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https://en.wikipedia.org/wiki/B.%20R.%20Bhat | Beliyar Ramdas Bhat was a professor and head of the department of statistics at Karnataka University for more than two decades. He was elected member of International Statistical Institute, and Fellow of Royal Statistical Society.
Bhat obtained a M.A. degree in mathematics from Madras University in 1954, and a M.A. degree in statistics from Karnataka University, prior to working on a Ph.D. from Berkeley in 1961. His advisor was David Blackwell. Bhat was elected member of Institute of Mathematical Statistics, and South African Statistical Association. He was the Secretary, Editor and President of Indian Society for Probability and Statistics. He was also the past President of the Section of Statistics of the Indian Science Congress Association.
Selected works
References
External links
University of California, Berkeley alumni
Scientists from Karnataka
Year of birth unknown
Fellows of the Royal Statistical Society
Elected Members of the International Statistical Institute
Academic staff of Karnatak University |
https://en.wikipedia.org/wiki/Spherical%20lune | In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, {2}θ, with dihedral angle θ. The word "lune" derives from luna, the Latin word for Moon.
Properties
Great circles are the largest possible circles (circumferences) of a sphere; each one divides the surface of the sphere into two equal halves. Two great circles always intersect at two polar opposite points.
Common examples of great circles are lines of longitude (meridians) on a sphere, which meet at the north and south poles.
A spherical lune has two planes of symmetry. It can be bisected into two lunes of half the angle, or it can be bisected by an equatorial line into two right spherical triangles.
Surface area
The surface area of a spherical lune is 2θ R2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles.
When this angle equals 2π radians (360°) — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon — the area formula for the spherical lune gives 4πR2, the surface area of the sphere.
Examples
A hosohedron is a tessellation of the sphere by lunes. A n-gonal regular hosohedron, {2,n} has n equal lunes of π/n radians. An n-hosohedron has dihedral symmetry Dnh, [n,2], (*22n) of order 4n. Each lune individually has cyclic symmetry C2v, [2], (*22) of order 4.
Each hosohedra can be divided by an equatorial bisector into two equal spherical triangles.
Astronomy
The visibly lighted portion of the Moon visible from the Earth is a spherical lune. The first of the two intersecting great circles is the terminator between the sunlit half of the Moon and the dark half. The second great circle is a terrestrial terminator that separates the half visible from the Earth from the unseen half. The spherical lune is a lighted crescent shape seen from Earth.
n-sphere lunes
Lunes can be defined on higher dimensional spheres as well.
In 4-dimensions a 3-sphere is a generalized sphere. It can contain regular digon lunes as {2}θ,φ, where θ and φ are two dihedral angles.
For example, a regular hosotope {2,p,q} has digon faces, {2}2π/p,2π/q, where its vertex figure is a spherical platonic solid, {p,q}. Each vertex of {p,q} defines an edge in the hosotope and adjacent pairs of those edges define lune faces. Or more specifically, the regular hosotope {2,4,3}, has 2 vertices, 8 180° arc edges in a cube, {4,3}, vertex figure between the two vertices, 12 lune faces, {2}π/4,π/3, between pairs of adjacent edges, and 6 hosohedral cells, {2,p}π/3.
References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, Florida: CRC Press, p. 130, 1987.
Harris, J. W. and Stocker, H. "Spherical Wedge." §4.8.6 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 108, 1998.
Gellert, W.; Gottwald, S.; Hellwich, M.; Kästner, H. |
https://en.wikipedia.org/wiki/V.%20S.%20Huzurbazar | Vasant Shankar Huzurbazar (15 September 1919 – 15 November 1991) was an Indian statistician from Kolhapur. Huzurbazar was the founder head of the department of statistics, University of Pune from 1953 to 1976. From 1979 to 1991, he served as professor at University of Denver, Colorado until his death. He served as visiting professor for two years to the Iowa State University in 1962.
In 1974, Huzurbazar was awarded the Padma Bhushan from Government of India for his contributions to the field of statistics.
In 1983 he was elected as a Fellow of the American Statistical Association.
Career
Huzurbazar completed his high school from Rajaram High school, Kolhapur. He did his B.Sc. from University of Mumbai and M.Sc. in statistics from Banaras Hindu University during 1940–1941. Huzurbazar earned his Ph.D. in statistics from University of Cambridge in 1950; his advisor was Harold Jeffreys. Huzurbazar worked in the Gauhati University, Lucknow University and also in the Bureau of Economics and Statistics of Government of Bombay.
Personal
Huzurbazar's two daughters, Snehalata V. Huzurbazar and Aparna V. Huzurbazar, both became notable statisticians.
Works
References
External links
University of Mumbai alumni
University of Denver faculty
Recipients of the Padma Bhushan in literature & education
1919 births
1991 deaths
Indian statisticians
20th-century Indian mathematicians
Academic staff of Savitribai Phule Pune University
Scientists from Maharashtra
Marathi people
People from Kolhapur
Alumni of the University of Cambridge
Fellows of the American Statistical Association |
https://en.wikipedia.org/wiki/Mat%C3%A9rn%20covariance%20function | In statistics, the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces. It is named after the Swedish forestry statistician Bertil Matérn. It specifies the covariance between two measurements as a function of the distance between the points at which they are taken. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the Matérn covariance is also isotropic.
Definition
The Matérn covariance between measurements taken at two points separated by d distance units is given by
where is the gamma function, is the modified Bessel function of the second kind, and ρ and are positive parameters of the covariance.
A Gaussian process with Matérn covariance is times differentiable in the mean-square sense.
Spectral density
The power spectrum of a process with Matérn covariance defined on is the (n-dimensional) Fourier transform of the Matérn covariance function (see Wiener–Khinchin theorem). Explicitly, this is given by
Simplification for specific values of ν
Simplification for ν half integer
When , the Matérn covariance can be written as a product of an exponential and a polynomial of order :
which gives:
for :
for :
for :
The Gaussian case in the limit of infinite ν
As , the Matérn covariance converges to the squared exponential covariance function
Taylor series at zero and spectral moments
The behavior for can be obtained by the following Taylor series (reference is needed, the formula below leads to division by zero in case ):
When defined, the following spectral moments can be derived from the Taylor series:
See also
Radial basis function
References
Geostatistics
Spatial analysis
Covariance and correlation |
https://en.wikipedia.org/wiki/Homological%20integration | In the mathematical fields of differential geometry and geometric measure theory, homological integration or geometric integration is a method for extending the notion of the integral to manifolds. Rather than functions or differential forms, the integral is defined over currents on a manifold.
The theory is "homological" because currents themselves are defined by duality with differential forms. To wit, the space of -currents on a manifold is defined as the dual space, in the sense of distributions, of the space of -forms on . Thus there is a pairing between -currents and -forms , denoted here by
Under this duality pairing, the exterior derivative
goes over to a boundary operator
defined by
for all . This is a homological rather than cohomological construction.
References
.
.
Definitions of mathematical integration
Measure theory |
https://en.wikipedia.org/wiki/Rational%20quadratic%20covariance%20function | In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields where multivariate statistical analysis is conducted on metric spaces. It is commonly used to define the statistical covariance between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the rational quadratic covariance function is also isotropic.
The rational quadratic covariance between two points separated by d distance units is given by
where α and k are non-negative parameters of the covariance.
References
Spatial analysis
Geostatistics
Covariance and correlation |
https://en.wikipedia.org/wiki/Arithmetic%20variety | In mathematics, an arithmetic variety is the quotient space of a Hermitian symmetric space by an arithmetic subgroup of the associated algebraic Lie group.
Kazhdan's theorem
Kazhdan's theorem says the following:
References
Further reading
See also
Arithmetic Chow groups
Arithmetic of abelian varieties
Abelian variety
Arithmetic geometry |
https://en.wikipedia.org/wiki/Eigengap | In linear algebra, the eigengap of a linear operator is the difference between two successive eigenvalues, where eigenvalues are sorted in ascending order.
The Davis–Kahan theorem, named after Chandler Davis and William Kahan, uses the eigengap to show how eigenspaces of an operator change under perturbation. In spectral clustering, the eigengap is often referred to as the spectral gap; although the spectral gap may often be defined in a broader sense than that of the eigengap.
See also
Eigenvalue perturbation
References
Linear algebra |
https://en.wikipedia.org/wiki/Sma%C3%AFl%20Diss | Smaïl Diss (born 2 December 1976) in Mostaganem is an Algerian former football player.
Honours
Won the Algerian Cup once with ES Sétif in 2012
National team statistics
References
External links
1976 births
Living people
Algerian men's footballers
Algeria men's international footballers
ES Sétif players
USM Blida players
Algerian Ligue Professionnelle 1 players
People from Mostaganem
ES Mostaganem players
2002 African Cup of Nations players
Men's association football defenders
21st-century Algerian people |
https://en.wikipedia.org/wiki/Omega%20and%20agemo%20subgroup | In mathematics, or more specifically group theory, the omega and agemo subgroups described the so-called "power structure" of a finite p-group. They were introduced in where they were used to describe a class of finite p-groups whose structure was sufficiently similar to that of finite abelian p-groups, the so-called, regular p-groups. The relationship between power and commutator structure forms a central theme in the modern study of p-groups, as exemplified in the work on uniformly powerful p-groups.
The word "agemo" is just "omega" spelled backwards, and the agemo subgroup is denoted by an upside-down omega.
Definition
The omega subgroups are the series of subgroups of a finite p-group, G, indexed by the natural numbers:
The agemo subgroups are the series of subgroups:
When i = 1 and p is odd, then i is normally omitted from the definition. When p is even, an omitted i may mean either i = 1 or i = 2 depending on local convention. In this article, we use the convention that an omitted i always indicates i = 1.
Examples
The dihedral group of order 8, G, satisfies: ℧(G) = Z(G) = [ G, G ] = Φ(G) = Soc(G) is the unique normal subgroup of order 2, typically realized as the subgroup containing the identity and a 180° rotation. However Ω(G) = G is the entire group, since G is generated by reflections. This shows that Ω(G) need not be the set of elements of order p.
The quaternion group of order 8, H, satisfies Ω(H) = ℧(H) = Z(H) = [ H, H ] = Φ(H) = Soc(H) is the unique subgroup of order 2, normally realized as the subgroup containing only 1 and −1.
The Sylow p-subgroup, P, of the symmetric group on p2 points is the wreath product of two cyclic groups of prime order. When p = 2, this is just the dihedral group of order 8. It too satisfies Ω(P) = P. Again ℧(P) = Z(P) = Soc(P) is cyclic of order p, but [ P, P ] = Φ(G) is elementary abelian of order pp−1.
The semidirect product of a cyclic group of order 4 acting non-trivially on a cyclic group of order 4,
has ℧(K) elementary abelian of order 4, but the set of squares is simply { 1, aa, bb }. Here the element aabb of ℧(K) is not a square, showing that ℧ is not simply the set of squares.
Properties
In this section, let G be a finite p-group of order |G| = pn and exponent exp(G) = pk. Then the omega and agemo families satisfy a number of useful properties.
General properties
Both Ωi(G) and ℧i(G) are characteristic subgroups of G for all natural numbers, i.
The omega and agemo subgroups form two normal series:
G = ℧0(G) ≥ ℧1(G) ≥ ℧2(G) ≥ ... ≥ ℧k−2(G) ≥ ℧k−1(G) > ℧k(G) = 1
G = Ωk(G) ≥ Ωk−1(G) ≥ Ωk−2(G) ≥ ... ≥ Ω2(G) ≥ Ω1(G) > Ω0(G) = 1
and the series are loosely intertwined: For all i between 1 and k:
℧i(G) ≤ Ωk−i(G), but
℧i−1(G) is not contained in Ωk−i(G).
Behavior under quotients and subgroups
If H ≤ G is a subgroup of G and N ⊲ G is a normal subgroup of G, then:
℧i(H) ≤ H ∩ ℧i(G)
℧i(N) ⊲ G
Ωi(N) ⊲ G
℧i(G/N) = ℧i(G)N/N
Ωi(G/N) ≥ Ωi(G)N/N
Relation to other important |
https://en.wikipedia.org/wiki/False%20positive%20rate | In statistics, when performing multiple comparisons, a false positive ratio (also known as fall-out or false alarm ratio) is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as the ratio between the number of negative events wrongly categorized as positive (false positives) and the total number of actual negative events (regardless of classification).
The false positive rate (or "false alarm rate") usually refers to the expectancy of the false positive ratio.
Definition
The false positive rate is
where is the number of false positives, is the number of true negatives and is the total number of ground truth negatives.
The level of significance that is used to test each hypothesis is set based on the form of inference (simultaneous inference vs. selective inference) and its supporting criteria (for example FWER or FDR), that were pre-determined by the researcher.
When performing multiple comparisons in a statistical framework such as above, the false positive ratio (also known as the false alarm ratio, as opposed to false positive rate / false alarm rate ) usually refers to the probability of falsely rejecting the null hypothesis for a particular test. Using the terminology suggested here, it is simply .
Since V is a random variable and is a constant (), the false positive ratio is also a random variable, ranging between 0–1.
The false positive rate (or "false alarm rate") usually refers to the expectancy of the false positive ratio, expressed by .
It is worth noticing that the two definitions ("false positive ratio" / "false positive rate") are somewhat interchangeable. For example, in the referenced article serves as the false positive "rate" rather than as its "ratio".
Classification of multiple hypothesis tests
Comparison with other error rates
While the false positive rate is mathematically equal to the type I error rate, it is viewed as a separate term for the following reasons:
The type I error rate is often associated with the a-priori setting of the significance level by the researcher: the significance level represents an acceptable error rate considering that all null hypotheses are true (the "global null" hypothesis). The choice of a significance level may thus be somewhat arbitrary (i.e. setting 10% (0.1), 5% (0.05), 1% (0.01) etc.)
As opposed to that, the false positive rate is associated with a post-prior result, which is the expected number of false positives divided by the total number of hypotheses under the real combination of true and non-true null hypotheses (disregarding the "global null" hypothesis). Since the false positive rate is a parameter that is not controlled by the researcher, it cannot be identified with the significance level.
Moreover, false positive rate is usually used regarding a medical test or diagnostic device (i.e. "the false positive rate of a certain diagnostic device is 1%"), while type I error is a term associate |
https://en.wikipedia.org/wiki/Proof%20of%20Fermat%27s%20Last%20Theorem%20for%20specific%20exponents | Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proven by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent n larger than 2. In the centuries following the initial statement of the result and before its general proof, various proofs were devised for particular values of the exponent n. Several of these proofs are described below, including Fermat's proof in the case n = 4, which is an early example of the method of infinite descent.
Mathematical preliminaries
Fermat's Last Theorem states that no three positive integers (a, b, c) can satisfy the equation an + bn = cn for any integer value of n greater than two. (For n equal to 1, the equation is a linear equation and has a solution for every possible a, b. For n equal to 2, the equation has infinitely many solutions, the Pythagorean triples.)
Factors of exponents
A solution (a, b, c) for a given n leads to a solution for all the factors of n: if h is a factor of n then there is an integer g such that n = gh. Then (ag, bg, cg) is a solution for the exponent h:
(ag)h + (bg)h = (cg)h.
Therefore, to prove that Fermat's equation has no solutions for n > 2, it suffices to prove that it has no solutions for n = 4 and for all odd primes p.
For any such odd exponent p, every positive-integer solution of the equation ap + bp = cp corresponds to a general integer solution to the equation ap + bp + cp = 0. For example, if (3, 5, 8) solves the first equation, then (3, 5, −8) solves the second. Conversely, any solution of the second equation corresponds to a solution to the first. The second equation is sometimes useful because it makes the symmetry between the three variables a, b and c more apparent.
Primitive solutions
If two of the three numbers (a, b, c) can be divided by a fourth number d, then all three numbers are divisible by d. For example, if a and c are divisible by d = 13, then b is also divisible by 13. This follows from the equation
bn = cn − an
If the right-hand side of the equation is divisible by 13, then the left-hand side is also divisible by 13. Let g represent the greatest common divisor of a, b, and c. Then (a, b, c) may be written as a = gx, b = gy, and c = gz where the three numbers (x, y, z) are pairwise coprime. In other words, the greatest common divisor (GCD) of each pair equals one
GCD(x, y) = GCD(x, z) = GCD(y, z) = 1
If (a, b, c) is a solution of Fermat's equation, then so is (x, y, z), since the equation
an + bn = cn = gnxn + gnyn = gnzn
implies the equation
xn + yn = zn.
A pairwise coprime solution (x, y, z) is called a primitive solution. Since every solution to Fermat's equation can be reduced to a primitive solution by dividing by their greatest common divisor g, Fermat's Last Theorem can be proven by demonstrating that no primitive solutions exist.
Even and odd
Integers can be divided into even and odd, those that are evenly divisible by two and those t |
https://en.wikipedia.org/wiki/Tom%20Van%20Vleck | Tom Van Vleck is an American computer software engineer.
Life and work
Van Vleck graduated from MIT in 1965 with a BS in Mathematics. He worked on CTSS at MIT, and co-authored its first email program with Noel Morris. In 1965, he joined Project MAC, the predecessor of the MIT Computer Science and Artificial Intelligence Laboratory. He worked on the development of Multics for more than 16 years at MIT and at Honeywell Information Systems. He has also worked at Tandem Computers, Taligent, CyberCash, Sun Microsystems, Encirq (an internet advertising company), and SPARTA (a computer security company).
He is also known as a computer security expert.
Notes
Bibliography
Operational changes for MR 4.0, T. H. Van Vleck, MULTICS OPERATING STAFF NOTE MOSN-A001, Honeywell, April 23, 1976
The Multics System Programming Process, Van Vleck, T. H. and Clingen, C. T.; Invited Paper, ICSE 1978, pp. 278–280
Getting the picture; it can be done, IEEE Computer, vol. 27, no. 5, pp. 112, May 1994
SPMA - Java Binary Enhancement Tool, DARPA Information Survivability Conference and Exposition - Volume II, pp. 152, April 2003; (DOI)
Self-Protecting Mobile Agents Obfuscation Report, L. D'Anna, B. Matt, A. Reisse, T. Van Vleck, S. Schwab, and P. LeBlanc, Report #03-015, Network Associates Laboratories, June 2003.
