form
(P×Z).Q.(L×B)
where
P is a bosonic abelian vector normal subalgebra of dimension d, normally identified with translations of spacetime. It is a vector representation of L.
Z is a scalar bosonic algebra in the center whose elements are called central charges.
Q is an abelian fermionic spinor subquotient algebra, and is a sum of N real spinor representations of L. (When the signature of spacetime is divisible by 4 there are two different spinor representations of L, so there is some ambiguity about the structure of Q as a representation of L.) The elements of Q, or rather their inverse images in the supersymmetry algebra, are called supercharges. The subalgebra (P×Z).Q is sometimes also called the supersymmetry algebra and is nilpotent of length at most 2, with the Lie bracket of two supercharges lying in P×Z.
L is a bosonic subalgebra, isomorphic to the Lorentz algebra in d dimensions, of dimension d(d–1)/2
B is a scalar bosonic subalgebra, given by the Lie algebra of some compact group, called the group of internal symmetries. It commutes with P,Z, and L, but may act non-trivially on the supercharges Q.
The terms "bosonic" and "fermionic" refer to even and odd subspaces of the superalgebra.
The terms "scalar", "spinor", "vector", refer to the behavior of subalgebras under the action of the Lorentz algebra |
https://en.wikipedia.org/wiki/RWTH%20Aachen%20Faculty%20of%20Mathematics%2C%20Computer%20science%2C%20and%20Natural%20sciences | The Faculty of Mathematics, Computer science, and Natural sciences is one of nine faculties at the RWTH Aachen University. It comprises five sections for mathematics, computer science, physics, chemistry and biology. The faculty was founded in 1880 and produced several notable individuals like Arnold Sommerfeld and Nobel laureates Philipp Lenard, Wilhelm Wien, Johannes Stark or Karl Ziegler. Peter Debye studied physics at the RWTH Aachen and won the Nobel Prize in 1936. Furthermore, Helmut Zahn and his team of the Institute for textile chemistry were the first who synthesised Insulin.
The faculty cooperates with Forschungszentrum Jülich and the 4 Fraunhofer Institutes in Aachen. Several projects are assisted by the Deutsche Forschungsgemeinschaft and the European Union. In the academic year 2019/20, approximately 9,700 students are enrolled in the faculty, which makes it the second largest faculty at the RWTH.
References
External links
Department of Mathematics (German version)
Department of Computer science (English version)
Department of Physics (English version)
Department of Chemistry (English version)
Department of Biology (German version)
RWTH Aachen University |
https://en.wikipedia.org/wiki/Steiner%20chain | In geometry, a Steiner chain is a set of circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where is finite and each circle in the chain is tangent to the previous and next circles in the chain. In the usual closed Steiner chains, the first and last (-th) circles are also tangent to each other; by contrast, in open Steiner chains, they need not be. The given circles and do not intersect, but otherwise are unconstrained; the smaller circle may lie completely inside or outside of the larger circle. In these cases, the centers of Steiner-chain circles lie on an ellipse or a hyperbola, respectively.
Steiner chains are named after Jakob Steiner, who defined them in the 19th century and discovered many of their properties. A fundamental result is Steiner's porism, which states:
If at least one closed Steiner chain of circles exists for two given circles and , then there is an infinite number of closed Steiner chains of circles; and any circle tangent to and in the same way is a member of such a chain.
The method of circle inversion is helpful in treating Steiner chains. Since it preserves tangencies, angles and circles, inversion transforms one Steiner chain into another of the same number of circles. One particular choice of inversion transforms the given circles and into concentric circles; in this case, all the circles of the Steiner chain have the same size and can "roll" around in the annulus between the circles similar to ball bearings. This standard configuration allows several properties of Steiner chains to be derived, e.g., its points of tangencies always lie on a circle. Several generalizations of Steiner chains exist, most notably Soddy's hexlet and Pappus chains.
