source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Hasse%E2%80%93Witt%20matrix
|
In mathematics, the Hasse–Witt matrix H of a non-singular algebraic curve C over a finite field F is the matrix of the Frobenius mapping (p-th power mapping where F has q elements, q a power of the prime number p) with respect to a basis for the differentials of the first kind. It is a g × g matrix where C has genus g. The rank of the Hasse–Witt matrix is the Hasse or Hasse–Witt invariant.
Approach to the definition
This definition, as given in the introduction, is natural in classical terms, and is due to Helmut Hasse and Ernst Witt (1936). It provides a solution to the question of the p-rank of the Jacobian variety J of C; the p-rank is bounded by the rank of H, specifically it is the rank of the Frobenius mapping composed with itself g times. It is also a definition that is in principle algorithmic. There has been substantial recent interest in this as of practical application to cryptography, in the case of C a hyperelliptic curve. The curve C is superspecial if H = 0.
That definition needs a couple of caveats, at least. Firstly, there is a convention about Frobenius mappings, and under the modern understanding what is required for H is the transpose of Frobenius (see arithmetic and geometric Frobenius for more discussion). Secondly, the Frobenius mapping is not F-linear; it is linear over the prime field Z/pZ in F. Therefore the matrix can be written down but does not represent a linear mapping in the straightforward sense.
Cohomology
The interpretation for sheaf cohomology is this: the p-power map acts on
H1(C,OC),
or in other words the first cohomology of C with coefficients in its structure sheaf. This is now called the Cartier–Manin operator (sometimes just Cartier operator), for Pierre Cartier and Yuri Manin. The connection with the Hasse–Witt definition is by means of Serre duality, which for a curve relates that group to
H0(C, ΩC)
where ΩC = Ω1C is the sheaf of Kähler differentials on C.
Abelian varieties and their p-rank
The p-rank of an abelian variety A over a field K of characteristic p is the integer k for which the kernel A[p] of multiplication by p has pk points. It may take any value from 0 to d, the dimension of A; by contrast for any other prime number l there are l2d points in A[l]. The reason that the p-rank is lower is that multiplication by p on A is an inseparable isogeny: the differential is p which is 0 in K. By looking at the kernel as a group scheme one can get the more complete structure (reference David Mumford Abelian Varieties pp. 146–7); but if for example one looks at reduction mod p of a division equation, the number of solutions must drop.
The rank of the Cartier–Manin operator, or Hasse–Witt matrix, therefore gives an upper bound for the p-rank. The p-rank is the rank of the Frobenius operator composed with itself g times. In the original paper of Hasse and Witt the problem is phrased in terms intrinsic to C, not relying on J. It is there a question of classifying the possible Artin–Schreier ext
|
https://en.wikipedia.org/wiki/Supersingular%20elliptic%20curve
|
In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ordinary and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory.
The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular. It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular. The primes p such that every supersingular elliptic curve in characteristic p can be defined over the prime subfield rather than are called supersingular primes.
Definition
There are many different but equivalent ways of defining supersingular elliptic curves that have been used. Some of the ways of defining them are given below. Let be a field with algebraic closure and E an elliptic curve over K.
The -valued points have the structure of an abelian group. For every n, we have a multiplication map . Its kernel is denoted by . Now assume that the characteristic of K is p > 0. Then one can show that either
for r = 1, 2, 3, ... In the first case, E is called supersingular. Otherwise it is called ordinary. In other words, an elliptic curve is supersingular if and only if the group of geometric points of order p is trivial.
Supersingular elliptic curves have many endomorphisms over the algebraic closure in the sense that an elliptic curve is supersingular if and only if its endomorphism algebra (over ) is an order in a quaternion algebra. Thus, their endomorphism algebra (over ) has rank 4, while the endomorphism group of every other elliptic curve has only rank 1 or 2. The endomorphism ring of a supersingular elliptic curve can have rank less than 4, and it may be necessary to take a finite extension of the base field K to make the rank of the endomorphism ring 4. In particular the endomorphism ring of an elliptic curve over a field of prime order is never of rank 4, even if the elliptic curve is supersingular.
Let G be the formal group associated to E. Since K is of positive characteristic, we can define its height ht(G), which is 2 if and only if E is supersingular and else is 1.
We have a Frobenius morphism , which induces a map in cohomol
|
https://en.wikipedia.org/wiki/Hasse%20invariant
|
In mathematics, Hasse invariant may refer to:
Hasse invariant of an algebra
Hasse invariant of an elliptic curve
Hasse invariant of a quadratic form
|
https://en.wikipedia.org/wiki/Hasse%20invariant%20of%20a%20quadratic%20form
|
In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt.
The quadratic form Q may be taken as a diagonal form
Σ aixi2.
Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras
(ai, aj) for i < j.
This is independent of the diagonal form chosen to compute it.
It may also be viewed as the second Stiefel–Whitney class of Q.
Symbols
The invariant may be computed for a specific symbol φ taking values in the group C2 = {±1}.
In the context of quadratic forms over a local field, the Hasse invariant may be defined using the Hilbert symbol, the unique symbol taking values in C2. The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.
For quadratic forms over a number field, there is a Hasse invariant ±1 for every finite place. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.
See also
Hasse–Minkowski theorem
References
Quadratic forms
|
https://en.wikipedia.org/wiki/Diagonal%20form
|
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is
for some given degree m.
Such forms F, and the hypersurfaces F = 0 they define in projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of Diophantine equations.
A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method.
Examples
is the unit circle in P2
is the unit hyperbola in P2.
gives the Fermat cubic surface in P3 with 27 lines. The 27 lines in this example are easy to describe explicitly: they are the 9 lines of the form (x : ax : y : by) where a and b are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.
gives a K3 surface in P3.
Homogeneous polynomials
Algebraic varieties
|
https://en.wikipedia.org/wiki/Enriques%E2%80%93Kodaira%20classification
|
In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
Max Noether began the systematic study of algebraic surfaces, and Guido Castelnuovo proved important parts of the classification. described the classification of complex projective surfaces. later extended the classification to include non-algebraic compact surfaces. The analogous classification of surfaces in positive characteristic was begun by and completed by ; it is similar to the characteristic 0 projective case, except that one also gets singular and supersingular Enriques surfaces in characteristic 2, and quasi-hyperelliptic surfaces in characteristics 2 and 3.
Statement of the classification
The Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic, properly quasi-elliptic, or general type surfaces.
For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on the global spherical shell conjecture, still unproved in 2009). For surfaces of general type not much is known about their explicit classification, though many examples have been found.
The classification of algebraic surfaces in positive characteristic (, ) is similar to that of algebraic surfaces in characteristic 0, except that there are no Kodaira surfaces or surfaces of type VII, and there are some extra families of Enriques surfaces in characteristic 2, and hyperelliptic surfaces in characteristics 2 and 3, and in Kodaira dimension 1 in characteristics 2 and 3 one also allows quasielliptic fibrations. These extra families can be understood as follows: In characteristic 0 these surfaces are the quotients of surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finite group schemes that are not étale.
Oscar Zariski constructed some surfaces in positive characteristic that are unirational but not rational, derived from inseparable extensions (Zariski surfaces). In positive characteristic Serre showed that may differ from , and Igusa showed that even when they are equal they may be greater than the irregularity (the dimension of the Picard variety).
Invariants of surfaces
Hodge numbers and Kodaira dimension
The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of various coherent sheaf cohomology groups. The basic
|
https://en.wikipedia.org/wiki/Small%20stellated%20dodecahedron
|
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol {,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeting at each vertex.
It shares the same vertex arrangement as the convex regular icosahedron. It also shares the same edge arrangement with the great icosahedron, with which it forms a degenerate uniform compound figure.
It is the second of four stellations of the dodecahedron (including the original dodecahedron itself).
The small stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the edges (1-faces) of the core polytope until a point is reached where they intersect.
Topology
If the pentagrammic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron, but with much taller isosceles triangle faces, with the height of the pentagonal pyramids adjusted so that the five triangles in the pentagram become coplanar. The critical angle is atan(2) above the dodecahedron face.
If we regard it as having 12 pentagrams as faces, with these pentagrams meeting at 30 edges and 12 vertices, we can compute its genus using Euler's formula
and conclude that the small stellated dodecahedron has genus 4. This observation, made by Louis Poinsot, was initially confusing, but Felix Klein showed in 1877 that the small stellated dodecahedron could be seen as a branched covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. In fact this Riemann surface, called Bring's curve, has the greatest number of symmetries of any Riemann surface of genus 4: the symmetric group acts as automorphisms
Images
In art
A small stellated dodecahedron can be seen in a floor mosaic in St Mark's Basilica, Venice by Paolo Uccello circa 1430. The same shape is central to two lithographs by M. C. Escher: Contrast (Order and Chaos) (1950) and Gravitation (1952).
Related polyhedra
Its convex hull is the regular convex icosahedron. It also shares its edges with the great icosahedron; the compound with both is the great complex icosidodecahedron.
There are four related uniform polyhedra, constructed as degrees of truncation. The dual is a great dodecahedron. The dodecadodecahedron is a rectification, where edges are truncated down to points.
The truncated small stellated dodecahedron can be considered a degenerate uniform polyhedron since edges and vertices coincide, but it is included for completeness. Visually, it looks like a regular dodecahedron on the surface, but it has 24 faces in overlapping pairs. The spikes are truncated until they reach the plane of the pentagram beneath them. The 24 faces are 12 pentagons from the truncated vertices and 12 decagons taking the form of doubly-wound pentagons overlapping the first 12 pentagons. The latter faces are form
|
https://en.wikipedia.org/wiki/Great%20stellated%20dodecahedron
|
In geometry, the great stellated dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol {,3}. It is one of four nonconvex regular polyhedra.
It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each vertex.
It shares its vertex arrangement, although not its vertex figure or vertex configuration, with the regular dodecahedron, as well as being a stellation of a (smaller) dodecahedron. It is the only dodecahedral stellation with this property, apart from the dodecahedron itself. Its dual, the great icosahedron, is related in a similar fashion to the icosahedron.
Shaving the triangular pyramids off results in an icosahedron.
If the pentagrammic faces are broken into triangles, it is topologically related to the triakis icosahedron, with the same face connectivity, but much taller isosceles triangle faces. If the triangles are instead made to invert themselves and excavate the central icosahedron, the result is a great dodecahedron.
The great stellated dodecahedron can be constructed analogously to the pentagram, its two-dimensional analogue, by attempting to stellate the n-dimensional pentagonal polytope which has pentagonal polytope faces and simplex vertex figures until it can no longer be stellated; that is, it is its final stellation.
Images
Related polyhedra
A truncation process applied to the great stellated dodecahedron produces a series of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a rectified great stellated dodecahedron. The process completes as a birectification, reducing the original faces down to points, and producing the great icosahedron.
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) pentagonal faces as truncations of the original pentagram faces, the latter forming a great dodecahedron inscribed within and sharing the edges of the icosahedron.
References
External links
Uniform polyhedra and duals
Polyhedral stellation
Regular polyhedra
Kepler–Poinsot polyhedra
|
https://en.wikipedia.org/wiki/Great%20icosahedron
|
In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.
The great icosahedron can be constructed analogously to the pentagram, its two-dimensional analogue, via the extension of the -dimensional simplex faces of the core -polytope (equilateral triangles for the great icosahedron, and line segments for the pentagram) until the figure regains regular faces. The grand 600-cell can be seen as its four-dimensional analogue using the same process.
Construction
The Great Icosahedron edge length is times the original icosahedron edge length.
Images
As a snub
The great icosahedron can be constructed as a uniform snub, with different colored faces and only tetrahedral symmetry: . This construction can be called a retrosnub tetrahedron or retrosnub tetratetrahedron, similar to the snub tetrahedron symmetry of the icosahedron, as a partial faceting of the truncated octahedron (or omnitruncated tetrahedron): . It can also be constructed with 2 colors of triangles and pyritohedral symmetry as, or , and is called a retrosnub octahedron.
Related polyhedra
It shares the same vertex arrangement as the regular convex icosahedron. It also shares the same edge arrangement as the small stellated dodecahedron.
A truncation operation, repeatedly applied to the great icosahedron, produces a sequence of uniform polyhedra. Truncating edges down to points produces the great icosidodecahedron as a rectified great icosahedron. The process completes as a birectification, reducing the original faces down to points, and producing the great stellated dodecahedron.
The truncated great stellated dodecahedron is a degenerate polyhedron, with 20 triangular faces from the truncated vertices, and 12 (hidden) doubled up pentagonal faces ({10/2}) as truncations of the original pentagram faces, the latter forming two great dodecahedra inscribed within and sharing the edges of the icosahedron.
References
(1st Edn University of Toronto (1938))
H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , 3.6 6.2 Stellating the Platonic solids, pp. 96–104
External links
Uniform polyhedra and duals
Kepler–Poinsot polyhedra
Regular polyhedra
Polyhedral stellation
Deltahedra
|
https://en.wikipedia.org/wiki/Fuzzy%20sphere
|
In mathematics, the fuzzy sphere is one of the simplest and most canonical examples of non-commutative geometry. Ordinarily, the functions defined on a sphere form a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated by spherical harmonics whose spin l is at most equal to some j. The terms in the product of two spherical harmonics that involve spherical harmonics with spin exceeding j are simply omitted in the product. This truncation replaces an infinite-dimensional commutative algebra by a -dimensional non-commutative algebra.
The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite-dimensional vector space.
Take the three j-dimensional matrices that form a basis for the j dimensional irreducible representation of the Lie algebra su(2). They satisfy the relations , where is the totally antisymmetric symbol with , and generate via the matrix product the algebra of j dimensional matrices. The value of the su(2) Casimir operator in this representation is
where I is the j-dimensional identity matrix.
Thus, if we define the 'coordinates'
where r is the radius of the sphere and k is a parameter, related to r and j by , then the above equation concerning the Casimir operator can be rewritten as
,
which is the usual relation for the coordinates on a sphere of radius r embedded in three dimensional space.
One can define an integral on this space, by
where F is the matrix corresponding to the function f.
For example, the integral of unity, which gives the surface of the sphere in the commutative case is here equal to
which converges to the value of the surface of the sphere if one takes j to infinity.
Notes
Jens Hoppe, "Membranes and Matrix Models", lectures presented during the summer school on ‘Quantum Field Theory – from a Hamiltonian Point of View’, August 2–9, 2000,
John Madore, An introduction to Noncommutative Differential Geometry and its Physical Applications, London Mathematical Society Lecture Note Series. 257, Cambridge University Press 2002
References
J. Hoppe, Quantum Theory of a Massless Relativistic Surface and a Two dimensional Bound State Problem. PhD thesis, Massachusetts Institute of Technology, 1982.
Mathematical quantization
Noncommutative geometry
|
https://en.wikipedia.org/wiki/Josif%20Shtokalo
|
Josif Zakharovich Shtokalo (; November 16, 1897 – January 5, 1987) was a famous Ukrainian mathematician. Shtokalo worked mainly in the areas of differential equations, operational calculus and the history of mathematics.
Investigation of the Stability of Lindstedt's Equation Using Shtokalo’s Method by Samuel Kohn contains a description of Shotkalo's method in English.
References
1897 births
1987 deaths
Soviet mathematicians
20th-century Ukrainian mathematicians
|
https://en.wikipedia.org/wiki/Zero%20point
|
Zero point may refer to:
The hypocenter of a nuclear explosion
Origin (mathematics), a fixed point of reference for a coordinate system
Zero Point (film), an Estonian film
Zero point (photometry), a calibration mechanism for magnitude in astronomy
Zero Point (South Georgia), a point in Possession Bay, South Georgia
Zero Point Interchange, a cloverleaf interchange in Islamabad, Pakistan, at the intersection of Islamabad Highway, Kashmir Highway and Khayaban-e-Suharwardy
Zero Point railway station, a railway station on the Pakistan–India border
Lingdian (band) (), sometimes translated in English as Zero Point, a Chinese band
"Zero Point", a song by Tori Amos, released on A Piano: The Collection
Zero Point, a fictional orb of energy in Fortnite Battle Royale
See also
Zero-point energy, the minimum energy a quantum mechanical system may have
Zero-point field, a synonym for the vacuum state in quantum field theory
Hofstadter zero-point, a special point associated with every plane triangle
Point of origin (disambiguation)
Triple zero (disambiguation)
Point Zero (disambiguation)
|
https://en.wikipedia.org/wiki/Mikhail%20Kravchuk
|
Mykhailo Pylypovych Kravchuk, also Krawtchouk () (September 27, 1892 – March 9, 1942), was a Soviet Ukrainian mathematician and the author of around 180 articles on mathematics.
He primarily wrote papers on differential equations and integral equations, studying both their theory and applications. His two-volume monograph on the solution of linear differential and integral equations by the method of moments was translated 1938–1942 by John Vincent Atanasoff who found this work useful in his computer-project (Atanasoff–Berry computer). His student Klavdiya Latysheva was the first Ukrainian woman to obtain a doctorate in the mathematical and physical sciences (1936).
Kravchuk held a mathematics chair at the Kyiv Polytechnic Institute. His course listeners included Sergey Korolev, Arkhip Lyulka, and Vladimir Chelomei, future leading rocket and jet engine designers. Kravchuk was arrested by the Soviet secret police on February 23, 1938 on political and spying charges. He was sentenced to 20 years of prison in September 1938. Kravchuk died in a Gulag camp in the Kolyma region on March 9, 1942. In September 1956 Kravchuk was posthumously acquitted of all charges.
He was restored as a member of the National Academy of Sciences of Ukraine posthumously in 1992. He is the eponym of the Kravchuk polynomials and Kravchuk matrix.
References
External links
MacTutor biography
Biography page(this uses the transliteration Mikhail Krawtchouk, which is phonetic for Francophones, and under which he published work)
Ukrainian biographical website
Krawtchouk Polynomials Home Page
I. Katchanovski, Krawtchouk's Mind Biographical article
Video about Mykhailo Pylypovych Kravchuk
S. Hrabovsy, Mykhailo Kravchuk, a mathematician, patriot and precursor of computers, Welcome to Ukraine, 4, 2003
N. Virchenko, Life and death of Mykhailo Kravchuk, a brilliant mathematician, Welcome to Ukraine, 2, 2008
INTERNATIONAL MATHEMATICAL KRAVCHUK CONFERENCE
Soviet mathematicians
Taras Shevchenko National University of Kyiv alumni
1892 births
1942 deaths
People from Volyn Oblast
People from Volhynian Governorate
Ukrainian people who died in Soviet detention
Full Members of the All-Ukrainian Academy of Sciences
Russian scientists
|
https://en.wikipedia.org/wiki/Ciprian%20Manolescu
|
Ciprian Manolescu (born December 24, 1978) is a Romanian-American mathematician, working in gauge theory, symplectic geometry, and low-dimensional topology. He is currently a professor of mathematics at Stanford University.
Biography
Manolescu completed his first eight classes at School no. 11 Mihai Eminescu and his secondary education at Ion Brătianu High School in Piteşti. He completed his undergraduate studies and PhD at Harvard University under the direction of Peter B. Kronheimer. He was the winner of the Morgan Prize, awarded jointly by AMS-MAA-SIAM, in 2002. His undergraduate thesis was on Finite dimensional approximation in Seiberg–Witten theory, and his PhD thesis topic was A spectrum valued TQFT from the Seiberg–Witten equations.
