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https://en.wikipedia.org/wiki/1990%20Swedish%20football%20Division%202 | Statistics of the Swedish football Division 2 for the 1990 season.
League standings
Division 2 Norra
Division 2 Östra
Division 2 Västra
Division 2 Södra
References
Sweden - List of final tables (Clas Glenning)
1990
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1991%20Swedish%20football%20Division%202 | Statistics of Swedish football Division 2 for the 1991 season.
Vårserier (Springseries)
Norra Norrland
Södra Norrland
Östra Svealand
Mellersta Svealand
Östra Götaland
Mellersta Götaland
Västra Götaland
Södra Götaland
Höstserier (Autumnseries)
Kvalettan Norra
Kvalettan Södra
Hösttvåan Norrland
Östra Svealand
Västra Svealand
Mellersta Götaland
Västra Götaland
Södra Götaland
Footnotes
References
Sweden - List of final tables (Clas Glenning)
1991
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1992%20Swedish%20football%20Division%202 | Statistics of Swedish football Division 2 for the 1992 season.
Vårserier (Springseries)
Norra Norrland
Södra Norrland
Östra Svealand
Mellersta Svealand
Östra Götaland
Mellersta Götaland
Västra Götaland
Södra Götaland
Höstserier (Autumnseries)
Kvalettan Norra
Kvalettan Södra
Hösttvåan Norrland
Östra Svealand
Västra Svealand
Östra Götaland
Västra Götaland
Södra Götaland
Footnotes
References
Sweden - List of final tables (Clas Glenning)
1992
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1993%20Swedish%20football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 1993 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
3
1993
Sweden
Sweden |
https://en.wikipedia.org/wiki/1994%20Swedish%20football%20Division%202 | The 1994 Swedish Division 2 season.
The following are the statistics of the Swedish football Division 2 for the 1994 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
Division 1 qualification play-off
1st round
2nd round
References
Sweden - List of final tables (Clas Glenning)
1994
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1995%20Swedish%20football%20Division%202 | Statistics of Swedish football Division 2 for the 1995 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
1995
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1996%20Swedish%20football%20Division%202 | Statistics of Swedish football Division 2 for the 1996 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
1996
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1997%20Swedish%20football%20Division%202 | Statistics of Swedish football Division 2 for the 1997 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
1997
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1998%20Swedish%20Football%20Division%202 | Statistics of Swedish football Division 2 in season 1998.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
1998
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/1999%20Swedish%20football%20Division%202 | Statistics of Swedish football Division 2 for the 1999 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden – List of final tables (Clas Glenning)
1999
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2000%20Swedish%20football%20Division%202 | Statistics of the Swedish football Division 2 for the 2000 season.
League standings
Division 2 Norrland
Division 2 Västra Svealand
Division 2 Östra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
Swedish Football Division 2 seasons
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2001%20Swedish%20football%20Division%202 | The following are the statistics of the Swedish football Division 2 in the season of 2001.
League standings
Division 2 Norrland
Division 2 Västra Svealand
Division 2 Östra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
Swedish Football Division 2 seasons
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2002%20Swedish%20football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 2002 season.
League standings
Division 2 Norrland
Division 2 Västra Svealand
Division 2 Östra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
Swedish Football Division 2 seasons
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2003%20Swedish%20football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 2003 season.
League standings
Division 2 Norrland
Division 2 Östra Svealand
Division 2 Västra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
Swedish Football Division 2 seasons
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2004%20Swedish%20football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 2004 season.
League standings
Division 2 Norrland
Division 2 Västra Svealand
Division 2 Östra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
Swedish Football Division 2 seasons
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2005%20Swedish%20football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 2005 season.
League standings 2005
Division 2 Norrland
Division 2 Norra Svealand
Division 2 Östra Svealand
Division 2 Mellersta Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (Clas Glenning)
Swedish Football Division 2 seasons
3
Sweden
Sweden |
https://en.wikipedia.org/wiki/2006%20Swedish%20Football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 2006 season.
League standings
Division 2 Norrland
Division 2 Norra Svealand
Division 2 Östra Svealand
Division 2 Mellersta Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (RSSSF)
Swedish Football Division 2 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/2007%20Swedish%20Football%20Division%202 | The following are the statistics of the Swedish football Division 2 for the 2007 season.
League standings
Division 2 Norrland
Division 2 Norra Svealand
Division 2 Östra Svealand
Division 2 Mellersta Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
References
Sweden - List of final tables (RSSSF)
Swedish Football Division 2 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/2008%20Swedish%20Football%20Division%202 | The following are the statistics of Swedish football Division 2 for the 2008 season.
League standings
Division 2 Norrland
Division 2 Norra Svealand
Division 2 Östra Svealand
Division 2 Östra Götaland
Division 2 Västra Götaland
Division 2 Södra Götaland
Player of the year awards
Ever since 2003 the online bookmaker Unibet have given out awards at the end of the season to the best players in Division 2. The recipients are decided by a jury of sportsjournalists, coaches and football experts. The names highlighted in green won the overall national award.
References
Sweden - List of final tables (RSSSF)
Swedish Football Division 2 seasons
4
Sweden
Sweden |
https://en.wikipedia.org/wiki/List%20of%20isotoxal%20polyhedra%20and%20tilings | In geometry, isotoxal polyhedra and tilings are defined by the property that they have symmetries taking any edge to any other edge. Polyhedra with this property can also be called "edge-transitive", but they should be distinguished from edge-transitive graphs, where the symmetries are combinatorial rather than geometric.
Regular polyhedra are isohedral (face-transitive), isogonal (vertex-transitive), and isotoxal (edge-transitive).
Quasiregular polyhedra are isogonal and isotoxal, but not isohedral; their duals are isohedral and isotoxal, but not isogonal.
The dual of an isotoxal polyhedron is also an isotoxal polyhedron. (See the Dual polyhedron article.)
Convex isotoxal polyhedra
The dual of a convex polyhedron is also a convex polyhedron.
There are nine convex isotoxal polyhedra based on the Platonic solids: the five (regular) Platonic solids, the two (quasiregular) common cores of dual Platonic solids, and their two duals.
The vertex figures of the quasiregular forms are (squares or) rectangles; the vertex figures of the duals of the quasiregular forms are (equilateral triangles and equilateral triangles, or) equilateral triangles and squares, or equilateral triangles and regular pentagons.
Isotoxal star-polyhedra
The dual of a non-convex polyhedron is also a non-convex polyhedron. (By contraposition.)
There are ten non-convex isotoxal polyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron: the five (quasiregular) hemipolyhedra based on the quasiregular octahedron, cuboctahedron, and icosidodecahedron, and their five (infinite) duals:
(*) Faces, edges, and intersection points are the same; only, some other of these intersection points, not at infinity, are considered as vertices.
There are sixteen non-convex isotoxal polyhedra based on the Kepler–Poinsot polyhedra: the four (regular) Kepler–Poinsot polyhedra, the six (quasiregular) common cores of dual Kepler–Poinsot polyhedra (including four hemipolyhedra), and their six duals (including four (infinite) hemipolyhedron-duals):
Finally, there are six other non-convex isotoxal polyhedra: the three quasiregular ditrigonal (3 | p q) star polyhedra, and their three duals:
Isotoxal tilings of the Euclidean plane
There are at least 5 polygonal tilings of the Euclidean plane that are isotoxal. (The self-dual square tiling recreates itself in all four forms.)
Isotoxal tilings of the hyperbolic plane
There are infinitely many isotoxal polygonal tilings of the hyperbolic plane, including the Wythoff constructions from the regular hyperbolic tilings {p,q}, and non-right (p q r) groups.
Here are six (p q 2) families, each with two regular forms, and one quasiregular form. All have rhombic duals of the quasiregular form, but only one is shown:
Here's 3 example (p q r) families, each with 3 quasiregular forms. The duals are not shown, but have isotoxal hexagonal and octagonal faces.
Isotoxal tilings of the sphere
All isotoxal polyhedra listed above can |
https://en.wikipedia.org/wiki/Somos%20sequence | In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but nevertheless many Somos sequences have the property that all of their members are integers.
Recurrence equations
For an integer number k larger than 1, the Somos-k sequence is defined by the equation
when k is odd, or by the analogous equation
when k is even, together with the initial values
ai = 1 for i < k.
For k = 2 or 3, these recursions are very simple (there is no addition on the right-hand side) and they define the all-ones sequence (1, 1, 1, 1, 1, 1, ...). In the first nontrivial case, k = 4, the defining equation is
while for k = 5 the equation is
These equations can be rearranged into the form of a recurrence relation, in which the value an on the left hand side of the recurrence is defined by a formula on the right hand side, by dividing the formula by an − k. For k = 4, this yields the recurrence
while for k = 5 it gives the recurrence
While in the usual definition of the Somos sequences, the values of ai for i < k are all set equal to 1, it is also possible to define other sequences by using the same recurrences with different initial values.
Sequence values
The values in the Somos-4 sequence are
1, 1, 1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209, 83313, 620297, 7869898, ... .
The values in the Somos-5 sequence are
1, 1, 1, 1, 1, 2, 3, 5, 11, 37, 83, 274, 1217, 6161, 22833, 165713, ... .
The values in the Somos-6 sequence are
1, 1, 1, 1, 1, 1, 3, 5, 9, 23, 75, 421, 1103, 5047, 41783, 281527, ... .
The values in the Somos-7 sequence are
1, 1, 1, 1, 1, 1, 1, 3, 5, 9, 17, 41, 137, 769, 1925, 7203, 34081, ... .
Integrality
The form of the recurrences describing the Somos sequences involves divisions, making it appear likely that the sequences defined by these recurrence will contain fractional values. Nevertheless, for k ≤ 7 the Somos sequences contain only integer values. Several mathematicians have studied the problem of proving and explaining this integer property of the Somos sequences; it is closely related to the combinatorics of cluster algebras.
For k ≥ 8 the analogously defined sequences eventually contain fractional values. For Somos-8 the first fractional value is the 19th term with value 420514/7.
For k < 7, changing the initial values (but using the same recurrence relation) also typically results in fractional values.
References
External links
Jim Propp's Somos Sequence Site
The Troublemaker Number. Numberphile video on the Somos sequences
Integer sequences
Recurrence relations |
https://en.wikipedia.org/wiki/Irrelevant%20ideal | In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. It corresponds to the origin in the affine space which cannot be mapped to a point in the projective space. More generally, a homogeneous ideal of a graded ring is called an irrelevant ideal if its radical contains the irrelevant ideal.
The terminology arises from the connection with algebraic geometry. If R = k[x0, ..., xn] (a multivariate polynomial ring in n+1 variables over an algebraically closed field k) graded with respect to degree, there is a bijective correspondence between projective algebraic sets in projective n-space over k and homogeneous, radical ideals of R not equal to the irrelevant ideal. More generally, for an arbitrary graded ring R, the Proj construction disregards all irrelevant ideals of R.
Notes
References
Sections 1.5 and 1.8 of
Commutative algebra
Algebraic geometry |
https://en.wikipedia.org/wiki/Connection%20%28algebraic%20framework%29 | Geometry of quantum systems (e.g.,
noncommutative geometry and supergeometry) is mainly
phrased in algebraic terms of modules and
algebras. Connections on modules are
generalization of a linear connection on a smooth vector bundle written as a Koszul connection on the
-module of sections of .
Commutative algebra
Let be a commutative ring
and an A-module. There are different equivalent definitions
of a connection on .
First definition
If is a ring homomorphism, a -linear connection is a -linear morphism
which satisfies the identity
A connection extends, for all to a unique map
satisfying . A connection is said to be integrable if , or equivalently, if the curvature vanishes.
Second definition
Let be the module of derivations of a ring . A
connection on an A-module is defined
as an A-module morphism
such that the first order differential operators on
obey the Leibniz rule
Connections on a module over a commutative ring always exist.
The curvature of the connection is defined as
the zero-order differential operator
on the module for all .
If is a vector bundle, there is one-to-one
correspondence between linear
connections on and the
connections on the
-module of sections of . Strictly speaking, corresponds to
the covariant differential of a
connection on .
Graded commutative algebra
The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a graded
commutative algebra. This is the case of
superconnections in supergeometry of
graded manifolds and supervector bundles.
Superconnections always exist.
Noncommutative algebra
If is a noncommutative ring, connections on left
and right A-modules are defined similarly to those on
modules over commutative rings. However
these connections need not exist.
In contrast with connections on left and right modules, there is a
problem how to define a connection on an
R-S-bimodule over noncommutative rings
R and S. There are different definitions
of such a connection. Let us mention one of them. A connection on an
R-S-bimodule is defined as a bimodule
morphism
which obeys the Leibniz rule
See also
Connection (vector bundle)
Connection (mathematics)
Noncommutative geometry
Supergeometry
Differential calculus over commutative algebras
Notes
References
External links
Connection (mathematics)
Noncommutative geometry |
https://en.wikipedia.org/wiki/Eberhard%20Becker | Eberhard Becker (born July 23, 1943 in Stavenhagen) is a German mathematician whose career was spent at the University of Dortmund. A very active researcher in algebra, he later became rector of the university there. During his term as rector, it was renamed the Technical University of Dortmund.
Education and career
Becker received his Ph.D. at the University of Hamburg in 1972 with the dissertation, "Contributions to the theory of semi-simple quadratic algebra" under advisor Hel Braun. He completed his habilitation at the University of Cologne in 1976. In 1979 Becker was appointed to the mathematics department at the University of Dortmund.
His research included work in the algebraic theory of quadratic forms and real algebraic geometry. His collaborators included Manfred Knebusch and Alex F. T. W. Rosenberg. He supervised over a dozen doctoral students, including Markus Schweighofer, Susanne Pumplün, and Thorsten Wörmann.
In the mid 1980s, Becker proved that the expression (1 + )/(2 + ) was a sum of 4th powers, 6th powers, 8th powers and so on. He offered a bottle of champaign to anyone who could find explicit representations of this form. Bruce Reznick came up with a solution in 1994.
After working as institute director, dean and member of the Senate at the University of Dortmund, Becker became rector there on April 30, 2002 following the resignation of Hans-Jürgen Klein. During his term in office the Senate decided, at his request, to change the name from “University of Dortmund” to “Technical University of Dortmund”. This decision was not without controversy, particularly in the humanities departments, but was confirmed at the crucial Senate meeting by two-thirds of the vote. On September 1, 2008 he was replaced by Ursula Gather.
On the occasion of his retirement on November 7, 2008, the faculty organized a celebratory colloquium for him. To mark his 80th birthday, a conference on Quadratic Forms and Real Algebra was held in October 2023 at the University of Dortmund.
References
External links
Eberhard Becker at the Mathematics Genealogy Project
Papers by Eberhard Becker 1994 - 2005
20th-century German mathematicians
University of Hamburg alumni
University of Cologne alumni
Academic staff of the Technical University of Dortmund
Algebraists
1943 births
Living people |
https://en.wikipedia.org/wiki/Buchholz%27s%20ordinal | In mathematics, ψ0(Ωω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem -CA0 of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of , the theory of finitely iterated inductive definitions, and of , a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set. Buchholz's ordinal is also the order type of the segment bounded by in Buchholz's ordinal notation . Lastly, it can be expressed as the limit of the sequence: , , , ...
Definition
, and for n > 0.
is the closure of under addition and the function itself (the latter of which only for and ).
is the smallest ordinal not in .
Thus, ψ0(Ωω) is the smallest ordinal not in the closure of under addition and the function itself (the latter of which only for and ).
References
G. Takeuti, Proof theory, 2nd edition 1987
K. Schütte, Proof theory, Springer 1977
Ordinal numbers
Proof theory |
https://en.wikipedia.org/wiki/Balanced%20boolean%20function | In mathematics and computer science, a balanced boolean function is a boolean function whose output yields as many 0s as 1s over its input set. This means that for a uniformly random input string of bits, the probability of getting a 1 is 1/2.
Examples of balanced boolean functions are the function that copies the first bit of its input to the output,
and the function that produces the exclusive or of the input bits.
Usage
Balanced boolean functions are primarily used in cryptography. If a function is not balanced, it will have a statistical bias, making it subject to cryptanalysis such as the correlation attack.
See also
Bent function
References
Balanced boolean functions that can be evaluated so that every input bit is unlikely to be read, Annual ACM Symposium on Theory of Computing
Boolean algebra |
https://en.wikipedia.org/wiki/Martina%20Hingis%20career%20statistics | This is a list of the main career statistics of Swiss former professional tennis player Martina Hingis.
Significant finals
Grand Slam finals
Singles: 12 finals (5 titles, 7 runner-ups)
Doubles: 16 finals (13 titles, 3 runner-ups)
By winning the 1998 US Open title, Hingis completed the doubles Career Grand Slam. She became the 17th female player in history to achieve this.
Mixed doubles: 7 finals (7 titles)
By winning the 2016 French Open title, Hingis completed the mixed doubles Career Grand Slam. She became the 7th female player in history to achieve this.
