source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/G%C3%A5rding%20domain
|
In mathematics, a Gårding domain is a concept in the representation theory of topological groups. The concept is named after the mathematician Lars Gårding.
Let G be a topological group and let U be a strongly continuous unitary representation of G in a separable Hilbert space H. Denote by g the family of all one-parameter subgroups of G. For each δ = { δ(t) | t ∈ R } ∈ g, let U(δ) denote the self-adjoint generator of the unitary one-parameter subgroup { U(δ(t)) | t ∈ R }. A Gårding domain for U is a linear subspace of H that is U(g)- and U(δ)-invariant for all g ∈ G and δ ∈ g and is also a domain of essential self-adjointness for U
Gårding showed in 1947 that, if G is a Lie group, then a Gårding domain for U consisting of infinitely differentiable vectors exists for each continuous unitary representation of G. In 1961, Kats extended this result to arbitrary locally compact topological groups. However, these results do not extend easily to the non-locally compact case because of the lack of a Haar measure on the group. In 1996, Danilenko proved the following result for groups G that can be written as the inductive limit of an increasing sequence G1 ⊆ G2 ⊆ ... of locally compact second countable subgroups:
Let U be a strongly continuous unitary representation of G in a separable Hilbert space H. Then there exist a separable nuclear Montel space F and a continuous, bijective, linear map J : F → H such that
the dual space of F, denoted by F∗, has the structure of a separable Fréchet space with respect to the strong topology on the dual pairing (F∗, F);
the image of J, im(J), is dense in H;
for all g ∈ G, U(g)(im(J)) = im(J);
for all δ ∈ g, U(δ)(im(J)) ⊆ im(J) and im(J) is a domain of essential self-adjointness for U(δ);
for all g ∈ G, J−1U(g)J is a continuous linear map from F to itself;
moreover, the map G → Lin(F; F) taking g to J−1U(g)J is continuous with respect to the topology on G and the weak operator topology on Lin(F; F).
The space F is known as a strong Gårding space for U and im(J) is called a strong Gårding domain for U. Under the above assumptions on G there is a natural Lie algebra structure on G, so it makes sense to call g the Lie algebra of G.
References
Unitary representation theory
|
https://en.wikipedia.org/wiki/Tensor-hom%20adjunction
|
In mathematics, the tensor-hom adjunction is that the tensor product and hom-functor form an adjoint pair:
This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.
General statement
Say R and S are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):
Fix an -bimodule and define functors and as follows:
Then is left adjoint to . This means there is a natural isomorphism
This is actually an isomorphism of abelian groups. More precisely, if is an -bimodule and is a -bimodule, then this is an isomorphism of -bimodules. This is one of the motivating examples of the structure in a closed bicategory.
Counit and unit
Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit
has components
given by evaluation: For
The components of the unit
are defined as follows: For in ,
is a right -module homomorphism given by
The counit and unit equations can now be explicitly verified. For in ,
is given on simple tensors of by
Likewise,
For in ,
is a right -module homomorphism defined by
and therefore
The Ext and Tor functors
The Hom functor commutes with arbitrary limits, while the tensor product functor commutes with arbitrary colimits that exist in their domain category. However, in general, fails to commute with colimits, and fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.
See also
Currying
Ext functor
Tor functor
Change of rings
References
Adjoint functors
Commutative algebra
|
https://en.wikipedia.org/wiki/Raising%20and%20lowering%20indices
|
In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.
Vectors, covectors and the metric
Mathematical formulation
Mathematically vectors are elements of a vector space over a field , and for use in physics is usually defined with or . Concretely, if the dimension of is finite, then, after making a choice of basis, we can view such vector spaces as or .
The dual space is the space of linear functionals mapping . Concretely, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by , so that is a linear map .
Then under a choice of basis , we can view vectors as an vector with components (vectors are taken by convention to have indices up). This picks out a choice of basis for , defined by the set of relations .
For applications, raising and lowering is done using a structure known as the (pseudo-)metric tensor (the 'pseudo-' refers to the fact we allow the metric to be indefinite). Formally, this is a non-degenerate, symmetric bilinear form
In this basis, it has components , and can be viewed as a symmetric matrix in with these components. The inverse metric exists due to non-degeneracy and is denoted , and as a matrix is the inverse to .
Raising and lowering vectors and covectors
Raising and lowering is then done in coordinates. Given a vector with components , we can contract with the metric to obtain a covector:
and this is what we mean by lowering the index. Conversely, contracting a covector with the inverse metric gives a vector:
This process is called raising the index.
Raising and then lowering the same index (or conversely) are inverse operations, which is reflected in the metric and inverse metric tensors being inverse to each other (as is suggested by the terminology):
where is the Kronecker delta or identity matrix.
Finite-dimensional real vector spaces with (pseudo-)metrics are classified up to signature, a coordinate-free property which is well-defined by Sylvester's law of inertia. Possible metrics on real space are indexed by signature . This is a metric associated to dimensional real space. The metric has signature if there exists a basis (referred to as an orthonormal basis) such that in this basis, the metric takes the form with positive ones and negative ones.
The concrete space with elements which are -vectors and this concrete realization of the metric is denoted , where the 2-tuple is meant to make it clear that the underlying vector space of is : equipping this vector space with the metric is what turns the space into .
Examples:
is a model for 3-dimensional space. The metric is equivalent to the standard dot product.
, equivalent to dimensional real space as an inner product space with . In Euclidean space, raising and lowering is not necessary due to vectors
|
https://en.wikipedia.org/wiki/Holbein%2C%20Saskatchewan
|
Holbein is an organized hamlet in Saskatchewan that lies within the Rural Municipality of Shellbrook No. 493.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Holbein had a population of 122 living in 48 of its 54 total private dwellings, a change of from its 2016 population of 109. With a land area of , it had a population density of in 2021.
References
External links
Holbein, Saskatchewan Community Profile
Designated places in Saskatchewan
Organized hamlets in Saskatchewan
Shellbrook No. 493, Saskatchewan
Division No. 16, Saskatchewan
|
https://en.wikipedia.org/wiki/Surface%20probability
|
In immunology, surface probability is the amount of reflection of an antigen's secondary or tertiary structure to the outside of the molecule.
A greater surface probability means that an antigen is more likely to be immunogenic (i.e. induce the formation of antibodies).
References
Immunology
|
https://en.wikipedia.org/wiki/Positive%20current
|
In mathematics, more particularly in complex geometry,
algebraic geometry and complex analysis, a positive current
is a positive (n-p,n-p)-form over an n-dimensional complex manifold,
taking values in distributions.
For a formal definition, consider a manifold M.
Currents on M are (by definition)
differential forms with coefficients in distributions; integrating
over M, we may consider currents as "currents of integration",
that is, functionals
on smooth forms with compact support. This way, currents
are considered as elements in the dual space to the space
of forms with compact support.
Now, let M be a complex manifold.
The Hodge decomposition
is defined on currents, in a natural way, the (p,q)-currents being
functionals on .
A positive current is defined as a real current
of Hodge type (p,p), taking non-negative values on all positive
(p,p)-forms.
Characterization of Kähler manifolds
Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.
Theorem: Let M be a compact complex manifold. Then M does not admit a Kähler structure if and only if M admits a non-zero positive (1,1)-current which is a (1,1)-part of an exact 2-current.
Note that the de Rham differential maps 3-currents to 2-currents, hence is a differential of a 3-current; if is a current of integration of a complex curve, this means that this curve is a (1,1)-part of a boundary.
When M admits a surjective map to a Kähler manifold with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry.
Corollary: In this situation, M is non-Kähler if and only if the homology class of a generic fiber of is a (1,1)-part of a boundary.
Notes
References
P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley.
J.-P. Demailly, $L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)
Complex manifolds
Several complex variables
|
https://en.wikipedia.org/wiki/Goormaghtigh%20conjecture
|
In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation
satisfying and are
and
Partial results
showed that, for each pair of fixed exponents and , this equation has only finitely many solutions. But this proof depends on Siegel's finiteness theorem, which is ineffective. showed that, if and with , , and , then is bounded by an effectively computable constant depending only on and . showed that for and odd , this equation has no solution other than the two solutions given above.
Balasubramanian and Shorey proved in 1980 that there are only finitely many possible solutions to the equations with prime divisors of and lying in a given finite set and that they may be effectively computed.
showed that, for each fixed and , this equation has at most one solution.
For fixed x (or y), equation has at most 15 solutions, and at most two unless x is either odd prime power times a power of two, or in the finite set {15, 21, 30, 33, 35, 39, 45, 51, 65, 85, 143, 154, 713}, in which case there are at most three solutions. Furthermore, there is at most one solution if the odd part of n is squareful unless n has at most two distinct odd prime factors or n is in a finite set {315, 495, 525, 585, 630, 693, 735, 765, 855, 945, 1035, 1050, 1170, 1260, 1386, 1530, 1890, 1925, 1950, 1953, 2115, 2175, 2223, 2325, 2535, 2565, 2898, 2907, 3105, 3150, 3325, 3465, 3663, 3675, 4235, 5525, 5661, 6273, 8109, 17575, 39151}.
Application to repunits
The Goormaghtigh conjecture may be expressed as saying that 31 (111 in base 5, 11111 in base 2) and 8191 (111 in base 90, 1111111111111 in base 2) are the only two numbers that are repunits with at least 3 digits in two different bases.
See also
Feit–Thompson conjecture
References
Goormaghtigh, Rene. L’Intermédiaire des Mathématiciens 24 (1917), 88
Diophantine equations
Conjectures
Unsolved problems in number theory
|
https://en.wikipedia.org/wiki/Busemann%27s%20theorem
|
In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler spaces.
Statement of the theorem
Let K be a convex body in n-dimensional Euclidean space Rn containing the origin in its interior. Let S be an (n − 2)-dimensional linear subspace of Rn. For each unit vector θ in S⊥, the orthogonal complement of S, let Sθ denote the (n − 1)-dimensional hyperplane containing θ and S. Define r(θ) to be the (n − 1)-dimensional volume of K ∩ Sθ. Let C be the curve {θr(θ)} in S⊥. Then C forms the boundary of a convex body in S⊥.
See also
Brunn–Minkowski inequality
Prékopa–Leindler inequality
References
Euclidean geometry
Geometric inequalities
Theorems in convex geometry
|
https://en.wikipedia.org/wiki/Vitale%27s%20random%20Brunn%E2%80%93Minkowski%20inequality
|
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets.
Statement of the inequality
Let X be a random compact set in Rn; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of Rn equipped with the Hausdorff metric. A random vector V : Ω → Rn is called a selection of X if Pr(V ∈ X) = 1. If K is a non-empty, compact subset of Rn, let
and define the set-valued expectation E[X] of X to be
Note that E[X] is a subset of Rn. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set X with ,
where "" denotes n-dimensional Lebesgue measure.
Relationship to the Brunn–Minkowski inequality
If X takes the values (non-empty, compact sets) K and L with probabilities 1 − λ and λ respectively, then Vitale's random Brunn–Minkowski inequality is simply the original Brunn–Minkowski inequality for compact sets.
References
Probabilistic inequalities
Theorems in measure theory
|
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Ukraine
|
Ukrainian Premier League
As of the end of the 2020–21 season, unless stated otherwise.
Team records
Titles
Most League titles:
16, Dynamo Kyiv
Most consecutive League titles:
9, Dynamo Kyiv (1992–93 – 2000–01)
Biggest title-winning margin:
23 points, 2019–20; Shakhtar Donetsk (82 points) over Dynamo Kyiv (59 points).
Smallest title-winning margin:
0 points:
1992–93, Dynamo Kyiv and Dnipro Dnipropetrovsk both finished on 44 points, but Dynamo Kyiv won the title with a superior goal difference: (+45) over (+31);
2005–06, Shakhtar Donetsk and Dynamo Kyiv both finished on 75 points, but Shakhtar won the title by winning the golden match.
Winning a title with most remaining games:
5 games, Shakhtar Donetsk (2019–20)
Wins
Most wins in a season: 27, Dynamo Kyiv (1999–2000)
26 games: 23, Dynamo Kyiv (2015–16)
28 games: 21, Shakhtar Donetsk (2013–14)
30 games: 27, Dynamo Kyiv (1999–2000)
32 games: 26, Shakhtar Donetsk (2019–20)
34 games: 25, Dynamo Kyiv (1994–95)
Fewest wins in a season: 0, Zirka Kirovohrad (1999–2000)
26 games: 2, Nyva Ternopil (2000–01)
28 games: 2, Metalurh Zaporizhzhia and Tavriya Simferopol (2013–14)
30 games: 0, Zirka Kirovohrad (1999–2000)
32 games: 2, Karpaty Lviv (2019–20)
34 games: 4, Zorya Luhansk (1995–96)
Losses
Most losses in a season: 26, Zorya Luhansk (1995–96)
26 games: 21, Nyva Ternopil (2000–01)
28 games: 22, Tavriya Simferopol (2013–14)
30 games: 22, SC Mykolaiv (1998–99)
32 games: 24, Volyn Lutsk (2016–17)
34 games: 26, Zorya Luhansk (1995–96)
Fewest losses in a season: 0, Dynamo Kyiv (1999–2000, 2006–07, 2014–15), Shakhtar Donetsk (2001–02)
26 games: 0, Shakhtar Donetsk (2001–02), Dynamo Kyiv (2014–15)
28 games: 3, Metalist Kharkiv (2013–14)
30 games: 0, Dynamo Kyiv (1999–2000, 2006–07)
32 games: 2, Shakhtar Donetsk (2016–17, 2019–20)
34 games: 1, Dynamo Kyiv (1993–94, 1994–95)
Points
Most points in a season: 84, Dynamo Kyiv (1999–2000)
26 games: 70, Dynamo Kyiv (2015–16)
28 games: 65, Shakhtar Donetsk (2013–14)
30 games: 84, Dynamo Kyiv (1999–2000)
32 games: 82, Shakhtar Donetsk (2019–20)
34 games: 83, Dynamo Kyiv (1994–95)
Fewest points in a season: 9, Zirka Kirovohrad (1999–2000), Nyva Ternopil (2000–01)
26 games: 9, Nyva Ternopil (2000–01)
28 games: 10, Tavriya Simferopol (2013–14)
30 games: 9, Zirka Kirovohrad (1999–2000)
32 games: 10, Volyn Lutsk (2016–17)
34 games: 16, Zorya Luhansk (1995–96)
Goals scored
Most goals scored in a season: 87, Dynamo Kyiv (1994–95)
26 games: 76, Shakhtar Donetsk (2015–16)
28 games: 65, Shakhtar Donetsk (2013–14)
30 games: 85, Dynamo Kyiv (1999–2000)
32 games: 80, Shakhtar Donetsk (2019–20)
34 games: 87, Dynamo Kyiv (1994–95)
Fewest goals scored in a season: 13, Hoverla Uzhhorod (2015–16)
26 games: 13, Hoverla Uzhhorod (2015–16)
28 games: 15, Tavriya Simferopol (2013–14)
30 games: 15, Borysfen Boryspil (2004–05)
32 games: 17, Volyn Lutsk (2016–17)
34 games: 16, Zorya Luhansk (1995–96)
Goals conceded
Most goals conceded in a season: 80, Zorya-MALS Luhansk (1995–96)
26 game
|
https://en.wikipedia.org/wiki/H%20tree
|
In fractal geometry, the H tree is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering.
Construction
An H tree can be constructed by starting with a line segment of arbitrary length, drawing two shorter segments at right angles to the first through its endpoints, and continuing in the same vein, reducing (dividing) the length of the line segments drawn at each stage by . A variant of this construction could also be defined in which the length at each iteration is multiplied by a ratio less than , but for this variant the resulting shape covers only part of its bounding rectangle, with a fractal boundary.
An alternative process that generates the same fractal set is to begin with a rectangle with sides in the ratio , and repeatedly bisect it into two smaller silver rectangles, at each stage connecting the two centroids of the two smaller rectangles by a line segment. A similar process can be performed with rectangles of any other shape, but the rectangle leads to the line segment size decreasing uniformly by a factor at each step while for other rectangles the length will decrease by different factors at odd and even levels of the recursive construction.
Properties
The H tree is a self-similar fractal; its Hausdorff dimension is equal to 2.
The points of the H tree come arbitrarily close to every point in a rectangle (the same as the starting rectangle in the constructing by centroids of subdivided rectangles). However, it does not include all points of the rectangle; for instance, the points on the perpendicular bisector of the initial line segment (other than the midpoint of this segment) are not included.
Applications
In VLSI design, the H tree may be used as the layout for a complete binary tree using a total area that is proportional to the number of nodes of the tree. Additionally, the H tree forms a space efficient layout for trees in graph drawing, and as part of a construction of a point set for which the sum of squared edge lengths of the traveling salesman tour is large. It is commonly used as a clock distribution network for routing timing signals to all parts of a chip with equal propagation delays to each part, and has also been used as an interconnection network for VLSI multiprocessors.
The planar H tree can be generalized to the three-dimensional structure via adding line segments on the direction perpendicular to the H tree plane. The resultant three-dimensional H tree has Hausdorff dimension equal to 3. The planar H tree and its three-dimensional version have been found to constitute artificial electromagnetic atoms in photonic crystals and metamaterials and might have potential applications in
|
https://en.wikipedia.org/wiki/Kalenderhane%20Mosque
|
Kalenderhane Mosque () is a former Eastern Orthodox church in Istanbul, converted into a mosque by the Ottomans. With high probability the church was originally dedicated to the Theotokos Kyriotissa. The building is sometimes referred to as Kalender Haneh Jamissi and St. Mary Diaconissa. This building represents one among the few extant examples of a Byzantine church with domed Greek cross plan.
Location
The mosque is located in the Fatih district of Istanbul, Turkey, in the picturesque neighborhood of Vefa, and lies immediately to the south of the easternmost extant section of the aqueduct of Valens, and less than one km to the southeast of the Vefa Kilise Mosque.
History
The first building on this site was a Roman bath, followed by a sixth-century (the dating was based on precise coin finds in stratigraphic excavation) hall church with an apse laying up against the Aqueduct of Valens. Later – possibly in the seventh century – a much larger church was built to the south of the first church. A third church, which reused the sanctuary and the apse (later destroyed by the Ottomans) of the second one, can be dated to the end of the twelfth century, during the late Comnenian period. It may date to between 1197 and 1204, since Constantine Stilbes alluded to its destruction in a fire in 1197. The church was surrounded by monastery buildings, which disappeared totally during the Ottoman period. After the Latin conquest of Constantinople, the building was used by the Crusaders as a Roman Catholic church, and partly officiated by Franciscan clergy.
After the conquest of Constantinople in 1453, the church was assigned personally by Mehmed II to the Kalenderi sect of the Dervish. The Dervishes used it as a zaviye and imaret (public kitchen), and the building has been known since as Kalenderhane ().
