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https://en.wikipedia.org/wiki/Graham%20Everest | Graham Robert Everest (14 December 1957 in Southwick, West Sussex – 30 July 2010) was a British mathematician working on arithmetic dynamics and recursive equations in number theory.
Life
Everest studied at Bedford College (now Royal Holloway College) of the University of London where he completed a Ph.D. in 1983 under the supervision of Colin J. Bushnell of King's College London (The distribution of normal integral generators in tame extensions of Q.) He joined the faculty of the University of East Anglia in 1983 as a lecturer and spent his academic career there.
He was ordained a priest in the Church of England in 2006. He died of prostate cancer on 30 July 2010, leaving behind his wife and three children.
Awards
In 1983 he became a member of the London Mathematical Society. In 2012 he was awarded the Lester Randolph Ford Award jointly with Thomas Ward for their work in diophantine equations.
Writing
With Thomas Ward: Introduction to Number Theory, Springer-Verlag 2005.
With Alf van der Poorten, Thomas Ward, and Igor Shparlinski: Recurrence sequences, American Mathematical Society 2003.
With Thomas Ward: Heights of polynomials and entropy in algebraic dynamics, Springer Verlag 1999.
References
1957 births
2010 deaths
Alumni of King's College London
Academics of the University of East Anglia
20th-century British mathematicians
21st-century British mathematicians
21st-century English Anglican priests |
https://en.wikipedia.org/wiki/Peter%20B.%20Gilkey | Peter Belden Gilkey (born February 27, 1946 in Utica, New York) is an American mathematician, working in differential geometry and global analysis.
Gilkey graduated from Yale University with a master 's degree in 1967 and received a doctoral degree in 1972 from the Harvard University under the supervision of Louis Nirenberg (Curvature and the Eigenvalues of the Laplacian for Geometrical Elliptic Complexes). From 1971 to 1972 he was an instructor in computer science at the New York University and from 1972 to 1974 was a lecturer at the University of California, Berkeley. From 1974 to 1980 he was assistant professor at Princeton University, he spent one year at U.S.C., and in 1981 he became associate professor and in 1985 professor at the University of Oregon.
He wrote a textbook on the Atiyah–Singer index theorem. In 1975 he was Sloan Fellow. He is a fellow of the American Mathematical Society.
Gilkey retired from the University of Oregon in June 2021 and is now a Professor Emeritus.
Writings
Online
With Tohru Eguchi, Andrew J. Hanson Gravitation, Gauge Theories and Differential Geometry, Physics Reports, Volume 66, 1980, pp. 213–393
Notes
External links
Homepage
20th-century American mathematicians
Fellows of the American Mathematical Society
1946 births
Living people
New York University faculty
University of California, Berkeley College of Letters and Science faculty
University of Oregon faculty
Yale University alumni
Harvard University alumni
21st-century American mathematicians |
https://en.wikipedia.org/wiki/Marcellus%20Neal | Marcellus Neal (1868–1939) was the first African-American graduate of Indiana University in the United States. He graduated in 1895 with a B.A. in Mathematics. The Neal-Marshall Center on the Indiana University Bloomington campus is jointly named for Neal and Frances Marshall, the first African-American female student to graduate from Indiana University.
Early years
Neal was born in 1868 in Lebanon, Tennessee, but the family later moved to Greenfield, Indiana. Neal excelled academically at the local high school, graduating with distinction.
Education and career
Neal's high school academic record earned him a scholarship to Indiana University, where he enrolled in the fall of 1891. He did well academically and graduated with a Bachelor of Arts degree in Mathematics on June 19, 1895. During his time at Indiana University, Neal was a member of the Married Folks' Club. His wife is listed as Mrs. M. Neal, in the Arbutus, the Indiana University yearbook. However, her name is listed as Caledonia in Frank O. Beck's book on race relations at Indiana University. She was a southern teacher, whose teaching skills were highly regarded. The Neals were not allowed to live on campus, enter the Indiana Memorial Union, or eat in establishments on campus.
For ten years after his graduation from IU, Neal did graduate work, held short-term teaching jobs, and traveled, spending time in both Europe and Canada. For nearly twenty-five years, he served as the head of the science department at Washington High School in Dallas, Texas. Neal and his wife left Dallas to live in Chicago, Illinois, where he worked at a Civil Service job. He also completed a treatise on science and wrote political editorials for African-American newspapers.
Death and legacy
On October 20, 1939, Marcellus Neal was struck by a driver in a hit and run at State and 37th Street in Chicago. He died of his injuries on November 6 and was buried in Mount Greenwood Cemetery, in Glenwood, Illinois.
The Neal-Marshall Alumni Club was founded in 1981 as a means of meeting the needs of black students, faculty, and staff, while also promoting African-American history at Indiana University. The organization was actually founded in Jackson, Mississippi, while other Indiana chapters soon followed in Gary, and Indianapolis.
The Neal-Marshall Black Culture Center opened on the Indiana University campus in 1969. The center was originally called the "Black House," but renamed the Black Culture Center in 1972. The Center was located in various buildings on campus, but moved into its new home at 109 North Jordan Avenue in 2001 after years of effort by students, alumni, and administration. In April 1995, the Indiana General Assembly passed a resolution to commemorate the 100th anniversary of Neal's graduation, and coinciding with the 175th anniversary of Indiana University.
References
External links
Neal-Marshall Black Culture Center
1868 births
1939 deaths
People from Lebanon, Tennessee
Indiana Unive |
https://en.wikipedia.org/wiki/List%20of%20Brisbane%20Roar%20FC%20managers | Brisbane Roar FC is an association football club based in Brisbane, Australia. This page consists of information and statistics about all previous and current Managers of the Roar.
Managerial history
Bleiberg
Miron Bleiberg was appointed as the then Queensland Roar's inaugural manager on 2 March 2005. Under pressure from the fans to deliver on his promises of attractive, attacking and successful football he resigned on 12 November 2006 following a poor start to the 2006–07 season. After much speculation, Bleiberg was replaced by former Australian national team coach, Frank Farina just three days after Bleiberg's resignation.
Farina
Frank Farina's arrival led to a mini-revival which saw the club narrowly miss out on what would have been the Roar's first finals appearance, on goal difference. The 2007–08 season, however, saw Farina make up for the shortfall of the previous season, qualifying for the finals for the first time in the club's history. A memorable performance in the second leg of the semi-final saw the Roar defeat arch rivals 2–0 (2–0 agg.) Sydney FC in front of a (then) club record 36,221 fans to qualify for the preliminary final against the Newcastle Jets. The Roar would controversially lose 3–2 to the Newcastle side, who would ultimately go on to win the Grand Final. Farina again qualified for the finals in 2008–09, where the Roar dispatched of Central Coast Mariners 4–2 on aggregate, however they ultimately lost, again in the preliminary final, to Adelaide United after failing to capitalise on their dominance. On 10 October 2009, Farina was arrested by Queensland Police for Driving under the influence. He was initially suspended by the Roar and asked to show cause as to why he should not be sacked for tarnishing the name of the club. It was announced that assistant manager, Rado Vidošić fwould step into a caretakers role until a decision had been made which would include the M1 Derby, which the Roar lost 1–0 at home. Farina was ultimately sacked on 14 October 2009 and was replaced by former Australian Under-20 national team coach, Ange Postecoglou.
Postecoglou
Ange Postecoglou arrived mid-season armed with the task of picking up the pieces of a season in tatters. The 2009–10 season ended as the worst in the club's short history, finishing second from the bottom. Postecoglou completed a turn-around in the 2010–11 season. He made wholesale changes to the squad, commencing with the replacement of the "old-guard" of Charlie Miller, Craig Moore and Danny Tiatto and brought in his own squad which was a mixture of youth and talented experience. Under his brand of possession/attacking football, he would lead the team to win the club's inaugural Championship and go on to complete the club's first Double by also wrapping up the Premiership in a memorable 2011 A-League Grand Final in front of a then club record 50,168 supporters. The club went on an Australian sporting record 36-match unbeaten run which commenced in the 2010–11 season and |
https://en.wikipedia.org/wiki/Robert%20Gompf | Robert Ernest Gompf (born 1957) is an American mathematician specializing in geometric topology.
Gompf received a Ph.D. in 1984 from the University of California, Berkeley under the supervision of Robion Kirby (An invariant for Casson handles, disks and knot concordants). He is now a professor at the University of Texas at Austin.
His research concerns the topology of 4-manifolds. In 1990, he demonstrated with Tomasz Mrowka that there is a simply connected irreducible 4-manifold that admits no complex structures. In 1995, he constructed new examples of simply connected compact symplectic 4-manifolds that are not homeomorphic or diffeomorphic to complex manifolds (Kähler manifolds).
He is a fellow of the American Mathematical Society. He was an invited speaker at the International Congress of Mathematicians in 1994 in Zurich (Smooth four-manifolds and symplectic topology).
Writings
With András I. Stipsicz: 4-manifolds and Kirby calculus, AMS 1999
A new construction of symplectic manifolds, Annals of Mathematics, Volume 142, 1995, p. 527–595
With Tomasz Mrowka Irreducible four manifolds need not be complex, Annals of Mathematics, Volume 138, 1993, p. 61–111
Handlebody construction of Stein surfaces, Annals of Mathematics, Volume 148, 1998, p. 619–693
Notes
External links
Homepage
1957 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Kasper%20Skaanes | Kasper Skaanes (born 19 March 1995) is a Norwegian professional footballer who plays for Sogndal, as a midfielder.
Career statistics
Club
References
1995 births
Living people
Men's association football midfielders
Norwegian men's footballers
SK Brann players
IK Start players
Eliteserien players
Norwegian First Division players
Footballers from Bergen |
https://en.wikipedia.org/wiki/Global%20analysis | In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis uses techniques in infinite-dimensional manifold theory and topological spaces of mappings to classify behaviors of differential equations, particularly nonlinear differential equations. These spaces can include singularities and hence catastrophe theory is a part of global analysis. Optimization problems, such as finding geodesics on Riemannian manifolds, can be solved using differential equations, so that the calculus of variations overlaps with global analysis. Global analysis finds application in physics in the study of dynamical systems and topological quantum field theory.
Journals
Annals of Global Analysis and Geometry
The Journal of Geometric Analysis
See also
Atiyah–Singer index theorem
Geometric analysis
Lie groupoid
Pseudogroup
Morse theory
Structural stability
Harmonic map
References
Further reading
Mathematics 241A: Introduction to Global Analysis
Fields of mathematical analysis
Manifolds |
https://en.wikipedia.org/wiki/List%20of%20C.D.%20FAS%20records%20and%20statistics | This article lists various statistics related to Club Deportivo FAS.
All stats accurate as of 2 April 2022.
Honours
As of 18 May 2022 FAS have won 18 Primera División and one CONCACAF Champions League trophies.
Domestic competitions
League
Primera División de Fútbol de El Salvador and predecessors
Champions (18): 1951–1952, 1953–1954, 1957–1958, 1961–1962, 1962, 1977–1978, 1978–1979, 1981, 1984, 1994–1995, 1995–1996, Clausura 2002, Apertura 2002, Apertura 2003, Apertura 2004, Clausura 2005, Apertura 2009. Clausura 2021 (record)
Minor Cups
American Airlines Cup
Champions (1) : 2002
Copa Salvadorean Classic Soccer Challenge
Runners up (1) : 2014
EDESSA Independence Cup
Runners up (1) : 2014
CONCACAF competitions
Official titles
CONCACAF Champions League and predecessors
Champions (1) : 1979
Copa Interamericana
Runners up (1) : 1979
UNCAF Club Championship
Runners up (1) : 1980
Individual awards
Award winners
Top Goalscorer (10)
The following players have won the Goalscorer titles while playing for FAS:
Omar Muraco (39) – 1957-58
Hector Dadderio (29) – 1959
Mario Monge (16) – 1961–62
David Arnoldo Cabrera (20) – 1981
David Arnoldo Cabrera (16) – 1983
Ever Hernandez (17) – 1984
Williams Reyes (14) – Clausura 2004
Nicolás Muñoz (12) – Apertura 2004
Néstor Ayala (12) – Apertura 2006
Williams Reyes (11) – 2009 Apertura
Luis Perea (14) – Clausura 2018
Bladimir Diaz (9) – Clausura 2022
Goalscorers
Most goals scored : 240 – David Arnoldo Cabrera
Most League goals: TBD –
Most League goals in a season: 26 – Omar Muraco, Primera División, 1958
Most goals scored by a FAS player in a match: 6 – David Arnoldo Cabrera v. UES (FAS 7-2 UES), 3 January 1980
Most goals scored by a FAS player in an International match: 4 – Alejandro de la Cruz Bentos & Williams Reyes v. Jalapa, 11 October 2002
Most goals scored in CONCACAF competition: 13 – tbd, tbd
All-time top goalscorers
Note: Players in bold text are still active with Club Deportivo FAS.
Historical goals
kroeker
Players
Appearances
Competitive, professional matches only including substitution, number of appearances as a substitute appears in brackets.
Last updated -
Other appearances records
Youngest first-team player: – TBD v TBD, Primera División, Day Month Year
Oldest post-Second World War player: – TBD v TBD, Primera División, Day Month Year
Most appearances in Primera División: TBD – TBD
Most appearances in Copa Presidente: TBD – TBD
Most appearances in International competitions: TBD – TBD
Most appearances in CONCACAF competitions: TBD – TBD
Most appearances in UNCAF competitions: TBD – TBD
Most appearances in CONCACAF Champions League: TBD – TBD
Most appearances in UNCAF Copa: TBD TBD
Most appearances in FIFA Club World Cup: 2
Zózimo
Most appearances as a foreign player in all competitions: TBD – TBD
Most appearances as a foreign player in Primera División: TBD – TBD
Most consecutive League appearances: TBD – |
https://en.wikipedia.org/wiki/Quarter%20order-6%20square%20tiling | In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232 orbifold notation, and can be seen as a half symmetry of *443 and *662, and quarter symmetry of *642.
Images
Projections centered on a vertex, triangle and hexagon:
Related polyhedra and tiling
See also
Square tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Order-6 tilings
Square tilings
Uniform tilings |
https://en.wikipedia.org/wiki/Munster%20Senior%20Hurling%20Championship%20records%20and%20statistics | This page details statistics of the Munster Senior Hurling Championship.