Olympics
Doubles (1 silver medal)
WTA Tour Championships
Singles: 4 finals (2 titles, 2 runner-ups)
(i) = Indoor
Doubles: 3 finals (3 titles)
Tier I / Premier Mandatory & Premier 5 finals
Singles: 27 finals (17 titles, 10 runner-ups)
Doubles: 35 (26 titles, 9 runner-ups)
WTA finals
Singles: 69 (43 titles, 25 runner-ups)
Doubles: 86 (64 titles, 22 runner-ups)
Team competition: 3 (1 title, 2 runner-ups)
ITF Circuit finals
Singles: 3 finals (2 titles, 1 runner-up)
Doubles: 1 final (1 title)
Junior Grand Slam tournament finals
Singles: 4 finals (3 titles, 1 runner-up)
Doubles: 1 final (1 title)
Performance timelines
Singles
1 Including Fed Cup 2015 (0–2); Fed Cup record at least World Group II: 10–2
2 If ITF women's circuit (Hardcourt: 12–2; Carpet: 6–1) and Fed Cup (10–2) participations are included, overall win–loss record stands at 548–135.
Doubles
Only Main Draw results in WTA Tour, Grand Slam Tournaments and Olympic Games (no Fed Cup) are included in win–loss records.
This table is current through the 2017 WTA Finals.
Mixed doubles
Fed Cup
Levels of Fed Cup in which Switzerland did not compete in a particular year are marked "Not Participating" or "NP".
WTA tour career earnings
*: As of: 20 March 2017
Career Grand Slam seedings
Head-to-head vs. top 10 ranked players
Top 10 wins
Longest winning streaks
37-match win streak (1997)
26-match Grand Slam win streak (1997–98)
External links
Hingis, Martina |
https://en.wikipedia.org/wiki/Kansas%20Academy%20of%20Mathematics%20and%20Science | The Kansas Academy of Mathematics and Science (KAMS) is a two-year, residential, early-entrance-to-college program for U.S. high school juniors and seniors who are academically talented in the areas of mathematics and science. Located on the Fort Hays State University campus in Hays, Kansas, students concurrently complete their last two years of high school, while earning over 60 college credits.
History
The Kansas Academy of Mathematics and Science was established by legislative action in 2006 by the Kansas Legislature. The establishment of this program stemmed from national concern regarding anticipated shortages of students who would be sufficiently well prepared in mathematical and scientific problem solving. Recognizing that American youth would need to compete in an increasingly technological global society, Kansas is the 16th state to create alternative educational programs that would attract students to the fields of mathematics and science as well as offer young students an accelerated education in these areas of study.
Fort Hays State University, part of the Kansas Board of Regents system, was chosen to host the Academy after an extensive bid process. After finding a home, a planning committee worked to develop and shape the Academy including securing resources, hiring staff, promoting the Academy throughout the state, and recruiting students. In August 2009, KAMS opened its doors to the first class of 26 students.
Academics
Admissions process
Information Sessions throughout the state and a Preview Day on the FHSU campus occur in the fall, and completed applications are asked to be completed by December 15. KAMS uses criteria such as ACT/SAT scores, cumulative GPA, class rank, teacher evaluations, personal interviews, essays, and short answer questions in its admissions process to select high school sophomores who are most likely to succeed in an academically challenging environment. KAMS currently selects approximately 40 Kansas students per class.
Curriculum
KAMS students are expected to earn over 60 college credit hours over the two-year academic program. Students must complete both years to successfully complete the program. Courses are taught by doctoral level professors at Fort Hays State University. Courses include the subject areas of calculus, geometry, chemistry, physics, biology, computer science, English/communication, history, leadership, and a unique global climate change course that all students take together. Students may also be able to take elective courses (such as band, music, theater, foreign language, etc.) as long as the required core curriculum is met.
Research requirement
All KAMS students are required to engage in a research project to successfully complete the program. Students are exposed to various research opportunities both on the FHSU campus and nationwide. Students select their topic and are assigned a mentor in the spring of their junior year. Work is completed throughout the rest of their KAM |
https://en.wikipedia.org/wiki/Euler%27s%20differential%20equation | In mathematics, Euler's differential equation is a first-order non-linear ordinary differential equation, named after Leonhard Euler. It is given by:
This is a separable equation and the solution is given by the following integral equation:
References
Eponymous equations of physics
Mathematical physics
Differential equations
Ordinary differential equations
Leonhard Euler |
https://en.wikipedia.org/wiki/Polyhedral%20graph | In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graph-theoretic terms, the polyhedral graphs are the 3-vertex-connected, planar graphs.
Characterization
The Schlegel diagram of a convex polyhedron represents its vertices and edges as points and line segments in the Euclidean plane, forming a subdivision of an outer convex polygon into smaller convex polygons (a convex drawing of the graph of the polyhedron). It has no crossings, so every polyhedral graph is also a planar graph. Additionally, by Balinski's theorem, it is a 3-vertex-connected graph.
According to Steinitz's theorem, these two graph-theoretic properties are enough to completely characterize the polyhedral graphs: they are exactly the 3-vertex-connected planar graphs. That is, whenever a graph is both planar and 3-vertex-connected, there exists a polyhedron whose vertices and edges form an isomorphic graph. Given such a graph, a representation of it as a subdivision of a convex polygon into smaller convex polygons may be found using the Tutte embedding.
Hamiltonicity and shortness
Tait conjectured that every cubic polyhedral graph (that is, a polyhedral graph in which each vertex is incident to exactly three edges) has a Hamiltonian cycle, but this conjecture was disproved by a counterexample of W. T. Tutte, the polyhedral but non-Hamiltonian Tutte graph. If one relaxes the requirement that the graph be cubic, there are much smaller non-Hamiltonian polyhedral graphs. The graph with the fewest vertices and edges is the 11-vertex and 18-edge Herschel graph, and there also exists an 11-vertex non-Hamiltonian polyhedral graph in which all faces are triangles, the Goldner–Harary graph.
More strongly, there exists a constant (the shortness exponent) and an infinite family of polyhedral graphs such that the length of the longest simple path of an -vertex graph in the family is .
Combinatorial enumeration
Duijvestijn provides a count of the polyhedral graphs with up to 26 edges; The number of these graphs with 6, 7, 8, ... edges is
1, 0, 1, 2, 2, 4, 12, 22, 58, 158, 448, 1342, 4199, 13384, 43708, 144810, ... .
One may also enumerate the polyhedral graphs by their numbers of vertices: for graphs with 4, 5, 6, ... vertices, the number of polyhedral graphs is
1, 2, 7, 34, 257, 2606, 32300, 440564, 6384634, 96262938, 1496225352, ... .
Special cases
A polyhedral graph is the graph of a simple polyhedron if it is cubic (every vertex has three edges), and it is the graph of a simplicial polyhedron if it is a maximal planar graph. The Halin graphs, graphs formed from a planar embedded tree by adding an outer cycle connecting all of the leaves of the tree, form another important subclass of the polyhedral graphs.
References
External links
Geometric graphs
Planar graphs |
https://en.wikipedia.org/wiki/Karin%20Erdmann | Karin Erdmann (born 1948) is a German mathematician specializing in the areas of algebra known as representation theory (especially modular representation theory) and homological algebra (especially Hochschild cohomology). She is notable for her work in modular representation theory which has been cited over 1500 times according to the Mathematical Reviews. Her nephew Martin Erdmann is professor for experimental particle physics at the RWTH Aachen University.
Education
She attended the Justus-Liebig-Universität Gießen and wrote her Ph.D. thesis on "2-Hauptblöcke von Gruppen mit Dieder-Gruppen als 2-Sylow-Gruppen" (Principal 2-blocks of groups with dihedral Sylow 2-subgroups) in 1976 under the direction of Gerhard O. Michler.
Professional career
Erdmann was a Fellow of Somerville College, Oxford. Erdmann is a university lecturer emeritus at the Mathematical Institute at the University of Oxford where she has had 25 doctoral students and 45 descendants. She has published over 115 papers and her work has been cited over 2000 times. She has contributed to the understanding of the representation theory of the symmetric group.
Honors
Erdmann was the inaugural Emmy Noether Lecturer of the German Mathematical Society in 2008.
Selected bibliography
References
External links
Mathematical Reviews author profile
Home page at Oxford
Living people
1948 births
20th-century German mathematicians
Women mathematicians
Fellows of Somerville College, Oxford
University of Giessen alumni
Algebraists
21st-century German mathematicians |
https://en.wikipedia.org/wiki/Effective%20Polish%20space | In mathematical logic, an effective Polish space is a complete separable metric space that has a computable presentation. Such spaces are studied in effective descriptive set theory and in constructive analysis. In particular, standard examples of Polish spaces such as the real line, the Cantor set and the Baire space are all effective Polish spaces.
Definition
An effective Polish space is a complete separable metric space X with metric d such that there is a countable dense set C = (c0, c1,...) that makes the following two relations on computable (Moschovakis 2009:96-7):
References
Yiannis N. Moschovakis, 2009, Descriptive Set Theory, 2nd edition, American Mathematical Society.
Computable analysis
Effective descriptive set theory
Computability theory |
https://en.wikipedia.org/wiki/Logicomix | Logicomix: An Epic Search for Truth is a graphic novel about the foundational quest in mathematics, written by Apostolos Doxiadis, author of Uncle Petros and Goldbach's Conjecture, and at the time Berkeley's theoretical computer scientist Christos Papadimitriou. Character design and artwork are by Alecos Papadatos and color is by Annie Di Donna. The book was originally written in English, and was translated into Greek by author Apostolos Doxiadis for the release in Greece, which preceded the UK and U.S. releases.
Plot
Set between the late 19th century and the present day, the graphic novel Logicomix is based on the story of the so-called "foundational quest" in mathematics.
Logicomix intertwines the philosophical struggles with the characters' own personal turmoil. These are in turn played out just upstage of the momentous historical events of the era and the ideological battles which gave rise to them. The narrator of the story is Bertrand Russell, who stands as an icon of many of these themes: a deeply sensitive and introspective man, Russell was not just a philosopher and pacifist, he was also one of the prominent figures in the foundational quest. Russell's life story, depicted by Logicomix, is itself a journey through the goals and struggles, and triumph and tragedy shared by many great thinkers of the 20th century: Georg Cantor, Ludwig Wittgenstein, G. E. Moore, Alfred North Whitehead, David Hilbert, Gottlob Frege, Henri Poincaré, Kurt Gödel, and Alan Turing.
A parallel tale, set in present-day Athens, records the creators’ disagreement on the meaning of the story, thus setting in relief the foundational quest as a quintessentially modern adventure. It is on the one hand a tragedy of the hubris of rationalism, which descends inextricably on madness, and on the other an origin myth of the computer.
Releases
In chronological order:
Greece – October 20, 2008, Ikaros Publications,
Netherlands – August 15, 2009, De Vliegende Hollander,
United Kingdom – September 7, 2009, Bloomsbury,
United States – September 29, 2009, Bloomsbury USA,
France – May 10, 2010, Vuibert,
Italy – June 10, 2010, Guanda,
Germany – August 30, 2010, Atrium-Verlag,
Finland – September 10, 2010, Avain,
Brazil – 2010, Martins Fontes,
Croatia (Stripologikon) - 2010, Logicomix Print Ltd. / Mate d.o.o.,
Spain (Logicomix. Una búsqueda épica de la verdad) – March 24, 2011, Ediciones Sins Entido,
Norway – 2010, Arneberg,
Poland – November 2011, W.A.B.,
Denmark – February 2012, Politisk Revy,
Czech Republic – September 2012, Dokořán,
Turkey – October 2012, Albatros Kitap,
Russia – March 2014, Kar'era Press,
Iran – March 2014, Fatemi Publishing Co.,
Israel – 2016, Aliyat Hagag press,
Reception
Jim Holt reviewed the book for the New York Times and says the story "is presented with real graphic verve. (Even though I’m a text guy, I couldn’t keep my eyes off the witty drawings.)" although he does note "one serious misstep" involving the overplaying of the im |
https://en.wikipedia.org/wiki/Project%20Mathematics%21 | Project Mathematics! (stylized as Project MATHEMATICS!), is a series of educational video modules and accompanying workbooks for teachers, developed at the California Institute of Technology to help teach basic principles of mathematics to high school students. In 2017, the entire series of videos was made available on YouTube.
Overview
The Project Mathematics! series of videos is a teaching aid for teachers to help students understand the basics of geometry and trigonometry. The series was developed by Tom M. Apostol and James F. Blinn, both from the California Institute of Technology. Apostol led the production of the series, while Blinn provided the computer animation used to depict the ideas beings discussed. Blinn mentioned that part of his inspiration was the Bell science series of films from the 1950s.
The material was designed for teachers to use in their curriculums and was aimed at grades 8 through 13. Workbooks are also available to accompany the videos and to assist teachers in presenting the material to their students. The videos are distributed as either 9 VHS videotapes or 3 DVDs, and include a history of mathematics and examples of how math is used in real world applications.
Video module descriptions
A total of nine educational video modules were created between 1988 and 2000. Another two modules, Teachers Workshop and Project MATHEMATICS! Contest, were created in 1991 for teachers and are only available on videotape. The content of the nine educational modules follows below.
The Theorem of Pythagoras
In 1988, The Theorem of Pythagoras was the first video produced by the series and reviews the Pythagorean theorem. For all right triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a2 + b2 = c2). The theorem is named after Pythagoras of ancient Greece. Pythagorean triples occur when all three sides of a right triangle are integers such as a = 3, b = 4 and c = 5. A clay tablet shows that the Babylonians knew of Pythagorean triples 1200 years before Pythagoras, but nobody knows if they knew the more-general Pythagorean theorem. The Chinese proof uses four similar triangles to prove the theorem.
Today, we know of the Pythagorean theorem because of Euclid's Elements, a set of 13 books on mathematics—from around 300 BCE—and the knowledge it contained has been used for more than 2000 years. Euclid's proof is described in book 1, proposition 47 and uses the idea of equal areas along with shearing and rotating triangles. In the dissection proof, the square of the hypotenuse is cut into pieces to fit into the other two squares. Proposition 31 in book 6 of Euclid's Elements describes the similarity proof, which states that the squares of each side can be replaced by shapes that are similar to each other and the proof still works.
The Story of Pi
The second module created was The Story of Pi, in 1989, and describes the mathematical constant pi and its history. The first letter in the Gree |
https://en.wikipedia.org/wiki/David%20Dickey | David Alan Dickey (born c. 1945) is an American statistician who has specialised in time series analysis. He is a William Neal Reynolds Professor in the Department of Statistics at North Carolina State University. The Dickey–Fuller test is named for him and Wayne Arthur Fuller.
David Dickey is listed as an ISI highly cited researcher by the ISI Highly Cited Database of the ISI Web of Knowledge. He is an elected Fellow (2000) of the American Statistical Association. He is from Ohio.
Selected works
References
External links
Home page at North Carolina State University
1940s births
Living people
Academics from Ohio
Miami University alumni
Iowa State University alumni
North Carolina State University faculty
American statisticians
Time series econometricians
Fellows of the American Statistical Association |
https://en.wikipedia.org/wiki/Complex%20normal%20distribution | In probability theory, the family of complex normal distributions, denoted or , characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: location parameter μ, covariance matrix , and the relation matrix . The standard complex normal is the univariate distribution with , , and .
An important subclass of complex normal family is called the circularly-symmetric (central) complex normal and corresponds to the case of zero relation matrix and zero mean: and . This case is used extensively in signal processing, where it is sometimes referred to as just complex normal in the literature.
Definitions
Complex standard normal random variable
The standard complex normal random variable or standard complex Gaussian random variable is a complex random variable whose real and imaginary parts are independent normally distributed random variables with mean zero and variance . Formally,
where denotes that is a standard complex normal random variable.
Complex normal random variable
Suppose and are real random variables such that is a 2-dimensional normal random vector. Then the complex random variable is called complex normal random variable or complex Gaussian random variable.
Complex standard normal random vector
A n-dimensional complex random vector is a complex standard normal random vector or complex standard Gaussian random vector if its components are independent and all of them are standard complex normal random variables as defined above.
That is a standard complex normal random vector is denoted .
Complex normal random vector
If and are random vectors in such that is a normal random vector with components. Then we say that the complex random vector
is a complex normal random vector or a complex Gaussian random vector.
Mean, covariance, and relation
The complex Gaussian distribution can be described with 3 parameters:
where denotes matrix transpose of , and denotes conjugate transpose.
Here the location parameter is a n-dimensional complex vector; the covariance matrix is Hermitian and non-negative definite; and, the relation matrix or pseudo-covariance matrix is symmetric. The complex normal random vector can now be denoted asMoreover, matrices and are such that the matrix
is also non-negative definite where denotes the complex conjugate of .
Relationships between covariance matrices
As for any complex random vector, the matrices and can be related to the covariance matrices of and via expressions
and conversely
Density function
The probability density function for complex normal distribution can be computed as
where and .