Definitions and types of tangency
The two given circles α and β cannot intersect; hence, the smaller given circle must lie inside or outside the larger. The circles are usually shown as an annulus, i.e., with the smaller given circle inside the larger one. In this configuration, the Steiner-chain circles are externally tangent to the inner given circle and internally tangent to the outer circle. However, the smaller circle may also lie completely outside the larger one (Figure 2). The black circles of Figure 2 satisfy the conditions for a closed Steiner chain: they are all tangent to the two given circles and each is tangent to its neighbors in the chain. In this configuration, the Steiner-chain circles have the same type of tangency to both given circles, either externally or internally tangent to both. If the two given circles are tangent at a point, the Steiner chain becomes an infinite Pappus chain, which is often discussed in the context of the arbelos (shoemaker's knife), a geometric figure made from three circles. There is no general name for a sequence of circles tangent to two given circles that intersect at two points.
Closed, open and multi-cyclic
The two given circles α and β touch the n circles of the |
https://en.wikipedia.org/wiki/Conditional%20mutual%20information | In probability theory, particularly information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.
Definition
For random variables , , and with support sets , and , we define the conditional mutual information as
This may be written in terms of the expectation operator: .
Thus is the expected (with respect to ) Kullback–Leibler divergence from the conditional joint distribution to the product of the conditional marginals and . Compare with the definition of mutual information.
In terms of PMFs for discrete distributions
For discrete random variables , , and with support sets , and , the conditional mutual information is as follows
where the marginal, joint, and/or conditional probability mass functions are denoted by with the appropriate subscript. This can be simplified as
In terms of PDFs for continuous distributions
For (absolutely) continuous random variables , , and with support sets , and , the conditional mutual information is as follows
where the marginal, joint, and/or conditional probability density functions are denoted by with the appropriate subscript. This can be simplified as
Some identities
Alternatively, we may write in terms of joint and conditional entropies as
This can be rewritten to show its relationship to mutual information
usually rearranged as the chain rule for mutual information
or
Another equivalent form of the above is
Another equivalent form of the conditional mutual information is
Like mutual information, conditional mutual information can be expressed as a Kullback–Leibler divergence:
Or as an expected value of simpler Kullback–Leibler divergences:
,
.
More general definition
A more general definition of conditional mutual information, applicable to random variables with continuous or other arbitrary distributions, will depend on the concept of regular conditional probability.
Let be a probability space, and let the random variables , , and each be defined as a Borel-measurable function from to some state space endowed with a topological structure.
Consider the Borel measure (on the σ-algebra generated by the open sets) in the state space of each random variable defined by assigning each Borel set the -measure of its preimage in . This is called the pushforward measure The support of a random variable is defined to be the topological support of this measure, i.e.
Now we can formally define the conditional probability measure given the value of one (or, via the product topology, more) of the random variables. Let be a measurable subset of (i.e. ) and let Then, using the disintegration theorem:
where the limit is taken over the open neighborhoods of , as they are allowed to become arbitrarily smaller with respect to set inclusion.
Finally we can define the conditional mutual information via Lebesgue integration:
where the integrand is the logarithm of a |
https://en.wikipedia.org/wiki/Ludwig%20Stickelberger | Ludwig Stickelberger (18 May 1850 – 11 April 1936) was a Swiss mathematician who made important contributions to linear algebra (theory of elementary divisors) and algebraic number theory (Stickelberger relation in the theory of cyclotomic fields).
Short biography
Stickelberger was born in Buch in the canton of Schaffhausen into a family of a pastor. He graduated from a gymnasium in 1867 and studied next in the University of Heidelberg. In 1874 he received a doctorate in Berlin under the direction of Karl Weierstrass for his work on the transformation of quadratic forms to a diagonal form. In the same year, he obtained his Habilitation from Polytechnicum in Zurich (now ETH Zurich). In 1879 he became an extraordinary professor in the Albert Ludwigs University of Freiburg. From 1896 to 1919 he worked there as a full professor, and from 1919 until his return to Basel in 1924 he held the title of a distinguished professor ("ordentlicher Honorarprofessor"). He was married in 1895, but his wife and son both died in 1918. Stickelberger died on 11 April 1936 and was buried next to his wife and son in Freiburg.
