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https://en.wikipedia.org/wiki/Tschirnhausen%20cubic | In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation
where is the secant function.
History
The curve was studied by von Tschirnhaus, de L'Hôpital, and Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by Raymond Clare Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan.
Other equations
Put . Then applying triple-angle formulas gives
giving a parametric form for the curve. The parameter t can be eliminated easily giving the Cartesian equation
.
If the curve is translated horizontally by 8a and the signs of the variables are changed, the equations of the resulting right-opening curve are
and in Cartesian coordinates
.
This gives the alternative polar form
.
Generalization
The Tschirnhausen cubic is a Sinusoidal spiral with n = −1/3.
References
J. D. Lawrence, A Catalog of Special Plane Curves. New York: Dover, 1972, pp. 87-90.
External links
"Tschirnhaus' Cubic" at MacTutor History of Mathematics archive
Tschirnhausen cubic at mathcurve.com
Cubic curves |
https://en.wikipedia.org/wiki/Skoda%E2%80%93El%20Mir%20theorem | The Skoda–El Mir theorem is a theorem of complex geometry,
stated as follows:
Theorem (Skoda, El Mir, Sibony). Let X be a complex manifold, and
E a closed complete pluripolar set in X. Consider a closed positive current on
which is locally integrable around E. Then the trivial extension of to X is closed on X.
Notes
References
J.-P. Demailly, L² vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)
Complex manifolds
Several complex variables
Theorems in geometry |
https://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert%20correspondence | In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generalizations of this.
The original setting appearing in Hilbert's twenty-first problem was for the Riemann sphere, where it was about the existence of systems of linear regular differential equations with prescribed monodromy representations.
First the Riemann sphere may be replaced by an arbitrary Riemann surface and then, in higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1.
There is a correspondence between certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions.
Such a result was proved for algebraic connections with regular singularities by Pierre Deligne (1970, generalizing existing work in the case of Riemann surfaces) and more generally for regular holonomic D-modules by Masaki Kashiwara (1980, 1984) and Zoghman Mebkhout (1980, 1984) independently.
In the setting of nonabelian Hodge theory, the Riemann-Hilbert correspondence provides a complex analytic isomorphism between two of the three natural algebraic structures on the moduli spaces, and so is naturally viewed as a nonabelian analogue of the comparison isomorphism between De Rham cohomology and singular/Betti cohomology.
Statement
Suppose that X is a smooth complex algebraic variety.
Riemann–Hilbert correspondence (for regular singular connections):
there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces on X. For X connected, the category of local systems is also equivalent to the category of complex representations of the fundamental group of X.
Thus such connections give a purely algebraic way to access the finite dimensional representations of the topological fundamental group.
The condition of regular singularities means that locally constant sections of the bundle (with respect to the flat connection) have moderate growth at points of Y − X, where Y is an algebraic compactification of X. In particular, when X is compact, the condition of regular singularities is vacuous.
More generally there is the
Riemann–Hilbert correspondence (for regular holonomic D-modules): there is a functor DR called the de Rham functor, that is an equivalence from the category of holonomic D-modules on X with regular singularities to the category of perverse sheaves on X.
By considering the irreducible elements of each category, this gives a 1:1 correspondence between isomorphism classes of
irreducible holonomic D-modules on X with regular singularities,
and
intersection cohomology complexes of irreducible closed subvarie |
https://en.wikipedia.org/wiki/Allen%27s%20interval%20algebra | Allen's interval algebra is a calculus for temporal reasoning that was introduced by James F. Allen in 1983.
The calculus defines possible relations between time intervals and provides a composition table that can be used as a basis
for reasoning about temporal descriptions of events.
Formal description
Relations
The following 13 base relations capture the possible relations between two intervals.
Using this calculus, given facts can be formalized and then used for automatic reasoning. Relations between intervals are formalized as sets of base relations.
The sentence
During dinner, Peter reads the newspaper. Afterwards, he goes to bed.
is formalized in Allen's Interval Algebra as follows:
In general, the number of different relations between n intervals, starting with n = 0, is 1, 1, 13, 409, 23917, 2244361... OEIS A055203. The special case shown above is for n = 2.
Composition of relations between intervals
For reasoning about the relations between temporal intervals, Allen's interval algebra provides a composition table. Given the relation between and and the relation between and , the composition table allows for concluding about the relation between and . Together with a converse operation, this turns Allen's interval algebra into a relation algebra.
For the example, one can infer .
Extensions
Allen's interval algebra can be used for the description of both temporal intervals and spatial configurations. For the latter use, the relations are interpreted as describing the relative position of spatial objects. This also works for three-dimensional objects by listing the relation for each coordinate separately.
The study of overlapping markup uses a similar algebra (see ). Its models have more variations depending on whether endpoints of document structures are permitted to be truly co-located, or merely [tangent].
Implementations
A simple java library implementing the concept of Allen's temporal relations and the path consistency algorithm
Java library implementing Allen's Interval Algebra (incl. data and index structures, e.g., interval tree)
OWL-Time Time Ontology in OWL an OWL-2 DL ontology of temporal concepts, for describing the temporal properties of resources in the world or described in Web pages.
GQR is a reasoner for Allen's interval algebra (and many others)
qualreas is a Python framework for qualitative reasoning over networks of relation algebras, such as RCC-8, Allen's interval algebra, and Allen's algebra integrated with Time Points and situated in either Left- or Right-Branching Time.
SparQ is a reasoner for Allen's interval algebra (and many others)
EveXL is a small domain-specific language for the detection of events that implements the Interval Algebra's operators via ASCII art patterns.
See also
Temporal logic
Logic
Region connection calculus
Spatial relation (analog)
Commonsense reasoning
References
Sources
Knowledge representation
Constraint programming |
https://en.wikipedia.org/wiki/Strange%20Geometry | Strange Geometry is the third studio album by English indie pop band The Clientele. The album was released on 30 August 2005 by Merge Records and on 5 September 2005 by Pointy Records. It was recorded in Walthamstow, London, and received generally positive reviews upon release.
The album's first single was "Since K Got Over Me", which was released on 22 August 2005 in limited quantities on 7" vinyl, backed with "Devil Got My Woman" and "I Believe It". The song "(I Can't Seem To) Make You Mine" originally appeared on a split single with The Relict in 2001, featuring additional vocals by Pam Berry.
The album cover features the 1963 painting The Viaduct by Paul Delvaux.
Track listing
Personnel
Credits for Strange Geometry adapted from album liner notes.
The Clientele
Alasdair MacLean – vocals, guitar, bouzouki, Rhodes piano, bells
James Hornsey – bass, piano, Hammond organ, percussion
Mark Keen – drums, vocals, piano, Hammond organ, percussion
Additional musicians
Nikki Gleed – violin (1st)
Sarah Squires – violin (2nd)
Hannah Stewart – cello
Charlie Stock – viola
Production
Brian O'Shaughnessy – production
References
External links
2005 albums
The Clientele albums
Merge Records albums |
https://en.wikipedia.org/wiki/List%20of%20AFL%20debuts%20in%202007 | This is a listing of Australian rules footballers who made their debut with a club during the 2007 Australian Football League season.
References
Australian rules football records and statistics
Australian rules football-related lists
2007 in Australian rules football
2007 Australian Football League season |
https://en.wikipedia.org/wiki/Yabroud | Yabroud or Yabrud () is a city in Syria, located in the Rif Dimashq (i.e. Damascus' countryside) governorate about north of the capital Damascus. According to the Syria Central Bureau of Statistics (CBS), Yabroud had a population of 25,891 in the 2004 census.
Etymology
The name Yabroud is said to have originated from an Aramaic word meaning "cold"; the city rests upon the Qalamoun Mountains slopes (Anti-Lebanon) at a height of 1,550 m.