Characteristic function
The characteristic function of complex normal distribution is given by
where the argument is an n-dimensional complex vector.
Properties
If is a complex normal n-vector, an m×n matrix, and a constant m-vector, then the linear transform will be distributed also complex-n |
https://en.wikipedia.org/wiki/Herschel%20graph | In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges. It is a polyhedral graph (the graph of a convex polyhedron), and is the smallest polyhedral graph that does not have a Hamiltonian cycle, a cycle passing through all its vertices. It is named after British astronomer Alexander Stewart Herschel, because of Herschel's studies of Hamiltonian cycles in polyhedral graphs (but not of this graph).
Definition and properties
The Herschel graph has three vertices of degree four (the three blue vertices aligned vertically in the center of the illustration) and eight vertices of degree three. Each two distinct degree-four vertices share two degree-three neighbors, forming a four-vertex cycle with these shared neighbors. There are three of these cycles, passing through six of the eight degree-three vertices (red in the illustration). Two more degree-three vertices (blue) do not participate in these four-vertex cycles; instead, each is adjacent to three of the six red vertices.
The Herschel graph is a polyhedral graph; this means that it is a planar graph, one that can be drawn in the plane with none of its edges crossing, and that it is 3-vertex-connected: the removal of any two of its vertices leaves a connected subgraph. It is a bipartite graph: when it is colored with five blue and six red vertices, as illustrated, each edge has one red endpoint and one blue endpoint.
It has order-6 dihedral symmetry, for a total of 12 members of its automorphism group. The degree-four vertices can be permuted arbitrarily, giving six permutations, and in addition, for each permutation of the degree-four vertices, there is a symmetry that keeps these vertices fixed and exchanges pairs of degree-three vertices.
Polyhedron
By Steinitz's theorem, every graph that is planar and 3-vertex-connected has a convex polyhedron with the graph as its skeleton. Because the Herschel graph has these properties, it can be represented in this way by a convex polyhedron, an enneahedron having polyhedron has nine quadrilaterals as its faces. This can be chosen so that each graph automorphism corresponds to a symmetry of the polyhedron, in which case three of the faces will be rhombi or squares, and the other six will be kites.
The dual polyhedron is a rectified triangular prism, which can be formed as the convex hull of the midpoints of the edges of a triangular prism. When constructed in this way, it has three square faces on the same planes as the square faces of the prism, two equilateral triangle faces on the planes of the triangular ends of the prism, and six more isosceles triangle faces. This polyhedron has the property that its faces cannot be numbered in such a way that consecutive numbers appear on adjacent faces, and such that the first and last numbers are also on adjacent faces, because such a numbering would necessarily correspond to a Hamiltonian cycle in the Herschel graph. Polyhedral face numberings |
https://en.wikipedia.org/wiki/Schreier%20coset%20graph | In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set S={si : i in I } of G, and a subgroup H ≤ G. The Schreier graph encodes the abstract structure of a group modulo an equivalence relation formed by the coset.
The graph is named after Otto Schreier, who used the term “Nebengruppenbild”. An equivalent definition was made in an early paper of Todd and Coxeter.
Description
The Schreier graph of a group G, a subgroup H, and a generating set S⊆G is denoted by Sch(G,H,S) or Sch(H\G,S). Its vertices are the right cosets Hg = {hg : h in H } for g in G, and its edges are of the form (Hg, Hgs) for g in G and s in S.
More generally, if X is a G-set, the Schreier graph of the action of G on X (with respect to S⊆G) is denoted by Sch(G,X,S) or Sch(X,S). Its vertices are the elements of X, and its edges are of the form (x,xs) for x in X and s in S. This includes the original Schreier coset graph definition, as H\G is a naturally a G-set with respect to multiplication from the right. From an algebraic-topological perspective, the graph Sch(X,S) has no distinguished vertex, whereas Sch(G,H,S) has the distinguished vertex H, and is thus a pointed graph.
The Cayley graph of the group G itself is the Schreier coset graph for H = {1G} .
A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma .
The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids. A subgroup H of a group G determines a covering morphism of groupoids and if S is a generating set for G then its inverse image under p is the Schreier graph of (G, S).
Applications
The graph is useful to understand coset enumeration and the Todd–Coxeter algorithm.
Coset graphs can be used to form large permutation representations of groups and were used by Graham Higman to show that the alternating groups of large enough degree are Hurwitz groups, .
Stallings' core graphs are retracts of Schreier graphs of free groups, and are an essential tool for computing with subgroups of a free group.
Every vertex-transitive graph is a coset graph.
References
Schreier graphs of the Basilica group Authors: Daniele D'Angeli, Alfredo Donno, Michel Matter, Tatiana Nagnibeda
Philip J. Higgins, Categories and Groupoids, van Nostrand, New York, Lecture Notes, 1971, Republished as TAC Reprint, 2005
Combinatorial group theory |
https://en.wikipedia.org/wiki/Special%20conformal%20transformation | In projective geometry, a special conformal transformation is a linear fractional transformation that is not an affine transformation. Thus the generation of a special conformal transformation involves use of multiplicative inversion, which is the generator of linear fractional transformations that is not affine.
In mathematical physics, certain conformal maps known as spherical wave transformations are special conformal transformations.
Vector presentation
A special conformal transformation can be written
It is a composition of an inversion (xμ → xμ/x2 = yμ), a translation (yμ → yμ − bμ = zμ), and another inversion (zμ → zμ/z2 = x′μ)
Its infinitesimal generator is
Special conformal transformations have been used to study the force field of an electric charge in hyperbolic motion.
Projective presentation
The inversion can also be taken to be multiplicative inversion of biquaternions B. The complex algebra B can be extended to P(B) through the projective line over a ring. Homographies on P(B) include translations:
The homography group G(B) includes of translations at infinity with respect to the embedding q → U(q:1);
The matrix describes the action of a special conformal transformation.
Group property
The translations form a subgroup of the linear fractional group acting on a projective line. There are two embeddings into the projective line of homogeneous coordinates: z → [z:1] and z → [1:z]. An addition operation corresponds to a translation in the first embedding. The translations to the second embedding are special conformal transformations, forming translations at infinity. Addition by these transformations reciprocates the terms before addition, then returns the result by another reciprocation. This operation is called the parallel operation. In the case of the complex plane the parallel operator forms an addition operation in an alternative field using infinity but excluding zero. The translations at infinity thus form another subgroup of the homography group on the projective line.
References
Projective geometry
Conformal mappings
Conformal field theory |
https://en.wikipedia.org/wiki/Janet%20Siefert | Janet Louise Siefert received her PhD from the University of Houston (1997) and is an Associate Research Professor in the Department of Statistics at Rice University since January, 1998. She is an origin of life researcher who received her training from Dr. George E. Fox, co-discoverer of the third domain of life.
Siefert's work deals with understanding life's origins through microbial ecology and genomic and metabolic evolution. She has worked for 20 years in the Cuatro Ciénegas Basin, Mexico, a desert oasis protected by the Mexican government and known to exhibit endemism equal to the Galapagos.
She is the first woman to serve as chairman of the Gordon Research Conference on the Origin of Life and the first female president (2008–2011) of ISSOL: International Astrobiology Society. She is a member of University of Washington's Virtual Planet Laboratory and was recognized as an Astrobiology Pioneer. Her work has been featured on Discovery, National Geographic, and the History Channel. Additionally, she heads up an international team of investigators funded by the National Science Foundation looking at the legacy of human occupation in archaeological sites as reflected in the microbial community.
References
External links
21st-century American biologists
Astrobiologists
American women biologists
Living people
Rice University fellows
University of Houston alumni
People from Cuatro Ciénegas
21st-century American women scientists
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Multiple%20correspondence%20analysis | In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables.
As an extension of correspondence analysis
MCA is performed by applying the CA algorithm to either an indicator matrix (also called complete disjunctive table – CDT) or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Analyzing the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display.
In the indicator matrix approach, associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). These associations are then represented graphically as "maps", which eases the interpretation of the structures in the data. Oppositions between rows and columns are then maximized, in order to uncover the underlying dimensions best able to describe the central oppositions in the data. As in factor analysis or principal component analysis, the first axis is the most important dimension, the second axis the second most important, and so on, in terms of the amount of variance accounted for. The number of axes to be retained for analysis is determined by calculating modified eigenvalues.
Details
Since MCA is adapted to make statistical conclusion out of categorical variables (such as multiple choices questions), the first thing one needs to do is to transform quantitative data (such as age, size, weight, day time, etc) into categories (using for instance statistical quantiles).
When the dataset is completely represented as categorical variables, one is able to build the corresponding so called completely disjunctive table. We denote this table . If persons answered a survey with multiple choices questions with 4 answers each, will have rows and columns.
More theoretically, assume is the completely disjunctive table of observations of categorical variables. Assume also that the -th variable have different levels (categories) and set . The table is then a matrix with all coe |
https://en.wikipedia.org/wiki/Exploratory%20factor%20analysis | In multivariate statistics, exploratory factor analysis (EFA) is a statistical method used to uncover the underlying structure of a relatively large set of variables. EFA is a technique within factor analysis whose overarching goal is to identify the underlying relationships between measured variables. It is commonly used by researchers when developing a scale (a scale is a collection of questions used to measure a particular research topic) and serves to identify a set of latent constructs underlying a battery of measured variables. It should be used when the researcher has no a priori hypothesis about factors or patterns of measured variables. Measured variables are any one of several attributes of people that may be observed and measured. Examples of measured variables could be the physical height, weight, and pulse rate of a human being. Usually, researchers would have a large number of measured variables, which are assumed to be related to a smaller number of "unobserved" factors. Researchers must carefully consider the number of measured variables to include in the analysis. EFA procedures are more accurate when each factor is represented by multiple measured variables in the analysis.
EFA is based on the common factor model. In this model, manifest variables are expressed as a function of common factors, unique factors, and errors of measurement. Each unique factor influences only one manifest variable, and does not explain correlations between manifest variables. Common factors influence more than one manifest variable and "factor loadings" are measures of the influence of a common factor on a manifest variable. For the EFA procedure, we are more interested in identifying the common factors and the related manifest variables.
EFA assumes that any indicator/measured variable may be associated with any factor. When developing a scale, researchers should use EFA first before moving on to confirmatory factor analysis (CFA). EFA is essential to determine underlying factors/constructs for a set of measured variables; while CFA allows the researcher to test the hypothesis that a relationship between the observed variables and their underlying latent exists.
EFA requires the researcher to make a number of important decisions about how to conduct the analysis because there is no one set method.
Fitting procedures
Fitting procedures are used to estimate the factor loadings and unique variances of the model (Factor loadings are the regression coefficients between items and factors and measure the influence of a common factor on a measured variable). There are several factor analysis fitting methods to choose from, however there is little information on all of their strengths and weaknesses and many don't even have an exact name that is used consistently. Principal axis factoring (PAF) and maximum likelihood (ML) are two extraction methods that are generally recommended. In general, ML or PAF give the best results, depending on whether data are |
https://en.wikipedia.org/wiki/Bessel%20potential | In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity.
If s is a complex number with positive real part then the Bessel potential of order s is the operator
where Δ is the Laplace operator and the fractional power is defined using Fourier transforms.
Yukawa potentials are particular cases of Bessel potentials for in the 3-dimensional space.
Representation in Fourier space
The Bessel potential acts by multiplication on the Fourier transforms: for each
Integral representations
When , the Bessel potential on can be represented by
where the Bessel kernel is defined for by the integral formula
Here denotes the Gamma function.
The Bessel kernel can also be represented for by
This last expression can be more succinctly written in terms of a modified Bessel function, for which the potential gets its name:
Asymptotics
At the origin, one has as ,
In particular, when the Bessel potential behaves asymptotically as the Riesz potential.
At infinity, one has, as ,
See also
Riesz potential
Fractional integration
Sobolev space
Fractional Schrödinger equation
Yukawa potential
References
Fractional calculus
Partial differential equations
Potential theory
Singular integrals |
https://en.wikipedia.org/wiki/T%281%29%20theorem | In mathematics, the T(1) theorem, first proved by , describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L2(Rn). The name T(1) theorem refers to a condition on the distribution T(1), given by the operator T applied to the function 1.
Statement
Suppose that T is a continuous operator from Schwartz functions on Rn to tempered distributions, so that T is given by a kernel K which is a distribution. Assume that the kernel is standard, which means that off the diagonal it is given by a function satisfying certain conditions.
Then the T(1) theorem states that T can be extended to a bounded operator on the Hilbert space L2(Rn) if and only if the following conditions are satisfied:
T(1) is of bounded mean oscillation (where T is extended to an operator on bounded smooth functions, such as 1).
T*(1) is of bounded mean oscillation, where T* is the adjoint of T.
T is weakly bounded, a weak condition that is easy to verify in practice.
References
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Tesla%20valve | A Tesla valve, called a valvular conduit by its inventor, is a fixed-geometry passive check valve. It allows a fluid to flow preferentially in one direction, without moving parts. The device is named after Nikola Tesla, who was awarded in 1920 for its invention. The patent application describes the invention as follows:
The interior of the conduit is provided with enlargements, recesses, projections, baffles, or buckets which, while offering virtually no resistance to the passage of the fluid in one direction, other than surface friction, constitute an almost impassable barrier to its flow in the opposite direction.
Tesla illustrated this with the drawing, showing one possible construction with a series of eleven flow-control segments, although any other number of such segments could be used as desired to increase or decrease the flow regulation effect.
With no moving parts, Tesla valves are much more resistant to wear and fatigue, especially in applications with frequent pressure reversal such as a pulsejet.
The Tesla valve is used in microfluidic applications and offers advantages such as scalability, durability, and ease of fabrication in a variety of materials. It is also used in macrofluidic applications and pulse jet engines.
One computational fluid dynamics simulation of Tesla valves with two and four segments showed that the flow resistance in the blocking (or reverse) direction was about 15 and 40 times greater, respectively, than the unimpeded (or forward) direction. This lends support to Tesla's patent assertion that in the valvular conduit in his diagram, a pressure ratio "approximating 200 can be obtained so that the device acts as a slightly leaking valve".
Steady flow experiments, including with the original design, however, show smaller ratios of the two resistances in the range of 2 to 4. It has also been shown that the device works better with pulsatile flows.
Diodicity
The valves are structures that have a higher pressure drop for the flow in one direction (reverse) than the other (forward). This difference in flow resistance causes a net directional flow rate in the forward direction in oscillating flows. The efficiency is often expressed in diodicity , being the ratio of directional resistances.
The flow resistance is defined, analogously to Ohm's law for electrical resistance, as the ratio of applied pressure drop and resulted flow rate:
where is the applied pressure difference between two ends of the conduit, and the flow rate.
The diodicity is then the ratio of the reversed flow resistance to the forward flow resistance:
. If , the conduit in question has diodic behavior.
Thus diodicity is also the ratio of pressure drops for identical flow rates:
where is the reverse flow pressure drop, and the forward flow pressure drop for flow rate .
Equivalently, diodicity could also be defined as ratio of dimensionless Hagen number or Darcy friction factor at the same Reynolds number.
See also
Coandă effect
Check |
https://en.wikipedia.org/wiki/Mikhail%20Menshikov | Mikhail Vasilyevich Menshikov (; born January 17, 1948) is a Russian-British mathematician with publications in areas ranging from probability to combinatorics. He currently holds the post of Professor in the University of Durham.
Menshikov has made a substantial contribution to percolation theory and the theory of random walks.
Menshikov was born in Moscow and went to school in Kharkov, Ukrainian SSR, Soviet Union. He studied at Moscow State University earning all his degrees up to Candidate of Sciences (1976) and Doctor of Sciences (1988). After briefly working in Zhukovsky, Menshikov worked in Moscow State University for many years. His career then took him to the University of Sao Paulo before becoming a professor at the University of Durham, where he currently lives.
External links
Personal webpage
20th-century British mathematicians
21st-century British mathematicians
Russian mathematicians
Probability theorists
1948 births
Living people
Soviet mathematicians
Academics of Durham University |
https://en.wikipedia.org/wiki/Group-based%20cryptography | Group-based cryptography is a use of groups to construct cryptographic primitives. A group is a very general algebraic object and most cryptographic schemes use groups in some way. In particular Diffie–Hellman key exchange uses finite cyclic groups. So the term group-based cryptography refers mostly to cryptographic protocols that use infinite non-abelian groups such as a braid group.