The Waqf (foundation) was endowed with several properties in Thrace, and many hamams in Istanbul and Galata. Some years later, Arpa Emini Mustafa Efendi built a Mektep (school) and a Medrese.
In 1746, Hacı Beşir Ağa (d. 1747), the Kizlar Ağası of the Topkapı Palace, built a mihrab, minbar and mahfil, completing the conversion of the building into a mosque. Ravaged by fire and damaged by earthquakes, the mosque was restored in 1855 and again between 1880 and 1890. It was abandoned in the 1930s, after the collapse of the minaret due to lightning, and the demolition of the Medrese.
The conservation of the building dates from the 1970s, when it was extensively restored and studied in a ten-year effort by Cecil L. Striker and Doğan Kuban, who restored its twelfth-century condition. Moreover, the minaret and the mihrab were rebuilt, which allowed the mosque to reopen for worship.
The restoration also provided a solution to the problem of the dedication of the church: while before it was thought that the church was named after Theotokos tēs Diakonissēs ("Virgin of the Deaconesses") or Christos ho Akatalēptos ("Christ the Inconceivable"), the discovery of a donor
|
https://en.wikipedia.org/wiki/Milman%27s%20reverse%20Brunn%E2%80%93Minkowski%20inequality
|
In mathematics, particularly, in asymptotic convex geometry, Milman's reverse Brunn–Minkowski inequality is a result due to Vitali Milman that provides a reverse inequality to the famous Brunn–Minkowski inequality for convex bodies in n-dimensional Euclidean space Rn. Namely, it bounds the volume of the Minkowski sum of two bodies from above in terms of the volumes of the bodies.
Introduction
Let K and L be convex bodies in Rn. The Brunn–Minkowski inequality states that
where vol denotes n-dimensional Lebesgue measure and the + on the left-hand side denotes Minkowski addition.
In general, no reverse bound is possible, since one can find convex bodies K and L of unit volume so that the volume of their Minkowski sum is arbitrarily large. Milman's theorem states that one can replace one of the bodies by its image under a properly chosen volume-preserving linear map so that the left-hand side of the Brunn–Minkowski inequality is bounded by a constant multiple of the right-hand side.
The result is one of the main structural theorems in the local theory of Banach spaces.
Statement of the inequality
There is a constant C, independent of n, such that for any two centrally symmetric convex bodies K and L in Rn, there are volume-preserving linear maps φ and ψ from Rn to itself such that for any real numbers s, t > 0
One of the maps may be chosen to be the identity.
Notes
References
Asymptotic geometric analysis
Euclidean geometry
Geometric inequalities
Theorems in measure theory
|
https://en.wikipedia.org/wiki/Regular%20Hadamard%20matrix
|
In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order be a square number. The excess, denoted E(H), of a Hadamard matrix H of order n is defined to be the sum of the entries of H. The excess satisfies the bound
|E(H)| ≤ n3/2. A Hadamard matrix attains this bound if and only if it is regular.
Parameters
If n = 4u2 is the order of a regular Hadamard matrix, then the excess is ±8u3 and the row and column sums all equal ±2u. It follows that each row has 2u2 ± u positive entries and 2u2 ∓ u negative entries. The orthogonality of rows implies that any two distinct rows have exactly u2 ± u positive entries in common. If H is interpreted as the incidence matrix of a block design, with 1 representing incidence and −1 representing non-incidence, then H corresponds to a symmetric 2-(v,k,λ) design with parameters (4u2, 2u2 ± u, u2 ± u). A design with these parameters is called a Menon design.
Construction
A number of methods for constructing regular Hadamard matrices are known, and some exhaustive computer searches have been done for regular Hadamard matrices with specified symmetry groups, but it is not known whether every even perfect square is the order of a regular Hadamard matrix. Bush-type Hadamard matrices are regular Hadamard matrices of a special form, and are connected with finite projective planes.
History and naming
Like Hadamard matrices more generally, regular Hadamard matrices are named after Jacques Hadamard. Menon designs are named after P Kesava Menon, and Bush-type Hadamard matrices are named after Kenneth A. Bush.
References
C.J. Colbourn and J.H. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd ed., CRC Press, Boca Raton, Florida., 2006.
W. D. Wallis, Anne Penfold Street, and Jennifer Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Springer-Verlag, Berlin 1972.
Matrices
|
https://en.wikipedia.org/wiki/Skills%20for%20Life
|
Skills for Life is a national lifelong learning strategy in England for improving adult skills, designed to help learners develop their reading, writing, maths, technical, and digital skills. It provides universal free education and training; including courses in digital, numeracy and transferable skills; traineeships; apprenticeships; and vocational qualifications for all adults (19 and over) in further education colleges and beyond.
The courses and qualifications provided by training providers can also be provided by employees and businesses, and can be taken in Skill Bootcamps at colleges and universities throughout England. It is linked with the National Careers Service.
Training courses
Training cources include classroom-based, on-the-job, online, and short cources. The full list can be found on the Skills for Life website.
Courses for jobs
Multiply (a new a new government-funded programme to help adults improve numeracy skills)
Skills Bootcamps
Returnerships (courses and training for over 50s)
Digital – Essential Skills
Numeracy – Essential Skills
English – Essential Skills (including ESOL)
Higher Technical Qualifications (HTQs)
Apprenticeships
The Skills Toolkit
Sector-based Work Academy Programme (SWAP)
Skills for Life (2001-2010)
Skills for Life was also a national strategy in England for improving adult literacy, language (ESOL) and numeracy skills and was established as part of the wider national skills srategy by the Labour Party from 2001-2010. The strategy set out how the government aimed to reach its Public Service Agreement (PSA) target to improve "the basic skill levels of 2.25 million adults between the launch of Skills for Life in 2001 and 2010, with a milestone of 1.5 million in 2007". This PSA target was part of the wider objective to "tackle the adult skills gaps", by increasing the number of adults with the skills required for employability and progression to higher levels of training. The rationale behind the Skills for Life strategy was to make Britain a more equal society and ‘close the gap’ by addressing issues that include area and neighbourhood deprivation, and educational attainment.
See also
T levels
Further education
National Careers Service
National Apprenticeship Service
Lifelong learning
Adult education
References
External links
Skills for Life website
Education in England
Adult education in the United Kingdom
|
https://en.wikipedia.org/wiki/Henry%20C.%20Yuen
|
Henry Che-Chuen Yuen (Chinese: 袁子春; born 7 April 1948, in Shanghai, China) is a founder and former CEO of Gemstar-TV Guide International. He has a PhD in applied mathematics from Caltech. He worked briefly at Caltech and New York University, then obtained a law degree from Loyola Law School.
He founded Gemstar in 1986. Called a "patent terrorist" for his aggressive litigation regarding the company's patent portfolio, he was also lauded as "the Bill Gates of TV", even negotiating a generous settlement with Microsoft for per-unit royalties and advertising revenue from Microsoft's WebTV and Ultimate TV set-top boxes. Commenting in 2001 on his approach to business, he remarked: "In business, where I am right now, the only rules that exist are the ones you make."
After he merged Gemstar with TV Guide - he became notorious for constructing irrational and obscure business plans that had no relevance and relationship with the realities of the media business.
He was fired from Gemstar in 2003, after the company revealed criminal manipulation of revenue recognition initiated by Yuen and other accounting problems. He was convicted of securities fraud in 2006, and ordered to pay $22 million in penalties. As of April 25, 2007, his whereabouts are unknown.
References
External links
Early profile
American businesspeople
Loyola Law School alumni
California Institute of Technology alumni
New York University faculty
Chinese emigrants to the United States
Living people
1948 births
American businesspeople convicted of crimes
|
https://en.wikipedia.org/wiki/Javier%20P%C3%A1ez
|
Javier Marcelo Páez (born September 23, 1975 in Merlo) is an Argentine football defender.
References
Javier Páez – Argentine Primera statistics at Fútbol XXI
Profile and statistics of Javier Páez on One.co.il
1975 births
Living people
Argentine men's footballers
Men's association football defenders
Club Atlético Independiente footballers
Deportivo Español footballers
Talleres de Córdoba footballers
Club Olimpo footballers
S.D. Quito footballers
Hapoel Tel Aviv F.C. players
Atlético Tucumán footballers
Chacarita Juniors footballers
Argentine Primera División players
Israeli Premier League players
Argentine expatriate men's footballers
Expatriate men's footballers in Israel
Expatriate men's footballers in Ecuador
Footballers from Buenos Aires Province
|
https://en.wikipedia.org/wiki/Coventry%20and%20Bedworth%20urban%20area
|
The Coventry/Bedworth Urban Area or Coventry Built-up area as defined by the Office for National Statistics had a population of 359,252 at the 2011 census, which made it the 16th largest conurbation in England and Wales by population. It is also one of the most densely populated. In the 2021 census the population of the urban area was recorded at 389,603.
Details
The largest settlement is Coventry (population 352,900) which is within the West Midlands County. Bedworth (population 30,648) and Binley Woods (population 2,665) are the other main parts of the conurbation and both lie in the county Warwickshire in the districts of Nuneaton and Bedworth and the Borough of Rugby respectively. There are no other towns in the conurbation.
There is a very small amount of green belt between the Coventry/Bedworth Urban Area and the Nuneaton Urban Area in the north -- however with the development of industrial and retail units in south Nuneaton, the two conurbations are virtually connected -- and also between Coventry and Kenilworth. A larger area of green belt exists between Coventry and the Greater West Midlands Urban Area which extends to about 5 miles.
References
Urban areas of England
Geography of the West Midlands (county)
Coventry
Geography of Warwickshire
Bedworth
|
https://en.wikipedia.org/wiki/Tuto
|
Livonir Ruschel, simply known as Tuto (born 2 July 1979 in Dionísio Cerqueira) is a Brazilian professional footballer.
Club statistics
External links
Player profile and statistics of Livonir Ruschel on One.co.il
1979 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
J2 League players
Japan Football League (1992–1998) players
Israeli Premier League players
Kawasaki Frontale players
FC Tokyo players
Urawa Red Diamonds players
Shimizu S-Pulse players
Omiya Ardija players
Shonan Bellmare players
Expatriate men's footballers in Japan
Associação Atlética Ponte Preta players
Associação Desportiva São Caetano players
Associação Chapecoense de Futebol players
Beitar Jerusalem F.C. players
Expatriate men's footballers in Israel
Men's association football forwards
|
https://en.wikipedia.org/wiki/Cristian%20Gonz%C3%A1lez%20%28footballer%2C%20born%201976%29
|
Cristian Mario González Aidinovich (born December 19, 1976) is an Uruguayan retired football player.
External links
Profile and statistics of Cristian Gonzalez on One.co.il
Profile at tenfieldigital
1976 births
Living people
Uruguayan men's footballers
Uruguay men's international footballers
Uruguayan Primera División players
Segunda División players
Israeli Premier League players
Liverpool F.C. (Montevideo) players
Defensor Sporting players
Peñarol players
UD Las Palmas players
Maccabi Tel Aviv F.C. players
F.C. Ashdod players
Club Atlético River Plate (Montevideo) players
Beitar Jerusalem F.C. players
Club Plaza Colonia de Deportes players
Deportivo Maldonado players
El Tanque Sisley players
Sud América players
Uruguayan expatriate men's footballers
Expatriate men's footballers in Spain
Expatriate men's footballers in Israel
Uruguayan expatriate sportspeople in Spain
Uruguayan expatriate sportspeople in Israel
Men's association football defenders
Footballers from Canelones Department
|
https://en.wikipedia.org/wiki/Quillen%20adjunction
|
In homotopy theory, a branch of mathematics, a Quillen adjunction between two closed model categories C and D is a special kind of adjunction between categories that induces an adjunction between the homotopy categories Ho(C) and Ho(D) via the total derived functor construction. Quillen adjunctions are named in honor of the mathematician Daniel Quillen.
Formal definition
Given two closed model categories C and D, a Quillen adjunction is a pair
(F, G): C D
of adjoint functors with F left adjoint to G such that F preserves cofibrations and trivial cofibrations or, equivalently by the closed model axioms, such that G preserves fibrations and trivial fibrations. In such an adjunction F is called the left Quillen functor and G is called the right Quillen functor.
Properties
It is a consequence of the axioms that a left (right) Quillen functor preserves weak equivalences between cofibrant (fibrant) objects. The total derived functor theorem of Quillen says that the total left derived functor
LF: Ho(C) → Ho(D)
is a left adjoint to the total right derived functor
RG: Ho(D) → Ho(C).
This adjunction (LF, RG) is called the derived adjunction.
If (F, G) is a Quillen adjunction as above such that
F(c) → d
with c cofibrant and d fibrant is a weak equivalence in D if and only if
c → G(d)
is a weak equivalence in C then it is called a Quillen equivalence of the closed model categories C and D. In this case the derived adjunction is an adjoint equivalence of categories so that
LF(c) → d
is an isomorphism in Ho(D) if and only if
c → RG(d)
is an isomorphism in Ho(C).
References
Philip S. Hirschhorn, Model Categories and Their Localizations, American Mathematical Soc., Aug 24, 2009 - Mathematics - 457 pages
Homotopy theory
Theory of continuous functions
Adjoint functors
|
https://en.wikipedia.org/wiki/Borell%E2%80%93Brascamp%E2%80%93Lieb%20inequality
|
In mathematics, the Borell–Brascamp–Lieb inequality is an integral inequality due to many different mathematicians but named after Christer Borell, Herm Jan Brascamp and Elliott Lieb.
The result was proved for p > 0 by Henstock and Macbeath in 1953. The case p = 0 is known as the Prékopa–Leindler inequality and was re-discovered by Brascamp and Lieb in 1976, when they proved the general version below; working independently, Borell had done the same in 1975. The nomenclature of "Borell–Brascamp–Lieb inequality" is due to Cordero-Erausquin, McCann and Schmuckenschläger, who in 2001 generalized the result to Riemannian manifolds such as the sphere and hyperbolic space.
Statement of the inequality in Rn
Let 0 < λ < 1, let −1 / n ≤ p ≤ +∞, and let f, g, h : Rn → [0, +∞) be integrable functions such that, for all x and y in Rn,
where
and .
Then
(When p = −1 / n, the convention is to take p / (n p + 1) to be −∞; when p = +∞, it is taken to be 1 / n.)
References
Geometric inequalities
Integral geometry
|
https://en.wikipedia.org/wiki/Birotunda
|
In geometry, a birotunda is any member of a family of dihedral-symmetric polyhedra, formed from two rotunda adjoined through the largest face. They are similar to a bicupola but instead of alternating squares and triangles, it alternates pentagons and triangles around an axis. There are two forms, ortho- and gyro-: an orthobirotunda has one of the two rotundas is placed as the mirror reflection of the other, while in a gyrobirotunda one rotunda is twisted relative to the other.
The pentagonal birotundas can be formed with regular faces, one a Johnson solid, the other a semiregular polyhedron:
pentagonal orthobirotunda,
pentagonal gyrobirotunda, which is also called an icosidodecahedron.
Other forms can be generated with dihedral symmetry and distorted equilateral pentagons.
Examples
See also
Gyroelongated pentagonal birotunda
Elongated pentagonal orthobirotunda
Elongated pentagonal gyrobirotunda
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
Johnson solids
|
https://en.wikipedia.org/wiki/Chandrashekhar%20Khare
|
Chandrashekhar B. Khare (born 1968) is a professor of mathematics at the University of California Los Angeles. In 2005, he made a major advance in the field of Galois representations and number theory by proving the level 1 Serre conjecture, and later a proof of the full conjecture with Jean-Pierre Wintenberger. He has been on the Mathematical Sciences jury for the Infosys Prize from 2015, serving as Jury Chair from 2020.
Professional career
Resident of Mumbai, India and completed his undergraduate education at Trinity College, Cambridge. He finished his thesis in 1995 under the supervision of Haruzo Hida at California Institute of Technology. His Ph.D. thesis was published in the Duke Mathematical Journal. He proved Serre's conjecture with Jean-Pierre Wintenberger, published in Inventiones Mathematicae.
He started his career as a Fellow at Tata Institute of Fundamental Research. As of the year 2021, he is a professor at the University of California, Los Angeles.
Awards and honors
Khare is the winner of the INSA Young Scientist Award (1999), Fermat Prize (2007), the Infosys Prize (2010), and the Cole Prize (2011).
He gave an invited talk at the International Congress of Mathematicians in 2010, on the topic of "Number Theory".
In 2012 he became a fellow of the American Mathematical Society and was elected as a Fellow of the Royal Society.
References
External links
Chandrashekhar Khare's homepage
Another proof for Fermat's last theorem
1968 births
Living people
Indian number theorists
21st-century Indian mathematicians
Institute for Advanced Study visiting scholars
University of California, Los Angeles faculty
American Hindus
20th-century Indian mathematicians
Fellows of the American Mathematical Society
Fellows of the Royal Society
American people of Marathi descent
American people of Indian descent
California Institute of Technology alumni
Alumni of the University of Cambridge
|
https://en.wikipedia.org/wiki/Unduloid
|
In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.
Formula
Let represent the normal Jacobi sine function and be the normal Jacobi elliptic function and let represent the normal elliptic integral of the first kind and represent the normal elliptic integral of the second kind. Let a be the length of the ellipse's major axis, and e be the eccentricity of the ellipse. Let k be a fixed value between 0 and 1 called the modulus.
Given these variables,
The formula for the surface of revolution that is the unduloid is then
Properties
One interesting property of the unduloid is that the mean curvature is constant. In fact, the mean curvature across the entire surface is always the reciprocal of twice the major axis length: 1/(2a).
Also, geodesics on an unduloid obey the Clairaut relation, and their behavior is therefore predictable.
Occurrence in material science
Unduloids are not a common pattern in nature. However, there are a few circumstances in which they form. First documented in 1970, passing a strong electric current through a thin (0.16—1.0mm), horizontally mounted, hard-drawn (non-tempered) silver wire will result in unduloids forming along its length.
This phenomenon was later discovered to also occur in molybdenum wire.
Unduloids have also been formed with ferrofluids. By passing a current axially through a cylinder coated with a viscous magnetic fluid film, the magnetic dipoles of the fluid interact with the magnetic field of the current, creating a droplet pattern along the cylinder’s length.
References
Surfaces
|
https://en.wikipedia.org/wiki/Propensity%20probability
|
The propensity theory of probability is a probability interpretation in which the probability is thought of as a physical propensity, disposition, or tendency of a given type of situation to yield an outcome of a certain kind, or to yield a long-run relative frequency of such an outcome.
Propensities are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate. Stable long-run frequencies are a manifestation of invariant single-case probabilities. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives. These single-case probabilities are known as propensities or chances.