Mathematical contributions
Stickelberger's obituary lists the total of 14 publications: his thesis (in Latin),
8 further papers that he authored which appeared during his lifetime, 4 joint papers with Georg Frobenius and a posthumously published paper written circa 1915. Despite this modest output, he is characterized there as "one of the sharpest among the pupils of Weierstrass" and a "mathematician of high rank". Stickelberger's thesis and several later papers streamline and complete earlier investigations of various authors, in a direct and elegant way.
Linear algebra
Stickelberger's work on the classification of pairs of bilinear and quadratic forms filled in important gaps in the theory earlier developed by Weierstrass and Darboux. Augmented with the contemporaneous work of Frobenius, it set the theory of elementary divisors upon a rigorous foundation. An important 1878 paper of Stickelberger and Frobenius gave the first complete treatment of the classification of finitely generated abelian groups and sketched the relation with the theory of modules that had just been developed by Dedekind.
Number theory
Three joint papers with Frobenius deal with the theory of elliptic functions. Today Stickelberger's name is most closely associated with his 1890 paper that established the Stickelberger relation for cyclotomic Gaussian sums. This generalized earlier work of Jacobi and Kummer and was later used by Hilbert in his formulation of the reciprocity laws in algebraic number fields. The Stickelberger relation also yields information about the structure of the class group of a cyclotomic field as a module over its abelian Galois group (cf Iwasawa theory).
References
Lothar Heffter, Ludwig Stickelberger, Jahresbericht der Deutschen Matematische Vereinigung, XLVII (1937), pp. 79–86
Ludwig Stickelberger, Ueber eine Verallgemeinerung der Kreistheilu |
https://en.wikipedia.org/wiki/Van%20Kampen%20diagram | In the mathematical area of geometric group theory, a Van Kampen diagram (sometimes also called a Lyndon–Van Kampen diagram ) is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group.
History
The notion of a Van Kampen diagram was introduced by Egbert van Kampen in 1933. This paper appeared in the same issue of American Journal of Mathematics as another paper of Van Kampen, where he proved what is now known as the Seifert–Van Kampen theorem. The main result of the paper on Van Kampen diagrams, now known as the van Kampen lemma can be deduced from the Seifert–Van Kampen theorem by applying the latter to the presentation complex of a group. However, Van Kampen did not notice it at the time and this fact was only made explicit much later (see, e.g.). Van Kampen diagrams remained an underutilized tool in group theory for about thirty years, until the advent of the small cancellation theory in the 1960s, where Van Kampen diagrams play a central role. Currently Van Kampen diagrams are a standard tool in geometric group theory. They are used, in particular, for the study of isoperimetric functions in groups, and their various generalizations such as isodiametric functions, filling length functions, and so on.
Formal definition
The definitions and notations below largely follow Lyndon and Schupp.
Let
(†)
be a group presentation where all r∈R are cyclically reduced words in the free group F(A). The alphabet A and the set of defining relations R are often assumed to be finite, which corresponds to a finite group presentation, but this assumption is not necessary for the general definition of a Van Kampen diagram. Let R∗ be the symmetrized closure of R, that is, let R∗ be obtained from R by adding all cyclic permutations of elements of R and of their inverses.
A Van Kampen diagram over the presentation (†) is a planar finite cell complex , given with a specific embedding with the following additional data and satisfying the following additional properties:
The complex is connected and simply connected.
Each edge (one-cell) of is labelled by an arrow and a letter a∈A.
Some vertex (zero-cell) which belongs to the topological boundary of is specified as a base-vertex.
For each region (two-cell) of , for every vertex on the boundary cycle of that region, and for each of the two choices of direction (clockwise or counter-clockwise), the label of the boundary cycle of the region read from that vertex and in that direction is a freely reduced word in F(A) that belongs to R∗.
Thus the 1-skeleton of is a finite connected planar graph Γ embedded in and the two-cells of are precisely the bounded complementary regions for this graph.