In early 2013, he released a paper detailing a disproof of the triangulation conjecture for manifolds of dimension 5 and higher. For this paper, he received the E. H. Moore Prize from the American Mathematical Society.
Awards and honors
He was among the recipients of the Clay Research Fellowship (2004–2008).
In 2012, he was awarded one of the ten prizes of the European Mathematical Society for his work on low-dimensional topology, and particularly for his role in the development of combinatorial Heegaard Floer homology.
He was elected as a member of the 2017 class of Fellows of the American Mathematical Society "for contributions to Floer homology and the topology of manifolds".
In 2018, he was an invited speaker at the International Congress of Mathematicians (ICM) in Rio de Janeiro.
In 2020, he received a Simons Investigator Award. The citation reads: "Ciprian Manolescu works in low-dimensional topology and gauge theory. His research is centered on constructing new versions of Floer homology and applying them to questions in topology. With collaborators, he showed that many Floer-theoretic invariants are algorithmically computable. He also developed a new variant of Seiberg-Witten Floer homology, which he used to prove the existence of non-triangulable manifolds in high dimensions."
Competitions
He has one of the best records ever in mathematical competitions:
He holds the sole distinction of writing three perfect papers at the International Mathematical Olympiad: Toronto, Canada (1995); Bombay, India (1996); Mar del Plata, Argentina (1997).
He placed in the top 5 on the William Lowell Putnam Mathematical Competition for college undergraduates in 1997, 1998, and 2000.
Selected works
References
External links
Manolescu's Stanford Page
The Clay Mathematics Institute page
21st-century Romanian mathematicians
21st-century American mathematicians
Topologists
Harvard University alumni
University of California, Los Angeles faculty
People from Alexandria, Romania
1978 births
Living people
Romanian emigrants to the United States
International Mathematical Olympiad participants
Geometers
Fellows of the American Mathematical Society
Putnam Fellows
|
https://en.wikipedia.org/wiki/Novikov%20conjecture
|
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965.
The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the higher signatures, which are certain numerical invariants of smooth manifolds, are homotopy invariants.
The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture.
Precise formulation of the conjecture
Let be a discrete group and its classifying space, which is an Eilenberg–MacLane space of type , and therefore unique up to homotopy equivalence as a CW complex. Let
be a continuous map from a closed oriented -dimensional manifold to , and
Novikov considered the numerical expression, found by evaluating the cohomology class in top dimension against the fundamental class , and known as a higher signature:
where is the Hirzebruch polynomial, or sometimes (less descriptively) as the -polynomial. For each , this polynomial can be expressed in the Pontryagin classes of the manifold's tangent bundle. The Novikov conjecture states that the higher signature is an invariant of the oriented homotopy type of for every such map and every such class , in other words, if is an orientation preserving homotopy equivalence, the higher signature associated to is equal to that associated to .
Connection with the Borel conjecture
The Novikov conjecture is equivalent to the rational injectivity of the assembly map in L-theory. The
Borel conjecture on the rigidity of aspherical manifolds is equivalent to the assembly map being an isomorphism.
References
John Milnor and James D. Stasheff, Characteristic Classes, Annals of Mathematics Studies 76, Princeton (1974).
Sergei P. Novikov, Algebraic construction and properties of Hermitian analogs of k-theory over rings with involution from the point of view of Hamiltonian formalism. Some applications to differential topology and to the theory of characteristic classes. Izv.Akad.Nauk SSSR, v. 34, 1970 I N2, pp. 253–288; II: N3, pp. 475–500. English summary in Actes Congr. Intern. Math., v. 2, 1970, pp. 39–45.
External links
Biography of Sergei Novikov
Novikov Conjecture Bibliography
Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 1
Novikov Conjecture 1993 Oberwolfach Conference Proceedings, Volume 2
2004 Oberwolfach Seminar notes on the Novikov Conjecture (pdf)
Scholarpedia article by S.P. Novikov (2010)
The Novikov Conjecture at the Manifold Atlas
Geometric topology
Homotopy theory
Conjectures
Unsolved problems in geometry
Surgery theory
|
https://en.wikipedia.org/wiki/Solution
|
Solution may refer to:
Solution (chemistry), a mixture where one substance is dissolved in another
Solution (equation), in mathematics
Numerical solution, in numerical analysis, approximate solutions within specified error bounds
Solution, in problem solving
Solution, in solution selling
Other uses
V-STOL Solution, an ultralight aircraft
Solution (band), a Dutch rock band
Solution (Solution album), 1971
Solution A.D., an American rock band
Solution (Cui Jian album), 1991
Solutions (album), a 2019 album by K.Flay
See also
The Solution (disambiguation)
|
https://en.wikipedia.org/wiki/Weil%27s%20conjecture%20on%20Tamagawa%20numbers
|
In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number of a simply connected simple algebraic group defined over a number field is 1. In this case, simply connected means "not having a proper algebraic covering" in the algebraic group theory sense, which is not always the topologists' meaning.
History
calculated the Tamagawa number in many cases of classical groups and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups: found examples where the Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.
Robert Langlands (1966) introduced harmonic analysis methods to show it for Chevalley groups. K. F. Lai (1980) extended the class of known cases to quasisplit reductive groups. proved it for all groups satisfying the Hasse principle, which at the time was known for all groups without E8 factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant E8 case (see strong approximation in algebraic groups), thus completing the proof of Weil's conjecture. In 2011, Jacob Lurie and Dennis Gaitsgory announced a proof of the conjecture for algebraic groups over function fields over finite fields.
Applications
used the Weil conjecture to calculate the Tamagawa numbers of all semisimple algebraic groups.
For spin groups, the conjecture implies the known Smith–Minkowski–Siegel mass formula.
See also
Tamagawa number
References
.
Further reading
Aravind Asok, Brent Doran and Frances Kirwan, "Yang-Mills theory and Tamagawa Numbers: the fascination of unexpected links in mathematics", February 22, 2013
J. Lurie, The Siegel Mass Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality posted June 8, 2012.
Conjectures
Theorems in group theory
Algebraic groups
Diophantine geometry
|
https://en.wikipedia.org/wiki/List%20of%20eponyms%20of%20special%20functions
|
This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym). Named symmetric functions, and other special polynomials, are included.
A
Niels Abel: Abel polynomials - Abelian function - Abel–Gontscharoff interpolating polynomial
Sir George Biddell Airy: Airy function
Waleed Al-Salam (1926–1996): Al-Salam polynomial - Al Salam–Carlitz polynomial - Al Salam–Chihara polynomial
C. T. Anger: Anger–Weber function
Kazuhiko Aomoto: Aomoto–Gel'fand hypergeometric function - Aomoto integral
Paul Émile Appell (1855–1930): Appell hypergeometric series, Appell polynomial, Generalized Appell polynomials
Richard Askey: Askey–Wilson polynomial, Askey–Wilson function (with James A. Wilson)
B
Ernest William Barnes: Barnes G-function
E. T. Bell: Bell polynomials
Bender–Dunne polynomial
Jacob Bernoulli: Bernoulli polynomial
Friedrich Bessel: Bessel function, Bessel–Clifford function
H. Blasius: Blasius functions
R. P. Boas, R. C. Buck: Boas–Buck polynomial
Böhmer integral
Erland Samuel Bring: Bring radical
de Bruijn function
Buchstab function
Burchnall, Chaundy: Burchnall–Chaundy polynomial
C
Leonard Carlitz: Carlitz polynomial
Arthur Cayley, Capelli: Cayley–Capelli operator
Celine's polynomial
Charlier polynomial
Pafnuty Chebyshev: Chebyshev polynomials
Elwin Bruno Christoffel, Darboux: Christoffel–Darboux relation
Cyclotomic polynomials
D
H. G. Dawson: Dawson function
Charles F. Dunkl: Dunkl operator, Jacobi–Dunkl operator, Dunkl–Cherednik operator
Dickman–de Bruijn function
E
Engel: Engel expansion
Erdélyi Artúr: Erdelyi–Kober operator
Leonhard Euler: Euler polynomial, Eulerian integral, Euler hypergeometric integral
F
V. N. Faddeeva: Faddeeva function (also known as the complex error function; see error function)
G
C. F. Gauss: Gaussian polynomial, Gaussian distribution, etc.
Leopold Bernhard Gegenbauer: Gegenbauer polynomials
Gottlieb polynomial
Gould polynomial
Christoph Gudermann: Gudermannian function
H
Wolfgang Hahn: Hahn polynomial, (with H. Exton) Hahn–Exton Bessel function
Philip Hall: Hall polynomial, Hall–Littlewood polynomial
Hermann Hankel: Hankel function
Heine: Heine functions
Charles Hermite: Hermite polynomials
Karl L. W. M. Heun (1859 – 1929): Heun's equation
J. Horn: Horn hypergeometric series
Adolf Hurwitz: Hurwitz zeta-function
J
Henry Jack (1917–1978) Dundee: Jack polynomial
F. H. Jackson: Jackson derivative Jackson integral
Carl Gustav Jakob Jacobi: Jacobi polynomial, Jacobi theta function
K
Joseph Marie Kampe de Feriet (1893–1982): Kampe de Feriet hypergeometric series
David Kazhdan, George Lusztig: Kazhdan–Lusztig polynomial
Lord Kelvin: Kelvin function
Kibble–Slepian formula
Kirchhoff: Kirchhoff polynomial
Tom H. Koornwinder: Koornwinder polynomial
Kostka polynomial, Kostka–Foulkes polynomial
Mikhail Kravchuk: Kravchuk polynomial
L
E
|
https://en.wikipedia.org/wiki/Whittaker%20function
|
In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by to make the formulas involving the solutions more symmetric. More generally, introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R).
Whittaker's equation is
It has a regular singular point at 0 and an irregular singular point at ∞.
Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by
The Whittaker function is the same as those with opposite values of , in other words considered as a function of at fixed and it is even functions. When and are real, the functions give real values for real and imaginary values of . These functions of play a role in so-called Kummer spaces.
Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models.
References
.
.
.
.
Further reading
Special hypergeometric functions
E. T. Whittaker
Special functions
|
https://en.wikipedia.org/wiki/Spatial%20frequency
|
In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components (as determined by the Fourier transform) of the structure repeat per unit of distance.
The SI unit of spatial frequency is the reciprocal metre (m-1), although cycles per meter (c/m) is also common. In image-processing applications, spatial frequency is often expressed in units of cycles per millimeter (c/mm) or also line pairs per millimeter (LP/mm).
In wave propagation, the spatial frequency is also known as wavenumber. Ordinary wavenumber is defined as the reciprocal of wavelength and is commonly denoted by or sometimes :
Angular wavenumber , expressed in radian per metre (rad/m), is related to ordinary wavenumber and wavelength by
Visual perception
In the study of visual perception, sinusoidal gratings are frequently used to probe the capabilities of the visual system, such as contrast sensitivity. In these stimuli, spatial frequency is expressed as the number of cycles per degree of visual angle. Sine-wave gratings also differ from one another in amplitude (the magnitude of difference in intensity between light and dark stripes), orientation, and phase.
Spatial-frequency theory
The spatial-frequency theory refers to the theory that the visual cortex operates on a code of spatial frequency, not on the code of straight edges and lines hypothesised by Hubel and Wiesel on the basis of early experiments on V1 neurons in the cat. In support of this theory is the experimental observation that the visual cortex neurons respond even more robustly to sine-wave gratings that are placed at specific angles in their receptive fields than they do to edges or bars. Most neurons in the primary visual cortex respond best when a sine-wave grating of a particular frequency is presented at a particular angle in a particular location in the visual field. (However, as noted by Teller (1984), it is probably not wise to treat the highest firing rate of a particular neuron as having a special significance with respect to its role in the perception of a particular stimulus, given that the neural code is known to be linked to relative firing rates. For example, in color coding by the three cones in the human retina, there is no special significance to the cone that is firing most strongly – what matters is the relative rate of firing of all three simultaneously. Teller (1984) similarly noted that a strong firing rate in response to a particular stimulus should not be interpreted as indicating that the neuron is somehow specialized for that stimulus, since there is an unlimited equivalence class of stimuli capable of producing similar firing rates.)
The spatial-frequency theory of vision is based on two physical principles:
Any visual stimulus can be represented by plotting the intensity of the light along lines running through it.
Any curve can be
|
https://en.wikipedia.org/wiki/Point%20groups%20in%20two%20dimensions
|
In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries (isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its elements are rotations and reflections, and every such group containing only rotations is a subgroup of the special orthogonal group SO(2), including SO(2) itself. That group is isomorphic to R/Z and the first unitary group, U(1), a group also known as the circle group.
The two-dimensional point groups are important as a basis for the axial three-dimensional point groups, with the addition of reflections in the axial coordinate. They are also important in symmetries of organisms, like starfish and jellyfish, and organism parts, like flowers.
Discrete groups
There are two families of discrete two-dimensional point groups, and they are specified with parameter n, which is the order of the group of the rotations in the group.
Intl refers to Hermann–Mauguin notation or international notation, often used in crystallography. In the infinite limit, these groups become the one-dimensional line groups.
If a group is a symmetry of a two-dimensional lattice or grid, then the crystallographic restriction theorem restricts the value of n to 1, 2, 3, 4, and 6 for both families. There are thus 10 two-dimensional crystallographic point groups:
C1, C2, C3, C4, C6,
D1, D2, D3, D4, D6
The groups may be constructed as follows:
Cn. Generated by an element also called Cn, which corresponds to a rotation by angle 2π/n. Its elements are E (the identity), Cn, Cn2, ..., Cnn−1, corresponding to rotation angles 0, 2π/n, 4π/n, ..., 2(n − 1)π/n.
Dn. Generated by element Cn and reflection σ. Its elements are the elements of group Cn, with elements σ, Cnσ, Cn2σ, ..., Cnn−1σ added. These additional ones correspond to reflections across lines with orientation angles 0, π/n, 2π/n, ..., (n − 1)π/n. Dn is thus a semidirect product of Cn and the group (E,σ).
All of these groups have distinct abstract groups, except for C2 and D1, which share abstract group Z2. All of the cyclic groups are abelian or commutative, but only two of the dihedral groups are: D1 ~ Z2 and D2 ~ Z2×Z2. In fact, D3 is the smallest nonabelian group.
For even n, the Hermann–Mauguin symbol nm is an abbreviation for the full symbol nmm, as explained below. The n in the H-M symbol denotes n-fold rotations, while the m denotes reflection or mirror planes.
More general groups
These groups are readily constructed with two-dimensional orthogonal matrices.
The continuous cyclic group SO(2) or C∞ and its subgroups have elements that are rotation matrices:
where SO(2) has any possible θ. Not surprisingly, SO(2) and its subgroups are all abelian; addition of rotation angles commutes.
For discrete cyclic groups Cn, elements Cnk = R(2πk/n)
The continuous dihedral group O(2) or D∞ and its subgroups with reflections have elements that include not only rotation matrices,
|
https://en.wikipedia.org/wiki/Berezinian
|
In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold.
Definition
The Berezinian is uniquely determined by two defining properties:
where str(X) denotes the supertrace of X. Unlike the classical determinant, the Berezinian is defined only for invertible supermatrices.
The simplest case to consider is the Berezinian of a supermatrix with entries in a field K. Such supermatrices represent linear transformations of a super vector space over K. A particular even supermatrix is a block matrix of the form
Such a matrix is invertible if and only if both A and D are invertible matrices over K. The Berezinian of X is given by
For a motivation of the negative exponent see the substitution formula in the odd case.
More generally, consider matrices with entries in a supercommutative algebra R. An even supermatrix is then of the form
where A and D have even entries and B and C have odd entries. Such a matrix is invertible if and only if both A and D are invertible in the commutative ring R0 (the even subalgebra of R). In this case the Berezinian is given by
or, equivalently, by
These formulas are well-defined since we are only taking determinants of matrices whose entries are in the commutative ring R0. The matrix
is known as the Schur complement of A relative to
An odd matrix X can only be invertible if the number of even dimensions equals the number of odd dimensions. In this case, invertibility of X is equivalent to the invertibility of JX, where
Then the Berezinian of X is defined as
Properties
The Berezinian of is always a unit in the ring R0.
where denotes the supertranspose of .
Berezinian module
The determinant of an endomorphism of a free module M can be defined as the induced action on the 1-dimensional highest exterior power of M. In the supersymmetric case there is no highest exterior power, but there is a still a similar definition of the Berezinian as follows.
Suppose that M is a free module of dimension (p,q) over R. Let A be the (super)symmetric algebra S*(M*) of the dual M* of M. Then an automorphism of M acts on the ext module
(which has dimension (1,0) if q is even and dimension (0,1) if q is odd))
as multiplication by the Berezinian.
See also
Berezin integration
References
Super linear algebra
Determinants
|
https://en.wikipedia.org/wiki/Harvey%20Friedman
|
Harvey Friedman (born 23 September 1948) is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary. In recent years, this has advanced to a study of Boolean relation theory, which attempts to justify large cardinal axioms by demonstrating their necessity for deriving certain propositions considered "concrete".
Friedman earned his Ph.D. from the Massachusetts Institute of Technology in 1967, with a dissertation on Subsystems of Analysis. His advisor was Gerald Sacks. Friedman received the Alan T. Waterman Award in 1984. He also assumed the title of Visiting Scientist at IBM.<ref>Barwise et al., Harvey Friedman's Research on the Foundations of Mathematics p.xiii. Studies in Logic and the Foundations of Mathematics, vol. 117, North-Holland Amsterdam</ref> He delivered the Tarski Lectures in 2007.
In 1967, Friedman was listed in the Guinness Book of World Records for being the world's youngest professor when he taught at Stanford University at age 18 as an assistant professor of philosophy.Dr. Harvey Martin Friedman - Distinctions He has also been a professor of mathematics and a professor of music. He officially retired in July 2012. In September 2013, he received an honorary doctorate from Ghent University.
Jordana Cepelewicz (2017) profiled Friedman in Nautilus as "The Man Who Wants to Rescue Infinity".
Friedman made headlines in the Italian newspaper La Repubblica for his manuscript A Divine Consistency Proof for Mathematics, which shows in detail how, starting from the hypothesis of the existence of God (in the sense of Gödel's ontological proof), it can be shown that mathematics, as formalized by the usual ZFC axioms, is consistent.
He invented and proved important theorems regarding the finite promise games and greedy clique sequences, and Friedman's grand conjecture bears his name.
Friedman is the brother of mathematician Sy Friedman.
See also
Friedman translation
References
Further reading
L. A. Harrington et al.'', eds., Harvey Friedman's research in the foundations of mathematics, Studies in Logic and the Foundations of Mathematics 117, Amsterdam, North-Holland Publishing Company (1985)
External links
Harvey Friedman's homepage at the Ohio State University
20th-century American mathematicians
21st-century American mathematicians
American logicians
Set theorists
Ohio State University faculty
1948 births
Living people
|
https://en.wikipedia.org/wiki/Torsion%20of%20a%20curve
|
In the differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.