In addition to explaining the emergence of stable relative frequencies, the idea of propensity is motivated by the desire to make sense of single-case probability attributions in quantum mechanics, such as the probability of decay of a particular atom at a particular moment.
History
A propensity theory of probability was given by Charles Sanders Peirce.
Karl Popper
A later propensity theory was proposed by philosopher Karl Popper, who had only slight acquaintance with the writings of Charles S. Peirce, however. Popper noted that the outcome of a physical experiment is produced by a certain set of "generating conditions". When we repeat an experiment, as the saying goes, we really perform another experiment with a (more or less) similar set of generating conditions. To say that a set of generating conditions G has propensity p of producing the outcome E means that those exact conditions, if repeated indefinitely, would produce an outcome sequence in which E occurred with limiting relative frequency p. Thus the propensity p for E to occur depends upon G:. For Popper then, a deterministic experiment would have propensity 0 or 1 for each outcome, since those generating conditions would have the same outcome on each trial. In other words, non-trivial propensities (those that differ from 0 and 1) imply something less than determinism and yet still causal dependence on the generating conditions.
Recent work
A number of other philosophers, including David Miller and Donald A. Gillies, have proposed propensity theories somewhat similar to Popper's, in that propensities are defined in terms of either long-run or infinitely long-run relative frequencies.
Other propensity theorists (e.g. Ronald Giere) do not explicitly define propensities at all, but rather see propensity as defined by the theoretical role it plays in science. They argue, for example, that physical magnitudes such as electrical charge cannot be explicitly defined either, in terms of more basic things, but only in terms of what they do (such as attracting and repelling other electrical charges). In a similar way, propensity is whatever fi
|
https://en.wikipedia.org/wiki/Werner%20Hildenbrand
|
Werner Hildenbrand (born 25 May 1936 in Göttingen) is a German economist and mathematician. He was educated at the University of Heidelberg, where he received his Diplom in mathematics, applied mathematics and physics in 1961. He continued his education at the University of Heidelberg and received his Ph.D. in mathematics in 1964 and his habilitation in economics and mathematics in 1968.
From 1969 to 2001, he was a professor of economics at the University of Bonn. He has held various visiting positions at, among others, the University of California, Berkeley and the University of Louvain. His research has focused on general equilibrium theory, and in particular on the existence and properties of the core of an economy.
Books
Core and Equilibria of a Large Economy, Princeton University Press, 1974.
Introduction to Equilibrium Analysis, with Alan Kirman, North-Holland, 1976.
Equilibrium Analysis, with Alan Kirman, North-Holland, 1988.
Market Demand: Theory and Empirical Evidence, Princeton University Press, 1994.
External links
Werner Hildenbrand's personal homepage at University of Bonn
References
1936 births
Fellows of the Econometric Society
Fellows of the American Academy of Arts and Sciences
General equilibrium theorists
German economists
20th-century German mathematicians
21st-century German mathematicians
Academic staff of the University of Bonn
Gottfried Wilhelm Leibniz Prize winners
Living people
|
https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Israel
|
This page details football records in Israel.
National team
See Israel national football team records.
League
Records in this section refer to Palestine League from its founding in 1931 to 1947, Israeli League from 1949 to 1950, Liga Alef from 1951 to 1955, Liga Leumit from 1955 to 1999 and to the Premier League since the 1999 season.
Titles
Most League titles: 23, Maccabi Tel Aviv (1936, 1937, 1941–42, 1946–47, 1949–50, 1951–52, 1953–54, 1955–56, 1957–58, 1966–68, 1969–70, 1971–72, 1976–77, 1978–79, 1991–92, 1994–95, 1995–96, 2002–03, 2012–13, 2013–14, 2014–15, 2018–19, 2019–20)
Most consecutive League titles: 5, Hapoel Petah Tikva (1958–59, 1959–60, 1960–61, 1961–62, 1962–63)
Top-flight Appearances
Most appearances: 70 seasons, Maccabi Tel Aviv (1949–present)
Most consecutive seasons in top-flight: 70 seasons, Maccabi Tel Aviv (1949–present)
Unbeaten runs
Longest unbeaten League run: 48, Maccabi Haifa (Liga Leumit, between 3 April 1993 and 10 October 1994)
Points
Most points in a season:
2 points for a win: 47, Maccabi Netanya (Liga Leumit, 1970–71)
3 points for a win: 95, Maccabi Haifa (Liga Leumit, 1993–94)
Fewest points in a season
2 points for a win: 0, Maccabi Nes Tziona (Israeli League, 1949–50)
3 points for a win: 10, joint record:
Beitar Netanya (Liga Leumit, 1986–87)
Maccabi Jaffa (Liga Leumit, 1998–99)
Team
Most top-flight goals scored in a season: 103, Maccabi Tel Aviv (Israeli League, 1949–50)
Promotion and change in position
Highest finish by a promoted club: 1st, joint record:
Hapoel Ramat Gan (Liga Leumit, 1963–64)
Beitar Jerusalem (Liga Leumit, 1992–93)
Lowest finish by the previous season's champions: 16th out of 16 (relegated), Hapoel Kfar Saba (Liga Leumit, 1982–83)
Goals
Individual
Most career league goals: 207, Alon Mizrahi (1989 to 2005)
Most goals in a season: 35, Eran Zahavi (36 matches, for Maccabi Tel Aviv in the (2015–16 Premier League)
Fastest goal: 6 seconds, Ilan Bakhar (for Maccabi Herzliya v. Hapoel Be'er Sheva, 1997–98 Liga Leumit)
Premier League – Since 1999–2000 season
Titles
Most titles: 7, Maccabi Haifa
Most consecutive title wins: 3, joint record:
Maccabi Haifa (2003–04, 2004–05, 2005–06)
Maccabi Tel Aviv (2012–13, 2013–14, 2014–15)
Hapoel Be'er Sheva (2015–16, 2016–17, 2017–18)
Wins
Most wins in a season: 27, Maccabi Tel Aviv (2018–19)
Fewest wins in a season: 3, join record:
Hapoel Petah Tikva (2006–07)
Hapoel Ramat Gan (2010–11)
Most Premier League wins: 340, Maccabi Haifa (1999–present)
Draws
Most draws in a season: 18, Hapoel Haifa (1999–2000)
Fewest draws in a season: 3, Maccabi Tel Aviv (2002–03)
Most Premier League draws: 16, Beitar Jerusalem (1999–present)
Losses
Most losses in a season: 30, Tzafririm Holon (2000–01)
Fewest losses in a season: 1, joint record:
Hapoel Tel Aviv (2009–10)
Maccabi Tel Aviv (2018–19, 2019–20)
Most Premier League losses: 235, F.C. Ashdod (1999–2014, 2015–present)
Premier League Appearances
Most appearances: 20 seasons, joint record:
Beitar Jerusalem
|
https://en.wikipedia.org/wiki/Annie%20Dale%20Biddle%20Andrews
|
Annie Dale Biddle Andrews (December 13, 1885 – April 14, 1940) was the first woman to earn a Ph.D. in mathematics from the University of California, Berkeley.
Early life and career
She was born in Hanford, California, the youngest daughter (and youngest of seven children) of Samuel Edward Biddle and Achsah Annie Biddle (née McQuidy).
She received her B.A. degree from the University of California in 1908. In 1911, she wrote her thesis, Constructive theory of the unicursal plane quartic by synthetic methods, under her maiden name, Annie Dale Biddle; it was published by the university in 1912. Her advisors were Derrick Norman Lehmer and Mellen Haskell. The paper proved to be very useful in its time as it was found that all algebraic surfaces correspond to a universal quartic having no double or triple points with distinct tangents.
She was a math instructor at the University of Washington from 1911 to 1912, after which she married Wilhelm Samuel Andrews. She worked as a math instructor at the University of California between 1915 and 1932 after being appointed as a teaching fellow there in 1914.
She presented a research paper at the meeting of the Journal of the American Mathematical Society in March 1933 in Palo Alto, California, entitled "The space quartic of the second kind by synthetic methods". The abstract of the paper was published later that year.
Personal life
From 1936 Andrews took an active interest in public affairs and charities, in addition to her mathematical research. She died on April 14, 1940, after two years of illness. She was survived by her husband and two children.
References
1885 births
1940 deaths
American women mathematicians
20th-century American mathematicians
20th-century women mathematicians
20th-century American women
|
https://en.wikipedia.org/wiki/Brascamp%E2%80%93Lieb%20inequality
|
In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on n-dimensional Euclidean space . It generalizes the Loomis–Whitney inequality and Hölder's inequality. The second is a result of probability theory which gives a concentration inequality for log-concave probability distributions. Both are named after Herm Jan Brascamp and Elliott H. Lieb.
The geometric inequality
Fix natural numbers m and n. For 1 ≤ i ≤ m, let ni ∈ N and let ci > 0 so that
Choose non-negative, integrable functions
and surjective linear maps
Then the following inequality holds:
where D is given by
Another way to state this is that the constant D is what one would obtain by restricting attention to the case in which each is a centered Gaussian function, namely .
Alternative forms
Consider a probability density function . This probability density function is said to be a log-concave measure if the function is convex. Such probability density functions have tails which decay exponentially fast, so most of the probability mass resides in a small region around the mode of . The Brascamp–Lieb inequality gives another characterization of the compactness of by bounding the mean of any statistic .
Formally, let be any derivable function. The Brascamp–Lieb inequality reads:
where H is the Hessian and is the Nabla symbol.
BCCT inequality
The inequality is generalized in 2008 to account for both continuous and discrete cases, and for all linear maps, with precise estimates on the constant.
Definition: the Brascamp-Lieb datum (BL datum)
.
.
.
are linear surjections, with zero common kernel: .
Call a Brascamp-Lieb datum (BL datum).
For any with , define
Now define the Brascamp-Lieb constant for the BL datum:
Discrete case
Setup:
Finitely generated abelian groups .
Group homomorphisms .
BL datum defined as
is the torsion subgroup, that is, the subgroup of finite-order elements.
With this setup, we have (Theorem 2.4, Theorem 3.12 )
Note that the constant is not always tight.
BL polytope
Given BL datum , the conditions for are
, and
for all subspace of ,
Thus, the subset of that satisfies the above two conditions is a closed convex polytope defined by linear inequalities. This is the BL polytope.
Note that while there are infinitely many possible choices of subspace of , there are only finitely many possible equations of , so the subset is a closed convex polytope.
Similarly we can define the BL polytope for the discrete case.
Relationships to other inequalities
The geometric Brascamp–Lieb inequality
The geometric Brascamp–Lieb inequality, first derived in 1976, is a special case of the general inequality. It was used by Keith Ball, in 1989, to provide upper bounds for volumes of central sections of cubes.
For i = 1, ..., m, let ci > 0 and let ui ∈ Sn−1 be a unit vector; suppose that ci and ui satisfy
for all x in Rn. Let fi ∈ L1(R; [0,
|
https://en.wikipedia.org/wiki/Vikraman%20Balaji
|
Vikraman Balaji is an Indian mathematician and is currently a professor at Chennai Mathematical Institute. He completed his doctorate in Mathematics under the supervision of C. S. Seshadri. His primary area of research is in algebraic geometry, representation theory and differential geometry. Balaji was awarded the 2006 Shanti Swarup Bhatnagar Award in Mathematical Sciences along with Indranil Biswas "for his outstanding contributions to moduli problems of principal bundles over algebraic varieties, in particular on the Uhlenbeck-Yau compactification of the Moduli Spaces of µ-semistable bundles."
He was elected Fellow of the Indian Academy of Sciences in 2007, Fellow of the Indian National Science Academy in 2015 and was awarded the J.C. Bose National Fellowship in 2009.
Selected publications
Notes
External links
CSIR R&D Highlights
2006 Bhatnagar Awards
20th-century Indian mathematicians
Living people
Fellows of the Indian Academy of Sciences
Fellows of the Indian National Science Academy
Year of birth missing (living people)
Recipients of the Shanti Swarup Bhatnagar Award in Mathematical Science
|
https://en.wikipedia.org/wiki/Principal%20axis%20theorem
|
In geometry and linear algebra, a principal axis is a certain line in a Euclidean space associated with an ellipsoid or hyperboloid, generalizing the major and minor axes of an ellipse or hyperbola. The principal axis theorem states that the principal axes are perpendicular, and gives a constructive procedure for finding them.
Mathematically, the principal axis theorem is a generalization of the method of completing the square from elementary algebra. In linear algebra and functional analysis, the principal axis theorem is a geometrical counterpart of the spectral theorem. It has applications to the statistics of principal components analysis and the singular value decomposition. In physics, the theorem is fundamental to the studies of angular momentum and birefringence.
Motivation
The equations in the Cartesian plane R2:
define, respectively, an ellipse and a hyperbola. In each case, the x and y axes are the principal axes. This is easily seen, given that there are no cross-terms involving products xy in either expression. However, the situation is more complicated for equations like
Here some method is required to determine whether this is an ellipse or a hyperbola. The basic observation is that if, by completing the square, the quadratic expression can be reduced to a sum of two squares then the equation defines an ellipse, whereas if it reduces to a difference of two squares then the equation represents a hyperbola:
Thus, in our example expression, the problem is how to absorb the coefficient of the cross-term 8xy into the functions u and v. Formally, this problem is similar to the problem of matrix diagonalization, where one tries to find a suitable coordinate system in which the matrix of a linear transformation is diagonal. The first step is to find a matrix in which the technique of diagonalization can be applied.
The trick is to write the quadratic form as
where the cross-term has been split into two equal parts. The matrix A in the above decomposition is a symmetric matrix. In particular, by the spectral theorem, it has real eigenvalues and is diagonalizable by an orthogonal matrix (orthogonally diagonalizable).
To orthogonally diagonalize A, one must first find its eigenvalues, and then find an orthonormal eigenbasis. Calculation reveals that the eigenvalues of A are
with corresponding eigenvectors
Dividing these by their respective lengths yields an orthonormal eigenbasis:
Now the matrix S = [u1 u2] is an orthogonal matrix, since it has orthonormal columns, and A is diagonalized by:
This applies to the present problem of "diagonalizing" the quadratic form through the observation that
Thus, the equation is that of an ellipse, since the left side can be written as the sum of two squares.
It is tempting to simplify this expression by pulling out factors of 2. However, it is important not to do this. The quantities
have a geometrical meaning. They determine an orthonormal coordinate system on R2. In other wo
|
https://en.wikipedia.org/wiki/Sebastian%20Isra%C3%ABl
|
Sebastian Israël (, born 21 December 1977) is a former French-Israeli footballer.
External links
Profile and statistics of Sebastian Israël on One.co.il
1977 births
Living people
Israeli Jews
21st-century French Jews
Israeli men's footballers
French men's footballers
Jewish French sportspeople
Sektzia Ness Ziona F.C. players
Hapoel Rishon LeZion F.C. players
Hapoel Marmorek F.C. players
Israeli Premier League players
Jewish men's footballers
Liga Leumit players
French emigrants to Israel
Men's association football midfielders
|
https://en.wikipedia.org/wiki/Skew-Hamiltonian%20matrix
|
In linear algebra, skew-Hamiltonian matrices are special matrices which correspond to skew-symmetric bilinear forms on a symplectic vector space.
Let V be a vector space, equipped with a symplectic form . Such a space must be even-dimensional. A linear map is called a skew-Hamiltonian operator with respect to if the form is skew-symmetric.
Choose a basis in V, such that is written as . Then a linear operator is skew-Hamiltonian with respect to if and only if its matrix A satisfies , where J is the skew-symmetric matrix
and In is the identity matrix. Such matrices are called skew-Hamiltonian.
The square of a Hamiltonian matrix is skew-Hamiltonian. The converse is also true: every skew-Hamiltonian matrix can be obtained as the square of a Hamiltonian matrix.
Notes
Matrices
Linear algebra
|
https://en.wikipedia.org/wiki/Homer%20E.%20Newell%20Jr.
|
Homer Edward Newell Jr. (March 11, 1915 – July 18, 1983) was a mathematics professor and author who became a powerful United States government science administrator—eventually rising to the number three position at the National Aeronautics and Space Administration (NASA). In the early 1960s, he either controlled or influenced virtually all non-military uncrewed space missions for the free world.
Early life and education
Newell was born March 11, 1915, in Holyoke, Massachusetts. He was educated in the public schools, graduating at the top of his class from Holyoke High in 1932. In a 1980 interview, he recalled that his interest in science arose from his grandfather Arthur J. Newell, an engineer for a local electrical equipment manufacturer, who had an extensive private library where Newell found books on astronomy and chemistry. Arthur also provided the money for his grandson's university education at Harvard University, where he graduated with a 1936 Bachelor of Arts in Math, and a 1937 Master of Arts in Teaching. He applied for a scholarship to pursue a Doctorate in Math, but Harvard did not award it. Instead, he completed his education at the University of Wisconsin–Madison, which awarded him a Math Ph.D. in 1940 with Rudolf Langer as thesis advisor.
Career
From 1940 to 1944, Newell was an instructor, and then assistant professor of mathematics at the University of Maryland. During World War II he also worked as a Civil Aeronautics Authority (CAA) ground instructor in air navigation, taught engineering classes for military cadets, and briefly taught astronomy. The additional work for the CAA and military ended in 1944, and Newell, who was unhappy as a professor, applied for positions at several organizations doing military research. He was offered a contract position at the Naval Research Laboratory (NRL)'s communications security section in 1944, and later that year became an NRL employee. In 1945, the communications security section became the rocket sonde section. Newell became successively head of the theoretical analysis subsection, associate head of the section, and by 1947 headed the section; which performed upper atmosphere research using rockets including German-built V2s, US-built Aerobees and eventually NRL's own Viking; mostly launched from the White Sands Missile Range.
In 1954, when President Dwight D. Eisenhower assigned NRL responsibility to launch satellites during the International Geophysical Year (IGY), Newell was promoted to Acting Superintendent of NRL's Atmosphere and Astrophysics division, with an additional assignment as science coordinator for Project Vanguard. In this position, Newell worked with the National Academy of Sciences to identify which experiments would be flown on Vanguard satellites. In the wake of the first two Soviet satellites, and the explosion of the first Vanguard on the launch pad, one of the experiment packages selected by Newell was switched to the U.S. Army's Explorer I satellite, which subse
|
https://en.wikipedia.org/wiki/Hermann%20Friedrich%20Waesemann
|
Hermann Friedrich Waesemann (6 June 1813 – 28 January 1879) was a German architect.
He was born in Danzig (Gdańsk), the son of an architect. He studied mathematics and science in Bonn from 1830 to 1832, before going to Berlin to study architecture at the Bauakademie. His main work is the Rotes Rathaus in Berlin.
Waesemann died in Berlin and is buried at the Friedhof II der Sophiengemeinde Berlin.
1813 births
1879 deaths
19th-century German architects
University of Bonn alumni
People from Gdańsk
|
https://en.wikipedia.org/wiki/Conditional%20variance
|
In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables.
Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of autoregressive conditional heteroskedasticity (ARCH) models.
Definition
The conditional variance of a random variable Y given another random variable X is
The conditional variance tells us how much variance is left if we use to "predict" Y.
Here, as usual, stands for the conditional expectation of Y given X,
which we may recall, is a random variable itself (a function of X, determined up to probability one).
As a result, itself is a random variable (and is a function of X).
Explanation, relation to least-squares
Recall that variance is the expected squared deviation between a random variable (say, Y) and its expected value.
The expected value can be thought of as a reasonable prediction of the outcomes of the random experiment (in particular, the expected value is the best constant prediction when predictions are assessed by expected squared prediction error). Thus, one interpretation of variance is that it gives the smallest possible expected squared prediction error. If we have the knowledge of another random variable (X) that we can use to predict Y, we can potentially use this knowledge to reduce the expected squared error. As it turns out, the best prediction of Y given X is the conditional expectation. In particular, for any measurable,
By selecting , the second, nonnegative term becomes zero, showing the claim.
Here, the second equality used the law of total expectation.
We also see that the expected conditional variance of Y given X shows up as the irreducible error of predicting Y given only the knowledge of X.
Special cases, variations
Conditioning on discrete random variables
When X takes on countable many values with positive probability, i.e., it is a discrete random variable, we can introduce , the conditional variance of Y given that X=x for any x from S as follows:
where recall that is the conditional expectation of Z given that X=x, which is well-defined for .
An alternative notation for is
Note that here defines a constant for possible values of x, and in particular, , is not a random variable.
The connection of this definition to is as follows:
Let S be as above and define the function as . Then, almost surely.
Definition using conditional distributions
The "conditional expectation of Y given X=x" can also be defined more generally
using the conditional distribution of Y given X (this exists in this case, as both here X and Y are real-valued).
In particular, letting be the (regular) conditional distribution of Y given X, i.e., (the intention is that almost surely over the support of X), we can define
This can, of course, be specialized to when Y is discrete itself (replacing the integrals
|
https://en.wikipedia.org/wiki/Confirmatory%20factor%20analysis
|
In statistics, confirmatory factor analysis (CFA) is a special form of factor analysis, most commonly used in social science research. It is used to test whether measures of a construct are consistent with a researcher's understanding of the nature of that construct (or factor). As such, the objective of confirmatory factor analysis is to test whether the data fit a hypothesized measurement model. This hypothesized model is based on theory and/or previous analytic research. CFA was first developed by Jöreskog (1969) and has built upon and replaced older methods of analyzing construct validity such as the MTMM Matrix as described in Campbell & Fiske (1959).
In confirmatory factor analysis, the researcher first develops a hypothesis about what factors they believe are underlying the measures used (e.g., "Depression" being the factor underlying the Beck Depression Inventory and the Hamilton Rating Scale for Depression) and may impose constraints on the model based on these a priori hypotheses. By imposing these constraints, the researcher is forcing the model to be consistent with their theory. For example, if it is posited that there are two factors accounting for the covariance in the measures, and that these factors are unrelated to each other, the researcher can create a model where the correlation between factor A and factor B is constrained to zero. Model fit measures could then be obtained to assess how well the proposed model captured the covariance between all the items or measures in the model. If the constraints the researcher has imposed on the model are inconsistent with the sample data, then the results of statistical tests of model fit will indicate a poor fit, and the model will be rejected. If the fit is poor, it may be due to some items measuring multiple factors. It might also be that some items within a factor are more related to each other than others.
For some applications, the requirement of "zero loadings" (for indicators not supposed to load on a certain factor) has been regarded as too strict. A newly developed analysis method, "exploratory structural equation modeling", specifies hypotheses about the relation between observed indicators and their supposed primary latent factors while allowing for estimation of loadings with other latent factors as well.
Statistical model
In confirmatory factor analysis, researchers are typically interested in studying the degree to which responses on a p x 1 vector of observable random variables can be used to assign a value to one or more unobserved variable(s) η. The investigation is largely accomplished by estimating and evaluating the loading of each item used to tap aspects of the unobserved latent variable. That is, y[i] is the vector of observed responses predicted by the unobserved latent variable , which is defined as:
,
where is the p x 1 vector of observed random variables, are the unobserved latent variables and is a p x k matrix with k equal to the number of latent va
|
https://en.wikipedia.org/wiki/Kobon%20triangle%20problem
|
The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number N(k) of nonoverlapping triangles whose sides lie on an arrangement of k lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.
Known upper bounds
Saburo Tamura proved that the number of nonoverlapping triangles realizable by lines is at most . G. Clément and J. Bader proved more strongly that this bound cannot be achieved when is congruent to 0 or 2 (mod 6). The maximum number of triangles is therefore at most one less in these cases. The same bounds can be equivalently stated, without use of the floor function, as:
Solutions yielding this number of triangles are known when is 3, 4, 5, 6, 7, 8, 9, 13, 15 or 17. For k = 10, 11 and 12, the best solutions known reach a number of triangles one less than the upper bound.
Known constructions
Given an optimal solution with k0 > 3 lines, other Kobon triangle solution numbers can be found for all ki-values where
by using the procedure by D. Forge and J. L. Ramirez Alfonsin. For example, the solution for k0 = 5 leads to the maximal number of nonoverlapping triangles for k = 5, 9, 17, 33, 65, ....
Examples
See also
Roberts's triangle theorem, on the minimum number of triangles that lines can form
References
External links
Johannes Bader, "Kobon Triangles"
Discrete geometry
Unsolved problems in geometry
Recreational mathematics
Triangles
|
https://en.wikipedia.org/wiki/Normal%20polytope
|
In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties.
Definition
Let be a lattice polytope. Let denote the lattice (possibly in an affine subspace of ) generated by the integer points in . Letting be an arbitrary lattice point in , this can be defined as
P is integrally closed if the following condition is satisfied:
such that .
P is normal if the following condition is satisfied:
such that .
The normality property is invariant under affine-lattice isomorphisms of lattice polytopes and the integrally closed property is invariant under an affine change of coordinates. Note sometimes in combinatorial literature the difference between normal and integrally closed is blurred.
Examples
The simplex in Rk with the vertices at the origin and along the unit coordinate vectors is normal. unimodular simplices are the smallest polytope in the world of normal polytopes. After unimodular simplices, lattice parallelepipeds are the simplest normal polytopes.
For any lattice polytope P and , cP is normal.
All polygons or two-dimensional polytopes are normal.
If A is a totally unimodular matrix, then the convex hull of the column vectors in A is a normal polytope.
The Birkhoff polytope is normal. This can easily be proved using Hall's marriage theorem.
In fact, the Birkhoff polytope is compressed, which is a much stronger statement.
All order polytopes are known to be compressed. This implies that these polytopes are normal.
Properties
A lattice polytope is integrally closed if and only if it is normal and L is a direct summand of d.
A normal polytope can be made into a full-dimensional integrally closed polytope by changing the lattice of reference from d to L and the ambient Euclidean space d to the subspace L.
If a lattice polytope can be subdivided into normal polytopes then it is normal as well.
If a lattice polytope in dimension d has lattice lengths greater than or equal to 4d(d + 1) then the polytope is normal.
If P is normal and φ:d → d is an affine map with φ(d) = d then φ(P) is normal.
Every k-dimensional face of a normal polytope is normal.
Proposition
P ⊂ d a lattice polytope. Let C(P)=+(P,1) ⊂ d+1 the following are equivalent:
P is normal.
The Hilbert basis of C(P) ∩ d+
|
https://en.wikipedia.org/wiki/Intraclass%20correlation
|
In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly units in the same group resemble each other. While it is viewed as a type of correlation, unlike most other correlation measures, it operates on data structured as groups rather than data structured as paired observations.
The intraclass correlation is commonly used to quantify the degree to which individuals with a fixed degree of relatedness (e.g. full siblings) resemble each other in terms of a quantitative trait (see heritability). Another prominent application is the assessment of consistency or reproducibility of quantitative measurements made by different observers measuring the same quantity.
Early ICC definition: unbiased but complex formula
The earliest work on intraclass correlations focused on the case of paired measurements, and the first intraclass correlation (ICC) statistics to be proposed were modifications of the interclass correlation (Pearson correlation).
Consider a data set consisting of N paired data values (xn,1, xn,2), for n = 1, ..., N. The intraclass correlation r originally proposed by Ronald Fisher is
where
Later versions of this statistic used the degrees of freedom 2N −1 in the denominator for calculating s2 and N −1 in the denominator for calculating r, so that s2 becomes unbiased, and r becomes unbiased if s is known.
The key difference between this ICC and the interclass (Pearson) correlation is that the data are pooled to estimate the mean and variance. The reason for this is that in the setting where an intraclass correlation is desired, the pairs are considered to be unordered. For example, if we are studying the resemblance of twins, there is usually no meaningful way to order the values for the two individuals within a twin pair. Like the interclass correlation, the intraclass correlation for paired data will be confined to the interval [−1, +1].
The intraclass correlation is also defined for data sets with groups having more than 2 values. For groups consisting of three values, it is defined as
where
As the number of items per group grows, so does the number of cross-product terms in this expression grows. The following equivalent form is simpler to calculate:
where K is the number of data values per group, and is the sample mean of the nth group. This form is usually attributed to Harris. The left term is non-negative; consequently the intraclass correlation must satisfy
For large K, this ICC is nearly equal to
which can be interpreted as the fraction of the total variance that is due to variation between groups. Ronald Fisher devotes an entire chapter to intraclass correlation in his classic book Statistical Methods for Research Workers.
For data from a population that is completely noise, Fisher's formula produces ICC values
|
https://en.wikipedia.org/wiki/Gauge%20function
|
In mathematics, gauge function may refer to
the gauge as used in the definition of the Henstock-Kurzweil integral, also known as the gauge integral;
in fractal geometry, a synonym for dimension function;
in control theory and dynamical systems, a synonym for Lyapunov candidate function;
in gauge theory, a synonym for gauge symmetry.
a type of Minkowski functional
|
https://en.wikipedia.org/wiki/Table%20Producing%20Language
|
Table Producing Language was an IBM mainframe program developed by the US Bureau of Labor Statistics for producing statistical tables. It has been superseded by the commercial product TPL Tables developed by QQQ Software.
References
External links
QQQ Software
IBM mainframe software
Statistical software
|
https://en.wikipedia.org/wiki/Scatter%20matrix
|
For the notion in quantum mechanics, see scattering matrix.
In multivariate statistics and probability theory, the scatter matrix is a statistic that is used to make estimates of the covariance matrix, for instance of the multivariate normal distribution.
Definition
Given n samples of m-dimensional data, represented as the m-by-n matrix, , the sample mean is
where is the j-th column of .
The scatter matrix is the m-by-m positive semi-definite matrix
where denotes matrix transpose, and multiplication is with regards to the outer product. The scatter matrix may be expressed more succinctly as
where is the n-by-n centering matrix.
Application
The maximum likelihood estimate, given n samples, for the covariance matrix of a multivariate normal distribution can be expressed as the normalized scatter matrix
When the columns of are independently sampled from a multivariate normal distribution, then has a Wishart distribution.
See also
Estimation of covariance matrices
Sample covariance matrix
Wishart distribution
Outer product—or X⊗X is the outer product of X with itself.
Gram matrix
References
Covariance and correlation
Matrices
|
https://en.wikipedia.org/wiki/Centering%20matrix
|
In mathematics and multivariate statistics, the centering matrix is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector.
Definition
The centering matrix of size n is defined as the n-by-n matrix
where is the identity matrix of size n and is an n-by-n matrix of all 1's.
For example
,
,
Properties
Given a column-vector, of size n, the centering property of can be expressed as
where is a column vector of ones and is the mean of the components of .
is symmetric positive semi-definite.
is idempotent, so that , for . Once the mean has been removed, it is zero and removing it again has no effect.
is singular. The effects of applying the transformation cannot be reversed.
has the eigenvalue 1 of multiplicity n − 1 and eigenvalue 0 of multiplicity 1.
has a nullspace of dimension 1, along the vector .
is an orthogonal projection matrix. That is, is a projection of onto the (n − 1)-dimensional subspace that is orthogonal to the nullspace . (This is the subspace of all n-vectors whose components sum to zero.)
The trace of is .
Application
Although multiplication by the centering matrix is not a computationally efficient way of removing the mean from a vector, it is a convenient analytical tool. It can be used not only to remove the mean of a single vector, but also of multiple vectors stored in the rows or columns of an m-by-n matrix .
The left multiplication by subtracts a corresponding mean value from each of the n columns, so that each column of the product has a zero mean. Similarly, the multiplication by on the right subtracts a corresponding mean value from each of the m rows, and each row of the product has a zero mean.
The multiplication on both sides creates a doubly centred matrix , whose row and column means are equal to zero.
The centering matrix provides in particular a succinct way to express the scatter matrix, of a data sample , where is the sample mean. The centering matrix allows us to express the scatter matrix more compactly as
is the covariance matrix of the multinomial distribution, in the special case where the parameters of that distribution are , and .
References
Data processing
Matrices
|
https://en.wikipedia.org/wiki/Nodoid
|
In differential geometry, a nodoid is a surface of revolution with constant nonzero mean curvature obtained by rolling a hyperbola along a fixed line, tracing the focus, and revolving the resulting nodary curve around the line.
References
External links
Wolfram Demonstrations: Delaunay Nodoids
Surfaces
|
https://en.wikipedia.org/wiki/Yuzo%20Kanemaru
|
is a male Japanese sprinter. He set his 400 metres personal best at the 2009 Osaka Grand Prix, finishing in 45.16 seconds.
Competition record
Statistics
Personal bests
References
External links
1987 births
Living people
People from Takatsuki, Osaka
Sportspeople from Osaka Prefecture
Japanese male sprinters
Olympic male sprinters
Olympic athletes for Japan
Athletes (track and field) at the 2008 Summer Olympics
Athletes (track and field) at the 2012 Summer Olympics
Athletes (track and field) at the 2016 Summer Olympics
Asian Games gold medalists for Japan
Asian Games silver medalists for Japan
Asian Games medalists in athletics (track and field)
Athletes (track and field) at the 2006 Asian Games
Athletes (track and field) at the 2010 Asian Games
Athletes (track and field) at the 2014 Asian Games
Medalists at the 2010 Asian Games
Medalists at the 2014 Asian Games
Universiade medalists in athletics (track and field)
FISU World University Games gold medalists for Japan
Universiade bronze medalists for Japan
Medalists at the 2009 Summer Universiade
World Athletics Championships athletes for Japan
Japan Championships in Athletics winners
Asian Athletics Championships winners
20th-century Japanese people
21st-century Japanese people
|
https://en.wikipedia.org/wiki/Pepa
|
Pepa may refer to:
Pepa, Democratic Republic of the Congo, a village
Pepa Airport, an airstrip
PEPA or Performance Evaluation Process Algebra, a stochastic process algebra
PEPA (drug), an ampakine drug that is a potential nootropic
Pepa (musical instrument), a flute-like musical instrument from Assam
or the Spanish Constitution of 1812
People with the nickname
Pepa (footballer) (born 1980), Portuguese footballer
Pepa (rapper), Jamaican-American rap & hip-hop artist, member of Salt-N-Pepa
Pepa Fernández (born 1965), Spanish journalist
Pepa Rus (born 1985), Spanish actress, humorist, and singer
People with the surname
Avni Pepa (born 1988), Kosovar footballer
Brunild Pepa (born 1990), Albanian footballer
See also
Joseph (name)
Pepe (disambiguation)
|
https://en.wikipedia.org/wiki/Deng%20Jinghuang
|
Deng Jinghuang (born 24 January 1985) is a former Chinese-born Hong Kong professional footballer who played as a left back or a centre back.
Career statistics
Club
As of 14 May 2008
International
As of 9 February 2011
External links
Deng Jinghuang at HKFA
Scaafc.com 球員資料 – 4. 鄧景煌
SCAA Official Blog 4號 鄧景煌 (Deng Jing Huang)
1985 births
Living people
Footballers from Guangzhou
Chinese men's footballers
Hong Kong men's footballers
Guangzhou F.C. players
Hong Kong First Division League players
Hong Kong Premier League players
South China AA players
Hong Kong Pegasus FC players
Expatriate men's footballers in Hong Kong
Hong Kong men's international footballers
Men's association football defenders
|
https://en.wikipedia.org/wiki/Liang%20Zicheng
|
Liang Zicheng (, born 18 March 1982) is a former Chinese-born Hong Kong professional footballer who played as a striker.
Career statistics
Club
As of 20 September 2008
External links
Liang Zicheng at HKFA
1982 births
Living people
Footballers from Guangzhou
Chinese men's footballers
Hong Kong men's footballers
Guangzhou F.C. players
Hong Kong First Division League players
Hong Kong Premier League players
South China AA players
Hong Kong Rangers FC players
Metro Gallery FC players
Kitchee SC players
Sun Hei SC players
Eastern Sports Club footballers
R&F (Hong Kong) players
Expatriate men's footballers in Hong Kong
Chinese expatriate men's footballers
Chinese expatriate sportspeople in Hong Kong
Men's association football forwards
Men's association football midfielders
|
https://en.wikipedia.org/wiki/Zhang%20Jianzhong
|
Zhang Jianzhong (, born 18 September 1985) is a Chinese former professional association football player.
Career statistics in Hong Kong
As of 3 September 2009
External links
Zhang Jianzhong at HKFA
SCAA Official Blog 16號 張健忠 (Zhang Jian Zhong)
1985 births
Living people
Chinese men's footballers
Footballers from Guangzhou
Men's association football goalkeepers
Hong Kong First Division League players
South China AA players
Expatriate men's footballers in Hong Kong
Sun Hei SC players
Chinese expatriate sportspeople in Hong Kong
21st-century Chinese people
|
https://en.wikipedia.org/wiki/Moving%20sofa%20problem
|
In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region with legs of unit width. The area thus obtained is referred to as the sofa constant. The exact value of the sofa constant is an open problem. The currently leading solution, by Joseph L. Gerver, has a value of approximately 2.2195 and is thought to be close to the optimal, based upon subsequent study and theoretical bounds.
History
The first formal publication was by the Austrian-Canadian mathematician Leo Moser in 1966, although there had been many informal mentions before that date.
Bounds
Work has been done on proving that the sofa constant (A) cannot be below or above certain values (lower bounds and upper bounds).
Lower
Lower bounds can be proven by finding a specific shape of high area and a path for moving it through the corner. An obvious lower bound is . This comes from a sofa that is a half-disk of unit radius, which can slide up one passage into the corner, rotate within the corner around the center of the disk, and then slide out the other passage.