By the choice of R∗ Condition 4 is equivalent to requiring that for each region of there is some boundary vertex of that region and some choice of direction (clockwise or counter-clockwise) such th |
https://en.wikipedia.org/wiki/Riemannian%20connection%20on%20a%20surface | In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.
Historical overview
After the classical work of Gauss on the differential geometry of surfaces and the subsequent emergence of the concept of Riemannian manifold initiated by Bernhard Riemann in the mid-nineteenth century, the geometric notion of connection developed by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early twentieth century represented a major advance in differential geometry. The introduction of parallel transport, covariant derivatives and connection forms gave a more conceptual and uniform way of understanding curvature, allowing generalisations to higher-dimensional manifolds; this is now the standard approach in graduate-level textbooks. It also provided an important tool for defining new topological invariants called characteristic classes via the Chern–Weil homomorphism.
Although Gauss was the first to study the differential geometry of surfaces in Euclidean space E3, it was not until Riemann's Habilitationsschrift of 1854 that the notion of a Riemannian space was introduced. Christoffel introduced his eponymous symbols in 1869. Tensor calculus was developed by Ricci, who published a systematic treatment with Levi-Civita in 1901. Covariant differentiation of tensors was given a geometric interpretation by who introduced the notion of parallel transport on surfaces. His discovery prompted Weyl and Cartan to introduce various notions of connection, including in particular that of affine connection. Cartan's approach was rephrased in the modern language of principal bundles by Ehresmann, after which the subject rapidly took its current form following contributions by Chern, Ambrose and Singer, Kobayashi, Nomizu, Lichnerowicz and others.
Connections on a surface can be defined in a variety of ways. The Riemannian connection or Levi-Civita connection is perhaps most easily understood in terms of lifting vector fields, considered as first order differential operators acting on functions on the manifold, to differential operators on sections of the frame bundle. In the case of an embedded surface, this lift is very simply described in terms of orthogonal projection. Indeed, the vector bundles associated with the frame bundle are all sub-bundl |
https://en.wikipedia.org/wiki/Edward%20W.%20Piotrowski | Edward W. Piotrowski (b. Rybnik, Poland, 1955) is head of the Applied Mathematics Group at the University of Białystok, Poland. He is notable for the analysis of quantum strategies, showing connections between the Kelly criterion, thermodynamics, and special theory of relativity. In the area of econophysics, he discovered extremal properties of fixed point profits of elementary merchant tactics. He has published in the areas of statistical physics, quantum game theory, and econophysics.
Education
He graduated in theoretical physics from the University of Silesia (Katowice) and earned his PhD and habilitation from the University of Silesia, under Andrzej Pawlikowski.
See also
Quantum Aspects of Life
External links
Piotrowski at Ideas
Piotrowski's homepage
Piotrowski at Scientific Commons
1955 births
Living people
University of Silesia in Katowice alumni
20th-century Polish physicists
Quantum physicists
Probability theorists
21st-century Polish physicists
Academic staff of the University of Białystok |
https://en.wikipedia.org/wiki/Ac%C4%B1d%C9%99r%C9%99 | Acıdərə (also, Adzhidere and Akhzhidara) is a village in the Shamakhi Rayon of Azerbaijan. The village forms part of the municipality of Göylər. According to Azerbaijan's State Statistics Committee, only one person lived in the village as of 2014.
References
Populated places in Shamakhi District |
https://en.wikipedia.org/wiki/Hej%20Matematik | Hej Matematik (literally "Hi Mathematics") is a Danish pop group, consisting of Søren Rasted and his nephew Nicolaj Rasted. In live shows, they are accompanied by Nikolaj Teinvig on guitar, Nicholas Findsen on bass, Mads Storm on keyboards and Peter Düring on drums.