Examples
Shpilrain–Zapata public-key protocols
Magyarik–Wagner public key protocol
Anshel–Anshel–Goldfeld key exchange
Ko–Lee et al. key exchange protocol
See also
Non-commutative cryptography
References
Further reading
Paul, Kamakhya; Goswami, Pinkimani; Singh, Madan Mohan. (2022). "ALGEBRAIC BRAID GROUP PUBLIC KEY CRYPTOGRAPHY", Jnanabha, Vol. 52(2) (2022), 218-223. ISSN 0304-9892 (Print) ISSN 2455-7463 (Online)
External links
Cryptography and Braid Groups page (archived version 7/17/2017)
Theory of cryptography
Braid groups |
https://en.wikipedia.org/wiki/Lemniscus | Lemniscus can refer to:
Lemniscus (anatomy)
In mathematics, a lemniscate |
https://en.wikipedia.org/wiki/Robert%20Rumely | Robert Scott Rumely (born 1952) is a professor of mathematics at the University of Georgia who specializes in number theory and arithmetic geometry. He is one of the inventors of the Adleman–Pomerance–Rumely primality test.
Life
Rumely was born on June 23, 1952, in Pullman, Washington. He graduated from Grinnell College in 1974, and completed his Ph.D. in 1978 at Princeton University under the supervision of Goro Shimura. After temporary positions at the Massachusetts Institute of Technology and Harvard University, he joined the University of Georgia faculty in 1981.
Rumely has taught a summer Research Experiences for Undergraduates program on the mathematics of paper folding.
Books
He is the author or co-author of four books:
Capacity Theory on Algebraic Curves (Lecture Notes in Mathematics 1378, 1989)
Existence of the Sectional Capacity (Memoirs of the American Mathematical Society 145, 2000)
Potential Theory and Dynamics on the Berkovich Projective Line (Mathematical Surveys and Monographs 159, 2010)
Capacity Theory with Local Rationality: The Strong Fekete-Szegö Theorem on Curves (Mathematical Surveys and Monographs 193, 2013)
Awards
In 2015 he was elected as a Fellow of the American Mathematical Society "for contributions to arithmetic potential theory, computational number theory, and arithmetic dynamics".
References
External links
20th-century American mathematicians
21st-century American mathematicians
University of Georgia faculty
Grinnell College alumni
Princeton University alumni
Fellows of the American Mathematical Society
1952 births
Living people
Massachusetts Institute of Technology School of Science faculty |
https://en.wikipedia.org/wiki/Relay%20network | A relay network is a broad class of network topology commonly used in wireless networks, where the source and destination are interconnected by means of some nodes. In such a network the source and destination cannot communicate to each other directly because the distance between the source and destination is greater than the transmission range of both of them, hence the need for intermediate node(s) to relay.
A relay network is a type of network used to send information between two devices, for e.g. server and computer, that are too far away to send the information to each other directly. Thus the network must send or "relay" the information to different devices, referred to as nodes, that pass on the information to its destination. A well-known example of a relay network is the Internet. A user can view a web page from a server halfway around the world by sending and receiving the information through a series of connected nodes.
In many ways, a relay network resembles a chain of people standing together. One person has a note he needs to pass to the girl at the end of the line. He is the sender, she is the recipient, and the people in between them are the messengers, or the nodes. He passes the message to the first node, or person, who passes it to the second and so on until it reaches the girl and she reads it.
The people might stand in a circle, however, instead of a line. Each person is close enough to reach the person on either side of him and across from him. Together the people represent a network and several messages can now pass around or through the network in different directions at once, as opposed to the straight line that could only run messages in a specific direction. This concept, the way a network is laid out and how it shares data, is known as network topology. Relay networks can use many different topologies, from a line to a ring to a tree shape, to pass along information in the fastest and most efficient way possible.
Often the relay network is complex and branches off in multiple directions to connect many servers and computers. Where two lines from two different computers or servers meet forms the nodes of the relay network. Two computer lines might run into the same router, for example, making this the node.
Wireless networks also take advantage of the relay network system. A laptop, for example, might connect to a wireless network which sends and receives information through another network and another until it reaches its destination. Even though not all parts of the network have physical wires, they still connect to other devices that function as the nodes.
This type of network holds several advantages. Information can travel long distances, even if the sender and receiver are far apart. It also speeds up data transmission by choosing the best path to travel between nodes to the receiver's computer. If one node is too busy, the information is simply routed to a different one. Without relay networks, sending an |
https://en.wikipedia.org/wiki/Gather/scatter%20%28vector%20addressing%29 | Gather/scatter is a type of memory addressing that at once collects (gathers) from, or stores (scatters) data to, multiple, arbitrary indices. Examples of its use include sparse linear algebra operations, sorting algorithms, fast Fourier transforms, and some computational graph theory problems. It is the vector equivalent of register indirect addressing, with gather involving indexed reads, and scatter, indexed writes. Vector processors (and some SIMD units in CPUs) have hardware support for gather and scatter operations, as do many input/output systems, allowing large data sets to be transferred to main memory more rapidly.
The concept is somewhat similar to vectored I/O, which is sometimes also referred to as scatter-gather I/O. This system differs in that it is used to map multiple sources of data from contiguous structures into a single stream for reading or writing. A common example is writing out a series of strings, which in most programming languages would be stored in separate memory locations.
Definitions
Gather
A sparsely populated vector holding non-empty elements can be represented by two densely populated vectors of length ; containing the non-empty elements of , and giving the index in where 's element is located. The gather of into , denoted , assigns with having already been calculated. Assuming no pointer aliasing between x[], y[],idx[], a C implementation is
for (i = 0; i < N; ++i)
x[i] = y[idx[i]];
Scatter
The sparse scatter, denoted is the reverse operation. It copies the values of into the corresponding locations in the sparsely populated vector , i.e. .
for (i = 0; i < N; ++i)
y[idx[i]] = x[i];
Support
Scatter/gather units were also a part of most vector computers, notably the Cray-1. In this case, the purpose was to efficiently store values in the limited resource of the vector registers. For instance, the Cray-1 had eight 64-word vector registers, so data that contained values that had no effect on the outcome, like zeros in an addition, were using up valuable space that would be better used. By gathering non-zero values into the registers, and scattering the results back out, the registers could be used much more efficiently, leading to higher performance. Such machines generally implemented two access models, scatter/gather and "stride", the latter designed to quickly load contiguous data. This basic layout was widely copied in later supercomputer designs, especially on the variety of models from Japan.
As microprocessor design improved during the 1990s, commodity CPUs began to add vector processing units. At first these tended to be simple, sometimes overlaying the CPU's general purpose registers, but over time these evolved into increasingly powerful systems that met and then surpassed the units in high-end supercomputers. By this time, scatter/gather instructions had been added to many of these designs.
x86-64 CPUs which support the AVX2 instruction set can gather 32-bit and 64-bit element |
https://en.wikipedia.org/wiki/Daniel%20Friedrich%20Hecht | Daniel Friedrich Hecht (8 July 1777 – 13 March 1833) was a German mathematician born in Sosa. He was a mine manager, then a teacher and finally a professor of mathematics. He is most notable for writing high school textbooks on math and geometry. He died in Saxony.
References
Further reading
M Koch, Biography in Dictionary of Scientific Biography (New York 1970–1990).
C Schiffner, Daniel Friedrich Hecht, Aus dem Leben alter Freiberger Bergstudenten I (Freiberg, 1935), 244–245.
1777 births
1833 deaths
People from Eibenstock
People from the Electorate of Saxony
German Lutherans
19th-century German mathematicians |
https://en.wikipedia.org/wiki/Exponentiated%20Weibull%20distribution | In statistics, the exponentiated Weibull family of probability distributions was introduced by Mudholkar and Srivastava (1993) as an extension of the Weibull family obtained by adding a second shape parameter.
The cumulative distribution function for the exponentiated Weibull distribution is
for x > 0, and F(x; k; λ; α) = 0 for x < 0. Here k > 0 is the first shape parameter, α > 0 is the second shape parameter and λ > 0 is the scale parameter of the distribution.
The density is
There are two important special cases:
α = 1 gives the Weibull distribution;
k = 1 gives the exponentiated exponential distribution.
Background
The family of distributions accommodates unimodal, bathtub shaped* and monotone failure rates. A similar distribution was introduced in 1984 by Zacks, called a Weibull-exponential distribution (Zacks 1984). Crevecoeur introduced it in assessing the reliability of ageing mechanical devices and showed that it accommodates bathtub shaped failure rates (1993, 1994). Mudholkar, Srivastava, and Kollia (1996) applied the generalized Weibull distribution to model survival data. They showed that the distribution has increasing, decreasing, bathtub, and unimodal hazard functions. Mudholkar, Srivastava, and Freimer (1995), Mudholkar and Hutson (1996) and Nassar and Eissa (2003) studied various properties of the exponentiated Weibull distribution. Mudholkar et al. (1995) applied the exponentiated Weibull distribution to model failure data. Mudholkar and Hutson (1996) applied the exponentiated Weibull distribution to extreme value data. They showed that the exponentiated Weibull distribution has increasing, decreasing, bathtub, and unimodal hazard rates. The exponentiated exponential distribution proposed by Gupta and Kundu (1999, 2001) is a special case of the exponentiated Weibull family. Later, the moments of the EW distribution were derived by Choudhury (2005). Also, M. Pal, M.M. Ali, J. Woo (2006) studied the EW distribution and compared it with the two-parameter Weibull and gamma distributions with respect to failure rate.
References
Further reading
Continuous distributions
Survival analysis |
https://en.wikipedia.org/wiki/Research%20Diagnostic%20Criteria | The Research Diagnostic Criteria (RDC) are a collection of influential psychiatric diagnostic criteria published in late 1970s under auspices of Statistics Section NY Psychiatric Institute, authors were Spitzer, R L; Endicott J; Robins E. PMID 1153649. As psychiatric diagnoses widely varied especially between the US and Europe, the purpose of the criteria was to allow diagnoses to be consistent in psychiatric research.
Some of the criteria were based on the earlier Feighner Criteria, although many new disorders were included; "The historical record shows that the small group of individuals who created the Feighner criteria instigated a paradigm shift that has had profound effects on the course of American and, ultimately, world psychiatry."
The RDC is important in the history of psychiatric diagnostic criteria as the DSM-III was based on many of the RDC descriptions, head of DSM III Edition was R L Spitzer.
See also
Diagnostic classification and rating scales used in psychiatry
Schedule for Affective Disorders and Schizophrenia
References
Psychiatric research
Classification of mental disorders |
https://en.wikipedia.org/wiki/Rufus%20Bowen | Robert Edward "Rufus" Bowen (23 February 1947 – 30 July 1978) was an internationally known professor in the Department of Mathematics at the University of California, Berkeley, who specialized in dynamical systems theory. Bowen's work dealt primarily with axiom A systems, but the methods he used while exploring topological entropy, symbolic dynamics, ergodic theory, Markov partitions, and invariant measures "have application far beyond the axiom A systems for which they were invented." The Bowen Lectures at the University of California, Berkeley, are given in his honor.
Life
Robert Edward Bowen was born in Vallejo, California, to Marie DeWinter Bowen, a school teacher, and Emery Bowen, a Travis Air Force Base budget officer, but he grew up fifteen miles away in Fairfield, California, where he attended the public schools and graduated from Armijo High School in 1964. His senior yearbook documents that he played two years of varsity basketball, was a member of the science, math, and language clubs, and was President of the senior class. During his first three years of high school, he finished 102nd, 7th, and 2nd among Californians in the MAA (Mathematical Association of America) mathematics test. In 1964, he finished second in the Westinghouse (now Intel) Science Talent Search in Washington, D.C. During his senior year in high school, his first published paper appeared in the American Mathematical Monthly.
As an undergraduate at the University of California, Berkeley, Bowen was a Putnam Fellow in 1964 and 1965. He earned his bachelor's degree from Berkeley where he received, on 15 June 1967, the University Medal as the most distinguished graduating senior. He also received the Dorothea Klumpke Roberts Prize (as top mathematics student) and the Mathematics Department Citation. At this time, Bowen was quoted as saying, "I'm slightly involved in political activity." He was "active in organizations devoted to preventing nuclear war."
Bowen married Carol Twito of Hayward on 6 March 1968. They had no children.
In 1970, Bowen completed his doctorate in mathematics at Berkeley under Stephen Smale, and joined the faculty as assistant professor in that year. At this time he began calling himself Rufus, the nickname he had been given because of his red hair and beard. He was an invited speaker at the 1974 International Congress of Mathematicians in Vancouver, British Columbia. He was promoted to full professorship in 1977.
Bowen's mature work dealt with dynamical systems theory, a field which Smale, Bowen's dissertation advisor, explored and broadened in the 1960s.
As studied by Smale, a dynamical system comprises a manifold and a smooth mapping ... As Poincaré emphasized, there is no general procedure for this, and therefore one must resort to describing average, typical, or most probable behavior. Bowen's work is an important part of the program of expressing these vague ideas in mathematically precise and useful ways.
Bowen died in Santa Ros |
https://en.wikipedia.org/wiki/Jeremy%20Gray | Jeremy John Gray (born 25 April 1947) is an English mathematician primarily interested in the history of mathematics.
Biography
Gray studied mathematics at the University of Oxford from 1966 to 1969, and then at Warwick University, obtaining his PhD in 1980 under the supervision of Ian Stewart and David Fowler. He has worked at the Open University since 1974, and became a lecturer there in 1978. He also lectured at the University of Warwick from 2002 to 2017, teaching a course on the history of mathematics.
Gray was a consultant on the television series, The Story of Maths, a co-production between the Open University and the BBC. He edits Archive for History of Exact Sciences.
In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2012 he became a fellow of the American Mathematical Society.
Books
Gray has been awarded prizes for his contributions to mathematics, including the Albert Leon Whiteman Memorial Prize from the American Mathematical Society in 2009, the Otto Neugebauer Prize of the European Mathematical Society in 2016, and the London Mathematical Society's Hirst Prize and Lectureship in 2018.
He has authored the following:
Ideas of space: Euclidean, non-Euclidean, and relativistic. Oxford University Press 1979, 2nd edition 1989, Spanish edition, Ideas de Espacio, Mondadori Espana, 1992. Romanian edition, Idei Spatiu, Editura All Educational, 1998.
Linear differential equations and group theory from Riemann to Poincaré Birkhäuser, 1986, 2nd edition with three new appendices and other additional material, 2000, Japanese edition 2002.
The Geometrical Work of Girard Desargues (with J. V. Field). Springer 1986.
Geometry (with D.A. Brannan and M. Esplen). Cambridge University Press 1999, 2nd edition 2012, Arabic edition 2001.
The Hilbert Challenge. Oxford University Press 2000, French translation Le défi de Hilbert Dunod, 2003, Taiwanese translation 2003, Spanish translation El reto de Hilbert Critica, 2004.
Janos Bolyai, non-Euclidean Geometry and the Nature of Space. Burndy Library, MIT, 2004.
Worlds out of Nothing; A Course on the History of Geometry in the 19th Century. Springer 2006, 2nd revised edition 2010.
Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton University Press 2008.
Henri Poincaré: a scientific biography. Princeton University Press 2012.
Hidden Harmony – Geometric Fantasies. The Rise of Complex Function Theory (with Umberto Bottazzini). Springer 2013.
The Real and the Complex: A History of Analysis in the 19th Century. Springer 2015.
A History of Abstract Algebra: From Algebraic Equations to Modern Algebra. Springer 2018.
Change and Variations: A History of Differential Equations to 1900. Springer 2021.
Books edited or co-edited:
The History of Mathematics; a Reader (with John Fauvel). Macmillan 1987.
L'Europe mathématique, Mathematical Europe (with C. Goldstein and J. Ritter). Edited with an introductory essay, Éditions de la maison d |
https://en.wikipedia.org/wiki/Gareth%20Roberts%20%28statistician%29 | Gareth Owen Roberts FRS FLSW (born 1964) is a statistician and applied probabilist. He is Professor of Statistics in the Department of Statistics and Director of the Centre for Research in Statistical Methodology (CRiSM) at the University of Warwick. He is an established authority on the stability of Markov chains, especially applied to Markov chain Monte Carlo (MCMC) theory methodology for a wide range of latent statistical models with applications in spatial statistics, infectious disease epidemiology and finance.
Education
Roberts was educated at Liverpool Blue Coat School and Jesus College, Oxford, graduating in 1985 in Mathematics and subsequently went on to complete in 1988 a PhD thesis on Some boundary hitting problems for diffusion processes under the supervision of Saul Jacka at the University of Warwick.
Career
Following his PhD, Roberts held various academic positions at the University of Nottingham, the University of Cambridge and Lancaster University before returning to the University of Warwick. He was a Fellow of St Catharine's College, Cambridge from 1992 to 1998.
Roberts is a talented tournament bridge player, whose achievements include winning the Great Northern Swiss Pairs in 1997, and the Garden Cities Trophy in 2008 and 2013.