In 1968, John Hammersley stated a lower bound of . This can be achieved using a shape resembling a telephone handset, consisting of two quarter-disks of radius 1 on either side of a 1 by rectangle from which a half-disk of radius has been removed.
In 1992, Joseph L. Gerver of Rutgers University described a sofa specified by 18 curve sections each taking a smooth analytic form. This further increased the lower bound for the sofa constant to approximately 2.2195 .
Upper
Hammersley stated an upper bound on the sofa constant of at most . Yoav Kallus and Dan Romik published a new upper bound in 2018, capping the sofa constant at . Their approach involves rotating the corridor (rather than the sofa) through a finite sequence of distinct angles (rather than continuously), and using a computer search to find translations for each rotated copy so that the intersection of all of the copies has a connected component with as large an area as possible. As they show, this provides a valid upper bound for the optimal sofa, which can be made more accurate by using more rotation angles. A set of five carefully-chosen rotation angles leads to the stated upper bound.
Ambidextrous sofa
A variant of the sofa problem asks the shape of largest area that can go round both left and right 90 degree corners in a corridor of unit width (where the left and right corners are spaced sufficiently far apart that one is fully negotiated before the other is encountered). A lower bound of area approximately 1.64495521 has been described by Dan Romik. His sofa is also described by 18 curve sections.
See also
Dirk Gently's Holistic Detective Agency – novel by Douglas Adams, a subplot of which revolves around such a problem.
Moser's worm problem
Square packing in a square
"
|
https://en.wikipedia.org/wiki/Swiss%20cheese%20%28mathematics%29
|
In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth.
More generally, a Swiss cheese may be all or part of Euclidean space Rn – or of an even more complicated manifold – with "holes" in it.
References
Complex analysis
|
https://en.wikipedia.org/wiki/List%20of%20Fenerbah%C3%A7e%20S.K.%20managers
|
This is a list of all managers of Fenerbahçe, including honours.
Managers
Statistics
Records
Nationalities
As of 11 June 2023.
Most games managed
As of June 2018.
References
Notes
Main
Fenerbahce
Managers
|
https://en.wikipedia.org/wiki/Islam%20in%20London
|
There were 1,318,755 Muslims reported in the 2021 census in the Greater London area. In the 2021 census Office for National Statistics, the proportion of Muslims in London had risen to 15% of the population, making Islam the second largest religion in the city after Christianity.
History
The first Muslims to settle in London were lascars, that is, Bengali and Yemeni sailors from the 19th century. Many Muslims from the Indian sub-continent served in the British Army and British Indian Army in the First and Second World Wars. In the wave of immigration that followed the Second World War, many Muslims emigrated to the UK from these Commonwealth countries and former colonies to satisfy labour shortages and seek new opportunities for themselves. Following the partition of India, many came from Pakistan especially the Punjab and Azad Kashmir in addition to the Indian state of Gujarat. This initial wave of immigration of the 1950s and 60s was followed by migrants from Cyprus, Sylhet in Bangladesh, formerly East Pakistan. Many Muslims also arrived from various other countries, although the percentage is far smaller than from South Asia. Amongst those from other countries, Muslims from Yemen, Somalia and Turkey have significant numbers, whereas those from Malaysia, Nigeria, Ghana and Kenya represent smaller fractions. Today, London's Muslims come from all over the world and there is a small but growing group of converts.
21st century
Following waves of immigration over the previous decades, London now has one of the most diverse array of Muslim communities in the world.
London's Muslims are geographically dispersed with settlements principally shaped by earlier patterns of immigration. The greatest concentration can be found in the east London boroughs of Tower Hamlets, Newham and Waltham Forest, where Bangladeshis, Pakistanis, and Indians tend to predominate.
Outside of east London, Bangladeshi Muslims have settled throughout the city, in boroughs like Camden, Southwark, and Hackney.
North London's Muslims are concentrated in the boroughs of Haringey, Islington, and Enfield, with older communities of Turkish Cypriots more recently being joined by Algerians, Somalis, and mainland Turks.
The traditional homeland of London's Arabic-speaking Muslims is in the City of Westminster, with the initial settlement around Edgware Road since spreading to Kensington & Chelsea, Hammersmith & Fulham, Brent, and Ealing. These five boroughs contain the highest proportion of Arabs in the UK, the majority of whom are Muslim - the recent 2021 census put the figure between 3 and 8%. In recent years, refugees and migrants from countries such as Eritrea, Ethiopia, Sudan, Afghanistan, Algeria, Morocco, and Yemen have joined these various communities, in many cases setting up their own mosques, such as the Iqraa Foundation in Harlesden.
Indian and Pakistani Muslims have settled in significant numbers further west in Hounslow and Southall, but in a much smaller proport
|
https://en.wikipedia.org/wiki/Amenable%20Banach%20algebra
|
In mathematics, specifically in functional analysis, a Banach algebra, A, is amenable if all bounded derivations from A into dual Banach A-bimodules are inner (that is of the form for some in the dual module).
An equivalent characterization is that A is amenable if and only if it has a virtual diagonal.
Examples
If A is a group algebra for some locally compact group G then A is amenable if and only if G is amenable.
If A is a C*-algebra then A is amenable if and only if it is nuclear.
If A is a uniform algebra on a compact Hausdorff space then A is amenable if and only if it is trivial (i.e. the algebra C(X) of all continuous complex functions on X).
If A is amenable and there is a continuous algebra homomorphism from A to another Banach algebra, then the closure of is amenable.
References
F.F. Bonsall, J. Duncan, "Complete normed algebras", Springer-Verlag (1973).
H.G. Dales, "Banach algebras and automatic continuity", Oxford University Press (2001).
B.E. Johnson, "Cohomology in Banach algebras", Memoirs of the AMS 127 (1972).
J.-P. Pier, "Amenable Banach algebras", Longman Scientific and Technical (1988).
Volker Runde, "Amenable Banach Algebras. A Panorama", Springer Verlag (2020).
Banach algebras
|
https://en.wikipedia.org/wiki/Prime%20k-tuple
|
In number theory, a prime -tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a -tuple , the positions where the -tuple matches a pattern in the prime numbers are given by the set of integers such that all of the values are prime. Typically the first value in the -tuple is 0 and the rest are distinct positive even numbers.
Named patterns
Several of the shortest k-tuples are known by other common names:
OEIS sequence covers 7-tuples (prime septuplets) and contains an overview of related sequences, e.g. the three sequences corresponding to the three admissible 8-tuples (prime octuplets), and the union of all 8-tuples. The first term in these sequences corresponds to the first prime in the smallest prime constellation shown below.
Admissibility
In order for a -tuple to have infinitely many positions at which all of its values are prime, there cannot exist a prime such that the tuple includes every different possible value modulo . For, if such a prime existed, then no matter which value of was chosen, one of the values formed by adding to the tuple would be divisible by , so there could only be finitely many prime placements (only those including itself). For example, the numbers in a -tuple cannot take on all three values 0, 1, and 2 modulo 3; otherwise the resulting numbers would always include a multiple of 3 and therefore could not all be prime unless one of the numbers is 3 itself. A -tuple that satisfies this condition (i.e. it does not have a for which it covers all the different values modulo ) is called admissible.
It is conjectured that every admissible -tuple matches infinitely many positions in the sequence of prime numbers. However, there is no admissible tuple for which this has been proven except the 1-tuple (0). Nevertheless, by Yitang Zhang's famous proof of 2013 it follows that there exists at least one 2-tuple which matches infinitely many positions; subsequent work showed that some 2-tuple exists with values differing by 246 or less that matches infinitely many positions.
Positions matched by inadmissible patterns
Although is not admissible it does produce the single set of primes, .
Some inadmissible -tuples have more than one all-prime solution. This cannot happen for a -tuple that includes all values modulo 3, so to have this property a -tuple must cover all values modulo a larger prime, implying that there are at least five numbers in the tuple. The shortest inadmissible tuple with more than one solution is the 5-tuple , which has two solutions: and , where all values mod 5 are included in both cases.
Prime constellations
The diameter of a -tuple is the difference of its largest and smallest elements. An admissible prime -tuple with the smallest possible diameter (among all admissible -tuples) is a prime constellation. For all this will always produce consecutive primes. (Recall that all are integers for which the values are prime.)
This
|
https://en.wikipedia.org/wiki/Go%20and%20mathematics
|
The game of Go is one of the most popular games in the world. As a result of its elegant and simple rules, the game has long been an inspiration for mathematical research. Shen Kuo, an 11th century Chinese scholar, estimated in his Dream Pool Essays that the number of possible board positions is around 10172. In more recent years, research of the game by John H. Conway led to the development of the surreal numbers and contributed to development of combinatorial game theory (with Go Infinitesimals being a specific example of its use in Go).
Computational complexity
Generalized Go is played on n × n boards, and the computational complexity of determining the winner in a given position of generalized Go depends crucially on the ko rules.
Go is “almost” in PSPACE, since in normal play, moves are not reversible, and it is only through capture that there is the possibility of the repeating patterns necessary for a harder complexity.
Without ko
Without ko, Go is PSPACE-hard. This is proved by reducing True Quantified Boolean Formula, which is known to be PSPACE-complete, to generalized geography, to planar generalized geography, to planar generalized geography with maximum degree 3, finally to Go positions.
Go with superko is not known to be in PSPACE. Though actual games seem never to last longer than moves, in general it is not known if there were a polynomial bound on the length of Go games. If there were, Go would be PSPACE-complete. As it currently stands, it might be PSPACE-complete, EXPTIME-complete, or even EXPSPACE-complete.
Japanese ko rule
Japanese ko rules state that only the basic ko, that is, a move that reverts the board to the situation one move previously, is forbidden. Longer repetitive situations are allowed, thus potentially allowing a game to loop forever, such as the triple ko, where there are three kos at the same time, allowing a cycle of 12 moves.
With Japanese ko rules, Go is EXPTIME-complete.
Superko rule
The superko rule (also called the positional superko rule) states that a repetition of any board position that has previously occurred is forbidden. This is the ko rule used in most Chinese and US rulesets.
It is an open problem what the complexity class of Go is under superko rule. Though Go with Japanese ko rule is EXPTIME-complete, both the lower and the upper bounds of Robson’s EXPTIME-completeness proof break when the superko rule is added.
It is known that it is at least PSPACE-hard, since the proof in of the PSPACE-hardness of Go does not rely on the ko rule, or lack of the ko rule. It is also known that Go is in EXPSPACE.
Robson showed that if the superko rule, that is, “no previous position may ever be recreated”, is added to certain two-player games that are EXPTIME-complete, then the new games would be EXPSPACE-complete. Intuitively, this is because an exponential amount of space is required even to determine the legal moves from a position, because the game history leading up to a position could be
|
https://en.wikipedia.org/wiki/Demography%20of%20Birmingham
|
The demography of Birmingham, England, is analysed by the Office for National Statistics and data produced for each of the wards that make up the city, and the overall city itself, which is the largest city proper in England as well as the core of the third most populous urban area, the West Midlands conurbation.
Population
Birmingham city's total population was 977,099 in 2001. The 2005 estimate for the population of the district of Birmingham was 1,001,200. This is the first time the population has broken the 1,000,000 barrier since 1996. This was a population increase of 0.9% (8,800) from 2004, higher than the 0.6% for the United Kingdom as a whole and 0.7% for England. It is believed to have been caused as a result of increased numbers of births, increased migration and a decrease in deaths in the district. The population of Birmingham is predicted to increase, though it cannot be predicted at certainty due to fluctuations in previous years in migration. The population in Birmingham is predicted to increase by 12.2% (121,500) from 992,100 in 2003 to 1,113,600 in 2028. This is an increase of around 4,000 - 5,000 each year until 2028.
The mid-year population estimates from previous years have showed a general decrease in the population of Birmingham from 1982 to 2002, before beginning to increase again up to 2005, with the increase from 2004 to 2005 being the largest population increase recorded. Though, in total, the overall decline in the population of Birmingham has been by just over 1%. The dependent population (0-14 and 65+) has declined since 2001 as the working population (15-64) has increased.
The ward with the lowest population following the boundary readjustments of 2004 was Ladywood with 14,801. Prior to the boundary readjustments, it had a population of 23,789. The ward with the highest population following the boundary readjustments of 2004 was Sparkbrook with a population of 31,485, which is an increase from 28,311 prior to the boundary adjustments.
Age
Birmingham has a young population compared to England as a whole. The mid-year population estimates of 2005 estimate that Birmingham has a younger age structure compared to England, with a higher proportion of the population of Birmingham being under the age of 34, and lower proportion being above the age of 35, than England. In the 20 to 24 year age group, the proportion in Birmingham is about 2% above the national figure.
23.4% of people were aged under 16, 57.7% were aged between 16 and 59, while 18.9% were aged over 60. The average age was 36, compared with 38.6 years for England.
The district of Ladywood was found to have the lowest proportion of people who are 60 years and over than all other districts. Sutton Coldfield district had the highest proportion of people aged 60 years and over. Hodge Hill was found to have the highest population of people from the age of 0 to 15 whilst the districts of Edgbaston and Sutton Coldfield had the lowest.
Population density
The p
|
https://en.wikipedia.org/wiki/Complement%20%28group%20theory%29
|
In mathematics, especially in the area of algebra known as group theory, a complement of a subgroup H in a group G is a subgroup K of G such that
Equivalently, every element of G has a unique expression as a product hk where h ∈ H and k ∈ K. This relation is symmetrical: if K is a complement of H, then H is a complement of K. Neither H nor K need be a normal subgroup of G.
Properties
Complements need not exist, and if they do they need not be unique. That is, H could have two distinct complements K1 and K2 in G.
If there are several complements of a normal subgroup, then they are necessarily isomorphic to each other and to the quotient group.
If K is a complement of H in G then K forms both a left and right transversal of H. That is, the elements of K form a complete set of representatives of both the left and right cosets of H.
The Schur–Zassenhaus theorem guarantees the existence of complements of normal Hall subgroups of finite groups.
Relation to other products
Complements generalize both the direct product (where the subgroups H and K are normal in G), and the semidirect product (where one of H or K is normal in G). The product corresponding to a general complement is called the internal Zappa–Szép product. When H and K are nontrivial, complement subgroups factor a group into smaller pieces.
Existence
As previously mentioned, complements need not exist.
A p-complement is a complement to a Sylow p-subgroup. Theorems of Frobenius and Thompson describe when a group has a normal p-complement. Philip Hall characterized finite soluble groups amongst finite groups as those with p-complements for every prime p; these p-complements are used to form what is called a Sylow system.
A Frobenius complement is a special type of complement in a Frobenius group.
A complemented group is one where every subgroup has a complement.
See also
Product of group subsets
References
Group theory
|
https://en.wikipedia.org/wiki/Haidar%20Aboodi
|
Haeder Aboudi () (born 1986) is an Iraqi former footballer who played as a defender for Najaf FC and the Iraq national football team.
Managerial statistics
Honours
Country
2002 Arab Police Championship: Champions
2006 Asian Games Silver medallist.
External links
Profil on www.goalzz.com
1986 births
Living people
Iraqi men's footballers
Iraqi expatriate men's footballers
Al-Najaf SC players
Expatriate men's footballers in the United Arab Emirates
Iraqi expatriate sportspeople in the United Arab Emirates
Amanat Baghdad SC players
Fujairah FC players
Asian Games medalists in football
Footballers at the 2006 Asian Games
UAE First Division League players
Asian Games silver medalists for Iraq
Men's association football defenders
Medalists at the 2006 Asian Games
Iraq men's international footballers
|
https://en.wikipedia.org/wiki/Sylvester%27s%20criterion
|
In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester.
Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant:
the upper left 1-by-1 corner of M,
the upper left 2-by-2 corner of M,
the upper left 3-by-3 corner of M,
M itself.
In other words, all of the leading principal minors must be positive. By using appropriate permutations of rows and columns of M, it can also be shown that the positivity of any nested sequence of n principal minors of M is equivalent to M being positive-definite.
An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors:
a Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.
Simple proof for special case
Suppose is Hermitian matrix . Let be the principal minor matrices, i.e. the upper left corner matrices. It will be shown that if is positive definite, then the principal minors are positive; that is, for all .
is positive definite. Indeed, choosing
we can notice that Equivalently, the eigenvalues of are positive, and this implies that since the determinant is the product of the eigenvalues.
To prove the reverse implication, we use induction. The general form of an Hermitian matrix is
,
where is an Hermitian matrix, is a vector and is a real constant.
Suppose the criterion holds for . Assuming that all the principal minors of are positive implies that , , and that is positive definite by the inductive hypothesis. Denote
then
By completing the squares, this last expression is equal to
where (note that exists because the eigenvalues of are all positive.)
The first term is positive by the inductive hypothesis. We now examine the sign of the second term. By using the block matrix determinant formula
on we obtain
, which implies .
Consequently,
Proof for general case
The previous proof is only for nonsingular Hermitian matrix with coefficients in , and therefore only for nonsingular real-symmetric matrices.
Theorem I: A real-symmetric matrix A has nonnegative eigenvalues if and only if A can be factored as A = BTB, and all eigenvalues are positive if and only if B is nonsingular.
Theorem II (The Cholesky decomposition): The symmetric matrix A possesses positive pivots if and only if A can be uniquely factored as A = RTR, where R is an upper-triangular matrix with positive diagonal entries. This is known as the Cholesky decomposition of A, and R is called the Cholesky factor of A.
Theorem III: Let Ak be the k × k leading principal submatrix of An×n. If A has an LU factorization A = LU, where L is a lower triangular matrix with a unit diagonal, then det(Ak) = u11u22 · · · ukk, and the k-th pivot i
|
https://en.wikipedia.org/wiki/Ahmed%20Al-Busafy
|
Ahmed Al Busafy (; born 1 September 1976) is an Omani former footballer. He played for Al-Seeb Club from 1999 to 2011 in the Omani League.
Club career statistics
International career
Ahmed was part of the first team squad of the Oman national football team till 2008. He was selected for the national team for the first time in 2002. He has represented the national team in the 2006 FIFA World Cup qualification.
National team career statistics
Goals for Senior National Team
Honours
Club
With Al-Seeb
Sultan Qaboos Cup (0): Runners-up 2003, 2005
Omani Federation Cup (1): 2007
Oman Super Cup (0): Runners-up 1999, 2004
References
External links
Ahmed Al-Busafy - GOALZZ.com
Ahmed Al-Busafy - KOOORA.com
1976 births
Living people
Omani men's footballers
Oman men's international footballers
Men's association football midfielders
Al-Seeb Club players
Oman Professional League players
|
https://en.wikipedia.org/wiki/Classifying%20space%20for%20O%28n%29
|
In mathematics, the classifying space for the orthogonal group O(n) may be constructed as the Grassmannian of n-planes in an infinite-dimensional real space .
It is analogous to the classifying space for U(n).