History
The city is known for its ancient caves, most notably the Iskafta cave (where, in 1930, a thirty-year-old German traveller and self-taught archeologist Alfred Rust made many important pre-historical findings), which dates back to a period known as Jabroudian culture, named after Yabroud; and the Yabroud temple, which was once Jupiter Yabroudiss temple but later became "Konstantin and Helena Cathedral". Yabroud is home of the oldest church in Syria. The Natufian archeological site Yabroud III is named for the town of Yabroud.
Yabroud was mentioned in the pottery tablets of Mesopotamia in the 1st century B.C., and Ptolemy's writings in the 2nd century A.D.
In 1838, its inhabitants were Sunni Muslim, Melkite Catholic and Greek Orthodox Christians.
During the Syrian Civil War the city was the center of the Battle of Yabroud in March 2014.
Notable people
The parents of former President of Argentina Carlos Menem were both born in Yabroud; they emigrated to Argentina before the end of World War I.
Antun Maqdisi (1914–2005), a Syrian philosopher, politician and human rights activist, died in Yabroud.
Elias Konsol (1914–1981), a Syrian-Argentine poet, writer and journalist of the Mahjar.
Gregory Atta lived for a time in Yabroud
Youssef Halaq (1939–2007), a Syrian writer and literary translator.
Zaki Konsol (1916–1994), a poet and writer.
Khaled M. Al Baradea (1934–2008), a poet and writer.
Asem Al Basha (born 1948), an artist and writer.
George Haswani, Syrian-Russian businessman.
Saeed Alnahhal, Syrian-Swedish journalist.
References
Gallery
Bibliography
Cities in Syria
Populated places in Yabroud District
Natufian sites
Emesene dynasty
Archaeological sites in Rif Dimashq Governorate
Christian communities in Syria |
https://en.wikipedia.org/wiki/Bent%20bond%20%28disambiguation%29 | The term bent bond generally refers to any form of bond in a structure that resembles the shape of a banana, but may also refer to:
Bent molecular geometry in VSEPR Theory and molecular geometry, a structure which contains one or two lone pairs.
Rarely, in Thermodynamics, as the application of heat to a system contributing towards the breaking of bonds. |
https://en.wikipedia.org/wiki/Yetter%E2%80%93Drinfeld%20category | In mathematics a Yetter–Drinfeld category is a special type of braided monoidal category. It consists of modules over a Hopf algebra which satisfy some additional axioms.
Definition
Let H be a Hopf algebra over a field k. Let denote the coproduct and S the antipode of H. Let V be a vector space over k. Then V is called a (left left) Yetter–Drinfeld module over H if
is a left H-module, where denotes the left action of H on V,
is a left H-comodule, where denotes the left coaction of H on V,
the maps and satisfy the compatibility condition
for all ,
where, using Sweedler notation, denotes the twofold coproduct of , and .
Examples
Any left H-module over a cocommutative Hopf algebra H is a Yetter–Drinfeld module with the trivial left coaction .
The trivial module with , , is a Yetter–Drinfeld module for all Hopf algebras H.
If H is the group algebra kG of an abelian group G, then Yetter–Drinfeld modules over H are precisely the G-graded G-modules. This means that
,
where each is a G-submodule of V.
More generally, if the group G is not abelian, then Yetter–Drinfeld modules over H=kG are G-modules with a G-gradation
, such that .
Over the base field all finite-dimensional, irreducible/simple Yetter–Drinfeld modules over a (nonabelian) group H=kG are uniquely given through a conjugacy class together with (character of) an irreducible group representation of the centralizer of some representing :
As G-module take to be the induced module of :
(this can be proven easily not to depend on the choice of g)
To define the G-graduation (comodule) assign any element to the graduation layer:
It is very custom to directly construct as direct sum of X´s and write down the G-action by choice of a specific set of representatives for the -cosets. From this approach, one often writes
(this notation emphasizes the graduation, rather than the module structure)
Braiding
Let H be a Hopf algebra with invertible antipode S, and let V, W be Yetter–Drinfeld modules over H. Then the map ,
is invertible with inverse
Further, for any three Yetter–Drinfeld modules U, V, W the map c satisfies the braid relation
A monoidal category consisting of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is called a Yetter–Drinfeld category. It is a braided monoidal category with the braiding c above. The category of Yetter–Drinfeld modules over a Hopf algebra H with bijective antipode is denoted by .
References
Hopf algebras
Quantum groups
Monoidal categories |
https://en.wikipedia.org/wiki/D%C3%AAnis%20Marques | Dênis Marques do Nascimento or simply Dênis Marques (born February 22, 1981), is a retired Brazilian football striker.
Club statistics
Flamengo career statistics
(Correct )
according to combined sources on the Flamengo official website.
Santa Cruz career statistics
Honours
Individual
Brazilian Cup Top Scorer: 2007
Club
Atlético Paranaense
Paraná State League: 2005
Flamengo
Brazilian Série A: 2009
Santa Cruz
Pernambuco State League: 2012, 2013
Campeonato Brasileiro Série C: 2013
References
External links
CBF
furacao
rubronegro
atleticopr
ipcdigital
1981 births
Living people
Footballers from Maceió
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Kuwait
Club Athletico Paranaense players
Brazilian expatriate sportspeople in Kuwait
Mogi Mirim Esporte Clube players
Brazilian expatriate sportspeople in Japan
CR Flamengo footballers
Santa Cruz Futebol Clube players
ABC Futebol Clube players
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
Omiya Ardija players
Men's association football forwards
Kuwait SC players |
https://en.wikipedia.org/wiki/Hochschild%20homology | In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by .
Definition of Hochschild homology of algebras
Let k be a field, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product of A with its opposite algebra. Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as Ae-modules. defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by
Hochschild complex
Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by
with boundary operator defined by
where is in A for all and . If we let
then , so is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M. Henceforth, we will write as simply .
Remark
The maps are face maps making the family of modules a simplicial object in the category of k-modules, i.e., a functor Δo → k-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δo is the opposite category of Δ. The degeneracy maps are defined by
Hochschild homology is the homology of this simplicial module.
Relation with the Bar complex
There is a similar looking complex called the Bar complex which formally looks very similar to the Hochschild complexpg 4-5. In fact, the Hochschild complex can be recovered from the Bar complex asgiving an explicit isomorphism.
As a derived self-intersection
There's another useful interpretation of the Hochschild complex in the case of commutative rings, and more generally, for sheaves of commutative rings: it is constructed from the derived self-intersection of a scheme (or even derived scheme) over some base scheme . For example, we can form the derived fiber productwhich has the sheaf of derived rings . Then, if embed with the diagonal mapthe Hochschild complex is constructed as the pullback of the derived self intersection of the diagonal in the diagonal product schemeFrom this interpretation, it should be clear the Hochschild homology should have some relation to the Kähler differentials since the Kähler differentials can be defined using a self-intersection from the diagonal, or more generally, the cotangent complex since this is the derived replacement for the Kähler differentials. We can recover the original definition of the Hochschild complex of a commutative -algebra by setting and Then, the Hochschild complex is quasi-isomorphic toIf is a flat -algebra, then there's the chain of isomorphismgiving an alterna |
https://en.wikipedia.org/wiki/Geometric%20Langlands%20correspondence | In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by function fields and applying techniques from algebraic geometry. The geometric Langlands correspondence relates algebraic geometry and representation theory.
The specific case of the geometric Langlands correspondence for general linear groups over function fields was proven by Laurent Lafforgue in 2002, where it follows as a consequence of Lafforgue's theorem.
History
In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory and representation theory. Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case. Establishing the Langlands correspondence in the number theoretic context has proven extremely difficult. As a result, some mathematicians have posed the geometric Langlands correspondence.