Definition
Let be a space curve parametrized by arc length and with the unit tangent vector . If the curvature of at a certain point is not zero then the principal normal vector and the binormal vector at that point are the unit vectors
respectively, where the prime denotes the derivative of the vector with respect to the parameter . The torsion measures the speed of rotation of the binormal vector at the given point. It is found from the equation
which means
As , this is equivalent to .
Remark: The derivative of the binormal vector is perpendicular to both the binormal and the tangent, hence it has to be proportional to the principal normal vector. The negative sign is simply a matter of convention: it is a byproduct of the historical development of the subject.
Geometric relevance: The torsion measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
Properties
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane.
The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and non-zero is a helix. The torsion is positive for a right-handed helix and is negative for a left-handed one.
Alternative description
Let be the parametric equation of a space curve. Assume that this is a regular parametrization and that the curvature of the curve does not vanish. Analytically, is a three times differentiable function of with values in and the vectors
are linearly independent.
Then the torsion can be computed from the following formula:
Here the primes denote the derivatives with respect to and the cross denotes the cross product. For , the formula in components is
Notes
References
Differential geometry
Curves
Curvature (mathematics)
ru:Дифференциальная геометрия кривых#Кручение
|
https://en.wikipedia.org/wiki/Torsion%20%28algebra%29
|
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element.
This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements.
This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules).
In the case of groups that are noncommutative, a torsion element is an element of finite order. Contrary to the commutative case, the torsion elements do not form a subgroup, in general.
Definition
An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e.,
In an integral domain (a commutative ring without zero divisors), every non-zero element is regular, so a torsion element of a module over an integral domain is one annihilated by a non-zero element of the integral domain. Some authors use this as the definition of a torsion element, but this definition does not work well over more general rings.
A module M over a ring R is called a torsion module if all its elements are torsion elements, and torsion-free if zero is the only torsion element. If the ring R is commutative then the set of all torsion elements forms a submodule of M, called the torsion submodule of M, sometimes denoted T(M). If R is not commutative, T(M) may or may not be a submodule. It is shown in that R is a right Ore ring if and only if T(M) is a submodule of M for all right R-modules. Since right Noetherian domains are Ore, this covers the case when R is a right Noetherian domain (which might not be commutative).
More generally, let M be a module over a ring R and S be a multiplicatively closed subset of R. An element m of M is called an S-torsion element if there exists an element s in S such that s annihilates m, i.e., In particular, one can take for S the set of regular elements of the ring R and recover the definition above.
An element g of a group G is called a torsion element of the group if it has finite order, i.e., if there is a positive integer m such that gm = e, where e denotes the identity element of the group, and gm denotes the product of m copies of g. A group is called a torsion (or periodic) group if all its elements are torsion elements, and a if its only torsion element is the identity element. Any abelian group may be viewed as a module
|
https://en.wikipedia.org/wiki/List%20of%20topology%20topics
|
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling and bending, but not tearing or gluing.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the and . Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.
This is a list of topology topics. See also:
Topology glossary
List of topologies
List of general topology topics
List of geometric topology topics
List of algebraic topology topics
List of topological invariants (topological properties)
Publications in topology
Topology and physics
Quantum topology
Topological defect
Topological entropy in physics
Topological order
Topological quantum field theory
Topological quantum number
Topological string theory
Topology of the universe
Topology and dynamical systems
Milnor–Thurston kneading theory
Topological conjugacy
Topological dynamics
Topological entropy
Topological mixing
Topology and computing
Computational topology
Digital topology
Network topology
Topological computing
Topological Quantum Computing
Topological quantum computer
Miscellaneous
Combinatorial topology
Counterexamples in Topology
Differential topology
Geometric topology
Geospatial topology
Grothendieck topology
Link (knot theory)
Listing number
Mereotopology
Noncommutative topology
Pointless topology
Set-theoretic topology
Topological combinatorics
Topological data analysis
Topological degree theory
Topological game
Topological graph theory
Topological K-theory
Topological modular forms
Topological skeleton
Topology optimization
Water, gas, and electricity
Topology topics
|
https://en.wikipedia.org/wiki/Douglas%20Wiens
|
Douglas Paul Wiens is a Canadian statistician; he is a professor in the Department of Mathematical and Statistical Sciences at the University of Alberta.
Wiens earned a B.Sc. in mathematics (1972), two master's degrees in mathematical logic (1974) and statistics (1979), and a Ph.D. in statistics (1982), all from the University of Calgary. As part of his work on mathematical logic, in connection with Hilbert's tenth problem, Wiens helped find a diophantine formula for the primes: that is, multivariate polynomial with the property that the positive values of this polynomial, over integer arguments, are exactly the prime numbers. Wiens and his co-authors won the Lester R. Ford award of the Mathematical Association of America in 1977 for their paper describing this result. His Ph.D. dissertation was entitled Robust Estimation for Multivariate Location and Scale in the Presence of Asymmetry and was supervised by John R. Collins. After receiving his Ph.D. in 1982, Wiens took a faculty position at Dalhousie University, and moved in 1987 to Alberta.
Wiens was editor-in-chief of The Canadian Journal of Statistics from 2004 to 2006 and program chair of the 2003 annual meeting of the Statistical Society of Canada. Along with the Ford award, Wiens received The Canadian Journal of Statistics Award in 1990 for his paper "Minimax-variance L- and R-estimators of location". In 2005 he was elected as a Fellow of the American Statistical Association.
References
External links
Home page at the University of Alberta
Canadian statisticians
Year of birth missing (living people)
Living people
University of Calgary alumni
Academic staff of the Dalhousie University
Academic staff of the University of Alberta
Fellows of the American Statistical Association
Place of birth missing (living people)
|
https://en.wikipedia.org/wiki/Daihachiro%20Sato
|
was a Japanese mathematician who was awarded the Lester R. Ford Award in 1976 for his work in number theory, specifically on his work in the Diophantine representation of prime numbers. His doctoral supervisor at the University of California, Los Angeles was Ernst G. Straus.
Biography
Sato was an only child born in Fujinomiya, Shizuoka, Japan on June 1, 1932. While still attending high school, Sato published his first mathematics research paper, which led to his acceptance at the Tokyo University of Education. There, Sato earned a B.S. in theoretical physics, a popular academic field at the time due to the recent Nobel Prize in Physics awarded in 1949 to Hideki Yukawa. Later, in 1965, Shin'ichirō Tomonaga, one of Dr. Sato's professors at this university, was also awarded a Nobel Prize in Physics.
Following his undergraduate degree in Japan, he switched his studies to mathematics, earning a M.Sc. and a Ph.D. from UCLA, and eventually became tenured at the University of Saskatchewan, Regina campus in Regina, Saskatchewan, Canada. Following his retirement in 1997 he was granted the position Professor Emeritus at the University of Regina which is what the Regina campus became in 1974. Subsequently, he further taught at the Tokyo University of Social Welfare from 2000 until 2006, after which he returned to Canada. He died at Ladner, British Columbia on May 28, 2008.
Sato's interests included integer valued entire functions, generalized interpolation by analytic functions, prime representing functions, and function theory. It is in the field of prime representing functions that Sato co-authored a paper with James P. Jones, Hideo Wada, and Douglas Wiens entitled "Diophantine Representation of the Set of Prime Numbers", which won them the Lester R. Ford Award in Mathematics in 1976.
Publications
—Dissertation: Ph.D.
— MathSciNet review: 0409325
References
1932 births
2008 deaths
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Number theorists
People from Fujinomiya, Shizuoka
University of California, Los Angeles alumni
|
https://en.wikipedia.org/wiki/International%20Mathematics%20Competition
|
The International Mathematics Competition (IMC) for University Students is an annual mathematics competition open to all undergraduate students of mathematics. Participating students are expected to be at most twenty three years of age at the time of the IMC. The IMC is primarily a competition for individuals, although most participating universities select and send one or more teams of students. The working language is English.
The IMC is a residential competition and all student participants are required to stay in the accommodation provided by the organisers. It aims to provide a friendly, comfortable and secure environment for university mathematics students to enjoy mathematics with their peers from all around the world, to broaden their world perspective and to be inspired to set mathematical goals for themselves that might not have been previously imaginable or thought possible. Notably, in 2018 Caucher Birkar (born Fereydoun Derakhshani), an Iranian Kurdish mathematician, who participated in the 7th IMC held at University College London in 2000, received mathematics' most prestigious award, the Fields Medal. He is now a professor at Tsinghua University and at the University of Cambridge. In 2022 a Kyiv-born mathematician, Maryna Viazovska, was also awarded the Fields Medal. She participated in the IMC as a student four times, in 2002, 2003, 2004 and 2005. She is now a Professor and the Chair of Number Theory at the Institute of Mathematics of the École Polytechnique Fédérale de Lausanne in Switzerland.
Students from over 200 universities from over 50 countries have participated over the first thirty competitions. At the 29th IMC in 2022 participants were awarded Individual Result Prizes, Fair Play Prizes and Most Efficient Team Leader Prizes.
University College London has been involved in the organisation of the IMC and Professor John E. Jayne has served as the President from the beginning in 1994. The IMC runs over five or six days during which the competitors sit two five-hour examinations, each with five questions (six until 2008) chosen by a panel and representatives from the participating universities. Problems are from the fields of Algebra, Analysis (Real and Complex), Combinatorics and Geometry.
History
The IMC began in 1994 in Plovdiv, Bulgaria, with 49 participants, mostly from Bulgaria, and was hosted by Plovdiv University "Paisii Hilendarski". The 2nd, 3rd and 4th IMC were also hosted by Plovdiv University "Paisii Hilendarski" in Plovdiv. From 1996 to 1999 the IMC was one of the activities of the Structural Joint European Union TEMPUS Project #S_JEP-11087-96, entitled "Modular Education in Mathematics and Informatics", which was the flag ship European Union TEMPUS Project in Bulgaria at the time, aimed at bringing Bulgaria's university mathematics and computing degree programs into line with those in the European Union in preparation for Bulgaria's entry into the European Union. University College London was the Contracto
|
https://en.wikipedia.org/wiki/Moduli%20scheme
|
In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept (algebraic spaces, algebraic stacks of Michael Artin).
History
Work of Grothendieck and David Mumford (see geometric invariant theory) opened up this area in the early 1960s. The more algebraic and abstract approach to moduli problems is to set them up as a representable functor question, then apply a criterion that singles out the representable functors for schemes. When this programmatic approach works, the result is a fine moduli scheme. Under the influence of more geometric ideas, it suffices to find a scheme that gives the correct geometric points. This is more like the classical idea that the moduli problem is to express the algebraic structure naturally coming with a set (say of isomorphism classes of elliptic curves).
The result is then a coarse moduli scheme. Its lack of refinement is, roughly speaking, that it doesn't guarantee for families of objects what is inherent in the fine moduli scheme. As Mumford pointed out in his book Geometric Invariant Theory, one might want to have the fine version, but there is a technical issue (level structure and other 'markings') that must be addressed to get a question with a chance of having such an answer.
Teruhisa Matsusaka proved a result, now known as Matsusaka's big theorem, establishing a necessary condition on a moduli problem for the existence of a coarse moduli scheme.
Examples
Mumford proved that if g > 1, there exists a coarse moduli scheme of smooth curves of genus g, which is quasi-projective. According to a recent survey by János Kollár, it "has a rich and intriguing intrinsic geometry which is related to major questions in many branches of mathematics and theoretical physics." Braungardt has posed the question whether Belyi's theorem can be generalised to varieties of higher dimension over the field of algebraic numbers, with the formulation that they are generally birational to a finite étale covering of a moduli space of curves.
Using the notion of stable vector bundle, coarse moduli schemes for the vector bundles on any smooth complex variety have been shown to exist, and to be quasi-projective: the statement uses the concept of semistability. It is possible to identify the coarse moduli space of special instanton bundles, in mathematical physics, with objects in the classical geometry of conics, in certain cases.
References
Notes
Moduli theory
Representable functors
|
https://en.wikipedia.org/wiki/Generic%20point
|
In algebraic geometry, a generic point P of an algebraic variety X is a point in a general position, at which all generic properties are true, a generic property being a property which is true for almost every point.
In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension d is a point such that the field generated by its coordinates has transcendence degree d over the field generated by the coefficients of the equations of the variety.
In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space X is a point whose closure is X.
Definition and motivation
A generic point of the topological space X is a point P whose closure is all of X, that is, a point that is dense in X.
The terminology arises from the case of the Zariski topology on the set of subvarieties of an algebraic set: the algebraic set is irreducible (that is, it is not the union of two proper algebraic subsets) if and only if the topological space of the subvarieties has a generic point.
Examples
The only Hausdorff space that has a generic point is the singleton set.
Any integral scheme has a (unique) generic point; in the case of an affine integral scheme (i.e., the prime spectrum of an integral domain) the generic point is the point associated to the prime ideal (0).
History
In the foundational approach of André Weil, developed in his Foundations of Algebraic Geometry, generic points played an important role, but were handled in a different manner. For an algebraic variety V over a field K, generic points of V were a whole class of points of V taking values in a universal domain Ω, an algebraically closed field containing K but also an infinite supply of fresh indeterminates. This approach worked, without any need to deal directly with the topology of V (K-Zariski topology, that is), because the specializations could all be discussed at the field level (as in the valuation theory approach to algebraic geometry, popular in the 1930s).
This was at a cost of there being a huge collection of equally generic points. Oscar Zariski, a colleague of Weil's at São Paulo just after World War II, always insisted that generic points should be unique. (This can be put back into topologists' terms: Weil's idea fails to give a Kolmogorov space and Zariski thinks in terms of the Kolmogorov quotient.)
In the rapid foundational changes of the 1950s Weil's approach became obsolete. In scheme theory, though, from 1957, generic points returned: this time à la Zariski. For example for R a discrete valuation ring, Spec(R) consists of two points, a generic point (coming from the prime ideal {0}) and a closed point or special point coming from the unique maximal ideal. For morphisms to Spec(R), the fiber above the special point is
|
https://en.wikipedia.org/wiki/Half%20range%20Fourier%20series
|
In mathematics, a half range Fourier series is a Fourier series defined on an interval instead of the more common , with the implication that the analyzed function should be extended to as either an even (f(-x)=f(x)) or odd function (f(-x)=-f(x)). This allows the expansion of the function in a series solely of sines (odd) or cosines (even). The choice between odd and even is typically motivated by boundary conditions associated with a differential equation satisfied by .
Example
Calculate the half range Fourier sine series for the function where .
Since we are calculating a sine series,
Now,
When n is odd,
When n is even,
thus
With the special case , hence the required Fourier sine series is
Fourier series
|
https://en.wikipedia.org/wiki/Puma%20language
|
Puma (Puma: पुमा Pumā) is a Kiranti language spoken by about 4,310 people (Central Bureau of Statistics report 2001) in Sagarmatha Zone, Nepal. The actual population may be somewhat higher. The same term ‘Puma’ refers both to the people and the language they speak [Sharma 2014].
The Himalayan Languages Project has produced the first grammatical sketch of Puma. Like other Kiranti languages, Puma has a maximum syllable form of (C) (G) V (C) (C) for open syllables and (N) C V C for closed syllables, where ‘G’ is a glide and ‘N’ is a nasal (Sharma 2014:92].
Locations
Puma is spoken in Diplung, Mauwabote, Devisthan, Pauwasera, and Chisapani VDC's of southern Khotang District, and in Beltar and Saunechour VDC's of Udayapur District, Nepal. It is also spoken in Ruwa Khola valley to Buwa Khola, and southward across the Dudh Koshi.
Education
Puma language is taught in 14 schools of Barahapokhari and Jantedhunga Rural Municipalities of Khotang district in Nepal since 2020.
References
Bibliography
Sharma, Narayan Prasad. "Morphosyntax of Puma, a Tibeto-Burman language of Nepal." PhD diss., SOAS, University of London, 2014. https://eprints.soas.ac.uk/18554/1/Sharma_3615.pdf
External links
The Chintang and Puma Documentation Project (DoBeS)
ELAR archive of Documentation of Puma
Kiranti languages
Languages of Koshi Province
Languages written in Devanagari
|
https://en.wikipedia.org/wiki/Gibbs%20state
|
In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system. For example, a stationary or steady-state distribution of a Markov chain, such as that achieved by running a Markov chain Monte Carlo iteration for a sufficiently long time, is a Gibbs state.
Precisely, suppose is a generator of evolutions for an initial state , so that the state at any later time is given by . Then the condition for to be a Gibbs state is
.
In physics there may be several physically distinct Gibbs states in which a system may be trapped, particularly at lower temperatures.
They are named after Josiah Willard Gibbs, for his work in determining equilibrium properties of statistical ensembles. Gibbs himself referred to this type of statistical ensemble as being in "statistical equilibrium".
See also
Gibbs algorithm
Gibbs measure
KMS state
References
Statistical mechanics
Stochastic processes
|
https://en.wikipedia.org/wiki/Gibbs%20algorithm
|
In statistical mechanics, the Gibbs algorithm, introduced by J. Willard Gibbs in 1902, is a criterion for choosing a probability distribution for the statistical ensemble of microstates of a thermodynamic system by minimizing the average log probability
subject to the probability distribution satisfying a set of constraints (usually expectation values) corresponding to the known macroscopic quantities. in 1948, Claude Shannon interpreted the negative of this quantity, which he called information entropy, as a measure of the uncertainty in a probability distribution. In 1957, E.T. Jaynes realized that this quantity could be interpreted as missing information about anything, and generalized the Gibbs algorithm to non-equilibrium systems with the principle of maximum entropy and maximum entropy thermodynamics.
Physicists call the result of applying the Gibbs algorithm the Gibbs distribution for the given constraints, most notably Gibbs's grand canonical ensemble for open systems when the average energy and the average number of particles are given. (See also partition function).
This general result of the Gibbs algorithm is then a maximum entropy probability distribution. Statisticians identify such distributions as belonging to exponential families.
References
Statistical mechanics
Particle statistics
Entropy and information
|
https://en.wikipedia.org/wiki/Moishezon%20manifold
|
In mathematics, a Moishezon manifold is a compact complex manifold such that the field of meromorphic functions on each component has transcendence degree equal the complex dimension of the component:
Complex algebraic varieties have this property, but the converse is not true: Hironaka's example gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or scheme. showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric. showed that any Moishezon manifold carries an algebraic space structure; more precisely, the category of Moishezon spaces (similar to Moishezon manifolds, but are allowed to have singularities) is equivalent with the category of algebraic spaces that are proper over .
References
Algebraic geometry
Analytic geometry
|
https://en.wikipedia.org/wiki/Kelly%20criterion
|
In probability theory, the Kelly criterion (or Kelly strategy or Kelly bet) is a formula for sizing a bet. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth, which is equivalent to maximizing the expected geometric growth rate. It assumes that the expected returns are known and is optimal for a bettor who values their wealth logarithmically. J. L. Kelly Jr, a researcher at Bell Labs, described the criterion in 1956. Under the stated assumptions, the Kelly criterion leads to higher wealth than any other strategy in the long run (i.e., the theoretical maximum return as the number of bets goes to infinity).