Anti-Phishing: Best Practices for Institutions and Consumers; Tally, Gregg; Thomas, Roshan; Van Vleck, Tom; McAfee Research, Technical Report # 04-004
Three Questions about Each Bug You Find, Software Engineering Notes 14:5:62-63 (July 1989)
Cleaning Up the Basement in the Dark, Software Engineering Notes, (April 1992)
--, ed., with David Walden, The Compatible Time Sharing System (1961-1973) Fiftieth Anniversary Commemorative Overview, (also at Multicians.org) IEEE Computer Society, 2011
The IBM 7094 and CTSS
The IBM 360/67 and CP/CMS
The IBM 7070
1401s I have Known
References
The Multics web site, Mirror of multicians.org
Tom Van Vleck's home page
Tom Van Vleck, "THVV Multics Bio", Multicians.org, most recent update 11/29/02
Jeffrey R. Yost, "An Interview with Thomas Van Vleck", 24 October 2012, Computer Security History Project, Charles Babbage Institute
Computer systems engineers
American software engineers
Software engineering researchers
Computer security academics
American computer programmers
Massachusetts Institute of Technology School of Science alumni
Multics people
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/List%20of%20transitive%20finite%20linear%20groups | In mathematics, especially in areas of abstract algebra and finite geometry, the list of transitive finite linear groups is an important classification of certain highly symmetric actions of finite groups on vector spaces.
The solvable finite 2-transitive groups were classified by Bertram Huppert. The classification of finite simple groups made possible the complete classification of finite doubly transitive permutation groups. This is a result by Christoph Hering. A finite 2-transitive group has a socle that is either a vector space over a finite field or a non-abelian primitive simple group; groups of the latter kind are almost simple groups and described elsewhere. This article provides a complete list of the finite 2-transitive groups whose socle is elementary abelian.
Let be a prime, and a subgroup of the general linear group acting transitively on the nonzero vectors of the d-dimensional vector space over the finite field with p elements.
Infinite classes
There are four infinite classes of finite transitive linear groups.
Notice that the exceptional group of Lie type G2(q) is usually constructed as the automorphism groups of the split octonions. Hence, it has a natural representation as a subgroup of the 7-dimensional orthogonal group O(7, q). If q is even, then the underlying quadratic form polarizes to a degenerate symplectic form. Factoring out with the radical, one obtains an isomorphism between O(7, q) and the symplectic group Sp(6, q). The subgroup of Sp(6, q) which corresponds to G2(q)′ is transitive.
In fact, for q>2, the group G2(q) = G2(q)′ is simple. If q=2 then G2(2)′ ≅ PSU(3,3) is simple with index 2 in G2(2).
Sporadic finite transitive linear groups
These groups are usually classified by some typical normal subgroup, this normal subgroup is denoted by G0 and are written in the third column of the table. The notation 21+4− stands for the extraspecial group of minus type of order 32 (i.e. the extraspecial group of order 32 with an odd number (namely one) of quaternion factor).
All but one of the sporadic transitive linear groups yield a primitive permutation group of degree at most 2499. In the computer algebra programs GAP and MAGMA, these groups can be accessed with the command PrimitiveGroup(p^d,k); where the number k is the primitive identification of . This number is given in the last column of the following table.
Seven of these groups are sharply transitive; these groups were found by Hans Zassenhaus and are also known as the multiplicative groups of the Zassenhaus near-fields. These groups are marked by a star in the table.
This list is not explicitly contained in Hering's paper. Many books and papers give a list of these groups, some of them an incomplete one. For example, Cameron's book misses the groups in line 11 of the table, that is, containing as a normal subgroup.
References
Permutation groups |
https://en.wikipedia.org/wiki/All-Ireland%20Senior%20Hurling%20Championship%20records%20and%20statistics | This page details statistics of the All-Ireland Senior Hurling Championship.
General performances
By province
Counties
Team results
Legend
– Champions
– Runners-up
– Semi-finals
– Quarter-finals/Preliminary quarter-finals/Round 2/Round 1
– Provincial Groups/Joe McDonagh Cup
– Relegated
L – Leinster Senior Hurling Championship
M – Munster Senior Hurling Championship
JM – Joe McDonagh Cup
CR – Christy Ring Cup
For each year, the number of teams eligible for the All-Ireland (in brackets) are shown.
By decade
The most successful team of each decade, judged by number of All-Ireland Senior Hurling Championship titles, is as follows:
1890s: 4 each for Cork (1890, 1892, 1893, 1894) and Tipperary (1895, 1896, 1898, 1899)
1900s: 4 for Kilkenny (1904, 1905, 1907, 1909)
1910s: 3 for Kilkenny (1911, 1912, 1913)
1920s: 3 each for Dublin (1920, 1924, 1927) and Cork (1926, 1928, 1929)
1930s: 4 for Kilkenny (1932, 1933, 1935, 1939)
1940s: 5 for Cork (1941, 1942, 1943, 1944, 1946)
1950s: 3 each for Tipperary (1950, 1951, 1958) and Cork (1952, 1953, 1954)
1960s: 4 for Tipperary (1961, 1962, 1964, 1965)
1970s: 4 each for Cork (1970, 1976, 1977, 1978) and Kilkenny (1972, 1974, 1975, 1979)
1980s: 3 for Galway (1980, 1987, 1988)
1990s: 2 each for Cork (1990, 1999), Kilkenny (1992, 1993), Offaly (1994, 1998) and Clare (1995, 1997)
2000s: 7 for Kilkenny (2000, 2002, 2003, 2006, 2007, 2008, 2009)
2010s: 4 for Kilkenny (2011, 2012, 2014, 2015)
2020s: 4 for Limerick (2020, 2021, 2022, 2023)
By semi-final appearances
As of 24 July 2023.
23 counties have reached an all-Ireland semi-final at least once.
Carlow, Meath and Westmeath are the most notable counties to never reach a semi-final.
Semi-final appearances (2001-)
Consecutive Wins
Quadruple
Cork (1941, 1942, 1943, 1944)
Kilkenny (2006, 2007, 2008, 2009)
Limerick (2020, 2021, 2022, 2023)
Treble
Cork (1892, 1893, 1894)
Tipperary (1898, 1899, 1900)
Kilkenny (1911, 1912, 1913)
Tipperary (1949, 1950, 1951)
Cork (1952, 1953, 1954)
Cork (1976, 1977, 1978)
Double
Tipperary (1895, 1896)
Cork (1902, 1903)
Kilkenny (1904, 1905)
Cork (1928, 1929)
Kilkenny (1932, 1933)
Wexford (1955, 1956)
Tipperary (1961, 1962)
Tipperary (1964, 1965)
Kilkenny (1974, 1975)
Kilkenny (1982, 1983)
Galway (1987, 1988)
Kilkenny (1992, 1993)
Kilkenny (2002, 2003)
Cork (2004, 2005)
Kilkenny (2011, 2012)
Kilkenny (2014, 2015)
Single
Tipperary (1887, 1906, 1908, 1916, 1925, 1930, 1937, 1945, 1958, 1971, 1989, 1991, 2001, 2010, 2016, 2019)
Kilkenny (1907, 1909, 1922, 1935, 1939, 1947, 1957, 1963, 1967, 1969, 1972, 1979, 2000)
Cork (1890, 1919, 1926, 1931, 1946, 1966, 1970, 1984, 1986, 1990, 1999)
Limerick (1897, 1918, 1921, 1934, 1936, 1940, 1973, 2018)
Dublin (1889, 1917, 1920, 1924, 1927, 1938)
Wexford (1910, 1960, 1968, 1996)
Offaly (1981, 1985, 1994, 1998)
Clare (1914, 1995, 1997, 2013)
Galway (1923, 1980, 2017)
Waterfor |
https://en.wikipedia.org/wiki/Javad%20Razzaghi | Javad Razzaghi (born 28 November 1982 in Tehran) is an Iranian professional footballer who currently plays for Shahrdari Bandar Abbas in the Azadegan League.
Club career
Club career statistics
Last Update 13 May 2022
Assist Goals
External links
Persian League Profile
Living people
1982 births
Iranian men's footballers
Iranian expatriate men's footballers
PAS Tehran F.C. players
F.C. Aboomoslem players
Esteghlal Ahvaz F.C. players
Persepolis F.C. players
F.C. Shahrdari Bandar Abbas players
SK Sturm Graz players
FC Admira Wacker Mödling players
FC DAC 1904 Dunajská Streda players
Slovak First Football League players
Expatriate men's footballers in Slovakia
Iranian expatriate sportspeople in Slovakia
Expatriate men's footballers in Austria
Iranian expatriate sportspeople in Austria
Men's association football midfielders
Footballers from Tehran |
https://en.wikipedia.org/wiki/Quartet%20distance | The quartet distance is a way of measuring the distance between two phylogenetic trees. It is defined as the number of subsets of four leaves that are not related by the same topology in both trees.
Computing the quartet distance
The most straightforward computation of the quartet distance would require time, where is the number of leaves in the trees.
For binary trees, better algorithms have been found to compute the distance in
time
time
and
time
Gerth Stølting Brodal et al. found an algorithm that takes time to compute the quartet distance between two multifurcating trees when is the maximum degree of the trees, which is accessible in C, perl, and the R package Quartet.
References
Computational phylogenetics
Bioinformatics algorithms |
https://en.wikipedia.org/wiki/Barry%20Mazur | Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology.
Life
Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. He was nonetheless accepted for graduate studies at Princeton University, from where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled "On embeddings of spheres." He then became a Junior Fellow at Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother of Joseph Mazur and the father of Alexander J. Mazur.
Work
His early work was in geometric topology. In an elementary fashion, he proved the generalized Schoenflies conjecture (his complete proof required an additional result by Marston Morse), around the same time as Morton Brown. Both Brown and Mazur received the Veblen Prize for this achievement. He also discovered the Mazur manifold and the Mazur swindle.
His observations in the 1960s on analogies between primes and knots were taken up by others in the 1990s giving rise to the field of arithmetic topology.
Coming under the influence of Alexander Grothendieck's approach to algebraic geometry, he moved into areas of diophantine geometry. Mazur's torsion theorem, which gives a complete list of the possible torsion subgroups of elliptic curves over the rational numbers, is a deep and important result in the arithmetic of elliptic curves. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves. This proof was carried in his seminal paper "Modular curves and the Eisenstein ideal".
The ideas of this paper and Mazur's notion of Galois deformations, were among the key ingredients in Wiles's proof of Fermat's Last Theorem. Mazur and Wiles had earlier worked together on the main conjecture of Iwasawa theory.
In an expository paper, Number Theory as Gadfly, Mazur describes number theory as a field which
He expanded his thoughts in the 2003 book Imagining Numbers and Circles Disturbed, a collection of essays on mathematics and narrative that he edited with writer Apostolos Doxiadis.
Awards and honors
Mazur was elected to the American Academy of Arts and Sciences in 1978. In 1982 he was elected a member of the National Academy of Sciences. Mazur was elected to the American Philosophical Society in 2001, and in 2012 he became a fellow of the American Mathematical Society.
Mazur has received the Veblen Prize in geometry (1966), the Cole Prize in number theory (1982), the Chauvenet Prize for exposit |
https://en.wikipedia.org/wiki/Mordell%E2%80%93Weil%20theorem | In mathematics, the Mordell–Weil theorem states that for an abelian variety over a number field , the group of K-rational points of is a finitely-generated abelian group, called the Mordell–Weil group. The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.
History
The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for to be finitely generated; and it shows that the rank is finite. This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point on E.
Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928. More abstract methods were required, to carry out a proof with the same basic structure. The second half of the proof needs some type of height function, in terms of which to bound the 'size' of points of . Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set of homogeneous coordinates. For an abelian variety, there is no a priori preferred representation, though, as a projective variety.
Both halves of the proof have been improved significantly by subsequent technical advances: in Galois cohomology as applied to descent, and in the study of the best height functions (which are quadratic forms).
Further results
The theorem leaves a number of questions still unanswered:
Calculation of the rank. This is still a demanding computational problem, and does not always have effective solutions.
Meaning of the rank: see Birch and Swinnerton-Dyer conjecture.
Possible torsion subgroups: Barry Mazur proved in 1978 that the Mordell–Weil group can have only finitely many torsion subgroups. This is the elliptic curve case of the torsion conjecture.
For a curve in its Jacobian variety as , can the intersection of with be infinite? Because of Faltings's theorem, this is false unless .
In the same context, can contain infinitely many torsion points of ? Because of the Manin–Mumford conjecture, proved by Michel Raynaud, this is false unless it is the elliptic curve case.
See also
Arithmetic geometry
Mordell–Weil group
References
Diophantine geometry
Elliptic curves
Abelian varieties
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Symmetric%20tensor | In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments:
for every permutation σ of the symbols Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies
The space of symmetric tensors of order r on a finite-dimensional vector space V is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics.
Definition
Let V be a vector space and
a tensor of order k. Then T is a symmetric tensor if
for the braiding maps associated to every permutation σ on the symbols {1,2,...,k} (or equivalently for every transposition on these symbols).
Given a basis {ei} of V, any symmetric tensor T of rank k can be written as
for some unique list of coefficients (the components of the tensor in the basis) that are symmetric on the indices. That is to say
for every permutation σ.
The space of all symmetric tensors of order k defined on V is often denoted by Sk(V) or Symk(V). It is itself a vector space, and if V has dimension N then the dimension of Symk(V) is the binomial coefficient
We then construct Sym(V) as the direct sum of Symk(V) for k = 0,1,2,...
Examples
There are many examples of symmetric tensors. Some include, the metric tensor, , the Einstein tensor, and the Ricci tensor, .
Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example: stress, strain, and anisotropic conductivity. Also, in diffusion MRI one often uses symmetric tensors to describe diffusion in the brain or other parts of the body.
Ellipsoids are examples of algebraic varieties; and so, for general rank, symmetric tensors, in the guise of homogeneous polynomials, are used to define projective varieties, and are often studied as such.
Given a Riemannian manifold equipped with its Levi-Civita connection , the covariant curvature tensor is a symmetric order 2 tensor over the vector space of differential 2-forms. This corresponds to the fact that, viewing , we have the symmetry between the first and second pairs of arguments in addition to antisymmetry within each pair: .
Symmetric part of a tensor
Suppose is a vector space over a field of characteristic 0. If is a tensor of order , then the symmetric part of is the symmetric tensor defined by
the summation extending over the symmetric group on k symbols. In terms of a basis, and employing the Einstein summation convention, if
then
The components of the tensor appearing on the right are often denoted by
with parentheses () around the indices being symmetrized. Square brackets [] are used to indicate anti-symmetrization.
Sy |
https://en.wikipedia.org/wiki/Mathematics%20and%20Science%20Academy%20UTB | The Mathematics and Science Academy (MSA), a high school located in Brownsville, Texas, was established by the 79th Texas Legislature in May 2005. It was designed as a commuter program at the University of Texas at Brownsville and Texas Southmost College (UTB) for high school-aged students who are :gifted in mathematics and science. Rather than complete their final two years of traditional high school at other public institutions, students of the Math and Science Academy are required to take classes taught by UTB/TSC faculty with regular UTB students, but are provided with more supervision and guidance than traditional college students. The students are not charged tuition, book fees, nor any other fees typically charged by the university, but transportation and food are not provided for students. Graduating from the Mathematics and Science Academy program grants you a Distinguished high school diploma as well as an Associates of Arts degree, if you complete the necessary college hours. MSA is only the second high school program of its kind; the University of North Texas hosts a similar program, the Texas Academy of Mathematics and Science.
History
The Inaugural Class began its courses in 2007, averaging 18 college hours per semester per student. They were engaged in actual college lectures with normal university students, despite being high school students themselves. Five students of the inaugural class were not able to meet to requirements to stay in Mathematics and Science Academy and were instead forced to go back to their respective high schools. The following year, the second year class of MSA students were admitted. The group of forty students, from all around the Rio Grande Valley, participated in a variety of difficult college classes as well, and most were able to score a GPA of 3.5 or higher on a 4.0 scale. The Mathematics and Science Academy's third year of students graduated in May 2011, with several of them having gained acceptance to post-secondary schools such as Dartmouth College, the Massachusetts Institute of Technology, Columbia University, Cornell University, Bowdoin College, the University of Texas, and other such prestigious schools. The school aims to remain small so as to be able to provide individualized attention to each student, though it has experienced growth throughout the past several years.
Requirements
In order to apply for a slot in the MSA, you must meet the following requirements:
Applicants must be residents of Texas.
The Scholastic Assessment Test (SAT I) must be taken in the sophomore year. The score should be competitive with those of Texas college-bound high school seniors planning to major in math, science, or engineering (1070 composite score for Math and Reading with a minimum of 500 in both areas).
Transcripts of all school work from the ninth through tenth grades, showing excellent grades (more A's than B's), preferably in Honors, GT, and AP courses.
Completion of Algebra I, Algebra II, and |
https://en.wikipedia.org/wiki/The%20Marian%20Finucane%20Show | The Marian Finucane Show was an Irish radio programme, presented by Marian Finucane. It aired Saturday - Sunday at 11:00 to 13:00. According to statistics from 2009, it was then the highest-rating weekend radio show in Ireland.
When Finucane was away, Rachael English or Brendan O'Connor presented the programme.
Finucane died on 2 January 2020.
History
The show started in 1999 after the retirement of veteran broadcaster Gay Byrne. She had moved from her Liveline slot which she had had since the late 1970s.
The programme stayed in that early morning slot from 09:00 to 10:00 until 2005. Then the programme was replaced by RTÉ 2fm DJ Ryan Tubridy. His new programme The Tubridy Show, was similar in format keeping old items such as her book club. Her programme was then moved to a weekend slot from 11:00 to 13:00.
In 2001, the political career of Fianna Fáil minister Joe Jacob was damaged when he was unable to explain to Finucane and her listeners what people should do in the event of a nuclear explosion at Sellafield. The show was discontinued when Finucane died suddenly on the 2 January 2020.
Format
The show began with Finucane's signature tune. She spoke over the music saying:Hello there, and a very good-morning to you!
When the tune stopped she previewed the newspapers of that day.
The programme continued on with interviews, human interest stories, consumer and lifestyle news as well as panel discussions on issues of the week. In the second hour, there were typically guests in studio or by telephone link. The show was a public forum for serious issues such as the Commission to Inquire into Child Abuse and Shell to Sea campaign.
At the end of the Sunday programme, entertainment journalists previewed the coming week in television and film. Michael Dwyer was a regular contributor to this slot.
The programme usually ended at around 12:55 and was followed by a weather forecast from Met Éireann.
References
External links
Irish talk radio shows
RTÉ Radio 1 programmes |
https://en.wikipedia.org/wiki/Probability%20of%20precipitation | Probability of precipitation (PoP) is a commonly used term referring to the likelihood of precipitation falling in a particular area over a defined period of time, which is commonly a day, half day, or hour.
The PoP measure is meaningless unless it is associated with an interval of time. Forecasts commonly use PoP defined over 12-hour periods (PoP12), though 6-hour periods (PoP6) and other measures are also published. A "daytime" PoP12 means from 6 am to 6 pm.
PoPs are generally not statistically independent. A good example of an event that has a strongly dependent hour-to-hour PoP is a hurricane. In that case, there may be a 1 in 5 chance of the hurricane hitting a given stretch of coast, but if it does arrive there will be rain for several hours, with the effect that a one-hour PoP for the same region and period would be similar: about 1 in 5. Localized thunderstorms may be less dependent, with the effect that the one-hour PoPs may be somewhat less than the one-day PoP.