Langlands correspondences can be formulated for global fields (as well as local fields), which are classified into number fields or global function fields. The classical Langlands correspondence is formulated for number fields. The geometric Langlands correspondence is instead formulated for global function fields, which in some sense have proven easier to deal with.
In 2002, the geometric Langlands correspondence was proven for general linear groups over a function field by Laurent Lafforgue.
Connection to physics
In a paper from 2007, Anton Kapustin and Edward Witten described a connection between the geometric Langlands correspondence and S-duality, a property of certain quantum field theories.
In 2018, when accepting the Abel Prize, Langlands delivered a paper reformulating the geometric program using tools similar to his original Langlands correspondence.
Notes
References
External links
Quantum geometric Langlands correspondence at nLab
Algebraic geometry
Langlands program
Representation theory |
https://en.wikipedia.org/wiki/Braided%20Hopf%20algebra | In mathematics, a braided Hopf algebra is a Hopf algebra in a braided monoidal category. The most common braided Hopf algebras are objects in a Yetter–Drinfeld category of a Hopf algebra H, particularly the Nichols algebra of a braided vector space in that category.
The notion should not be confused with quasitriangular Hopf algebra.
Definition
Let H be a Hopf algebra over a field k, and assume that the antipode of H is bijective. A Yetter–Drinfeld module R over H is called a braided bialgebra in the Yetter–Drinfeld category if
is a unital associative algebra, where the multiplication map and the unit are maps of Yetter–Drinfeld modules,
is a coassociative coalgebra with counit , and both and are maps of Yetter–Drinfeld modules,
the maps and are algebra maps in the category , where the algebra structure of is determined by the unit and the multiplication map
Here c is the canonical braiding in the Yetter–Drinfeld category .
A braided bialgebra in is called a braided Hopf algebra, if there is a morphism of Yetter–Drinfeld modules such that
for all
where in slightly modified Sweedler notation – a change of notation is performed in order to avoid confusion in Radford's biproduct below.
Examples
Any Hopf algebra is also a braided Hopf algebra over
A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra .
The tensor algebra of a Yetter–Drinfeld module is always a braided Hopf algebra. The coproduct of is defined in such a way that the elements of V are primitive, that is
The counit then satisfies the equation for all
The universal quotient of , that is still a braided Hopf algebra containing as primitive elements is called the Nichols algebra. They take the role of quantum Borel algebras in the classification of pointed Hopf algebras, analogously to the classical Lie algebra case.
Radford's biproduct
For any braided Hopf algebra R in there exists a natural Hopf algebra which contains R as a subalgebra and H as a Hopf subalgebra. It is called Radford's biproduct, named after its discoverer, the Hopf algebraist David Radford. It was rediscovered by Shahn Majid, who called it bosonization.
As a vector space, is just . The algebra structure of is given by
where , (Sweedler notation) is the coproduct of , and is the left action of H on R. Further, the coproduct of is determined by the formula
Here denotes the coproduct of r in R, and is the left coaction of H on
References
Andruskiewitsch, Nicolás and Schneider, Hans-Jürgen, Pointed Hopf algebras, New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.
Hopf algebras |
https://en.wikipedia.org/wiki/Spectral%20element%20method | In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method (see Development History)
Discussion
The spectral method expands the solution in trigonometric series, a chief advantage being that the resulting method is of a very high order.
This approach relies on the fact that trigonometric polynomials are an orthonormal basis for .
The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy.
Such polynomials are usually orthogonal Chebyshev polynomials or very high order Lagrange polynomials over non-uniformly spaced nodes.
In SEM computational error decreases exponentially as the order of approximating polynomial increases, therefore a fast convergence of solution to the exact solution is realized with fewer degrees of freedom of the structure in comparison with FEM.
In structural health monitoring, FEM can be used for detecting large flaws in a structure, but as the size of the flaw is reduced there is a need to use a high-frequency wave. In order to simulate the propagation of a high-frequency wave, the FEM mesh required is very fine resulting in increased computational time. On the other hand, SEM provides good accuracy with fewer degrees of freedom.
Non-uniformity of nodes helps to make the mass matrix diagonal, which saves time and memory and is also useful for adopting a central difference method (CDM).
The disadvantages of SEM include difficulty in modeling complex geometry, compared to the flexibility of FEM.
Although the method can be applied with a modal piecewise orthogonal polynomial basis, it is most often implemented with a nodal tensor product Lagrange basis. The method gains its efficiency by placing the nodal points at the Legendre-Gauss-Lobatto (LGL) points and performing the Galerkin method integrations with a reduced Gauss-Lobatto quadrature using the same nodes. With this combination, simplifications result such that mass lumping occurs at all nodes and a collocation procedure results at interior points.
The most popular applications of the method are in computational fluid dynamics and modeling seismic wave propagation.
A-priori error estimate
The classic analysis of Galerkin methods and Céa's lemma holds here and it can be shown that, if is the solution of the weak equation, is the approximate solution and :
where is related to the discretization of the domain (ie. element length), is independent from , and is no larger than the degree of the piecewise polynomial basis. Similar results can be obtained to bound the error in stronger topologies. If
As we increase , we can |
https://en.wikipedia.org/wiki/Israel%20national%20football%20team%20records%20and%20statistics | This article lists various football records in relation to the Israel national football team.
Records in this section refer to Eretz Israel football team from its first official game in 1934 to 1948 and to the Israel national football team since Israel Declaration of Independence in 1948.
The page is updated where necessary after each Israel match, and is correct as of 15 November 2015.
Appearances
Most appearances: Yossi Benayoun, 102; 18 November 1998 – 9 October 2017
Longest Israel career: Yossi Benayoun, 18 years 325 days; 18 November 1998 – 9 October 2017
Shortest Israel career: Ze'ev Haimovich, 3 minutes, 17 October 2007 vs Belarus
Youngest player: Gai Assulin, 16 years 350 days; 26 March 2008, vs. Chile
Oldest player: Yossi Benayoun, 37 years 157 days; 9 October 2017, vs. Spain
Most appearances as a substitute: Yossi Benayoun, 33
Most times substituted off: Eyal Berkovic, 44
Most appearances as a substitute without ever starting a game: Ofer Shitrit, 6
Goals
First goal (as Eretz Israel): Avraham Nudelman; 16 March 1934, 1–7 vs. Egypt
First goal (as Israel): Shmuel Ben-Dror; 26 September 1948, 1–3 vs. USA Olympic Team
Most goals: Eran Zahavi, 33; 2 September 2010 – 12 November 2021
Most goals in a match: Mordechai Spiegler, 4; 25 September 1968, 4–0 vs. USA
Youngest scorer: Ben Sahar, 17 years 206 days; 28 March 2007, vs. Estonia
Oldest scorer: Avi Nimni, 33 years 231 days; 7 September 2005, vs. Faroe Islands
Youngest player to score a Hat-trick: Shlomi Arbeitman; 18 years 270 days, 18 February 2004, vs. Azerbaijan
Scoring in most consecutive matches: Yehoshua Feigenbaum, 5; 28 May 1974 – 9 October 1974
Most goals on debut: Shlomi Arbeitman, 3; 18 February 2004, 6–0 vs. Azerbaijan
Most different goalscorers in one match: 6, 18 January 1991, 7–0 vs. Estonia
Most goals scored in a single game: 9; 28 February 2001, 9–0 vs. Chinese Taipei on 23 March 1988
Most goals scored during the first half: 5 goals, 18 January 1999 vs. Estonia and 6 September 1974 vs. Philippines
Most goals scored during the second half: 7, vs Chinese Taipei on 23 March 1988
Highest scoring draw 3–3 vs United States on 15 September 1968, vs Greece on 12 March 1969, vs Wales on 8 February 1989, vs Croatia on 9 February 2005 and vs Portugal on 22 March 2013
Largest defeat: 7–1 vs Egypt on 16 March 1934 and vs Germany on 13 February 2002
Largest defeat at home: 5-0 vs Denmark on 13 November 1999, 2–7 vs Argentina on 4 May 1986
Most goals conceded during a home game: 7, vs Argentina on 4 May 1986
Most goals conceded during the first half: 4, vs Egypt on 16 March 1934
vs Yugoslavia on 21 August 1949
Most goals conceded during the second half: 5, vs Italy on 4 Novamber 1961, Argentina 4 May 1986
Managers
Most AFC Asian Cup wins: Yosef Merimovich, 1
Most matches as coach: Shlomo Scharf, 82
Most matches won as coach: Shlomo Scharf, 31
Most matches draws as coach: Shlomo Scharf, 18
Youngest coach: Avram Grant, 47
Notes
External links
RSSS |
https://en.wikipedia.org/wiki/Lift%20%28mathematics%29 | In category theory, a branch of mathematics, given a morphism f: X → Y and a morphism g: Z → Y, a lift or lifting of f to Z is a morphism h: X → Z such that . We say that f factors through h.