The practical use of the formula has been demonstrated for gambling, and the same idea was used to explain diversification in investment management. In the 2000s, Kelly-style analysis became a part of mainstream investment theory and the claim has been made that well-known successful investors including Warren Buffett and Bill Gross use Kelly methods. Also see Intertemporal portfolio choice.
Gambling formula
Where losing the bet involves losing the entire wager, the Kelly bet is:
where:
is the fraction of the current bankroll to wager.
is the probability of a win.
is the probability of a loss ().
is the proportion of the bet gained with a win. E.g., if betting $10 on a 2-to-1 odds bet (upon win you are returned $30, winning you $20), then .
As an example, if a gamble has a 60% chance of winning (, ), and the gambler receives 1-to-1 odds on a winning bet (), then to maximize the long-run growth rate of the bankroll, the gambler should bet 20% of the bankroll at each opportunity ().
If the gambler has zero edge, i.e., if , then the criterion recommends for the gambler to bet nothing.
If the edge is negative () the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in American roulette, the bettor is offered an even money payoff () on red, when there are 18 red numbers and 20 non-red numbers on the wheel (). The Kelly bet is , meaning the gambler should bet one-nineteenth of their bankroll that red will not come up. There is no explicit anti-red bet offered with comparable odds in roulette, so the best a Kelly gambler can do is bet nothing.
Investment formula
A more general form of the Kelly formula allows for partial losses, which is relevant for investments:
where:
is the fraction of the assets to apply to the security.
is the probability that the investment increases in value.
is the probability that the investment decreases in value ().
is the fraction that is lost in a negative outcome. If the security price falls 10%, then
is the fraction that is gained in a positive outcome. If the security price rises 10%, then .
Note that the Kelly criterion is valid only for known outcome probabilities, which is not the case with investments. Risk averse investors should not invest the full Kelly fraction.
The general form c
|
https://en.wikipedia.org/wiki/Divergence%20%28disambiguation%29
|
Divergence is a mathematical function that associates a scalar with every point of a vector field.
Divergence, divergent, or variants of the word, may also refer to:
Mathematics
Divergence (computer science), a computation which does not terminate (or terminates in an exceptional state)
Divergence, the defining property of divergent series; series that do not converge to a finite limit
Divergence, a result of instability of a dynamical system in stability theory
Statistics
Divergence (statistics), a measure of dissimilarity between probability measures
Bregman divergence
f-divergence
Jensen–Shannon divergence
Kullback–Leibler divergence, also known as the "information divergence" in probability theory and information theory
Rényi's divergence
Science
Divergence (eye), the simultaneous outward movement of both eyes away from each other
Divergence (optics), the angle formed between spreading rays of light
Beam divergence, the half-angle of the cone formed by a beam of light as it propagates and spreads out
Divergence problem, an anomaly between the instrumental record and temperatures calculated using some tree ring proxies
Divergent boundary, a linear feature that exists between tectonic plates that are moving away from each other
Evolutionary divergence, the accumulation of differences between populations of closely related species
Genetic divergence, the process in which two or more populations of an ancestral species accumulate independent genetic changes
Infrared divergence, due to contributions of objects with low energy
Ultraviolet divergence, due to contributions of objects with very high energy
Arts
Multimedia
Divergent (book series), a series of young adult science fiction adventure novels by Veronica Roth, published between 2011 and 2018
Divergent (novel), the first novel in the series, 2011
The Divergent Series, a film trilogy based on the novels, released 2014–16
Divergent (film), the 2014 film based on the novel
Divergent (soundtrack), the soundtrack to the 2014 film
Books
Divergence (novel), a 1991 novel by Charles Sheffield
Divergence, a 2007 novel by Tony Ballantyne
Film
Diverge (film), a 2016 American sci-fi film
Divergence (film), a 2005 film from Hong Kong
Television
Diverged (The Walking Dead), an episode of the television series The Walking Dead
"Divergence" (Star Trek: Enterprise), a fourth-season episode of Star Trek: Enterprise
Music
Divergence (album), a 1972 album by Solution
See also
Convergence (disambiguation)
|
https://en.wikipedia.org/wiki/Fredholm%20alternative
|
In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
Linear algebra
If V is an n-dimensional vector space and is a linear transformation, then exactly one of the following holds:
For each vector v in V there is a vector u in V so that . In other words: T is surjective (and so also bijective, since V is finite-dimensional).
A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:
Either: A x = b has a solution x
Or: AT y = 0 has a solution y with yTb ≠ 0.
In other words, A x = b has a solution if and only if for any y such that AT y = 0, it follows that yTb = 0 .
Integral equations
Let be an integral kernel, and consider the homogeneous equation, the Fredholm integral equation,
and the inhomogeneous equation
The Fredholm alternative is the statement that, for every non-zero fixed complex number , either the first equation has a non-trivial solution, or the second equation has a solution for all .
A sufficient condition for this statement to be true is for to be square integrable on the rectangle (where a and/or b may be minus or plus infinity). The integral operator defined by such a K is called a Hilbert–Schmidt integral operator.
Functional analysis
Results about Fredholm operators generalize these results to complete normed vector spaces of infinite dimensions; that is, Banach spaces.
The integral equation can be reformulated in terms of operator notation as follows. Write (somewhat informally)
to mean
with the Dirac delta function, considered as a distribution, or generalized function, in two variables.
Then by convolution, induces a linear operator acting on a Banach space of functions
given by
with given by
In this language, the Fredholm alternative for integral equations is seen to be analogous to the Fredholm alternative for finite-dimensional linear algebra.
The operator given by convolution with an kernel, as above, is known as a Hilbert–Schmidt integral operator.
Such operators are always compact. More generally, the Fredholm alternative is valid when is any compact operator. The Fredholm alternative may be restated in the following form: a nonzero either is an eigenvalue of or lies in the domain of the resolvent
Elliptic partial differential equations
The Fredholm alternative can be applied to solving linear elliptic boundary value problems. The basic result is: if the equation and the appropriate Banach spaces have been set up correctly, then either
(1) The homogeneous equation has a nontrivial solution, or
(2) The inhomogeneous equation can be solved uniquely for
|
https://en.wikipedia.org/wiki/Urdaneta%20Municipality%2C%20Miranda
|
Urdaneta is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2016 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 167,768. The town of Cúa is the municipal seat of the Urdaneta Municipality. The municipality is one of several in Venezuela named "Urdaneta Municipality" in honour of Venezuelan independence hero Rafael Urdaneta.
History
The first establishments of Cúa dates from the pre-Columbian period, being the first founders the Quiriquires natives, has like nickname The Tuy Pearl, because its location at the borders of the Tuy River. After the officially foundation on October 6, 1690 by fray Manuel de Alesson, under the invocation of Our Lady of the Rosary of Cúa, the first inhabitants came from different regions motivated by the agriculture, due to the fertility and strategic location of the valley. However, this town was founded initially at the site known like Marín in 1633, this first village was destroyed in its totality by a violent earthquake that affected a great part of the Tuy Valleys.
The name of Cúa, according to some historians, has it origin from the Cumanagotos natives, from the Carib language that means Crab, meaning that Cúa is the place where the crab abounds. Others affirm that its name is associate with Apacuana, a brave native woman from the region, that fought against the Spaniards. Others think that that name was giving in honor to the native Cue, ally of the Spaniards, that helped in the foundation and consolidation of the town.
Demographics
The Urdaneta Municipality, according to a 2016 population estimate by the National Institute of Statistics of Venezuela, has a population of 167,768 (up from 114,221 in 2000). This amounts to 4.6% of the state's population. The municipality's population density is .
Government
The mayor of the Urdaneta Municipality is Jorge H. Castro, re-elected on October 31, 2004 with 42% of the vote. The municipality is divided into Cúa parishes; Cúa and Nueva Cúa.
Transportation
Buses are the main means of mass transportation, operated by several companies on normal streets and avenues:
bus; large buses.
buseta; medium size buses.
microbus or colectivo; vans or minivans.
IAFE; train services to and from Caracas and Charallave.
Main avenues
Perimetral avenue
Monseñor Pellín avenue
José María Carreño street
El Rosario street
El Carmen street
Lecumberry street
San Rafael street
Juan España street
Zamora street
Notable natives
José María Carreño
Evencio Castellanos
María Teresa Castillo
Baudilio Díaz
Pancho Prin
Victor Guillermo Ramos Rangel
Cristóbal Rojas
Ezequiel Zamora
References
External links
urdaneta-miranda.gob.ve
Municipalities of Miranda (state)
|
https://en.wikipedia.org/wiki/Urdaneta%20Municipality%2C%20Aragua
|
Urdaneta is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 21,271. The town of Barbacoas is the shire town of the Urdaneta Municipality.
Name
The municipality is one of several in Venezuela named "Urdaneta Municipality" in honour of Venezuelan independence hero Rafael Urdaneta.
Demographics
The Urdaneta Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 21,270 (up from 19,339 in 2000). This amounts to 1.3% of the state's population. The municipality's population density is .
Government
The mayor of the Urdaneta Municipality is Sotero González, re-elected on October 31, 2004, with 32% of the vote. The municipality is divided into four parishes; Capital Urdaneta, Las Peñitas, San Francisco de Cara, and Taguay.
References
External links
urdaneta-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Kodaira%27s%20classification
|
In mathematics, Kodaira's classification is either
The Enriques–Kodaira classification, a classification of complex surfaces, or
Kodaira's classification of singular fibers, which classifies the possible fibers of an elliptic fibration.
|
https://en.wikipedia.org/wiki/Bol%C3%ADvar%20Municipality%2C%20Aragua
|
Bolívar Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 38,047. The town of San Mateo is the shire town of the Bolívar Municipality.
Name
The municipality is one of several in Venezuela named "Bolívar Municipality" in honour of Venezuelan independence hero Simón Bolívar.
Geography
The municipality is mountainous in the center and north, but in the south it is flat due to the depression formed by Lake Valencia. The temperature generally varies between 24.5 °C and 30 °C, while annual precipitation averages 900 mm.
Economy
Agriculture and industry are the primary sources of income for the municipality, which contains some 3% of the industry of Aragua state. In agriculture, the municipality stands out for its production of papaya and cassava, in which it ranks 2nd and 3rd, respectively, in the state.
Demographics
The Bolívar Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 42,295 (up from 39,260 in 2000). This amounts to 2.5% of the state's population. The municipality's population density is .
Government
The mayor of the Bolívar Municipality is Freddy Arenas, elected on November 23, 2008, with 61% of the vote. He replaced César Augusto Barrera Ramirez shortly after the elections. The municipality is divided into one parish; Capital Bolívar.
References
External links
bolivar-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Camatagua%20Municipality
|
The Camatagua Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 16,627. The town of Camatagua is the shire town of the Camatagua Municipality.
Demographics
The Camatagua Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 17,674 (up from 15,691 in 2000). This amounts to 1% of the state's population. The municipality's population density is .
Government
The mayor of the Camatagua Municipality is Rafael González, re-elected on November 23, 2008, with 42% of the vote. The municipality is divided into two parishes; Capital Camatagua and Carmen de Cura.
See also
Camatagua
Aragua
Municipalities of Venezuela
References
External links
camatagua-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Libertador%20Municipality%2C%20Aragua
|
The Libertador Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 114,355. The town of Palo Negro is the municipal seat of the Libertador Municipality.
Name
The municipality is one of a number in Venezuela named "Libertador Municipality", in honour of Venezuelan independence hero Simón Bolívar.
Demographics
The Libertador Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 87,520 (up from 78,427 in 2000). This amounts to 5.3% of the state's population. The municipality's population density is .
Government
The mayor of the Libertador Municipality is Gonzalo Díaz, re-elected on October 31, 2004, with 47% of the vote. The municipality is divided into two parishes; Capital Libertador and San Martín de Porres (created on January 30, 1995).
References
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Jos%C3%A9%20Rafael%20Revenga%20Municipality
|
The José Rafael Revenga Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 48,800. The town of El Consejo is the shire town of the José Rafael Revenga Municipality. The municipality is named for José Rafael Revenga.
Demographics
The José Rafael Revenga Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 49,593 (up from 43,513 in 2000). This amounts to 3% of the state's population. The municipality's population density is .
Government
The mayor of the José Rafael Revenga Municipality is Francisco Martínez, elected on November 23, 2008, with 60% of the vote. He replaced Juan Pablo Perdomo Piñero shortly after the elections. The municipality is divided into one parish; Capital José Rafael Revenga.
References
External links
joserafaelrevenga-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Jos%C3%A9%20F%C3%A9lix%20Ribas%20Municipality%2C%20Aragua
|
The José Félix Ribas Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 143,501. The town of La Victoria is the shire town of the José Félix Ribas Municipality.
History
La Victoria is famous for the independence battle of February 12, 1814, where José Félix Ribas led a young and inexperienced army that succeeded in halting the royalist troops of José Tomás Boves. Venezuela celebrates "Youth Day" every February 12 in La Victoria, with a ceremony is usually presided by the President of the Republic.
Demographics
The José Félix Ribas Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 156,139 (up from 137,581 in 2000). This amounts to 9.3% of the state's population. The municipality's population density is .
Government
The mayor of the José Félix Ribas Municipality is Juan Carlos Sánchez, elected on November 23, 2008, with 48% of the vote. He replaced Rosa León Brabo shortly after the elections. The municipality is divided into five parishes; Capital José Félix Ribas, Castor Nieves Ríos, Las Guacamayas, Pao de Zárate, and Zuata.
See also
La Victoria
Aragua
Municipalities of Venezuela
References
External links
josefelixribas-aragua.gov.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Jos%C3%A9%20Angel%20Lamas%20Municipality
|
The José Angel Lamas Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 32,981. The town of Santa Cruz is the municipal seat of the José Angel Lamas Municipality. The municipality is named for the Venezuelan composer José Ángel Lamas.
Demographics
The José Angel Lamas Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 33,513 (up from 28,263 in 2000). This amounts to 2% of the state's population. The municipality's population density is .
Government
The mayor of the José Angel Lamas Municipality is Ybis Pérez, elected on November 23, 2008, with 61% of the vote. She replaced Nancy López shortly after the elections. The municipality contains one parish; Capital José Angel Lamas.
References
External links
joseangellamas-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Girardot%20Municipality%2C%20Aragua
|
The Girardot Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua. According to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 407,109. The city of Maracay is the shire town of the Girardot Municipality.
History
The city of Maracay was officially established on March 5, 1701, by Bishop Diego de Baños y Sotomayor in the valleys of Tocopio and Tapatapa (what is known today as the central valley of Aragua) in northern Venezuela. According to the most accepted explanation, it was named after a local indigenous chief, and refers to the "Maracayo" (Felis mitis), a small tiger. Alternative etymologies cite a local aromatic tree called Mara.
Maracay experienced rapid growth during Juan Vicente Gómez's dictatorship (1908 - 1935). Gómez saw Maracay as a suitable place to make his residence during his rule, and ordered the construction of an Arc of Triumph, a bull plaza (a near replica of the one in Seville, Spain), an opera house, a zoo, and, most notably, the Hotel Jardín (Garden Hotel), a majestic, tourist attraction with very large gardens. The city is home to the Mausoleo de Gómez (Gómez's mausoleum), where the dictator's remains are stored.
Geography
The mountains on the north side of Maracay, that separate it from the coast, make up the Henri Pittier National Park, named after the Swiss naturalist that studied them. The park is a very lush rainforest, with a great variety of ferns. Two very winding roads cut through the park over the mountains to the coast. One, beginning at the North-Central part of the city known as Urbanización El Castaño, goes to the beach town of Choroní. The other, beginning at the North-Western part of the city known as Urbanización El Limón, goes to Ocumare de la Costa and the beaches of Cata and Catica.
Maracay & the Military
Maracay is a city heavily influenced by the military. Maracay is the cradle of Venezuelan aviation, and it is home to the two largest Air Force bases in the country. The Venezuelan F-16 fighter planes are stationed here, as well as the new Sukhoi-30MKEs acquired by the Venezuelan Government.
Other military facilities include the Fourth Armored Division of the Army and the Venezuelan Paratroopers main base and training center.
It is also home to the government-owned ammunition and weapons factory (CAVIM) that produces the Venezuelan version of the FN FAL (Fusil Automatique Leger - Light Automatic Rifle) rifle and will produce the newly acquired AK-103s; as well as the ammunition for both models.
Economy
One of the most important cities in Venezuela, Maracay is primarily an industrial and commercial center, the city produces paper, textiles, chemicals, tobacco, cement, cattle derived foods, such as milk or meat conserves, as well as soap and perfumes.
Even though it is an industrial center, the surroundings of Maracay live of an intensive agriculture, where sugarcane, tobacco, coff
|
https://en.wikipedia.org/wiki/San%20Casimiro%20Municipality
|
The San Casimiro Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 25,540. The town of San Casimiro is the shire town of the San Casimiro Municipality.
Demographics
The San Casimiro Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 26,030 (up from 23,212 in 2000). This amounts to 1.6% of the state's population. The municipality's population density is .
Government
The mayor of the San Casimiro Municipality is Johnny Martinez, elected on October 31, 2004, with 44% of the vote. He replaced Luis Rodriguez shortly after the elections. The municipality is divided into four parishes; Capital San Casimiro, Güiripa, Ollas de Caramacate, and Valle Morín.
See also
San Casimiro
Aragua
Municipalities of Venezuela
References
External links
sancasimiro-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/San%20Sebasti%C3%A1n%20Municipality
|
The San Sebastián Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 23,279. The town of San Sebastián is the shire town of the San Sebastián Municipality.
Demographics
The San Sebastián Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 22,906 (up from 20,096 in 2000). This amounts to 1.4% of the state's population. The municipality's population density is .
Government
The mayor of the San Sebastián Municipality is Carlos Guillermo Miranda Escobar, elected on October 31, 2004, with 43% of the vote. He replaced Enrique Barrios shortly after the elections. The municipality is divided into one parishes; Capital San Sebastián.
See also
San Sebastián
Aragua
Municipalities of Venezuela
References
External links
sansebastiandelosreyes-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Santiago%20Mari%C3%B1o%20Municipality
|
The Santiago Mariño Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 211,010. The town of Turmero is the shire town of the Santiago Mariño Municipality and Chuao where some of the finest cocoa beans in the world are produced. The municipality is named for Venezuelan independence hero Santiago Mariño.
Demographics
The Santiago Mariño Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 191,731 (up from 165,436 in 2000). This amounts to 11.5% of the state's population. The municipality's population density is .
Government
The mayor of the Santiago Mariño Municipality is Francisco J. Gerratana, elected on October 31, 2004 with 62% of the vote. He replaced Efren Rodriguez shortly after the elections. The municipality is divided into five parishes; Capital Santiago Mariño, Arévalo Aponte (separated from Capital Santiago Mariño parish effective January 30, 1995), Chuao, Samán de Güere, and Alfredo Pacheco Miranda (separated from Samán de Güere parish effective December 16, 1997).
Chuao
Chuao is a small village founded in the 16th century famous in the world for its cacao plantations. The village is surrounded by mountains and dense rainforests to the south Caribbean Sea near the Henri Pittier National Park. There is no road access and visitors must come by boat from the town of Puerto Colombia along the coast, or by foot, crossing the mountains and the luxurious cloud forest from Turmero near Maracay.