Definitions
U.S. National Weather Service
According to the U.S. National Weather Service (NWS), PoP is the probability of exceedance that more than of precipitation will fall in a single spot, averaged over the forecast area.
The NWS also provides hourly forecasts. The hourly PoP can be similar to the daily PoP and vary little, or it can vary dramatically.
Other US forecasters
AccuWeather's definition is based on the probability at the forecast area's official rain gauge. There is also a probability of precipitation for every location in the United States for every minute for the next two hours. This is also known as a minute-cast. The Weather Channel's definition may include precipitation amounts below 0.01 inch (0.254 mm) and includes the chance of precipitation 3 hours before or after the forecast period. This latter change was described as less objective and more consumer-centric. The Weather Channel has an observed wet bias – the probability of precipitation is exaggerated in some cases.
Environment Canada
Environment Canada reports a chance of precipitation (COP) that is defined as "The chance that measurable precipitation (0.2 mm of rain or 0.2 cm of snow) will fall on any random point of the forecast region during the forecast period." The values are rounded to 10% increments, but are never rounded to 50%.
UK Met Office
The UK's Met Office reports a POP that is rounded to 5% and is based on a minimum threshold of 0.1 mm of precipitation.
Alternative expressions
The probability of precipitation can also be expressed using descriptive terms instead of numerical values. For instance, the NWS might describe a precipitation forecast with terms such as "slight chance" meaning 20% certainty and "scattered" meaning 30–50% areal coverage. The precise meaning of these terms varies.
The UK's Met Office replaced descriptive terms, such as "likely", with percentage chance of precipitation in November 2011.
Public understanding
Probability of precipitation may |
https://en.wikipedia.org/wiki/Near%20sets | In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common. Descriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets.
The underlying assumption with descriptively close sets is that such sets contain elements that have location and measurable features such as colour and frequency of occurrence. The description of the element of a set is defined by a feature vector. Comparison of feature vectors provides a basis for measuring the closeness of descriptively near sets. Near set theory provides a formal basis for the observation, comparison, and classification of elements in sets based on their closeness, either spatially or descriptively. Near sets offer a framework for solving problems based on human perception that arise in areas such as image processing, computer vision as well as engineering and science problems.
Near sets have a variety of applications in areas such as topology, pattern detection and classification, abstract algebra, mathematics in computer science, and solving a variety of problems based on human perception that arise in areas such as image analysis, image processing, face recognition, ethology, as well as engineering and science problems. From the beginning, descriptively near sets have proved to be useful in applications of topology, and visual pattern recognition , spanning a broad spectrum of applications that include camouflage detection, micropaleontology, handwriting forgery detection, biomedical image analysis, content-based image retrieval, population dynamics, quotient topology, textile design, visual merchandising, and topological psychology.
As an illustration of the degree of descriptive nearness between two sets, consider an example of the Henry colour model for varying degrees of nearness
between sets of picture elements in pictures (see, e.g., §4.3). The two pairs of ovals in Fig. 1 and Fig. 2 contain coloured segments. Each segment in the figures corresponds to an equivalence class where all pixels in the class have similar descriptions, i.e., picture elements with similar colours. The ovals in Fig.1 are closer to each other descriptively than the ovals in Fig. 2.
History
It has been observed that the simple concept of nearness unifies various concepts of topological structures inasmuch as the category Near of all nearness spaces and nearness preserving maps contains categories sTop (symmetric topological spaces and continuous maps), Prox (proximity spaces and -maps), Unif (uniform spaces and uniformly continuous maps) and Cont (contiguity spaces and contiguity maps) as embedded full subcategories. The categories and are shown to be full supercategories of various well-known categories, including the c |
https://en.wikipedia.org/wiki/Spherical%20wedge | In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base). The angle between the radii lying within the bounding semidisks is the dihedral . If is a semidisk that forms a ball when completely revolved about the z-axis, revolving only through a given produces a spherical wedge of the same angle . Beman (2008) remarks that "a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon." A spherical wedge of radians (180°) is called a hemisphere, while a spherical wedge of radians (360°) constitutes a complete ball.
The volume of a spherical wedge can be intuitively related to the definition in that while the volume of a ball of radius is given by , the volume a spherical wedge of the same radius is given by
Extrapolating the same principle and considering that the surface area of a sphere is given by , it can be seen that the surface area of the lune corresponding to the same wedge is given by
Hart (2009) states that the "volume of a spherical wedge is to the volume of the sphere as the number of degrees in the [angle of the wedge] is to 360". Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if is the volume of the sphere and is the volume of a given spherical wedge,
Also, if is the area of a given wedge's lune, and is the area of the wedge's sphere,
See also
Spherical cap
Spherical segment
Ungula
Notes
A. A distinction is sometimes drawn between the terms "sphere" and "ball", where a sphere is regarded as being merely the outer surface of a solid ball. It is common to use the terms interchangeably, as the commentaries of both Beman (2008) and Hart (2008) do.
References
Spherical geometry |
https://en.wikipedia.org/wiki/Ungula | In solid geometry, an ungula is a region of a solid of revolution, cut off by a plane oblique to its base. A common instance is the spherical wedge. The term ungula refers to the hoof of a horse, an anatomical feature that defines a class of mammals called ungulates.
The volume of an ungula of a cylinder was calculated by Grégoire de Saint Vincent. Two cylinders with equal radii and perpendicular axes intersect in four double ungulae. The bicylinder formed by the intersection had been measured by Archimedes in The Method of Mechanical Theorems, but the manuscript was lost until 1906.
A historian of calculus described the role of the ungula in integral calculus:
Grégoire himself was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lies of plane figures. The ungula, however, proved a valuable source of inspiration for those who followed him, and who saw in it a means of representing and transforming integrals in many ingenious ways.
Cylindrical ungula
A cylindrical ungula of base radius r and height h has volume
,.
Its total surface area is
,
the surface area of its curved sidewall is
,
and the surface area of its top (slanted roof) is
.
Proof
Consider a cylinder bounded below by plane and above by plane where k is the slope of the slanted roof:
.
Cutting up the volume into slices parallel to the y-axis, then a differential slice, shaped like a triangular prism, has volume
where
is the area of a right triangle whose vertices are, , , and ,
and whose base and height are thereby and , respectively.
Then the volume of the whole cylindrical ungula is
which equals
after substituting .
A differential surface area of the curved side wall is
,
which area belongs to a nearly flat rectangle bounded by vertices , , , and , and whose width and height are thereby and (close enough to) , respectively.
Then the surface area of the wall is
where the integral yields , so that the area of the wall is
,
and substituting yields
.
The base of the cylindrical ungula has the surface area of half a circle of radius r: , and the slanted top of the said ungula is a half-ellipse with semi-minor axis of length r and semi-major axis of length , so that its area is
and substituting yields
. ∎
Note how the surface area of the side wall is related to the volume: such surface area being , multiplying it by gives the volume of a differential half-shell, whose integral is , the volume.
When the slope k equals 1 then such ungula is precisely one eighth of a bicylinder, whose volume is . One eighth of this is .
Conical ungula
A conical ungula of height h, base radius r, and upper flat surface slope k (if the semicircular base is at the bottom, on the plane z = 0) has volume
where
is the height of the cone from which the ungula has been cut out, and
.
The surface area of the curved sidewall is
.
As a consistency chec |
https://en.wikipedia.org/wiki/Irrigation%20statistics | This page shows statistical data on irrigation of agricultural lands worldwide.
Irrigation is the artificial abstraction of water from a source followed by the distribution of it at scheme level aiming at application at field level to enhance crop production when rainfall is scarce.
Irrigated area
The appended table gives an overview of irrigated areas in the world in 2003
Only the countries with more than 10 million ha of irrigated land are mentioned.
(*) Including India, China and Pakistan
There are 4 countries with 5 to 10 million ha irrigated land: Iran (7.7), Mexico (6.3), Turkey (5.1), and Thailand (5.0).
The 16 countries with 2 to 5 million ha irrigated land are: Bangladesh (4.7), Indonesia (4.5), Russia (4.5), Uzbekistan (4.3), Spain (3.8), Brazil (3.5), Iraq (3.5), Egypt (3.4), Romania (3.0), Vietnam (3.0), Italy (2.8), France (2.6), Japan (2.6), Australia (2.6), Ukraine (2.3), and Kazakhstan (2.1)
See also List of countries by irrigated land area
Area per application method at field level
94% of the application methods of irrigation water at field level is of the category surface irrigation, whereby the water is spread over the field by gravity.
Of the remaining 6%, the majority is irrigated by methods requiring energy, expensive hydraulic pressure techniques and pipe systems like sprinkler irrigation and drip irrigation, for the major part in the USA. The source of irrigation water in these cases often is groundwater from aquifers. However, the exploitation of aquifers can also be combined with surface irrigation at field level.
In relatively small areas one applies subirrigation whereby the water infiltrates into the soil below the soil surface from pipes or ditches. This category includes tidal irrigation used in the lower part of rivers where the tidal influence is felt by permitting the river water to enter ditches at high tide and allowing it to infiltrate from there into the soil
In relative rare cases one uses labor-intensive methods like irrigation with watering-cans and by filling dug-in porous pots (pitcher irrigation) from where the water enters the soil by capillary suction.
Surface irrigation can be divided into the following types, based on the method by which water is spread over the field after it has been admitted through the inlet:
spate irrigation (in Pakistan called Rod Koh), which may occur in hilly regions in dry zones where small rivers produce spate floods; ditches and bunds are built to guide the water to the fields to be irrigated; the number of fields irrigated at each flood event depends on the duration and intensity of the flood. The sailaba system in Balochistan is an example
flood-plain irrigation, which may occur in dry zones in larger river plains, where the river has high discharges during a short season only. Bunds are constructed to retain the river floods and the lands are being planted to crops when the floods recede (flood recession cropping). The molapos in the Okavango inland delta |
https://en.wikipedia.org/wiki/Locally%20catenative%20sequence | In mathematics, a locally catenative sequence is a sequence of words in which each word can be constructed as the concatenation of previous words in the sequence.
Formally, an infinite sequence of words w(n) is locally catenative if, for some positive integers k and i1,...ik:
Some authors use a slightly different definition in which encodings of previous words are allowed in the concatenation.
Examples
The sequence of Fibonacci words S(n) is locally catenative because
The sequence of Thue–Morse words T(n) is not locally catenative by the first definition. However, it is locally catenative by the second definition because
where the encoding μ replaces 0 with 1 and 1 with 0.
References
Formal languages
Combinatorics on words |
https://en.wikipedia.org/wiki/SigmaNet | SigmaNet is the Academic Network Laboratory of the University of Latvia Institute of Mathematics and Computer Science. It is also the Latvian NREN (National Research and Education Network), providing Internet services for the academic community in Latvia, including connectivity to the European network GÉANT, hosting, e-mail, data centre services, and grid resources.
History and re-organization
The Laboratory began operation in 1992 when it was known as LATNET. In 2006 the LATNET CERT team was established, dealing with computer security incidents mostly in the Latvian academic network.
In 2008 due to internal changes, the Latvian NREN was renamed to SigmaNet, but the CERT team was moved under the Network Solutions department (NIC) and placed in the country top level domain .lv registry. The name of the CERT team was changed to CERT NIC.LV. The constituency of the team was widened to include not only the academic network, but also other constituencies that have concluded cooperation agreement on incident response services.
Main activities
Research is one of the main SigmaNet’s activities. SigmaNet is participating actively in various European Commission, Structural funds and community funded projects. SigmaNet’s goal is to become the leading institution for implementation of research projects in the areas of computer network design and applications.
SigmaNet is also very active in various research projects (BalticGrid-II, GÉANT2, etc.) and task forces (TF-Storage, TF-CSIRT) related to network technologies and services, security, inclusiveness and other areas. It has been dealing with security related problems since 1992 when first activities in networking started.
SigmaNet is also one of the largest hosting companies in Latvia that offers e-mail services, web hosting, server collocation, Ultra DSL internet connection, virtual private server hosting, data storage and related services.
References
Communications in Latvia
Internet in Latvia
National research and education networks |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20FC%20O%C8%9Belul%20Gala%C8%9Bi%20season |
Match results
Friendlies
Liga I
League table
Results by round
Results summary
Matches
Cupa României
Players
Squad statistics
Transfers
In
Out
Club
Coaching staff
Kit
|
|
References
See also
2009-10
Romanian football clubs 2009–10 season |
https://en.wikipedia.org/wiki/Mads%20Jessen | Mads S. Jessen (born 14 October 1989) is a Danish professional football attacking midfielder, who last played for Hobro IK.
References
External links
Career statistics at Danmarks Radio
1989 births
Living people
Danish men's footballers
Sønderjyske Fodbold players
Danish Superliga players
Men's association football midfielders
Men's association football forwards
Hobro IK players |
https://en.wikipedia.org/wiki/M/M/c%20queue | In queueing theory, a discipline within the mathematical theory of probability, the M/M/c queue (or Erlang–C model) is a multi-server queueing model. In Kendall's notation it describes a system where arrivals form a single queue and are governed by a Poisson process, there are servers, and job service times are exponentially distributed. It is a generalisation of the M/M/1 queue which considers only a single server. The model with infinitely many servers is the M/M/∞ queue.
Model definition
An M/M/c queue is a stochastic process whose state space is the set {0, 1, 2, 3, ...} where the value corresponds to the number of customers in the system, including any currently in service.
Arrivals occur at rate according to a Poisson process and move the process from state to +1.
Service times have an exponential distribution with parameter . If there are fewer than jobs, some of the servers will be idle. If there are more than jobs, the jobs queue in a buffer.
The buffer is of infinite size, so there is no limit on the number of customers it can contain.
The model can be described as a continuous time Markov chain with transition rate matrix
on the state space The model is a type of birth–death process. We write = /() for the server utilization and require < 1 for the queue to be stable. represents the average proportion of time which each of the servers is occupied (assuming jobs finding more than one vacant server choose their servers randomly).
The state space diagram for this chain is as below.
Stationary analysis
Number of customers in the system
If the traffic intensity is greater than one then the queue will grow without bound but if server utilization then the system has a stationary distribution with probability mass function
where is the probability that the system contains customers.
The probability that an arriving customer is forced to join the queue (all servers are occupied) is given by
which is referred to as Erlang's C formula and is often denoted C(, /) or E2,(/). The average number of customers in the system (in service and in the queue) is given by
Busy period of server
The busy period of the M/M/c queue can either refer to:
full busy period: the time period between an arrival which finds −1 customers in the system until a departure which leaves the system with −1 customers
partial busy period: the time period between an arrival which finds the system empty until a departure which leaves the system again empty.
Write = min( t: jobs in the system at time 0+ and − 1 jobs in the system at time ) and () for the Laplace–Stieltjes transform of the distribution of . Then
For > , has the same distribution as .
For = ,
For < ,
Response time
The response time is the total amount of time a customer spends in both the queue and in service. The average response time is the same for all work conserving service disciplines and is
Customers in first-come, first-served discipline
The customer either experie |
https://en.wikipedia.org/wiki/Giorgi%20Krasovski | Georgi Krasovski (born 20 December 1979 in Poti, Georgian SSR) is a retired Georgian/Polish professional football player who played as a defender.
Career statistics
References
External links
Profile at Armenian football federation
1979 births
Living people
People from Poti
Sportspeople from Samegrelo-Zemo Svaneti
Men's footballers from Georgia (country)
Expatriate men's footballers from Georgia (country)
FC Ararat Yerevan players
Mes Sarcheshme players
Ulisses FC players
Expatriate men's footballers in Azerbaijan
Expatriate men's footballers in Ukraine
Expatriate men's footballers in Uzbekistan
Expatriate men's footballers in Armenia
Expatriate men's footballers in Iran
Expatriate sportspeople from Georgia (country) in Azerbaijan
Armenian Premier League players
Gabala SC players
Men's association football defenders
FC Kolkheti-1913 Poti players
FC Torpedo Kutaisi players
FK Andijon players
FC Gandzasar Kapan players |
https://en.wikipedia.org/wiki/M.%20C.%20Chakrabarti | Mukund Chand Chakrabarti (died 1972) a statistician from Bengal of the British India was the founder head of the department of statistics, University of Mumbai India. He nurtured the department from its birth in 1948 until he died in 1972. The department of mathematics at University of Mumbai was established later in 1963 under the guidance of Professor S. S. Shrikhande.
Chakrabarti was known for his work in design of experiments. He guided a number of students for their Ph.D.s in statistics from University of Mumbai. He was also associated with University of Pune where his notes on design of experiments were taught and he used to come as external examiner for the practical examination.
In 1972, Chakrabarti died of a heart attack in Mumbai.
Select work
A note on skewness and kurtosis, MC Chakrabarti – Bull Calcutta Soc Math, 1946
M. C. Chakrabarti, On the C-matrix in design of experiments, J. Indian Statist. Assoc. 1 (1963), 8-23.
On the use of incidence matrices of designs in sampling from finite populations, MC Chakrabarti – Journal of Indian Statistical Association, 1963
On the ratio of the mean deviation to standard deviation, MC Chakrabarti – Calcutta Statistical Association Bulletin
[BOOK] Mathematics of design and analysis of experiments, MC Chakrabarti – 1962 – orton.catie.ac.cr
References
External links
Encyclopaedia of Design Theory: Bibliography
Review: M. C. Chakrabarti, Mathematics of Design and Analysis of Experiments
1972 deaths
Indian statisticians
People from Dhaka
University of Dhaka alumni
Year of birth missing
Bengali mathematicians
20th-century Indian mathematicians |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20F.C.%20Copenhagen%20season | This article shows statistics of individual players for the football club F.C. Copenhagen. It also lists all matches that F.C. Copenhagen will play in the 2009–10 season.
Events
Players
Squad information
This section show the squad as currently, considering all players who are confirmedly moved in and out (see section Players in / out).
Squad stats
Starting 11
This section shows the most used players for each position considering a 4-4-2 formation.
Players in / out
In
Out
Club
Coaching staff
Kit
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|
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Other information
Competitions
Overall
Danish Superliga
Classification
Results summary
Results by round
Matches
Competitive
Friendlies
References
External links
F.C. Copenhagen official website
2009-10
Danish football clubs 2009–10 season |
https://en.wikipedia.org/wiki/Regular%20surface | In mathematics, regular surface may refer to:
Regular surface (differential geometry)
Non-singular algebraic variety of dimension two |
https://en.wikipedia.org/wiki/RNF | RNF may refer to:
Aircraft Accident Investigation Board (Iceland) (Rannsóknarnefnd flugslysa)
Ring-sum normal form, in Boolean algebra
Rassemblement National Français ('French National Rally'), a French political party 1954–1957
Royal Northumberland Fusiliers, former British army regiment |
https://en.wikipedia.org/wiki/Connected%20ring | In mathematics, especially in the field of commutative algebra, a connected ring is a commutative ring A that satisfies one of the following equivalent conditions:
A possesses no non-trivial (that is, not equal to 1 or 0) idempotent elements;
the spectrum of A with the Zariski topology is a connected space.