A basic example in topology is lifting a path in one topological space to a path in a covering space. For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval [0,1]. We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have
Lifts are ubiquitous; for example, the definition of fibrations (see Homotopy lifting property) and the valuative criteria of separated and proper maps of schemes are formulated in terms of existence and (in the last case) uniqueness of certain lifts.
In algebraic topology and homological algebra, tensor product and the Hom functor are adjoint; however, they might not always lift to an exact sequence. This leads to the definition of the Ext functor and the Tor functor.
Algebraic logic
The notations of first-order predicate logic are streamlined when quantifiers are relegated to established domains and ranges of binary relations. Gunther Schmidt and Michael Winter have illustrated the method of lifting traditional logical expressions of topology to calculus of relations in their book Relational Topology.
They aim "to lift concepts to a relational level making them point free as well as quantifier free, thus
liberating them from the style of first order predicate logic and approaching the clarity of algebraic reasoning."
For example, a partial function M corresponds to the inclusion where denotes the identity relation on the range of M. "The notation for quantification is hidden and stays deeply incorporated in the typing of the relational operations (here transposition and composition) and their rules."
Circle maps
For maps of a circle, the definition of a lift to the real line is slightly different (a common application is the calculation of rotation number). Given a map on a circle, , a lift of , , is any map on the real line, , for which there exists a projection (or, covering map), , such that .
See also
Covering space
Projective module
Formally smooth map satisfies an infinitesimal lifting property.
Lifting property in categories
Monsky–Washnitzer cohomology lifts p-adic varieties to characteristic zero.
SBI ring allows idempotents to be lifted above the Jacobson radical.
Ikeda lift
Miyawaki lift of Siegel modular forms
Saito–Kurokawa lift of modular forms
Rotation number uses a lift of a homeomorphism of the circle to the real line.
Arithmetic geometry |
https://en.wikipedia.org/wiki/Integral%20of%20secant%20cubed | The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
where is the inverse Gudermannian function, the integral of the secant function.
There are a number of reasons why this particular antiderivative is worthy of special attention:
The technique used for reducing integrals of higher odd powers of secant to lower ones is fully present in this, the simplest case. The other cases are done in the same way.
The utility of hyperbolic functions in integration can be demonstrated in cases of odd powers of secant (powers of tangent can also be included).
This is one of several integrals usually done in a first-year calculus course in which the most natural way to proceed involves integrating by parts and returning to the same integral one started with (another is the integral of the product of an exponential function with a sine or cosine function; yet another the integral of a power of the sine or cosine function).
This integral is used in evaluating any integral of the form
where is a constant. In particular, it appears in the problems of:
rectifying the parabola and the Archimedean spiral
finding the surface area of the helicoid.
Derivations
Integration by parts
This antiderivative may be found by integration by parts, as follows:
where
Then
Next add to both sides:
using the integral of the secant function,
Finally, divide both sides by 2:
which was to be derived. A possible mnemonic is: "The integral of secant cubed is the average of the derivative and integral of secant".
Reduction to an integral of a rational function
where , so that . This admits a decomposition by partial fractions:
Antidifferentiating term-by-term, one gets
Hyperbolic functions
Integrals of the form: can be reduced using the Pythagorean identity if is even or and are both odd. If is odd and is even, hyperbolic substitutions can be used to replace the nested integration by parts with hyperbolic power-reducing formulas.
Note that follows directly from this substitution.
Higher odd powers of secant
Just as the integration by parts above reduced the integral of secant cubed to the integral of secant to the first power, so a similar process reduces the integral of higher odd powers of secant to lower ones. This is the secant reduction formula, which follows the syntax:
Even powers of tangents can be accommodated by using binomial expansion to form an odd polynomial of secant and using these formulae on the largest term and combining like terms.
See also
Lists of integrals
Notes
References
Integral calculus |
https://en.wikipedia.org/wiki/Institut%20Henri%20Poincar%C3%A9 | The Henri Poincaré Institute (or IHP for Institut Henri Poincaré) is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrondissement of Paris, on the Sainte-Geneviève Hill.
History
Just after World War I, mathematicians Émile Borel in France and George Birkhoff in the United States persuaded French and American sponsors (Edmond de Rothschild and the Rockefeller Foundation respectively) to fund the building of a centre for lectures and international exchanges in mathematics and theoretical physics. The Institute was inaugurated on 17 November 1928 and named after French mathematician Henri Poincaré (1854–1912).
The institute's objective has been to promote mathematical physics, and it soon became a meeting place for the French scientific community. In the 1990s, the IHP became a thematic institute modeled on Berkeley's Mathematical Sciences Research Institute (MSRI).
Organization
The IHP's governing board has about 35 members. There are no permanent researchers other than the director and the deputy director. From 2009 to 2017, the institute was headed by mathematician Cédric Villani, Fields Medals laureate in 2010, as director. French cosmologist Jean-Philippe Uzan was his deputy director. Since 2018, Sylvie Benzoni is the institute's director.
Together with the Institut des Hautes Études Scientifiques (IHES), the Centre International de Rencontres Mathématiques (CIRM) and the Centre International de Mathématiques Pures et Appliquées (CIMPA) it is a member of the Carmin LabEx (Laboratory of Excellence), which aims at facilitating exchanges between mathematicians by building infrastructures for pooling skills and information.
Scientific activities
As a venue for national and international mathematical exchanges, the Institute organizes "thematic quarters" (three-month programmes on specific topics), intensive short-time collaborations, PhD training courses, conferences and seminars in mathematics or related fields, such as physics, biology or computer science. Topics are selected by the IHP's Scientific Steering Committee. The IHP welcomes around 11,000 mathematicians each year.
In 2013, the Institute launched the "Poincaré Chair", a research program designed to foster the international careers of young researchers. Numerous seminars or series of lectures take place at the IHP, such as the Bourbaki and Bourbaphy Seminars, the History of Mathematics Seminar, as well as some more specialised lectures in algebra, number theory, mathematical physics and elliptic curves.
The institute also publishes four international scientific journals, the Annals of the Institut Henri Poincaré (Journal of Theoretical and Mathematical Physics, Probability and Statistics, Non Linear Analysis, and Combinatorics, Physics and their Interactions).
General public
The Institute co-organises and sponsors numerous, scientific and cultural events aimed at |
https://en.wikipedia.org/wiki/Al-Hajar%20al-Aswad | Al-Hajar al-Aswad () is a Syrian city just south of the centre of Damascus in the Darayya District of the Rif Dimashq Governorate.
According to the Syria Central Bureau of Statistics (CBS), Al-Hajar al-Aswad had a population of 84,948 in the 2004 census, making it the 13th largest city per geographical entity in Syria.