General performances
By county
Teams
By Semi-Final/Top 4 Appearances (since 2015)
By decade
The most successful team of each decade, judged by number of Munster Senior Hurling Championship titles, is as follows:
1890s: 4 each for Cork (1890-92-93-94) and Tipperary (1895-96-98-99)
1900s: 6 for Cork (1901-02-03-04-05-07)
1910s: 3 each for Limerick (1910-11-18), Cork (1912-15-19) and Tipperary (1913-16-17)
1920s: 5 for Cork (1920-26-27-28-29)
1930s: 4 for Limerick (1933-34-35-36)
1940s: 5 for Cork (1942-43-44-46-47)
1950s: 4 for Cork (1952-53-54-56)
1960s: 7 for Tipperary (1960-61-62-64-65-67-68)
1970s: 7 for Cork (1970-72-75-76-77-78-79)
1980s: 5 for Cork (1982-83-84-85-86)
1990s: 3 each for Clare (1995-97-98) and Cork (1990-92-99)
2000s: 4 for Cork (2000-03-05-06)
2010s: 4 for Tipperary (2011-12-15-16)
2020s: 4 for Limerick (2020-21-22-23)
Finishing positions
Most championships
54, Cork (1888, 1890, 1892, 1893, 1894, 1901, 1902, 1903, 1904, 1905, 1907, 1912, 1915, 1919, 1920, 1926, 1927, 1928, 1929, 1931, 1939, 1942, 1943, 1944, 1946, 1947, 1952, 1953, 1954, 1956, 1966, 1969, 1970, 1972, 1975, 1976, 1977, 1978, 1979, 1982, 1983, 1984, 1985, 1986, 1990, 1992, 1999, 2000, 2003, 2005, 2006, 2014, 2017, 2018)
Most second-place finishes
30, Cork (1896, 1897, 1898, 1906, 1909, 1910, 1913, 1914, 1916, 1921, 1932, 1940, 1941, 1948, 1950, 1951, 1957, 1959, 1960, 1961, 1964, 1965, 1968, 1980, 1987, 1988, 1991, 2004, 2010, 2013)
Most third-place finishes
2, Cork (2019, 2022)
Most fourth-place finishes
1, Tipperary (2018)
1, Clare (2019)
1, Waterford (2022)
1, Cork (2023)
Most fifth-place finishes
3, Waterford (2018, 2019, 2023)
Most semi-final finishes
00, 000 (0000)
Most quarter-final finishes
00, 000 (0000)
Other records
Biggest wins
The most one sided Munster finals since 1896 when goals were made equal to three points:
31 points – 1918: Limerick 11-03 – 1-02 Clare
31 points – 1982: Cork 5-31 – 3-06 Waterford
28 points – 1893: Cork 5-13 – 0-00 Limerick
27 points – 1903: Cork 5-16 – 1-01 Waterford
26 points – 1905: Cork 7-12 – 1-04 Limerick
23 points – 1899: Tipperary 5-16 – 0-08 Clare
22 points – 1900: Tipperary 6-11 – 2-01 Kerry
22 points – 1972: Cork 6-18 – 2-08 Clare
21 points – 2016: Tipperary 5-19 (34) – (13 0-13 Waterford
21 points – 1896: Tipperary 7-09 – 2-03 Cork
21 points – 2011: Tipperary 7-19 – 0-19 Waterford
20 points – 1962: Tipperary 5-14 – 2-03 Waterford
19 points – 1915: Cork 8-02 – 2-01 Clare
19 points – 1983: Cork 3-22 – 0-12 Waterford
18 points – 1993: Tipperary 3-27 – 2-12 Clare
18 points – 1965: Tipperary 4-11 – 0-05 Cork
The most one sided games from the semi-final and quarter-final stages of the championship:
42 points – 1923: Cork 13-04 – 0-01 Waterford
41 points – 1904: Tipperary 8-21 – 0-04 Waterford
34 points – 2000: Cork 2-32 – 0-04 Kerry
34 points – 1953: Clare 10 |
https://en.wikipedia.org/wiki/Regular%20scheme | In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.
For an example of a regular scheme that is not smooth, see Geometrically regular ring#Examples.
See also
Étale morphism
Dimension of an algebraic variety
Glossary of scheme theory
Smooth completion
References
Algebraic geometry
Scheme theory |
https://en.wikipedia.org/wiki/Bimorphic | Bimorphic can refer to:
Bimorphism, a type of mapping in mathematics
Bimorph, a piezoelectric cantilever with two active layers |
https://en.wikipedia.org/wiki/Monomial%20ideal | In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field.
A toric ideal is an ideal generated by differences of monomials (provided the ideal is a prime ideal). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or projective toric variety, possibly non-normal.
Definitions and Properties
Let be a field and be the polynomial ring over with n variables .
A monomial in is a product for an n-tuple of nonnegative integers.
The following three conditions are equivalent for an ideal :
is generated by monomials,
If , then , provided that is nonzero.
is torus fixed, i.e, given , then is fixed under the action for all .
We say that is a monomial ideal if it satisfies any of these equivalent conditions.
Given a monomial ideal , is in if and only if every monomial ideal term of is a multiple of one the .
Proof:
Suppose and that is in . Then , for some .
For all , we can express each as the sum of monomials, so that can be written as a sum of multiples of the . Hence, will be a sum of multiples of monomial terms for at least one of the .
Conversely, let and let each monomial term in be a multiple of one of the in . Then each monomial term in can be factored from each monomial in . Hence is of the form for some , as a result .
The following illustrates an example of monomial and polynomial ideals.
Let then the polynomial is in , since each term is a multiple of an element in , i.e., they can be rewritten as and both in . However, if , then this polynomial is not in , since its terms are not multiples of elements in .
Monomial Ideals and Young Diagrams
A monomial ideal can be interpreted as a Young diagram. Suppose , then can be interpreted in terms of the minimal monomials generators as , where and . The minimal monomial generators of can be seen as the inner corners of the Young diagram. The minimal generators would determine where we would draw the staircase diagram.
The monomials not in lie inside the staircase, and these monomials form a vector space basis for the quotient ring .
Consider the following example.
Let be a monomial ideal. Then the set of grid points corresponds to the minimal monomial generators in . Then as the figure shows, the pink Young diagram consists of the monomials that are not in . The points in the inner corners of the Young diagram, allow us to identify the minimal monomials in as seen in the green boxes. Hence, .
In general, to any set of grid points, we can associate a Young diagram, so that the monomial ideal is constructed by determining the inner corners that make up the staircase diagram; likewise, given a monomial ideal, we can make up the Young diagram by looking at the and representing them as the inner corners of the Young diagram. The coordinates of the inner corners would represent the powers of the minimal monomials in . Thus, monomia |
https://en.wikipedia.org/wiki/Majres | Majres is one of the villages surrounding the towns of Sodfa in the Asyut Governorate, Egypt. According to statistics from the year 2006, the total population in Majres was 11858 people, of which 6093 are men and 5765 women.
References
Villages in Egypt |
https://en.wikipedia.org/wiki/Kardous | Kardous village is one of the villages of sodfa in Asyut Governorate, Egypt. According to statistics from the year 2006, the total population in Kardous was 8268 people, 4389 men and 3879 women.
References
Villages in Egypt |
https://en.wikipedia.org/wiki/Hilary%20Priestley | Hilary Ann Priestley is a British mathematician. She is a professor at the University of Oxford and a Fellow of St Anne's College, Oxford, where she has been Tutor in Mathematics since 1972.
Hilary Priestley introduced ordered separable topological spaces; such topological spaces are now usually called Priestley spaces in her honour. The term "Priestley duality" is also used for her application of these spaces in the representation theory of distributive lattices.
Books
References
External links
Hilary Priestley home page
Professor Hilary Priestley profile at the Mathematical Institute, University of Oxford
Professor Hilary Ann Priestley profile at St Anne's College, Oxford
Hilary Priestley on ResearchGate
Year of birth missing (living people)
Living people
Place of birth missing (living people)
Alumni of the University of Oxford
British women mathematicians
20th-century English mathematicians
21st-century English mathematicians
Algebraists
Lattice theorists
Fellows of St Anne's College, Oxford
20th-century women mathematicians
21st-century women mathematicians |
https://en.wikipedia.org/wiki/Thaine%27s%20theorem | In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by . Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem , to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem .
Formulation
Let and be distinct odd primes with not dividing . Let be the Galois group of over , let be its group of units, let be the subgroup of cyclotomic units, and let be its class group. If annihilates then it annihilates .
References
See in particular Chapter 14 (pp. 91–94) for the use of Thaine's theorem to prove Mihăilescu's theorem, and Chapter 16 "Thaine's Theorem" (pp. 107–115) for proof of a special case of Thaine's theorem.
See in particular Chapter 15 (pp. 332–372) for Thaine's theorem (section 15.2) and its application to the Mazur–Wiles theorem.
Cyclotomic fields
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Saw%20Swee%20Hock | Saw Swee Hock HFLSE (; sometimes written Saw Swee-Hock) (1931 – 16 February 2021) was a Singaporean leading expert in statistics, population, and economics, and was a noted philanthropist.
Education
Saw received his BA in 1956 and MA in 1960 from the University of Malaya in Singapore, now the National University of Singapore (NUS). He earned his PhD in statistics from the London School of Economics (LSE) in 1963.
Career
He began his academic career at the University of Malaya in Kuala Lumpur (1963-1969). He subsequently became founding professor of statistics at the University of Hong Kong (1969-1971) and professor of statistics at NUS (1975-1991). He was a professorial fellow at the Institute of Southeast Asian Studies in Singapore and a member of the NUS board of trustees.
Saw has been a member of more than 30 advisory panels and committees, including the United Nations Committee on Salary Adjustments and the International Statistical Institute, and was the first chairman of the National Statistical Commission of Singapore. He has held visiting positions in, among others, Princeton, Stanford, Cambridge and LSE. He has written or edited 49 books, 31 book chapters and over 110 articles on statistics, demography and economics.
Philanthropy
Saw was also known for his philanthropy. He was named one of the "48 Heroes of Philanthropy" in the Asia-Pacific Region by Forbes Asia Magazine in 2014. His largest reported donation was of US$24 million in 2011 to his alma mater, NUS, to establish the Saw Swee Hock School of Public Health. He also contributed £2 million to his other alma mater, LSE, for the student centre building that bears his name and also supported the Saw Swee Hock Southeast Asia Centre. He has made contributions to universities in Singapore, China, Hong Kong and the UK. His family made its money by investing in bungalows, in part with an inheritance from his businessman father-in-law.
Swee Hock’s contributions to the field of education include an endowment in 2004 to establish 12 annual bursaries for students studying for MBAs. He has also endowed several professorships at NUS including a donation in 2002 for the establishment of the Saw Swee Hock Professor of Statistics in the Department of Statistics and Applied Probability at NUS.
In 2016, Saw established the Saw Swee Hock Study Award at Yale-NUS College with three students from the Class of 2018 as recipients of the inaugural award.
Honors and awards
Saw has been awarded numerous honors in recognition of his academic achievements and his philanthropy. He was named honorary professor of Statistics at the University of Hong Kong, honorary professor at Xiamen University, and President's Honorary Professor of Statistics at NUS. He was made an honorary fellow at LSE and an honorary university fellow of the University of Hong Kong. He was conferred the Outstanding Service Award by NUS, the Singapore President's Award for Philanthropy, and the Public Service Medal by the Singapore Go |
https://en.wikipedia.org/wiki/Polygon%20covering | In geometry, a covering of a polygon is a set of primitive units (e.g. squares) whose union equals the polygon. A polygon covering problem is a problem of finding a covering with a smallest number of units for a given polygon. This is an important class of problems in computational geometry. There are many different polygon covering problems, depending on the type of polygon being covered. An example polygon covering problem is: given a rectilinear polygon, find a smallest set of squares whose union equals the polygon.
In some scenarios, it is not required to cover the entire polygon but only its edges (this is called polygon edge covering) or its vertices (this is called polygon vertex covering).
In a covering problem, the units in the covering are allowed to overlap, as long as their union is exactly equal to the target polygon. This is in contrast to a packing problem, in which the units must be disjoint and their union may be smaller than the target polygon, and to a polygon partition problem, in which the units must be disjoint and their union must be equal to the target polygon.
A polygon covering problem is a special case of the set cover problem. In general, the problem of finding a smallest set covering is NP-complete, but for special classes of polygons, a smallest polygon time.
A covering of a polygon is a collection of maximal units, possibly overlapping, whose union equals .
A minimal covering is a covering that does not contain any other covering (i.e. it is a local minimum).
A minimum covering is a covering with a smallest number of units (i.e. a global minimum). Every minimum covering is minimal, but not vice versa.
Covering a rectilinear polygon with squares
A rectilinear polygon can always be covered with a finite number of vertices of the polygon. The algorithm uses a local optimization approach: it builds the covering by iteratively selecting maximal squares that are essential to the cover (i.e., contain uncovered points not covered by other maximal squares) and then deleting from the polygon the points that become unnecessary (i.e., unneeded to support future squares). Here is a simplified pseudo-code of the algorithm:
While the polygon P is not empty:
Select a continuator square s in P.
If the balcony of s is not yet covered, then add s to the covering.
Remove the balcony of s from P.
If what remains of s is a one-knob continuator, then remove from P a certain rectangle adjacent to the knob, taking care to leave a sufficient security distance for future squares.
For polygons which may contain holes, finding a minimum such covering is NP-hard. This sharp difference between hole-free and general polygons can be intuitively explained based on the following analogy between maximal squares in a rectilinear polygon and nodes in an undirected graph:
Some maximal squares have a continuous intersection with the boundary of the polygon; when they are removed, the remaining polygon remains connected. Such squares are |
https://en.wikipedia.org/wiki/Iterated%20forcing | In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcing can construct models where Martin's axiom holds and the continuum is any given regular cardinal.
In iterated forcing, one has a transfinite sequence Pα of forcing notions indexed by some ordinals α, which give a family of Boolean-valued models VPα. If α+1 is a successor ordinal then Pα+1 is often constructed from Pα using a forcing notion in VPα, while if α is a limit ordinal then Pα is often constructed as some sort of limit (such as the direct limit) of the Pβ for β<α.
A key consideration is that, typically, it is necessary that is not collapsed. This is often accomplished by the use of a preservation theorem such as:
Finite support iteration of c.c.c. forcings (see countable chain condition) are c.c.c. and thus preserve .
Countable support iterations of proper forcings are proper (see Fundamental Theorem of Proper Forcing) and thus preserve .
Revised countable support iterations of semi-proper forcings are semi-proper and thus preserve .
Some non-semi-proper forcings, such as Namba forcing, can be iterated with appropriate cardinal collapses while preserving using methods developed by Saharon Shelah.
References
Sources
External links
Forcing (mathematics) |
https://en.wikipedia.org/wiki/C.%20Stanley%20Ogilvy | Charles Stanley Ogilvy (1913–2000) was an American mathematician, sailor, and author. He was a professor of mathematics at Hamilton College (New York), and a frequent competitor at the Star World Championships. His many books include works on both mathematics and sailing.
Sailing
Ogilvy grew up sailing near New Rochelle, New York, on the mainland side of the Long Island Sound. Beginning in 1931 he crewed for Howard McMichael on the two-man Star class Grey Fox, and in 1934 he bought the boat and renamed it the Jay. He won over 47 regattas, and was a frequent competitor in the Star World Championships; his best finishes were second in 1947 (crewing for Hilary Smart) and third in 1949 and 1951 (both with his own boat, Flame). Later, he also sailed Etchells.
Ogilvy belonged to the Larchmont Yacht Club for 62 years, and served as its historian. He was the first vice president of the International Star Class Yacht Racing Association, edited its publications for many years, and also served as historian for the class.
In 1990, Ogilvy was the second recipient of the Harry Nye Memorial Trophy of the International Star Class Yacht Racing Association, in recognition of his contributions to Star class sailing. The C. Stanley Ogilvy Masters Trophy, an antique sextant awarded to a sailor over the age of 50, was named in his honor and has been presented annually by the Etchells World Championships since 1999.