Algebraic topology
|
https://en.wikipedia.org/wiki/Katherine%20Hannigan
|
Katherine Hannigan (born 1962) is a children's and young adults' writer.
Biography
Hannigan was born in Lockport, New York in 1962. She has undergraduate degrees in mathematics, education, and painting, and a Master of Fine Arts in studio art. She has worked as assistant professor of art and design and as an education coordinator for Head Start. She currently lives in a small town in Iowa.
Works
Ida B. (2004)
Emmaline and the Bunny - (2009)
True... (sort of) - (2011)
Dirt + Warter = Mud - (2016)
Awards
2004 Josette Frank Award, Ida B
2004 Mitten Award, Ida B
References
External links
KatherineHannigan.com
Living people
1962 births
21st-century American novelists
American children's writers
American women novelists
American women children's writers
21st-century American women writers
People from Lockport, New York
|
https://en.wikipedia.org/wiki/Lyre%20arm
|
A lyre arm is an element of design in furniture, architecture and the decorative arts, wherein a shape is employed to emulate the geometry of a lyre; the original design of this element is from the Classical Greek period, simply reflecting the stylistic design of the musical instrument. One of the earliest uses extant of the lyre design in the Christian era is a 6th-century AD gravestone with lyre design in double volute form. In a furniture context, the design is often associated with a scrolling effect of the arms of a chair or sofa. The lyre arm design arises in many periods of furniture, including Neoclassical schools and in particular the American Federal Period and the Victorian era. Well known designers who employed this stylistic element include the noted New York City furniture designer Duncan Phyfe.
The term lyre chair is a closely associated design element also originating in motif from the Greek Classical period and appearing often in chair backs starting circa 1700 AD. In the lyre chair, the splat features a pair of single lyre scrolls with bilateral symmetry. This particular splat chair back was a favourite motif employed by the well known English furniture designer Thomas Sheraton. Sometimes a chair of this design is called a lyre back chair.
In musical apparatus
Not surprisingly the lyre motif has been used through history as an element of music stand and other musical appurtenance design. Perhaps most commonly the lyre design has been used for centuries as the backing of sheet music stands. As an example of the lyre design in other musical furniture, one highly ornate piano described in the 1902 catalog of the collection of the New York Metropolitan Museum of Art was depicted as: "having in the centre a lyre supporting the pedals".
Other use of the lyre design
Beyond the use of the lyre design in chairs, this motif is common in other decorative applications for furniture and other contents' accessories. In prehistoric Celtic design, the lyre is present in a number of works including a well preserved scabbard found in Antrim, Northern Ireland and now preserved in the Ulster Museum; this artifact has a bilaterally symmetric double lyre design. For example, in the Empire Period the lyre was commonly applied to mirrors, especially in the American Federal Period. In London in the late 18th century, Thomas Sheraton illustrated the lyre design for use in table supports. Another example of lyre supports in a table design is illustrated in History Of Furniture: Ancient to 19th Century, showing a small ebony table. Lockwood also documents that Sheraton enjoyed using a painted form of the lyre on furniture elements as decoration. Lockwood further illustrates a lyre supported games table from circa 1820 believed to have been produced by Duncan Phyfe.
In fiction
Numerous references exist to the lyre arm or lyre chair in fictional literature, the lyre design being associated with historical splendour and opulent living circumstances. In t
|
https://en.wikipedia.org/wiki/Gilbert%20Baumslag
|
Gilbert Baumslag (April 30, 1933 – October 20, 2014) was a Distinguished Professor at the City College of New York, with joint appointments in mathematics, computer science, and electrical engineering. He was director of the Center for Algorithms and Interactive Scientific Software, which grew out of the MAGNUS computational group theory project he also headed. Baumslag was also the organizer of the New York Group Theory Seminar.
Baumslag graduated from the University of the Witwatersrand in South Africa with a B.Sc. Honours (Masters) and D.Sc. He earned his Ph.D. from the University of Manchester in 1958; his thesis, written under the direction of Bernhard Neumann, was titled Some aspects of groups with unique roots. His contributions include the Baumslag–Solitar groups and parafree groups.
Baumslag was a visiting scholar at the Institute for Advanced Study in 1968–69. In 2012, he became a fellow of the American Mathematical Society.
Works
Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. I. The groups, Transactions of the American Mathematical Society 129 (1967), 308–321.
Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. II. Properties, Transactions of the American Mathematical Society 142 (1969), 507–538.
Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201.
Notes
External links
Gilbert Baumslag, at CCNY
Center for Algorithms and Interactive Scientific Software, at CCNY
New York Group Theory Seminar, at The CUNY Graduate Center
20th-century American mathematicians
21st-century American mathematicians
Group theorists
City College of New York faculty
1933 births
2014 deaths
Alumni of the University of Manchester
University of the Witwatersrand alumni
Institute for Advanced Study visiting scholars
Fellows of the American Mathematical Society
|
https://en.wikipedia.org/wiki/Antiparallel%20lines
|
In geometry, two lines and are antiparallel with respect to a given line if they each make congruent angles with in opposite senses. More generally, lines and are antiparallel with respect to another pair of lines and if they are antiparallel with respect to the angle bisector of and
In any cyclic quadrilateral, any two opposite sides are antiparallel with respect to the other two sides.
Relations
The line joining the feet to two altitudes of a triangle is antiparallel to the third side. (any cevians which 'see' the third side with the same angle create antiparallel lines)
The tangent to a triangle's circumcircle at a vertex is antiparallel to the opposite side.
The radius of the circumcircle at a vertex is perpendicular to all lines antiparallel to the opposite sides.
References
A.B. Ivanov, Encyclopaedia of Mathematics -
Weisstein, Eric W. "Antiparallel." From MathWorld—A Wolfram Web Resource.
Elementary geometry
|
https://en.wikipedia.org/wiki/Hemispherical%20photography
|
Hemispherical photography, also known as canopy photography, is a technique to estimate solar radiation and characterize plant canopy geometry using photographs taken looking upward through an extreme wide-angle lens or a fisheye lens (Rich 1990). Typically, the viewing angle approaches or equals 180-degrees, such that all sky directions are simultaneously visible. The resulting photographs record the geometry of visible sky, or conversely the geometry of sky obstruction by plant canopies or other near-ground features. This geometry can be measured precisely and used to calculate solar radiation transmitted through (or intercepted by) plant canopies, as well as to estimate aspects of canopy structure such as leaf area index. Detailed treatments of field and analytical methodology have been provided by Paul Rich (1989, 1990) and Robert Pearcy (1989).
History
The hemispherical lens (also known as a fisheye or whole-sky lens) was originally designed by Robin Hill (1924) to view the entire sky for meteorological studies of cloud formation. Foresters and ecologists conceived of using photographic techniques to study the light environment in forests by examining the canopy geometry. In particular, Evans and Coombe (1959) estimated sunlight penetration through forest canopy openings by overlaying diagrams of the sun track on hemispherical photographs. Later, Margaret Anderson (1964, 1971) provided a thorough theoretical treatment for calculating the transmission of direct and diffuse components of solar radiation through canopy openings using hemispherical photographs. At that time hemispherical photograph analysis required tedious manual scoring of overlays of sky quadrants and the track of the sun. With the advent of personal computers, researchers developed digital techniques for rapid analysis of hemispherical photographs (Chazdon and Field 1987, Rich 1988, 1989, 1990, Becker et al. 1989). In recent years, researchers have started using digital cameras in favor of film cameras, and algorithms are being developed for automated image classification and analysis. Various commercial software programs have become available for hemispherical photograph analysis, and the technique has been applied for diverse uses in ecology, meteorology, forestry, and agriculture.
Applications
Hemispherical photography has been used successfully in a broad range of applications involving microsite characterization and estimation of the solar radiation near the ground and beneath plant canopies. For example, hemispherical photography has been used to characterize winter roosting sites for monarch butterflies (Weiss et al. 1991), effects of forest edges (Galo et al. 1991), influence of forest treefall gaps on tree regeneration (Rich et al. 1993), spatial and temporal variability of light in tropical rainforest understory (Clark et al. 1996), impacts of hurricanes on forest ecology (Bellingham et al. 1996), leaf area index for validation of remote sensing (Ch
|
https://en.wikipedia.org/wiki/Shlomo%20Sawilowsky
|
Shlomo S. Sawilowsky (1954 - 11 January 2021) was a professor of educational statistics and Distinguished Faculty Fellow at Wayne State University in Detroit, Michigan, where he has received teaching, mentoring, and research awards.
Academic career
Sawilowsky obtained his Ph.D. in 1985 at the University of South Florida. He was inducted into the USF chapter of the Phi Kappa Phi honor society on May 17, 1981, when he received his M.A. In 2008 Sawilowsky served as president of the American Educational Research Association Special Interest Group/Educational Statisticians. He served as an Assistant Dean in the College of Education at WSU. Along with Miodrag Lovric (Serbia) and C. R. Rao (India), he was nominated for the 2013 Nobel Peace Prize for his contributions to the International Encyclopedia of Statistical Science.
Contributions to applied statistics and social/behavioral sciences
In 2000, the AMSTAT News, a publication of the American Statistical Association, described Professor Sawilowsky's award of Distinguished Faculty Fellow "in recognition of Sawilowsky's outstanding scholarly achievements in applied statistics, psychometrics, and experimental design in education and psychology."
Applied statistics
He is the author of a statistics textbook that presents statistical methods via Monte Carlo simulation methods, editor of a volume on real data analysis published by the American Educational Research Association SIG/Educational Statisticians, and author of over a hundred articles in applied statistics and social sciences journals. Sawilowsky has also authored 24 entries in statistics encyclopedias.
His presentation titled "The Rank Transform," with co-author R. Clifford Blair, was awarded the 1985 Florida Educational Research Association & 1986 American Educational Research Association State/Regions Distinguished Paper Award. Many of his publications are related to rank-based nonparametric statistics. For example, an examination of the robustness and comparative power properties of the rank transform statistic was called a "major Monte Carlo study". Hettmansperger and McKean stated that Sawilowsky provided "an excellent review of nonparametric approaches to testing for interaction" (p. 254-255).
Sawilowsky's Monte Carlo work has been cited as an exemplar for designing simulation studies. His work has been cited on a variety of statistical issues, such as
demonstrating sequential procedures of testing underlying assumptions of parametric tests, commonly recommended in textbooks and statistics software user manuals, "increases the rate of Type I error";
rounding down degrees of freedom when using tabled critical values decreases statistical power;
alternatives to the winsorized sample standard deviation can be invoked to increase the statistical power of Yuen's confidence interval;
maximum likelihood methods (e.g., one-step Huber) are superior to trimming in constructing robust estimators;
using effect sizes obtained when the null hypothesi
|
https://en.wikipedia.org/wiki/Bowyer%E2%80%93Watson%20algorithm
|
In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be also used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation.
Description
The Bowyer–Watson algorithm is an incremental algorithm. It works by adding points, one at a time, to a valid Delaunay triangulation of a subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a star-shaped polygonal hole which is then re-triangulated using the new point. By using the connectivity of the triangulation to efficiently locate triangles to remove, the algorithm can take O(N log N) operations to triangulate N points, although special degenerate cases exist where this goes up to O(N2).
History
The algorithm is sometimes known just as the Bowyer Algorithm or the Watson Algorithm. Adrian Bowyer and David Watson devised it independently of each other at the same time, and each published a paper on it in the same issue of The Computer Journal (see below).
Pseudocode
The following pseudocode describes a basic implementation of the Bowyer-Watson algorithm. Its time complexity is . Efficiency can be improved in a number of ways. For example, the triangle connectivity can be used to locate the triangles which contain the new point in their circumcircle, without having to check all of the triangles - by doing so we can decrease time complexity to . Pre-computing the circumcircles can save time at the expense of additional memory usage. And if the points are uniformly distributed, sorting them along a space filling Hilbert curve prior to insertion can also speed point location.
function BowyerWatson (pointList)
// pointList is a set of coordinates defining the points to be triangulated
triangulation := empty triangle mesh data structure
add super-triangle to triangulation // must be large enough to completely contain all the points in pointList
for each point in pointList do // add all the points one at a time to the triangulation
badTriangles := empty set
for each triangle in triangulation do // first find all the triangles that are no longer valid due to the insertion
if point is inside circumcircle of triangle
add triangle to badTriangles
polygon := empty set
for each triangle in badTriangles do // find the boundary of the polygonal hole
for each edge in triangle do
if edge is not shared by any other triangles in badTriangles
add edge to polygon
for each triangle in badTriangles do // remove them from the data structure
remove triangle from triangulation
for each edge in polygon do // re-triangulate the polygonal hole
newTri := form a triangle from edge to point
add newTri to triangulation
|
https://en.wikipedia.org/wiki/Abdel%20Hamid%20Bassiouny
|
Abdel Hamid Bassiouny (; born 15 December 1971) is an Egyptian footballer. He previously played in Egypt for Kafr El-Sheikh, Zamalek, Ismaily and Haras El-Hodood.
Managerial statistics
References
External links
Abdul-Hamid Bassiouny at Footballdatabase
1971 births
Living people
Zamalek SC players
Egyptian men's footballers
1999 FIFA Confederations Cup players
Ismaily SC players
Haras El Hodoud SC players
Egyptian Premier League players
Egyptian Premier League managers
Haras El Hodoud SC managers
People from Kafr El Sheikh Governorate
Men's association football forwards
Egyptian football managers
Egypt men's international footballers
Oman Professional League managers
Egyptian expatriate football managers
Mirbat SC managers
Egyptian expatriate sportspeople in Oman
Expatriate football managers in Oman
Tala'ea El Gaish SC managers
Ghazl El Mahalla SC managers
Smouha SC managers
|
https://en.wikipedia.org/wiki/Tamer%20Abdel%20Hamid
|
Tamer Abdel Hamid (; born 27 October 1975) is an Egyptian retired footballer who played as a defensive midfielder.
Career statistics
International
International goals
Scores and results list Egypt's goal tally first.
Honours
Zamalek
Egyptian Premier League: 2000–01, 2002–03, 2003–04
Egypt Cup: 2001–02, 2007–08
Egyptian Super Cup: 2001, 2002
CAF Champions League: 2002
CAF Super Cup: 2003
UAFA Club Cup: 2003
Saudi-Egyptian Super Cup: 2003
External links
Zamalek SC players
Egyptian men's footballers
1975 births
Living people
2004 African Cup of Nations players
People from Mansoura, Egypt
Egyptian Premier League players
Men's association football midfielders
Egypt men's international footballers
|
https://en.wikipedia.org/wiki/Secondary%20measure
|
In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.
Introduction
Under certain assumptions that we will specify further, it is possible to obtain the existence of a secondary measure and even to express it.
For example, if one works in the Hilbert space L2([0, 1], R, ρ)
with
in the general case, or:
when ρ satisfies a Lipschitz condition.
This application φ is called the reducer of ρ.
More generally, μ et ρ are linked by their Stieltjes transformation with the following formula:
in which c1 is the moment of order 1 of the measure ρ.
These secondary measures, and the theory around them, lead to some surprising results, and make it possible to find in an elegant way quite a few traditional formulas of analysis, mainly around the Euler Gamma function, Riemann Zeta function, and Euler's constant.
They also allowed the clarification of integrals and series with a tremendous effectiveness, though it is a priori difficult.
Finally they make it possible to solve integral equations of the form
where g is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.
The broad outlines of the theory
Let ρ be a measure of positive density on an interval I and admitting moments of any order. We can build a family {Pn} of orthogonal polynomials for the inner product induced by ρ. Let us call {Qn} the sequence of the secondary polynomials associated with the family P. Under certain conditions there is a measure for which the family Q is orthogonal. This measure, which we can clarify from ρ is called a secondary measure associated initial measure ρ.
When ρ is a probability density function, a sufficient condition so that μ, while admitting moments of any order can be a secondary measure associated with ρ is that its Stieltjes Transformation is given by an equality of the type:
a is an arbitrary constant and c1 indicating the moment of order 1 of ρ.
For a = 1 we obtain the measure known as secondary, remarkable since for n ≥ 1 the norm of the polynomial Pn for ρ coincides exactly with the norm of the secondary polynomial associated Qn when using the measure μ.
In this paramount case, and if the space generated by the orthogonal polynomials is dense in L2(I, R, ρ), the operator Tρ defined by
creating the secondary polynomials can be furthered to a linear map connecting space L2(I, R, ρ) to L2(I, R, μ) and becomes isometric if limited to the hyperplane Hρ of the orthogonal functions with P0 = 1.
For unspecified functions square integrable for ρ we obtain the more general formula of covariance:
The theory continues by introducing the concept of reducible measure, meaning that the quotient ρ/μ is element of L2(I, R, μ). The following results are then established:
The reducer φ of ρ
|
https://en.wikipedia.org/wiki/Minimax%20eversion
|
In geometry, minimax eversions are a class of sphere eversions, constructed by using half-way models.
It is a variational method, and consists of special homotopies (they are shortest paths with respect to Willmore energy); contrast with Thurston's corrugations, which are generic.
The original method of half-way models was not optimal: the regular homotopies passed through the midway models, but the path from the round sphere to the midway model was constructed by hand, and was not gradient ascent/descent.
Eversions via half-way models are called tobacco-pouch eversions by Francis and Morin.
Half-way models
A half-way model is an immersion of the sphere in , which is so-called because it is the half-way point of a sphere eversion. This class of eversions has time symmetry: the first half of the regular homotopy goes from the standard round sphere to the half-way model, and the second half (which goes from the half-way model to the inside-out sphere) is the same process in reverse.
Explanation
Rob Kusner proposed optimal eversions using the Willmore energy on the space of all immersions of the sphere in .
The round sphere and the inside-out round sphere are the unique global minima for Willmore energy, and a minimax eversion is a path connecting these by passing over a saddle point (like traveling between two valleys via a mountain pass).
Kusner's half-way models are saddle points for Willmore energy, arising (according to a theorem of Bryant) from certain complete minimal surfaces in 3-space; the minimax eversions consist of gradient ascent from the round sphere to the half-way model, then gradient descent down (gradient descent for Willmore energy is called Willmore flow). More symmetrically, start at the half-way model; push in one direction and follow Willmore flow down to a round sphere; push in the opposite direction and follow Willmore flow down to the inside-out round sphere.
There are two families of half-way models (this observation is due to Francis and Morin):
odd order: generalizing Boy's surface: 3-fold, 5-fold, etc., symmetry; half-way model is a double-covered projective plane (generically 2-1 immersed sphere).
even order: generalizing Morin surface: 2-fold, 4-fold, etc., symmetry; half-way model is a generically 1-1 immersed sphere, and a twist by half a symmetry interchanges sheets of the sphere
History
The first explicit sphere eversion was by Shapiro and Phillips in the early 1960s, using Boy's surface as a half-way model. Later Morin discovered the Morin surface and used it to construct other sphere eversions. Kusner conceived the minimax eversions in the early 1980s: historical details.