History
During the Syrian Civil War, on 26 July 2012, fighting was reported in the Al-Hajar al-Aswad suburb of the capital, a place described as home to thousands of poor refugees from the Israeli-occupied Golan Heights who were at the forefront of the movement against Assad. The Free Syrian Army had withdrawn to the southern suburb of Al-Hajar al-Aswad with the suburb being shelled by Government forces and an activist in the area said that there were still ongoing clashes in the south of the city. On 27 July 2012, the army took it back. On 30 October 2012, clashes broke out in Al-Hajar Al-Aswad between rebels and the army, spreading into the adjacent Yarmuk Palestinian camp.
On 19 November, rebels seized the headquarters of an army battalion and air defense base on the edge of the suburb, making it the nearest military base to Central Damascus to fall under rebel control. In January 2014, reports indicated that opposition fighters fleeing the fallen towns are concentrated in the remaining strongholds, particularly Al-Hajar al-Aswad.
The district became a hotspot for Islamic State of Iraq and the Levant militant activity, whom controlled large areas of the district and used it for a staging ground for their assault on Yarmouk Camp in 2015.
The entire location of Al Hajar al Aswad was captured from ISIL by the Syrian Arab Army (SAA) on 16 May 2018. Yarmouk Camp still remains under ISIL control. The SAA has been attacking both locations as part of an offensive that started on 1 May 2018.
In popular culture
In 2022, Al-Hajar al-Aswad served as a filming location for the Chinese action film Home Operation that dramatizes the 2015 evacuation of hundreds of Chinese citizens and other citizens from Yemen.
References
External links
Populated places in Darayya District
Cities in Syria |
https://en.wikipedia.org/wiki/Persymmetric%20matrix | In mathematics, persymmetric matrix may refer to:
a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or
a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.
The first definition is the most common in the recent literature. The designation "Hankel matrix" is often used for matrices satisfying the property in the second definition.
Definition 1
Let A = (aij) be an n × n matrix. The first definition of persymmetric requires that
for all i, j.
For example, 5 × 5 persymmetric matrices are of the form
This can be equivalently expressed as AJ = JAT where J is the exchange matrix.
A third way to express this is seen by post-multiplyingg AJ = JAT with J on both sides, showing that AT rotated 180 degrees is identical to A. A = JATJ.
A symmetric matrix is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes called bisymmetric matrices.
Definition 2
The second definition is due to Thomas Muir. It says that the square matrix A = (aij) is persymmetric if aij depends only on i + j. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form
A persymmetric determinant is the determinant of a persymmetric matrix.
A matrix for which the values on each line parallel to the main diagonal are constant is called a Toeplitz matrix.
See also
Centrosymmetric matrix
References
Determinants
Matrices |
https://en.wikipedia.org/wiki/List%20of%20towns%20in%20Chile | This article contains a list of towns in Chile.
A town is defined by Chile's National Statistics Institute (INE) as an urban entity possessing between 2,001 and 5,000 inhabitants—or between 1,001 and 2,000 inhabitants if 50% or more of its population is economically active in secondary and/or tertiary activities. This list is based on a June 2005 report by the INE based on the 2002 census, which registered 274 towns across the country, however only 269 of them are shown here. (Note: The higher number is based on the number given in the regional summary provided by the INE report. The lower number is based on a manual count of the report. The discrepancies are found in the Valparaíso Region (report: 31 / manual count: 28), the O'Higgins Region (report: 39 / manual count: 38) and the Los Ríos and Los Lagos Region combined (report: 31 / manual count: 30).)
List of towns by region (269)
Arica and Parinacota Region (1)
Putre
Tarapacá Region (3)
Pica
Collaguasi
La Tirana
Antofagasta Region (4)
Cerro Moreno
Juan López
Hornitos
San Pedro de Atacama
Atacama Region (7)
Bahía Inglesa
Loreto
Puerto Viejo
El Salado
Flamenco
Portal del Inca
Freirina
Coquimbo Region (14)
Las Tacas
Tongoy
Guanaqueros
Puerto Velero
La Higuera
Canela Baja
Pichidangui
Quilimarí Alto
Chillepín
Guamalata
La Chimba
Sotaquí
Chañaral Alto
Punitaqui
Valparaíso Region (28)
Laguna Verde
Quintay
San Juan Bautista
Maitencillo
Puchuncaví
Hanga Roa
San Rafael
Placilla
Valle Hermoso
Los Quinquelles
Pichicuy
Los Molles
Artificio
Papudo
Pullalli
Chincolco
Petorca
Zapallar
La Laguna de Zapallar
Catapilco
San Pedro
El Yeco
Mirasol
Las Brisas
Algarrobal-Punta El Olivo
Curimón
Panquehue
El Llano
O'Higgins Region (38)
La Compañía
Coinco
Coltauco
Loreto-Molino
Parral de Purén
El Manzano
Sewell
Coya
Pelequén
Malloa
Angostura
La Punta
Olivar Alto
Pichidegua
Rosario
Esmeralda
Los Lirios
El Tambo
Rastrojos
Cáhuil
La Estrella
Costa de Sol
Litueche
Marchihue
La Boca
La Vega de Pupuya
Paredones
Bucalemu
Angostura
Chépica
Auquinco
Tinguiririca
San Enrique de Romeral
Lolol
Cunaco
Peralillo
Población
Placilla
Maule Region (35)
Panguilemo
Huilquilemu
Santa Olga
Los Pellines
Curepto
Empedrado
Maule
Chacarillas
Pelarco
Pencahue
Cumpeo
San Rafael
Chanco
Pelluhue
Quilicura
Sarmiento
Villa Los Niches
San Alberto
Licantén
Iloca
Itahue Uno
Rauco
Romeral
Sagrada Familia
Villa Prat
Lago Vichuquén
Llico
Vara Gruesa
Las Obras
Colbún
Panimávida
Retiro
Copihue
Bobadilla
Yerbas Buenas
Ñuble Region (22)
Campanario
Cobquecura
El Carmen
El Emboque
Las Mariposas
Ninhue
Ñipas
Pemuco
Pinto
Quinchamalí
Quiriquina
Ranguelmo
Recinto-Los Lleuques
San Fabián de Alico
San Gregorio
San Ignacio
Santa Clara
San Nicolás
Treguaco
Portezuelo
Pueblo Seco
Puente Ñuble
Villa Illinois
Biobío Region (27)
Florida
Monte Aguila
Talcamávida
Caleta Tumbes
Dichato
Rafael
Santa Rosa
Laraquete
Ramadillas
Carampangue
Contulmo
Curanilahue
Antiguala
Tirúa
San Carlos de Purén
Millantú
Santa Fé
Virquenco
Antuco
Negrete
Coihue
Quidico
Quilaco |
https://en.wikipedia.org/wiki/Parabolic%20Lie%20algebra | In algebra, a parabolic Lie algebra is a subalgebra of a semisimple Lie algebra satisfying one of the following two conditions:
contains a maximal solvable subalgebra (a Borel subalgebra) of ;
the Killing perp of in is the nilradical of .
These conditions are equivalent over an algebraically closed field of characteristic zero, such as the complex numbers. If the field is not algebraically closed, then the first condition is replaced by the assumption that
contains a Borel subalgebra of
where is the algebraic closure of .
See also
Generalized flag variety
Bibliography
.
Lie algebras |
https://en.wikipedia.org/wiki/Flat%20topology | In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (faithfully flat descent). The term flat here comes from flat modules.
There are several slightly different flat topologies, the most common of which are the fppf topology and the fpqc topology. fppf stands for , and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat and of finite presentation. fpqc stands for , and in this topology, a morphism of affine schemes is a covering morphism if it is faithfully flat. In both categories, a covering family is defined be a family which is a cover on Zariski open subsets. In the fpqc topology, any faithfully flat and quasi-compact morphism is a cover. These topologies are closely related to descent. The "pure" faithfully flat topology without any further finiteness conditions such as quasi compactness or finite presentation is not used much as is not subcanonical; in other words, representable functors need not be sheaves.