Education and career
Ogilvy went to the Berkshire School, then did his undergraduate studies at Williams College. During World War II, his bad eyesight preventing him from serving in the Navy; instead he became the commander of a rescue boat on the Pacific Front for the U.S. Army. After earning an M.A. from Cambridge University and an M.S. at Columbia University, and doing additional studies at Princeton University, Ogilvy finished his graduate studies with a PhD in mathematics from Syracuse University in 1954. His thesis, supervised by Walter R. Baum, was entitled An Investigation of Some Properties of Asymptotic Lines on Surfaces of Negative Gaussian Curvature.
Ogilvy began his teaching career at Trinity College (Connecticut), and joined the faculty of Hamilton College (New York) in 1953. He chaired the mathematics department beginning in 1969, and was a fellow of the American Association for the Advancement of Science. He remained at Hamilton until 1974, when he retired so that he could spend more of his time sailing.
Ogilvy died on June 21, 2000, in Mamaroneck, New York.
Books
Ogilvy wrote many books on both mathematics and sailing, which were translated into several other languages.
They include:
Successful Yacht Racing (Norton, 1951)
Through the Mathescope (Oxford Univ. Press, 1956). Later republished as Excursions in Mathematics.
Tomorrow's Math: Unsolved Problems for the Amateur (Oxford Univ. Press, 1962)
Thoughts on Small Boat Racing (Van Nostrand, 1966)
Excursions in Number Theory (with John T. Anderson, Oxford Univ. Press, 1966)
Excu |
https://en.wikipedia.org/wiki/Enrico%20del%20Rosario | Enrico Domenic Tolentino del Rosario (born 21 March 1997) is a Northern Mariana Islander professional footballer who plays for the Northern Mariana Islands national team.
Career statistics
International
References
1997 births
Living people
People from Saipan
Northern Mariana Islands men's footballers
Men's association football midfielders
Northern Mariana Islands men's international footballers
Alderson Broaddus Battlers men's soccer players
Northern Kentucky Norse men's soccer players
Expatriate men's footballers in the Philippines |
https://en.wikipedia.org/wiki/Artin%20transfer%20%28group%20theory%29 | In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups (with a prime number p), which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers.
Transversals of a subgroup
Let be a group and be a subgroup of finite index
Definitions. A left transversal of in is an ordered system of representatives for the left cosets of in such that
Similarly a right transversal of in is an ordered system of representatives for the right cosets of in such that
Remark. For any transversal of in , there exists a unique subscript such that , resp. . Of course, this element with subscript which represents the principal coset (i.e., the subgroup itself) may be, but need not be, replaced by the neutral element .
Lemma. Let be a non-abelian group with subgroup . Then the inverse elements of a left transversal of in form a right transversal of in . Moreover, if is a normal subgroup of , then any left transversal is also a right transversal of in .
Proof. Since the mapping is an involution of we see that:
For a normal subgroup we have for each .
We must check when the image of a transversal under a homomorphism is also a transversal.
Proposition. Let be a group homomorphism and be a left transversal of a subgroup in with finite index The following two conditions are equivalent:
is a left transversal of the subgroup in the image with finite index
Proof. As a mapping of sets maps the union to another union:
but weakens the equality for the intersection to a trivial inclusion:
Suppose for some :
then there exists elements such that
Then we have:
Conversely if then there exists such that But the homomorphism maps the disjoint cosets to equal cosets:
Remark. We emphasize the important equivalence of the proposition in a formula:
Permutation representation
Suppose is a left transversal of a subgroup of finite index in a group . A fixed element |
https://en.wikipedia.org/wiki/Annali%20di%20Matematica%20Pura%20ed%20Applicata | The Annali di Matematica Pura ed Applicata (Annals of Pure and Applied Mathematics) is a bimonthly peer-reviewed scientific journal covering all aspects of pure and applied mathematics. The journal was established in 1850 under the title of Annali di scienze matematiche e fisiche (Annals of Mathematics and Physics), and changed to its current title in 1858: it was the first Italian periodical devoted to mathematics and written in Italian. The founding editors-in-chief were Barnaba Tortolini and Francesco Brioschi.
It is currently published by Springer Science+Business Media and the editor-in-chief is Graziano Gentili (University of Florence).
Abstracting and indexing
The journal is abstracted and indexed in:
According to the Journal Citation Reports, the journal has a 2020 impact factor of 0.969.
Notes
References
.
External links
Mathematics journals
Publications established in 1850
English-language journals
Springer Science+Business Media academic journals
Bimonthly journals |
https://en.wikipedia.org/wiki/Stephanie%20B.%20Alexander | Stephanie Brewster Brewer Taylor Alexander is an American mathematician, a professor emerita of mathematics at the University of Illinois at Urbana–Champaign. Her research concerns differential geometry and metric spaces.
Education and career
Alexander earned her Ph.D. from UIUC in 1967, under the supervision of Richard L. Bishop, with a thesis entitled Reducibility of Euclidean Immersions of Low Codimensions. After joining the UIUC faculty as a half-time instructor, she became a regular faculty member in 1972. She retired in 2009.
Books
With Vitali Kapovitch and Anton Petrunin, Alexander is the author of the book An Invitation to Alexandrov Geometry: CAT(0) Spaces (Springer, 2019).
Recognition
At Illinois, Alexander won the Luckman Distinguished Undergraduate Teaching Award and the William Prokasy Award for Excellence in Undergraduate Teaching in 1993.
In 2014 she was elected as a fellow of the American Mathematical Society "for contributions to geometry, for high-quality exposition, and for exceptional teaching of mathematics."
References
External links
Home page
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
University of Illinois College of Liberal Arts and Sciences alumni
University of Illinois Urbana-Champaign faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
Differential geometers
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Jane%20Piore%20Gilman | Jane Piore Gilman (born 1945) is an American mathematician, a distinguished professor of mathematics at Rutgers University. Her research concerns topology and group theory.
Education and career
Gilman is one of three children of physicist Emanuel R. Piore. She did her undergraduate studies at the University of Chicago, graduating in 1965, and received her Ph.D. from Columbia University in 1971. Her thesis, supervised by Lipman Bers, was entitled Relative Modular Groups in Teichmüller Spaces. She worked for a year as an instructor at Stony Brook University before joining Rutgers in 1972.
Books
Gilman is the author of a monograph on the problem of testing whether pairs of elements of PSL(2,R) (the group of orientation-preserving isometries of the hyperbolic plane) generate a Fuchsian group (a discrete subgroup of PSL(2,R)). It is Two-generator Discrete Subgroups of PSL(2, R) (Memoirs of the American Mathematical Society 117, 1995). With Irwin Kra and Rubí E. Rodríguez she is the co-author of a graduate-level textbook on complex analysis, Complex Analysis: In the Spirit of Lipman Bers (Graduate Texts in Mathematics 245, Springer, 2007; 2nd ed., 2013).
Recognition
In 2014 she was elected as a fellow of the American Mathematical Society "for contributions to topology and group theory, and for service to her department and the larger community."
References
1945 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Columbia University alumni
Stony Brook University faculty
Rutgers University faculty
American people of Lithuanian descent
Fellows of the American Mathematical Society
American women mathematicians
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Irena%20Peeva | Irena Vassileva Peeva is a professor of mathematics at Cornell University, specializing in commutative algebra. She disproved the Eisenbud–Goto regularity conjecture jointly with Jason McCullough.
Education and career
Peeva did her graduate studies at Brandeis University, earning a Ph.D. in 1995 under the supervision of David Eisenbud with a thesis entitled Free Resolutions. She was a postdoctoral researcher at the University of California, Berkeley and a C. L. E. Moore instructor at the Massachusetts Institute of Technology
before joining the Cornell Department of Mathematics faculty in 1998.
Peeva is an editor of the Transactions of AMS.
Books
Peeva is the author of:
Graded Syzygies (Springer, 2011).
Minimal Free Resolutions over Complete Intersections (with David Eisenbud, Springer, 2016).
Recognition
In 2014 Peeva was elected as a fellow of the American Mathematical Society "for contributions to commutative algebra and its applications."
In 2019/2020 and in 2012/2013 Peeva was a Simons Foundation Fellow. During 1999-2001 she was a Sloan Foundation Fellow and was a Sloan Doctoral Dissertation Fellow in 1994/1995.
References
External links
Home page at Cornell University
Irena Peeva in the Oberwolfach photo collection
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Brandeis University alumni
Massachusetts Institute of Technology School of Science faculty
Cornell University faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
Algebraists
21st-century American women |
https://en.wikipedia.org/wiki/Richard%20L.%20Bishop | Richard Lawrence Bishop (August 12, 1931 – December 18, 2019) was an American mathematician who specialized in differential geometry and taught at the University of Illinois at Urbana–Champaign.
Bishop went to Case Institute of Technology as an undergraduate, earning a B.S. in 1954. Next he earned his Ph.D. from the Massachusetts Institute of Technology in 1959, and immediately joined the UIUC faculty, where he stayed until his retirement in 1997. His thesis, On Imbeddings and Holonomy, was supervised by Isadore Singer. At UIUC, his doctoral students included future UIUC colleague Stephanie B. Alexander. He is the author of Geometry of Manifolds (with Richard J. Crittenden, AMS Chelsea Publishing, 1964, translated into Russian 1967 and reprinted 2001) and Tensor Analysis on Manifolds (with Samuel I. Goldberg, Macmillan, 1968, reprinted by Dover Books on Mathematics, 1980).
In 2013, Bishop became one of the inaugural fellows of the American Mathematical Society.
The Bishop–Gromov inequality in Riemannian geometry, one form of which appeared in his book with Crittenden, is named after him and Mikhail Gromov, who gave an improved formulation of Bishop's result. He introduced the "Bishop frame" of curves in Euclidean space, an alternative to the better-known Frenet frame. With Barrett O'Neill he made foundational contributions to the study of convex functions and convex sets in Riemannian geometry and their applications in the study of negative sectional curvature, including to the geometry of warped products.
Notable publications
R.L. Bishop and B. O'Neill. Manifolds of negative curvature. Trans. Amer. Math. Soc. 145 (1969), 1–49. ;
Richard L. Bishop. There is more than one way to frame a curve. Amer. Math. Monthly 82 (1975), 246–251.
Richard L. Bishop and Richard J. Crittenden. Geometry of manifolds. Reprint of the 1964 original. AMS Chelsea Publishing, Providence, RI, 2001. xii+273 pp. ;
References
1931 births
2019 deaths
20th-century American mathematicians
21st-century American mathematicians
Case Western Reserve University alumni
Massachusetts Institute of Technology alumni
University of Illinois Urbana-Champaign faculty
Fellows of the American Mathematical Society
People from Allegan, Michigan |
https://en.wikipedia.org/wiki/2014%20Dhivehi%20League | Statistics of Dhivehi League for the 2014 season. 2014 Dhivehi League started on June 16.
Teams
BG Sports Club
Club All Youth Linkage
Club Eagles
Club Valencia
Mahibadhoo SC
Maziya S&RC
New Radiant SC
Victory SC
Personnel
Note: Flags indicate national team as has been defined under FIFA eligibility rules. Players may hold more than one non-FIFA nationality.
Managerial changes
* Anil continues as assistant coach.
League table
Format: In Round 1 and Round 2, all eight teams play against each other. Top six teams after Round 2 play against each other in Round 3. Teams with most total points after Round 3 are crowned the Dhivehi League champions and qualify for the AFC Cup. The top four teams qualify for the President's Cup. Bottom two teams after Round 2 play against top two teams of Second Division in Dhivehi League Qualification for places in next year's Dhivehi League.
Standings of round 1
Standings of round 2
Standings of round 3
Final standings
Positions by round
The table lists the positions of teams after each week of matches.
Matches
Round 1 matches
A total of 28 matches will be played in this round.
Round 2 matches
A total of 20 matches will be played in this round.
Round 3 matches
A total of 15 matches will be played in this round.
Season statistics
Hat-tricks
4 Player scored 4 goals
5 Player scored 5 goals
Promotion/relegation playoff for 2015 Dhivehi League
Matches
A total of 6 matches will be played in this round. Top 2 reams will be promoted to 2015 Dhivehi Premier League and the bottom 2 teams relegated to the 2015 Second Division Football Tournament.
References
Dhivehi League seasons
Maldives
Maldives
1 |
https://en.wikipedia.org/wiki/Limiting%20case | The phrase limiting case has several different meanings in:
Limiting case (mathematics)
Limiting case (philosophy of science) |
https://en.wikipedia.org/wiki/Tensor%20product%20of%20representations | In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already knows a few.
Definition
Group representations
If are linear representations of a group , then their tensor product is the tensor product of vector spaces with the linear action of uniquely determined by the condition that
for all and . Although not every element of is expressible in the form , the universal property of the tensor product operation guarantees that this action is well defined.
In the language of homomorphisms, if the actions of on and are given by homomorphisms and , then the tensor product representation is given by the homomorphism given by
,
where is the tensor product of linear maps.
One can extend the notion of tensor products to any finite number of representations. If V is a linear representation of a group G, then with the above linear action, the tensor algebra is an algebraic representation of G; i.e., each element of G acts as an algebra automorphism.
Lie algebra representations
If and are representations of a Lie algebra , then the tensor product of these representations is the map given by
,
where is the identity endomorphism. This is called the Kronecker sum, defined in Matrix addition#Kronecker_sum and Kronecker product#Properties.
The motivation for the use of the Kronecker sum in this definition comes from the case in which and come from representations and of a Lie group . In that case, a simple computation shows that the Lie algebra representation associated to is given by the preceding formula.
Action on linear maps
If and are representations of a group , let denote the space of all linear maps from to . Then can be given the structure of a representation by defining
for all . Now, there is a natural isomorphism
as vector spaces; this vector space isomorphism is in fact an isomorphism of representations.
The trivial subrepresentation consists of G-linear maps; i.e.,
Let denote the endomorphism algebra of V and let A denote the subalgebra of consisting of symmetric tensors. The main theorem of invariant theory states that A is semisimple when the characteristic of the base field is zero.
Clebsch–Gordan theory
The general problem
The tensor product of two irreducible representations of a group or Lie algebra is usually not irreducible. It is therefore of interest to attempt to decompose into irreducible pieces. This decomposition problem is known as the Clebsch–Gordan problem.
The SU(2) case
The prototypical example of this problem is the case of the rotation group SO(3)—or its double cover, the special unitary group SU(2). The irreducible representations of SU(2) are described by a parameter , whose possible values are
(The dimension of the representati |
https://en.wikipedia.org/wiki/Tony%20Auden | Anthony "Tony" Auden is an Australian meteorologist.
He is currently weather presenter on Seven News Brisbane.
Career
Auden studied a Bachelor of Science, majoring in Meteorology and Mathematics at Monash University which he completed in 2005. He later gained employment with the Bureau of Meteorology and completed a Postgraduate Diploma in Meteorology at the head office in Melbourne.
In January 2005, Auden moved to Brisbane and joined the Bureau of Meteorology. He has extensive experience in weather having covered Cyclone Yasi, The Gap's severe thunderstorm in 2008, bushfires, floods, dangerous surf and tide events, dust storms and heat waves.
In December 2013, Auden joined Seven News Brisbane as weather presenter replacing John Schluter. He presented the weather on The Daily Edition from 2016 until 2020.
He also presents weather updates on The Latest: Seven News, and regularly appears on Sunrise.