References
Bending Energy and the Minimax Eversions (in John M. Sullivan's "The Optiverse" and Other Sphere Eversions)
Differential topology
|
https://en.wikipedia.org/wiki/1905%E2%80%9306%20Belgian%20First%20Division
|
Statistics of Belgian First Division in the 1905–06 season.
Overview
It was contested by 10 teams, and Union Saint-Gilloise won the championship.
League standings
Results
See also
1905–06 in Belgian football
References
Belgian Pro League seasons
Belgian First Division, 1913-14
1905–06 in Belgian football
|
https://en.wikipedia.org/wiki/1906%E2%80%9307%20Belgian%20First%20Division
|
Statistics of Belgian First Division in the 1906–07 season.
Overview
It was contested by 10 teams, and Union Saint-Gilloise won the championship.
League standings
Results
See also
1906–07 in Belgian football
References
Belgian Pro League seasons
Belgian First Division, 1913-14
1906–07 in Belgian football
|
https://en.wikipedia.org/wiki/Lifting-line%20theory
|
The Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing based on its geometry. It is also known as the Lanchester–Prandtl wing theory.
The theory was expressed independently by Frederick W. Lanchester in 1907, and by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk.
In this model, the bound vortex loses strength along the whole wingspan because it is shed as a vortex-sheet from the trailing edge, rather than just as a single vortex from the wing-tips.
Introduction
It is difficult to predict analytically the overall amount of lift that a wing of given geometry will generate.
When analyzing a three-dimensional finite wing, the first approximation to understanding is to consider slicing the wing into cross-sections and analyzing each cross-section independently as a wing in a two-dimensional world. Each of these slices is called an airfoil, and it is easier to understand an airfoil than a complete three-dimensional wing.
One might expect that understanding the full wing simply involves adding up the independently calculated forces from each airfoil segment. However, it turns out that this approximation is grossly incorrect: on a real wing, the lift over each wing segment (local lift per unit span, or ) does not correspond simply to what two-dimensional analysis predicts. In reality, the local amount of lift on each cross-section is not independent and is strongly affected by neighboring wing sections.
The lifting-line theory corrects some of the errors in the naive two-dimensional approach by including some of the interactions between the wing slices. It produces the lift distribution along the span-wise direction, based on the wing geometry (span-wise distribution of chord, airfoil, and twist) and flow conditions (, , ).
Principle
The lifting-line theory applies the concept of circulation and the Kutta–Joukowski theorem,
so that instead of the lift distribution function, the unknown effectively becomes the distribution of circulation over the span, .
Modeling the local lift (unknown and sought-after) with the local circulation (also unknown) allows us to account for the influence of one section over its neighbors. In this view, any span-wise change in lift is equivalent to a span-wise change of circulation. According to Helmholtz's theorems, a vortex filament cannot begin or terminate in the air. Any span-wise change in lift can be modeled as the shedding of a vortex filament down the flow, behind the wing.
This shed vortex, whose strength is the derivative of the (unknown) local wing circulation distribution, , influences the flow left and right of the wing section.
This sideways influence (upwash on the outboard, downwash on the inboard) is the key to the lifting-line theory. Now, if the change in lift distribution is known at given lift section, it is possible to predict how that section influences the lift over its neighbors: the
|
https://en.wikipedia.org/wiki/1907%E2%80%9308%20Belgian%20First%20Division
|
Statistics of Belgian First Division in the 1907–08 season.
Overview
It was contested by 10 teams, and Racing Club de Bruxelles won the championship.
There was no relegation, as the First Division was extended the following season from 10 clubs to 12.
League standings
Results
See also
1907–08 in Belgian football
References
Belgian Pro League seasons
Belgian First Division, 1913-14
1907–08 in Belgian football
|
https://en.wikipedia.org/wiki/National%20Administrative%20Department%20of%20Statistics
|
The National Administrative Department of Statistics (), commonly referred to as DANE, is the Colombian Administrative Department responsible for the planning, compilation, analysis and dissemination of the official statistics of Colombia. DANE is responsible for conducting the National Population and Housing census every ten years, among several other studies.
DANE offers more than 100 statistical operations on industrial, economic, agricultural, population and quality of life aspects aimed at supporting decision-making in the country.
Since 2022, the director is Beatriz Piedad Urdinola Contreras.
See also
Administrative Department of Security
National Planning Department
Geographic Institute Agustín Codazzi
References
External links
Official website
Colombia
Demographics of Colombia
National Administrative Department of Statistics
Government agencies established in 1953
1953 establishments in Colombia
|
https://en.wikipedia.org/wiki/Geometry%20Wars%3A%20Galaxies
|
Geometry Wars: Galaxies is a multidirectional shooter video game developed by Bizarre Creations and Kuju Entertainment, and published by Vivendi Games for the Wii and Nintendo DS in 2007. As the first Geometry Wars game to be released on non-Microsoft platforms, Galaxies is a spin-off of Geometry Wars, which was originally included as a bonus game within Project Gotham Racing 2 on Microsoft's Xbox console. This updated version includes a single-player campaign mode, several multiplayer modes, Geometry Wars: Retro Evolved (previously released on the Xbox Live Arcade online service), and support for online leaderboards. The Wii version supports widescreen and 480p progressive scan display.
The soundtrack was composed by Chris Chudley from Audioantics who created the music for all of the Geometry Wars series.
Gameplay
The object of Geometry Wars: Galaxies is to survive as long as possible and score as many points as possible by destroying a constant, generally increasing swarm of enemies. The game takes place in a closed, two-dimensional playfield, and the game ends when the player loses his/her last life. The player controls a claw-shaped "ship" that can move and fire simultaneously in any direction, with movement and firing being controlled independently via the Wii and DS' unique motion controls. A limited supply of screen-clearing bombs are available at the press of a button to instantly eliminate all enemies and missiles in an emergency. The Wii version utilizes the Wii Remote pointer and buttons to aim and fire, while assigning movement and bombs to the Nunchuk's analog stick and buttons respectively. An alternate control scheme based on the Classic Controller is available, mimicking the original twin-stick control of the original Geometry Wars, in which bombs are deployed with shoulder buttons. The DS version allows players to use the directional pad, face buttons and/or the touchscreen to move and aim, while using shoulder buttons to deploy bombs.
In addition to the updated control schemes, Galaxies introduces new enemies, multiplayer support, and several changes to the scoring system.
Galaxies mode
Galaxies is the game's main mode, which presents the player with a series of ten solar systems, each with a number of different planets featuring different challenges or styles of play. All of the enemies from Retro Evolved appear in Galaxies, along with some new enemies with a variety of different behaviours.
Some planets play very similarly to Geometry Wars: Retro Evolved, while others introduce moving obstacles, narrow tunnels, enemy carriers that split into smaller enemies when destroyed, and other gameplay mechanics. Most levels start the player with three lives and three bombs, but some require the player to play with only one life and/or no bombs, and the player may not be able to earn extra lives or bombs in those levels. Each planet sets a Bronze, Silver and Gold score level, challenging the player to reach those scores before runni
|
https://en.wikipedia.org/wiki/Amr%20Ghoneim
|
Amr Ghoneim (Arabic:عمرو غنيم) is a former tennis player
Rankings
Career High ATP ranking - Singles: 261 (30-Oct-00)
Career High Stanford ATP Doubles Ranking: 320 (13-Nov-00)
Davis Cup Statistics
He has the all-time Egyptian records for Davis cup ties played: 29
He has the all-time Egyptian records for Davis cup years played: 13
External links
Egyptian male tennis players
Egyptian tennis coaches
Year of birth missing (living people)
Living people
African Games medalists in tennis
African Games silver medalists for Egypt
African Games bronze medalists for Egypt
Competitors at the 1991 All-Africa Games
Competitors at the 1995 All-Africa Games
|
https://en.wikipedia.org/wiki/Phoenix%20Suns%20all-time%20roster
|
The following is a list of players, both past and current, who have appeared in at least one regular season or playoff game for the Phoenix Suns NBA franchise.
All statistics and awards listed were during the player's tenure with the Suns only. All statistics are accurate as of the end of the 2022–23 season.
Players
A to B
|-
|align="left"| || align="center"|F/C || align="left"|Baylor || align="center"|1 || align="center"| || 10 || 123 || 25 || 8 || 17 || 12.3 || 2.5 || 0.8 || 1.7 || align=center|
|-
|align="left" bgcolor="#FFCC00"|+ (#33) || align="center"|F/C || align="left"|Oklahoma || align="center" bgcolor="#CFECEC"|13 || align="center"|– || bgcolor="#CFECEC"|988 || bgcolor="#CFECEC"|27,203 || bgcolor="#CFECEC"|6,937 || 4,012 || 13,910 || 27.5 || 7.0 || 4.1 || 14.1 || align=center|
|-
|align="left"| || align="center"|G/F || align="left"|Syracuse || align="center"|1 || align="center"| || 62 || 711 || 106 || 45 || 359 || 11.5 || 1.7 || 0.7 || 5.8 || align=center|
|-
|align="left"| || align="center"|G || align="left"|BYU || align="center"|3 || align="center"|– || 222 || 5,092 || 454 || 650 || 2,124 || 22.9 || 2.0 || 2.9 || 9.6 || align=center|
|-
|align="left"| || align="center"|G || align="left"|Creighton || align="center"|1 || align="center"| || 15 || 47 || 10 || 6 || 9 || 3.1 || 0.7 || 0.4 || 0.6 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|UNLV || align="center"|2 || align="center"|– || 155 || 2,212 || 616 || 59 || 692 || 14.3 || 4.0 || 0.4 || 4.5 || align=center|
|-
|align="left"| || align="center"|F || align="left"|California || align="center"|1 || align="center"| || 15 || 278 || 45 || 17 || 56 || 18.5 || 3.0 || 1.1 || 3.7 || align=center|
|-
|align="left"| || align="center"|F || align="left"|Illinois || align="center"|1 || align="center"| || 1 || 6 || 1 || 1 || 1 || 6.0 || 1.0 || 1.0 || 1.0 || align=center|
|-
|align="left"| || align="center"|F || align="left"|UCLA || align="center"|1 || align="center"| || 26 || 884 || 145 || 87 || 258 || 34.0 || 5.6 || 3.3 || 9.9 || align=center|
|-
|align="left"| || align="center"|C || align="left"|Santa Clara || align="center"|4 || align="center"|– || 309 || 7,596 || 1,655 || 846 || 1,873 || 24.6 || 5.4 || 2.7 || 6.1 || align=center|
|-
|align="left" bgcolor="#CCFFCC"|x || align="center"|C || align="left"|Arizona || align="center"|5 || align="center"|– || 303 || 9,282 || 3,152 || 495 || 5,046 || 30.6 || 10.4 || 1.6 || 16.7 || align=center|
|-
|align="left"| || align="center"|F/C || align="left"|Rutgers || align="center"|1 || align="center"| || 65 || 869 || 210 || 42 || 288 || 13.4 || 3.2 || 0.6 || 4.4 || align=center|
|-
|align="left"| || align="center"|G || align="left"|UCLA || align="center"|2 || align="center"|– || 73 || 698 || 126 || 43 || 241 || 9.6 || 1.7 || 0.6 || 3.3 || align=center|
|-
|align="left"| || align="center"|G || align="left"|UNLV || align="center"|2 || align="center"|– || 69 || 813 || 58 || 85 || 346 || 11.8 || 0.8 || 1.2 || 5.0 || align=center|
|-
|
https://en.wikipedia.org/wiki/LCP%20theory
|
In chemistry, ligand close packing theory (LCP theory), sometimes called the ligand close packing model describes how ligand – ligand repulsions affect the geometry around a central atom. It has been developed by R. J. Gillespie and others from 1997 onwards and is said to sit alongside VSEPR which was originally developed by R. J. Gillespie and R Nyholm. The inter-ligand distances in a wide range of molecules have been determined. The example below shows a series of related molecules:
The consistency of the interligand distances (F-F and O-F) in the above molecules is striking and this phenomenon is repeated across a wide range of molecules and forms the basis for LCP theory.
Ligand radius
From a study of known structural data a series of inter-ligand distances has been determined and it has been found that there is a constant inter-ligand radius for a given central atom. The table below shows the inter-ligand radius (pm) for some of the period 2 elements:
The ligand radius should not be confused with the ionic radius.
Treatment of lone pairs
In LCP theory a lone pair is treated as a ligand. Gillespie terms the lone pair a lone pair domain and states that these lone pair domains push the ligands together until they reach the interligand distance predicted by the relevant inter-ligand radii. An example demonstrating this is shown below, where the F-F distance is the same in the AF3 and AF4+ species :
LCP and VSEPR
LCP and VSEPR make very similar predictions as to geometry but LCP theory has the advantage that predictions are more quantitative particularly for the second period elements, Be, B, C, N, O, F. Ligand -ligand repulsions are important when
the central atom is small e.g. period 2, (Be, B, C, N, O)
the ligands are only weakly electronegative compared to the central atom
the ligands are large compared to the central atom
there are 5 or more ligands around the central atom
References
Chemistry theories
Molecular geometry
Stereochemistry
Quantum chemistry
|
https://en.wikipedia.org/wiki/Radical%20of%20a%20Lie%20algebra
|
In the mathematical field of Lie theory, the radical of a Lie algebra is the largest solvable ideal of
The radical, denoted by , fits into the exact sequence
.
where is semisimple. When the ground field has characteristic zero and has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of that is isomorphic to the semisimple quotient via the restriction of the quotient map
A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra.
Definition
Let be a field and let be a finite-dimensional Lie algebra over . There exists a unique maximal solvable ideal, called the radical, for the following reason.
Firstly let and be two solvable ideals of . Then is again an ideal of , and it is solvable because it is an extension of by . Now consider the sum of all the solvable ideals of . It is nonempty since is a solvable ideal, and it is a solvable ideal by the sum property just derived. Clearly it is the unique maximal solvable ideal.
Related concepts
A Lie algebra is semisimple if and only if its radical is .
A Lie algebra is reductive if and only if its radical equals its center.
See also
Levi decomposition
References
Lie algebras
|
https://en.wikipedia.org/wiki/Baumslag%E2%80%93Solitar%20group
|
In the mathematical field of group theory, the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group theory and geometric group theory as (counter)examples and test-cases. They are given by the group presentation
For each integer and , the Baumslag–Solitar group is denoted . The relation in the presentation is called the Baumslag–Solitar relation.
Some of the various are well-known groups. is the free abelian group on two generators, and is the fundamental group of the Klein bottle.
The groups were defined by Gilbert Baumslag and Donald Solitar in 1962 to provide examples of non-Hopfian groups. The groups contain residually finite groups, Hopfian groups that are not residually finite, and non-Hopfian groups.
Linear representation
Define
The matrix group generated by and is a homomorphic image of , via the homomorphism induced by
It is worth noting that this will not, in general, be an isomorphism. For instance if is not residually finite (i.e. if it is not the case that , , or ) it cannot be isomorphic to a finitely generated linear group, which is known to be residually finite by a theorem of Anatoly Maltsev.
See also
Binary tiling
Solv geometry
Notes
References
Gilbert Baumslag and Donald Solitar, Some two-generator one-relator non-Hopfian groups, Bulletin of the American Mathematical Society 68 (1962), 199–201.
Combinatorial group theory
Infinite group theory
|
https://en.wikipedia.org/wiki/Hopfian%20group
|
In mathematics, a Hopfian group is a group G for which every epimorphism
G → G
is an isomorphism. Equivalently, a group is Hopfian if and only if it is not isomorphic to any of its proper quotients. A group G is co-Hopfian if every monomorphism
G → G
is an isomorphism. Equivalently, G is not isomorphic to any of its proper subgroups.
Examples of Hopfian groups
Every finite group, by an elementary counting argument.
More generally, every polycyclic-by-finite group.
Any finitely generated free group.
The group Q of rationals.
Any finitely generated residually finite group.
Any word-hyperbolic group.
Examples of non-Hopfian groups
Quasicyclic groups.
The group R of real numbers.
The Baumslag–Solitar group B(2,3).
Properties
It was shown by that it is an undecidable problem to determine, given a finite presentation of a group, whether the group is Hopfian. Unlike the undecidability of many properties of groups this is not a consequence of the Adian–Rabin theorem, because Hopficity is not a Markov property, as was shown by .
References
External links
Non-Hopf group in the Encyclopedia of Mathematics
Infinite group theory
Properties of groups
|
https://en.wikipedia.org/wiki/Conditional%20change%20model
|
The conditional change model in statistics is the analytic procedure in which change scores are regressed on baseline values, together with the explanatory variables of interest (often including indicators of treatment groups). The method has some substantial advantages over the usual two-sample t-test recommended in textbooks.
References
Regression models
|
https://en.wikipedia.org/wiki/List%20of%20active%20synagogues%20in%20Poland
|
Before the Nazi German invasion of Poland in 1939, almost every Polish town had a synagogue or a Jewish house of prayer of some kind. The 1939 statistics recorded the total of 1,415 Jewish communities in the country just before the outbreak of war, each composed of at least 100 members (Gruber, 1995). Every one of them owned at least one synagogue and a Jewish cemetery nearby. Approximately 9.8% of all believers in Poland were Jewish (according to 1931 census).
The list of actives synagogues in Poland cannot possibly include the hundreds of synagogue buildings which still stand today in about 250 cities and towns across the country – seventy years after the Holocaust in Poland which claimed the lives of over 90% of Polish Jewry. Devoid of their original hosts, many synagogue buildings house libraries and smaller museums as in Kraków, Łańcut, Włodawa, Tykocin, Zamość, Radzanów, but many more serve as apartment buildings, shops, gyms and whatever else community needs require. This isn't necessarily bad however, because the synagogues which remain empty are usually worse off due to lack of maintenance.