In the Chuao plantation there are currently pure Criollo and hybrid varieties of cacao being grown. Criollo beans from Chuao are of very high quality, and are considered Venezuela's finest beans together with Porcelana Blanca beans from Lake Maracaibo (another genetically pure variety of Criollo). Amedei, an Italian chocolate maker, and Chocolate NAIVE, a Lithuanian bean-to-bar chocolate maker, offer chocolate bars made with Chuao cacao. Naive was the winner of the European gold medal at the International Chocolate Awards 2014.
In November 2000, the cacao beans coming from Chuao region were awarded an appellation of origin under the title "Cacao de Chuao" (from Spanish Cacao de Chuao) effectively making this one of the most expensive and sought after types of cacao.
Gallery
References
External links
santiagomarino-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Santos%20Michelena%20Municipality
|
The Santos Michelena Municipality is one of the 18 municipalities (municipios) that make up the Venezuelan state of Aragua. According to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 38,574. The town of Las Tejerías is the shire town of the Santos Michelena Municipality. The municipality is named for the Venezuelan politician Santos Michelena.
Demographics
The Santos Michelena Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 44,409 (up from 38,638 in 2000). This amounts to 2.7% of the state's population. The municipality's population density is .
Government
The mayor of the Santos Michelena Municipality is Reinaldo Lorca, re-elected on October 31, 2004 with 32% of the votes. The municipality is divided into two parishes: Capital Santos Michelena and Tiara.
References
External links
santosmichelena-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Sucre%20Municipality%2C%20Aragua
|
Sucre Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 114,509. The town of Cagua is the shire town of the Sucre Municipality.
Name
The municipality is one of several in Venezuela named "Sucre Municipality" in honour of Venezuelan independence hero Antonio José de Sucre.
History
The town of Cagua was first established as Cagua la Vieja in 1620 and was originally made up of Spaniards. The origin of the name Cagua, comes from the native word Cahiua, which means snail. Cagua was rebuilt in its current location in 1622 under the new name of Nuestra Señora del Rosario de Cagua, which was later changed to San Jose de Cagua during the 18th century. The city is now known as just Cagua.
Cagua is a small city with several squares and has a tropical climate. It's ones of the most important cities of Aragua because of its proximity to Maracay, the state capital and Caracas. Cagua has been, for a long time, an important industrial zone.
One of its most important places are Casa Guipuzcoana and the Mountain "El Empalado", where original natives were burned alive by Spanish colonizers; this mountain is rich in marble and typical plants such as Araguaney, Samán, Bucare, and Apamate.
Sites of interest
Sucre square
Casa Guipuzcoana
Marble Mines
Empalado Mountain
Old Mill
Saint Joseph Church
Taiguaiguay Lake
Demographics
The Sucre Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 120,302 (up from 109,723 in 2000). This amounts to 7.2% of the state's population. The municipality's population density is .
Government
The mayor of the Sucre Municipality is Wilson Coy, elected on November 21, 2021, with 39.58% of the vote. He replaced Miriam Pardo shortly after the elections. The municipality is divided into two parishes; Capital Sucre and Bella Vista.
References
External links
sucre-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Tovar%20Municipality%2C%20Aragua
|
The Tovar Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 14,161. The town of Colonia Tovar is the shire town of the Tovar Municipality.
History
The village of La Colonia Tovar (The Tovar Colony) is located about 60 km from Caracas. The town was named after Martín Tovar y Ponte who donated the land over 100 years ago. The town is mainly known for its Germanic characteristics, culture, and a dessert called "golfiado", which is very similar to a Cinnamon roll. Founded in 1843 by German settlers, the city remained isolated from the rest of the country for decades, a factor that permitted the inhabitants to keep their culture and traditions. The majority of its residents are descendants of Germans and have a Northern European appearance. The Alemannic dialect of German, known as Alemán Coloniero ("Colonial German"), is nearly extinct.
Today, 6,000 people live in the main village, up from 1,300 in 1963.
Due to the cool climate and pleasant surrounding countryside, it is a popular week-end destination for many visitors from Caracas. Many houses are weekend retreats and second homes. There is a wide range of hotels, restaurants and tourist facilities, many of which are only open on the weekend.
The town and surrounding mountainous countryside have a superficial resemblance to Southern Germany. The visitor will occasionally see Germanic oddities such as waitresses in traditional Bavarian dress hawking "torta selva negra" and the odd "D" plate on some cars, but will be unlikely to find anyone who understands German fluently.
Geography
Colonia Tovar, at about 2,000 meters above sea level, has an annual average temperature of 16 degrees Celsius (61 degrees Fahrenheit), and at night it can drop to around 5 °C.
Economy
The cost of living is higher here than elsewhere in Venezuela. The town depends mostly on tourism and peach agriculture. The surrounding cloud forests are protected by the Pico Codazzi Natural Monument, though some signs of deforestation are visible due to the high tourist demand from Caracas, and the extraction of tree fern, sold dry for growing orchids. The town is situated on a steep hillside, making the street layout difficult and poorly suited to the heavy weekend traffic. During the week the town can be virtually deserted.
Demographics
The Tovar Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 17,429 (up from 14,798 in 2000). This amounts to 1% of the state's population. The municipality's population density is .
Government
The mayor of the Tovar Municipality is Esteban Bocaranda, elected on October 31, 2004, with 33% of the vote. He replaced Alfredo Durr shortly after the elections. The municipality is divided into one parish; Capital Tovar.
See also
La Coloni
|
https://en.wikipedia.org/wiki/Zamora%20Municipality%2C%20Aragua
|
Zamora Municipality is one of the 18 municipalities (municipios) that makes up the Venezuelan state of Aragua and, according to the 2011 census by the National Institute of Statistics of Venezuela, the municipality has a population of 144,759. The town of Villa de Cura is the municipal seat of the Zamora Municipality.
Name
The municipality is one of several named "Zamora Municipality" for the 19th century Venezuelan soldier Ezequiel Zamora.
History
The city of Villa de Cura was founded on May 25, 1722.
Demographics
The Zamora Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 145,742 (up from 127,595 in 2000). This amounts to 8.8% of the state's population. The municipality's population density is .
Government
The mayor of the Zamora Municipality is Aldo Lovera, elected on November 23, 2008, with 55% of the vote. He replaced Stefano Mangione shortly after the elections. The municipality is divided into five parishes; Capital Zamora, Magdaleno, San Francisco de Asís, Valles de Tucutunemo, and Augusto Mijares (separated from San Francisco de Asís parish effective 16/12/97).
References
External links
zamora-aragua.gob.ve
Municipalities of Aragua
|
https://en.wikipedia.org/wiki/Homography
|
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation.
Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term homography, which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term "projective transformation" originated in these abstract constructions. These constructions divide into two classes that have been shown to be equivalent. A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
For sake of simplicity, unless otherwise stated, the projective spaces considered in this article are supposed to be defined over a (commutative) field. Equivalently Pappus's hexagon theorem and Desargues's theorem are supposed to be true. A large part of the results remain true, or may be generalized to projective geometries for which these theorems do not hold.
Geometric motivation
Historically, the concept of homography had been introduced to understand, explain and study visual perspective, and, specifically, the difference in appearance of two plane objects viewed from different points of view.
In three-dimensional Euclidean space, a central projection from a point O (the center) onto a plane P that does not contain O is the mapping that sends a point A to the intersection (if it exists) of the line OA and the plane P. The projection is not defined if the point A belongs to the plane passing through O and parallel to P. The notion of projective space was originally introduced by extending the Euclidean space, that is, by adding points at infinity to it, in order to define the projection for every point except O.
Given another plane Q, which does not contain O, the restriction to Q of the above projection is called a perspectivity.
Wi
|
https://en.wikipedia.org/wiki/Bust/waist/hip%20measurements
|
Bust/waist/hip measurements (informally called 'body measurements' or ′vital statistics′) are a common method of specifying clothing sizes. They match the three inflection points of the female body shape. In human body measurement, these three sizes are the circumferences of the bust, waist and hips; usually rendered as xx–yy–zz in inches, or centimeters. The three sizes are used mostly in fashion, and almost exclusively in reference to women, who, compared to men, are more likely to have a narrow waist relative to their hips.
Measurements and perception
Breast volume will have an effect on the perception of a woman's figure even when bust/waist/hip measurements are nominally the same. Brassière band size is measured below the breasts, not at the bust. A woman with measurements of 36A–27–38 will have a different presentation than a woman with measurements of 34C–27–38. These women have ribcage circumferences differing by 2 inches, but when breast tissue is included the measurements are the same at 38 inches. The result is that the latter woman will appear "bustier" than the former due to the apparent difference in bust to hip ratios (narrower shoulders, more prominent breasts) even though they are both technically 38–27–38.
Height will also affect the presentation of the figure. A woman who is 36–24–36 (91.5–61–91.5) at tall looks different from a woman who is 36–24–36 at tall. Since the latter woman's figure has greater distance between measuring points, she will likely appear thinner than her former counterpart, again, even though they share the same measurements.
The specific proportions of 36–24–36 inches (90-60-90 centimeters) have frequently been given as the "hourglass" proportions for women since at least the 1960s (these measurements are, for example, the title of a hit instrumental by The Shadows).
See also
Female body shape
Physical attractiveness
Waist–hip ratio
Waist-to-height ratio
References
Sizes in clothing
Body shape
|
https://en.wikipedia.org/wiki/Cercado%20Province%20%28Beni%29
|
Cercado is a province located in northwestern Bolivia in Beni Department. It has an area of 12,276 km ² with a population estimated by the National Institute of Statistics of Bolivia for 2006 of 94,221 and a density of 7.67 people / km ². Its capital is the city of Trinidad.
Subdivision
Cercado Province is divided into two municipalities which are partly further subdivided into cantons.
References
Provinces of Beni Department
|
https://en.wikipedia.org/wiki/Cartan%27s%20criterion
|
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on defined by the formula
where tr denotes the trace of a linear operator. The criterion was introduced by .
Cartan's criterion for solvability
Cartan's criterion for solvability states:
A Lie subalgebra of endomorphisms of a finite-dimensional vector space over a field of characteristic zero is solvable if and only if whenever
The fact that in the solvable case follows from Lie's theorem that puts in the upper triangular form over the algebraic closure of the ground field (the trace can be computed after extending the ground field). The converse can be deduced from the nilpotency criterion based on the Jordan–Chevalley decomposition (for the proof, follow the link).
Applying Cartan's criterion to the adjoint representation gives:
A finite-dimensional Lie algebra over a field of characteristic zero is solvable if and only if (where K is the Killing form).
Cartan's criterion for semisimplicity
Cartan's criterion for semisimplicity states:
A finite-dimensional Lie algebra over a field of characteristic zero is semisimple if and only if the Killing form is non-degenerate.
gave a very short proof that if a finite-dimensional Lie algebra (in any characteristic) has a non-degenerate invariant bilinear form and no non-zero abelian ideals, and in particular if its Killing form is non-degenerate, then it is a sum of simple Lie algebras.
Conversely, it follows easily from Cartan's criterion for solvability that a semisimple algebra (in characteristic 0) has a non-degenerate Killing form.
Examples
Cartan's criteria fail in characteristic ; for example:
the Lie algebra is simple if k has characteristic not 2 and has vanishing Killing form, though it does have a nonzero invariant bilinear form given by .
the Lie algebra with basis for and bracket [ai,aj] = (i−j)ai+j is simple for but has no nonzero invariant bilinear form.
If k has characteristic 2 then the semidirect product gl2(k).k2 is a solvable Lie algebra, but the Killing form is not identically zero on its derived algebra sl2(k).k2.
If a finite-dimensional Lie algebra is nilpotent, then the Killing form is identically zero (and more generally the Killing form vanishes on any nilpotent ideal). The converse is false: there are non-nilpotent Lie algebras whose Killing form vanishes. An example is given by the semidirect product of an abelian Lie algebra V with a 1-dimensional Lie algebra acting on V as an endomorphism b such that b is not nilpotent and Tr(b2)=0.
In characteristic 0, every reductive Lie algebra (one that is a sum of abelian and simple Lie algebras) has a non-degenerate invariant symmetric bilinear form. However the converse is false: a Lie algebra with a non-degenerate invariant symmetric bilinear form
|
https://en.wikipedia.org/wiki/Brauer%27s%20theorem%20on%20induced%20characters
|
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, within representation theory of a finite group.
Background
A precursor to Brauer's induction theorem was Artin's induction theorem, which states that |G| times the trivial character of G is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of G. Brauer's theorem removes the factor |G|,
but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups.
Another result between Artin's induction theorem and Brauer's induction theorem, also due to Brauer and also known as Brauer's theorem or Brauer's lemma is the fact that the regular representation of G can be written as where the are positive rationals and the are induced from characters of cyclic subgroups of G. Note that in Artin's theorem the characters are induced from the trivial character of the cyclic group, while here they are induced from arbitrary characters (in applications to Artin's L functions it is important that the groups are cyclic and hence all characters are linear giving that the corresponding L functions are analytic).
Statement
Let G be a finite group and let Char(G) denote the subring of the ring of complex-valued class functions of G consisting of integer combinations of irreducible characters. Char(G) is known as the character ring of G, and its elements are known as virtual characters (alternatively, as generalized characters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of G is again a character of G. Its multiplication is given by the elementwise product of class functions.
Brauer's induction theorem shows that the character ring can be generated (as an abelian group) by induced characters of the form , where H ranges over subgroups of G and λ ranges over linear characters (having degree 1) of H.
In fact, Brauer showed that the subgroups H could be chosen from a very
restricted collection, now called Brauer elementary subgroups. These are direct products of cyclic groups and groups whose order is a power of a prime.
Proofs
The proof of Brauer's induction theorem exploits the ring structure of Char(G) (most proofs also make use of a slightly larger ring, Char*(G), which consists of -combinations of irreducible characters, where ω is a primitive complex |G|-th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal I(G) of Char(G), so the proof reduces to showing that the trivial character is in I(G). Several
|
https://en.wikipedia.org/wiki/Three-dimensional%20space
|
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space. More general three-dimensional spaces are called 3-manifolds.
The term may also refer colloquially to a subset of space, a three-dimensional region (or 3D domain), a solid figure.
Technically, a tuple of numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted and can be identified to the pair formed by a -dimensional Euclidean space and a Cartesian coordinate system.
When , this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear). It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width/breadth, height/depth, and length.
History
Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regular Platonic solids in a sphere.
In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.
In the 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton's development of the quaternions. In fact, it was Hamilton who coined the terms scalar and vector, and they were first defined within his geometric framework for quaternions. Three dimensional space could then be described by quaternions which had vanishing scalar component, that is, . While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements , as well as the dot product and cros
|
https://en.wikipedia.org/wiki/Complex%20torus
|
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian product of some number N circles). Here N must be the even number 2n, where n is the complex dimension of M.
All such complex structures can be obtained as follows: take a lattice Λ in a vector space V isomorphic to Cn considered as real vector space; then the quotient group
is a compact complex manifold. All complex tori, up to isomorphism, are obtained in this way. For n = 1 this is the classical period lattice construction of elliptic curves. For n > 1 Bernhard Riemann found necessary and sufficient conditions for a complex torus to be an algebraic variety; those that are varieties can be embedded into complex projective space, and are the abelian varieties.
The actual projective embeddings are complicated (see equations defining abelian varieties) when n > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for n = 1. Computer algebra can handle cases for small n reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.
Definition
One way to define complex tori is as a compact connected complex Lie group . These are Lie groups where the structure maps are holomorphic maps of complex manifolds. It turns out that all such compact connected Lie groups are commutative, and are isomorphic to a quotient of their Lie algebra whose covering map is the exponential map of a Lie algebra to its associated Lie group. The kernel of this map is a lattice and .
Conversely, given a complex vector space and a lattice of maximal rank, the quotient complex manifold has a complex Lie group structure, and is also compact and connected. This implies the two definitions for complex tori are equivalent.
Period matrix of a complex torus
One way to describe a g-dimensional complex torus is by using a matrix whose columns correspond to a basis of the lattice expanded out using a basis of . That is, we write
so
We can then write the torus as
If we go in the reverse direction by selecting a matrix , it corresponds to a period matrix if and only if the corresponding matrix constructed by adjoining the complex conjugate matrix to , so
is nonsingular. This guarantees the column vectors of span a lattice in hence must be linearly independent vectors over .
Example
For a two-dimensional complex torus, it has a period matrix of the form
for example, the matrix
forms a period matrix since the associated period matrix has determinant 4.
Normalized period matrix
For any complex torus of dimension it has a period matrix of the form
where is the identity matrix and where . We can get this from taking a change of basis of the vector space giving a block matrix of the form above. The condition for follows from
|
https://en.wikipedia.org/wiki/Hilbert%20modular%20variety
|
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. More generally, a Hilbert modular variety is an algebraic variety obtained by taking a quotient of a product of multiple copies of the upper half-plane by a Hilbert modular group.
Hilbert modular surfaces were first described by using some unpublished notes written by David Hilbert about 10 years before.
Definitions
If R is the ring of integers of a real quadratic field, then
the Hilbert modular group SL2(R) acts on the product H×H of two copies of the upper half plane H.
There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces:
The surface X is the quotient of H×H by SL2(R); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups.
The surface X* is obtained from X by adding a finite number of points corresponding to the cusps of the action. It is compact, and has not only the quotient singularities of X, but also singularities at its cusps.
The surface Y is obtained from X* by resolving the singularities in a minimal way. It is a compact smooth algebraic surface, but is not in general minimal.
The surface Y0 is obtained from Y by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal.
There are several variations of this construction:
The Hilbert modular group may be replaced by some subgroup of finite index, such as a congruence subgroup.
One can extend the Hilbert modular group by a group of order 2, acting on the Hilbert modular group via the Galois action, and exchanging the two copies of the upper half plane.
Singularities
showed how to resolve the quotient singularities, and showed how to resolve their cusp singularities.
Classification of surfaces
The papers , and identified their type in the classification of algebraic surfaces. Most of them are surfaces of general type, but several are rational surfaces or blown up K3 surfaces or elliptic surfaces.
Examples
gives a long table of examples.
The Clebsch surface blown up at its 10 Eckardt points is a Hilbert modular surface.
Associated to a quadratic field extension
Given a quadratic field extension for there is an associated Hilbert modular variety obtained from compactifying a certain quotient variety and resolving its singularities. Let denote the upper half plane and let act on viawhere the are the Galois conjugates. The associated quotient variety is denotedand can be compactified to a variety , called the cusps, which are in bijection with the ideal classes in . Resolving its singularities gives the variety called the Hilbert modular variety of the field extension. From the Bailey-Borel compactification theorem, there is an embedding of this surface into a projective space.
See
|
https://en.wikipedia.org/wiki/Complex%20measure
|
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number.
Definition
Formally, a complex measure on a measurable space is a complex-valued function
that is sigma-additive. In other words, for any sequence of disjoint sets belonging to , one has
As for any permutation (bijection) , it follows that converges unconditionally (hence absolutely).