Examples and non-examples
Connectedness defines a fairly general class of commutative rings. For example, all local rings and all (meet-)irreducible rings are connected. In particular, all integral domains are connected. Non-examples are given by product rings such as Z × Z; here the element (1, 0) is a non-trivial idempotent.
Generalizations
In algebraic geometry, connectedness is generalized to the concept of a connected scheme.
References
Commutative algebra
Ring theory |
https://en.wikipedia.org/wiki/Irreducible%20ring | In mathematics, especially in the field of ring theory, the term irreducible ring is used in a few different ways.
A (meet-)irreducible ring is a ring in which the intersection of two non-zero ideals is always non-zero.
A directly irreducible ring is a ring which cannot be written as the direct sum of two non-zero rings.
A subdirectly irreducible ring is a ring with a unique, non-zero minimum two-sided ideal.
A ring with an irreducible spectrum is a ring whose spectrum is irreducible as a topological space.
"Meet-irreducible" rings are referred to as "irreducible rings" in commutative algebra. This article adopts the term "meet-irreducible" in order to distinguish between the several types being discussed.
Meet-irreducible rings play an important part in commutative algebra, and directly irreducible and subdirectly irreducible rings play a role in the general theory of structure for rings. Subdirectly irreducible algebras have also found use in number theory.
This article follows the convention that rings have multiplicative identity, but are not necessarily commutative.
Definitions
The terms "meet-reducible", "directly reducible" and "subdirectly reducible" are used when a ring is not meet-irreducible, or not directly irreducible, or not subdirectly irreducible, respectively.
The following conditions are equivalent for a commutative ring R:
R is meet-irreducible;
the zero ideal in R is irreducible, i.e. the intersection of two non-zero ideals of A always is non-zero.
The following conditions are equivalent for a ring R:
R is directly irreducible;
R has no central idempotents except for 0 and 1.
The following conditions are equivalent for a ring R:
R is subdirectly irreducible;
when R is written as a subdirect product of rings, then one of the projections of R onto a ring in the subdirect product is an isomorphism;
The intersection of all non-zero ideals of R is non-zero.
The following conditions are equivalent for a commutative ring R:
the spectrum of R is irreducible.
R possesses exactly one minimal prime ideal (this prime ideal may be the zero ideal);
Examples and properties
If R is subdirectly irreducible or meet-irreducible, then it is also directly irreducible, but the converses are not true.
All integral domains are meet-irreducible, but not all integral domains are subdirectly irreducible (e.g. Z). In fact, a commutative ring is a domain if and only if it is both meet-irreducible and reduced.
A commutative ring is a domain if and only if its spectrum is irreducible and it is reduced.
The quotient ring Z/4Z is a ring which has all three senses of irreducibility, but it is not a domain. Its only proper ideal is 2Z/4Z, which is maximal, hence prime. The ideal is also minimal.
The direct product of two non-zero rings is never directly irreducible, and hence is never meet-irreducible or subdirectly irreducible. For example, in Z × Z the intersection of the non-zero ideals {0} × Z and Z × {0} is equal to the zero ide |
https://en.wikipedia.org/wiki/Ernesto%20Estrada%20%28scientist%29 | Ernesto Estrada (born 2 May 1966) is a Cuban-Spanish scientist. He has been Senior ARAID Researcher at the Institute of Mathematics and Applications at the University of Zaragoza, Spain since 2019. Before that he was the chair in Complexity Science, and full professor at the Department of Mathematics and Statistics of the University of Strathclyde, Glasgow, United Kingdom. He is known by his contributions in different disciplines, including mathematical chemistry and complex network theory.
Birth and education
Estrada was born in the city of Sancti Spiritus, in the central region of Cuba. Since the age of 11 he studied in a school which specialized in exact sciences. He later studied for a technical degree in Analytical chemistry in the technological institute IPQI Mártires de Girón Havana. At the age of 18 years, and before entering the university, he presented his first scientific paper in an international congress together with his mentor, Dr. Jose F. Fernández-Bertrán. The paper was about the detection of polyatomic anions in matrices of NaCl using Infrared spectroscopy. Between 1985 and 1990, he studied chemical sciences at the Central University of Las Villas in Santa Clara, Cuba, where he obtained his degree in only 4 of the 5 years established for the program. In the first years after graduation, Estrada investigated on the organic synthesis and the use of spectroscopy for the characterization of new chemical entities with pharmacological activity. This research introduced him to the world of Computational chemistry due to the requirement of using efficient methods for drug design and drug discovery. He is one of the co-authors of the patent for the bactericide and fungicide drug Furvina. In 1997, he obtained his PhD in Mathematical chemistry under the direction of Luis A. Montero Cabrera on the topic of "Graph Theory Applied to Molecular Design".
Academic career
After completing his PhD, Estrada spent some time as Research fellow at the University of Valencia, Spain with Prof. Jorge Galvez working on drug design and at the Hebrew University of Jerusalem with Prof. David Avnir working on molecular symmetry numbers and rotational partition functions. In 2000, he emigrated to Spain. Between 2002 and 2003, Estrada worked as a scientist at the Safety and Environmental Assurance Centre, Unilever in Colworth, U.K. He then obtained a position as "Ramón y Cajal" researcher at the University of Santiago de Compostela, Spain. Between 2008 and 2018, Estrada occupied the chair in Complexity Science at the University of Strathclyde. In 2019 he became ARAID Researcher at the University of Zaragoza. In 2021 he incorporated to the IFISC research staff.
Research and achievements
Estrada has been a major contributor in the study of complex networks, where he has developed several approaches to investigate the network topology and network dynamics. An index introduced by him in 1999 to characterize the degree of protein folding, and then generalized to t |
https://en.wikipedia.org/wiki/Conrad%20Wolfram | Conrad Wolfram (born 10 June 1970) is a British technologist and businessman known for his work in information technology and mathematics education reform. In June 2020, Wolfram released his first book, The Math(s) Fix: An Education Blueprint for the AI Age.
Education and early life
Born in Oxford, England, in 1970, Wolfram was educated at Dragon School and Eton College where he learned to program on a BBC Micro. He was an undergraduate student at Pembroke College, Cambridge where he studied the Natural Sciences tripos graduating with a Master of Arts degree from the University of Cambridge.
Career
Wolfram has been a proponent of Computer-Based Math—a reform of mathematics education to "rebuild the curriculum assuming computers exist."
and is the founder of computerbasedmath.org.
He argues, "There are a few cases where it is important to do calculations by hand, but these are small fractions of cases. The rest of the time you should assume that students should use a computer just like everyone does in the real world." And that "School mathematics is very disconnected from mathematics used to solve problems in the real world". In an interview with the Guardian he described the replacement of hand calculation by computer use as "democratising expertise". He argues that "A good guide to how and what you should do with a computer in the classroom is what you'd do with it outside. As much as possible, use real-world tools in the classroom in an open-ended way not special education-only closed-ended approaches."
In 2009, he spoke about education reform at the TEDx Conference at the EU Parliament. and again at TED Global 2010 where he argued that "Maths should be more practical and more conceptual, but less mechanical," and that "Calculating is the machinery of math - a means to an end."
In August 2012, he was a member of the judging panel at the Festival of Code, the culmination of Young Rewired State 2012. Wolfram is also part of Flooved advisory board.
On 10 June 2020, Wolfram released his first book, The Math(s) Fix: An Education Blueprint for the AI Age. The book summarises Wolfram's thoughts on the current state of mathematics education and sets out a vision for a new core subject based on computational thinking.
Wolfram research
Wolfram co-founded Wolfram Research Europe Ltd. in 1991 and remains its CEO. In 1996, he additionally became Strategic and International Director of Wolfram Research, Inc., making him also responsible for Wolfram Research Asia Ltd, and communications such as the wolfram.com website.
Wolfram Research was founded by his brother Stephen Wolfram, the maker of Mathematica software and the Wolfram Alpha knowledge engine.
Wolfram has led the effort to move the use of Mathematica from pure computation system to development and deployment engine, instigating technology such as the Mathematica Player family and web Mathematica and by pushing greater automation within the system.
He has also led the focus on interactive p |
https://en.wikipedia.org/wiki/Niklas%20Andersson%20%28ice%20hockey%2C%20born%201986%29 | Niklas Andersson (born March 5, 1986) is a professional Swedish ice hockey player. He is currently a defenceman for Alba-Volán in the EBEL.
Career statistics
External links
1986 births
Fehérvár AV19 players
Almtuna IS players
Djurgårdens IF Hockey players
Living people
Swedish ice hockey defencemen
Ice hockey people from Stockholm
Swedish expatriate sportspeople in Hungary
Swedish expatriate ice hockey people
Expatriate ice hockey players in Hungary |
https://en.wikipedia.org/wiki/Victoria%20Azarenka%20career%20statistics | This is a list of the main career statistics of Belarusian professional tennis player Victoria Azarenka. To date, she has won 34 WTA Tour level tournaments (21 in singles, ten in doubles and three in mixed doubles). Her most major titles are two Grand Slam singles titles (back-to-back Australian Open in 2012 and 2013). From the same tier, she also has four finals in doubles as well as two mixed doubles (2007 US Open and 2008 French Open). Qualified a couple of times at the year-end WTA Finals, she reached one final in 2011 when she lost to Petra Kvitová.
She is also successful at the WTA 1000 tournaments, winning 10 in singles (six as Mandatory and four as non-Mandatory). In doubles, she won four (two Mandatory and two non-Mandatory). In 2016, in singles she achieved Sunshine Double after winning Indian Wells and Miami Open in the same year. Among other achievements, she was also successful at the national tournaments, playing for Belarus. At the 2012 Summer Olympics in London, she is the bronze medalist in women's singles and gold medalist in mixed doubles with Max Mirnyi.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments, Fed Cup/Billie Jean King Cup and Olympic Games are included in win–loss records.
Singles
Current after the 2023 Italian Open.
Doubles
Current through the 2022 Cincinnati Masters.
Mixed doubles
Grand Slam finals
Singles: 5 (2 titles, 3 runner-ups)
Doubles: 4 (4 runner-ups)
Mixed doubles: 4 (2 titles, 2 runner-ups)
Other significant finals
Olympics finals
Singles: 1 (bronze medal)
Mixed doubles: 1 (gold medal)
Year-end championships
Singles finals: 1 (1 runner-up)
WTA 1000 finals
Singles: 15 (10 titles, 5 runner-ups)
Doubles: 8 (5 titles, 3 runner-ups)
WTA career finals
Singles: 41 (21 titles, 20 runner-ups)
Doubles: 21 (10 titles, 11 runner–ups)
ITF Circuit finals
Singles: 3 (1 title, 2 runner–ups)
Doubles: 4 (3 titles, 1 runner–up)
Junior Grand Slam tournament finals
Singles: 2 (2 titles)
Doubles: 4 (4 titles)
WTA Tour career earnings
Current after the 2022 WTA Finals.
Career Grand Slam statistics
Career Grand Slam seedings
The tournaments won by Azarenka are in boldface, and advanced into finals by Azarenka are in italics.
Best Grand Slam results details
Grand Slam winners are in boldface, and runner–ups are in italics.
Record against other players
No. 1 wins
Record against top 10 players
She has a record against players who were, at the time the match was played, ranked in the top 10.
Double bagel matches (6–0, 6–0)
Winning streaks
Victoria Azarenka has one 20+-match win streak: 26 (2012)
26-match win streak 2012
Azarenka's 26 match winning streak was the best start to a WTA Tour season since Martina Hingis won 37 in a row in 1997.
Notes
References
External links
Victoria Azarenka
Tennis career statistics |
https://en.wikipedia.org/wiki/Sperminator | Sperminator may refer to:
Cecil Jacobson, an American former fertility doctor who used his own sperm to impregnate his patients without informing them.
Ari Nagel, an American maths professor and sperm donor. |
https://en.wikipedia.org/wiki/Motides | Motides (; ) is a small village in Cyprus, to the east of Lapithos. De facto, it is under the control of Northern Cyprus.
According to the Statistics Office of Northern Cyprus, in 2011 there were 180 inhabitants in the village.
References
Communities in Kyrenia District
Populated places in Girne District |
https://en.wikipedia.org/wiki/Vieta%20jumping | In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it can be used to produce new solutions of a quadratic Diophantine equation from known ones. There exist multiple variations of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas.
History
Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic forms. For example, it was used in the analysis of Markov equation back in 1879 and in the 1953 paper of Mills.
In 1988, the method came to the attention to mathematical olympiad problems in the light of the first olympiad problem to use it in a solution that was proposed for the International Mathematics Olympiad and assumed to be the most difficult problem on the contest:
Let and be positive integers such that divides . Show that is the square of an integer.
Arthur Engel wrote the following about the problem's difficulty:
Among the eleven students receiving the maximum score for solving this problem were Ngô Bảo Châu, Ravi Vakil, Zvezdelina Stankova, and Nicușor Dan. Emanouil Atanassov (from Bulgaria) solved the problem in a paragraph and received a special prize.
Standard Vieta jumping
The concept of standard Vieta jumping is a proof by contradiction, and consists of the following four steps:
Assume toward a contradiction that some solution () exists that violates the given requirements.
Take the minimal such solution according to some definition of minimality.
Replace some by a variable x in the formulas, and obtain an equation for which is a solution.
Using Vieta's formulas, show that this implies the existence of a smaller solution, hence a contradiction.
Example
Problem #6 at IMO 1988: Let and be positive integers such that divides . Prove that is a perfect square.
Fix some value that is a non-square positive integer. Assume there exist positive integers for which .
Let be positive integers for which and such that is minimized, and without loss of generality assume .
Fixing , replace with the variable to yield . We know that one root of this equation is . By standard properties of quadratic equations, we know that the other root satisfies and .
The first expression for shows that is an integer, while the second expression implies that since is not a perfect square. From it further follows that , and hence is a positive integer. Finally, implies that , hence , and thus , which contradicts the minimality of .
Constant descent Vieta jumping
The method of constant descent Vieta jumping is used when we wish to prove a statement regarding a constant having something to do with the relation between and . Unlike standard Vieta jumping, constant descent is not a proof by contradiction, and it con |
https://en.wikipedia.org/wiki/Hurwitz%27s%20theorem%20%28complex%20analysis%29 | In mathematics and in particular the field of complex analysis, Hurwitz's theorem is a theorem associating the zeroes of a sequence of holomorphic, compact locally uniformly convergent functions with that of their corresponding limit. The theorem is named after Adolf Hurwitz.
Statement
Let {fk} be a sequence of holomorphic functions on a connected open set G that converge uniformly on compact subsets of G to a holomorphic function f which is not constantly zero on G. If f has a zero of order m at z0 then for every small enough ρ > 0 and for sufficiently large k ∈ N (depending on ρ), fk has precisely m zeroes in the disk defined by |z − z0| < ρ, including multiplicity. Furthermore, these zeroes converge to z0 as k → ∞.
Remarks
The theorem does not guarantee that the result will hold for arbitrary disks. Indeed, if one chooses a disk such that f has zeroes on its boundary, the theorem fails. An explicit example is to consider the unit disk D and the sequence defined by
which converges uniformly to f(z) = z − 1. The function f(z) contains no zeroes in D; however, each fn has exactly one zero in the disk corresponding to the real value 1 − (1/n).
Applications
Hurwitz's theorem is used in the proof of the Riemann mapping theorem, and also has the following two corollaries as an immediate consequence:
Let G be a connected, open set and {fn} a sequence of holomorphic functions which converge uniformly on compact subsets of G to a holomorphic function f. If each fn is nonzero everywhere in G, then f is either identically zero or also is nowhere zero.
If {fn} is a sequence of univalent functions on a connected open set G that converge uniformly on compact subsets of G to a holomorphic function f, then either f is univalent or constant.
Proof
Let f be an analytic function on an open subset of the complex plane with a zero of order m at z0, and suppose that {fn} is a sequence of functions converging uniformly on compact subsets to f. Fix some ρ > 0 such that f(z) ≠ 0 in 0 < |z − z0| ≤ ρ. Choose δ such that |f(z)| > δ for z on the circle |z − z0| = ρ. Since fk(z) converges uniformly on the disc we have chosen, we can find N such that |fk(z)| ≥ δ/2 for every k ≥ N and every z on the circle, ensuring that the quotient fk′(z)/fk(z) is well defined for all z on the circle |z − z0| = ρ. By Weierstrass's theorem we have uniformly on the disc, and hence we have another uniform convergence:
Denoting the number of zeros of fk(z) in the disk by Nk, we may apply the argument principle to find
In the above step, we were able to interchange the integral and the limit because of the uniform convergence of the integrand. We have shown that Nk → m as k → ∞. Since the Nk are integer valued, Nk must equal m for large enough k.
This proof contains a gap towards the end. Indeed, one should still argue why the functions do not have any other zeroes or poles inside the small disc.
See also
Rouché's theorem
References
John B. Conway. Functions o |
https://en.wikipedia.org/wiki/Hurwitz%27s%20theorem%20%28number%20theory%29 | In number theory, Hurwitz's theorem, named after Adolf Hurwitz, gives a bound on a Diophantine approximation. The theorem states that for every irrational number ξ there are infinitely many relatively prime integers m, n such that
The condition that ξ is irrational cannot be omitted. Moreover the constant is the best possible; if we replace by any number and we let (the golden ratio) then there exist only finitely many relatively prime integers m, n such that the formula above holds.
The theorem is equivalent to the claim that the Markov constant of every number is larger than .
References
Diophantine approximation
Theorems in number theory |
https://en.wikipedia.org/wiki/Eckmann%E2%80%93Hilton%20duality | In the mathematical disciplines of algebraic topology and homotopy theory, Eckmann–Hilton duality in its most basic form, consists of taking a given diagram for a particular concept and reversing the direction of all arrows, much as in category theory with the idea of the opposite category. A significantly deeper form argues that the fact that the dual notion of a limit is a colimit allows us to change the Eilenberg–Steenrod axioms for homology to give axioms for cohomology. It is named after Beno Eckmann and Peter Hilton.
Discussion
An example is given by currying, which tells us that for any object , a map is the same as a map , where is the exponential object, given by all maps from to . In the case of topological spaces, if we take to be the unit interval, this leads to a duality between and , which then gives a duality between the reduced suspension , which is a quotient of , and the loop space , which is a subspace of . This then leads to the adjoint relation , which allows the study of spectra, which give rise to cohomology theories.
We can also directly relate fibrations and cofibrations: a fibration is defined by having the homotopy lifting property, represented by the following diagram
and a cofibration is defined by having the dual homotopy extension property, represented by dualising the previous diagram:
The above considerations also apply when looking at the sequences associated to a fibration or a cofibration, as given a fibration we get the sequence
and given a cofibration we get the sequence
and more generally, the duality between the exact and coexact Puppe sequences.