Awards and honours
1995 – Raybould Fellowship
1997 – Royal Statistical Society's Guy medal in Bronze
1999 – Rollo Davidson Prize of the University of Cambridge
2004 – ISI highly cited researcher (Mathematics: Ranked 16th )
2008 – Royal Statistical Society's Guy medal in Silver
2009 – Institute of Mathematical Statistics Medallion
2010 – Editor of Journal of the Royal Statistical Society (Series B)
2013 – Elected a Fellow of the Royal Society
2019 - Elected a Fellow of the Learned Society of Wales
His nomination to become a Fellow of the Royal Society (FRS) in 2013 reads:
2015 – Wolfson Research Merit Award from the Royal Society
References
Living people
English statisticians
Alumni of the University of Warwick
Alumni of Jesus College, Oxford
Fellows of St Catharine's College, Cambridge
Fellows of the Royal Society
Academics of the University of Nottingham
Date of birth missing (living people)
1964 births
Mathematical statisticians
Computational statisticians
Fellows of the Learned Society of Wales |
https://en.wikipedia.org/wiki/Balinski%27s%20theorem | In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional convex polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional convex polyhedron or polytope (its skeleton), then the resulting graph is at least d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices (together with their incident edges) are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair.
Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961, although the three-dimensional case dates back to the earlier part of the 20th century and the discovery of Steinitz's theorem that the graphs of three-dimensional polyhedra are exactly the three-connected planar graphs.
Balinski's proof
Balinski proves the result based on the correctness of the simplex method for finding the minimum or maximum of a linear function on a convex polytope (the linear programming problem). The simplex method starts at an arbitrary vertex of the polytope and repeatedly moves towards an adjacent vertex that improves the function value; when no improvement can be made, the optimal function value has been reached.
If S is a set of fewer than d vertices to be removed from the graph of the polytope, Balinski adds one more vertex v0 to S and finds a linear function ƒ that has the value zero on the augmented set but is not identically zero on the whole space. Then, any remaining vertex at which ƒ is non-negative (including v0) can be connected by simplex steps to the vertex with the maximum value of ƒ, while any remaining vertex at which ƒ is non-positive (again including v0) can be similarly connected to the vertex with the minimum value of ƒ. Therefore, the entire remaining graph is connected.
References
Polyhedral combinatorics
Graph connectivity
Theorems in discrete geometry
Theorems in graph theory |
https://en.wikipedia.org/wiki/Sepideh%20Farsi | Sepideh Farsi (; born 1965) is an Iranian film director.
Early years
Farsi left Iran in 1984 and went to Paris to study mathematics. However, eventually she was drawn to the visual arts and initially experimented in photography before making her first short films. A main theme of her works is identity. She still visits Tehran each year.
Awards/Recognition
Farsi was a Member of the Jury of the Locarno International Film Festival in Best First Feature in 2009. She won the FIPRESCI Prize (2002), Cinéma du Réel and Traces de Vie prize (2001) for "Homi D. Sethna, filmmaker" and Best documentary prize in Festival dei Popoli (2007) for "HARAT".
Recent News
One of her latest films is called Tehran Bedun-e Mojavvez (Tehran Without Permission). The 83-minute documentary shows life in Iran's crowded capital city of Tehran, facing international sanctions over its nuclear ambitions and experiencing civil unrest. It was shot entirely with a Nokia camera phone because of the government restrictions over shooting a film. The film shows various aspects of city life including following women at the hairdressers talking of the latest fads, young men speaking of drugs, prostitution and other societal problems, and the Iranian rapper “Hichkas”. The dialogue is in Persian with English and Arabic subtitles. In December 2009, Tehran Without Permission was shown at the Dubai International Film Festival.
Filmography
The Siren (2021)
Red Rose (2014)
Cloudy Greece (2013)
Zir-e āb / The house under the water (2010)
Tehrān bedun-e mojavvez / Tehran without permission (2009)
If it were Icarus (2008)
Harāt (2007)
Negāh / The Gaze (2006)
Khāb-e khāk / Dreams of Dust (2003)
Safar-e Mariam / The journey of Maryam (2002)
Mardān-e ātash / Men of Fire (2001)
Homi D. Sethna, filmmaker (2000)
Donyā khāne-ye man ast / The world is my home (1999)
Khab-e āb / Water dreams (1997)
Bād-e shomāl / Northwind (1993)
References
External links
Iranian women film directors
Iranian film directors
Iranian documentary filmmakers
Iranian expatriates in France
1965 births
Living people
Iranian diaspora film people
Women documentary filmmakers |
https://en.wikipedia.org/wiki/Babaj%20i%20Bok%C3%ABs | Babaj i Bokës is a village in Gjakova Municipality, Kosovo. According to the Kosovo Agency of Statistics (KAS) estimate from the 2011 census, there were 595 people residing in Babaj i Bokës, with Albanians constituting the majority of the population.
References
Villages in Gjakova |
https://en.wikipedia.org/wiki/John%20Lott%20%28mathematician%29 | John William Lott (born January 12, 1959) is a professor of Mathematics at the University of California, Berkeley. He is known for contributions to differential geometry.
Academic history
Lott received his B.S. from the Massachusetts Institute of Technology in 1978 and M.A. degrees in mathematics and physics from University of California, Berkeley. In 1983, he received a Ph.D. in mathematics under the supervision of Isadore Singer. After postdoctoral positions at Harvard University and the Institut des Hautes Études Scientifiques, he joined the faculty at the University of Michigan. In 2009, he moved to University of California, Berkeley.
Among his awards and honors:
Sloan Research Fellowship (1989-1991)
Alexander von Humboldt Fellowship (1991-1992)
U.S. National Academy of Sciences Award for Scientific Reviewing (with Bruce Kleiner)
Mathematical contributions
A 1985 article of Dominique Bakry and Michel Émery introduced a generalized Ricci curvature, in which one adds to the usual Ricci curvature the hessian of a function. In 2003, Lott showed that much of the standard comparison geometry results for the Ricci tensor extend to the Bakry-Émery setting. For instance, if is a closed and connected Riemannian manifold with positive Bakry-Émery Ricci tensor, then the fundamental group of must be finite; if instead the Bakry-Émery Ricci tensor is negative, then the isometry group of the Riemannian manifold must be finite. The comparison geometry of the Bakry-Émery Ricci tensor was taken further in an influential article of Guofang Wei and William Wylie. Additionally, Lott showed that if a Riemannian manifold with smooth density arises as a collapsed limit of Riemannian manifolds with a uniform upper bound on diameter and sectional curvature and a uniform lower bound on Ricci curvature, then the lower bound on Ricci curvature is preserved in the limit as a lower bound on Bakry-Émery's Ricci curvature. In this sense, the Bakry-Émery Ricci tensor is shown to be natural in the context of Riemannian convergence theory.
In 2002 and 2003, Grigori Perelman posted two papers to the arXiv which claimed to provide a proof for William Thurston's geometrization conjecture, using Richard Hamilton's theory of Ricci flow. Perelman's papers attracted immediate attention for their bold claims and the fact that some of their results were quickly verified. However, due to Perelman's abbreviated style of presentation of highly technical material, many mathematicians were unable to understand much of his work, especially in his second paper. Beginning in 2003, Lott and Bruce Kleiner posted a series of annotations of Perelman's work to their websites, which was finalized in a 2008 publication. Their article was most recently updated for corrections in 2013. In 2015, Kleiner and Lott were awarded the Award for Scientific Reviewing from the National Academy of Sciences of the United States for their work. Other well-known expositions of Perelman's work are due to Hua |
https://en.wikipedia.org/wiki/TAPAs%20model%20checker | TAPAs is a tool for specifying and analyzing concurrent systems. Its aim is to support teaching of process algebras. Systems are described as process algebra terms that are then mapped to labeled transition systems (LTSs). Properties can be verified by checking equivalences between concrete and abstract system descriptions or by model checking temporal formulas (expressed as μ-calculus or ACTL) over the obtained LTS. A key feature of TAPAs that makes it particularly suited for teaching is that it maintains a consistent graphical and textual representation of each system. After a change in the graphic notation, the textual representation is updated immediately; but after textual modifications, the update of the graphical representation has to be manually triggered.
In TAPAs, concurrent systems are described by means of processes, which are nondeterministic descriptions of system behaviors, and process systems, which are obtained by process compositions. Notably, processes can be defined in terms of other processes or process systems. Processes and process systems are composed by using the operators of a given process algebra. Currently, TAPAs supports two process algebras: CCSP and PEPA.
CCSP (= CCS + CSP) is obtained from CCS by considering some operators of CSP. After creating a CCSP process system, the user can analyze it using one of the following tools.
Equivalence Checker: allows to compare pairs of automata using a choice of equivalence (bisimulation, branching bisimulation, or decorated traces)
Model checker: given a model of a system, test automatically whether this model meets a given specification
Simulator: following one possible execution path through the system and presenting the resulting execution trace to the user.
PEPA (Performance Evaluation Process Algebra) is a stochastic process algebra designed for modeling computer and communication systems introduced by Jane Hillston in the 1990s. The language extends classical process algebras such as Milner's CCS and Hoare's CSP by introducing probabilistic branching and timing of transitions. Rates are drawn from the exponential distribution and PEPA models are finite state, so they give rise to a stochastic process---specifically a continuous-time Markov process (CTMC). Thus the language can be used to study quantitative properties of models of computer and communication systems such as throughput, utilization and response time as well as qualitative properties such as freedom from deadlock. The language is formally defined using a structured operational semantics in the style invented by Gordon Plotkin.
TAPAS is the result of collective work, beginning in 1990 with a tool named JACK by IEI CNR of Pisa. The work was continued by ISTI-CNR of Pisa. The new TAPAs version was developed at the Dipartimento Sistemi ed Informatica of the University of Florence.
See also
List of Model Checking Tools
References
F. Calzolai, R. De Nicola, M. Loreti, F. Tiezzi. TAPAs: a Tool for the |
https://en.wikipedia.org/wiki/Polymake | Polymake is software for the algorithmic treatment of convex polyhedra.
Albeit primarily a tool to study the combinatorics and the geometry of convex polytopes and polyhedra, it is by now also capable of dealing with simplicial complexes, matroids, polyhedral fans, graphs, tropical objects, toric varieties and other objects.
Polymake has been cited in over 100 recent articles indexed by Zentralblatt MATH as can be seen from its entry in the swMATH database.
Special Features
modular
Polymake was originally designed as a research tool for studying aspects of polytopes. As such, polymake uses many third party software packages for specialized computations, thereby providing a common interface and bridge between different tools. A user can easily (and unknowingly) switch between using different software packages in the process of computing properties of a polytope.
rule based computation
Polymake internally uses a server-client model where the server holds information about each object (e.g., a polytope) and the clients sends requests to compute properties. The server has the job of determining how to complete each request from information already known about each object using a rule based system. For example, there are many rules on how to compute the facets of a polytope. Facets can be computed from a vertex description of the polytope, and from a (possibly redundant) inequality description. Polymake builds a dependency graph outlining the steps to process each request and selects the best path via a Dijkstra type algorithm.
scripting
Polymake can be used within a perl script. Moreover, users can extend polymake and define new objects, properties, rules for computing properties, and algorithms.
Polymake applications
Polymake divides its collection of functions and objects into 10 different groups called applications. They behave like C++ namespaces. The polytope application was the first one developed and it is the largest.
Common application
This application contains many "helper" functions used in other applications.
Fan application
The Fan application contains functions for polyhedral complexes (which generalize simplicial complexes), planar drawings of 3-polytopes, polyhedral fans, and subdivisions of points or vectors.
Fulton application
This application deals with normal toric varieties. The name of this application is from the book "Introduction to Toric Varieties" by William Fulton.
Graph application
The graph application is for manipulating directed and undirected graphs. Some the standard graph functions exist (like for adjacency and cliques) together with combinatorial functions like computing the lattice represented by a directed acyclic graph.
Group application
The group application focuses on finite permutation groups. Basic properties of a group can be calculated like characters and conjugacy classes. Combined with a polytope, this application can compute properties associated with a group acting on a polytope b |
https://en.wikipedia.org/wiki/Argand%20system | In mathematics, an nth-order Argand system (named after French mathematician Jean-Robert Argand) is a coordinate system constructed around the nth roots of unity. From the origin, n axes extend such that the angle between each axis and the axes immediately before and after it is 360/n degrees. For example, the number line is the 2nd-order Argand system because the two axes extending from the origin represent 1 and −1, the 2nd roots of unity. The complex plane (sometimes called the Argand plane, also named after Argand) is the 4th-order Argand system because the 4 axes extending from the origin represent 1, i, −1, and −i, the 4th roots of unity.
References
Flanigan, Francis J., Complex Variables: Harmonic and Analytic Functions, Dover, 1983,
Jones, Phillip S., "Argand, Jean-Robert", Dictionary of Scientific Biography 237–240, Charles Scribner's Sons, 1970,
Mathematical structures |
https://en.wikipedia.org/wiki/Martyn%20Cundy | Henry Martyn Cundy (23 December 1913 – 25 February 2005) was a mathematics teacher and professor in Britain and Malawi as well as a singer, musician and poet. He was one of the founders of the School Mathematics Project to reform O level and A level teaching. Through this he had a big effect on maths teaching in Britain and especially in Africa.
Education and career
Cundy attended Monkton Combe School and then read mathematics at Trinity College, Cambridge, where he earned a PhD in quantum theory in 1938. In 1937, Cundy was awarded the Cambridge University Rayleigh Prize for Mathematical Physics (now known as the Rayleigh-Knight Prize) for an essay entitled "Motion in a Tetrahedral Field". Others awarded the Rayleigh Prize include Royal Society Fellows Alan Turing and Fred Hoyle; instead of acquiring a University position, Cundy initially chose work at the secondary school level. He taught at the Sherborne School from 1938 to 1966 and became prominently involved in the reform of school mathematics teaching in Great Britain. Secondary school Mathematics teachers became aware of Cundy after the appearance of his and his co-author A.P. Rollett's Mathematical Models, in continuous publication since 1952. A book focusing on the model construction of many of the regular polyhedra and other mathematical objects, Mathematical Models has remained "an inspiration for generations of mathematics teachers". Cundy was Deputy Director of the School Mathematics Project between 1967 and 1968. In 1968 he became Chair of Mathematics at the University of Malawi, and held the post until 1975.
Cundy spent many years publishing dozens of articles in The Mathematical Gazette, including at age 89 the "Article of the Year" for 2003, entitled "A Journey round the Triangle: Lester's Circle, Kiepert's Hyperbola and a Configuration from Morley".
Personal life
He married Kathleen Ethel ("Kittie") Hemmings in 1939 and had three children, including Ian Cundy, successively Bishop of Lewes and of Peterborough.
Martyn Cundy was a devout Christian and especially notable for his ecumenical views toward worship. In 1932 he was secretary of the Cambridge University Prayer Fellowship. Subsequently he served as a Methodist lay preacher and after taking up his position at the University of Malawi, an elder in the Malawi Presbyterian Church.
In Malawi Cundy learned to speak the Chewa language and he and his wife Kittie were active members of the university community there. The Cundys were enthusiastic trekkers and together they contributed a walking guide to the Zomba Massif.
On returning to the U.K. in 1975, the Cundys settled in Kendal and became active in the church community there. Martyn Cundy continued with his contributions to mathematics, religion and pedagogy for the remainder of his life.
Publications
The Faith of a Christian (London: Inter-Varsity Press, 1950).
Mathematical Models, with A.P. Rollett (London: Oxford University Press, 1952).
More than fifty articles in the |
https://en.wikipedia.org/wiki/Anders%20Bj%C3%B6rner | Anders Björner (born 17 December 1947) received his Ph.D. from Stockholm University in 1979, under Bernt Lindström. He is a Swedish professor of mathematics, in the Department of Mathematics at the Royal Institute of Technology, Stockholm, Sweden. His research interests are in combinatorics, as well as the related areas of algebra, geometry, topology, and computer science.
His other positions included being director of the Mittag-Leffler Institute and editor-in-chief of Acta Mathematica.
Björner is a recognized expert in algebraic and topological combinatorics. He is a 1983 recipient of the Pólya Prize, and is a member of the Royal Swedish Academy of Sciences since 1999.
Books
Oriented Matroids (with Michel Las Vergnas, Bernd Sturmfels, N. White and Günter M. Ziegler), Cambridge University Press, 1993. Second Edition 1999, 560 pages.
Combinatorics of Coxeter Groups (with F. Brenti), Graduate Texts in Mathematics, Vol. 231, Springer-Verlag, New York, 2005, 367 pages.
Chapter "Topological Methods" in Handbook of Combinatorics, (eds. Ronald L. Graham, M. Grötschel and László Lovász), North-Holland, Amsterdam, 1995, pp. 1819–1872.
References
External links
1947 births
Living people
21st-century Swedish mathematicians
Academic staff of the KTH Royal Institute of Technology
Members of the Royal Swedish Academy of Sciences
Directors of the Mittag-Leffler Institute
Stockholm University alumni
Combinatorialists
20th-century Swedish mathematicians |
https://en.wikipedia.org/wiki/Constant%20%28mathematics%29 | In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings:
A fixed and well-defined number or other non-changing mathematical object. The terms mathematical constant or physical constant are sometimes used to distinguish this meaning.