Integration with respect to a complex measure
One can define the integral of a complex-valued measurable function with respect to a complex measure in the same way as the Lebesgue integral of a real-valued measurable function with respect to a non-negative measure, by approximating a measurable function with simple functions. Just as in the case of ordinary integration, this more general integral might fail to exist, or its value might be infinite (the complex infinity).
Another approach is to not develop a theory of integration from scratch, but rather use the already available concept of integral of a real-valued function with respect to a non-negative measure. To that end, it is a quick check that the real and imaginary parts μ1 and μ2 of a complex measure μ are finite-valued signed measures. One can apply the Hahn-Jordan decomposition to these measures to split them as
and
where μ1+, μ1−, μ2+, μ2− are finite-valued non-negative measures (which are unique in some sense). Then, for a measurable function f which is real-valued for the moment, one can define
as long as the expression on the right-hand side is defined, that is, all four integrals exist and when adding them up one does not encounter the indeterminate ∞−∞.
Given now a complex-valued measurable function, one can integrate its real and imaginary components separately as illustrated above and define, as expected,
Variation of a complex measure and polar decomposition
For a complex measure μ, one defines its variation, or absolute value, |μ| by the formula
where A is in Σ and the supremum runs over all sequences of disjoint sets (An)n whose union is A. Taking only finite partitions of the set A into measurable subsets, one obtains an equivalent definition.
It turns out that |μ| is a non-negative finite measure. In the same way as a complex number can be represented in a polar form, one has a polar decomposition for a complex measure: There exists a measurable function θ with real values such that
meaning
for any absolutely integrable measurable function f, i.e., f satisfying
One can use the Radon–Nikodym theorem to prove that the variation is a measure and the existence of the polar decomposition.
The space of complex measures
The sum of two complex measures is a complex measure, as is the product of a complex measure by a complex number. That is to say, the set of all complex measures on a measure space (X, Σ) forms a vector space over the complex numbers.
|
https://en.wikipedia.org/wiki/Free%20Boolean%20algebra
|
In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that:
Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and
The generators are as independent as possible, in the sense that there are no relationships among them (again in terms of finite expressions using the Boolean operations) that do not hold in every Boolean algebra no matter which elements are chosen.
A simple example
The generators of a free Boolean algebra can represent independent propositions. Consider, for example, the propositions "John is tall" and "Mary is rich". These generate a Boolean algebra with four atoms, namely:
John is tall, and Mary is rich;
John is tall, and Mary is not rich;
John is not tall, and Mary is rich;
John is not tall, and Mary is not rich.
Other elements of the Boolean algebra are then logical disjunctions of the atoms, such as "John is tall and Mary is not rich, or John is not tall and Mary is rich". In addition there is one more element, FALSE, which can be thought of as the empty disjunction; that is, the disjunction of no atoms.
This example yields a Boolean algebra with 16 elements; in general, for finite n, the free Boolean algebra with n generators has 2n atoms, and therefore elements.
If there are infinitely many generators, a similar situation prevails except that now there are no atoms. Each element of the Boolean algebra is a combination of finitely many of the generating propositions, with two such elements deemed identical if they are logically equivalent.
Another way to see why the free Boolean algebra on an n-element set has elements is to note that each element is a function from n bits to one. There are possible inputs to such a function and the function will choose 0 or 1 to output for each input, so there are possible functions.
Category-theoretic definition
In the language of category theory, free Boolean algebras can be defined simply in terms of an adjunction between the category of sets and functions, Set, and the category of Boolean algebras and Boolean algebra homomorphisms, BA. In fact, this approach generalizes to any algebraic structure definable in the framework of universal algebra.
Above, we said that a free Boolean algebra is a Boolean algebra with a set of generators that behave a certain way; alternatively, one might start with a set and ask which algebra it generates. Every set X generates a free Boolean algebra FX defined as the algebra such that for every algebra B and function f : X → B, there is a unique Boolean algebra homomorphism f′ : FX → B that extends f. Diagrammatically,
where iX is the inclusion, and the dashed arrow denotes uniqueness. The idea is that once one chooses where to send the elements of X, the laws for Boolean algebra homomorphisms determine where to send everything else in the free algebra FX. If FX contained elements inexpressible as combinatio
|
https://en.wikipedia.org/wiki/Atom%20%28measure%20theory%29
|
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure. A measure which has no atoms is called non-atomic or atomless.
Definition
Given a measurable space and a measure on that space, a set in is called an atom if
and for any measurable subset with
the set has measure zero, i.e. .
If is an atom, all the subsets in the -equivalence class of are atoms, and is called an atomic class. If is a -finite measure, there are countably many atomic classes.
Examples
Consider the set X = {1, 2, ..., 9, 10} and let the sigma-algebra be the power set of X. Define the measure of a set to be its cardinality, that is, the number of elements in the set. Then, each of the singletons {i}, for i = 1, 2, ..., 9, 10 is an atom.
Consider the Lebesgue measure on the real line. This measure has no atoms.
Atomic measures
A -finite measure on a measurable space is called atomic or purely atomic if every measurable set of positive measure contains an atom. This is equivalent to say that there is a countable partition of formed by atoms up to a null set. The assumption of -finitude is essential. Consider otherwise the space where denotes the counting measure. This space is atomic, with all atoms being the singletons, yet the space is not able to be partitioned into the disjoint union of countably many disjoint atoms, and a null set since the countable union of singletons is a countable set, and the uncountability of the real numbers shows that the complement would have to be uncountable, hence its -measure would be infinite, in contradiction to it being a null set. The validity of the result for -finite spaces follows from the proof for finite measure spaces by observing that the countable union of countable unions is again a countable union, and that the countable unions of null sets are null.
Discrete measures
A -finite atomic measure is called discrete if the intersection of the atoms of any atomic class is non empty.
It is equivalent to say that is the weighted sum of countably many Dirac measures, that is, there is a sequence of points in , and a sequence of positive real numbers (the weights) such that , which means that for every . We can choose each point to be a common point of the atoms
in the -th atomic class.
A discrete measure is atomic but the inverse implication fails: take , the -algebra of countable and co-countable subsets, in countable subsets and in co-countable subsets. Then there is a single atomic class, the one formed by the co-countable subsets. The measure is atomic but the intersection of the atoms in the unique atomic class is empty and can't be put as a sum of Dirac measures.
If every atom is equivalent to a singleton, then is discrete iff it is atomic. In this case the above are the atomic singletons, so they are unique. Any finite measure in a separable metric space provided with the Borel sets satisf
|
https://en.wikipedia.org/wiki/Ski%20geometry
|
Ski geometry is the shape of the ski. Described in the direction of travel, the front of the ski, typically pointed or rounded, is the tip, the middle is the waist and the rear is the tail. Skis have four aspects that define their basic performance: length, width, sidecut and camber. Skis also differ in more minor ways to address certain niche roles. For instance, skis for moguls are much softer to absorb shocks from the quick and sharp turns of the moguls and skis for powder are much wider to provide more "float" in deeper, softer snow.
Length and width
The length and width of the ski define its total surface area, which provides some indication of the ski's float, or ability to remain on top of the snow instead of sinking into it. Cross-country skis must be narrow to reduce drag, and thus must be long to produce the required float. Alpine skis are generally not designed to reduce drag, and tend to be shorter and wider. Skis used in downhill race events are longer, with a subtle side cut, built for speed and wide turns. Slalom skis, as well as many recreational skis, are shorter with a greater side cut to facilitate tighter, easier turns. For off-piste skis the trend is towards wider skis that better float on top of powder snow. The ski width of all-mountain and off-piste skis has generally increased since the 1990s when 85 mm width was considered a wide powderski. From 2010 and onwards, many well known ski manufacturers sell all-round freeride skis for the general public starting in the 90mm range and going up to 120 mm or more.
Tips and tails
The tip of the ski often strikes the snow and is normally curled upward in order to ride over it. Tips were pointed for much of the history of skiing, but the introduction of wider shaped skis has led to a change to more rounded shapes.
Tails were, and often remain, straight cut. For freestyle skiing, where the skier is often skiing backwards, it is common to have a "twin-tip" design with the tail of the ski rounded and curled up like the nose so that it skis the same in both directions.
One design note that makes a periodic comeback is the "swallowtail" design, where a notch, often V-shaped, is cut out of the rear of the ski. This makes the tail into two independent fingers. When turning, only one edge of the ski is in contact with the snow, and in a traditional ski design, this pressure causes both the turning force as well as a torsional force on the ski, making it want to flatten out on the snow and lose the edge. The swallowtail allows the two tips to move independently, reducing this torsional force and, in theory, keeping the edge in firm contact.
Camber & rocker
Camber is the ski's shape as viewed from the side. Typically skis are designed so that when the tip and tail are on the ground, the waist is in the air. Without camber, when the skier's weight is applied at the waist, the weight would be distributed on the surface closest to the foot, diminishing along the length. Camber distributes
|
https://en.wikipedia.org/wiki/Schubert%20calculus
|
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in particular its algorithmic aspects are still of current interest. The term Schubert calculus is sometimes used to mean the enumerative geometry of linear subspaces of a vector space, which is roughly equivalent to describing the cohomology ring of Grassmannians. Sometimes it is used to mean the more general enumerative geometry of algebraic varieties that are homogenous spaces of simple Lie groups. Even more generally, Schubert calculus is often understood to encompass the study of analogous questions in generalized cohomology theories.
The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For further details see Schubert variety.
The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumerative questions. A key result is that the Schubert cells (or rather, the classes of their Zariski closures, the Schubert cycles or Schubert varieties) span the whole cohomology ring.
The combinatorial aspects mainly arise in relation to computing intersections of Schubert cycles. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (as block traingular matrices).
Putting Schubert's system on a rigorous footing was Hilbert's fifteenth problem.
Construction
Schubert calculus can be constructed using the Chow ring
of the Grassmannian, where the generating cycles are represented by geometrically defined data. Denote the Grassmannian of -planes in a fixed -dimensional vector space as , and its Chow ring as . Note that the Grassmannian is sometimes denoted if the vector space isn't explicitly given or as if the ambient space and its -dimensional subspaces are replaced by their projectizations. Choosing an (arbitrary) complete flag
to each weakly decreasing -tuple of integers , where
i.e., to each partition of weight
whose Young diagram fits into the rectangular one , we associate a Schubert variety (or Schubert cycle) , defined as
This is the closure, in the Zariski topology, of the Schubert cell
which is used when considering cellular homology instead of the Chow ring. The latter are disjoint affine spaces, of dimension , whose union is .
Since the homology class , called a Schuber
|
https://en.wikipedia.org/wiki/J-homomorphism
|
In mathematics, the J-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of .
Definition
Whitehead's original homomorphism is defined geometrically, and gives a homomorphism
of abelian groups for integers q, and . (Hopf defined this for the special case .)
The J-homomorphism can be defined as follows.
An element of the special orthogonal group SO(q) can be regarded as a map
and the homotopy group ) consists of homotopy classes of maps from the r-sphere to SO(q).
Thus an element of can be represented by a map
Applying the Hopf construction to this gives a map
in , which Whitehead defined as the image of the element of under the J-homomorphism.
Taking a limit as q tends to infinity gives the stable J-homomorphism in stable homotopy theory:
where is the infinite special orthogonal group, and the right-hand side is the r-th stable stem of the stable homotopy groups of spheres.
Image of the J-homomorphism
The image of the J-homomorphism was described by , assuming the Adams conjecture of which was proved by , as follows. The group is given by Bott periodicity. It is always cyclic; and if r is positive, it is of order 2 if r is 0 or 1 modulo 8, infinite if r is 3 or 7 modulo 8, and order 1 otherwise . In particular the image of the stable J-homomorphism is cyclic. The stable homotopy groups are the direct sum of the (cyclic) image of the J-homomorphism, and the kernel of the Adams e-invariant , a homomorphism from the stable homotopy groups to . If r is 0 or 1 mod 8 and positive, the order of the image is 2 (so in this case the J-homomorphism is injective). If r is 3 or 7 mod 8, the image is a cyclic group of order equal to the denominator of , where is a Bernoulli number. In the remaining cases where r is 2, 4, 5, or 6 mod 8 the image is trivial because is trivial.
{| class="wikitable" style="text-align: center; background-color:white"
|-
! style="text-align:right;width:10%" | r
! style="width:5%" | 0
! style="width:5%" | 1
! style="width:5%" | 2
! style="width:5%" | 3
! style="width:5%" | 4
! style="width:5%" | 5
! style="width:5%" | 6
! style="width:5%" | 7
! style="width:5%" | 8
! style="width:5%" | 9
! style="width:5%" | 10
! style="width:5%" | 11
! style="width:5%" | 12
! style="width:5%" | 13
! style="width:5%" | 14
! style="width:5%" | 15
! style="width:5%" | 16
! style="width:5%" | 17
|-
! style="text-align:right" |
| 1 || 2 || 1 || || 1 || 1 || 1 || || 2 || 2 || 1 || || 1 || 1 || 1 || || 2 || 2
|-
! style="text-align:right" |
| 1 || 2 || 1 || 24 || 1 || 1 || 1 || 240 || 2 || 2 || 1 || 504 || 1 || 1 || 1 || 480 || 2 || 2
|-
! style="text-align:right" |
| || 2 || 2 || 24 || 1 || 1 || 2 || 240 || 22 || 23 || 6 || 504 || 1 || 3 || 22 || 480×2 || 22 || 24
|-
! style="text-align:right" |
| || || || 1⁄6 || || || || −1⁄30 || || || || 1⁄42 || || || || −1⁄30 || ||
|}
Applications
introduced the gr
|
https://en.wikipedia.org/wiki/Reflection%20map
|
Reflection map may refer to:
Reflection mapping in computer graphics
A reflection (mathematics), specifically
an element of a reflection group
an element of a Weyl group
Reflection map (logic optimization), a conventional Gray code Karnaugh map in logic optimization
|
https://en.wikipedia.org/wiki/Cayley%E2%80%93Bacharach%20theorem
|
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane . The original form states:
Assume that two cubics and in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes through any eight of the points also passes through the ninth point.
A more intrinsic form of the Cayley–Bacharach theorem reads as follows:
Every cubic curve over an algebraically closed field that passes through a given set of eight points also passes through (counting multiplicities) a ninth point which depends only on .
A related result on conics was first proved by the French geometer Michel Chasles and later generalized to cubics by Arthur Cayley and Isaak Bacharach.
Details
If seven of the points lie on a conic, then the ninth point can be chosen on that conic, since will always contain the whole conic on account of Bézout's theorem. In other cases, we have the following.
If no seven points out of are co-conic, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) (with multiplicity for double points) has dimension two.
In that case, every cubic through also passes through the intersection of any two different cubics through , which has at least nine points (over the algebraic closure) on account of Bézout's theorem. These points cannot be covered by only, which gives us .
Since degenerate conics are a union of at most two lines, there are always four out of seven points on a degenerate conic that are collinear. Consequently:
If no seven points out of lie on a non-degenerate conic, and no four points out of lie on a line, then the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) has dimension two.
On the other hand, assume are collinear and no seven points out of are co-conic. Then no five points of and no three points of are collinear. Since will always contain the whole line through on account of Bézout's theorem, the vector space of cubic homogeneous polynomials that vanish on (the affine cones of) is isomorphic to the vector space of quadratic homogeneous polynomials that vanish (the affine cones of) , which has dimension two.
Although the sets of conditions for both dimension two results are different, they are both strictly weaker than full general positions: three points are allowed to be collinear, and six points are allowed to lie on a conic (in general two points determine a line and five points determine a conic). For the Cayley–Bacharach theorem, it is necessary to have a family of cubics passing through the nine points, rather than a single one.
According to Bézout's theorem, two different cubic curves over an algebraically closed field which have no common irreducible component meet in exactly nine points (counted with multiplicity). The Cayley–Bacharach theorem thus asserts that the last point
|
https://en.wikipedia.org/wiki/Pitch%20axis
|
Pitch axis may refer to:
In music
Pitch axis (music), the center about which a melody is inverted
Pitch axis theory, a musical technique used in constructing chord progressions
In mathematics and engineering
Aircraft principal axes, the axes of an airplane in flight
Yaw, pitch, and roll, a specific kind of Euler angles
|
https://en.wikipedia.org/wiki/Enumerative%20geometry
|
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
History
The problem of Apollonius is one of the earliest examples of enumerative geometry. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. In general, the problem for three given circles has eight solutions, which can be seen as 23, each tangency condition imposing a quadratic condition on the space of circles. However, for special arrangements of the given circles, the number of solutions may also be any integer from 0 (no solutions) to six; there is no arrangement for which there are seven solutions to Apollonius' problem.
Key tools
A number of tools, ranging from the elementary to the more advanced, include:
Dimension counting
Bézout's theorem
Schubert calculus, and more generally characteristic classes in cohomology
The connection of counting intersections with cohomology is Poincaré duality
The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. The study of quantum cohomology, Gromov–Witten invariants and mirror symmetry gave a significant progress in Clemens conjecture.
Enumerative geometry is very closely tied to intersection theory.
Schubert calculus
Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of Hermann Schubert. He introduced it for the purpose the Schubert calculus, which has proved of fundamental geometrical and topological value in broader areas. The specific needs of enumerative geometry were not addressed until some further attention was paid to them in the 1960s and 1970s (as pointed out for example by Steven Kleiman). Intersection numbers had been rigorously defined (by André Weil as part of his foundational programme 1942–6, and again subsequently), but this did not exhaust the proper domain of enumerative questions.
Fudge factors and Hilbert's fifteenth problem
Naïve application of dimension counting and Bézout's theorem yields incorrect results, as the following example shows. In response to these problems, algebraic geometers introduced vague "fudge factors", which were only rigorously justified decades later.
As an example, count the conic sections tangent to five given lines in the projective plane. The conics constitute a projective space of dimension 5, taking their six coefficients as homogeneous coordinates, and five points determine a conic, if the points are in general linear position, as passing through a given point imposes a linear condition. Similarly, tangency to a given line L (tangency is intersection with multiplicity two) is one quadratic condition, so determined a quadric in P5. However the linear system of divisors consisting of all such quadrics is not without a base locus. In fact each such quadric contains the Verone
|
https://en.wikipedia.org/wiki/Pleiku%20Airport
|
Pleiku Airport () is a regional airport located near the city of Pleiku within Gia Lai Province in southern Vietnam.
Airlines and destinations
Statistics
History
Pleiku Airport was little more than an undeveloped air strip in December 1962 when it was designated by the Republic of Vietnam Air Force (VNAF) as Air Base 62. It was expanded during the Vietnam War and became a major air base for the VNAF and United States Air Force activities, but never reached the saturation and population proportions of the major air bases of the coastal lowlands. After 1975, it was developed into a civil airport.
See also
List of airports in Vietnam
References
External links
Trip To Pleiku - 2006
Pleiku
Buildings and structures in Gia Lai province
Airports in Vietnam
|
https://en.wikipedia.org/wiki/363%20%28number%29
|
363 (three hundred [and] sixty-three) is the natural number following 362 and preceding 364.
In mathematics
It is an odd, composite, positive, real integer, composed of a prime (3) and a prime squared (112).
363 is a deficient number and a perfect totient number.
363 is a palindromic number in bases 3, 10, 11 and 32.
363 is a repdigit (BB) in base 32.