Active synagogues in Poland
The Union of Jewish Religious Communities in Poland (ZGWŻ) with branches in nine metropolitan centres helps the descendants of the Holocaust survivors in the process of recovery and restoration of synagogue buildings once owned by the Jewish Kehilla (קהלה), and nationalized in Communist Poland. The list of active and rededicated synagogues in the country include:
Warsaw
Nożyk Synagogue
Chabad Lubavitch Synagogue (pl)
Beit Warszawa Synagogue - Związek Postępowych Gmin Żydowskich w Polsce
Masorti Synagogue - Synagoga konserwatywna w Warszawie
Kraków
Remuh Synagogue
Tempel Synagogue
Izaak Synagogue
See also Synagogues of Kraków for a more complete list of 124 synagogue buildings with description of selected historical monuments of Jewish sacred architecture in the city
Łódź
Reicher Synagogue (pl)
Pomorska Street Synagogue (pl)
Lublin
Chewra Nosim (Chevrat Nossim) Synagogue (pl)
Chachmei Lublin Yeshiva Synagogue (Yeshivat Chachamei Lublin) (pl)
New Jewish Cemetery Synagogue (pl)
Wrocław
White Stork Synagogue
Small Synagogue (pl)
Gdańsk
New Synagogue (pl)
Oświęcim Synagogue (Chevrah Lomdei Mishnayot Synagogue, pl), Oświęcim
Bajs Nusn (Beys Nusn) Synagogue (pl), Nowy Sącz
Bobowa Synagogue (pl), Bobowa
Leżajsk Synagogue (pl), Leżajsk
Lelów Synagogue (pl), Lelów
Bielsko-Biała Synagogue (pl), Bielsko-Biała
Gliwice Synagogue (pl), Gliwice
Katowice Synagogue (pl), Katowice
Legnica Synagogue (pl), Legnica
Poznań Synagogue, Poznań
Szczecin Synagogue, Szczecin
Wałbrzych Synagogue, Wałbrzych
Żary Synagogue, Żary
Włodawa Synagogue (Wlodowa Synagogue) in Włodawa
Notes
External links
Chabad-Lubavitch Centers in Poland
9 Illustrious Synagogues You Can Visit in Poland
Synagogues
Poland
|
https://en.wikipedia.org/wiki/Markov%20partition
|
A Markov partition in mathematics is a tool used in dynamical systems theory, allowing the methods of symbolic dynamics to be applied to the study of hyperbolic dynamics. By using a Markov partition, the system can be made to resemble a discrete-time Markov process, with the long-term dynamical characteristics of the system represented as a Markov shift. The appellation 'Markov' is appropriate because the resulting dynamics of the system obeys the Markov property. The Markov partition thus allows standard techniques from symbolic dynamics to be applied, including the computation of expectation values, correlations, topological entropy, topological zeta functions, Fredholm determinants and the like.
Motivation
Let be a discrete dynamical system. A basic method of studying its dynamics is to find a symbolic representation: a faithful encoding of the points of by sequences of symbols such that the map becomes the shift map.
Suppose that has been divided into a number of pieces which are thought to be as small and localized, with virtually no overlaps. The behavior of a point under the iterates of can be tracked by recording, for each , the part which contains . This results in an infinite sequence on the alphabet which encodes the point. In general, this encoding may be imprecise (the same sequence may represent many different points) and the set of sequences which arise in this way may be difficult to describe. Under certain conditions, which are made explicit in the rigorous definition of a Markov partition, the assignment of the sequence to a point of becomes an almost one-to-one map whose image is a symbolic dynamical system of a special kind called a shift of finite type. In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system .
Formal definition
A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles such that
For any pair of points , that
for
If and , then
Here, and are the unstable and stable manifolds of x, respectively, and simply denotes the interior of .
These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system. It is this property of the covering that merits the 'Markov' appellation. The resulting dynamics is that of a Markov shift; that this is indeed the case is due to theorems by Yakov Sinai (1968) and Rufus Bowen (1975), thus putting symbolic dynamics on a firm footing.
Variants of the definition are found, corresponding to conditions on the geometry of the pieces .
Examples
Markov partitions have been constructed in several situations.
Anosov diffeomorphisms of the torus.
Dynamical billiards, in which case the covering is countable.
Markov partitions make homoclinic and heteroclinic
|
https://en.wikipedia.org/wiki/EWM
|
EWM may refer to:
Edinburgh Woollen Mill, a British retailer
Ellsworth–Whitmore Mountains, in Antarctica
Exploding wire method
European Women in Mathematics
|
https://en.wikipedia.org/wiki/Polar%20homology
|
In complex geometry, a polar homology is a group which captures holomorphic invariants of a complex manifold in a similar way to usual homology of a manifold in differential topology. Polar homology was defined by B. Khesin and A. Rosly in 1999.
Definition
Let M be a complex projective manifold. The space of polar k-chains is a vector space over defined as a quotient , with and vector spaces defined below.
Defining Ak
The space is freely generated by the triples , where X is a smooth, k-dimensional complex manifold, a holomorphic map, and is a rational k-form on X, with first order poles on a divisor with normal crossing.
Defining Rk
The space is generated by the following relations.
if .
provided that
where
for all and the push-forwards are considered on the smooth part of .
Defining the boundary operator
The boundary operator is defined by
,
where are components of the polar divisor of , res is the Poincaré residue, and are restrictions of the map f to each component of the divisor.
Khesin and Rosly proved that this boundary operator is well defined, and satisfies . They defined the polar cohomology as the quotient .
Notes
B. Khesin, A. Rosly, Polar Homology and Holomorphic Bundles Phil. Trans. Roy. Soc. Lond. A359 (2001) 1413-1428
Complex manifolds
Several complex variables
Homology theory
|
https://en.wikipedia.org/wiki/Secondary%20polynomials
|
In mathematics, the secondary polynomials associated with a sequence of polynomials orthogonal with respect to a density are defined by
To see that the functions are indeed polynomials, consider the simple example of Then,
which is a polynomial provided that the three integrals in (the moments of the density ) are convergent.
See also
Secondary measure
Polynomials
|
https://en.wikipedia.org/wiki/Ramaswamy%20S.%20Vaidyanathaswamy
|
Ramaswamy S. Vaidyanathaswamy (1894–1960) was an Indian mathematician who wrote the first textbook of point-set topology in India.
Life
He was born in India on 24 October 1894.
Vaidyanathaswamy studied Mathematics at the University of Edinburgh in Scotland, under Prof Edmund Taylor Whittaker graduating around 1914. He then did postgraduate studies at the University of Cambridge under Prof H. F. Baker. After his return to India, he was a professor at the University of Madras, and after his retirement was associated with the Indian Statistical Institute in Calcutta.
He contributed extensively to point-set topology, and wrote a well-known textbook on the subject (and the first such textbook published in India), "Set Topology", which was first published in 1947. A second edition, published in 1960, was reprinted by Dover Publications in 1999.
He was elected a fellow of the Royal Society of Edinburgh in 1924. His proposers were Herbert Westren Turnbull, Edmund Taylor Whittaker, Ralph Allan Sampson and James Hartley Ashworth.
He was president of the Indian Mathematical Society from 1940 to 1942.
Selected publications
References
Welcome to University of Madras
20th-century Indian mathematicians
Fellows of the Royal Society of Edinburgh
1894 births
1960 deaths
Presidents of the Indian Mathematical Society
Alumni of the University of Edinburgh
Expatriates from British India in the United Kingdom
|
https://en.wikipedia.org/wiki/Koreans%20in%20Germany
|
Koreans in Germany numbered 31,248 individuals , according to the statistics of South Korea's Ministry of Foreign Affairs and Trade. Though they are now only the 14th-largest Korean diaspora community worldwide, they remain the second-largest in Western Europe, behind the rapidly growing community of Koreans in the United Kingdom. As of 2010, Germany has been hosting the second-largest number of Koreans residing in Western Europe, if one excludes Korean sojourners (students and general sojourners).
The largest community of Koreans is situated in the Frankfurt-Rhine Main Area, with 5,300 residents. This area also contains German and European headquarters of large Korean companies such as Kia Motors, Hyundai, Samsung Electronics, LG International, SK Group, Nexen Tire.
History
South Koreans
Some students, nurses, and industrial trainees from South Korea had already been in West Germany in the late 1950s. However, mass migration did not begin until the 1960s, when West Germany invited nurses and miners from South Korea to come as Gastarbeiter; their recruitment of labourers specifically from South Korea was driven not just by economic necessity, but also by a desire to demonstrate support for a country that, like Germany, had been divided by ideology. The first group of miners arrived on 16 December 1963, under a programme paid for largely by the South Korean government; German enterprises were not responsible for travel costs, but only for wages and language training. They had high levels of education compared with other Gastarbeiter of the same era; over 60% had completed high school or tertiary education. Nurses began arriving in large numbers in 1966. Koreans were one of the few non-European groups recruited; West German migration policy generally excluded workers of African and Asian origin during the 1950s through 1970s. After living in Germany, some Koreans migrated onwards to the United States under the relaxed entrance standards of the Immigration and Nationality Act of 1965. Though the South Korean workers came on limited-term contracts and most initially planned to return home, in the end, half of the workers enlisted ended up remaining in Germany. Throughout the 1970s, they staged protests demanding the right to stay, citing their contributions to the economy and health care system; in the end, the West German government refrained from expelling those whose work contracts had expired, instead letting them move on to other work.
North and South Korea vied for influence among the Korean community in West Germany during the 1960s and 1970s; North Korea sent operatives to West Germany disguised as professors in order to recruit among the Korean community there. In 1967, South Korea forcibly extradited, without the consent of the West German government, a number of Koreans suspected of spying for the North, the most famous of whom was composer and later German citizen Isang Yun. They were tortured to extract false confessions, and six we
|
https://en.wikipedia.org/wiki/Plate%20trick
|
In mathematics and physics, the plate trick, also known as Dirac's string trick, the belt trick, or the Balinese cup trick, is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, a total rotation of 720 degrees, does. Mathematically, it is a demonstration of the theorem that SU(2) (which double-covers SO(3)) is simply connected. To say that SU(2) double-covers SO(3) essentially means that the unit quaternions represent the group of rotations twice over. A detailed, intuitive, yet semi-formal articulation can be found in the article on tangloids.
Demonstrations
Resting a small plate flat on the palm, it is possible to perform two rotations of one's hand while keeping the plate upright. After the first rotation of the hand, the arm will be twisted, but after the second rotation it will end in the original position. To do this, the hand makes one rotation passing over the elbow, twisting the arm, and then another rotation passing under the elbow untwists it.
In mathematical physics, the trick illustrates the quaternionic mathematics behind the spin of spinors. As with the plate trick, these particles' spins return to their original state only after two full rotations, not after one.
The belt trick
The same phenomenon can be demonstrated using a leather belt with an ordinary frame buckle, whose prong serves as a pointer. The end opposite the buckle is clamped so it cannot move. The belt is extended without a twist and the buckle is kept horizontal while being turned clockwise one complete turn (360°), as evidenced by watching the prong. The belt will then appear twisted, and no maneuvering of the buckle that keeps it horizontal and pointed in the same direction can undo the twist. Obviously a 360° turn counterclockwise would undo the twist. The surprise element of the trick is that a second 360° turn in the clockwise direction, while apparently making the belt even more twisted, does allow the belt to be returned to its untwisted state by maneuvering the buckle under the clamped end while always keeping the buckle horizontal and pointed in the same direction.
Mathematically, the belt serves as a record, as one moves along it, of how the buckle was transformed from its original position, with the belt untwisted, to its final rotated position. The clamped end always represents the null rotation. The trick demonstrates that a path in rotation space (SO(3)) that produces a 360 degree rotation is not homotopic to a null rotation, but a path that produces a double rotation (720°) is null-homotopic.
Belt trick has been witnessed in 1-d Classical Heisenberg model as a breather solution.
See also
Anti-twister mechanism
Spin–statistics theorem
Orientation entanglement
Tangloids
References
External links
Animation of the Dirac belt trick, including the path through SU(2)
Animation of the Dirac belt trick,
|
https://en.wikipedia.org/wiki/Xcas
|
Xcas is a user interface to Giac, which is an open source computer algebra system (CAS) for Windows, macOS and Linux among many other platforms. Xcas is written in C++. Giac can be used directly inside software written in C++.
Xcas has compatibility modes with many popular algebra systems like WolframAlpha, Mathematica, Maple, or MuPAD. Users can use Giac/Xcas to develop formal algorithms or use it in other software. Giac is used in SageMath for calculus operations. Among other things, Xcas can solve equations (Figure 3) and differential equations (Figure 4) and draw graphs. There is a forum for questions about Xcas.
CmathOOoCAS, an OpenOffice.org plugin which allows formal calculation in Calc spreadsheet and Writer word processing, uses Giac to perform calculations.
Features
Here is a brief overview of what Xcas is able to do:
Xcas has the ability of a scientific calculator that provides show input and writes pretty print
Xcas works also as a spreadsheet;
computer algebra;
2D geometry in the plane;
3D geometry in space;
spreadsheet;
statistics;
regression (exponential, linear, logarithmic, logistic, polynomial, power)
programming;
solve equations even with complex roots (Figure 2);
solving trigonometric equations
solve differential equations (Figure 3);
draw graphs;
calculate differential (or derivative) of functions (Figure 2);
calculate antiderivative of functions (Figure 2);
calculate area and integral calculus;
linear algebra
Example Xcas commands:
Produce mixed fractions: propfrac(42/15) gives 2 +
Calculate square root: sqrt(4) = 2
Draw a vertical line in coordinate system: line(x=1) draws the vertical line in the output window
Draw graph: plot(function) (for example, plot(3 * x^2 - 5) produces a plot of
Calculate average: mean([3, 4, 2]) is 3
Calculate variance: variance([3, 4, 2]) is
Calculate standard deviation: stddev([3, 4, 2]) is
Calculate determinant of a matrix: is
Calculate local extrema of a function: extrema(-2*cos(x)-cos(x)^2,x) is [0, π]
Calculate cross product of two vectors: cross([1, 2, 3], [4, 3, 2]) is
Calculate permutations: nPr()
Calculate combinations: nCr()
Solve equation: solve(equation,x)
Factoring Polynomials: factor(polynomial,x) or cfactor(polynomial,x)
Differentiation of function: diff(function,x)
Calculate indefinite integrals/antiderivatives: int(function,x)
Calculate definite integrals/area under the curve of a function: int(function,x,lowerlimit,upperlimit)
Calculate definite integral aka solid of revolution - finding volume by rotation (around the x-axis): int(pi*function^2,x,lowerlimit,upperlimit)
Calculate definite integral aka solid of revolution - finding volume by rotation (around the y-axis) for a decreasing function: int(2*pi*x*function,x,lowerlimit,upperlimit)
Separation of variables: split((x+1)*(y-2),[x,y]) produces
desolve differential equation (the derivatives are written as y or y): desolve(differential equation,y)
Supported operating systems
|
https://en.wikipedia.org/wiki/Combinatorial%20commutative%20algebra
|
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra and combinatorics, and frequently uses methods of one to address problems arising in the other. Less obviously, polyhedral geometry plays a significant role.
One of the milestones in the development of the subject was Richard Stanley's 1975 proof of the Upper Bound Conjecture for simplicial spheres, which was based on earlier work of Melvin Hochster and Gerald Reisner. While the problem can be formulated purely in geometric terms, the methods of the proof drew on commutative algebra techniques.
A signature theorem in combinatorial commutative algebra is the characterization of h-vectors of simplicial polytopes conjectured in 1970 by Peter McMullen. Known as the g-theorem, it was proved in 1979 by Stanley (necessity of the conditions, algebraic argument) and by Louis Billera and Carl W. Lee (sufficiency, combinatorial and geometric construction). A major open question was the extension of this characterization from simplicial polytopes to simplicial spheres, the g-conjecture, which was resolved in 2018 by Karim Adiprasito.
Important notions of combinatorial commutative algebra
Square-free monomial ideal in a polynomial ring and Stanley–Reisner ring of a simplicial complex.
Cohen–Macaulay ring.
Monomial ring, closely related to an affine semigroup ring and to the coordinate ring of an affine toric variety.
Algebra with a straightening law. There are several versions of those, including Hodge algebras of Corrado de Concini, David Eisenbud, and Claudio Procesi.
See also
Algebraic combinatorics
Polyhedral combinatorics
Zero-divisor graph
References
A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory:
The first book is a classic (first edition published in 1983):
Very influential, and well written, textbook-monograph:
Additional reading:
A recent addition to the growing literature in the field, contains exposition of current research topics:
Commutative algebra
Algebraic geometry
Algebraic combinatorics
|
https://en.wikipedia.org/wiki/H-vector
|
In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito.
Stanley introduced a generalization of the h-vector, the toric h-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to hold. A different, more combinatorial, generalization of the h-vector that has been extensively studied is the flag h-vector of a ranked poset. For Eulerian posets, it can be more concisely expressed by means of a noncommutative polynomial in two variables called the cd-index.
Definition
Let Δ be an abstract simplicial complex of dimension d − 1 with fi i-dimensional faces and f−1 = 1. These numbers are arranged into the f-vector of Δ,
An important special case occurs when Δ is the boundary of a d-dimensional convex polytope.
For k = 0, 1, …, d, let
The tuple
is called the h-vector of Δ. In particular, , , and , where is the Euler characteristic of . The f-vector and the h-vector uniquely determine each other through the linear relation
from which it follows that, for ,
In particular, . Let R = k[Δ] be the Stanley–Reisner ring of Δ. Then its Hilbert–Poincaré series can be expressed as
This motivates the definition of the h-vector of a finitely generated positively graded algebra of Krull dimension d as the numerator of its Hilbert–Poincaré series written with the denominator (1 − t)d.
The h-vector is closely related to the h*-vector for a convex lattice polytope, see Ehrhart polynomial.
Recurrence relation
The -vector can be computed from the -vector by using the recurrence relation
.
and finally setting for . For small examples, one can use this method to compute -vectors quickly by hand by recursively filling the entries of an array similar to Pascal's triangle. For example, consider the boundary complex of an octahedron. The -vector of is . To compute the -vector of , construct a triangular array by first writing s down the left edge and the -vector down the right edge.
(We set just to make the array triangular.) Then, starting from the top, fill each remaining entry by subtracting its upper-left neighbor from its upper-right neighbor. In this way, we generate the following array:
The entries of the bottom row (apart from the final ) are the entries of the -vector. Hence, the -vecto
|
https://en.wikipedia.org/wiki/Stieltjes%20transformation
|
In mathematics, the Stieltjes transformation of a measure of density on a real interval is the function of the complex variable defined outside by the formula
Under certain conditions we can reconstitute the density function starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density is continuous throughout , one will have inside this interval
Connections with moments of measures
If the measure of density has moments of any order defined for each integer by the equality
then the Stieltjes transformation of admits for each integer the asymptotic expansion in the neighbourhood of infinity given by
Under certain conditions the complete expansion as a Laurent series can be obtained:
Relationships to orthogonal polynomials
The correspondence defines an inner product on the space of continuous functions on the interval .
If is a sequence of orthogonal polynomials for this product, we can create the sequence of associated secondary polynomials by the formula
It appears that is a Padé approximation of in a neighbourhood of infinity, in the sense that
Since these two sequences of polynomials satisfy the same recurrence relation in three terms, we can develop a continued fraction for the Stieltjes transformation whose successive convergents are the fractions .
The Stieltjes transformation can also be used to construct from the density an effective measure for transforming the secondary polynomials into an orthogonal system. (For more details see the article secondary measure.)
See also
Orthogonal polynomials
Secondary polynomials
Secondary measure
References
Integral transforms
Continued fractions
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.