A function whose value remains unchanged (i.e., a constant function). Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question.
For example, a general quadratic function is commonly written as:
where , and are constants (coefficients or parameters), and a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is
which makes the function-argument status of (and by extension the constancy of , and ) clear. In this example , and are coefficients of the polynomial. Since occurs in a term that does not involve , it is called the constant term of the polynomial and can be thought of as the coefficient of . More generally, any polynomial term or expression of degree zero (no variable) is a constant.
Constant function
A constant may be used to define a constant function that ignores its arguments and always gives the same value. A constant function of a single variable, such as , has a graph of a horizontal line parallel to the x-axis. Such a function always takes the same value (in this case 5), because the variable does not appear in the expression defining the function.
Context-dependence
The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus:
"Constant" means not depending on some variable; not changing as that variable changes. In the first case above, it means not depending on h; in the second, it means not depending on x. A constant in a narrower context could be regarded as a variable in a broader context.
Notable mathematical constants
Some values occur frequently in mathematics and are conventionally denoted by a specific symbol. These standard symbols and their values are called mathematical constants. Examples include:
0 (zero).
1 (one), the natural number after zero.
(pi), the constant representing the ratio of a circle's circumference to its diameter, approximately equal to 3.141592653589793238462643.
, approximately equal to 2.718281828459045235360287.
, the imaginary unit such that .
(square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688.
(golden ratio), approximately equal to 1.618033988749894848204586, or algebraically, .
Constants in calculus
In calculus, constants are treated in several different ways depending on the operation. For example, the derivative (rate of change) of a constant function is zero. This is because constants, by definition, do not change. Their derivative is hence zero.
Con |
https://en.wikipedia.org/wiki/Continuum%20%28topology%29 | In the mathematical field of point-set topology, a continuum (plural: "continua") is a nonempty compact connected metric space, or, less frequently, a compact connected Hausdorff space. Continuum theory is the branch of topology devoted to the study of continua.
Definitions
A continuum that contains more than one point is called nondegenerate.
A subset A of a continuum X such that A itself is a continuum is called a subcontinuum of X. A space homeomorphic to a subcontinuum of the Euclidean plane R2 is called a planar continuum.
A continuum X is homogeneous if for every two points x and y in X, there exists a homeomorphism h: X → X such that h(x) = y.
A Peano continuum is a continuum that is locally connected at each point.
An indecomposable continuum is a continuum that cannot be represented as the union of two proper subcontinua. A continuum X is hereditarily indecomposable if every subcontinuum of X is indecomposable.
The dimension of a continuum usually means its topological dimension. A one-dimensional continuum is often called a curve.
Examples
An arc is a space homeomorphic to the closed interval [0,1]. If h: [0,1] → X is a homeomorphism and h(0) = p and h(1) = q then p and q are called the endpoints of X; one also says that X is an arc from p to q. An arc is the simplest and most familiar type of a continuum. It is one-dimensional, arcwise connected, and locally connected.
The topologist's sine curve is a subset of the plane that is the union of the graph of the function f(x) = sin(1/x), 0 < x ≤ 1 with the segment −1 ≤ y ≤ 1 of the y-axis. It is a one-dimensional continuum that is not arcwise connected, and it is locally disconnected at the points along the y-axis.
The Warsaw circle is obtained by "closing up" the topologist's sine curve by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose homotopy groups are all trivial, but it is not a contractible space.
An n-cell is a space homeomorphic to the closed ball in the Euclidean space Rn. It is contractible and is the simplest example of an n-dimensional continuum.
An n-sphere is a space homeomorphic to the standard n-sphere in the (n + 1)-dimensional Euclidean space. It is an n-dimensional homogeneous continuum that is not contractible, and therefore different from an n-cell.
The Hilbert cube is an infinite-dimensional continuum.
Solenoids are among the simplest examples of indecomposable homogeneous continua. They are neither arcwise connected nor locally connected.
The Sierpinski carpet, also known as the Sierpinski universal curve, is a one-dimensional planar Peano continuum that contains a homeomorphic image of any one-dimensional planar continuum.
The pseudo-arc is a homogeneous hereditarily indecomposable planar continuum.
Properties
There are two fundamental techniques for constructing continua, by means of nested intersections and inverse limits.
If {Xn} is a nested family of continua, i.e. Xn ⊇ Xn+1, then their intersecti |
https://en.wikipedia.org/wiki/Continuum%20%28set%20theory%29 | In the mathematical field of set theory, the continuum means the real numbers, or the corresponding (infinite) cardinal number, denoted by . Georg Cantor proved that the cardinality is larger than the smallest infinity, namely, . He also proved that is equal to , the cardinality of the power set of the natural numbers.
The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers, , or alternatively, that .
Linear continuum
According to Raymond Wilder (1965), there are four axioms that make a set C and the relation < into a linear continuum:
C is simply ordered with respect to <.
If [A,B] is a cut of C, then either A has a last element or B has a first element. (compare Dedekind cut)
There exists a non-empty, countable subset S of C such that, if x,y ∈ C such that x < y, then there exists z ∈ S such that x < z < y. (separability axiom)
C has no first element and no last element. (Unboundedness axiom)
These axioms characterize the order type of the real number line.
See also
Aleph null
Suslin's problem
Transfinite number
References
Bibliography
Raymond L. Wilder (1965) The Foundations of Mathematics, 2nd ed., page 150, John Wiley & Sons.
Set theory
Infinity |
https://en.wikipedia.org/wiki/Cosocle | In mathematics, the term cosocle (socle meaning pedestal in French) has several related meanings.
In group theory, a cosocle of a group G, denoted by Cosoc(G), is the intersection of all maximal normal subgroups of G. If G is a quasisimple group, then Cosoc(G) = Z(G).
In the context of Lie algebras, a cosocle of a symmetric Lie algebra is the eigenspace of its structural automorphism that corresponds to the eigenvalue +1. (A symmetric Lie algebra decomposes into the direct sum of its socle and cosocle.)
In the context of module theory, the cosocle of a module over a ring R is defined to be the maximal semisimple quotient of the module.
See also
Socle
Radical of a module
References
Group theory
Module theory |
https://en.wikipedia.org/wiki/Tangent%20indicatrix | In differential geometry, the tangent indicatrix of a closed space curve is a curve on the unit sphere intimately related to the curvature of the original curve. Let be a closed curve with nowhere-vanishing tangent vector . Then the tangent indicatrix of is the closed curve on the unit sphere given by .
The total curvature of (the integral of curvature with respect to arc length along the curve) is equal to the arc length of .
References
Solomon, B. "Tantrices of Spherical Curves." American Mathematical Monthly 103, 30–39, 1996.
Differential geometry
Spherical geometry |
https://en.wikipedia.org/wiki/Steven%20Gaal | Steven Alexander Gaal (February 22, 1924 – March 17, 2016) (also known as István Sándor Gál or I. S. Gál) was a Hungarian-American mathematician and Professor of Mathematics at the University of Minnesota—Minneapolis.
Education
Gaal received his Ph.D. under Frigyes Riesz and Lipót Fejér in 1947, although at the time, graduate study in Hungary did not exist in the formal way it is thought of today. There were no formalities of preliminary exams or qualifying exams, no thesis advisor or tuition. After World War I, Hungary was dismembered and the Austro-Hungarian monarchy ended. Two thirds of Hungary's territory was given to other states, some existing, others created and since vanished. Under these conditions, only few higher education faculty could be appointed and students had to learn only from books or one or two old sick professors. Nevertheless, doctoral students managed to create publishable theses. Gaal's thesis problem had its origin in a letter Paul Erdős wrote to Pál Turán, in which he mentions a prize problem posed by the Netherlands Mathematical Society. Gaal solved it and with Erdős jointly published the solution.
Career
Gaal later went to Paris, where he was employed by the CNRS (Centre National del la Recherche Scientifique) at the rank of attaché de recherché. His supervisor was Jean Favard with higher supervisor Jacques Hadamard. Gaal met many leading French mathematicians at the CNRS, including Jean Leray and both Élie and Henri Cartan. After emigrating to the United States, he held positions at Yale and Princeton before joining the faculty of the School of Mathematics at the University of Minnesota. Atle Selberg was instrumental in bringing Gaal to the Institute for Advanced Study in Princeton, New Jersey. While in Princeton, Gaal met Albert Einstein, though the two did not work together. It also was in Paris that Gaal had first met Paul Erdős. Seven years later, they wrote two more joint papers. Over the years, Gaal met Erdős on a number of other occasions, including his last visit to Minneapolis on the invitation of Carleton College, who sponsored his visit. Robert Langlands has cited Gaal's influence in his early investigations of zeta functions and Eisenstein series. Gaal's former wife, Lisl Gaal (originally Lisl Novak), is an accomplished mathematician in her own right and is well known for her text Classical Galois Theory. In 2004, Gaal was honored at the Hungarian Academy of Sciences 80th anniversary as one of the "big five" most distinguished Hungarian mathematicians. The other honorees included John Horvath, János Aczél, Ákos Császár and László Fuchs. Gaal gave a talk entitled "When is a Fibonacci sequence periodic?"
Books
(453 pp.)
(208 pp.)
References
External links
Erdős, P., and Gál, I. S. (1955). On the law of the iterated logarithm. I. Proc. Konikl. Akad. Wetensch Ser A 58 65–84, a joint paper by Gaal and Erdős. and On the law of the iterated logarithm. II
Mathematical Institute, Budapest Universit |
https://en.wikipedia.org/wiki/Pol%20Bueso | Pol Bueso Paradís (born 27 April 1985 in Moncofa, Province of Castellón, Valencian Community) is a Spanish professional footballer who plays as a centre-back for CD Arenteiro.
Career statistics
Club
References
External links
1985 births
Living people
People from Plana Baixa
Footballers from the Province of Castellón
Spanish men's footballers
Men's association football defenders
Segunda División players
Segunda División B players
Tercera División players
Segunda Federación players
CD Castellón footballers
AD Ceuta footballers
Valencia CF Mestalla footballers
UD Salamanca players
Albacete Balompié players
Gimnàstic de Tarragona footballers
UCAM Murcia CF players
Hércules CF players
Pontevedra CF footballers |
https://en.wikipedia.org/wiki/Cycle%20basis | In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles.
A fundamental cycle basis may be formed from any spanning tree or spanning forest of the given graph, by selecting the cycles formed by the combination of a path in the tree and a single edge outside the tree. Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time.
In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual graph.
Definitions
A spanning subgraph of a given graph G has the same set of vertices as G itself but, possibly, fewer edges. A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has even degree (its number of incident edges). Every simple cycle in a graph is an Eulerian subgraph, but there may be others. The cycle space of a graph is the collection of its Eulerian subgraphs. It forms a vector space over the two-element finite field. The vector addition operation is the symmetric difference of two or more subgraphs, which forms another subgraph consisting of the edges that appear an odd number of times in the arguments to the symmetric difference operation.
A cycle basis is a basis of this vector space in which each basis vector represents a simple cycle. It consists of a set of cycles that can be combined, using symmetric differences, to form every Eulerian subgraph, and that is minimal with this property. Every cycle basis of a given graph has the same number of cycles, which equals the dimension of its cycle space. This number is called the circuit rank of the graph, and it equals where is the number of edges in the graph, is the number of vertices, and is the number of connected components.
Special cycle bases
Several special types of cycle bases have been studied, including the fundamental cycle bases, weakly fundamental cycle bases, sparse (or 2-) cycle bases, and integral cycle bases.
Induced cycles
Every graph has a cycle basis in which every cycle is an induced cycle. In a 3-vertex-connected graph, there always exists a basis consisting of peripheral cycles, cycles whose removal does not separate the remaining graph. In any graph other than one formed by adding one edge to a cycle, a peripheral cycle must be an induced cycle.
Fundamental cycles
If is a spanning tree or spanning forest of a given graph , and is an edge that does not belong to , then the fundamental cycle defined by is the simple cycle consisting of together with the path in connecting the endpoints of . There are exactly fundamental cycles, one for each edge that does not belong to . Each of them is linearly inde |
https://en.wikipedia.org/wiki/George%20Garratly | George Garratly (October 1888 – 1929) was a footballer who played in the Football League for Wolverhampton Wanderers. Garratly guested for Stoke in 1918–19, making ten appearances.
Career statistics
Source:
References
1888 births
1929 deaths
Footballers from Walsall
Men's association football defenders
English men's footballers
Bloxwich Strollers F.C. players
Walsall F.C. players
Wolverhampton Wanderers F.C. players
Hednesford Town F.C. players
English Football League players
Stoke City F.C. wartime guest players |
https://en.wikipedia.org/wiki/Assumed%20mean | In statistics the assumed mean is a method for calculating the arithmetic mean and standard deviation of a data set. It simplifies calculating accurate values by hand. Its interest today is chiefly historical but it can be used to quickly estimate these statistics. There are other rapid calculation methods which are more suited for computers which also ensure more accurate results than the obvious methods.
Example
First:
The mean of the following numbers is sought:
219, 223, 226, 228, 231, 234, 235, 236, 240, 241, 244, 247, 249, 255, 262
Suppose we start with a plausible initial guess that the mean is about 240. Then the deviations from this "assumed" mean are the following:
−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22
In adding these up, one finds that:
22 and −21 almost cancel, leaving +1,
15 and −17 almost cancel, leaving −2,
9 and −9 cancel,
7 + 4 cancels −6 − 5,
and so on. We are left with a sum of −30. The average of these 15 deviations from the assumed mean is therefore −30/15 = −2. Therefore, that is what we need to add to the assumed mean to get the correct mean:
correct mean = 240 − 2 = 238.
Method
The method depends on estimating the mean and rounding to an easy value to calculate with. This value is then subtracted from all the sample values. When the samples are classed into equal size ranges a central class is chosen and the count of ranges from that is used in the calculations. For example, for people's heights a value of 1.75m might be used as the assumed mean.
For a data set with assumed mean x0 suppose:
Then
or for a sample standard deviation using Bessel's correction:
Example using class ranges
Where there are a large number of samples a quick reasonable estimate of the mean and standard deviation can be got by grouping the samples into classes using equal size ranges. This introduces a quantization error but is normally accurate enough for most purposes if 10 or more classes are used.
For instance with the exception,
167.8 175.4 176.1 166 174.7 170.2 178.9 180.4 174.6 174.5 182.4 173.4 167.4 170.7 180.6 169.6 176.2 176.3 175.1 178.7 167.2 180.2 180.3 164.7 167.9 179.6 164.9 173.2 180.3 168 175.5 172.9 182.2 166.7 172.4 181.9 175.9 176.8 179.6 166 171.5 180.6 175.5 173.2 178.8 168.3 170.3 174.2 168 172.6 163.3 172.5 163.4 165.9 178.2 174.6 174.3 170.5 169.7 176.2 175.1 177 173.5 173.6 174.3 174.4 171.1 173.3 164.6 173 177.9 166.5 159.6 170.5 174.7 182 172.7 175.9 171.5 167.1 176.9 181.7 170.7 177.5 170.9 178.1 174.3 173.3 169.2 178.2 179.4 187.6 186.4 178.1 174 177.1 163.3 178.1 179.1 175.6
The minimum and maximum are 159.6 and 187.6 we can group them as follows rounding the numbers down. The class size (CS) is 3. The assumed mean is the centre of the range from 174 to 177 which is 175.5. The differences are counted in classes.
The mean is then estimated to be
which is very close to the actual mean of 173.846.
The standard deviation is estimated as
References
Means |
https://en.wikipedia.org/wiki/Titer%20Khel | Titer Khel is a Marwat village on the Indus Highway in Khyber Pakhtunkhwa, Pakistan.
Topology
Titer Khel is surrounded by sand dunes.
Populated places in Khyber Pakhtunkhwa |
https://en.wikipedia.org/wiki/Directorate%20General%20of%20Budget%2C%20Accounting%20and%20Statistics | The Directorate General of Budget, Accounting and Statistics (DGBAS; ) is a branch of the Executive Yuan of the Republic of China (Taiwan), performs the role of both a comptroller for the government and census bureau.
History
The DGBAS was established in April 1931 under the Nationalist Government. In May 1948, DGBAS was elevated to the Ministry of Budget, Accounting and Statistics and was placed under the Executive Yuan. In November 1973, the Organization Act of DGBAS was revised and came into force. It was again revised in May 1983. In line with Executive Yuan restructuring policy, DBGAS was reorganized in February 2012. The functions of its departments were reviewed and the Electronic Data Processing Center was merged into DGBAS.