This also allows us to relate homotopy and cohomology: we know that homotopy groups are homotopy classes of maps from the n-sphere to our space, written , and we know that the sphere has a single nonzero (reduced) cohomology group. On the other hand, cohomology groups are homotopy classes of maps to spaces with a single nonzero homotopy group. This is given by the Eilenberg–MacLane spaces and the relation
A formalization of the above informal relationships is given by Fuks duality.
See also
Model category
References
Duality theories
Algebraic topology |
https://en.wikipedia.org/wiki/Anna%20Kournikova%20career%20statistics | This is a list of the main career statistics of retired professional tennis player Anna Kournikova.
Significant finals
Singles
WTA Tier I
Doubles
Grand Slam
WTA Tour Championships
WTA Tier I
Mixed doubles
Grand Slam
WTA Tour finals
Singles (4)
Runners-up (4)
Doubles (28)
Wins (16)
Runners-up (12)
ITF Circuit finals
Singles: 2 (2–0)
Doubles (0–1)
Singles performance timeline
Doubles performance timeline
Head vs. Head Record
Jennifer Capriati 3-1
Arantxa Sánchez Vicario 3-2
Daniela Hantuchová 2-0
Elena Dementieva 1-0
Nadia Petrova 1-0
Anastasia Myskina 0-1
Dinara Safina 0-1
Amélie Mauresmo 1-2
Steffi Graf 1-2
Jelena Dokic 0-2
Dominique Monami 0-2
Serena Williams 0-2
Kim Clijsters 1-4
Justine Henin 0-4
Monica Seles 1-5
Lindsay Davenport 3-7
Venus Williams 0-8
Martina Hingis 1-11
WTA Tour career earnings
References
External links
Official website of Anna Kournikova
Anna Kournikova at the WTA Tour's official website
Kournikova, Anna |
https://en.wikipedia.org/wiki/Thomas%20M.%20Liggett | Thomas Milton Liggett (March 29, 1944 – May 12, 2020) was a mathematician at the University of California, Los Angeles. He worked in probability theory, specializing in interacting particle systems.
Early life
Thomas Milton Liggett was born on March 29, 1944, in Danville, Kentucky. Liggett moved at the age of two with his missionary parents to Latin America, where he was educated in Bueno Aires, Argentina and San Juan, Puerto Rico. He graduated from Oberlin College with a Bachelor of Arts in 1965, where he was influenced towards probability by Samuel Goldberg (b. 1925), an ex-student of William Feller. He moved to Stanford, taking classes with Kai Lai Chung, and writing his thesis, Weak Convergence of Conditioned Sums of Independent Random Vectors, in 1969 with advisor Samuel Karlin on problems associated with the invariance principle. He graduated with a Master of Science in 1966 and a Doctor of Philosophy in 1969.
Career
Liggett joined the faculty at UCLA in 1969, where he spent his entire career. He became a professor in the mathematics department in 1976, and served as department chair from 1991 to 1994. He retired in 2011, but remained active within the department. He was the advisor of Norman Matloff.
Liggett had contributed to numerous areas of probability theory, including subadditive ergodic theory, random graphs, renewal theory, and was best known for his pioneering work on interacting particle systems, including the contact process, the voter model, and the exclusion process. His two books in this field have been influential.
Liggett was the managing editor of the Annals of Probability from 1985–1987. He held a Sloan Research Fellowship from 1973–1977, and a Guggenheim Fellowship from 1997–1998. He was the Wald Memorial Lecturer of the Institute of Mathematical Statistics in 1996, and was elected to the National Academy of Sciences in 2008. He had been elected to the American Academy of Arts & Sciences in 2012, and in 2012 he also became a fellow of the American Mathematical Society.
Personal life
Liggett married Christina Marie Goodale on August 19, 1972. They had two children, Timothy and Amy. Liggett died on May 12, 2020, in Los Angeles.
Notes
External links
2020 deaths
1944 births
20th-century American mathematicians
21st-century American mathematicians
People from Danville, Kentucky
Oberlin College alumni
Stanford University alumni
University of California, Los Angeles faculty
Probability theorists
Fellows of the American Mathematical Society
Members of the United States National Academy of Sciences
Annals of Probability editors |
https://en.wikipedia.org/wiki/Tart%C3%A1 | Vinícius Silva Soares, usually known simply as Tartá (Rio de Janeiro, April 13, 1989) is a Brazilian footballer.
Career statistics
References
External links
1989 births
Living people
Footballers from Rio de Janeiro (city)
Brazilian men's footballers
Brazilian expatriate men's footballers
Fluminense FC players
Club Athletico Paranaense players
Esporte Clube Vitória players
Kashima Antlers players
Criciúma Esporte Clube players
Goiás Esporte Clube players
Joinville Esporte Clube players
Ulsan Hyundai FC players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
K League 1 players
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in South Korea
Brazilian expatriate sportspeople in South Korea
Brazilian expatriate sportspeople in Thailand
Ubon United F.C. players
Boavista Sport Club players
Men's association football midfielders |
https://en.wikipedia.org/wiki/C.%20N.%20S.%20Iyengar | C. N. S. Iyengar (- died 1972) was an Indian professor of mathematics and the founder head of the department of mathematics, Karnatak University, Dharwar. The department was started in the year 1956 under the leadership of Iyengar.
Iyengar received a D.Sc. (c.c) from Calcutta University, Calcutta. After retiring from the Central College of Bangalore University, he joined the Karnataka University, Dharwar in 1956 and retired from there in 1965. He had contributed extensively to the field of differential geometry and Riemannian geometry. Iyengar wrote a book The History of Ancient Indian Mathematics (1967 - World Press)
References
1972 deaths
Kannada people
People from Dharwad
Academic staff of Bangalore University
University of Calcutta alumni
Year of birth missing
Academic staff of Karnatak University
20th-century Indian mathematicians
Scientists from Bangalore |
https://en.wikipedia.org/wiki/Conic%20bundle | In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form:
Conic bundles can be considered a Severi–Brauer surface, or, more precisely, a Châtelet surface. This can be a double covering of a ruled surface. Through an isomorphism, it can be associated with a symbol in the second Galois cohomology of the field . In practice, it is more commonly observed as a surface with a well-understood divisor class group, and the simplest cases share with Del Pezzo surfaces the property of being a rational surface. But many problems of contemporary mathematics remain open, notably (for those examples which are not rational) the question of unirationality.
A naive point of view
In order to properly express a conic bundle, the initial step involves simplifying the quadratic form on the left side. This can be achieved through an alteration, as such:
In a second step, it should be placed in a projective space in order to complete the surface at infinity.
To achieve this, we write the equation in homogeneous coordinates and expresses the first visible part of the fiber:
That is not enough to complete the fiber as non-singular (smooth and proper), and then glue it to infinity by a change of classical maps.
Seen from infinity, (i.e. through the change ), the same fiber (excepted the fibers and ), written as the set of solutions where appears naturally as the reciprocal polynomial of . Details are below about the map-change .
The fiber c
Going a little further, while simplifying the issue, limit to cases where the field is of characteristic zero and denote by any integer except zero. Denote by P(T) a polynomial with coefficients in the field , of degree 2m or 2m − 1, without multiple root. Consider the scalar a.
One defines the reciprocal polynomial by , and the conic bundle Fa,P as follows:
Definition
is the surface obtained as "gluing" of the two surfaces and of equations
and
along the open sets by isomorphisms
and .
One shows the following result:
Fundamental property
The surface Fa,P is a k smooth and proper surface, the mapping defined by
by
and the same on gives to Fa,P a structure of conic bundle over P1,k.
See also
Algebraic surface
Intersection number (algebraic geometry)
List of complex and algebraic surfaces
References
Algebraic geometry
Algebraic varieties |
https://en.wikipedia.org/wiki/Manohar%20Vartak | Manohar N. Vartak (29 November 1926–1997) was a Professor Emeritus of Mathematics and Statistics at the Indian Institute of Technology, Mumbai. Professor Vartak was one of founder members of the Department of Mathematics and the head of the department (1973–1977). He was superannuated in 1986. Vartak was specialized in balanced incomplete block designs, graph theory and operations research.
Vartak received his PhD (1961) in statistics from University of Mumbai under the guidance of professor M. C. Chakrabarti. Vartak guided numerous students for their PhDs.
References
Mathematics Department – I.I.T, Mumbai
Indian combinatorialists
University of Mumbai alumni
Scientists from Mumbai
Academic staff of IIT Bombay
1997 deaths
1926 births
20th-century Indian mathematicians |
https://en.wikipedia.org/wiki/Vector%20%28mathematics%20and%20physics%29 | In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.
Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.
The term vector is also used, in some contexts, for tuples, which are finite sequences of numbers of a fixed length.
Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called a Euclidean vector space, and a vector space formed by tuples is called a coordinate vector space.
Many vector spaces are considered in mathematics, such as extension field, polynomial rings, algebras and function spaces. The term vector is generally not used for elements of these vectors spaces, and is generally reserved for geometric vectors, tuples, and elements of unspecified vector spaces (for example, when discussing general properties of vector spaces).
Vectors in Euclidean geometry
Vector spaces
Vectors in algebra
Every algebra over a field is a vector space, but elements of an algebra are generally not called vectors. However, in some cases, they are called vectors, mainly due to historical reasons.
Vector quaternion, a quaternion with a zero real part
Multivector or -vector, an element of the exterior algebra of a vector space.
Spinors, also called spin vectors, have been introduced for extending the notion of rotation vector. In fact, rotation vectors represent well rotations locally, but not globally, because a closed loop in the space of rotation vectors may induce a curve in the space of rotations that is not a loop. Also, the manifold of rotation vectors is orientable, while the manifold of rotations is not. Spinors are elements of a vector subspace of some Clifford algebra.
Witt vector, an infinite sequence of elements of a commutative ring, which belongs to an algebra over this ring, and has been introduced for handling carry propagation in the operations on p-adic numbers.
Data represented by vectors
The set of tuples of real numbers has a natural structure of vector space defined by component-wise addition and scalar multiplication. It is common to call these tuples vectors, even in contexts where vector-space operations do not apply. More generally, when some data can be represented naturally by vectors, they are often called vectors even when addition and scalar multiplication of vectors are not valid operations on these data. Here are some examples.
|
https://en.wikipedia.org/wiki/Keith%20Williams%20%28bodybuilder%29 | Keith Anthony Lamar Williams (born May 16, 1973) is an American bodybuilder and former American football player. He and his wife reside in Lawrence, Kansas.
Physical statistics
Age: 45
Height: 5'10”
Weight: Contest 245-250 lbs (111–113 kg or 17 stone, 7 lbs to 17 stone, 9 lbs) Off Season 287
NPC Contest Record
September 2008 NPC North Americans - Superheavyweight -4th
July 2008 NPC Flex Wheeler Classic, Superheavyweight -Class and Overall Winner
June 2008 NPC JR. Nationals, Super heavyweight division placed - 2nd
July 2007 USA Championships Las Vegas, NV Super heavyweight division placed - 8th
June 2007 Jr. National Championships Chicago, IL Super heavyweight division - 4th
July 2006 USA Championships Las Vegas, NV Super heavyweight division - 11th
July 2006 Mr. Minnesota/ Mr. Midwest Super heavyweight Champion
June 2006 Junior Nationals Chicago, IL Heavyweight division - 4th
April 2005 Gopher State Competition MN Heavyweight Champion
Football
(5-11, 220, 4.20)
Positions: RB, KR, Special Teams Specialist, CB
Minnesota Vikings
Signed February 2002 as DB
Green Bay Packers
(CB/KR) Signed: April 12, 1999
St. Cloud State University:
1995: All-Conference in football for SCSU North Central Conference All-American
References
Packers, retrieved
Vikings, Retrieved June 14, 2009.
"Keith Williams - Junior Nationals Bodybuilding, Fitness & Figure Championships 2008". Muscular Development Online Magazine, Retrieved .
St. Cloud State University,
1973 births
Living people
American bodybuilders |
https://en.wikipedia.org/wiki/D%C3%B6wletmyrat%20Ata%C3%BDew | Döwletmyrat Ataýew (born March 16, 1983) is a professional Turkmen football player whose last known club was Shurtan Guzar.
Career statistics
International career statistics
Goals for Senior National Team
References
External links
Turkmenistan men's footballers
Living people
1983 births
Navbahor Namangan players
FC Shurtan Guzar players
FC Aşgabat players
Karvan FK players
Turkmenistan expatriate men's footballers
Expatriate men's footballers in Uzbekistan
Turkmenistan expatriate sportspeople in Uzbekistan
Expatriate men's footballers in Azerbaijan
Turkmenistan expatriate sportspeople in Azerbaijan
Turkmenistan men's international footballers
Men's association football forwards |
https://en.wikipedia.org/wiki/Sigur%C3%B0ur%20Helgason%20%28mathematician%29 | Sigurdur Helgason (born 30 September 1927; Icelandic: Sigurður) is an Icelandic mathematician whose research has been devoted to the geometry and analysis on symmetric spaces. In particular, he has used new integral geometric methods to establish fundamental existence theorems for differential equations on symmetric spaces as well as some new results on the representations of their isometry groups. He also introduced a Fourier transform on these spaces and proved the principal theorems for this
transform, the inversion formula, the Plancherel theorem and the analog of the Paley–Wiener theorem.
He was born in Akureyri, Iceland. In 1954, he earned a PhD from Princeton University under Salomon Bochner. Since 1965, Helgason has been a professor of mathematics at the Massachusetts Institute of Technology.
He was winner of the 1988 Leroy P. Steele Prize for Seminal Contributions for his books Groups and Geometric Analysis and Differential Geometry, Lie Groups and Symmetric Spaces. This was followed by the 2008 book Geometric Analysis on Symmetric Spaces. On May 31, 1996 Helgason received an honorary doctorate from the Faculty of
Science and Technology at Uppsala University, Sweden.
He has been a fellow of the American Academy of Arts and Sciences since 1970. In 2012, he became a fellow of the American Mathematical Society.
Selected works
Articles
Books
Differential geometry and symmetric spaces. Academic Press 1962, AMS 2001
Analysis on Lie groups and homogeneous spaces. AMS 1972
Differential geometry, Lie groups and symmetric spaces. Academic Press 1978, 7th edn. 1995
The Radon Transform. Birkhäuser, 1980, 2nd edn. 1999
Topics in harmonic analysis on homogeneous spaces. Birkhäuser 1981
Groups and geometric analysis: integral geometry, invariant differential operators and spherical functions. Academic Press 1984, AMS 1994
Geometric analysis on symmetric spaces. AMS 1994, 2nd. edn. 2008
References
Sources
External links
Sigurdur Helgason – Publications – MIT Mathematics
1927 births
Living people
Sigurdur Helgason
Differential geometers
Massachusetts Institute of Technology School of Science faculty
Fellows of the American Mathematical Society
20th-century mathematicians
21st-century mathematicians
Sigurdur Helgason
Sigurdur Helgason
Princeton University alumni |
https://en.wikipedia.org/wiki/American%20Time%20Use%20Survey | The American Time Use Survey (ATUS), sponsored by the Bureau of Labor Statistics (BLS) and conducted by the United States Census Bureau (USCB), is a time-use survey which provides measures of the amounts of time people spend on various activities, including working, leisure, childcare, and household activities. The survey has been conducted annually since 2003.
Methodology
Eligible survey participants are households that have completed all eight months of the Current Population Survey (CPS). Of the eligible households, those representing a range of demographic characteristics are selected to participate in the survey. Between 2–5 months after the household's eighth and final CPS interview, one randomly-selected person of at least fifteen years of age is selected from each household to be interviewed for the ATUS and asked questions about their time use.
Sample size
Since December 2003, the ATUS sample has been 2,190 households per month (approximately 26,400 households per year). The ATUS sample was initially 3,375 households per month (approximately 40,500 households per year), but was reduced to lower costs. The selected households are categorized into one of twelve strata based on race/ethnicity (Hispanic, Non-Hispanic black, Non-Hispanic non-black) and household type (child under age six, child between age six and age seventeen, single adult no children, two or more adults no children).
Data
The ATUS data includes:
Time spent by the civilian population in primary activities, including daily averages by age, sex, race, Hispanic or Latino ethnicity, marital status, and educational attainment, averages for weekdays vs weekends and holidays, and averages by time of day
Time spent by employed persons working and working at home or at their workplace, by full-time and part-time status and sex, jobholding status, educational attainment, class of worker, earnings, industry, occupation, time of day, and day of week
Time spent by married mothers and married fathers in primary activities, including averages by employment status
Time spent by over age eighteen civilian population in primary activities by age of youngest household child and sex, for employed vs non-employed
Time spent in leisure and sports by selected characteristics
From 2005–2010, the ATUS included questions relating to overnight trips. In January 2011, the overnight trips questions were replaced by questions relating to eldercare.
Modules
The ATUS sometimes includes special questions, called modules, at the end of the interview. The ATUS added an Eating & Health module from 2006–2008 and 2014–2016, a Well-Being Module in 2010, 2012, and 2013, and a Leave module, relating to workers' access to leave, in 2011.
Data uses
ATUS data is used by the Bureau of Economic Analysis (BEA) to account for the value of household production, the Bureau of Transportation Statistics (BTS) in their Passenger Travel: Facts and Figures report, and the Economic Research Service (ERS) to examine ho |
https://en.wikipedia.org/wiki/Constructible%20set%20%28topology%29 | In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure.
They are used particularly in algebraic geometry and related fields. A key result known as Chevalley's theorem
in algebraic geometry shows that the image of a constructible set is constructible for an important class of mappings
(more specifically morphisms) of algebraic varieties (or more generally schemes).
In addition, a large number of "local" geometric properties of schemes, morphisms and sheaves are (locally) constructible.
Constructible sets also feature in the definition of various types of constructible sheaves in algebraic geometry
and intersection cohomology.
Definitions
A simple definition, adequate in many situations, is that a constructible set is a finite union of locally closed sets. (A set is locally closed if it is the intersection of an open set and closed set.)
However, a modification and another slightly weaker definition are needed to have definitions that behave better with "large" spaces:
Definitions: A subset of a topological space is called retrocompact if is compact for every compact open subset . A subset of is constructible if it is a finite union of subsets of the form where both and are open and retrocompact subsets of .
A subset is locally constructible if there is a cover of consisting of open subsets with the property that each is a constructible subset of .
Equivalently the constructible subsets of a topological space are the smallest collection of subsets of that (i) contains all open retrocompact subsets and (ii) contains all complements and finite unions (and hence also finite intersections) of sets in it. In other words, constructible sets are precisely the Boolean algebra generated by retrocompact open subsets.
In a locally noetherian topological space, all subsets are retrocompact, and so for such spaces the simplified definition given first above is equivalent to the more elaborate one. Most of the commonly met schemes in algebraic geometry (including all algebraic varieties) are locally Noetherian, but there are important constructions that lead to more general schemes.