References
External links
Year of birth missing (living people)
Living people
Australian meteorologists
Seven News presenters |
https://en.wikipedia.org/wiki/Lax%20functor | In category theory, a discipline within mathematics, the notion of lax functor between bicategories generalizes that of functors between categories.
Let C,D be bicategories. We denote composition in diagrammatic order. A lax functor P from C to D, denoted , consists of the following data:
for each object x in C, an object ;
for each pair of objects x,y ∈ C a functor on morphism-categories, ;
for each object x∈C, a 2-morphism in D;
for each triple of objects, x,y,z ∈C, a 2-morphism in D that is natural in f: x→y and g: y→z.
These must satisfy three commutative diagrams, which record the interaction between left unity, right unity, and associativity between C and D. See http://ncatlab.org/nlab/show/pseudofunctor.
A lax functor in which all of the structure 2-morphisms, i.e. the and above, are invertible is called a pseudofunctor.
Category theory |
https://en.wikipedia.org/wiki/Applied%20Mathematics%20and%20Mechanics%20%28English%20Edition%29 | Applied Mathematics and Mechanics (English Edition) is a peer-reviewed journal of mechanics, established in 1980 by Wei-zang Chien in 1980. Chien was the editor-in-chief from 1980 to 2002 and subsequently an honorary editor-in-chief. Xingming Guo is the editor-in-chief now. In 1980, it was quarterly, became bimonthly in 1981, and then monthly in 1985.
Abstracting and indexing
The journal is abstracted and indexed in the following bibliographic databases:
According to the Journal Citation Reports, it has a 2016 impact factor of 1.205.
References
External links
Academic journals established in 1980
Monthly journals
English-language journals |
https://en.wikipedia.org/wiki/Formal%20ball | In topology, a formal ball is an extension of the notion of ball to allow unbounded and negative radius. The concept of formal ball was introduced by Weihrauch and Schreiber in 1981 and the negative radius case (the generalized formal ball) by Tsuiki and Hattori in 2008.
Specifically, if is a metric space and the nonnegative real numbers, then an element of is a formal ball. Elements of are known as generalized formal balls.
Formal balls possess a partial order defined by if , identical to that defined by set inclusion.
Generalized formal balls are interesting because this partial order works just as well for as for , even though a generalized formal ball with negative radius does not correspond to a subset of .
Formal balls possess the Lawson topology and the Martin topology.
References
K. Weihrauch and U. Schreiber 1981. "Embedding metric spaces into CPOs". Theoretical computer science, 16:5-24.
H. Tsuiki and Y. Hattori 2008. "Lawson topology of the space of formal balls and the hyperbolic topology of a metric space". Theoretical computer science, 405:198-205
Y. Hattori 2010. "Order and topological structures of posets of the formal balls on metric spaces". Memoirs of the Faculty of Science and Engineering. Shimane University. Series B 43:13-26
Topology |
https://en.wikipedia.org/wiki/Linda%20Weiser%20Friedman | Linda Weiser Friedman (born 1953) is an author and academic. She is a Professor of Statistics and Computer Information Systems at Baruch College and the CUNY Graduate Center. Friedman holds a PhD in Operations Research from New York University Polytechnic School of Engineering and a B.A degree in Statistics / Biology from Baruch College. Her research and teaching interests are varied and include business statistics, object-oriented programming, humor studies, Jewish studies, online education, social media, and all things technology. Her most recent book is God Laughed: Sources of Jewish Humor (), which Publishers Weekly called a "lighthearted but thoughtful study". She also writes fiction and poetry, and lives in the Borough Park neighborhood of Brooklyn.
Early life
Linda Weiser Friedman was born in New York City to Norman and Marion Weiser, and grew up in the Brooklyn neighborhoods of Crown Heights and Midwood. She attended Crown Heights Yeshiva (when it was still located at 310 Crown St.) and Esther Schoenfeld High School (in the Lower East Side). She married Hershey H. Friedman in 1972; they have five adult children.
Books
Hershey H. Friedman and Linda Weiser Friedman, God Laughed: Sources of Jewish Humor, Transaction Publishers, 2014.
Linda Weiser Friedman, The Simulation Metamodel, Kluwer Academic, Norwell, Massachusetts, 1996.
Linda Weiser Friedman, Comparative Programming Languages: Generalizing the Programming Function. Englewood Cliffs, NJ: Prentice Hall, 1991.
Deadly Stakes, coauthored with Hershey H. Friedman under the pseudonym H. Fred Wiser, Walker Publishing, 1989.
References
External links
Official website
SSRN author page
Google Scholar page
1953 births
21st-century American Jews
Jewish American academics
Jewish scientists
Baruch College faculty
CUNY Graduate Center faculty
Polytechnic Institute of New York University alumni
Baruch College alumni
American Orthodox Jews
People from Borough Park, Brooklyn
People from Crown Heights, Brooklyn
People from Midwood, Brooklyn
Living people |
https://en.wikipedia.org/wiki/G%C3%B6del%20operation | In mathematical set theory, a set of Gödel operations is a finite collection of operations on sets that can be used to construct the constructible sets from ordinals. introduced the original set of 8 Gödel operations 𝔉1,...,𝔉8 under the name fundamental operations. Other authors sometimes use a slightly different set of about 8 to 10 operations, usually denoted G1, G2,...
Definition
used the following eight operations as a set of Gödel operations (which he called fundamental operations):
The second expression in each line gives Gödel's definition in his original notation, where the dot means intersection, V is the universe, E is the membership relation, and so on.
uses the following set of 10 Gödel operations.
Properties
Gödel's normal form theorem states that if φ(x1,...xn) is a formula in the language of set theory with all quantifiers bounded, then the function {(x1,...,xn) ∈ X1×...×Xn | φ(x1, ..., xn)) of X1, ..., Xn is given by a composition of some Gödel operations. This result is closely related to Jensen's rudimentary functions.
References
Inline references
Constructible universe |
https://en.wikipedia.org/wiki/Principalization%20%28algebra%29 | In the mathematical field of algebraic number theory, the concept of principalization refers to a situation when, given an extension of algebraic number fields, some ideal (or more generally fractional ideal) of the ring of integers of the smaller field isn't principal but its extension to the ring of integers of the larger field is. Its study has origins in the work of Ernst Kummer on ideal numbers from the 1840s, who in particular proved that for every algebraic number field there exists an extension number field such that all ideals of the ring of integers of the base field (which can always be generated by at most two elements) become principal when extended to the larger field. In 1897 David Hilbert conjectured that the maximal abelian unramified extension of the base field, which was later called the Hilbert class field of the given base field, is such an extension. This conjecture, now known as principal ideal theorem, was proved by Philipp Furtwängler in 1930 after it had been translated from number theory to group theory by Emil Artin in 1929, who made use of his general reciprocity law to establish the reformulation. Since this long desired proof was achieved by means of Artin transfers of non-abelian groups with derived length two, several investigators tried to exploit the theory of such groups further to obtain additional information on the principalization in intermediate fields between the base field and its Hilbert class field. The first contributions in this direction are due to Arnold Scholz and Olga Taussky in 1934, who coined the synonym capitulation for principalization. Another independent access to the principalization problem via Galois cohomology of unit groups is also due to Hilbert and goes back to the chapter on cyclic extensions of number fields of prime degree in his number report, which culminates in the famous Theorem 94.
Extension of classes
Let be an algebraic number field, called the base field, and let be a field extension of finite degree. Let and denote the ring of integers, the group of nonzero fractional ideals and its subgroup of principal fractional ideals of the fields respectively. Then the extension map of fractional ideals
is an injective group homomorphism. Since , this map induces the extension homomorphism of ideal class groups
If there exists a non-principal ideal (i.e. ) whose extension ideal in is principal (i.e. for some and ), then we speak about principalization or capitulation in . In this case, the ideal and its class are said to principalize or capitulate in . This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel of the class extension homomorphism.
More generally, let be a modulus in , where is a nonzero ideal in and is a formal product of pair-wise different real infinite primes of . Then
is the ray modulo , where is the group of nonzero fractional ideals in relatively prime to and the condition |
https://en.wikipedia.org/wiki/Kim%20Jin-sung%20%28baseball%29 | Kim Jin-sung (born March 7, 1985) is South Korean professional baseball pitcher for the NC Dinos of the KBO League.
References
External links
Career statistics and player information from Korea Baseball Organization
Kim Jin-sung at ncdinos.com
NC Dinos players
KBO League pitchers
South Korean baseball players
Baseball players from Seoul
1985 births
Living people |
https://en.wikipedia.org/wiki/Stable%20range%20condition | In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring is the smallest integer such that whenever in generate the unit ideal (they form a unimodular row), there exist some in such that the elements for also generate the unit ideal.
If is a commutative Noetherian ring of Krull dimension , then the stable range of is at most (a theorem of Bass).
Bass stable range
The Bass stable range condition refers to precisely the same notion, but for historical reasons it is indexed differently: a ring satisfies if for any in generating the unit ideal there exist in such that for generate the unit ideal.
Comparing with the above definition, a ring with stable range satisfies . In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension satisfies . (For this reason, one often finds hypotheses phrased as "Suppose that satisfies Bass's stable range condition ...")
Stable range relative to an ideal
Less commonly, one has the notion of the stable range of an ideal in a ring . The stable range of the pair is the smallest integer such that for any elements in that generate the unit ideal and satisfy mod and mod for , there exist in such that for also generate the unit ideal. As above, in this case we say that satisfies the Bass stable range condition .
By definition, the stable range of is always less than or equal to the stable range of .
References
Charles Weibel, The K-book: An introduction to algebraic K-theory
H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
External links
Bass' stable range condition for principal ideal domains
K-theory |
https://en.wikipedia.org/wiki/Limit%20of%20distributions | In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This notion generalizes a limit of a sequence of functions; a limit as a distribution may exist when a limit of functions does not.
The notion is a part of distributional calculus, a generalized form of calculus that is based on the notion of distributions, as opposed to classical calculus, which is based on the narrower concept of functions.
Definition
Given a sequence of distributions , its limit is the distribution given by
for each test function , provided that distribution exists. The existence of the limit means that (1) for each , the limit of the sequence of numbers exists and that (2) the linear functional defined by the above formula is continuous with respect to the topology on the space of test functions.
More generally, as with functions, one can also consider a limit of a family of distributions.
Examples
A distributional limit may still exist when the classical limit does not. Consider, for example, the function:
Since, by integration by parts,
we have: . That is, the limit of as is .
Let denote the distributional limit of as , if it exists. The distribution is defined similarly.
One has
Let be the rectangle with positive orientation, with an integer N. By the residue formula,
On the other hand,
Oscillatory integral
See also
Distribution (number theory)
References
Demailly, Complex Analytic and Differential Geometry
Generalized functions
Schwartz distributions |
https://en.wikipedia.org/wiki/Furniture%20industry%20in%20Pakistan | According to the World Trade Organization's statistics, Pakistan's exports of wooden furniture amounted to $51 million in 2011. The furniture industry in Pakistan has been expanding. Many new furniture companies are now joining the furniture industry.
Our handmade furniture is in high demand by high-end customers. Pakistani furniture manufacturers have expertise in this area, due to the type of wood used namely 'sheesham' (rosewood) which adds to this furniture's demand. The leading wood-furniture-making areas of Pakistan are Chiniot, Gujrat, Peshawar, Lahore and Karachi.
Impact of increased consumer spending
Consumer spending increased by an average of 26% - per the Bloomberg's report, which was published on November 21, 2013. It's not only that the Fast-moving consumer goods (FMCGs) companies get benefits from increased consumer spending; the furniture manufacturers also benefit from it.
Pakistan's furniture exports
As per World Trade Organization's statistics, the wooden furniture exports of Pakistan reportedly amounted to almost $51 million in the calendar year (CY) 2011. Whereas, in 2009-2010 the furniture exports of Pakistan ranged between $25 million to $30 million (excluding undocumented exports of an approximate amount of $10 million). However, according to the All Pakistan Furniture Exporters Association (APFEA)] founder, Turhan Baig Muhammad, these exports represent a very small portion of the total furniture business of the country. According to him, the local furniture market is almost 50 times stronger than that of the exports. Keeping this in view, the approximate total furniture sale of the country is more than $2.5 billion.
Fears for the furniture industry of Pakistan
Total world trade of furniture is estimated to be $23.2 billion. Wood furniture accounts for 77 percent, metal furniture 17 percent, and plastic furniture 6 percent of the total. In 2010, the share of Pakistan in the international furniture market is trivial. Even though the country takes pride in having a history of craftsmanship, it does not share a significant position in the international wood furniture market.
Domestic furniture industry is suffering because furniture exports to Pakistan from some other countries have increased, whereas, the high cost of furniture-making business is increasing problems for the local furniture manufacturers. The prices of raw materials which includes timber, colour paints, chipboard, polish materials, and foams have increased manifold. Due to unchecked deforestation, timber production of the country is also suffering.
Wood furniture industry of Pakistan is categorized as being small because old obsolete machinery is used in this industry, which is the reason for higher cost and low output. Traditional wood furniture in Pakistan is heavy and bulky. Pakistani furniture industry needs to go to light-weight and moveable furniture to be exported to the world market where demand is high to meet the needs of offices, shopping ma |
https://en.wikipedia.org/wiki/National%20Numeracy | National Numeracy is an independent charity (registered no. 1145669 in England and Wales) based in Brighton, UK, that promotes the importance of numeracy and "everyday maths".
The charity was founded in 2012; its chair is Perdita Fraser and vice chair Andy Haldane. Its current chief executive is Sam Sims, who replaced Mike Ellicock in 2020.
The charity aims to challenge negative attitudes towards maths and promotes effective approaches to improving functional numeracy skills. Chris Humphries, former chair of National Numeracy and a former chief executive of the UK Commission for Employment and Skills, said: "It is simply inexcusable for anyone to say: 'I can't do maths.' It is a peculiarly British disease which we aim to eradicate." The charity's Theory of Change is detailed on their website.
National Numeracy has been critical of the UK mathematics curriculum, claiming that it is flawed and requires radical improvement to ensure that everyone leaves compulsory education with essential numeracy skills.
National Numeracy is supported by a number of celebrities, including Rachel Riley, financial journalist Martin Lewis of Money Saving Expert, author, television presenter and mathematics teacher Bobby Seagull, financial writer Iona Bain, Strictly Come Dancing's Katya Jones, Great British Bake Off 2020 winner Peter Sawkins, and the poet and comedian Harry Baker. It is also supported by organisations, including TP ICAP, KPMG, Experian, Ufi VocTech Trust, Garfield Weston Foundation and the Edge Foundation.
History of National Numeracy
A 2010 report commissioned by Lord Moser from New Philanthropy Capital recommended the creation of a national numeracy trust. The report, which focused on low levels of numeracy in the UK, showed how charities and funders can help people to be confidently numerate. These problems are a focus of National Numeracy's strategy. National Numeracy was legally registered as a charity in January 2012 with the press launch of the charity in March 2012.
Work and campaigns
On 30 October 2014 National Numeracy CEO Mike Ellicock was featured on an edition of ITV's Tonight documentary programme The Trouble With Numbers. Mike Ellicock spoke about cultural and attitudinal problems preventing people from succeeding in maths.