Unfortunately the terminology for flat topologies is not standardized. Some authors use the term "topology" for a pretopology, and there are several slightly different pretopologies sometimes called the fppf or fpqc (pre)topology, which sometimes give the same topology.
Flat cohomology was introduced by Grothendieck in about 1960.
The big and small fppf sites
Let X be an affine scheme. We define an fppf cover of X to be a finite and jointly surjective family of morphisms
(φa : Xa → X)
with each Xa affine and each φa flat, finitely presented. This generates a pretopology: for X arbitrary, we define an fppf cover of X to be a family
(φa : Xa → X)
which is an fppf cover after base changing to an open affine subscheme of X. This pretopology generates a topology called the fppf topology. (This is not the same as the topology we would get if we started with arbitrary X and Xa and took covering families to be jointly surjective families of flat, finitely presented morphisms.) We write Fppf for the category of schemes with the fppf topology.
The small fppf site of X is the category O(Xfppf) whose objects are schemes U with a fixed morphism U → X which is part of some covering family. (This does not imply that the morphism is flat, finitely presented.) The morphisms are morphisms of schemes compatible with the fixed maps to X. The large fppf site of X is the category Fppf/X, that is, the category of schemes with a fixed map to X, considered with the fppf topology.
"Fppf" is an abbreviation for "fidèlement plate de présentation finie", that is, "faithfully flat and of finite presentation". Every surjective family of flat and finitely presented morphisms is a covering family for this topology, hence the name. The definition of the fppf pretopology can also be given with an extra quasi-finiteness condition; it follows from Corollary 1 |
https://en.wikipedia.org/wiki/London%20Buses%20route%20205 | London Buses route 205 is a Transport for London contracted bus route in London, England. Running between Paddington and Bow Church station, it is operated by Stagecoach London.
2015 statistics from Transport for London stated that this route was responsible for the most injuries to cyclists of any TfL bus route in London.
History
Route 205 commenced operating on 31 August 2002, replacing the former SL1 (StationLink 1) service, which had begun as an accessible route called Carelink for disabled people operated by National Bus Company owned Beeline. This route was withdrawn in 1988, and it became a London Transport contracted route. It was initially operated by London General, but in 1992 the contract was won by Thorpes. For a short period the route continued to be branded as Stationlink.
Route 205 was introduced as part improvements in preparation for the introduction of London congestion charge in February 2003. It connects Paddington, Marylebone, Euston, King's Cross and Liverpool Street termini stations, as well as many London Underground stations following the northern part of the Circle line. A route 705, linking stations on the southern section of the Circle Line, was also created but later withdrawn. The contract to operate the new route was won by Metroline.
It was extended from Whitechapel to Mile End tube station on 16 June 2007, and was converted into 24-hour service at the same time. Seven new Scania N230UDs arrived in summer 2007 to increase the frequency of the route.
Upon being re-tendered, on 29 August 2009 the route passed to East London Bus Company. At the same time the route was extended further east, from Mile End to Bow Church. The contract required 25 new buses.
On 31 August 2013, the night service on this 24 hour route was withdrawn and replaced by night bus route N205, the existing night-time services on the 205 was simply re-numbered as N205 and extended to Leyton, Downsell Road via Stratford.
Upon being re-tendered the route was retained by Stagecoach London with a new contract to commence on 30 August 2014 with new Alexander Dennis Enviro400Hs.
In 2018, Transport for London consulted on reducing the frequency of the route.
Current route
Route 205 operates via these primary locations:
Paddington station
Edgware Road station
Marylebone station
Baker Street station
Regent's Park station
Great Portland Street station
Euston Square station
Euston bus station for Euston station
St Pancras International
King's Cross station
Angel station
Old Street station
Liverpool Street station
Aldgate station
Aldgate East station
Whitechapel station
Stepney Green station
Mile End station
Bow Road station
Bow Church station
References
External links
London Bus Routes gallery
Timetable
Bus routes in London
Transport in the London Borough of Camden
Transport in the London Borough of Islington
Transport in the City of Westminster
Transport in the London Borough of Tower Hamlets |
https://en.wikipedia.org/wiki/Change%20detection | In statistical analysis, change detection or change point detection tries to identify times when the probability distribution of a stochastic process or time series changes. In general the problem concerns both detecting whether or not a change has occurred, or whether several changes might have occurred, and identifying the times of any such changes.
Specific applications, like step detection and edge detection, may be concerned with changes in the mean, variance, correlation, or spectral density of the process. More generally change detection also includes the detection of anomalous behavior: anomaly detection.
Offline change point detection it is assumed that a sequence of length is available and the goal is to identify whether any change point(s) occurred in the series. This is an example of post hoc analysis and is often approached using hypothesis testing methods. By contrast, online change point detection is concerned with detecting change points in an incoming data stream.
Background
A time series measures the progression of one or more quantities over time. For instance, the figure above shows the level of water in the Nile river between 1870 and 1970. Change point detection is concerned with identifying whether, and if so when, the behavior of the series changes significantly. In the Nile river example, the volume of water changes significantly after a dam was built in the river. Importantly, anomalous observations that differ from the ongoing behavior of the time series are not generally considered change points as long as the series returns to its previous behavior afterwards.
Mathematically, we can describe a time series as an ordered sequence of observations . We can write the joint distribution of a subset of the time series as . If the goal is to determine whether a change point occurred at a time in a finite time series of length , then we really ask whether equals . This problem can be generalized to the case of more than one change point.
Algorithms
Online change detection
Using the sequential analysis ("online") approach, any change test must make a trade-off between these common metrics:
False alarm rate
Misdetection rate
Detection delay
In a Bayes change-detection problem, a prior distribution is available for the change time.
Online change detection is also done using streaming algorithms.
Offline change detection
Basseville (1993, Section 2.6) discusses offline change-in-mean detection with hypothesis testing based on the works of Page and Picard and maximum-likelihood estimation of the change time, related to two-phase regression.
Other approaches employ clustering based on maximum likelihood estimation,, use optimization to infer the number and times of changes, via spectral analysis, or singular spectrum analysis.
Statistically speaking, change detection is often considered as a model selection problem. Models with more changepoints fit data better but with more parameters. The best trade-off can be fo |
https://en.wikipedia.org/wiki/T-group%20%28mathematics%29 | In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups:
Every simple group is a T-group.
Every quasisimple group is a T-group.
Every abelian group is a T-group.
Every Hamiltonian group is a T-group.
Every nilpotent T-group is either abelian or Hamiltonian, because in a nilpotent group, every subgroup is subnormal.
Every normal subgroup of a T-group is a T-group.
Every homomorphic image of a T-group is a T-group.
Every solvable T-group is metabelian.
The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups G with an abelian normal Hall subgroup H of odd order such that the quotient group G/H is a Dedekind group and H is acted upon by conjugation as a group of power automorphisms by G.
A PT-group is a group in which permutability is transitive. A finite T-group is a PT-group.
References
Properties of groups |
https://en.wikipedia.org/wiki/Bapat%E2%80%93Beg%20theorem | In probability theory, the Bapat–Beg theorem gives the joint probability distribution of order statistics of independent but not necessarily identically distributed random variables in terms of the cumulative distribution functions of the random variables. Ravindra Bapat and Beg published the theorem in 1989, though they did not offer a proof. A simple proof was offered by Hande in 1994.
Often, all elements of the sample are obtained from the same population and thus have the same probability distribution. The Bapat–Beg theorem describes the order statistics when each element of the sample is obtained from a different statistical population and therefore has its own probability distribution.
Statement
Let be independent real valued random variables with cumulative distribution functions respectively . Write for the order statistics. Then the joint probability distribution of the order statistics (with and ) is
where
is the permanent of the given block matrix. (The figures under the braces show the number of columns.)