In any (not necessarily noetherian) topological space, every constructible set contains a dense open subset of its closure.
Terminology: The definition given here is the one used by the first edition of EGA and the Stacks Project. In the second edition of EGA constructible sets (according to the definition above) are called "globally constructible" while the word "constructible" is reserved for what are called locally constructible above.
Chevalley's theorem
A major reason for the importance of constructible sets in algebraic geometry is that the image of a (locally) constructible set is also (locally) constructible for a large class of maps (or "morphisms"). The key result is:
Chevalley's theorem. If is a finitely presented morphism of schemes and is a locally constructible subset, then is |
https://en.wikipedia.org/wiki/Boris%20Becker%20career%20statistics | This is a list of the main career statistics of professional tennis player Boris Becker.
Performance timelines
Singles
Doubles
Grand Slam finals
Singles: 10 (6–4)
Grand Prix / ATP year-end championships finals
Singles: 8 (3–5)
WCT year-end championships
Singles: 2 (1–1)
Grand Slam Cup
Singles: 1 (1–0)
ATP Super 9 finals (since 1990)
Singles: 11 (5–6)
Note: before the ATP took over running the men's professional tour in 1990 the Grand Prix Tour had a series of events that were precursors to the Masters Series known as the Grand Prix Super Series.
Olympic finals
Doubles: 1 (1–0)
Career finals
Singles: 77 (49 titles, 28 runner-ups)
Doubles: 27 (15 titles, 12 runner-ups)
Team competition: 6 (5 titles, 1 runner-up)
Top 10 wins
Becker has a 121–65 record (65.1%) against players who were, at the time the match was played, ranked in the top 10.
Record against No. 1 players
Becker's match record against players who have been ranked world No. 1.
External links
Becker, Boris |
https://en.wikipedia.org/wiki/Ivan%20Lendl%20career%20statistics | This is a list of the main career statistics of professional tennis player Ivan Lendl.
Grand Slam finals
Singles: 19 finals (8 titles, 11 runners-up)
Grand Prix year-end championships finals
Singles: 9 finals (5 titles, 4 runners-up)
Note: Lendl formerly held the record for the most final appearances at 9, until Federer broke it with his 10th final appearance in 2015. Roger Federer broke a tie with Lendl and Pete Sampras by claiming his sixth year-ending championship on November 27, 2011.
WCT year-end championships finals
Singles: 3 finals (2 titles, 1 runner-up)
Grand Prix Super Series / ATP Super 9 finals
Singles: 33 (22 titles, 11 runner-ups)
Singles performance timeline
Tournaments statistics
ATP win–loss includes WCT tournaments which were run outside Volvo Grand Prix and ATP Computer Ranking system during 1982-1984, also includes team events (Davis Cup, World Team Cup in Düsseldorf).
Career finals
Singles titles (94)
Runners-ups (52)
Other (non-ATP, invitational & special events) singles finals
Here are Lendl's tournament finals that are not included in the statistics on the Association of Tennis Professionals website. It includes non-ATP tournaments such as special, invitational and exhibition events.
Other singles titles - Draw at least 8 players (38)
Other singles titles - Draw less than 8 players (19)
Below are Lendl's winnings on exhibition tournaments (usually 4-men's draw)
Career doubles finals listed by ATP (16)
Doubles titles (6)
1979 (1): Berlin (CL) / (w/Kirmayr)
1980 (1): Barcelona (CL) / (w/Denton)
1984 (1): Wembley (IC) / (w/Gomez)
1985 (1): Stuttgart Outdoor (CL) / (w/Smid)
1986 (1): Fort Myers (H) / (w/Gomez)
1987 (1): Adelaide (G) / (w/Scanlon)
Doubles runners-up (10)
1979 (1): Florence (CL) / (w/Slozil)
1980 (2): Indianapolis (CL) / (w/Fibak), Cincinnati (H) / (w/Fibak)
1983 (1): San Francisco (IC) / (w/Van Patten)
1986 (1): Tokyo Indoor (IC) / (w/Gomez)
1988 (1): Monte Carlo (CL) / (w/Leconte)
1990 (1): Queen's Club (G) / (w/Leconte)
1990 (1): Sydney Indoor (IH) / (w/Edberg)
1992 (1): Barcelona (CL) / (w/Novacek)
1993 (1): Marseille (IC) / (w/Van Rensburg)
click on the year link expands all Lendl's doubles matches for the respective year listed on ATP website
Record against top players
Career prize money statistics
Top 10 wins
References
Lendl, Ivan |
https://en.wikipedia.org/wiki/Mats%20Wilander%20career%20statistics | This is a list of the main career statistics of Swedish former professional tennis player Mats Wilander whose career ran from 1980 until 1996.
Grand Slam finals
Singles: 11 (7 titles, 4 runner-ups)
Doubles: 3 (1 titles, 2 runner-ups)
Grand Prix year-end championships finals
Singles: 1 (0 titles, 1 runner-ups)
Doubles: 1 (0 titles, 1 runner-ups)
ATP career finals
Singles: 59 (33 titles, 26 runner-ups)
Doubles: 18 (7 titles, 11 runner-ups)
Singles performance timeline
Top 10 wins
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Stefan%20Edberg%20career%20statistics | This is a list of the main career statistics of former professional tennis player Stefan Edberg.
Grand Slam finals
Singles: 11 finals (6–5)
Doubles: 5 finals (3–2)
Grand Prix / ATP year-end championships finals
Singles: 2 (1–1)
WCT year-end championships finals
Singles: 1 (0–1)
ATP Super 9 finals (since 1990)
Singles: 9 (4–5)
Doubles: 2 finals (1–1)
Note: before the ATP took over running the men's professional tour in 1990 the Grand Prix Tour had a series of events that were precursors to the Masters Series known as the Grand Prix Super Series.
Performance timelines
Singles
Doubles
Career finals
Singles: 77 (41 titles, 36 runner-ups)
Doubles: 29 (18–11)
Junior career finals
Grand Slam finals
Singles: 4 (4–0)
1 The 1983 Australian Open was held in December.
Head-to-head record
Edberg's record against top 10 ranked players
Boris Becker 10–25
Ivan Lendl 14–13
Michael Chang 12–9
Mats Wilander 9–11
Brad Gilbert 15–4
Goran Ivanišević 9–10
Jakob Hlasek 15–1
Michael Stich 6–10
Miloslav Mečíř 10–5
Pete Sampras 6–8
Guy Forget 7–6
John McEnroe 6–7
Emilio Sánchez 9–3
Jimmy Connors 6–6
Thomas Muster 10-0
Anders Järryd 9–2
Aaron Krickstein 7–4
Jonas Svensson 10–0
Pat Cash 8–2
Jim Courier 4–6
Sergi Bruguera 6–3
Johan Kriek 6–3
Petr Korda 4–5
Andre Agassi 3–6
Kevin Curren 7–1
Jimmy Arias 7–0
Henri Leconte 6–1
Richard Krajicek 3–4
Todd Martin 3–4
Yannick Noah 6–0
Tim Mayotte 5–1
Andriy Medvedev 4–2
Cédric Pioline 4–2
Andrés Gómez 4–0
Wayne Ferreira 3–1
Karel Nováček 3–1
Patrick Rafter 3–0
Yevgeny Kafelnikov 1–2
Joakim Nyström 0–3
Tim Henman 2–0
Alberto Mancini 2–0
Henrik Sundström 1–1
José Luis Clerc 1–0
Carlos Moyá 1–0
Marcelo Ríos 1–0
Greg Rusedski 1–0
Eliot Teltscher 1–0
Jonas Björkman 0–1
Top 10 wins
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Bj%C3%B6rn%20Borg%20career%20statistics | This is a list of the main career statistics and records of retired Swedish professional tennis player Björn Borg. His professional career spanned from 1973 until 1984 with a brief comeback between 1991 and 1993.
Grand Slam finals
Singles: 16 (11 titles, 5 runner-ups)
Grand Prix year-end championships finals
Singles: 4 (2 titles, 2 runner-ups)
WCT year-end championship finals
Singles: 4 (1 title, 3 runner-ups)
Grand Prix Super Series finals
Singles: 20 (15 titles, 5 runner-ups)
Note: before the ATP took over running the men's professional tour in 1990 the Grand Prix Tour had a series of events that were precursors to the Masters Series known during some years as the Grand Prix Super Series.
Singles performance timeline
Career finals
Singles titles (66)
Runner-ups (27)
Singles titles – invitational tournaments exhibitions, or special events (35)
Non-ATP, exhibition, invitational, or special events – draw ≥ 8 (9)
Non-ATP, exhibition, invitational, or special events – draw < 8 (26)
Records and statistics
Youngest to win
In 1972 Borg became the youngest winner of a Davis Cup match at age 15.
Borg won his 11th Grand Slam singles title in 1981 aged 25 years and one day, the youngest male to reach that number of titles. By comparison, Roger Federer won his 11th aged 25 years and 324 days; Rafael Nadal was aged 26 years and 8 days; Pete Sampras won his 11th at almost age 27, Novak Djokovic at age 28, Roy Emerson at age 30, and Rod Laver at age 31.
Statistics
Borg's 66 official ATP career titles as listed on the Association of Tennis Professionals (ATP) website places him eighth on the Open Era list behind Jimmy Connors (109), Roger Federer (103), Ivan Lendl (94), Rafael Nadal (86), Novak Djokovic (81), John McEnroe (77), Rod Laver (72).
In 1979, Borg became the first tennis player to earn more than one million dollars in prize money in a single season.
On the list of open era winning streaks, Borg is first and second (49 consecutive tour matches in 1978, 48 in 1979–1980). The only other men with winning streaks of at least 40 matches are Guillermo Vilas (46), Ivan Lendl (44), Novak Djokovic (43), John McEnroe (42), and Roger Federer (41).
Borg holds third place for most consecutive wins on clay, with 46 victories in 1977–79. Only Rafael Nadal with 81 and Vilas with 53 have won more consecutive clay court matches.
Borg won 19 consecutive points on serve in the fifth set on two occasions: his 1980 Wimbledon final against McEnroe and his 1980 US Open quarterfinal against Roscoe Tanner.
Borg retired in 1983 with $3.6 million in career prize money, a record at the time.
According to the match scores listed on the ATP website, Borg bageled his opponents (sets won 6–0) 131 times in his career, compared to Federer's 93 bagels from 1999 through 2019 Basel.
Borg was inducted into the International Tennis Hall of Fame in 1987 at 30 years of age.
In 1999, Borg was elected the best Swedish sportsman ever by a jury in his home country. His tennis ri |
https://en.wikipedia.org/wiki/Double%20layer%20potential | In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions. Thus a double layer potential is a scalar-valued function of given by
where ρ denotes the dipole distribution, ∂/∂ν denotes the directional derivative in the direction of the outward unit normal in the y variable, and dσ is the surface measure on S.
More generally, a double layer potential is associated to a hypersurface S in n-dimensional Euclidean space by means of
where P(y) is the Newtonian kernel in n dimensions.
See also
Single layer potential
Potential theory
Electrostatics
Laplacian of the indicator
References
.
.
.
.
Potential theory |
https://en.wikipedia.org/wiki/Rod%20Laver%20career%20statistics | This is a list of the main career statistics of Australian former tennis player Rod Laver whose playing career ran from 1956 until 1977. He played as an amateur from 1956 until the end of 1962 when he joined Jack Kramer's professional circuit. As a professional he was banned from playing the Grand Slam tournaments as well as other tournaments organized by the national associations of the International Lawn Tennis Federation (ILTF). In 1968, with the advent of the Open Era, the distinction between amateurs and professionals disappeared and Laver was again able to compete in most Grand Slam events until the end of his career in 1977. During his career he won eleven Grand Slam tournaments, eight Pro Slam tournaments and five Davis Cup titles.
Grand Slam finals
Singles: 17 finals (11 titles, 6 runner-ups)
Doubles: 12 finals (6 titles, 6 runner-ups)
Mixed doubles: 5 finals (3 titles, 2 runner-ups)
Pro Slam finals
Before the Open Era.
Singles * : 14 (8 titles, 6 runners-up)
Performance timeline
Laver joined Professional tennis in 1963 and was unable to compete in the Grand Slams until the start of the Open Era at the 1968 French Open.
Season-ending championships
Masters Grand Prix (1 runner-up)
Tokyo 1970
Standings
WCT (2 runners-up)
Singles titles (198)
Overview
Amateur career (53 titles)
Professional career: before the Open Era (72 titles)
* 3 titles listed by the ATP website
Professional career: Open Era (73 titles)
Notes and sources for this section
This list of 198 singles titles from 1956 through 1976 may be incomplete.
Association of Tennis Professionals website
International Tennis Federation (1970). BP Yearbook of World Tennis 1970. London. Edited by Barrett, John.
International Tennis Federation (1971). World of Tennis '71. London. Edited by Barrett, John.
International Tennis Federation (1972). World of Tennis '72. London. Edited by Barrett, John.
International Tennis Federation (1973). World of Tennis '73. London. Edited by Barrett, John.
International Tennis Federation (1974). World of Tennis '74. London. Edited by Barrett, John.
International Tennis Federation (1975). World of Tennis '75. London. Edited by Barrett, John.
International Tennis Federation (1976). World of Tennis '76. London. Edited by Barrett, John.
McCauley, Joe (2003). The History of Professional Tennis. London.
Sutter, Michel (1992). Vainqueurs-Winners 1946–1991. Paris. (forewords by Arthur Ashe and Mark Miles).
Singles runner-ups during the Open Era (27)
As listed on the website of the Association of Tennis Professionals.
Doubles finals during the Open Era
Davis Cup
Laver won 16 out of 20 Davis Cup singles matches and all four of his doubles. Laver was a member of the victorious Australian Davis Cup teams in 1959, 1960, 1961, 1962 and 1973.
See also
Laver–Rosewall rivalry
Notes
References
External links
Laver, Rod |
https://en.wikipedia.org/wiki/Antithetic%20variates | In statistics, the antithetic variates method is a variance reduction technique used in Monte Carlo methods. Considering that the error in the simulated signal (using Monte Carlo methods) has a one-over square root convergence, a very large number of sample paths is required to obtain an accurate result. The antithetic variates method reduces the variance of the simulation results.
Underlying principle
The antithetic variates technique consists, for every sample path obtained, in taking its antithetic path — that is given a path to also take . The advantage of this technique is twofold: it reduces the number of normal samples to be taken to generate N paths, and it reduces the variance of the sample paths, improving the precision.
Suppose that we would like to estimate
For that we have generated two samples
An unbiased estimate of is given by
And
so variance is reduced if is negative.
Example 1
If the law of the variable X follows a uniform distribution along [0, 1], the first sample will be , where, for any given i, is obtained from U(0, 1). The second sample is built from , where, for any given i: . If the set is uniform along [0, 1], so are . Furthermore, covariance is negative, allowing for initial variance reduction.
Example 2: integral calculation
We would like to estimate
The exact result is . This integral can be seen as the expected value of , where
and U follows a uniform distribution [0, 1].
The following table compares the classical Monte Carlo estimate (sample size: 2n, where n = 1500) to the antithetic variates estimate (sample size: n, completed with the transformed sample 1 − ui):
{| cellspacing="1" border="1"
|
| align="right" | Estimate
| align="right" | standard error
|-
| Classical Estimate
| align="right" | 0.69365
| align="right" | 0.00255
|-
| Antithetic Variates
| align="right" | 0.69399
| align="right" | 0.00063
|}
The use of the antithetic variates method to estimate the result shows an important variance reduction.
See also
Control variates
References
Variance reduction
Computational statistics
Monte Carlo methods |
https://en.wikipedia.org/wiki/Phi%20coefficient | In statistics, the phi coefficient (or mean square contingency coefficient and denoted by φ or rφ) is a measure of association for two binary variables.
In machine learning, it is known as the Matthews correlation coefficient (MCC) and used as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975.
Introduced by Karl Pearson, and also known as the Yule phi coefficient from its introduction by Udny Yule in 1912 this measure is similar to the Pearson correlation coefficient in its interpretation.
Definition
A Pearson correlation coefficient estimated for two binary variables will return the phi coefficient.
Two binary variables are considered positively associated if most of the data falls along the diagonal cells. In contrast, two binary variables are considered negatively associated if most of the data falls off the diagonal.
If we have a 2×2 table for two random variables x and y
where n11, n10, n01, n00, are non-negative counts of numbers of observations that sum to n, the total number of observations. The phi coefficient that describes the association of x and y is
Phi is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2×2).
The phi coefficient can also be expressed using only , , , and , as
Maximum values
Although computationally the Pearson correlation coefficient reduces to the phi coefficient in the 2×2 case, they are not in general the same. The Pearson correlation coefficient ranges from −1 to +1, where ±1 indicates perfect agreement or disagreement, and 0 indicates no relationship. The phi coefficient has a maximum value that is determined by the distribution of the two variables if one or both variables can take on more than two values. See Davenport and El-Sanhury (1991) for a thorough discussion.
Machine learning
The MCC is defined identically to phi coefficient, introduced by Karl Pearson, also known as the Yule phi coefficient from its introduction by Udny Yule in 1912. Despite these antecedents which predate Matthews's use by several decades, the term MCC is widely used in the field of bioinformatics and machine learning.
The coefficient takes into account true and false positives and negatives and is generally regarded as a balanced measure which can be used even if the classes are of very different sizes. The MCC is in essence a correlation coefficient between the observed and predicted binary classifications; it returns a value between −1 and +1. A coefficient of +1 represents a perfect prediction, 0 no better than random prediction and −1 indicates total disagreement between prediction and observation. However, if MCC equals neither −1, 0, or +1, it is not a reliable indicator of how similar a predictor is to random guessing because MCC is dependent on the dataset. MCC is closely related to the chi-square statistic for a 2×2 contingency table
where n is th |
https://en.wikipedia.org/wiki/Gotem | Gotem is a small town in the Limburg province of Belgium and is part of the municipality of Borgloon.
Statistics
Population: 282 (1 January 2002)
Postal code: 3840
Coordinates: 50° 47' 60" N, 5° 17' 60" E
Elevation: 58 meters
Points of interest
Kasteel Fonteinhof
Church of Saint Denis
References
Populated places in Limburg (Belgium) |
https://en.wikipedia.org/wiki/Big%20Ideas%20Learning | Big Ideas Learning, LLC is an educational publisher in the United States. The company's headquarters is located in Erie, Pennsylvania. It publishes mathematics textbooks and instructional technology materials.
Big Ideas Learning is a privately owned Limited liability company.
History
The origins of Big Ideas Learning go back to 1980, when mathematics textbook author Ron Larson started a small company called Larson Texts. The company became incorporated in Pennsylvania in 1992 and became Larson Texts, Inc.