In 2014, National Numeracy launched the National Numeracy Challenge, a free online tool which allows users to assess their numeracy level and access resources to help them improve. By December 2020, 340,000 people had registered on the National Numeracy Challenge.
During the 2014-15 FA Cup season, BBC Sport and BBC Learning worked with National Numeracy on Maths of the Day, a series of films shown across the BBC, as well as accompanying content on the BBC iWonder website, exploring maths in football. The films featured former footballer and commentator Robbie Savage and Countdown co-host Rachel Riley among others. In March 2015, there was also a Maths of the Day live event on BBC Radio 5 Live in which Nati |
https://en.wikipedia.org/wiki/Breakthrough%20Prize%20in%20Mathematics | The Breakthrough Prize in Mathematics is an annual award of the Breakthrough Prize series announced in 2013.
It is funded by Yuri Milner and Mark Zuckerberg and others. The annual award comes with a cash gift of $3 million. The Breakthrough Prize Board also selects up to three laureates for the New Horizons in Mathematics Prize, which awards $100,000 to early-career researchers. Starting in 2021 (prizes announced in September 2020), the $50,000 Maryam Mirzakhani New Frontiers Prize is also awarded to a number of women mathematicians who have completed their PhDs within the past two years.
Motivation
The founders of the prize have stated that they want to help scientists to be perceived as celebrities again, and to reverse a 50-year "downward trend". They hope that this may make "more young students aspire to be scientists".
Laureates
New Horizons in Mathematics Prize
The past laureates of the New Horizons in Mathematics prize were:
2016
André Arroja Neves
Larry Guth
(prize was rejected by Peter Scholze)
2017
Geordie Williamson
Benjamin Elias
Hugo Duminil-Copin
Mohammed Abouzaid
2018
Zhiwei Yun
Wei Zhang
Maryna Viazovska
Aaron Naber
2019
Chenyang Xu
Karim Adiprasito
June Huh
Kaisa Matomäki
Maksym Radziwill
2020
Tim Austin
Emmy Murphy
Xinwen Zhu
2021
Bhargav Bhatt – "For outstanding work in commutative algebra and arithmetic algebraic geometry, particularly on the development of p-adic cohomology theories."
Aleksandr Logunov – "For novel techniques to study solutions to elliptic equations, and their application to long-standing problems in nodal geometry."
Song Sun – "For many groundbreaking contributions to complex differential geometry, including existence results for Kähler–Einstein metrics and connections with moduli questions and singularities."
2022
Aaron Brown and Sebastian Hurtado Salazar – "For contributions to the proof of Zimmer's conjecture."
Jack Thorne – "For transformative contributions to diverse areas of algebraic number theory, and in particular for the proof, in collaboration with James Newton, of the automorphy of all symmetric powers of a holomorphic modular newform."
Jacob Tsimerman – "For outstanding work in analytic number theory and arithmetic geometry, including breakthroughs on the André–Oort and Griffiths conjecture
2023
Ana Caraiani – "For diverse transformative contributions to the Langlands program, and in particular for work with Peter Scholze on the Hodge-Tate period map for Shimura varieties and its applications."
Ronen Eldan – "For the creation of the stochastic localization method, that has led to significant progress in several open problems in high-dimensional geometry and probability, including Jean Bourgain's slicing problem and the KLS conjecture."
James Maynard – "For multiple contributions to analytic number theory, and in particular to the distribution of prime numbers."
2024
Roland Bauerschmidt, New York University – "For outstanding contributions to probability theory and the development of renormal |
https://en.wikipedia.org/wiki/2013%20Third%20Division%20Football%20Tournament | Statistics of Third Division Football Tournament in the 2013 season. Tournament started on August 19.
Teams
51 teams are competition in the 2013 Third Division Football Tournament, and these teams were divided into 17 groups of 3 teams. Winner of each group and the best 1 team among the group runners-up will be advanced into the second round. Between the 18 teams in the second round, a knock out format will be used.
Group 1
Our Recreation Club
Kuda Henveyru United
Club Middle Bros
Group 2
Tent Sports Club
Zeal Sports Club
Teenage Juniors
Group 3
Maamigili Youth Recreation Club
LQ Sports
Club 010
Group 4
Youth Revolution Club
Thoddoo Football Club
Raising Stars for Vilufushi Recreation
Group 5
Society Alifushi for Youth
Vidha
Lagoons Sports Club
Group 6
Sports Club Veloxia
Huravee Inivative for Youth
Falcon Sports Club
Group 7
Naivadhoo Trainers Sports Club
Biss Buru Sports
Decagon Sports Club
Group 8
Kelaa Nalhi Sports
Sports Club Rivalsa
Buru Sports
Group 9
Ilhaar
Club Amigos
Veyru Sports Club
Group 10
Club PK
West Sports Club
The Futsal Town
Group 11
Muiveyo Friends Club
Stelco Recreation Club
MS Helping Hand Sports Academy
Group 12
Red Line Club
Nazaki Sports
Vaikaradhoo Football Club
Group 13
Sent Sports Club
The Vakko Sports Club
UN Friends
Group 14
Hinnavaru Youth Society
CBL Sports
Sealand Sports Club
Group 15
Club New Oceans
New Star Sports Club
Offu Football Club
Group 16
LT Sports
Aim
Valiant Sports Club
Group 17
Fiyoree Sports Club
TC Sports
The Bows Sports Club
Group stage
Group 1
Group 2
Group 3
Group 4
Group 5
Group 6
Group 7
Group 8
Group 9
Group 10
Final
Awards
References
Maldivian Third Division Football Tournament seasons
3 |
https://en.wikipedia.org/wiki/Hausdorff%20gap | In mathematics, a Hausdorff gap consists roughly of two collections of sequences of integers, such that there is no sequence lying between the two collections. The first example was found by . The existence of Hausdorff gaps shows that the partially ordered set of possible growth rates of sequences is not complete.
Definition
Let be the set of all sequences of non-negative integers, and define to mean .
If is a poset and and are cardinals, then a -pregap in is a set of elements for and a set of elements for such that:
The transfinite sequence is strictly increasing;
The transfinite sequence is strictly decreasing;
Every element of the sequence is less than every element of the sequence .
A pregap is called a gap if it satisfies the additional condition:
There is no element greater than all elements of and less than all elements of .
A Hausdorff gap is a -gap in such that for every countable ordinal and every natural number there are only a finite number of less than such that for all we have .
There are some variations of these definitions, with the ordered set replaced by a similar set. For example, one can redefine to mean for all but finitely many . Another variation introduced by is to replace by the set of all subsets of , with the order given by if has only finitely many elements not in but has infinitely many elements not in .
Existence
It is possible to prove in ZFC that there exist Hausdorff gaps and -gaps where is the cardinality of the smallest unbounded set in , and that there are no -gaps. The stronger open coloring axiom can rule out all types of gaps except Hausdorff gaps and those of type with .
References
External links
Descriptive set theory
Order theory
Integer sequences
General topology |
https://en.wikipedia.org/wiki/From%20A%20to%20B | From A to B may refer to:
From A to B in geometry
From A to B (New Musik album)
From A to B (Octopus album)
From A to B (film), a 2015 Emirati film |
https://en.wikipedia.org/wiki/Teaching%20Mathematics%20and%20Its%20Applications | Teaching Mathematics and Its Applications is a quarterly peer-reviewed academic journal in the field of mathematics education. The Journal was established in 1982 and is published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. The editors-in-chief are Duncan Lawson (Newman University, Birmingham), Chris Sangwin (University of Edinburgh), and Anne Watson (University of Oxford).
The journal is abstracted and indexed in the British Education Index, Education Research Abstracts, Educational Management Abstracts, Educational Technology Abstracts, MathEduc Database, and ProQuest databases.
See also
List of mathematics education journals
References
External links
Submission website
Institute of Mathematics and its Applications
English-language journals
Mathematics education in the United Kingdom
Mathematics education journals
Oxford University Press academic journals
Academic journals established in 1982
Triannual journals
1982 establishments in England |
https://en.wikipedia.org/wiki/Marc%20Luyckx%20Ghisi | Marc Luyckx Ghisi was born 20 April 1942 in Louvain, Belgium. He lives with his wife Isabelle near Brussels.
Initially, he studied mathematics, philosophy and theology (Ph.D.) and became a Catholic priest. He presented a doctorate in Rome (Pontifical Oriental Institute), in Russian and Greek theology, on Nikolai Berdyaev's early writings in Russian, since his discovery of Marxism until his conversion to orthodoxy" (Pontifical Oriental Institute, Rome).
After his marriage, he was for ten years (1990-1999), member of the Forward Studies Unit of the European Commission, created by Jacques Delors, where he focused on the meaning of European integration and created the programme The soul of Europe. He had the opportunity to travel a lot and meet worldwide government officials and advisors in Europe, the U.S., China, Japan, or in India. Some were aware of the shift of civilization in which we are engaged globally, but these visionaries were a minority.
In the Forward Studies Unit he invited many thinkers such as US sociologist Paul H. Ray (cultural creatives), Edgar Morin (a leading French philosopher of the paradigm shift), Hazel Henderson (author of numerous books on the win-win economy and the new green sustainable economy), Rinaldo Brutoco (CEO of World Business Academy), Avon Mattison (founder of Pathways to Peace), Harlan Cleveland, (President of the World Academy of Art and Science), Prof. Ziauddin Sardar.
He was Dean of the Cotrugli Business School in Zagreb and Belgrade (2005-2009). For 8 years he has also been a member of the Auroville International Advisory Council in South India. He is a Fellow of the World Business Academy, a member of the Club of Rome-EU,
a member of the World Futures Studies Federation and is Honorary President of Eurotas, European Transpersonal Association.
Individual work
A Congress in the European Commission in 1998: The transmodern Hypothesis: A new dialogue between cultures seems possible
In May 1998, the "Forward Studies Unit" organized with the World Academy of Art and Science
a Congress on "Civilizations and Governance", in the Brussels headquarters of the European Commission. We were presenting the hypothesis of a worldwide shift to a transmodern world. This hypothesis has been published in "World Affairs", in the "Integral Review" and in "Futures". And this seems to be worldwide. There are indications that the same shift to a transmodern vision of religion and culture is happening around the world: inside the Muslim world (Ziauddin Sardar), in China (Nicanor Perlas), and in South America (Leonardo Boff). And in this transmodern vision, a new type of dialogue between religions and civilizations is increasingly coming up, from the bottom of our societies. The minutes of this International Congress can be found on "The Future of Religions"
Willis Harman: The tool change is the vision of consciousness
As a member of the Forward Studies Unit, I had the great honour to meet with a genius thinker in the Si |
https://en.wikipedia.org/wiki/Euclides%20Danicus | Euclides Danicus (the Danish Euclid) is one of three books of mathematics written by Georg Mohr. It was published in 1672 simultaneously in Copenhagen and Amsterdam, in Danish and Dutch respectively. It contains the first proof of the Mohr–Mascheroni theorem, which states that every geometric construction that can be performed using a compass and straightedge can also be done with compass alone.
The book is divided into two parts. In the first part, Mohr shows how to perform all of the constructions of Euclid's Elements using a compass alone. In the second part, he includes some other specific constructions, including some related to the mathematics of the sundial.
Euclides Danicus languished in obscurity, possibly caused by its choice of language, until its rediscovery in 1928 in a bookshop in Copenhagen. Until then, the Mohr–Mascheroni theorem had been credited to Lorenzo Mascheroni, who published a proof in 1797, independently of Mohr's work. Soon after the rediscovery of Mohr's book, publications about it by Florian Cajori and Nathan Altshiller Court made its existence much more widely known. The Danish version was republished in facsimile in 1928 by the Royal Danish Academy of Sciences and Letters, with a foreword by Johannes Hjelmslev, and a German translation was published in 1929.
Only eight copies of the original publication of the book are known to survive. In 2005, one of these original copies was sold at auction, to Fry's Electronics, for what Gerald L. Alexanderson calls a "ridiculously low price": US$13,000.
References
Wikicommons has a copy of the original: https://commons.wikimedia.org/wiki/File:Georg_Mohr%27s_Euclides_Danicus.pdf
Further reading
.
1672 books
Danish non-fiction books
Dutch non-fiction books
Mathematics books |
https://en.wikipedia.org/wiki/Posner%27s%20theorem | In algebra, Posner's theorem states that given a prime polynomial identity algebra A with center Z, the ring is a central simple algebra over , the field of fractions of Z. It is named after Ed Posner.
References
Edward C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), pp. 180–183.
Theorems in ring theory |
https://en.wikipedia.org/wiki/Central%20polynomial | In algebra, a central polynomial for n-by-n matrices is a polynomial in non-commuting variables that is non-constant but yields a scalar matrix whenever it is evaluated at n-by-n matrices. That such polynomials exist for any square matrices was discovered in 1970 independently by Formanek and Razmyslov. The term "central" is because the evaluation of a central polynomial has the image lying in the center of the matrix ring over any commutative ring. The notion has an application to the theory of polynomial identity rings.
Example: is a central polynomial for 2-by-2-matrices. Indeed, by the Cayley–Hamilton theorem, one has that for any 2-by-2-matrices x and y.
See also
Generic matrix ring
References
Ring theory |
https://en.wikipedia.org/wiki/Geometric%20Poisson%20distribution | In probability theory and statistics, the geometric Poisson distribution (also called the Pólya–Aeppli distribution) is used for describing objects that come in clusters, where the number of clusters follows a Poisson distribution and the number of objects within a cluster follows a geometric distribution. It is a particular case of the compound Poisson distribution.
The probability mass function of a random variable N distributed according to the geometric Poisson distribution is given by
where λ is the parameter of the underlying Poisson distribution and θ is the parameter of the geometric distribution.
The distribution was described by George Pólya in 1930. Pólya credited his student Alfred Aeppli's 1924 dissertation as the original source. It was called the geometric Poisson distribution by Sherbrooke in 1968, who gave probability tables with a precision of four decimal places.
The geometric Poisson distribution has been used to describe systems modelled by a Markov model, such as biological processes or traffic accidents.
See also
Poisson distribution
Compound Poisson distribution
Geometric distribution
References
Bibliography
Further reading
Poisson distribution |
https://en.wikipedia.org/wiki/Representation%20on%20coordinate%20rings | In mathematics, a representation on coordinate rings is a representation of a group on coordinate rings of affine varieties.
Let X be an affine algebraic variety over an algebraically closed field k of characteristic zero with the action of a reductive algebraic group G. G then acts on the coordinate ring of X as a left regular representation: . This is a representation of G on the coordinate ring of X.
The most basic case is when X is an affine space (that is, X is a finite-dimensional representation of G) and the coordinate ring is a polynomial ring. The most important case is when X is a symmetric variety; i.e., the quotient of G by a fixed-point subgroup of an involution.
Isotypic decomposition
Let be the sum of all G-submodules of that are isomorphic to the simple module ; it is called the -isotypic component of . Then there is a direct sum decomposition:
where the sum runs over all simple G-modules . The existence of the decomposition follows, for example, from the fact that the group algebra of G is semisimple since G is reductive.
X is called multiplicity-free (or spherical variety) if every irreducible representation of G appears at most one time in the coordinate ring; i.e., .