Independent identically distributed case
In the case when the variables are independent and identically distributed with cumulative probability distribution function for all i the theorem reduces to
Remarks
No assumption of continuity of the cumulative distribution functions is needed.
If the inequalities x1 < x2 < ... < xk are not imposed, some of the inequalities "may be redundant and the probability can be evaluated after making the necessary reduction."
Complexity
Glueck et al. note that the Bapat‒Beg formula is computationally intractable, because it involves an exponential number of permanents of the size of the number of random variables. However, when the random variables have only two possible distributions, the complexity can be reduced to . Thus, in the case of two populations, the complexity is polynomial in for any fixed number of statistics .
References
Probability theorems
Theorems in statistics |
https://en.wikipedia.org/wiki/Chris%20Hardwick%20%28speedcuber%29 | Christopher Michael Hardwick (born December 6, 1983) is an American competitive speedcuber.
Born in Merritt Island, Florida, he attended the North Carolina School of Science and Mathematics (class of 2002) and University of North Carolina at Chapel Hill (class of 2005).
He is known especially for his blindfolded world record solution times of Rubik's Revenge and Rubik's Professor's Cube puzzles, though he started out as a top one-handed cuber. Chris holds the former world record for the blindfolded solve time of the Rubik's Professor's Cube with 15 minutes 22 seconds.
Hardwick has made a number of television appearances demonstrating the Rubik's Cube, including MTV in 2002, Canada AM and Much Music in the fall of 2003, discussing the 2003 Rubik's Cube World Championships. His home videos have also appeared on numerous online video sites including CollegeHumor and Digg. A home video of Hardwick solving a 3x3x3 Rubik's Cube one-handed appeared on VH1 in 2006 for Web Junk 20. Axe deodorant parodied another one of Hardwick's home videos in their 2006 South African "Get a Girlfriend" commercial campaign.
Chris Hardwick, the comedian, referenced this particular Chris Hardwick on Web Soup in 2009 when a comment about the Rubik's Cube was made. The two met during the comedian's show at the Barrymore Theatre in Madison, Wisconsin, on April 11, 2015.
References
External links
Chris Hardwick's listing at the World Cube Association
Chris Hardwick's webpage
American speedcubers
Living people
1983 births
People from Merritt Island, Florida |
https://en.wikipedia.org/wiki/Domino%20tiling | In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares.
Height functions
For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the vertices of the grid. For instance, draw a chessboard, fix a node with height 0, then for any node there is a path from to it. On this path define the height of each node (i.e. corners of the squares) to be the height of the previous node plus one if the square on the right of the path from to is black, and minus one otherwise.
More details can be found in .
Thurston's height condition
describes a test for determining whether a simply-connected region, formed as the union of unit squares in the plane, has a domino tiling. He forms an undirected graph that has as its vertices the points (x,y,z) in the three-dimensional integer lattice, where each such point is connected to four neighbors: if x + y is even, then (x,y,z) is connected to (x + 1,y,z + 1), (x − 1,y,z + 1), (x,y + 1,z − 1), and (x,y − 1,z − 1), while if x + y is odd, then (x,y,z) is connected to (x + 1,y,z − 1), (x − 1,y,z − 1), (x,y + 1,z + 1), and (x,y − 1,z + 1). The boundary of the region, viewed as a sequence of integer points in the (x,y) plane, lifts uniquely (once a starting height is chosen) to a path in this three-dimensional graph. A necessary condition for this region to be tileable is that this path must close up to form a simple closed curve in three dimensions, however, this condition is not sufficient. Using more careful analysis of the boundary path, Thurston gave a criterion for tileability of a region that was sufficient as well as necessary.
Counting tilings of regions
The number of ways to cover an rectangle with dominoes, calculated independently by and , is given by
When both m and n are odd, the formula correctly reduces to zero possible domino tilings.
A special case occurs when tiling the rectangle with n dominoes: the sequence reduces to the Fibonacci sequence.
Another special case happens for squares with m = n = 0, 2, 4, 6, 8, 10, 12, ... is
These numbers can be found by writing them as the Pfaffian of an skew-symmetric matrix whose eigenvalues can be found explicitly. This technique may be applied in many mathematics-related subjects, for example, in the classical, 2-dimensional computation of the dimer-dimer correlator function in statistical mechanics.
The number of tilings of a region is very sensitive to boundary conditions, and can change dramatically with apparently insignificant changes in the shape of the region. This is illustrated by the number of tilings of an Aztec diamond of order n, where the number |
https://en.wikipedia.org/wiki/Hott | HOTT may refer to:
Mathematics:
Homotopy type theory
Games:
Halls of the Things, an early video game
Hordes of the Things (wargame)
Entertainment:
"Hanging on the Telephone", a song by the power pop band The Nerves, also recorded by Blondie
Hour of the Time, a shortwave radio show
Other:
Hot Topic's former NASDAQ ticker symbol |
https://en.wikipedia.org/wiki/Fox%20derivative | In mathematics, the Fox derivative is an algebraic construction in the theory of free groups which bears many similarities to the conventional derivative of calculus. The Fox derivative and related concepts are often referred to as the Fox calculus, or (Fox's original term) the free differential calculus. The Fox derivative was developed in a series of five papers by mathematician Ralph Fox, published in Annals of Mathematics beginning in 1953.
Definition
If G is a free group with identity element e and generators gi, then the Fox derivative with respect to gi is a function from G into the integral group ring which is denoted , and obeys the following axioms:
, where is the Kronecker delta
for any elements u and v of G.
The first two axioms are identical to similar properties of the partial derivative of calculus, and the third is a modified version of the product rule. As a consequence of the axioms, we have the following formula for inverses
for any element u of G.
Applications
The Fox derivative has applications in group cohomology, knot theory, and covering space theory, among other areas of mathematics.
See also
Alexander polynomial
Free group
Ring (mathematics)
Integral domain
References
Geometric topology
Combinatorial group theory |
https://en.wikipedia.org/wiki/Filipinos%20in%20Japan | Filipinos in Japan (, Zainichi Firipinjin, ) formed a population of 309,943 in June 2023 individuals, making them Japan's fourth-largest foreign community, according to the statistics of the Philippines. Their population reached as high as 245,518 in 1998, but fell to 144,871 individuals in 2000 before beginning to recover slightly when Japan cracked down on human trafficking. In 2006, Japanese/Filipino marriages were the most frequent of all international marriages in Japan. As of 2016, the Filipino population in Japan was 237,103 according to the Ministry of Justice. Filipinos in Japan formed a population of 325,000 individuals at year-end 2020, making them Japan's third-largest foreign community along with Vietnamese, according to the statistics of the Philippine Global National Inquirer and the Ministry of Justice. In December 2021, the number of Filipinos in Japan was estimated at 276,615.
According to figures published by the Central Bank of the Philippines, overseas Filipino workers in Japan remitted more than US$1 billion between 1990 and 1999; one newspaper described the contributions of overseas workers as a "major source of life support for the Philippines' ailing economy." Though most Filipinos in Japan are short-term residents, the history of their community extends back further; during the Japanese occupation of the Philippines, some Filipino students studied in Japanese universities.
Media
There is a magazine called Kumusta! (クムスタ). Junta Shimozawa publishes and edits the Japanese portion and his spouse Hermie edits the Tagalog version. In 1996 it had a weekly circulation of 30,000, and its website was to appear in March of that month.