In 2008, the owners of Larson Texts formed a separate publishing company called Big Ideas Learning. Big Ideas Learning develops and publishes mathematics textbooks. The name of the company is related to the 2006 "Focal Point" recommendations of the National Council of Teachers of Mathematics. In September 2006, NCTM released Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence. In the Focal Points, NCTM identifies what it believes to be the most important mathematical topics ("the big ideas") for each grade level, including the related ideas, concepts, skills, and procedures that form the foundation for understanding and lasting learning.
In 2017 Big Ideas Learning announced a new partnership with National Geographic Learning (NGL), a Cengage company. National Geographic Learning is the sole distributor of the Big Ideas Math programs.
Publications
The first publication for Big Ideas Learning was a series of middle school mathematics textbooks that implemented the NCTM's focal point curriculum. Each book had a national edition and a Florida edition, which was submitted for adoption in the state of Florida. In the spring of 2010, Big Ideas Math was adopted by over half of Florida's 67 counties. The series won the 2010 Textbook Excellence Award ("Texty") for excellence in textbook publishing in the Elementary-High School Division for Mathematics.
In 2010, Big Ideas Learning published a series of middle school mathematics textbooks that implemented the Virginia Standards of Learning. In 2011, Big Ideas Learning published a series of middle school mathematics textbooks that implemented the Common Core State Standards. The series won the 2012 Textbook Excellence Award ("Texty") for excellence in textbook publishing in the Elementary-High School Division for Mathematics. In 2013, Big Ideas Math: A Common Core Curriculum Algebra 1 received the award for Most Promising Textbook from the Text and Academic Authors Association.
In 2014, Big Ideas Learning debuted the Big Ideas Math Algebra 1, Geometry, and Algebra 2 Common Core high school mathematics curriculum. The company also announced that it will be releasing the Big Ideas Math Course 1, Course 2, and Course 3 Common Core integrated high school mathematics curriculum in the spring of 2015.
Larson, Ron; Laurie Boswell (2010), Big Ideas Math Green, Big Ideas Learning
Larson, Ron; Laurie Boswell (2010), Big Ideas Math Red, Big Ideas Learning
Larso |
https://en.wikipedia.org/wiki/Frank%20William%20Land | Frank William Land (9 January 1911 in Edmonton, Middlesex – 2 June 1990 in Wrexham, Wales) was a populariser of mathematics and a professor of mathematics at Hull University.
He was lecturer at the College of St Mark and St John ('Marjohn') in Chelsea with Cyril Bibby, whom he later followed to Hull where he worked with Bill Cockcroft. For a time during the Second World War he taught at the Royal Grammar School, High Wycombe.
Marcus du Sautoy describes Land's book The Language of Mathematics as seminal in his intellectual development.
One of his children is the neurobiologist Professor Michael F. Land.
Publications
Land F.W. (1962) The Language of Mathematics John Murray, ASIN B0013FK5TW
References
20th-century British mathematicians
1911 births
1990 deaths |
https://en.wikipedia.org/wiki/Finitely%20presented | In mathematics, finitely presented may refer to:
finitely presented group
finitely presented monoid
finitely presented module
finitely presented algebra
finitely presented scheme, a global version of a finitely presented algebra
See also finitely generated (disambiguation). |
https://en.wikipedia.org/wiki/Royal%20Mathematical%20School | Royal Mathematical School is a branch of Christ's Hospital, founded by Charles II. It is currently Christ's Hospital's Maths Department.
History
It was established so that potential sailors could learn navigation and mathematicians could train at the school. Samuel Pepys was closely involved in the foundation, in 1673, with Jonas Moore. The School was integrated into Christ's Hospital, with boys who were pupils being selected aged 11 or 12 and prepared for a career in the Royal Navy.
There was a short-lived new mathematical school within Christ's Hospital, backed by Isaac Newton, and taught by Humphry Ditton; it ran from 1706 to 1715, when Ditton died, but then was closed down. James Hodgson was master of the Royal Mathematical School from 1709 to 1755, with John Robertson as assistant towards the end of his life. Other masters included James Dodson and William Wales. The master of the early 1760s, Daniel Harris, wrote with John Bevis the manual "Hints for Running the Lines" used by Charles Mason and Jeremiah Dixon for their survey of the Mason–Dixon line.
See also
Christ's Hospital
Samuel Pepys
The Salters School of Chemistry
References
N. Plumley, The Royal Mathematical School within Christ's Hospital: The early years. — Its aims and achievements, Vistas in Astronomy, Volume 20, Part 1, 1976, Pages 51–56.
Notes
Christ's Hospital |
https://en.wikipedia.org/wiki/Forest%20Hall%20School | Forest Hall School is a coeducational secondary school located in Stansted Mountfitchet, Essex, England.
In 2016 64% of students gained at least 5 GCSEs at A*-C including Maths and English. This figure had increased since 2013, when 28% of students achieved at least 5 GCSE grades A*-C including English and Maths. 67% of students also achieved A*-C grades on both English and Mathematics.
The school was originally called Mountfitchet High School before being renamed Mountfitchet Maths and Computing College. The school was re-branded as Forest Hall School in September 2013, named after the road it is situated. The school converted to academy status in February 2015 and is now sponsored by BMAT.
References
External links
Official website
Secondary schools in Essex
Academies in Essex
Stansted Mountfitchet |
https://en.wikipedia.org/wiki/Continuity%20set | In measure theory, a branch of mathematics, a continuity set of a measure μ is any Borel set B such that
where is the (topological) boundary of B. For signed measures, one asks that
The class of all continuity sets for given measure μ forms a ring.
Similarly, for a random variable X, a set B is called continuity set if
Continuity set of a function
The continuity set C(f) of a function f is the set of points where f is continuous.
References
Measure theory |
https://en.wikipedia.org/wiki/Completeness%20of%20atomic%20initial%20sequents | In sequent calculus, the completeness of atomic initial sequents states that initial sequents (where is an arbitrary formula) can be derived from only atomic initial sequents (where is an atomic formula). This theorem plays a role analogous to eta expansion in lambda calculus, and dual to cut-elimination and beta reduction. Typically it can be established by induction on the structure of , much more easily than cut-elimination.
References
Gaisi Takeuti. Proof theory. Volume 81 of Studies in Logic and the Foundation of Mathematics. North-Holland, Amsterdam, 1975.
Anne Sjerp Troelstra and Helmut Schwichtenberg. Basic Proof Theory. Edition: 2, illustrated, revised. Published by Cambridge University Press, 2000.
Theorems in the foundations of mathematics
Proof theory |
https://en.wikipedia.org/wiki/Funk%20transform | In the mathematical field of integral geometry, the Funk transform (also known as Minkowski–Funk transform, Funk–Radon transform or spherical Radon transform) is an integral transform defined by integrating a function on great circles of the sphere. It was introduced by Paul Funk in 1911, based on the work of . It is closely related to the Radon transform. The original motivation for studying the Funk transform was to describe Zoll metrics on the sphere.
Definition
The Funk transform is defined as follows. Let ƒ be a continuous function on the 2-sphere S2 in R3. Then, for a unit vector x, let
where the integral is carried out with respect to the arclength ds of the great circle C(x) consisting of all unit vectors perpendicular to x:
Inversion
The Funk transform annihilates all odd functions, and so it is natural to confine attention to the case when ƒ is even. In that case, the Funk transform takes even (continuous) functions to even continuous functions, and is furthermore invertible.
Spherical harmonics
Every square-integrable function on the sphere can be decomposed into spherical harmonics
Then the Funk transform of f reads
where for odd values and
for even values. This result was shown by .
Helgason's inversion formula
Another inversion formula is due to .
As with the Radon transform, the inversion formula relies on the dual transform F* defined by
This is the average value of the circle function ƒ over circles of arc distance p from the point x. The inverse transform is given by
Generalization
The classical formulation is invariant under the rotation group SO(3). It is also possible to formulate the Funk transform in a manner that makes it invariant under the special linear group SL(3,R) . Suppose that ƒ is a homogeneous function of degree −2 on R3. Then, for linearly independent vectors x and y, define a function φ by the line integral
taken over a simple closed curve encircling the origin once. The differential form
is closed, which follows by the homogeneity of ƒ. By a change of variables, φ satisfies
and so gives a homogeneous function of degree −1 on the exterior square of R3,
The function Fƒ : Λ2R3 → R agrees with the Funk transform when ƒ is the degree −2 homogeneous extension of a function on the sphere and the projective space associated to Λ2R3 is identified with the space of all circles on the sphere. Alternatively, Λ2R3 can be identified with R3 in an SL(3,R)-invariant manner, and so the Funk transform F maps smooth even homogeneous functions of degree −2 on R3\{0} to smooth even homogeneous functions of degree −1 on R3\{0}.
Applications
The Funk-Radon transform is used in the Q-Ball method for Diffusion MRI introduced by .
It is also related to intersection bodies in convex geometry.
Let be a star body with radial function .
Then the intersection body IK of K has the radial function .
See also
Radon transform
Spherical mean
References
.
.
.
.
Integral geometry
Integral tran |
https://en.wikipedia.org/wiki/Rapha%C3%ABl%20Rouquier | Raphaël Alexis Marcel Rouquier (born 9 December 1969) is a French mathematician and
a professor of mathematics at UCLA.
Education
Rouquier was born in Étampes, France.
Rouquier studied at the École Normale Supérieure from 1988 to 1989 and from 1989 to 1990 for a DEA in mathematics under the direction of Michel Broué, where he continued to study for his PhD. Rouquier spent the second year of his PhD study at the University of Cambridge under the supervision of J. G. Thompson.
Career
He was hired by the CNRS in 1992 where he completed his PhD (1992) and Habilitation (1998–1999). He was appointed director of research there in 2003. From 2005 to 2006 he was Professor of Representation Theory at the Department of Pure Mathematics at the University of Leeds before moving to the University of Oxford as the Waynflete Professor of Pure Mathematics. In 2012, he moved to UCLA.
Awards and honors
He was awarded the Whitehead Prize in 2006 and the Adams Prize in 2009 for contributions to representation theory. He was awarded the Elie Cartan Prize in 2009.
In 2012 he became a fellow of the American Mathematical Society. In 2015 he became a Simons Investigator.
He gave the Peccot Lectures at Collège de France in 2000, the Whittemore Lectures at Yale University in 2005, an Algebra Section lecture at the International Congress of Mathematicians in 2006, the Albert Lectures at the University of Chicago in 2008, the Moursund Lectures at the University of Oregon in 2013, the Simons Lectures at MIT in 2013, the CBMS Lectures in 2014 and the Ellis Kolchin Memorial Lecture at Columbia University in 2016.
Notes
External links
Raphaël Rouquier at Oxford
Raphaël Rouquier at UCLA
1969 births
Living people
École Normale Supérieure alumni
People from Étampes
20th-century French mathematicians
21st-century French mathematicians
Fellows of Magdalen College, Oxford
Fellows of the American Mathematical Society
Waynflete Professors of Pure Mathematics
Whitehead Prize winners
Academics of the University of Leeds
Simons Investigator
Research directors of the French National Centre for Scientific Research |
https://en.wikipedia.org/wiki/Arf%20ring | In mathematics, an Arf ring was defined by to be a 1-dimensional commutative semi-local Macaulay ring satisfying some extra conditions studied by .
References
Commutative algebra |
https://en.wikipedia.org/wiki/Denis%20Knitel | Denis Aleksandrovich Knitel (; born 29 November 1977) is a Tajikistani professional footballer who also holds Russian citizenship.
Career statistics
International
Statistics accurate as of 8 September 2016
International goals
Scores and results list Tajikistan's goal tally first.
Honours
Varzob Dushanbe
Tajik League (3): 1998, 1999, 2000
Tajik Cup (1): 1999
References
1977 births
Living people
Tajikistani men's footballers
Tajikistani expatriate men's footballers
Tajikistan men's international footballers
FC Zhemchuzhina-Sochi players
Tajikistani people of German descent
Men's association football defenders
FC Kuzbass Kemerovo players
Expatriate men's footballers in Russia |
https://en.wikipedia.org/wiki/Chris%20Evert%20career%20statistics | This is a list of the main career statistics of former professional tennis player Chris Evert.
Significant finals
Grand Slam finals
Singles: 34 finals (18 titles, 16 runners-up)
Evert played in a total of 56 grand slams in her career. From her debut as a 16-year-old at the 1971 US Open, she reached the semifinals or better in her first 34 grand slam events. Overall, she reached 54 quarterfinals, 52 semifinals, and 34 finals. Her only 2 quarterfinal losses were both at the US Open, in 1987 to Lori McNeil and in 1989, when the US Open served as her farewell from tournament play, to Zina Garrison. She lost before the quarterfinals 2 times, both in the third round, at the 1983 Wimbledon where she lost to Kathy Jordan and at the 1988 French Open where she lost to Arantxa Sánchez Vicario. Her 299 grand slam singles match wins is 3rd best in the Open Era.
Doubles: 4 finals (3 titles, 1 runner-up)
Mixed doubles: 1 final (1 runner-up)
Year-end championships finals
Singles: 8 (4 titles, 4 runners-up)
Grand Slam singles tournament timeline
Notes:
The Australian Open was held twice in 1977, in January and December.
Evert retired in September 1989 after playing in the U.S Open, at which time she was ranked world No.4.
WTA singles finals
Titles: (157)
Runner-ups: (73)
WTA Tour career earnings
Record against other top players
Evert's win–loss record against certain players who have been ranked World No. 10 or higher is as follows:
Players who have been ranked World No. 1 are in boldface.
Virginia Wade 40–6
/ Martina Navratilova 37–43
Evonne Goolagong 26–13
Virginia Ruzici 24–0
Sue Barker 23–1
Betty Stöve 22–0
Rosemary Casals 22–1
Wendy Turnbull 21–1
/ Hana Mandlíková 21–7
Pam Shriver 19–3
Billie Jean King 19–7
Kerry Reid 18–2
/ Manuela Maleeva 17–2
/ Helena Suková 17–2
Andrea Jaeger 17–3
Sylvia Hanika 16–1
Dianne Fromholtz 16–3
Olga Morozova 15–0
Bettina Bunge 14–0
Mima Jaušovec 14–0
Françoise Dürr 13–0
Claudia Kohde-Kilsch 13–0
Bonnie Gadusek 12–0
Kathy Jordan 12–3
Carling Bassett-Seguso 9–0
Jo Durie 9–0
Kathy Rinaldi 9–0
Zina Garrison 9–2
Margaret Court 9–4
Tracy Austin 8–9
Barbara Potter 8–0
Mary Joe Fernández 7–0
Kathy Horvath 7–0
Lisa Bonder 7–1
Stephanie Rehe 6–0
Greer Stevens 6–1
Gabriela Sabatini 6–3
Nancy Richey 6–5
Steffi Graf 6–8
Catarina Lindqvist 4–0
Kathy May 4–0
/ Jana Novotná 3–0
Nathalie Tauziat 3–0
Andrea Temesvári 3–0
Lori McNeil 3–2
Katerina Maleeva 2–0
Conchita Martínez 2–0
Kristien Shaw 2–0
// Monica Seles 2–1
Barbara Paulus 1–1
Arantxa Sánchez Vicario 1–1
/ Natasha Zvereva 1–1
125-match clay court winning streak from August 1973 to May 1979
1973:
US Clay Court Championships, IN. (d. Pat Bostrom 6–0 6–0; Isabel Fernandez 6–3 6–4; Pat Pretorius 6–2 6–1; Linda Tuero 6–0 6–0; Veronica Burton 6–4 6–3 in final);
Columbus, GA. (d. Janet Haas 6–1 6–0; Francoise Durr 6–0 6–2; Laurie Fleming 6–4 6–0; Rosie Casals 6–3 7–6; won by default over Margaret Court – defaults not counted |
https://en.wikipedia.org/wiki/Evonne%20Goolagong%20Cawley%20career%20statistics | This is a list of the main career statistics of former Australian tennis player Evonne Goolagong Cawley. During her career, which lasted from 1967 to 1983, Goolagong won seven singles titles at a Grand Slam event and was a runner-up in 11 occasions. In addition, she won five Grand Slam doubles titles, partnering Margaret Court, Peggy Michel and Helen Gourlay, as well as one mixed doubles title with Kim Warwick. In total she won 84 singles titles, 53 doubles titles and 6 mixed doubles titles. She achieved the No. 1 singles ranking for a two-week period in April–May 1976, although this was only officially recognized in 2007. She was a member of the Australian Federation Cup teams that won the cup in 1971, 1973 and 1974.
Major finals
Grand Slam finals
Singles: 18 finals (7 titles, 11 runners-up)
Women's doubles: 6 finals (5 titles, 1 runner-up)
Note: The shared women's doubles title at the Australian Open in 1977 (December) isn't traditionally counted in Goolagong's win total because the finals were never played. Otherwise, she would have 14 Grand Slam titles, 6 Grand Slam women's doubles titles, and 7 Grand Slam women's doubles finals.
Mixed doubles: 2 finals (1 title, 1 runner-up)
Year-End Championships finals
Singles: 3 finals (2 titles, 1 runner-up)
Grand Slam singles tournament timeline
Note: The Australian Open was held twice in '77, in January and December. Goolagong won the December edition. She was seeded #4 for the 1980 US Open Championships, but withdrew from the tournament before play began.
Career titles (Open Era)
The Open Era began on 22 April 1968.
Singles (84 titles)
Doubles (53 titles)
*Due to bad weather, the 1976 Australian Open doubles final was played for one special set.
Most Titles w/ Doubles Partners:
10 w/ Patricia Edwards
7 w/ Margaret Court
7 w/ Peggy Michel
6 w/ Helen Gourlay
6 w/ Janet Young
Mixed doubles (6 titles)
5 November 1970 – Australian Hardcourt Championships (w/Bob Giltinan) defeated Patricia Edwards and Ross Case 6–2, 7-5
4 June 1972 – French Open (w/Kim Warwick) defeated Françoise Dürr and Jean-Claude Barclay 6-2 6-4
14 January 1973 – New Zealand Open (w/Ross Case) defeated Janet Young and Dick Crealy 6–1, 6-3
27 November 1973 – South African Open (w/Jürgen Fassbender) defeated Ilana Kloss and Bernard Mitton 6-3 6-2
19 January 1974 – New Zealand Hard Court Championships (w/Russell Simpson) defeated Sue Barker and John Lloyd 6–3, 6-4
11 October 1980 – Hilton Head, South Carolina, World Couples Championships (w/Stan Smith) defeated Andrea Jaeger and Dick Stockton 7-5 6-4
Federation Cup
Goolagong played in 26 ties in the then-named Federation Cup between 1970 and 1982. Of the seven years she played, the Australian team won in three: 1971, 1973 and 1974. She accumulated a total of 35 wins over her career, which ties with Margaret Court and Dianne Balestrat as the third-most ever from an Australian. She holds a 22–3 singles record and a 13–2 record in doubles.