For example, is multiplicity-free as -module. More precisely, given a closed subgroup H of G, define
by setting and then extending by linearity. The functions in the image of are usually called matrix coefficients. Then there is a direct sum decomposition of -modules (N the normalizer of H)
,
which is an algebraic version of the Peter–Weyl theorem (and in fact the analytic version is an immediate consequence.) Proof: let W be a simple -submodules of . We can assume . Let be the linear functional of W such that . Then .
That is, the image of contains and the opposite inclusion holds since is equivariant.
Examples
Let be a B-eigenvector and X the closure of the orbit . It is an affine variety called the highest weight vector variety by Vinberg–Popov. It is multiplicity-free.
The Kostant–Rallis situation
See also
Algebra representation
Notes
References
Group theory
Representation theory
Representation theory of groups |
https://en.wikipedia.org/wiki/Fixed-point%20subgroup | In algebra, the fixed-point subgroup of an automorphism f of a group G is the subgroup of G:
More generally, if S is a set of automorphisms of G (i.e., a subset of the automorphism group of G), then the set of the elements of G that are left fixed by every automorphism in S is a subgroup of G, denoted by GS.
For example, take G to be the group of invertible n-by-n real matrices and (called the Cartan involution). Then is the group of n-by-n orthogonal matrices.
To give an abstract example, let S be a subset of a group G. Then each element s of S can be associated with the automorphism , i.e. conjugation by s. Then
;
that is, the centralizer of S.
References
Algebraic groups |
https://en.wikipedia.org/wiki/Arrangement%20%28space%20partition%29 | In discrete geometry, an arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions, induced by a finite collection of geometric objects, which are usually of dimension one less than the dimension of the space, and often of the same type as each other, such as hyperplanes or spheres.
Definition
For a set of objects in , the cells in the arrangement
are the connected components of sets of the form
for subsets of . That is, for each the cells are the connected components of the points that belong to every object in and do not belong to any other object. For instance the cells of an arrangement of lines in the Euclidean plane are of three types:
Isolated points, for which is the subset of all lines that pass through the point.
Line segments or rays, for which is a singleton set of one line. The segment or ray is a connected component of the points that belong only to that line and not to any other line of
Convex polygons (possibly unbounded), for which is the empty set, and its intersection (the empty intersection) is the whole space. These polygons are the connected components of the subset of the plane formed by removing all the lines in .
Types of arrangement
Of particular interest are the arrangements of lines and arrangements of hyperplanes.
More generally, geometers have studied arrangements of other types of curves in the plane, and of other more complicated types of surface. Arrangements in complex vector spaces have also been studied; since complex lines do not partition the complex plane into multiple connected components, the combinatorics of vertices, edges, and cells does not apply to these types of space, but it is still of interest to study their symmetries and topological properties.
Applications
An interest in the study of arrangements was driven by advances in computational geometry, where the arrangements were unifying structures for many problems. Advances in study of more complicated objects, such as algebraic surfaces, contributed to "real-world" applications, such as motion planning and computer vision.
References
Computational geometry
Discrete geometry |
https://en.wikipedia.org/wiki/Arrangement%20%28disambiguation%29 | In music, an arrangement is a reconceptualization of a previously composed work.
Arrangement may also refer to:
Mathematics
Arrangement (space partition), a partition of the space by a set of objects of a certain type
Arrangement of hyperplanes
Arrangement of lines
Vertex arrangement, in geometry
Arrangement as a permutation or partial permutation in combinatorics
Social arrangements
Marriage arrangement
Sugar baby, a person who receives material benefits in exchange for company
Other uses
Flower arrangement
Korean flower arrangement
Ikebana, Japanese flower arrangement
Sentence arrangement, the location of ideas and the placement of emphasis within a sentence
Arrangements (album), a 2022 album by Preoccupations
Sorting, any process of arranging items systematically
See also
Arraignment, part of the criminal law process in various jurisdictions
The Arrangement (disambiguation) |
https://en.wikipedia.org/wiki/Breno%20Lorran | Breno Lorran da Silva Talvares (born 6 March 1995), or simply Breno Lorran, is a Brazilian professional footballer who plays as a left back for Marília Atlético Clube.
Career statistics
References
External links
Breno at Portal Oficial do Grêmio
1995 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Men's association football fullbacks
Grêmio Foot-Ball Porto Alegrense players
Vitória S.C. players
São Bernardo Futebol Clube players
Grêmio Esportivo Brasil players
Goiás Esporte Clube players
Figueirense FC players
Londrina Esporte Clube players
Clube Náutico Capibaribe players
Associação Atlética Aparecidense players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Expatriate men's footballers in Portugal
Footballers from São Paulo (state)
People from Teodoro Sampaio, São Paulo |
https://en.wikipedia.org/wiki/2014%E2%80%9315%20F.C.%20Copenhagen%20season | This article shows statistics of individual players for the football club F.C. Copenhagen. It also lists all matches that F.C. Copenhagen played in the 2014–15 season.
Players
Squad information
This section show the squad as currently, considering all players who are confirmedly moved in and out (see section Players in / out).
Squad stats
Players in / out
In
Out
Club
Coaching staff
Kit
|
|
|
Other information
Competitions
Overall
Danish Superliga
League table
Results summary
Results by round
UEFA Champions League
Third qualifying round
Play-off round
Results summary
UEFA Europa League
Group stage
Results summary
Matches
Competitive
References
External links
F.C. Copenhagen official website
2014-15
Danish football clubs 2014–15 season |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Rayo%20Vallecano%20season | The 2013–14 season was the 89th season in Rayo's history and the 15th in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
La Liga
League table
Matches
Kickoff times are in CET.
Copa del Rey
Round of 32
Round of 16
References
Rayo Vallecano seasons
Rayo |
https://en.wikipedia.org/wiki/Generic%20matrix%20ring | In algebra, a generic matrix ring is a sort of a universal matrix ring.
Definition
We denote by a generic matrix ring of size n with variables . It is characterized by the universal property: given a commutative ring R and n-by-n matrices over R, any mapping extends to the ring homomorphism (called evaluation) .
Explicitly, given a field k, it is the subalgebra of the matrix ring generated by n-by-n matrices , where are matrix entries and commute by definition. For example, if m = 1 then is a polynomial ring in one variable.
For example, a central polynomial is an element of the ring that will map to a central element under an evaluation. (In fact, it is in the invariant ring since it is central and invariant.)
By definition, is a quotient of the free ring with by the ideal consisting of all p that vanish identically on all n-by-n matrices over k.
Geometric perspective
The universal property means that any ring homomorphism from to a matrix ring factors through . This has a following geometric meaning. In algebraic geometry, the polynomial ring is the coordinate ring of the affine space , and to give a point of is to give a ring homomorphism (evaluation) (either by the Hilbert nullstellensatz or by the scheme theory). The free ring plays the role of the coordinate ring of the affine space in the noncommutative algebraic geometry (i.e., we don't demand free variables to commute) and thus a generic matrix ring of size n is the coordinate ring of a noncommutative affine variety whose points are the Spec's of matrix rings of size n (see below for a more concrete discussion.)
The maximal spectrum of a generic matrix ring
For simplicity, assume k is algebraically closed. Let A be an algebra over k and let denote the set of all maximal ideals in A such that . If A is commutative, then is the maximal spectrum of A and is empty for any .
References
Algebraic structures |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Getafe%20CF%20season | The 2013–14 season was the 31st season in Getafe's history and the 10th in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
La Liga
Copa del Rey
References
Getafe CF seasons
Getafe |
https://en.wikipedia.org/wiki/Elliptic%20algebra | In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2. If the cubic divisor happens to be an elliptic curve, then the algebra is called a Sklyanin algebra. The notion is studied in the context of noncommutative projective geometry.
References
Algebraic structures
Algebraic logic |
https://en.wikipedia.org/wiki/Weil%E2%80%93Brezin%20Map | In mathematics, the Weil–Brezin map, named after André Weil and Jonathan Brezin, is a unitary transformation that maps a Schwartz function on the real line to a smooth function on the Heisenberg manifold. The Weil–Brezin map gives a geometric interpretation of the Fourier transform, the Plancherel theorem and the Poisson summation formula. The image of Gaussian functions under the Weil–Brezin map are nil-theta functions, which are related to theta functions. The Weil–Brezin map is sometimes referred to as the Zak transform, which is widely applied in the field of physics and signal processing; however, the Weil–Brezin Map is defined via Heisenberg group geometrically, whereas there is no direct geometric or group theoretic interpretation from the Zak transform.
Heisenberg manifold
The (continuous) Heisenberg group is the 3-dimensional Lie group that can be represented by triples of real numbers with multiplication rule
The discrete Heisenberg group is the discrete subgroup of whose elements are represented by the triples of integers. Considering acts on on the left, the quotient manifold is called the Heisenberg manifold.
The Heisenberg group acts on the Heisenberg manifold on the right. The Haar measure on the Heisenberg group induces a right-translation-invariant measure on the Heisenberg manifold. The space of complex-valued square-integrable functions on the Heisenberg manifold has a right-translation-invariant orthogonal decomposition:
where
.
Definition
The Weil–Brezin map is the unitary transformation given by
for every Schwartz function , where convergence is pointwise.
The inverse of the Weil–Brezin map is given by
for every smooth function on the Heisenberg manifold that is in .
Fundamental unitary representation of the Heisenberg group
For each real number , the fundamental unitary representation of the Heisenberg group is an irreducible unitary representation of on defined by
.
By Stone–von Neumann theorem, this is the unique irreducible representation up to unitary equivalence satisfying the canonical commutation relation
.
The fundamental representation of on and the right translation of on are intertwined by the Weil–Brezin map
.
In other words, the fundamental representation on is unitarily equivalent to the right translation on through the Wei-Brezin map.
Relation to Fourier transform
Let be the automorphism on the Heisenberg group given by
.
It naturally induces a unitary operator , then the Fourier transform
as a unitary operator on .
Plancherel theorem
The norm-preserving property of and , which is easily seen, yields the norm-preserving property of the Fourier transform, which is referred to as the Plancherel theorem.
Poisson summation formula
For any Schwartz function ,
.
This is just the Poisson summation formula.
Relation to the finite Fourier transform
For each , the subspace can further be decomposed into right-translation-invariant orthogonal subspaces
where
.
The left |
https://en.wikipedia.org/wiki/100%20prisoners%20problem | The 100 prisoners problem is a mathematical problem in probability theory and combinatorics. In this problem, 100 numbered prisoners must find their own numbers in one of 100 drawers in order to survive. The rules state that each prisoner may open only 50 drawers and cannot communicate with other prisoners. At first glance, the situation appears hopeless, but a clever strategy offers the prisoners a realistic chance of survival.
Danish computer scientist Peter Bro Miltersen first proposed the problem in 2003.
Problem
The 100 prisoners problem has different renditions in the literature. The following version is by Philippe Flajolet and Robert Sedgewick:
The director of a prison offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 drawers. The director randomly puts one prisoner's number in each closed drawer. The prisoners enter the room, one after another. Each prisoner may open and look into 50 drawers in any order. The drawers are closed again afterwards. If, during this search, every prisoner finds their number in one of the drawers, all prisoners are pardoned. If even one prisoner does not find their number, all prisoners die. Before the first prisoner enters the room, the prisoners may discuss strategy — but may not communicate once the first prisoner enters to look in the drawers. What is the prisoners' best strategy?
If every prisoner selects 50 drawers at random, the probability that a single prisoner finds their number is 50%. Therefore, the probability that all prisoners find their numbers is the product of the single probabilities, which is ()100 ≈ , a vanishingly small number. The situation appears hopeless.
Solution
Strategy
Surprisingly, there is a strategy that provides a survival probability of more than 30%. The key to success is that the prisoners do not have to decide beforehand which drawers to open. Each prisoner can use the information gained from the contents of every drawer they already opened to decide which one to open next. Another important observation is that this way the success of one prisoner is not independent of the success of the other prisoners, because they all depend on the way the numbers are distributed.
To describe the strategy, not only the prisoners, but also the drawers, are numbered from 1 to 100; for example, row by row starting with the top left drawer. The strategy is now as follows:
Each prisoner first opens the drawer labeled with their own number.
If this drawer contains their number, they are done and were successful.
Otherwise, the drawer contains the number of another prisoner, and they next open the drawer labeled with this number.
The prisoner repeats steps 2 and 3 until they find their own number, or fail because the number is not found in the first fifty opened drawers.
If the prisoner could continue indefinitely this way, they would inevitably loop back to the drawer they started with, forming a permutation cyc |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Celta%20de%20Vigo%20season | The 2013–14 Celta de Vigo season was the club's 90th season in its history and the 48th in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
La Liga
Copa del Rey
Round of 32
References
RC Celta de Vigo seasons
Celta |
https://en.wikipedia.org/wiki/Imprecise%20Dirichlet%20process | In probability theory and statistics, the Dirichlet process (DP) is one of the most popular Bayesian nonparametric models. It was
introduced by Thomas Ferguson as a prior over probability distributions.
A Dirichlet process is completely defined by its parameters: (the base distribution or base measure) is an arbitrary distribution and (the concentration parameter) is a positive real number (it is often denoted as ).
According to the Bayesian paradigm these parameters should be chosen based on the available prior information on the domain.
The question is: how should we choose the prior parameters of the DP, in particular the infinite dimensional one , in case of lack of prior information?
To address this issue, the only prior that has been proposed so far is the limiting DP obtained for , which has been introduced under
the name of Bayesian bootstrap by Rubin; in fact it can be proven that the Bayesian bootstrap is asymptotically equivalent to the frequentist bootstrap introduced by Bradley Efron.
The limiting Dirichlet process has been criticized on diverse grounds. From an a-priori point of view, the main
criticism is that taking is far from leading to a noninformative prior.
Moreover, a-posteriori, it assigns zero probability to any set that does not include the observations.
The imprecise Dirichlet process has been proposed to overcome these issues. The basic idea is to fix but do not choose any precise base measure .
More precisely, the imprecise Dirichlet process (IDP) is defined as follows:
where is the set of all probability measures. In other words, the IDP is the set of all Dirichlet processes (with a fixed ) obtained
by letting the base measure to span the set of all probability measures.
Inferences with the Imprecise Dirichlet Process
Let a probability distribution on (here is a standard Borel space with Borel -field ) and assume that .
Then consider a real-valued bounded function defined on . It is well known that the expectation of with respect to the Dirichlet process is
One of the most remarkable properties of the DP priors is that the posterior distribution of is again a DP.
Let be an independent and identically distributed sample from and , then the posterior distribution of given the observations is
where is an atomic probability measure (Dirac's delta) centered at . Hence, it follows
that
Therefore, for any fixed , we can exploit the previous equations to derive prior and posterior expectations.
In the IDP can span the set of all distributions . This implies that we will get a different prior and posterior expectation of for any choice of . A way to characterize inferences for the IDP is by computing lower and upper bounds for the expectation of w.r.t. .