Organizational structure
The Board of Comptrollers
Department of Planning
Department of General Fund Budget
Department of Special Fund Budget
Department of Accounting and Financial Reporting
Department of Statistics
Department of Census
Department of Information Management
Secretariat
Department of Personnel
Civil Service Ethics Office
BAS Office
BAS Training Center
Laws and Regulations Committee
National Income Statistics Review Committee
List of leaders
(April 1931 – October 1946)
(October 1946 – November 1948)
Pang Songzhou (龐松舟) (November 1948 – December 1949)
Wang Yao (王燿) (December 1949 – March 1950)
Chen Liang (陳良) (March 1950 – August 1950)
Pang Songzhou (龐松舟) (August 1950 – July 1958)
Chen Ching-yu (陳慶瑜) (July 1958 – March 1963)
Chang Daoming (張導民) (March 1963 – January 1969)
(January 1969 – June 1978)
Ching Shih-yi (鍾時益) (June 1978 – 11 January 1987)
Yu Chien-ming (于建民) (12 January 1987 – 26 February 1993)
Wang Kun (汪錕) (27 February 1993 – 9 June 1996)
(10 June 1996 – 19 May 2000)
Lin Chuan (20 May 2000 – 1 December 2002)
Liu San-chi (劉三錡) (2 December 2002 – 19 May 2004)
Hsu Jan-yau (20 May 2004 – 19 May 2008)
Shih Su-mei (20 May 2008 – 19 May 2016)
Chu Tzer-ming (20 May 2016 –)
Transportation
The building is accessible within walking distance South from Xiaonanmen Station of the Taipei Metro.
See also
Executive Yuan
References
External links
1931 establishments in China
Government agencies established in 1931
Executive Yuan
Government of Taiwan
Statistical organizations |
https://en.wikipedia.org/wiki/Rodri%20%28footballer%2C%20born%201985%29 | Rodrigo Ángel Gil Torres (born 25 April 1985 in Cehegín, Murcia), known as Rodri, is a Spanish footballer who plays for Orihuela CF as a midfielder.
Club statistics
References
External links
1985 births
Living people
Spanish men's footballers
Footballers from the Region of Murcia
Men's association football midfielders
Segunda División players
Segunda División B players
Tercera División players
Real Murcia Imperial players
Real Murcia CF players
Real Madrid C footballers
Real Madrid Castilla footballers
Elche CF players
Orihuela CF players
RSD Alcalá players
UCAM Murcia CF players
Burgos CF footballers
Cypriot First Division players
Doxa Katokopias FC players
AC Omonia players
Spanish expatriate men's footballers
Expatriate men's footballers in Cyprus
Spanish expatriate sportspeople in Cyprus |
https://en.wikipedia.org/wiki/Extractor | Extractor may refer to:
Extractor (firearms)
Extractor (mathematics)
Extractor (screws), a tool used to remove broken screws
Randomness extractor
Soxhlet extractor
Exhaust manifold |
https://en.wikipedia.org/wiki/Florian%20Pop | Florian Pop (born 1952 in Zalău) is a Romanian mathematician, a professor of mathematics at the University of Pennsylvania.
Pop received his Ph.D. in 1987 and his habilitation in 1991, both from the University of Heidelberg. He has been a member of the Institute for Advanced Study in Princeton, and (from 1996 to 2003) a professor at the University of Bonn prior to joining the University of Pennsylvania faculty.
Pop's research concerns algebraic geometry, arithmetic geometry, anabelian geometry, and Galois theory. call his habilitation thesis, concerning the characterization of certain fields by their absolute Galois groups, a "milestone".
In 1996, Pop was awarded the Gay-Lussac–von Humboldt Prize for Mathematics, and in 2003 he was awarded the Romanian Order of Merit. In 2012 he became a fellow of the American Mathematical Society.
References
Selected publications
1952 births
Living people
20th-century Romanian mathematicians
Academic staff of the University of Bonn
Institute for Advanced Study visiting scholars
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
Heidelberg University alumni
People from Zalău
Fellows of the American Mathematical Society
21st-century Romanian mathematicians
Arithmetic geometers
Romanian emigrants to the United States |
https://en.wikipedia.org/wiki/List%20of%20Galatasaray%20S.K.%20records%20and%20statistics | Below are statistics and records related to Galatasaray.
Honours
Domestic competitions
Süper Lig
Winners (23) (record): 1961–62, 1962–63, 1968–69, 1970–71, 1971–72, 1972–73, 1986–87, 1987–88, 1992–93, 1993–94, 1996–97, 1997–98, 1998–99, 1999–2000, 2001–02, 2005–06, 2007–08, 2011–12, 2012–13, 2014–15, 2017–18, 2018–19, 2022–23
Runners-up (11): 1959, 1960–61, 1965–66, 1974–75, 1978–79, 1985–86, 1990–91, 2000–01, 2002–03, 2013–14, 2020–21
Turkish Cup
Winners (18) (record): 1962–63, 1963–64, 1964–65, 1965–66, 1972–73, 1975–76, 1981–82, 1984–85, 1990–91, 1992–93, 1995–96, 1998–99, 1999–2000, 2004–05, 2013–14, 2014–15, 2015–16, 2018–19
Runners-up (5): 1968–69, 1979–80, 1993–94, 1994–95, 1997–98
Turkish Super Cup
Winners (16) (record): 1966, 1969, 1972, 1982, 1987, 1988, 1991, 1993, 1996, 1997, 2008, 2012, 2013, 2015, 2016, 2019
Runners-up (9): 1971, 1973, 1976, 1985, 1994, 1998, 2006, 2014, 2018
Turkish National Division
Winners (1): 1939
Runners-up (5): 1937, 1940, 1941, 1943, 1950
Turkish Football Championship
Runners-up (1): 1949
Atatürk Cup
Runners-up (1): 2000
Prime Minister's Cup
Winners (5): 1975, 1979, 1986, 1990, 1995
Runners-up (2): 1980, 1989
International competitions
UEFA Cup
Winners (1): 1999–2000
UEFA Super Cup
Winners (1): 2000
European Cup / UEFA Champions League
Semi-finalist (1): 1988–89
Regional competitions
Istanbul Football League
Winners (15): 1908–09, 1909–10, 1910–11, 1914–15, 1915–16, 1921–22, 1924–25, 1925–26, 1926–27, 1928–29, 1930–31, 1948–49, 1954–55, 1955–56, 1957–58
Istanbul Football Cup
Winners (2): 1941–42, 1942–43 (shared-record)
Istanbul Shield
Winners (1): 1932–33
Doubles and Trebles
Doubles
Süper Lig and Turkish Cup: 1962–63, 1972–73, 1998–99, 2014–15
Domestic trebles
Süper Lig, Turkish Cup and TFF Super Cup: 1992–93, 2018–19
International trebles
Süper Lig, Turkish Cup and UEFA Europa League: 1999–2000
Other
Turkish Amateur Football Championship
Winners (1): 1952
TSYD Cup
Winners (12): 1963, 1966, 1967, 1970, 1977, 1981, 1987, 1991, 1992, 1997, 1998, 1999 (shared-record)
Runners-up (9): 1965, 1969, 1971, 1973, 1976, 1979, 1980, 1986, 1991
Atatürk Gazi Cup
Winners (1): 1928
50. Anniversary Cup
Winners (1): 1973
Emirates Cup
Winners (1): 2013
Uhrencup
Winners (1): 2016
Union Club Cup
Winners (1): 1909
Team records
Süper Lig
Wins
Most wins in a Süper Lig season (current season in bold)
Most home wins in a Süper Lig season
Most away wins in a Süper Lig season
Most consecutive wins in a Süper Lig season
Most consecutive home wins in a Süper Lig season
Most consecutive home wins league
Most consecutive away wins league
Record wins
8–0, Galatasaray – Altınordu, 1959–1960, 24 October 1959
0–8, Ankaragücü – Galatasaray, 1992–1993, 30 May 1993
9–2, Galatasaray – Adana Demirspor, 1983–1984, 11 December 1983
1–8, Altay – Galatasaray, 1996–1997, 27 October 1996
7–0, Galatasaray – Erzurumspor, 2000–2001, 19 August 2000
0–7, Karabükspor – Galatasaray, 2017–2018, |
https://en.wikipedia.org/wiki/Francis%20B.%20Hildebrand | Francis Begnaud Hildebrand (1915 – 29 November 2002) was an American mathematician. He was a Professor of mathematics at the Massachusetts Institute of Technology (MIT) from 1940 until 1984. Hildebrand was known for his many influential textbooks in mathematics and numerical analysis.
Education and career
Hildebrand received his bachelor's degree in 1936 and a master's degree in 1938 from Washington & Jefferson College, both in mathematics. He then received his Ph.D. degree from Massachusetts Institute of Technology in 1940 under the supervision of Prescott Durand Crout. He also received an honorary doctorate from Washington and Jefferson College in 1969. During World War II, he worked for two years in the Radiation Laboratory.
At MIT, he taught 18.075 and 18.076, the classes on advanced calculus for engineering students. The big green textbook from these classes (originally Advanced Calculus for Engineers, later Advanced Calculus for Applications) was a fixture in engineers' offices for decades.
Books
Hildebrand had authored many influential textbooks in mathematics, including
Advanced Calculus for Engineers, Prentice Hall, 1948.
Methods of Applied Mathematics, Prentice Hall, 1952.
Advanced Calculus for Applications, Prentice Hall, 1964.
Introduction to Numerical Analysis, 2 ed., Dover Publications, 1987 (First edition in 1956).
References
External links
20th-century American mathematicians
21st-century American mathematicians
Massachusetts Institute of Technology School of Science faculty
Massachusetts Institute of Technology alumni
Washington & Jefferson College alumni
1915 births
2002 deaths |
https://en.wikipedia.org/wiki/Generatrix | In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent.
Examples
A cone can be generated by moving a line (the generatrix) fixed at the future apex of the cone along a closed curve (the directrix); if that directrix is a circle perpendicular to the line connecting its center to the apex, the motion is rotation around a fixed axis and the resulting shape is a circular cone.
The generatrix of a cylinder, a limiting case of a cone, is a line that is kept parallel to some axis.
See also
Surface of revolution
References
Elementary geometry
Computer graphics |
https://en.wikipedia.org/wiki/Toroidal%20polyhedron | In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
Variations in definition
Toroidal polyhedra are defined as collections of polygons that meet at their edges and vertices, forming a manifold as they do. That is, each edge should be shared by exactly two polygons, and at each vertex the edges and faces that meet at the vertex should be linked together in a single cycle of alternating edges and faces, the link of the vertex. For toroidal polyhedra, this manifold is an orientable surface. Some authors restrict the phrase "toroidal polyhedra" to mean more specifically polyhedra topologically equivalent to the (genus 1) torus.
In this area, it is important to distinguish embedded toroidal polyhedra, whose faces are flat polygons in three-dimensional Euclidean space that do not cross themselves or each other, from abstract polyhedra, topological surfaces without any specified geometric realization. Intermediate between these two extremes are polyhedra formed by geometric polygons or star polygons in Euclidean space that are allowed to cross each other.
In all of these cases the toroidal nature of a polyhedron can be verified by its orientability and by its Euler characteristic being non-positive. The Euler characteristic generalizes to V − E + F = 2 − 2N, where N is the number of holes.
Császár and Szilassi polyhedra
Two of the simplest possible embedded toroidal polyhedra are the Császár and Szilassi polyhedra.
The Császár polyhedron is a seven-vertex toroidal polyhedron with 21 edges and 14 triangular faces. It and the tetrahedron are the only known polyhedra in which every possible line segment connecting two vertices forms an edge of the polyhedron. Its dual, the Szilassi polyhedron, has seven hexagonal faces that are all adjacent to each other, hence providing the existence half of the theorem that the maximum number of colors needed for a map on a (genus one) torus is seven.
The Császár polyhedron has the fewest possible vertices of any embedded toroidal polyhedron, and the Szilassi polyhedron has the fewest possible faces of any embedded toroidal polyhedron.
Stewart toroids
A special category of toroidal polyhedra are constructed exclusively by regular polygon faces, without crossings, and with a further restriction that adjacent faces may not lie in the same plane as each other. These are called Stewart toroids, named after Bonnie Stewart, who studied them intensively. They are analogous to the Johnson solids in the case of convex polyhedra; however, unlike the Johnson solids, there are infinitely many Stewart toroids. They include also toroidal deltahedra, polyhedra whose faces are all equilateral triangles.
A restricted class of Stewart toroids, also defined by Stewart, are the quasi-convex toroidal polyhedra. These are Stewart toroids that include all of the edges of their convex hulls |
https://en.wikipedia.org/wiki/List%20of%20Athletic%20Bilbao%20records%20and%20statistics | Athletic Bilbao is a Spanish football club. This article contains historical and current statistics and records pertaining to the club.
Records
One of the three clubs that have never been relegated from La Liga, along with Barcelona and Real Madrid.
Five-time La Liga and Copa del Rey winner in the same season.
First team to be permanently awarded the original Liga trophy.
Won La Liga unbeaten (1929–30 season, joint record).
Record for biggest Liga win, Athletic Bilbao 12–1 Barcelona (1930–31 season).
Record for most goals scored in a league match (includes both sides), 14 with Athletic Bilbao 9–5 Racing Santander (1932–33 season).
Record for most goals scored in a league match as visitors, Osasuna 1–8 Athletic Bilbao (1958–59 season, joint record).
Record for a draw with the most goals in a league match, Atlético 6–6 Athletic Bilbao (1949–50 season).
All-time La Liga top goalscorers (as a fraction) in a season (home); with 5.44 goals per match (1930–31 season).
All-time La Liga goalscorer (as a fraction) in a season (away); with 3.11 goals per match (1931–32 season).
All-time La Liga goalscorer (as a fraction) in a season; with 4.06 goals per match (1930–31 season).
Biggest away victories (in percentages) in a season with 8 out of 9, 89% (1932–33 season).
Biggest number of away points (percentage) in a season with 16 out of 18, 89% (1932–33 season).
Of the current Primera División teams, heaviest defeats inflicted at home ground against Barcelona, Real Betis, Espanyol and Celta Vigo (also Sporting Gijón, Zaragoza, Tenerife, Salamanca, Mérida and Lleida).
Of the current Primera División teams, heaviest La Liga defeats inflicted away against Barcelona (0–6), Real Madrid (0–6), Espanyol (1–5) and Osasuna (1–8).
Leading goalscorer in a single league match, Bata: seven goals (Athletic Bilbao 12–1 Barcelona, 1930–31 season, joint record).
Player with most appearances in La Liga: Zubizarreta, 622 (the first 169 matches with Athletic Bilbao).
Team with greatest number of Copa del Rey final appearances (disputed): 35.
Four permanently awarded league trophies (joint record).
Most goals scored in a Copa del Rey competition: 12 against Celta Vigo.
Player with most Copa del Rey final wins: Agustín Gaínza, 7.
Players with most Copa del Rey finals (disputed): Agustín Gaínza and Jose Mari Belauste, both have 9 confirmed wins.
All-time leading goalscorer in the history of the Copa del Rey: Telmo Zarra, 81 goals.
Leading goalscorer in a Copa del Rey match: Agustín Gaínza, eight goals (Athletic Bilbao 12–1 Celta Vigo, 1946–47 season).
Player with most Copa del Rey matches: Agustín Gaínza, 99.
Extended honours
Titles
(*) International friendly tournaments not included.
Notes
Awards
Amberes Trophy (2): 1953, 1959
Martini&Rossi Trophy (1): 1956
Pichichi Trophy (12): 1930, 1931, 1932, 1940, 1945, 1946, 1947, 1950, 1951, 1953, 1968, 1975.
Zamora Trophy (6): 1930, 1934, 1936, 1941, 1947, 1970.
Players
Games only include professiona |
https://en.wikipedia.org/wiki/Hans%20Werner%20Ballmann | Hans Werner Ballmann (known as Werner Ballmann; born 11 April 1951) is a German mathematician. His area of research is differential geometry with focus on geodesic flows, spaces of negative curvature as well as spectral theory of Dirac operators
Ballmann earned his doctorate from the University of Bonn in 1979, under the supervision of Wilhelm Klingenberg.
He currently is a professor at the University of Bonn, and the managing director of the Max Planck Institute for Mathematics in Bonn, Germany, since 2007. He has advised 16 doctoral students at Bonn, including Christian Bär and Anna Wienhard.
He is a member of the German Academy of Sciences Leopoldina since 2007, and a member of the scientific committee of the Mathematical Research Institute of Oberwolfach since 2004.
Selected works
Lectures on spaces of non positive curvatures (PDF; 818 kB), DMV Seminar, Birkhäuser 1995
Spaces of non positive curvature, Jahresbericht DMV, vol. 103, 2001, pp. 52–65
Der Satz von Lusternik und Schnirelmann, Bonner Mathematische Schriften, vol. 102, 1978, pp. 1–25
with M. Brin: Orbihedra of nonpositive curvature. Publications Mathématiques de l'IHÉS No. 82 (1995), 169–209 (1996).