The Mertens function returns 0.
Any subset of its digits is divisible by three.
363 is the sum of nine consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59).
363 is the sum of five consecutive powers of 3 (3 + 9 + 27 + 81 + 243).
363 can be expressed as the sum of three squares in four different ways: 112 + 112 + 112, 52 + 72 + 172, 12 + 12 + 192, and 132 + 132 + 52.
363 cubits is the solution given to Rhind Mathematical Papyrus question 50 – find the side length of an octagon with the same area as a circle 9 khet in diameter .
References
Integers
|
https://en.wikipedia.org/wiki/Elliptic%20surface
|
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change.
The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context). The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in F-theory.
Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and smooth 4-manifolds. They are similar to (have analogies with, that is), elliptic curves over number fields.
Examples
The product of any elliptic curve with any curve is an elliptic surface (with no singular fibers).
All surfaces of Kodaira dimension 1 are elliptic surfaces.
Every complex Enriques surface is elliptic, and has an elliptic fibration over the projective line.
Kodaira surfaces
Dolgachev surfaces
Shioda modular surfaces
Kodaira's table of singular fibers
Most of the fibers of an elliptic fibration are (non-singular) elliptic curves. The remaining fibers are called singular fibers: there are a finite number of them, and each one consists of a union of rational curves, possibly with singularities or non-zero multiplicities (so the fibers may be non-reduced schemes). Kodaira and Néron independently classified the possible fibers, and Tate's algorithm can be used to find the type of the fibers of an elliptic curve over a number field.
The following table lists the possible fibers of a minimal elliptic fibration. ("Minimal" means roughly one that cannot be factored through a "smaller" one; precisely, the singular fibers should contain no smooth rational curves with self-intersection number −1.) It gives:
Kodaira's symbol for the fiber,
André Néron's symbol for the fiber,
The number of irreducible components of the fiber (all rational except for type I0)
The intersection matrix of the components. This is either a 1×1 zero matrix, or an affine Cartan matrix, whose Dynkin diagram is given.
The multiplicities of each fiber are indicated in the Dynkin diagram.
This table can be found as follows. Geometric arguments show that the intersection matrix of the components of the fiber must be negative semidefinite, connected, symmetric, and have no diagonal entries equal to −1 (by minimality). Such a matrix must be 0 or a multiple of the Cartan matrix of an affine Dynkin diagram of type ADE.
The intersection matrix determines the fiber type with three exceptions:
If the intersection ma
|
https://en.wikipedia.org/wiki/ARGUS%20distribution
|
In physics, the ARGUS distribution, named after the particle physics experiment ARGUS, is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.
Definition
The probability density function (pdf) of the ARGUS distribution is:
for . Here and are parameters of the distribution and
where and are the cumulative distribution and probability density functions of the standard normal distribution, respectively.
Cumulative distribution function
The cumulative distribution function (cdf) of the ARGUS distribution is
.
Parameter estimation
Parameter c is assumed to be known (the kinematic limit of the invariant mass distribution), whereas χ can be estimated from the sample X1, …, Xn using the maximum likelihood approach. The estimator is a function of sample second moment, and is given as a solution to the non-linear equation
.
The solution exists and is unique, provided that the right-hand side is greater than 0.4; the resulting estimator is consistent and asymptotically normal.
Generalized ARGUS distribution
Sometimes a more general form is used to describe a more peaking-like distribution:
where Γ(·) is the gamma function, and Γ(·,·) is the upper incomplete gamma function.
Here parameters c, χ, p represent the cutoff, curvature, and power respectively.
The mode is:
The mean is:
where M(·,·,·) is the Kummer's confluent hypergeometric function.
The variance is:
p = 0.5 gives a regular ARGUS, listed above.
References
Further reading
Experimental particle physics
Continuous distributions
|
https://en.wikipedia.org/wiki/No%20decision
|
A no decision (sometimes written no-decision) is one of either of two sports statistics scenarios; one in baseball and softball, and the other in boxing and related combat sports.
Baseball and softball
A starting pitcher who leaves a game without earning either a win or a loss is said to have received a no decision. Major League Baseball (MLB) rules specify that a starting pitcher, in order to earn a win, must pitch at least five innings, leaving the game with a lead that their team "does not relinquish". There is no innings requirement for a starting pitcher to earn a loss, simply that the pitcher allows a run that gives the winning team a lead that they do not relinquish. When a starting pitcher does not earn a win or a loss, it is a no decision, and the outcome of the game does not affect the starting pitcher's win–loss record, as a relief pitcher will receive the win or loss.
Attributing wins, losses, and no decisions can be complex, such as when a starting pitcher leaves a game mid-inning with runners on base, as runs scored by those runners would still be considered the starting pitcher's responsibility. Further, if a starting pitcher leaves a game while losing (colloquially, that pitcher is said to be "on the hook"), he or she will receive a no decision if their team comes back to tie the score or take the lead, regardless of the final outcome. Box scores for completed games indicate who the winning and losing pitchers are, as determined by the official scorer; the absence of a win or loss designation for a starting pitcher indicates a no decision.
Examples
Assume in these examples that each starting pitcher exits the game at the end of the 6th inning.
Red Starter gets the win since, during the time that he or she was in the game, the Red team established a lead that was never relinquished. Conversely, Blue Starter gets the loss. While the run that provided the winning margin could be viewed as having been scored in the eighth inning, after both starters had left the game, the first-inning lead was never relinquished (the Red team was always in the lead).
Red Starter and Blue Starter each get a no decision, as the 4–0 lead established in the first inning was later relinquished (at the end of the eighth inning, the score was tied).
The above examples highlight how events that happen after starting pitchers have left the game can affect whether they receive a decision (win or loss) or no decision. This is one reason that wins and losses are generally viewed by baseball statisticians as being an unreliable indicator of pitching effectiveness.
MLB records
In Major League Baseball (MLB), the record for the most no decisions by a starting pitcher in a single season (dating back to at least 1908) is 20, held by Bert Blyleven in 1979 and Zach Davies in 2022. Tommy John has the all-time record of 188 career no decisions. The starting staff of the 1918 Boston Red Sox recorded only three no decisions, the fewest of any MLB team dating back t
|
https://en.wikipedia.org/wiki/Fenchel%27s%20theorem
|
In differential geometry, Fenchel's theorem is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least . Equivalently, the average curvature is at least , where is the length of the curve. The only curves of this type whose total absolute curvature equals and whose average curvature equals are the plane convex curves. The theorem is named after Werner Fenchel, who published it in 1929.
The Fenchel theorem is enhanced by the Fáry–Milnor theorem, which says that if a closed smooth simple space curve is nontrivially knotted, then the total absolute curvature is greater than .
Proof
Given a closed smooth curve with unit speed, the velocity is also a closed smooth curve. The total absolute curvature is its length .
The curve does not lie in an open hemisphere. If so, then there is such that , so , a contradiction. This also shows that if lies in a closed hemisphere, then , so is a plane curve.
Consider a point such that curves and have the same length. By rotating the sphere, we may assume and are symmetric about the axis through the poles. By the previous paragraph, at least one of the two curves and intersects with the equator at some point . We denote this curve by . Then .
We reflect across the plane through , , and the north pole, forming a closed curve containing antipodal points , with length . A curve connecting has length at least , which is the length of the great semicircle between . So , and if equality holds then does not cross the equator.
Therefore, , and if equality holds then lies in a closed hemisphere, and thus is a plane curve.
References
; see especially equation 13, page 49
Theorems in differential geometry
Theorems in plane geometry
Theorems about curves
Curvature (mathematics)
|
https://en.wikipedia.org/wiki/Fenchel%27s%20duality%20theorem
|
In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel.
Let ƒ be a proper convex function on Rn and let g be a proper concave function on Rn. Then, if regularity conditions are satisfied,
where ƒ * is the convex conjugate of ƒ (also referred to as the Fenchel–Legendre transform) and g * is the concave conjugate of g. That is,
Mathematical theorem
Let X and Y be Banach spaces, and be convex functions and be a bounded linear map. Then the Fenchel problems:
satisfy weak duality, i.e. . Note that are the convex conjugates of f,g respectively, and is the adjoint operator. The perturbation function for this dual problem is given by .
Suppose that f,g, and A satisfy either
f and g are lower semi-continuous and where is the algebraic interior and , where h is some function, is the set , or
where are the points where the function is continuous.
Then strong duality holds, i.e. . If then supremum is attained.
One-dimensional illustration
In the following figure, the minimization problem on the left side of the equation is illustrated. One seeks to vary x such that the vertical distance between the convex and concave curves at x is as small as possible. The position of the vertical line in the figure is the (approximate) optimum.
The next figure illustrates the maximization problem on the right hand side of the above equation. Tangents are drawn to each of the two curves such that both tangents have the same slope p. The problem is to adjust p in such a way that the two tangents are as far away from each other as possible (more precisely, such that the points where they intersect the y-axis are as far from each other as possible). Imagine the two tangents as metal bars with vertical springs between them that push them apart and against the two parabolas that are fixed in place.
Fenchel's theorem states that the two problems have the same solution. The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents.
See also
Legendre transformation
Convex conjugate
Moreau's theorem
Wolfe duality
Werner Fenchel
References
Theorems in analysis
Convex optimization
|
https://en.wikipedia.org/wiki/Franz%20Taurinus
|
Franz Adolph Taurinus (15 November 1794 – 13 February 1874) was a German mathematician who is known for his work on non-Euclidean geometry.
Life
Franz Taurinus was the son of Julius Ephraim Taurinus, a court official of the Count of Erbach-Schönberg, and Luise Juliane Schweikart. He studied law in Heidelberg, Gießen and Göttingen. He lived as a private scholar in Cologne.
Hyperbolic geometry
Taurinus corresponded with his uncle Ferdinand Karl Schweikart (1780–1859), who was a law professor in Königsberg, among other things about mathematics. Schweikart examined a model (after Giovanni Girolamo Saccheri and Johann Heinrich Lambert) in which the parallel postulate is not satisfied, and in which the sum of three angles of a triangle is less than two right angles (which is now called hyperbolic geometry). While Schweikart never published his work (which he called "astral geometry"), he sent a short summary of its main principles by letter to Carl Friedrich Gauß.
Motivated by the work of Schweikart, Taurinus examined the model of geometry on a "sphere" of imaginary radius, which he called "logarithmic-spherical" (now called hyperbolic geometry). He published his "theory of parallel lines" in 1825 and "Geometriae prima elementa" in 1826. For instance, in his "Geometriae prima elementa" on p. 66, Taurinus defined the hyperbolic law of cosines
When solved for and using hyperbolic functions, it has the form
Taurinus described his logarithmic-spherical geometry as the "third system" besides Euclidean geometry and spherical geometry, and pointed out that infinitely many systems exist depending on an arbitrary constant. While he noticed that no contradictions can be found in his logarithmic-spherical geometry, he remained convinced of the special role of Euclidean geometry. According to Paul Stäckel and Friedrich Engel, as well as Zacharias, Taurinus must be given credit as a founder of non-Euclidean trigonometry (together with Gauss), but his contributions cannot be considered as being on the same level as those of the main founders of non-Euclidean geometry, Nikolai Lobachevsky and János Bolyai.
Taurinus corresponded with Gauss about his ideas in 1824. In his reply, Gauss mentioned some of his own ideas on the subject, and encouraged Taurinus to further investigate this topic, but he also told Taurinus not to publicly cite Gauss. When Taurinus sent his works to Gauss, the latter didn't respond – according to Stäckel that was probably due to the fact that Taurinus mentioned Gauss in the prefaces of his books. In addition, Taurinus sent some copies of his "Geometriae prima elementa" to friends and authorities (Stäckel reported a positive reply by Georg Ohm). Dissatisfied with the lack of recognition, Taurinus burnt the remaining copies of that book – the only copy found by Stäckel and Engel was in the library of the University of Bonn. In 2015, another copy of the "Geometriae prima elementa" was digitized and made freely available online by the Univ
|
https://en.wikipedia.org/wiki/Clairaut%27s%20formula
|
Clairaut's formula may refer to:
Clairaut's equation (mathematical analysis)
Clairaut's relation (differential geometry)
Clairaut's theorem (calculus)
Clairaut's theorem (gravity)
|
https://en.wikipedia.org/wiki/George%20F.%20Pinder
|
George Francis Pinder (born 1942) is an American environmental engineer who is Professor of Civil and Environmental Engineering with a secondary appointment in Mathematics and Statistics at the University of Vermont. He also served as a professional witness in various notable environmental cases including Love Canal and the Woburn groundwater contamination incident.
He was elected a member of the National Academy of Engineering in 2010 for leadership in groundwater modeling applied to diverse problems in water resources. He is founding editor of the journal "Advances in Water Resources", and he served as editor-in-chief of the journal Numerical Methods for Partial Differential Equations.
Pinder's principal research interest is in the development of numerical methods to solve complex problems pertaining to groundwater contamination and supply. He has published approximately one hundred and thirty five papers in refereed journals in the area of quantitative analysis of subsurface flow and transport, as well as twelve books.
In popular culture
Pinder was featured as a character in the movie A Civil Action, based on the Woburn toxic waste case and starring John Travolta. He was portrayed by British actor Stephen Fry. He and his wife Phyllis were also featured in the book on which the film is based.
References
External links
George Pinder at the University of Vermont
Official George Pinder Website
Environmental engineers
American civil engineers
Engineering educators
University of Vermont faculty
Living people
Members of the United States National Academy of Engineering
Mathematicians from Vermont
1942 births
Love Canal
|
https://en.wikipedia.org/wiki/Semi-differentiability
|
In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifically, the function f is said to be right differentiable at a point a if, roughly speaking, a derivative can be defined as the function's argument x moves to a from the right, and left differentiable at a if the derivative can be defined as x moves to a from the left.
One-dimensional case
In mathematics, a left derivative and a right derivative are derivatives (rates of change of a function) defined for movement in one direction only (left or right; that is, to lower or higher values) by the argument of a function.
Definitions
Let f denote a real-valued function defined on a subset I of the real numbers.
If is a limit point of and the one-sided limit
exists as a real number, then f is called right differentiable at a and the limit ∂+f(a) is called the right derivative of f at a.
If is a limit point of and the one-sided limit
exists as a real number, then f is called left differentiable at a and the limit ∂–f(a) is called the left derivative of f at a.
If is a limit point of and and if f is left and right differentiable at a, then f is called semi-differentiable at a.
If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a symmetric derivative, which equals the arithmetic mean of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.
Remarks and examples
A function is differentiable at an interior point a of its domain if and only if it is semi-differentiable at a and the left derivative is equal to the right derivative.
An example of a semi-differentiable function, which is not differentiable, is the absolute value function , at a = 0. We find easily
If a function is semi-differentiable at a point a, it implies that it is continuous at a.
The indicator function 1[0,∞) is right differentiable at every real a, but discontinuous at zero (note that this indicator function is not left differentiable at zero).
Application
If a real-valued, differentiable function f, defined on an interval I of the real line, has zero derivative everywhere, then it is constant, as an application of the mean value theorem shows. The assumption of differentiability can be weakened to continuity and one-sided differentiability of f. The version for right differentiable functions is given below, the version for left differentiable functions is analogous.
Differential operators acting to the left or the right
Another common use is to describe derivatives treated as binary operators in infix notation, in which the derivatives is to be applied either to the left or right operands. This is useful, for example, when defining generalizations of the Poisson bracket. For a pair of functions f and g, t
|
https://en.wikipedia.org/wiki/Serre%20spectral%20sequence
|
In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation.
Cohomology spectral sequence
Let be a Serre fibration of topological spaces, and let F be the (path-connected) fiber. The Serre cohomology spectral sequence is the following:
Here, at least under standard simplifying conditions, the coefficient group in the -term is the q-th integral cohomology group of F, and the outer group is the singular cohomology of B with coefficients in that group.
Strictly speaking, what is meant is cohomology with respect to the local coefficient system on B given by the cohomology of the various fibers. Assuming for example, that B is simply connected, this collapses to the usual cohomology. For a path connected base, all the different fibers are homotopy equivalent. In particular, their cohomology is isomorphic, so the choice of "the" fiber does not give any ambiguity.
The abutment means integral cohomology of the total space X.
This spectral sequence can be derived from an exact couple built out of the long exact sequences of the cohomology of the pair , where is the restriction of the fibration over the p-skeleton of B. More precisely, using this notation,
f is defined by restricting each piece on to , g is defined using the coboundary map in the long exact sequence of the pair, and h is defined by restricting to
There is a multiplicative structure
coinciding on the E2-term with (−1)qs times the cup product, and with respect to which the differentials are (graded) derivations inducing the product on the -page from the one on the -page.
Homology spectral sequence
Similarly to the cohomology spectral sequence, there is one for homology:
where the notations are dual to the ones above.
Example computations
Hopf fibration
Recall that the Hopf fibration is given by . The -page of the Leray–Serre Spectral sequence reads
The differential goes down and right. Thus the only differential which is not necessarily is , because the rest have domain or codomain 0 (since they are on the E2-page). In particular, this sequence degenerates at E2 = E∞. The E3-page reads
The spectral sequence abuts to i.e. Evaluating at the interesting parts, we have and Knowing the cohomology of both are zero, so the differential is an isomorphism.
Sphere bundle on a complex projective variety
Given a complex n-dimensional projective variety there is a canonical family of line bundles for coming from the embedding . This is given by the global sections which send
If we construct a rank vector bundle which is a finite whitney sum of vector bundles we c
|
https://en.wikipedia.org/wiki/Gibbs%20measure
|
In mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems.
The canonical ensemble gives the probability of the system X being in state x (equivalently, of the random variable X having value x) as
Here, is a function from the space of states to the real numbers; in physics applications, is interpreted as the energy of the configuration x. The parameter is a free parameter; in physics, it is the inverse temperature. The normalizing constant is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble. Traditional approaches in statistical physics studied the limit of intensive properties as the size of a finite system approaches infinity (the thermodynamic limit). When the energy function can be written as a sum of terms that each involve only variables from a finite subsystem, the notion of a Gibbs measure provides an alternative approach. Gibbs measures were proposed by probability theorists such as Dobrushin, Lanford, and Ruelle and provided a framework to directly study infinite systems, instead of taking the limit of finite systems.
A measure is a Gibbs measure if the conditional probabilities it induces on each finite subsystem satisfy a consistency condition: if all degrees of freedom outside the finite subsystem are frozen, the canonical ensemble for the subsystem subject to these boundary conditions matches the probabilities in the Gibbs measure conditional on the frozen degrees of freedom.
The Hammersley–Clifford theorem implies that any probability measure that satisfies a Markov property is a Gibbs measure for an appropriate choice of (locally defined) energy function. Therefore, the Gibbs measure applies to widespread problems outside of physics, such as Hopfield networks, Markov networks, Markov logic networks, and boundedly rational potential games in game theory and economics.
A Gibbs measure in a system with local (finite-range) interactions maximizes the entropy density for a given expected energy density; or, equivalently, it minimizes the free energy density.
The Gibbs measure of an infinite system is not necessarily unique, in contrast to the canonical ensemble of a finite system, which is unique. The existence of more than one Gibbs measure is associated with statistical phenomena such as symmetry breaking and phase coexistence.