A-priori these bounds are:
the lower (upper) bound is obtained by a probability measure that puts all the mass on the infimum (supremum) of , i.e., with (or respectively with ). From the above expressions of the lower and up |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Elche%20CF%20season | The 2013–14 season was the 91st season in Elche’s history and the 20th in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
La Liga
Copa del Rey
References
Elche CF seasons
Elche CF |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20RCD%20Espanyol%20season | The 2013–14 RCD Espanyol season was the club's 113th season in its history and its 79th in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
La Liga
League table
Copa del Rey
References
RCD Espanyol seasons
Espanyol
Espanyol |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Levante%20UD%20season | The 2013–14 season was the 105th season in Levante’s history and the 9th in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
Primera División
League table
Copa del Rey
References
Levante UD seasons
Levante UD |
https://en.wikipedia.org/wiki/Cohen%E2%80%93Hewitt%20factorization%20theorem | In mathematics, the Cohen–Hewitt factorization theorem states that if is a left module over a Banach algebra with a left approximate unit , then an element of can be factorized as a product (for some and ) whenever . The theorem was introduced by and .
References
Banach algebras
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Elena%20Vesnina%20career%20statistics | This is a list of the main career statistics of Russian professional tennis player Elena Vesnina.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup/Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Notes
The first Premier 5 event of the year has switched back and forth between the Dubai Tennis Championships and the Qatar Total Open since 2009. Dubai was classified as a Premier 5 event from 2009 to 2011 before being succeeded by Doha for the 2012–2014 period. In 2015, Dubai regained its Premier 5 status while Doha was demoted to Premier status. The two tournaments have since alternated status every year.
In 2014, the Toray Pan Pacific Open was downgraded to a Premier event and replaced by the Wuhan Open.
Doubles
Notes
The first Premier 5 event of the year has switched back and forth between the Dubai Tennis Championships and the Qatar Total Open since 2009. Dubai was classified as a Premier 5 event from 2009 to 2011 before being succeeded by Doha for the 2012–2014 period. In 2015, Dubai regained its Premier 5 status while Doha was demoted to Premier status. The two tournaments have since alternated status every year.
In 2014, the Toray Pan Pacific Open was downgraded to a Premier event and replaced by the Wuhan Open.
Mixed doubles
Grand Slam finals
Women's doubles: 11 (3 titles, 8 runner-ups)
Mixed doubles: 5 (1 title, 4 runner-ups)
Other significant finals
Olympic finals
Doubles: 2 (1 gold medal)
Mixed doubles: 1 (silver medal)
Year-end championships
Doubles: 2 (1 title, 1 runner–up)
WTA 1000 tournaments / Premier Mandatory / Premier 5 / Tier I
Singles: 1 (title)
Doubles: 17 (8 titles, 9 runner-ups)
WTA career finals
Singles: 10 (3 titles, 7 runner-ups)
Doubles: 45 (19 titles, 26 runner-ups)
ITF Circuit finals
Singles: 6 (2 titles, 4 runner–ups)
Doubles: 6 (6 titles)
WTA Tour career earnings
As of 31 May 2021
Fed Cup participation
This table is current through the 2017 Fed Cup
Singles (3–3)
Doubles (10–5)
Record against top-10 players
Vesnina's match record against players who have been ranked in the top 10 with those who have been No. 1 in boldface
Roberta Vinci 5–3
Venus Williams 4–3
Li Na 3–1
Dominika Cibulková 4–3
Andrea Petkovic 3–3
Eugenie Bouchard 2–0
Kimiko Date-Krumm 2–0
Lucie Šafářová 2–0
Patty Schnyder 2–0
Svetlana Kuznetsova 2–1
Francesca Schiavone 2–1
Ana Ivanovic 2–2
Elina Svitolina 2–2
Samantha Stosur 2–3
Marion Bartoli 2–4
Flavia Pennetta 2–4
Caroline Wozniacki 2–6
Ekaterina Makarova 2–7
Belinda Bencic 1–0
Sara Errani 1–1
Daniela Hantuchová 1–1
Simona Halep 1–1
Amélie Mauresmo 1–1
Timea Bacsinszky 1–2
Maria Sharapova 1–2
Angelique Kerber 1–3
Anna Chakvetadze 1–5
Agnieszka Radwańska 1–5
Vera Zvonareva 1–6
Jelena Janković 1–7
Mary Pierce 0–1
Dinara Safina 0–1
Carla Suárez Navarro 0–1
Nadia Petrova 0–2
Elena Dementieva 0–3
Justine Henin 0–3
Maria Kirilenko |
https://en.wikipedia.org/wiki/George%20Cybenko | George V. Cybenko is the Dorothy and Walter Gramm Professor of
Engineering at Dartmouth and a fellow of the IEEE and SIAM.
Education
Cybenko obtained his BA in mathematics from the University of Toronto in 1974 and received his PhD from Princeton in applied mathematics of electrical and computer engineering in 1978 under Bede Liu.
Work
Cybenko served as an advisor for the Defense Science Board and the Air Force Scientific Advisory Board, among several other government panels. He was the founding editor-in-chief of Security & Privacy and also of Computing in Science & Engineering, both IEEE technical magazines.
His current research interests are distributed information, control systems, and signal processing, with a focus on applications to security and infrastructure protection.
He is known for proving the universal approximation theorem for artificial neural networks with sigmoid activation functions.
Awards
SIAM Fellow (2020), "for contributions to theory and algorithms in signal processing, artificial neural networks, and distributed computing systems."
SPIE Eric A. Lehrfeld Award (2016), for "work in cyber security including developing algorithms, analysis techniques, and tools to improve the state of the art in many areas, including computational behavior analysis, adversarial deception detection and dynamics, disclosure risk, and covert channels, and for his efforts in support of the SPIE Defense + Commercial Sensing symposium".
US Air Force Commander’s Service Award (2016)
IEEE Fellow (1998), "for contributions to algorithms and theory of artificial neural networks in signal processing, and to theory and systems software for distributed and parallel computing."
References
External links
Homepage at Dartmouth
American computer scientists
20th-century American mathematicians
21st-century American mathematicians
American electrical engineers
Princeton University School of Engineering and Applied Science alumni
Fellow Members of the IEEE
Living people
Dartmouth College faculty
Place of birth missing (living people)
Year of birth missing (living people)
Fellows of the Society for Industrial and Applied Mathematics
University of Toronto alumni |
https://en.wikipedia.org/wiki/J%C3%B3nsson%E2%80%93Tarski%20algebra | In mathematics, a Jónsson–Tarski algebra or Cantor algebra is an algebraic structure encoding a bijection from an infinite set onto the product . They were introduced by . , named them after Georg Cantor because of Cantor's pairing function and Cantor's theorem that an infinite set has the same number of elements as . The term Cantor algebra is also occasionally used to mean the Boolean algebra of all clopen subsets of the Cantor set, or the Boolean algebra of Borel subsets of the reals modulo meager sets (sometimes called the Cohen algebra).
The group of order-preserving automorphisms of the free Jónsson–Tarski algebra on one generator is the Thompson group .
Definition
A Jónsson–Tarski algebra of type 2 is a set with a product from to and two 'projection' maps and from to , satisfying , , and . The definition for type > 2 is similar but with projection operators.
Example
If is any bijection from to then it can be extended to a unique Jónsson–Tarski algebra by letting be the projection of onto the th factor.
References
Algebraic structures |
https://en.wikipedia.org/wiki/Dmitry%20Matveyevich%20Smirnov | Dmitry Matveyevich Smirnov (; 27 October 1919 in Shilovo, Seredskii District,
Ivanovo Oblast, Soviet Union – 14 April 2005) was a Soviet mathematician working in group theory and Jónsson–Tarski algebras.
References
Bibliography
Soviet mathematicians
1919 births
2005 deaths
Russian mathematicians
Group theorists |
https://en.wikipedia.org/wiki/Geometric%20transformation | In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both or both — such that the function is bijective so that its inverse exists. The study of geometry may be approached by the study of these transformations.
Classifications
Geometric transformations can be classified by the dimension of their operand sets (thus distinguishing between, say, planar transformations and spatial transformations). They can also be classified according to the properties they preserve:
Displacements preserve distances and oriented angles (e.g., translations);
Isometries preserve angles and distances (e.g., Euclidean transformations);
Similarities preserve angles and ratios between distances (e.g., resizing);
Affine transformations preserve parallelism (e.g., scaling, shear);
Projective transformations preserve collinearity;
Each of these classes contains the previous one.
Möbius transformations using complex coordinates on the plane (as well as circle inversion) preserve the set of all lines and circles, but may interchange lines and circles.
Conformal transformations preserve angles, and are, in the first order, similarities.
Equiareal transformations, preserve areas in the planar case or volumes in the three dimensional case. and are, in the first order, affine transformations of determinant 1.
Homeomorphisms (bicontinuous transformations) preserve the neighborhoods of points.
Diffeomorphisms (bidifferentiable transformations) are the transformations that are affine in the first order; they contain the preceding ones as special cases, and can be further refined.
Transformations of the same type form groups that may be sub-groups of other transformation groups.
Opposite group actions
Many geometric transformations are expressed with linear algebra. The bijective linear transformations are elements of a general linear group. The linear transformation A is non-singular. For a row vector v, the matrix product vA gives another row vector w = vA.
The transpose of a row vector v is a column vector vT, and the transpose of the above equality is Here AT provides a left action on column vectors.
In transformation geometry there are compositions AB. Starting with a row vector v, the right action of the composed transformation is w = vAB. After transposition,
Thus for AB the associated left group action is In the study of opposite groups, the distinction is made between opposite group actions because commutative groups are the only groups for which these opposites are equal.
See also
Coordinate transformation
Erlangen program
Symmetry (geometry)
Motion
Reflection
Rigid transformation
Rotation
Topology
Transformation matrix
References
Further reading
Dienes, Z. P.; Golding, E. W. (1967) . Geometry Through Transformations (3 vols.): Geometry of Distortion, |
https://en.wikipedia.org/wiki/Random%20algebra | In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra was studied by John von Neumann in 1935 (in work later published as ) who showed that it is not isomorphic to the Cantor algebra of Borel sets modulo meager sets. Random forcing was introduced by .
See also
Random number
References
Boolean algebra
Forcing (mathematics) |
https://en.wikipedia.org/wiki/2013%E2%80%9314%20Real%20Valladolid%20season | The 2013–14 season was the 86th season in Real Valladolid ’s history and the 42nd in the top-tier.
Squad
As June, 2014..
Squad and statistics
|}
Transfers
Competitions
Overall
La Liga
Copa del Rey
References
Real Valladolid seasons
Valladolid |
https://en.wikipedia.org/wiki/Journal%20of%20Group%20Theory | The Journal of Group Theory is a bimonthly peer-reviewed mathematical journal covering all aspects of group theory. It was established in 1998 and is published by Walter de Gruyter. The editor-in-chief is Chris Parker (University of Birmingham).
Abstracting and indexing
The journal is abstracted and indexed in:
Its 2018 MCQ is 0.48. According to the Journal Citation Reports, the journal has a 2018 impact factor of 0.47, and the 5-year impact factor is 0.52.
References
External links
Mathematics journals
Bimonthly journals
De Gruyter academic journals
Academic journals established in 1998
English-language journals |
https://en.wikipedia.org/wiki/Basem%20Ali | Basem Ali (; born 27 October 1988) is an Egyptian international footballer who plays as a right back for Egyptian club El Gouna, on loan from Al Ahly.
Career statistics
Club
References
External links
1988 births
Living people
Egyptian men's footballers
Footballers from Cairo
Men's association football fullbacks
Asyut Petroleum SC players
Al Mokawloon Al Arab SC players
Al Ahly SC players
El Gouna FC players
Egyptian Premier League players
Egypt men's international footballers |
https://en.wikipedia.org/wiki/Ulam%20matrix | In mathematical set theory, an Ulam matrix is an array of subsets of a cardinal number with certain properties. Ulam matrices were introduced by Stanislaw Ulam in his 1930 work on measurable cardinals: they may be used, for example, to show that a real-valued measurable cardinal is weakly inaccessible.
Definition
Suppose that κ and λ are cardinal numbers, and let be a -complete filter on . An Ulam matrix is a collection of subsets of indexed by such that
If then and are disjoint.
For each , the union over of the sets , is in the filter .
References
Set theory |
https://en.wikipedia.org/wiki/Global%20Health%20Observatory | The Global Health Observatory (GHO) is a public health observatory established by the World Health Organization (WHO) to share data on global health, including statistics by country and information about specific diseases and health measures. The GHO tracks important information like "Response to the Millennium Development Goals".
History
The GHO was formed in around 2010 from the ashes of the WHO Statistical Information System, which was "upgraded... to provide you with more data, more tools, more analysis and more reports."
In December 2012, the WHO announced that it was making improvements in its GHO to improve its accessibility and usability by "specialists such as statisticians, epidemiologists, economists and public health researchers as well as anyone with an interest in global health."
Themes
The GHO website is organized around themes. For each theme, key statistics are presented on the associated webpage, and more detailed data and reports are available for download. The themes include:
Millennium Development Goals
Estimates of mortality and global health
Health systems
Public health and environment
Health Equity Monitor
International Health Regulations Monitoring framework
Urban health
Women and health
Noncommunicable diseases
Substance use and mental health
Infectious diseases
Injuries and violence
Reception and impact
The GHO has been listed by many libraries and dataset listings as a go-to source for information on health statistics. The GHO has also been cited in work of the Centers for Disease Control and Prevention in the United States.
GHO data has also been cited in academic studies on various aspects of global health, particularly for cross-country comparisons.
See also
The World Bank data sets
Gapminder, which compiles data on a number of indicators, including health indicators, from a variety of sources
Human Mortality Database, which includes information on mortality and causes of mortality, but is restricted to data built from official records
References
External links
World Health Organization
Scientific databases
Health informatics |
https://en.wikipedia.org/wiki/Melvyn%20B.%20Nathanson | Melvyn Bernard Nathanson (born October 10, 1944, in Philadelphia, Pennsylvania) is an American mathematician, specializing in number theory, and a Professor of Mathematics at Lehman College and The Graduate Center (City University of New York). His principal work is in additive and combinatorial number theory. He is the author of over 200 research papers in mathematics, and author or editor of 27 books.
Education
Nathanson graduated from Central High School in 1961 and from the University of Pennsylvania in 1965 with a BA in philosophy. He was a graduate student in biophysics at Harvard University in 1965–66, then moved to the University of Rochester, where he received a PhD in mathematics in 1972. During the academic year 1969–70 he was a visiting research student in the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.
Professional life
Nathanson was on the faculty of Southern Illinois University, Carbondale from 1971 to 1981. He was Professor of Mathematics and Dean of the Graduate School of Rutgers-Newark from 1981 to 1986, and Provost and Vice President of Academic Affairs at Lehman College (CUNY) from 1986 to 1991. He has been Professor of Mathematics at Lehman College and The Graduate Center (CUNY) since 1986. He held visiting positions at Harvard University in 1977–78, Rockefeller University in 1981–83, Tel Aviv University in Spring, 2001, and Princeton University in Fall, 2008.
In 1974–75 Nathanson was Assistant to André Weil in the School of Mathematics of the Institute for Advanced Study. Nathanson subsequently spent the academic years 1990–91 and 1999–2000, and the Fall, 2007, term at the Institute. He served as President of the Association of Members of the Institute for Advanced Study (AMIAS) from 1998 to 2012.