Win |
https://en.wikipedia.org/wiki/Margaret%20Court%20career%20statistics | This is a list of the main career statistics of Australian former tennis player Margaret Court. She won 64 Grand Slam events (24 singles, 19 doubles, 21 mixed doubles), which is a record for a male or female player. Her 24 Grand Slam singles titles and 21 in mixed doubles are also all-time records for both sexes. She achieved a career Grand Slam in singles, doubles, and mixed doubles. She is one of three women to have achieved the calendar year Grand Slam in singles (alongside Steffi Graf and Maureen Connolly), and is the only woman to have achieved the mixed doubles Grand Slam, which she did twice.
Court won more than half of the Grand Slam singles tournaments she played (24 of 47). She won 192 singles titles before and after the Open Era, an all-time record. Her career singles win-loss record was 1,177–106, for a winning percentage of 91.74 percent on all surfaces (hard, clay, grass, carpet), is also an all-time record.
She won at least 100 singles matches in 1965 (113–8), 1968 (107–12), 1969 (104–6), 1970 (110–6), and 1973 (108–6). She won more than 80 percent of her singles matches against top 10 players (297–73) and was the year-end top ranked player seven times.
Grand Slam performance timeline
Singles
Doubles
Mixed doubles
Note: The shared mixed doubles titles at the Australian Championships/Open in 1965 and 1969 are not always counted in Court's Grand Slam win total because the finals were never played. The Australian Open does officially count them as joint victories. Otherwise, she would have 21 Grand Slam mixed doubles titles, which is reflected in the above table.
Grand Slam finals
Singles: 29 finals (24 titles, 5 runner-ups)
Women's doubles: 33 finals (19 titles, 14 runner-ups)
Mixed doubles: 25 finals (21 titles, 4 runner-ups)
Win-Loss singles record in Majors
Australian Championships/Australian Open
Court's overall win-loss record at the Australian Championships/Australian Open was 61–3 (95.3%) in 14 years (1959–1966, 1968–1971, 1973, 1975). (Her win total includes one walkover but does not include any first round byes.) Her only losses were to Martina Navratilova in 1975, Billie Jean King in 1968, and Mary Carter Reitano in 1959.
French Championships/French Open
Court's overall win-loss record at the French Championships/French Open was 47–5 (90.3%) in 10 years (1961–1966, 1969–1971, 1973). (Her win total includes three walkovers but does not include any first round byes.) Her only losses were to Gail Chanfreau in 1971, Nancy Richey in 1966, Lesley Turner Bowrey in 1965, Věra Pužejová Suková in 1963, and Ann Haydon-Jones in 1961.
United States Championships/United States Open
Court's overall win-loss record at the United States Championships/United States Open was 51–6 (89.5%) in 11 years (1961–1965, 1968–1970, 1972–1973, 1975). (Her win total does not include any first round byes.) Her only losses were to Martina Navratilova in 1975, Billie Jean King in 1972, Maria Bueno in 1968 and 1963, Karen Hantze Susman in 196 |
https://en.wikipedia.org/wiki/John%20Newcombe%20career%20statistics | This is a list of the main career statistics of professional tennis player John Newcombe.
Grand Slam finals
Singles: 10 (7 titles, 3 runner-ups)
Men's doubles (21)
Wins (17)
A sudden-death tie-break instead of fifth set
Runner-ups (4)
Mixed doubles (3)
Wins (2)
Runner-up
Open Era finals
As listed by the ATP website.
Singles: 71 (41 titles, 30 runner-ups)
Doubles: 55 (33 titles, 22 runner-ups)
Grand Slam tournament performance timeline
Singles
Source: ITF
References
External links
Tennis career statistics |
https://en.wikipedia.org/wiki/Chebyshev%20iteration | In numerical linear algebra, the Chebyshev iteration is an
iterative method for determining the solutions of a system of linear equations. The method is named after Russian mathematician Pafnuty Chebyshev.
Chebyshev iteration avoids the computation of inner products as is necessary for the other nonstationary methods. For some distributed-memory architectures these inner products are a bottleneck with respect to efficiency. The price one pays for avoiding inner products is that the method requires enough knowledge about spectrum of the coefficient matrix A, that is an upper estimate for the upper eigenvalue and lower estimate for the lower eigenvalue. There are modifications of the method for nonsymmetric matrices A.
Example code in MATLAB
function [x] = SolChebyshev002(A, b, x0, iterNum, lMax, lMin)
d = (lMax + lMin) / 2;
c = (lMax - lMin) / 2;
preCond = eye(size(A)); % Preconditioner
x = x0;
r = b - A * x;
for i = 1:iterNum % size(A, 1)
z = linsolve(preCond, r);
if (i == 1)
p = z;
alpha = 1/d;
else if (i == 2)
beta = (1/2) * (c * alpha)^2
alpha = 1/(d - beta / alpha);
p = z + beta * p;
else
beta = (c * alpha / 2)^2;
alpha = 1/(d - beta / alpha);
p = z + beta * p;
end;
x = x + alpha * p;
r = b - A * x; %(= r - alpha * A * p)
if (norm(r) < 1e-15), break; end; % stop if necessary
end;
end
Code translated from
and.
See also
Iterative method. Linear systems
List of numerical analysis topics. Solving systems of linear equations
Jacobi iteration
Gauss–Seidel method
Modified Richardson iteration
Successive over-relaxation
Conjugate gradient method
Generalized minimal residual method
Biconjugate gradient method
Iterative Template Library
IML++
References
External links
Templates for the Solution of Linear Systems
Chebyshev Iteration. From MathWorld
Chebyshev Iteration. Implementation on Go language
Numerical linear algebra
Iterative methods |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb. It also lists all matches that Dinamo Zagreb played in the 2009–10 season.
Events
Pre-season
13 June: Midfielder Mirko Hrgović leaves the club after his contract is terminated on mutual consent. Hrgović spent a single season with the club after being transferred from JEF United in summer 2008.
19 June: The club releases official confirmation that the 24-year-old Chilean midfielder Pedro Morales contracted swine influenza while visiting Chile. Morales is expected to stay in Chile and join the team upon recovery.
19 June: Albanian midfielder Emiljano Vila signs for the club from Teuta Durrës for a yet undisclosed fee, on recommendation from Josip Kuže, former manager of Dinamo and Albania.
26 June: Greek international Dimitrios Papadopoulos signs for the club from Italian side Lecce on a three-year contract.
Season
14 December: Media reports confirm that the managing board had decided to terminate contracts with Dimitrios Papadopoulos and Leandro Cufré and that they would be free to find new clubs in the winter transfer period. The board also decided to end Denis Glavina's loan and send him back to Vorskla. The board also said cleared a possible transfer of Mario Mandžukić in case Dinamo receive a good offer for him, and the media cited rumours that Dodo would be brought from Inter Zaprešić to replace him.
First-team squad
Current squad
1 player has Croatian citizenship
Kit
|
|
|
Squad statistics
Updated 1 October 2009.
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: 2009–10 Prva HNL article
UEFA Europa League
Group A
Matches
Competitive
Last updated 13 May 2010Sources: Prva-HNL.hr, Sportske novosti, Sportnet.hr
Friendlies
Player seasonal records
Competitive matches only. Updated to games played 13 May 2010.
Key
Goalscorers
Source: Competitive matchesA: Sivonjić was loaned out to Inter Zaprešić in the winter transfer period.B: Papadopoulos' contract was terminated just before the winter break, after which he signed for Celta Vigo.C: Lovren was transferred out to Lyon in January 2010.D: Chago was loaned out to Istra 1961 in the winter transfer period.E: Dodo was brought in from Inter Zaprešić in the winter transfer period.
Notes
1. : Second leg of the UEFA Champions League second qualifying round against Pyunik had to be played behind closed doors due to unruly behaviour by Dinamo supporters at their last European away match against Udinese
References
External links
Dinamo Zagreb official website
2009-10
Croatian football clubs 2009–10 season
2009-10
2009–10 UEFA Europa League participants seasons |
https://en.wikipedia.org/wiki/K.%20B.%20Reid | Kenneth Brooks Reid, Jr. is a graph theorist and the founder faculty (Head 1989) professor at California State University, San Marcos (CSUSM). He specializes in combinatorial mathematics. He is known for his work in tournaments, frequency partitions and aspects of voting theory. He is known (with E. T. Parker) on a disproof of a conjecture on tournaments by Erdős and Moser.
He received his Ph.D. on a dissertation called "Structure in Finite Graphs" from the University of Illinois at Urbana-Champaign in 1968, his advisor was E. T. Parker. Reid is a professor emeritus at Louisiana State University (1968–1989) and has guided students for their Ph.D.s at Baton Rouge.
He is a professor emeritus at CSUSM.
Selected work
[Book] Disproof of a conjecture of Erdos and Moser on tournaments, KB Reid, ET Parker, ILLINOIS UNIV URBANA - 1964 - oai.dtic.mil
Domination graphs of tournaments and digraphs, DC Fisher, JR Lundgren, SK Merz, KB Reid - Congressus Numerantium, 1995 - citeseerx.ist.psu.edu
Tournaments, K.B. Reid, L. W. Beineke - Selected topics in graph theory, 1978
References
Reid's page at CSUSM
Graph theorists
Year of birth missing (living people)
Living people
University of Illinois Urbana-Champaign alumni
California State University San Marcos faculty
San Marcos |
https://en.wikipedia.org/wiki/Hannan%20Medal | The Hannan Medal in the Mathematical Sciences is awarded every two years by the Australian Academy of Science to recognize achievements by Australians in the fields of pure mathematics, applied and computational mathematics, and statistical science.
This medal commemorates the work of the late Edward J. Hannan, FAA, for his achievements in time series analysis.
Winners
Source:
See also
List of mathematics awards
Notes
External links
Hannan Medal site of the Australian Academy of Science
Mathematics awards
Australian science and technology awards
Awards established in 1994
Australian Academy of Science Awards
1994 establishments in Australia |
https://en.wikipedia.org/wiki/Arf%20semigroup | In mathematics, Arf semigroups are certain subsets of the non-negative integers closed under addition, that were studied by . They appeared as the semigroups of values of Arf rings.
A subset of the integers forms a monoid if it includes zero, and if every two elements in the subset have a sum that also belongs to the subset. In this case, it is called a "numerical semigroup".
A numerical semigroup is called an Arf semigroup if, for every three elements x, y, and z with z = min(x, y, and z), the semigroup also contains the element .
For instance, the set containing zero and all even numbers greater than 10 is an Arf semigroup.
References
.
Semigroup theory |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20PFC%20CSKA%20Sofia%20season | The 2009–10 season was PFC CSKA Sofia's 62nd consecutive season in A Group. This article shows player statistics and all matches (official and friendly) that the club have and will play during the 2009–10 season.
Team Kit
The team kit for the 2009–10 season is produced by Uhlsport and sponsored by Globul since 7 September 2009. The club introduced a new third kit at the game against Derry City
Players
Squad information
Appearances for competitive matches only
|-
|colspan="14"|Players sold or loaned out after the start of the season:
|}
As of game played start of season
Players in/out
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Player seasonal records
Competitive matches only. Updated to games played 31 May 2010.
Key
Goalscorers
Pre-season and friendlies
Pre-season
On-season (autumn)
Mid-season
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
Europa League
Third qualifying round
Play-off round
Group stage
See also
PFC CSKA Sofia
References
External links
CSKA Official Site
Bulgarian A Professional Football Group
UEFA Profile
PFC CSKA Sofia seasons
Cska Sofia |
https://en.wikipedia.org/wiki/Largest%20empty%20rectangle | In computational geometry, the largest empty rectangle problem, maximal empty rectangle problem or maximum empty rectangle problem, is the problem of finding a rectangle of maximal size to be placed among obstacles in the plane. There are a number of variants of the problem, depending on the particularities of this generic formulation, in particular, depending on the measure of the "size", domain (type of obstacles), and the orientation of the rectangle.
The problems of this kind arise e.g., in electronic design automation, in design and verification of physical layout of integrated circuits.
A maximal empty rectangle is a rectangle which is not contained in another empty rectangle. Each side of a maximal empty rectangle abuts an obstacle (otherwise the side may be shifted outwards, increasing the empty rectangle). An application of this kind is enumeration of "maximal white rectangles" in image segmentation R&D of image processing and pattern recognition. In the contexts of many algorithms for largest empty rectangles, "maximal empty rectangles" are candidate solutions to be considered by the algorithm, since it is easily proven that, e.g., a maximum-area empty rectangle is a maximal empty rectangle.
Classification
In terms of size measure, the two most common cases are the largest-area empty rectangle and largest-perimeter empty rectangle.
Another major classification is whether the rectangle is sought among axis-oriented or arbitrarily oriented rectangles.
Special cases
Maximum-area square
The case when the sought rectangle is an axis-oriented square may be treated using Voronoi diagrams in metrics for the corresponding obstacle set, similarly to the largest empty circle problem. In particular, for the case of points within rectangle an optimal algorithm of time complexity is known.
Domain: rectangle containing points
A problem first discussed by Naamad, Lee and Hsu in 1983 is stated as follows: given a rectangle A containing n points, find a largest-area rectangle with sides parallel to those of A which lies within A and does not contain any of the given points. Naamad, Lee and Hsu presented an algorithm of time complexity , where s is the number of feasible solutions, i.e., maximal empty rectangles. They also proved that and gave an example in which s is quadratic in n. Afterwards a number of papers presented better algorithms for the problem.
Domain: line segment obstacles
The problem of empty isothetic rectangles among isothetic line segments was first considered in 1990. Later a more general problem of empty isothetic rectangles among non-isothetic obstacles was considered.
Generalizations
Higher dimensions
In 3-dimensional space, algorithms are known for finding a largest maximal empty isothetic cuboid problem, as well as for enumeration of all maximal isothetic empty cuboids.
See also
Largest empty sphere
Minimum bounding box, Minimum bounding rectangle
References
Geometric algorithms |
https://en.wikipedia.org/wiki/Idodi | Idodi is an administrative Division in the Iringa Rural District of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,675 people in the ward, from 10,202 in 2012.
The villages of Mahuninga, Makifu, Tungamalenga, Mapogoro, Kitisi and Malinzanga all fall within Idodi Division. The Division contains two Wards: Tungamalenga and Mlowa.
Idodi Division is immediately south of Ruaha National Park, Tanzania's largest Park.
Villages / vitongoji
The ward has 4 villages and 22 vitongoji.
Idodi
Ilamba
Mbuyuni “A”
Mbuyuni “B”
Mjimwema “A”
Mjimwema “B”
Msimbi
Mapogoro
Idindiga
Kibaoni
Kisiwani
Kitanewa
Lungemba
Mapogoro
Kitisi
Kitisi
Nyamnango
Tungamalenga
Darajani
Kinyali
Malunde
Mbuyuni
Mlimani
Msembe
Ofisini
Zahanati
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Ifunda | Ifunda is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,765 people in the ward, from 12,199 in 2012.
Villages / vitongoji
The ward has 6 villages and 30 vitongoji.
Ifunda
Kibaoni A
Kilimahewa A
Kipera
Mgondo
Ulolage
Utibesa
Bandabichi
Bandabichi
Ifunda Sekondari
Ihagaha
Kibaoni B
Kilimahewa B
Kivalali A
Kivalali B
Mlafu
Isupilo
Kibena
Isenuka
Kwa Mama Fred
Kalonga
Kitasengwa
Lutitili
Ubalanzi
Ulyangwada
Mfukulembe
Igulumiti B
Igulumti A
Lyasa
Ndolela
Udumka
Ikungu
Ofisini
Utulo
Mibikimitali
Masimike
Mibikimitali
Ulangala
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Ilolo%20Mpya | Ilolo Mpya is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,672 people in the ward, from 6,376 in 2012.
Villages / vitongoji
The ward has 4 villages and 17 vitongoji.
Ilolompya
Ilolo
Magangamatitu
Mllimani
Luganga
Barabarambili
Kihesa
Mawande
Motomoto
Mtakuja
Sadani
Ukwega
Uwanjani
Magozi
Isegelele
Kimalanongwa
Magozi
Mkombilenga
Chamamba
Mtakuja
Muungano
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Itunundu | Itunundu is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,833 people in the ward, from 000 in 2012.
Villages / vitongoji
The ward has 3 villages and 16 vitongoji.
Itunundu
Changalawe
Ikolongo
Isele
Kibuegele
Kivukokalo
Majengo
Mbuyuni
Mkwajuni
Kimande
Gundamnani
Kikuluhe
Kimande
Mjimwema
Mwaitenga
Mbuyuni
Igodikafu
Mbuyuni
Ndolela
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Izazi | Izazi is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 5,526 people in the ward, from 5,281 in 2012.
Villages / vitongoji
The ward has 3 villages and 17 vitongoji.
Makuka
Itemagwe
Magombwe
Majengo
Makuka A
Makuka B
Mondomela
Nyamahato
Izazi
Barabarani
Chekechea
Ihanyi
Izazi Madukani
Kiwanjani
Sokoni
Mnadani
Kilamba Kitali
Mabati
Magungu
Mjimwema
Mnadani
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Kalenga | Kalenga is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics reported its population to be 7,286, up from 6,963 in 2012.
Kalenga, which is situated along the side-lines of the Great Ruaha River, is one among the historical villages of "Iringa". It is known for being the residence of the famous Chief Mtwa Mkwawa of the Hehe tribe, who resisted German colonization. Mkwawa fortified the village with a wall 4 meters high and 5 kilometers in circumference. The town was stormed by a German force in 1894, and the fortifications were destroyed. Mkwawa continued to resist until 1898, when he was finally hunted down by the Germans and committed suicide. His head was cut off and sent to Germany, but eventually returned in 1954; his skull is now on display in a small museum in Kalenga.
Villages and vitongoji
The ward has 3 villages and 16 vitongoji.
Tosamaganga
Ipamba
Irangi
Mabanda
Tosa kilimani
Unyangwila
Kalenga
Galinoma
Igawa
Ilundimembe
Kidope
Lipuli
Maktaba
Mwambao
Wangi
Isakalilo
Isakalilo A
Isakalilo B
Isakalilo C
Gallery
References
Wards of Iringa Region |
https://en.wikipedia.org/wiki/Kihorogota | Kihorogota is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,044 people in the ward, from 7,688 in 2012.
Villages / vitongoji
The ward has 8 villages and 28 vitongoji.
Igula
Ilogombe
Madibila
Maganga
Nyambala
Ismani (T)
Lugolola
Lwang’a
Lyanika
Mtiulaya
Ivangwa
Ivangwa
Mdendami
Kihorogota
Danida
Kihorogota
Mgugumbalo
Njiapanda
Mikong'wi
Kichangani
Lupembe
Shuleni
Utitili
Ndolela
Kibaoni
Ndolela
Ofisini
Ngano
Godauni A
Godauni B
Lyaveya
Mwang’ingo
Uhominyi
Tumaini
Ubena
Uhominyi
References
Wards of Iringa Region |
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