References
External links
Homepage at MPIM
Author profile in the database zbMATH
Living people
20th-century German mathematicians
21st-century German mathematicians
1951 births
University of Bonn alumni
Academic staff of the University of Bonn
Academic staff of ETH Zurich
Max Planck Institute directors |
https://en.wikipedia.org/wiki/Barbier | Barbier may refer to:
Barbier (surname)
Barbier (crater), a feature on the Moon
Barbier reaction, a reaction in organic chemistry
Barbier's theorem in mathematics |
https://en.wikipedia.org/wiki/Vladimir%20Miklyukov | Vladimir Michaelovich Miklyukov (, also spelled Miklioukov or Mikljukov) (8 January 1944 – October 2013) was a Russian educator in mathematics, and head of the Superslow Process workgroup based at Volgograd State University.
Biography
In 1970, as a student of Georgy D. Suvorov at Donetsk National University, he defended his Ph.D. thesis Theory of Quasiconformal Mappings in Space. In 1981 Miklyukov and his family moved to Volgograd. He was transferred to the newly built Volgograd State University where he became chairman of the Department of Mathematical Analysis and Theory of Functions.
His scientific research focused on geometrical analysis. At the same time, he was studying zero mean curvature surfaces in Euclidean and pseudo-Euclidean spaces, nonlinear elliptic type partial differential equations and quasiregular mappings of Riemannian manifolds. The main results of that work were related to the following groups of questions:
The external geometrical structure of zero mean curvature surfaces in Euclidean and pseudo-euclidean spaces; spacelike tubes and bands of zero mean curvature, their stability and instability with respect to small deformations, their life-time, branches, connections between branch points and Lorentz invariant characteristics of surfaces;
Phragmén-Lindelöf type theorems for differential forms; Ahlfors type theorems for differential forms with finite or infinite number of different asymptotic tracts; generalizations of Wiman theorem of forms, applications to quasiregular mappings on manifolds; applications of isoperimetric methods to the Phragmén–Lindelöf principle for quasiregular mappings on manifolds.
From 1998-2000 Miklyukov was a visiting professor at Brigham Young University. In 2004 he concentrated on studying of the mathematical theory of superslow processes and differential forms in micro- and nanoflows, and founded the Laboratory of Superslow Processes. In 2009 Miklyukov was named a Distinguished Scientist of Russian Federation.
Publications
References
External links
of Milyukov
Vladimir Miklyukov's obituary
1944 births
2013 deaths
Mathematical analysts
Mathematics educators
20th-century Russian mathematicians
21st-century Russian mathematicians
Soviet mathematicians |
https://en.wikipedia.org/wiki/Product%20of%20experts | Product of experts (PoE) is a machine learning technique. It models a probability distribution by combining the output from several simpler distributions.
It was proposed by Geoffrey Hinton in 1999, along with an algorithm for training the parameters of such a system.
The core idea is to combine several probability distributions ("experts") by multiplying their density functions—making the PoE classification similar to an "and" operation. This allows each expert to make decisions on the basis of a few dimensions without having to cover the full dimensionality of a problem.
This is related to (but quite different from) a mixture model, where several probability distributions are combined via an "or" operation, which is a weighted sum of their density functions.
The experts may be understood as each being responsible for enforcing a constraint in a high-dimensional space. A data point is considered likely iff none of the experts say that the point violates a constraint.
To optimize it, he proposed the contrastive divergence minimization algorithm. This algorithm is most often used for learning restricted Boltzmann machines.
See also
Mixture of experts
Boltzmann machine
References
External links
Product of experts article in Scholarpedia
Geoffrey Hinton's articles on PoE
Machine learning |
https://en.wikipedia.org/wiki/Surinam%2C%20Mauritius | Surinam is a village located in the Savanne District of Mauritius. According to the Statistics Mauritius census in 2011, the population was 10,507.
Nightingale College was a college in Surinam, Mauritius founded on 1 July 1964 by Seewooparsad Goolab. It was first located at Dr Sauzier's residence in Souillac. In 1965 it moved to L'Eglise St Jacques and in 1968 to a location near Souillac Hospital. It moved to Royal Road, Surinam in March 1970, where it remained until its takeover by the Ministry of Education.
See also
Districts of Mauritius
List of places in Mauritius
References
Savanne District
Populated places in Mauritius
Schools in Mauritius |
https://en.wikipedia.org/wiki/Polymorphic%20recursion | In computer science, polymorphic recursion (also referred to as Milner–Mycroft typability or the Milner–Mycroft calculus) refers to a recursive parametrically polymorphic function where the type parameter changes with each recursive invocation made, instead of staying constant. Type inference for polymorphic recursion is equivalent to semi-unification and therefore undecidable and requires the use of a semi-algorithm or programmer-supplied type annotations.
Example
Nested datatypes
Consider the following nested datatype:
data Nested a = a :<: (Nested [a]) | Epsilon
infixr 5 :<:
nested = 1 :<: [2,3,4] :<: [[5,6],[7],[8,9]] :<: Epsilon
A length function defined over this datatype will be polymorphically recursive, as the type of the argument changes from Nested a to Nested [a] in the recursive call:
length :: Nested a -> Int
length Epsilon = 0
length (_ :<: xs) = 1 + length xs
Note that Haskell normally infers the type signature for a function as simple-looking as this, but here it cannot be omitted without triggering a type error.
Higher-ranked types
Applications
Program analysis
In type-based program analysis polymorphic recursion is often essential in gaining high precision of the analysis. Notable examples of systems employing polymorphic recursion include Dussart, Henglein and Mossin's binding-time analysis and the Tofte–Talpin region-based memory management system. As these systems assume the expressions have already been typed in an underlying type system (not necessary employing polymorphic recursion), inference can be made decidable again.
Data structures, error detection, graph solutions
Functional programming data structures often use polymorphic recursion to simplify type error checks and solve problems with nasty "middle" temporary solutions that devour memory in more traditional data structures such as trees. In the two citations that follow, Okasaki (pp. 144–146) gives a CONS example in Haskell wherein the polymorphic type system automatically flags programmer errors. The recursive aspect is that the type definition assures that the outermost constructor has a single element, the second a pair, the third a pair of pairs, etc. recursively, setting an automatic error finding pattern in the data type. Roberts (p. 171) gives a related example in Java, using a Class to represent a stack frame. The example given is a solution to the Tower of Hanoi problem wherein a stack simulates polymorphic recursion with a beginning, temporary and ending nested stack substitution structure.
See also
Higher-ranked polymorphism
Notes
Further reading
Richard Bird and Lambert Meertens (1998). "Nested Datatypes".
C. Vasconcellos, L. Figueiredo, C. Camarao (2003). "Practical Type Inference for Polymorphic Recursion: an Implementation in Haskell". Journal of Universal Computer Science.
L. Figueiredo, C. Camarao. "Type Inference for Polymorphic Recursive Definitions: a Specification in Haskell".
External links
Standard ML |
https://en.wikipedia.org/wiki/Mathematics%20education%20in%20the%20United%20States | Mathematics education in the United States varies considerably from one state to the next, and even within a single state. However, with the adoption of the Common Core Standards in most states and the District of Columbia beginning in 2010, mathematics content across the country has moved into closer agreement for each grade level. The SAT, a standardized university entrance exam, has been reformed to better reflect the contents of the Common Core. However, many students take alternatives to the traditional pathways, including accelerated tracks. As of 2023, twenty-seven states require students to pass three math courses before graduation from high school, and seventeen states and the District of Columbia require four.
Compared to other developed countries in the Organisation for Economic Co-operation and Development (OECD), the average level of mathematical literacy of American students is mediocre. As in many other countries, math scores dropped even further during the COVID-19 pandemic. Secondary-school algebra proves to be the turning point of difficulty many students struggle to surmount, and as such, many students are ill-prepared for collegiate STEM programs, or future high-skilled careers. Meanwhile, the number of eighth-graders enrolled in Algebra I has fallen between the early 2010s and early 2020s. Across the United States, there is a shortage of qualified mathematics instructors. Despite their best intentions, parents may transmit their mathematical anxiety to their children, who may also have school teachers who fear mathematics. About one in five American adults are functionally innumerate. While an overwhelming majority agree that mathematics is important, many, especially the young, are not confident of their own mathematical ability.
Curricular content and standards
Each U.S. state sets its own curricular standards, and details are usually set by each local school district. Although there are no federal standards, since 2015 most states have based their curricula on the Common Core State Standards in mathematics. The stated goal of the Common Core mathematics standards is to achieve greater focus and coherence in the curriculum. This is largely in response to the criticism that American mathematics curricula are "a mile wide and an inch deep." The National Council of Teachers of Mathematics published educational recommendations in mathematics education in 1989 and 2000 which have been highly influential, describing mathematical knowledge, skills and pedagogical emphases from kindergarten through high school. The 2006 NCTM Curriculum Focal Points have also been influential for its recommendations of the most important mathematical topics for each grade level through grade 8. Many states either did not accept, or never adopted, the Common Core standards, but instead brought their own state standards into closer alignment with the Common Core. There has been considerable disagreement on the style and content of mathematics teach |
https://en.wikipedia.org/wiki/Mean%20dependence | In probability theory, a random variable is said to be mean independent of random variable if and only if its conditional mean equals its (unconditional) mean for all such that the probability density/mass of at , , is not zero. Otherwise, is said to be mean dependent on .
Stochastic independence implies mean independence, but the converse is not true.; moreover, mean independence implies uncorrelatedness while the converse is not true. Unlike stochastic independence and uncorrelatedness, mean independence is not symmetric: it is possible for to be mean-independent of even though is mean-dependent on .
The concept of mean independence is often used in econometrics to have a middle ground between the strong assumption of independent random variables () and the weak assumption of uncorrelated random variables
Further reading
References
Independence (probability theory) |
https://en.wikipedia.org/wiki/Domino%20%28mathematics%29 | In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one free domino.
Since it has reflection symmetry, it is also the only one-sided domino (with reflections considered distinct). When rotations are also considered distinct, there are two fixed dominoes: The second one can be created by rotating the one above by 90°.
In a wider sense, the term domino is sometimes understood to mean a tile of any shape.
Packing and tiling
Dominos can tile the plane in a countably infinite number of ways. The number of tilings of a 2×n rectangle with dominoes is , the nth Fibonacci number.
Domino tilings figure in several celebrated problems, including the Aztec diamond problem in which large diamond-shaped regions have a number of tilings equal to a power of two, with most tilings appearing random within a central circular region and having a more regular structure outside of this "arctic circle", and the mutilated chessboard problem, in which removing two opposite corners from a chessboard makes it impossible to tile with dominoes.
See also
Dominoes, a set of domino-shaped gaming pieces
Tatami, Japanese domino-shaped floor mats
References
Polyforms |
https://en.wikipedia.org/wiki/Borja%20Rubiato | Borja Rubiato Martínez (born 13 November 1984 in Madrid) is a Spanish professional footballer who plays for Real Ávila CF as a striker.
Club statistics
References
External links
HKFA profile
1984 births
Living people
Footballers from Madrid
Spanish men's footballers
Men's association football forwards
Segunda División players
Segunda División B players
Tercera División players
Las Rozas CF players
CA Osasuna B players
Getafe CF B players
CD Cobeña players
Atlético Madrid B players
Cádiz CF players
SD Huesca footballers
Real Oviedo players
Zamora CF footballers
RSD Alcalá players
Marbella FC players
CF Trival Valderas players
CD Olímpic de Xàtiva footballers
CD Mensajero players
CD Ebro players
Arandina CF players
Real Ávila CF players
North American Soccer League (2011–2017) players
San Antonio Scorpions players
Iraq Stars League players
Erbil SC players
Hong Kong First Division League players
Kitchee SC players
Spanish expatriate men's footballers
Expatriate men's soccer players in the United States
Expatriate men's footballers in Iraq
Expatriate men's footballers in Hong Kong
Spanish expatriate sportspeople in the United States
Spanish expatriate sportspeople in Iraq
Spanish expatriate sportspeople in Hong Kong |
https://en.wikipedia.org/wiki/Roman%20Bezrukavnikov | Roman Bezrukavnikov (born 1973) is an American mathematician born in Moscow. He is a mathematics professor at the Massachusetts Institute of Technology and the chief research fellow at the HSE International Laboratory of Representation Theory and Mathematical Physics who specializes in representation theory and algebraic geometry.
He graduated from Moscow State School 57 mathematical class in 1990, and earned an M.A. at Brandeis University in 1994. He received his Ph.D. in mathematics from Tel Aviv University in 1998 under the supervision of Joseph N. Bernstein.
Bezrukavnikov was a visiting scholar at the Institute for Advanced Study in 1996-1998 and again in 2007–2008. He was a Dickson Instructor at the University of Chicago in 1999-2001. In 2001 he was awarded a Clay Research Fellowship, and in 2004, he won a Sloan Research Fellowship from the Alfred P. Sloan Foundation. He was awarded a Simons Fellowship in Mathematics by the Simons Foundation in 2014, and again in 2020.
References
External links
Faculty page at Massachusetts Institute of Technology
1973 births
Living people
21st-century American mathematicians
Brandeis University alumni
Tel Aviv University alumni
Massachusetts Institute of Technology School of Science faculty
Institute for Advanced Study visiting scholars |
https://en.wikipedia.org/wiki/Galician%20derby | The Galician derby () is the name given to any association football match contested between Celta Vigo and Deportivo La Coruña, the two biggest clubs in Galicia.
Head-to-head statistics
Updated 5 May 2018
League matches
Cup matches
Head-to-head ranking in La Liga (1929–2023)
• Summary: Celta with 18 higher finishes and Deportivo with 17 higher finishes (only including seasons in which both teams played in La Liga).
References
Football rivalries in Spain
RC Celta de Vigo
Deportivo de La Coruña
Sport in Galicia (Spain)
Recurring sporting events established in 1928 |
https://en.wikipedia.org/wiki/Tahltan%20Indian%20Reserve%20No.%2010 | Tahltan Indian Reserve No. 10, referred to by Statistics Canada for census purposes as Tahtlan 10, is an Indian reserve of the Tahltan First Nation located one mile north of the confluence of the Klastline River with the Stikine in the Stikine Country of the northwestern British Columbia Interior of Canada.
See also
Tahltan, British Columbia (Tahltan IR No. 1)
List of Indian reserves in British Columbia
References
Stikine Country
Indian reserves in British Columbia
Tahltan |
https://en.wikipedia.org/wiki/David%20R%C3%ADos%20Insua | David Ríos Insua (born June 21, 1964 in Madrid) is a Spanish mathematician, and son and disciple of Sixto Ríos, the "father of Spanish statistics." He is currently also the youngest Fellow of the Spanish Royal Academy of Sciences (de la Real Academia de Ciencias Exactas, Físicas y Naturales, RAC), which he joined in 2008. He received a PhD in Computational Sciences at the University of Leeds. He is Full Professor of the Statistics and Operations Research Department at Rey Juan Carlos University (URJC), and he has been Vice-dean of New Technologies and International Relationships at URJC (2002–2009). He has worked in fields such as Bayesian inference in neuronal networks, MCMC methods in decision analysis, Bayesian robustness or adversarial risk analysis. He has also worked in applied areas such as Electronic Democracy, reservoirs management, counterterrorism model and many others. He is married and has two daughters.
Career
He obtained his master's degree in Mathematics at Universidad Complutense de Madrid (UCM, 1987), with Extraordinary and National Awards.
He did his PhD studies at Manchester and Leeds where he obtained his PhD degree in 1990.
He has been a lecturer and researcher at Universidad Politécnica de Madrid (UPM), Duke University, Purdue University, Université Paris-Dauphine, International Institute for Applied Systems Analysis (IIASA), Consiglio Nazionale delle Ricerche-Istituto di Matematica Applicata e Tecnologie Informatiche (CNR-IMATI), Statistical and Applied Mathematical Sciences Institute (SAMSI), where he was director of the Risk Analysis, Extreme Events and Decision Theory program.
He was director of the Towards Electronic Democracy (TED) program de la European Science Foundation (ESF).
He is current codirector of the Algorithmic Decision Theory (ALGODEC) program cofunded by the European Cooperation in Science and Technology (COST) and the European Science Foundation (ESF).
He has supervised 15 PhD theses.
He has supervised over 40 cofunded projects.
Awards
Extraordinary and National master's degree Award.
UPM Award for young researchers.
Ramiro Melendreras Award of Sociedad de Estadística e Investigación Operativa (SEIO) for young researchers.
Peccei Award from International Institute for Applied Systems Analysis (IIASA) for young researchers.
WirsboURJC Research Award.
Everis Award from Capital Semilla.
SRA Award for best Decision Analysis paper.
Publications
He is author of more than 95 refereed scientific papers in international journals.
He is the author of 15 books or monographs, including Sensitivity Analysis in Multiobjective Decision Making (Springer), Statistical Decision Theory (Kendall's), Robust Bayesian Analysis (Springer), E-Democracy: A GDN perspective (Springer), and Bayesian Analysis of Stochastic Processes (Wiley).
He is or has been associated editor of journals such as the Journal of Computational and Graphical Statistics, Group Decision and Negotiation, Journal of Multicriteria Decisi |
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