Statistical physics
The set of Gibbs measures on a system is always convex, so there is either a unique Gibbs measure (in which case the system is said to be "ergodic"), or there are infinitely many (and the system is called "nonergodic"). In the nonergodic case, the Gibbs measures can be expressed as the set of convex combinations of a much smaller number of sp
|
https://en.wikipedia.org/wiki/Burr%E2%80%93Erd%C5%91s%20conjecture
|
In mathematics, the Burr–Erdős conjecture was a problem concerning the Ramsey number of sparse graphs. The conjecture is named after Stefan Burr and Paul Erdős, and is one of many conjectures named after Erdős; it states that the Ramsey number of graphs in any sparse family of graphs should grow linearly in the number of vertices of the graph.
The conjecture was proven by Choongbum Lee. Thus it is now a theorem.
Definitions
If G is an undirected graph, then the degeneracy of G is the minimum number p such that every subgraph of G contains a vertex of degree p or smaller. A graph with degeneracy p is called p-degenerate. Equivalently, a p-degenerate graph is a graph that can be reduced to the empty graph by repeatedly removing a vertex of degree p or smaller.
It follows from Ramsey's theorem that for any graph G there exists a least integer
, the Ramsey number of G, such that any complete graph on at least vertices whose edges are coloured red or blue contains a monochromatic copy of G. For instance, the Ramsey number of a triangle is 6: no matter how the edges of a complete graph on six vertices are colored red or blue, there is always either a red triangle or a blue triangle.
The conjecture
In 1973, Stefan Burr and Paul Erdős made the following conjecture:
For every integer p there exists a constant cp so that any p-degenerate graph G on n vertices has Ramsey number at most cp n.
That is, if an n-vertex graph G is p-degenerate, then a monochromatic copy of G should exist in every two-edge-colored complete graph on cp n vertices.
Known results
Before the full conjecture was proved, it was first settled in some special cases. It was proven for bounded-degree graphs by ; their proof led to a very high value of cp, and improvements to this constant were made by and . More generally, the conjecture is known to be true for p-arrangeable graphs, which includes graphs with bounded maximum degree, planar graphs and graphs that do not contain a subdivision of Kp. It is also known for subdivided graphs, graphs in which no two adjacent vertices have degree greater than two.
For arbitrary graphs, the Ramsey number is known to be bounded by a function that grows only slightly superlinearly. Specifically, showed that there exists a constant cp such that, for any p-degenerate n-vertex graph G,
Notes
References
.
.
.
.
.
.
.
.
.
Graph theory
Ramsey theory
Conjectures that have been proved
Burr–Erdős conjecture
Theorems in graph theory
|
https://en.wikipedia.org/wiki/Masayoshi%20Nagata
|
Masayoshi Nagata (Japanese: 永田 雅宜 Nagata Masayoshi; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra.
Work
Nagata's compactification theorem shows that varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a group.
In 1959 he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension.
Nagata's conjecture on curves concerns the minimum degree of a plane curve specified to have given multiplicities at given points; see also Seshadri constant. Nagata's conjecture on automorphisms concerns the existence of wild automorphisms of polynomial algebras in three variables. Recent work has solved this latter problem in the affirmative.
Selected works
References
1927 births
2008 deaths
20th-century Japanese mathematicians
21st-century Japanese mathematicians
People from Ōbu, Aichi
Academic staff of Kyoto University
Nagoya University alumni
Algebraists
|
https://en.wikipedia.org/wiki/Seshadri%20constant
|
In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle L at a point P on an algebraic variety. It was introduced by Demailly to measure a certain rate of growth, of the tensor powers of L, in terms of the jets of the sections of the Lk. The object was the study of the Fujita conjecture.
The name is in honour of the Indian mathematician C. S. Seshadri.
It is known that Nagata's conjecture on algebraic curves is equivalent to the assertion that for more than nine general points, the Seshadri constants of the projective plane are maximal. There is a general conjecture for algebraic surfaces, the Nagata–Biran conjecture.
Definition
Let be a smooth projective variety, an ample line bundle on it, a point of , = { all irreducible curves passing through }.
.
Here, denotes the intersection number of and , measures how many times passing through .
Definition: One says that is the Seshadri constant of at the point , a real number. When is an abelian variety, it can be shown that is independent of the point chosen, and it is written simply .
References
Algebraic varieties
Vector bundles
Mathematical constants
|
https://en.wikipedia.org/wiki/Nagata%E2%80%93Biran%20conjecture
|
In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces.
Statement
Let X be a smooth algebraic surface and L be an ample line bundle on X of degree d. The Nagata–Biran conjecture states that for sufficiently large r the Seshadri constant satisfies
References
.
. See in particular page 3 of the pdf.
Algebraic surfaces
Conjectures
|
https://en.wikipedia.org/wiki/Fujita%20conjecture
|
In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved . It is named after Takao Fujita, who formulated it in 1985.
Statement
In complex geometry, the conjecture states that for a positive holomorphic line bundle L on a compact complex manifold M, the line bundle KM ⊗ L⊗m (where KM is a canonical line bundle of M) is
spanned by sections when m ≥ n + 1 ;
very ample when m ≥ n + 2,
where n is the complex dimension of M.
Note that for large m the line bundle KM ⊗ L⊗m is very ample by the standard Serre's vanishing theorem (and its complex analytic variant). Fujita conjecture provides an explicit bound on m, which is optimal for projective spaces.
Known cases
For surfaces the Fujita conjecture follows from Reider's theorem. For three-dimensional algebraic varieties, Ein and Lazarsfeld in 1993 proved the first part of the Fujita conjecture, i.e. that m≥4 implies global generation.
See also
Ohsawa–Takegoshi L2 extension theorem
References
.
.
.
.
.
External links
supporting facts to fujita conjecture
Algebraic geometry
Complex manifolds
Conjectures
Unsolved problems in geometry
|
https://en.wikipedia.org/wiki/Approximation%20in%20algebraic%20groups
|
In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups G over global fields k.
History
proved strong approximation for some classical groups.
Strong approximation was established in the 1960s and 1970s, for semisimple simply-connected algebraic groups over global fields. The results for number fields are due to and ; the function field case, over finite fields, is due to and . In the number field case Platonov also proved a related result over local fields called the Kneser–Tits conjecture.
Formal definitions and properties
Let G be a linear algebraic group over a global field k, and A the adele ring of k. If S is a non-empty finite set of places of k, then we write AS for the ring of S-adeles and AS for the product of the completions ks, for s in the finite set S. For any choice of S, G(k) embeds in G(AS) and G(AS).
The question asked in weak approximation is whether the embedding of G(k) in G(AS) has dense image. If the group G is connected and k-rational, then it satisfies weak approximation with respect to any set S . More generally, for any connected group G, there is a finite set T of finite places of k such that G satisfies weak approximation with respect to any set S that is disjoint with T . In particular, if k is an algebraic number field then any connected group G satisfies weak approximation with respect to the set S = S∞ of infinite places.
The question asked in strong approximation is whether the embedding of G(k) in G(AS) has dense image, or equivalently whether the set
G(k)G(AS)
is a dense subset in G(A). The main theorem of strong approximation states that a non-solvable linear algebraic group G over a global field k has strong approximation for the finite set S if and only if its radical N is unipotent, G/N is simply connected, and each almost simple component H of G/N has a non-compact component Hs for some s in S (depending on H).
The proofs of strong approximation depended on the Hasse principle for algebraic groups, which for groups of type E8 was only proved several years later.
Weak approximation holds for a broader class of groups, including adjoint groups and inner forms of Chevalley groups, showing that the strong approximation property is restrictive.
See also
Superstrong approximation
References
Algebraic groups
Diophantine geometry
|
https://en.wikipedia.org/wiki/American%20Mathematical%20Association%20of%20Two-Year%20Colleges
|
The American Mathematical Association of Two-Year Colleges (AMATYC) is an organization dedicated to the improvement of education in the first two years of college mathematics in the United States and Canada. AMATYC hosts an annual conference, summer institutes, workshops and mentoring for teachers in and outside math, and a semiannual math competition. AMATYC publishes one refereed journal, MathAMATYC Educator, and issues position statements on matters of mathematics education.
The math competition is held in spring and fall semester each year and is limited to problems in precalculus. Only students enrolled in two-year colleges are eligible to participate. Only students who haven't received any degree/diploma, including within or outside of the U.S, can enter the competition.
AMATYC was founded in 1974. Its office is at Southwest Tennessee Community College in Memphis, Tennessee.
AMATYC is divided into eight regions: Central, Mid-Atlantic, Midwest, Northeast, Northwest, Southeast, Southwest, and West. A vice president is assigned to each region.
See also
Mathematical Association of America
National Council of Teachers of Mathematics
External links
Student Math League home page
Mathematics education in the United States
1974 establishments in Tennessee
Organizations based in Memphis, Tennessee
Organizations established in 1974
|
https://en.wikipedia.org/wiki/Twiddle
|
Twiddle or twiddling may refer to:
Twiddle (band), an American rock band
Twiddle factor, used in fast Fourier transforms in mathematics
Thumb twiddling, action of the hands
Twiddly bits, English idiom
Tilde character ( ~ ), sometimes referred to as "twiddle" or "squiggle"
Mr Twiddle, zookeeper character in Wally Gator animated TV series
"Twiddling", the constant fine-tuning of online platforms that is part of the enshittification process
See also
Bit twiddler (disambiguation), for various uses in computing
Twiddler, a one-handed input device
|
https://en.wikipedia.org/wiki/List%20of%20complex%20and%20algebraic%20surfaces
|
This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification.
Kodaira dimension −∞
Rational surfaces
Projective plane
Quadric surfaces
Cone (geometry)
Cylinder
Ellipsoid
Hyperboloid
Paraboloid
Sphere
Spheroid
Rational cubic surfaces
Cayley nodal cubic surface, a certain cubic surface with 4 nodes
Cayley's ruled cubic surface
Clebsch surface or Klein icosahedral surface
Fermat cubic
Monkey saddle
Parabolic conoid
Plücker's conoid
Whitney umbrella
Rational quartic surfaces
Châtelet surfaces
Dupin cyclides, inversions of a cylinder, torus, or double cone in a sphere
Gabriel's horn
Right circular conoid
Roman surface or Steiner surface, a realization of the real projective plane in real affine space
Tori, surfaces of revolution generated by a circle about a coplanar axis
Other rational surfaces in space
Boy's surface, a sextic realization of the real projective plane in real affine space
Enneper surface, a nonic minimal surface
Henneberg surface, a minimal surface of degree 15
Bour's minimal surface, a surface of degree 16
Richmond surfaces, a family of minimal surfaces of variable degree
Other families of rational surfaces
Coble surfaces
Del Pezzo surfaces, surfaces with an ample anticanonical divisor
Hirzebruch surfaces, rational ruled surfaces
Segre surfaces, intersections of two quadrics in projective 4-space
Unirational surfaces of characteristic 0
Veronese surface, the Veronese embedding of the projective plane into projective 5-space
White surfaces, the blow-up of the projective plane at points by the linear system of degree- curves through those points
Bordiga surfaces, the White surfaces determined by families of quartic curves
Non-rational ruled surfaces
Class VII surfaces
Vanishing second Betti number:
Hopf surfaces
Inoue surfaces; several other families discovered by Inoue have also been called "Inoue surfaces"
Positive second Betti number:
Enoki surfaces
Inoue–Hirzebruch surfaces
Kato surfaces
Kodaira dimension 0
K3 surfaces
Kummer surfaces
Tetrahedroids, special Kummer surfaces
Wave surface, a special tetrahedroid
Plücker surfaces, birational to Kummer surfaces
Weddle surfaces, birational to Kummer surfaces
Smooth quartic surfaces
Supersingular K3 surfaces
Enriques surfaces
Reye congruences, the locus of lines that lie on two out of three general quadric surfaces in projective space
Abelian surfaces
Horrocks–Mumford surfaces, surfaces of degree 10 in projective 4-space that are the zero locus of sections of the rank-two Horrocks–Mumford bundle
Other classes of dimension-0 surfaces
Non-classical Enriques surfaces, a variation on the notion of Enriques surfaces that only exist in characteristic two
Hyperelliptic surfaces or bielliptic surfaces; quasi-hyperelliptic surfaces are a variation of this notion that exist only in characteristics two and three
Ko
|
https://en.wikipedia.org/wiki/A-League%20Men%20records%20and%20statistics
|
The A-League Men is an Australian professional league for association football clubs. At the top of the Australian soccer league system, it is the country's primary soccer competition and is contested by 12 clubs. The competition was formed in April 2004, following a number of issues including financial problems in the National Soccer League. Those records and statistics of the A-League Men are listed below. All updated as of 6 March 2023.
Team records
Titles
Most Premiership titles: 4, Sydney FC
Most Championship titles: 5, Sydney FC
Most consecutive Premiership title wins: 2, Sydney FC (2016–17, 2017–18); Melbourne City (2020–21, 2021–22)
Most consecutive Championship title wins: 2, Brisbane Roar (2011, 2012), Sydney FC (2019, 2020)
Biggest Premiership title winning margin: 17 points, 2016–17; Sydney FC (66 points) over Melbourne Victory (49 points)
Smallest Premiership title winning margin: 0 points and same goal difference, 2008–09; Melbourne Victory over Adelaide United through more goals scored.
Points
Most points in a season: 66, Sydney FC (2016–17)
Most home points in a season: 33, Brisbane Roar (2010–11, 2013–14)
Most away points in a season: 34, Sydney FC (2014–15)
Fewest points in a season: 6, New Zealand Knights (2005–06)
Fewest home points in a season: 2, New Zealand Knights (2005–06)
Fewest away points in a season: 4, New Zealand Knights (2005–06)
Most points in a season without winning the league: 57, Central Coast Mariners (2010–11)
Fewest points in a season while winning the league: 34, Central Coast Mariners (2007–08)
Wins
Most wins in total: 202, Sydney FC
Most wins in a season: 20, Sydney FC (2016–17, 2017–18)
Most home wins in a season: 10
Brisbane Roar (2010–11, 2013–14)
Sydney FC (2016–17)
Most away wins in a season: 10
Western Sydney Wanderers (2012–13)
Sydney FC (2016–17)
Fewest wins in a season: 1, New Zealand Knights (2005–06)
Fewest home wins in a season: 0, New Zealand Knights (2005–06)
Fewest away wins in a season: 1, New Zealand Knights (2005–06)
Most consecutive wins: 10, Western Sydney Wanderers (13 January 2013 – 16 March 2013)
Most consecutive home wins: 8, Melbourne Victory (21 November 2008 – 28 February 2009)
Most consecutive away wins: 8, Melbourne Victory (9 September 2006 – 30 December 2006)
Most consecutive games without a win: 19
New Zealand Knights (18 September 2005 – 27 August 2006)
Melbourne City (24 February 2013 – 17 January 2014)
Most consecutive home games without a win: 11, New Zealand Knights (2 September 2005 – 27 August 2006)
Most consecutive away games without a win: 22, Melbourne City (2 February 2012 – 16 February 2014)
Defeats
Most defeats in total: 183, Central Coast Mariners
Most defeats in a season: 20, Central Coast Mariners (2015–16)
Most home defeats in a season: 9, North Queensland Fury (2010–11)
Most away defeats in a season: 12, Melbourne City (2012–13)
Fewest defeats in a season: 1
Brisbane Roar (2010–11)
Sydney FC (2016–17)
Fewest home defeats in a season: 0
Brisbane Roa
|
https://en.wikipedia.org/wiki/Standard%20normal%20table
|
In statistics, a standard normal table, also called the unit normal table or Z table, is a mathematical table for the values of , the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities.
Normal and standard normal distribution
Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The standard normal distribution, represented by , is the normal distribution having a mean of 0 and a standard deviation of 1.
Conversion
If is a random variable from a normal distribution with mean and standard deviation , its Z-score may be calculated from by subtracting and dividing by the standard deviation:
If is the mean of a sample of size from some population in which the mean is and the standard deviation is , the standard error is
If is the total of a sample of size from some population in which the mean is and the standard deviation is , the expected total is and the standard error is
Reading a Z table
Formatting / layout
tables are typically composed as follows:
The label for rows contains the integer part and the first decimal place of .
The label for columns contains the second decimal place of .
The values within the table are the probabilities corresponding to the table type. These probabilities are calculations of the area under the normal curve from the starting point (0 for cumulative from mean, negative infinity for cumulative and positive infinity for complementary cumulative) to .
Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table.
To find a negative value such as -0.83, one could use a cumulative table for negative z-values which yield a probability of 0.20327.
But since the normal distribution curve is symmetrical, probabilities for only positive values of are typically given. The user might have to use a complementary operation on the absolute value of , as in the example below.
Types of tables
tables use at least three different conventions:
Cumulative from mean gives a probability that a statistic is between 0 (mean) and . Example: .
Cumulative gives a probability that a statistic is less than . This equates to the area of the distribution below . Example: .
Complementary cumulative gives a probability that a statistic is greater than . This equates to the area of the distribution above .
Example: Find . Since this is the portion of the
|
https://en.wikipedia.org/wiki/Crunode
|
In mathematics, a crunode (archaic) or node is a point where a curve intersects itself so that both branches of the curve have distinct tangent lines at the point of intersection. A crunode is also known as an ordinary double point.
For a plane curve, defined as the locus of points , where is a smooth function of variables and ranging over the real numbers, a crunode of the curve is a singularity of the function , where both partial derivatives and vanish. Further the Hessian matrix of second derivatives will have both positive and negative eigenvalues.
See also
Singular point of a curve
Acnode
Cusp
Tacnode
Saddle point
References
Curves
Algebraic curves
es:Punto singular de una curva#Crunodos
|
https://en.wikipedia.org/wiki/Pad%C3%A9%20approximant
|
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series.
The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel–Padé analysis.
The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method. Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the "incomplete two-point Padé approximation", in which the ordinary Padé approximation improves the method truncating a Taylor series.
Definition
Given a function f and two integers m ≥ 0 and n ≥ 1, the Padé approximant of order [m/n] is the rational function
which agrees with f(x) to the highest possible order, which amounts to
Equivalently, if is expanded in a Maclaurin series (Taylor series at 0), its first terms would equal the first terms of , and thus
When it exists, the Padé approximant is unique as a formal power series for the given m and n.
The Padé approximant defined above is also denoted as
Computation
For given x, Padé approximants can be computed by Wynn's epsilon algorithm and also other sequence transformations from the partial sums
of the Taylor series of f, i.e., we have
f can also be a formal power series, and, hence, Padé approximants can also be applied to the summation of divergent series.
One way to compute a Padé approximant is via the extended Euclidean algorithm for the polynomial greatest common divisor. The relation
is equivalent to the existence of some factor such that
which can be interpreted as the Bézout identity of one step in the computation of the extended greatest common divisor of the polynomials and .
Recall that, to compute the greatest common divisor of two polynomials p and q, one computes via long division the remainder sequence
k = 1, 2, 3, ... with , until . For the Bézout identities of the extended greatest common divisor one computes simultaneously the two polynomial sequences
to obtain in each step the Bézout identity
For the [m/n] approx
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.