In 1972–73 Nathanson became the first American mathematician to receive an IREX fellowship to spend a year in the former USSR, where he worked with I. M. Gel'fand at Moscow State University. In 1977 the National Academy of Sciences selected him to spend another year in Moscow on its exchange agreement with the USSR Academy of Sciences. An international brouhaha ensued when the Soviet government refused to allow him to re-enter the country. He spent the academic year 1977–78 in the mathematics department at Harvard University, where he also worked in the Program for Science and International Affairs, and contributed to the book Nuclear Nonproliferation: The Spent Fuel Problem.
Nathanson is the author/editor/translator of several books and articles on Soviet art and politics, including Komar/Melamid: Two Soviet Dissident Artists, and Grigori Freiman, It Seems I am a Jew: A Samizdat Essay on Anti-Semitism in Soviet Mathematics, both published by Southern Illinois University Press.
Nathanson was a frequent collaborator with Paul Erdős, with whom he wrote 19 papers in number theory. He also organizes the Workshop on Combinatorial and Additive Number Theory, which has been he |
https://en.wikipedia.org/wiki/Cantor%20algebra | In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete.
The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is the free Boolean algebra on a countable number of generators. Up to isomorphism, this is the only nontrivial Boolean algebra that is both countable and atomless.
The complete Cantor algebra is the complete Boolean algebra of Borel subsets of the reals modulo meager sets . It is isomorphic to the completion of the countable Cantor algebra. (The complete Cantor algebra is sometimes called the Cohen algebra, though "Cohen algebra" usually refers to a different type of Boolean algebra.) The complete Cantor algebra was studied by von Neumann in 1935 (later published as ), who showed that it is not isomorphic to the random algebra of Borel subsets modulo measure zero sets.
References
Forcing (mathematics)
Boolean algebra |
https://en.wikipedia.org/wiki/Maharam%20algebra | In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by .
Definitions
A continuous submeasure or Maharam submeasure on a Boolean algebra is a real-valued function m such that
and if .
If , then .
.
If is a decreasing sequence with greatest lower bound 0, then the sequence has limit 0.
A Maharam algebra is a complete Boolean algebra with a continuous submeasure.
Examples
Every probability measure is a continuous submeasure, so as the corresponding Boolean algebra of measurable sets modulo measure zero sets is complete, it is a Maharam algebra.
solved a long-standing problem by constructing a Maharam algebra that is not a measure algebra, i.e., that does not admit any countably additive strictly positive finite measure.
References
Boolean algebra |
https://en.wikipedia.org/wiki/Eugenie%20Bouchard%20career%20statistics | This is a list of the main career statistics of professional Canadian tennis player, Eugenie Bouchard. To date, Bouchard has won one WTA singles title at the 2014 Nuremberg Cup. Other highlights of Bouchard's career thus far include a runner-up finish at the 2014 Wimbledon Championships, semifinal appearances at the 2014 Australian Open and 2014 French Open and a quarterfinal run at the 2015 Australian Open. Bouchard achieved a career high singles ranking of world No. 5 on October 20, 2014.
Career achievements
Bouchard advanced to her first career singles final at the 2013 HP Open following a straight sets victory over Kurumi Nara, but lost to former US Open champion Samantha Stosur in three sets in the championship match. The following year, Bouchard reached her first Grand Slam semifinal at the Australian Open, defeating former world No. 1 Ana Ivanovic (who had upset the reigning world No. 1 and heavy favourite, Serena Williams) en route before losing in straight sets to the eventual champion, Li Na. During the clay court season, Bouchard won her first WTA singles title at the Nuremberg Cup, defeating Karolína Plíšková in the final in three sets before reaching her second consecutive major semifinal at the French Open, where she lost in three sets to the eventual champion, Maria Sharapova. In July, Bouchard became the first Canadian player to reach a Grand Slam final in singles when she defeated world No. 3, Simona Halep, in the semifinals of the Wimbledon Championships. However, she lost in the final to sixth seed and 2011 champion, Petra Kvitová. In September, Bouchard reached her first WTA Premier 5 final at the Wuhan Open, but was again defeated by Kvitová.
Performance timelines
Singles
Current through the 2023 Guadalajara Open.
Notes
Bouchard's 2015 US Open withdrawal in the fourth round does not count as a loss.
The first Premier 5 event of the year has switched back and forth between the Qatar Ladies Open and the Dubai Tennis Championships since 2009. Dubai was classified as a Premier 5 event from 2009 to 2011 before being succeeded by Doha for the 2012–2014 period. Since 2015, the two tournaments alternate between Premier 5 and Premier status every year.
In 2014, the Pan Pacific Open was downgraded to a Premier event and replaced by the Wuhan Open.
Only WTA Tour main draw (incl. major tournaments) and Olympics results are considered.
Doubles
Mixed doubles
Grand Slam tournament finals
Singles: 1 (runner-up)
Other significant finals
WTA 1000
Singles: 1 (runner-up)
WTA career finals
Singles: 8 (1 title, 7 runner-ups)
Doubles: 5 (1 title, 4 runner-ups)
ITF Circuit finals
Singles: 7 (6 titles, 1 runner-up)
Doubles: 4 (1 title, 3 runner-ups)
Junior Grand Slam finals
Singles: 1 (title)
Doubles: 2 (2 titles)
Singles Grand Slam seedings
The tournaments won by Bouchard are in boldface, while italics indicates Bouchard was the runner-up.
Coaches
Career prize money
*as of January 29, 2018
Head-to-head records
Wins ove |
https://en.wikipedia.org/wiki/Collapsing%20algebra | In mathematics, a collapsing algebra is a type of Boolean algebra sometimes used in forcing to reduce ("collapse") the size of cardinals. The posets used to generate collapsing algebras were introduced by Azriel Lévy in 1963.
The collapsing algebra of λω is a complete Boolean algebra with at least λ elements but generated by a countable number of elements. As the size of countably generated complete Boolean algebras is unbounded, this shows that there is no free complete Boolean algebra on a countable number of elements.
Definition
There are several slightly different sorts of collapsing algebras.
If κ and λ are cardinals, then the Boolean algebra of regular open sets of the product space κλ is a collapsing algebra. Here κ and λ are both given the discrete topology. There are several different options for the topology of
κλ. The simplest option is to take the usual product topology. Another option is to take the topology generated by open sets consisting of functions whose value is specified on less than λ elements of λ.
References
Boolean algebra
Forcing (mathematics) |
https://en.wikipedia.org/wiki/Laurynas%20Kulikas | Laurynas Kulikas (born 13 April 1994) is a footballer who plays as a forward for Holstein Kiel II.
Career statistics
References
External links
1994 births
Living people
Russian people of Lithuanian descent
German men's footballers
Russian men's footballers
Men's association football forwards
2. Bundesliga players
Regionalliga players
FC St. Pauli II players
FC St. Pauli players
VfL Bochum players
VfL Bochum II players
Hamburger SV II players
FC Eintracht Norderstedt 03 players
TSV Steinbach Haiger players
VfR Neumünster players
Holstein Kiel II players
Footballers from Kiel |
https://en.wikipedia.org/wiki/Alfred%20Aeppli | Alfred Aeppli was a Swiss mathematician. The Pólya–Aeppli distribution in probability theory and statistics is named after him and his doctoral advisor George Pólya.
Life and work
Alfred Aeppli was born in Zürich on 15 July 1894 to Alfred Aeppli and Rosa Aeppli-Gehring. He went to a primary school in Zürich and the canton's Industrial School, where he received his matura in the summer of 1913. Afterwards, Aeppli studied at the Eidgenössische Technische Hochschule (ETH Zürich) at the department for higher teachers of mathematics and physics. In the winter semester of 1914–1915 he was on leave for military service. After receiving his Diplom, he worked at a private school in Germany for a year and returned to the ETH in the spring of 1919 as a research assistant of Arthur Hirsch.
Aeppli earned his doctorate in 1924 under the supervision of George Pólya and Hermann Weyl. He came up with the Pólya–Aeppli distribution in his doctoral dissertation. This discovery was published by Pólya in 1930, and he credited its discovery to his student Aeppli. The Pólya–Aeppli distribution, now also known as the geometric Poisson distribution, is a particular case of the compound Poisson distribution, and is used to describe objects that come in clusters, where the number of clusters follows a Poisson distribution and the number of objects within a cluster follows a geometric distribution.
References
20th-century Swiss mathematicians
ETH Zurich alumni
Probability theorists
1894 births
Year of death missing
Scientists from Zürich |
https://en.wikipedia.org/wiki/Naval%20Research%20Logistics | Naval Research Logistics is a peer-reviewed scientific journal that publishes papers in the field of logistics, especially those in the areas of operations research, applied statistics, and quantitative modeling. It was established in 1954 and is published by John Wiley & Sons. Its current editor is Ming Hu.
External links
Statistics journals
Computational statistics journals
Mathematics journals
Academic journals established in 1954
English-language journals |
https://en.wikipedia.org/wiki/Cyclic%20algebra | In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple algebras.
Definition
Let A be a finite-dimensional central simple algebra over a field F. Then A is said to be cyclic if it contains a strictly maximal subfield E such that E/F is a cyclic field extension (i.e., the Galois group is a cyclic group).
See also
Factor system#Cyclic algebras - cyclic algebras described by factor systems.
Brauer group#Cyclic algebras - cyclic algebras are representative of Brauer classes.
References
Algebra |
https://en.wikipedia.org/wiki/Biro%20Biro | Diego Santos Gama Camilo (born 22 November 1994), commonly known as Biro Biro, is a Brazilian professional footballer who plays as a forward.
Career statistics
Club
Other includes Brazilian state competitions and national super cups.
References
External links
Biro Biro at playmakerstats.com (English version of ogol.com.br)
1994 births
Living people
Footballers from Rio de Janeiro (state)
Brazilian men's footballers
Men's association football forwards
Campeonato Brasileiro Série A players
China League One players
Fluminense FC players
Associação Atlética Ponte Preta players
Shanghai Shenxin F.C. players
São Paulo FC players
Nova Iguaçu FC players
Botafogo de Futebol e Regatas players
Brazilian expatriate men's footballers
Expatriate men's footballers in China
Brazilian expatriate sportspeople in China |
https://en.wikipedia.org/wiki/Geometry%20Dash | Geometry Dash is a series of music platforming video games developed by Swedish developer Robert "RobTop" Topala. The game was released on 13 August 2013 on iOS and Android, and the Steam version on 22 December 2014. In Geometry Dash, players control the movement of an icon and navigate along music-based levels, while avoiding obstacles such as spikes that instantly destroy the icon on impact.
Geometry Dash consists of 21 official levels. It has a level creation system, where players can create their own custom levels, share them online and play levels designed by other players. In addition to the official levels, certain user-created levels have been featured in in-game content. In-game currency, such as stars, coins, orbs or diamonds, can be obtained from various sources, such as official levels, user-created levels or chests.
In addition to the original game, three other spin-off games in the series have been made: Geometry Dash Meltdown, Geometry Dash World, and Geometry Dash SubZero.
Gameplay
Geometry Dash can be played with a touchscreen, keyboard, mouse, or controller. The player manipulates the movement of their icon through input in the form of pressing or holding to reach the end of a level. If the player crashes into an obstacle, such as a spike, a wall or a sawblade, the level restarts from the beginning. There is also a "practice mode" in which a player may place checkpoints to survey or practice a level, but is unable to collect coins or gain progress for normal mode. The timing and rhythm of the in-game music are key parts of the game, often in relation to each other.
During gameplay, the player's icon takes the form of one of seven different game modes, each of which behaves differently with each interaction. Player movement is further complicated by portals that allow the player to change between seven game modes, reverse gravity, change the size of their icon, mirror the direction of their movement, change their speed, or teleport. Furthermore, pads and orbs can be used to move the player in varying directions or change gravity.
There are 21 official levels in the full version of Geometry Dash, 18 of which are unlocked upon installation. The other three require a specific number of secret coins hidden within all the official levels. Each level grants rewards upon completion. Official levels steadily advance in difficulty. Levels are categorized into six difficulty ratings: Easy, Normal, Hard, Harder, Insane, and Demon (which also has five sub-categories: Easy Demon, Medium Demon, Hard Demon, Insane Demon, and Extreme Demon). Players can earn achievements that unlock rewards, such as icons or colours. Players may also utilize three shops that use an in-game currency to acquire icons or colours.
The full version of the game features the ability to upload and download user-created levels. The creator must complete their level with all user-placed coins in normal mode before it can be uploaded: a process known as verificatio |
https://en.wikipedia.org/wiki/Suslin%20algebra | In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin.
The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslin trees or Suslin lines.
See also
Andrei Suslin
References
Boolean algebra
Forcing (mathematics)
Independence results |
https://en.wikipedia.org/wiki/Suslin%20operation | In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers.
The Suslin operation was introduced by and . In Russia it is sometimes called the A-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A).
Definitions
A Suslin scheme is a family of subsets of a set indexed by finite sequences of non-negative integers. The Suslin operation applied to this scheme produces the set
Alternatively, suppose we have a Suslin scheme, in other words a function from finite sequences of positive integers to sets . The result of the Suslin operation is the set
where the union is taken over all infinite sequences
If is a family of subsets of a set , then is the family of subsets of obtained by applying the Suslin operation to all collections as above where all the sets are in .
The Suslin operation on collections of subsets of has the property that . The family is closed under taking countable unions or intersections, but is not in general closed under taking complements.
If is the family of closed subsets of a topological space, then the elements of are called Suslin sets, or analytic sets if the space is a Polish space.
Example
For each finite sequence , let be the infinite sequences that extend .
This is a clopen subset of .
If is a Polish space and is a continuous function, let .
Then is a Suslin scheme consisting of closed subsets of and .
References
Descriptive set theory |
https://en.wikipedia.org/wiki/Besirin | Besirin () is a Syrian village located in the Hama Subdistrict of the Hama District in the Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Besirin had a population of 4,697 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Bibliography
Populated places in Hama District |
https://en.wikipedia.org/wiki/Atshan | Atshan () is a Syrian village located in the Suran Subdistrict in Hama District. According to the Syria Central Bureau of Statistics (CBS), Atshan had a population of 1,809 in the 2004 census.
References
Populated places in Hama District |
https://en.wikipedia.org/wiki/Rodrigo%20Sabi%C3%A1 | Rodrigo Augusto Sabiá (born 3 September 1992) is a Brazilian professional footballer who plays as a centre back for Paulista, on loan from Grêmio.
Career statistics
References
External links
Rodrigo Sabiá profile. Portal Oficial do Grêmio.
1992 births
Living people
Brazilian men's footballers
Grêmio Foot-Ball Porto Alegrense players
Grêmio Osasco Audax Esporte Clube players
Campeonato Brasileiro Série A players
Men's association football defenders
People from Mogi Guaçu
Footballers from São Paulo (state) |
https://en.wikipedia.org/wiki/Tell%20Qartal | Tell Qartal () is a Syrian village located in the Subdistrict of the Hama District in the Hama Governorate. According to the Syria Central Bureau of Statistics (CBS), Tell Qartal had a population of 2,079 in the 2004 census. Its inhabitants are predominantly Sunni Muslims.
References
Bibliography
Populated places in Hama District |
https://en.wikipedia.org/wiki/Hama%20Subdistrict | Hama Subdistrict () is a Syrian nahiyah (subdistrict) located in Hama District in Hama. According to the Syria Central Bureau of Statistics (CBS), Hama Subdistrict had a population of 467254 in the 2004 census.
References
Hama
Hama District |
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