History
2005–08: Formation and Vi burde ses noget mere
Hej Matematik was founded in 2005 by Aqua member Søren Rasted and his nephew Nicolaj Rasted during Aqua's and Søren's own solo projects' LazyB hiatus. In 2007, Hej Matematik provided backing vocals on TV-2's album For dig ku' jeg gøre alting. The group's own debut album was Vi burde ses noget mere, released in 2008, preceded by the two single releases "Gymnastik" and "Centerpubben" in 2007. The album peaked #4 in the Danish album chart, while "Centerpubben" became a moderate success, peaking #22 in the Danish single chart. The music video for "Centerpubben" is known for having Søren's fellow band members of Aqua on board, who were reunited in a music video for the first time since their sudden split in 2001. In 2008, the two more singles "Du & jeg", peaking #32 in the Danish single chart, and the promotional single "Vi ka' alt vi to", a ballad featuring an uncredited vocal performance by Søren's fellow Aqua-bandmember and wife Lene. In 2008, the single "Walkmand", which features a vocal sample by Michael Hardinger from his 1981s recording "Walk, Mand", has been released. Due to the huge success of "Walkmand", which made it to #3 in the Danish single chart and won the Danish "Hit of the Year" at Zulu Awards 2009, an annual award-ceremony held by Denmark's largest commercial television station TV 2, the 11-track debut album has been re-released in a special edition with "Walkmand" as a bonus track in the 12th place.
2009–11: Alt går op i 6
Hej Matematik's release of "Party i provinsen", the 2009 lead single of their up-and-coming second album, proved to be successful and reached #2 in the Danish single chart. The second studio album Alt går op i 6, released on 25 January 2010, became a similar success, reaching platinum status and peaking #5 in the Danish album chart. Alt går op i 6 produced two follow up singles "Legendebørn" and "Maskinerne", both released in 2010, with the latter mentioned being extra remixed by the Danish DJ Kato for the single release. "Maskinerne", labelled "Kato på Maskinerne" on the single, became a minor success peaking #38 in the Danish single chart. In 2011, a promotional single-only "The loser sign", has been exclusively released and promoted through the Danish TV show "Natholdet". For the song's music video, fans were invited to send in a clip of themselves, presenting a loser sign resulting in a clip music video. "The loser sign" is also known for being the group's first release, not sung in Danish language, but completely in English, and the last release through Copenhagen Records.
2012–13: Hej lights 2012
With their new contract with ArtPeople and Labelland, in 2012, the two singles "Livet i plas |
https://en.wikipedia.org/wiki/Martin%20Beneke | Martin Beneke (born 1966) is a German physicist.
Biography
Beneke studied Physics, Mathematics and Philosophy at the University of Konstanz, University of Cambridge and University of Heidelberg. In 1993 he received his doctorate at the Technical University of Munich on the structure of perturbative series in higher order and habilitated in Heidelberg in 1998.
At the age of 33 Beneke became head of the Chair of Theoretical Physics (Department E) at the RWTH Aachen University in 1999. In 2008 Martin Beneke was awarded the Leibniz Prize in the amount of 2.5 Million Euro. The research done by Beneke considerably contributes to the verification of theoretical concepts of elementary particle physics, to the indication of variations and to the identification of new structures.
Awards
Thawani Prize 1989
Otto Hahn Medal 1994
Gottfried Wilhelm Leibniz Prize 2008
External links
Lehrstuhl von Martin Beneke, Technische Universität München
Martin Beneke, RWTH Aachen
Gottfried Wilhelm Leibniz Prize 2008 (in German)
RWTH Physicist Martin Beneke Receives Highly-renowned International Prize
1966 births
Living people
21st-century German physicists
RWTH Aachen University alumni
Academic staff of the Technical University of Munich
Heidelberg University alumni
Alumni of the University of Cambridge
Gottfried Wilhelm Leibniz Prize winners |
https://en.wikipedia.org/wiki/%C6%8Flm%C9%99kolu | Əlməkolu (also, Almagovlu and Almakuyuly) is a village in the Siazan Rayon of Azerbaijan. The village forms part of the municipality of Dağ Quşçu. According to Azerbaijan's State Statistics Committee, only four people lived in the village as of 2014.
References
Populated places in Siyazan District |
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