Notable people
Entertainment
Ruby Moreno, actress
Mokomichi Hayami, actor, chef, TV presenter, entrepreneur, and model
Miho Nishida, actress and model
Nicole Abe, fashion model
Noriyuki Abe, film director
Sayaka Akimoto, actress and singer
Ayame Goriki, actress, singer, and model
Stan "Xtra" Fukase, drag queen and social media influencer
Hiromi, fashion model
Elaiza Ikeda, fashion model and actress
Mark Ishii, voice actor
Rie Kaneko, model and singer
Loveli, fashion model
Rika Mamiya, model and singer
Megumi Nakajima, voice actress and singer
Maiko Nakamura, singer
Chieko Ochi, singer, model and actress
Aiko Otake, model
Reimy, musician
Rikako Sasaki, singer
Alan Shirahama, actor and DJ
Anna Suda, actress and dancer
Kiara Takahashi
Maryjun Takahashi, actress and model
Yu Takahashi, actress
Zawachin, television personality
Anna Mima, singer
Ruben Aquino, animator
, model, gravure idol and actress
, model, gravure idol and actress
, singer, dancer and idol for BXW
, singer, dancer and idol on Produce 101 Japan (season 2) and idol for boy group, TOZ
, singer, dancer and idol on Produce 101 Japan (season 2)
Yuri Komagata, singer and voice actress
Yuki Kimura, model, gravure idol and actress
Yuki Sonoda (screen name: Yana Fuentes), model, actress, Miss Universe Ja |
https://en.wikipedia.org/wiki/Layer%20cake%20representation | In mathematics, the layer cake representation of a non-negative, real-valued measurable function defined on a measure space is the formula
for all , where denotes the indicator function of a subset and denotes the super-level set
The layer cake representation follows easily from observing that
and then using the formula
The layer cake representation takes its name from the representation of the value as the sum of contributions from the "layers" : "layers"/values below contribute to the integral, while values above do not.
It is a generalization of Cavalieri's principle and is also known under this name.
An important consequence of the layer cake representation is the identity
which follows from it by applying the Fubini-Tonelli theorem.
An important application is that for can be written as follows
which follows immediately from the change of variables in the layer cake representation of .
See also
Symmetric decreasing rearrangement
References
Real analysis |
https://en.wikipedia.org/wiki/Pr%C3%A9kopa%E2%80%93Leindler%20inequality | In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians András Prékopa and László Leindler.
Statement of the inequality
Let 0 < λ < 1 and let f, g, h : Rn → [0, +∞) be non-negative real-valued measurable functions defined on n-dimensional Euclidean space Rn. Suppose that these functions satisfy
for all x and y in Rn. Then
Essential form of the inequality
Recall that the essential supremum of a measurable function f : Rn → R is defined by
This notation allows the following essential form of the Prékopa–Leindler inequality: let 0 < λ < 1 and let f, g ∈ L1(Rn; [0, +∞)) be non-negative absolutely integrable functions. Let
Then s is measurable and
The essential supremum form was given by Herm Brascamp and Elliott Lieb. Its use can change the left side of the inequality. For example, a function g that takes the value 1 at exactly one point will not usually yield a zero left side in the "non-essential sup" form but it will always yield a zero left side in the "essential sup" form.
Relationship to the Brunn–Minkowski inequality
It can be shown that the usual Prékopa–Leindler inequality implies the Brunn–Minkowski inequality in the following form: if 0 < λ < 1 and A and B are bounded, measurable subsets of Rn such that the Minkowski sum (1 − λ)A + λB is also measurable, then
where μ denotes n-dimensional Lebesgue measure. Hence, the Prékopa–Leindler inequality can also be used to prove the Brunn–Minkowski inequality in its more familiar form: if 0 < λ < 1 and A and B are non-empty, bounded, measurable subsets of Rn such that (1 − λ)A + λB is also measurable, then
Applications in probability and statistics
Log-concave distributions
The Prékopa–Leindler inequality is useful in the theory of log-concave distributions, as it can be used to show that log-concavity is preserved by marginalization and independent summation of log-concave distributed random variables. Since, if have pdf , and are independent, then is the pdf of , we also have that the convolution of two log-concave functions is log-concave.
Suppose that H(x,y) is a log-concave distribution for (x,y) ∈ Rm × Rn, so that by definition we have
and let M(y) denote the marginal distribution obtained by integrating over x:
Let y1, y2 ∈ Rn and 0 < λ < 1 be given. Then equation () satisfies condition () with h(x) = H(x,(1 − λ)y1 + λy2), f(x) = H(x,y1) and g(x) = H(x,y2), so the Prékopa–Leindler inequality applies. It can be written in terms of M as
which is the definition of log-concavity for M.
To see how this implies the preservation of log-convexity by independent sums, suppose that X and Y are independent random variables with log-concave distribution. Since the product of two log-concave functions is log-concave, the joint distribution of (X,Y) is also log-co |
https://en.wikipedia.org/wiki/Calendrical%20calculation | A calendrical calculation is a calculation concerning calendar dates. Calendrical calculations can be considered an area of applied mathematics.
Some examples of calendrical calculations:
Converting a Julian or Gregorian calendar date to its Julian day number and vice versa .
The number of days between two dates, which is simply the difference in their Julian day numbers.
The dates of moveable holidays, like Christian Easter (the calculation is known as Computus) followed up by Ascension Thursday and Pentecost or Advent Sundays, or the Jewish Passover, for a given year.
Converting a date between different calendars. For instance, dates in the Gregorian calendar can be converted to dates in the Islamic calendar with the Kuwaiti algorithm.
Calculating the day of the week.
Calendrical calculation is one of the five major Savant syndrome characteristics.
Examples
Numerical methods were described in the Journal of the Department of Mathematics, Open University, Milton Keynes, Buckinghamshire (M500) in 1997 and 1998. The following algorithm gives the number of days (d) in month m of year y. The value of m is given on the right of the month in the following list:
January 11 February 12 March 1 April 2 May 3 June 4 July 5 August 6 September 7 October 8 November 9 December 10.
The algorithm enables a computer to print calendar and diary pages for past or future sequences of any desired length from the reform of the calendar, which in England was 3/14 September 1752. The article Date of Easter gives algorithms for calculating the date of Easter. Combining the two enables the page headers to show any fixed or movable festival observed on the day, and whether it is a bank holiday.
The algorithm utilises the integral or floor function: thus is that part of the number x which lies to the left of the decimal point. It is only necessary to work through the complete function when calculating the length of February in a year which is divisible by 100 without remainder. When calculating the length of February in any other year it is only necessary to evaluate the terms to the left of the fifth + sign. When calculating the length of any other month it is only necessary to evaluate the terms to the left of the third - sign.
To find the length of, for example, February 2000 the calculation is
See also
Calendrical Calculations
References
Edward M. Reingold and Nachum Dershowitz. Calendrical Calculations: The Ultimate Edition. Cambridge University Press; (2018).
Calendar algorithms |
https://en.wikipedia.org/wiki/Deligne%E2%80%93Lusztig%20theory | In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by .
used these representations to find all representations of all finite simple groups of Lie type.
Motivation
Suppose that G is a reductive group defined over a finite field, with Frobenius map F.
Ian G. Macdonald conjectured that there should be a map from general position characters of F-stable maximal tori to irreducible representations of (the fixed points of F). For general linear groups this was already known by the work of . This was the main result proved by Pierre Deligne and George Lusztig; they found a virtual representation for all characters of an F-stable maximal torus, which is irreducible (up to sign) when the character is in general position.
When the maximal torus is split, these representations were well known and are given by parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to G). The representations of parabolic induction can be constructed using functions on a space, which can be thought of as elements of a suitable zeroth cohomology group. Deligne and Lusztig's construction is a generalization of parabolic induction to non-split tori using higher cohomology groups. (Parabolic induction can also be done with tori of G replaced by Levi subgroups of G, and there is a generalization of Deligne–Lusztig theory to this case too.)
Vladimir Drinfeld proved that the discrete series representations of SL2(Fq) can be found in the ℓ-adic cohomology groups
of the affine curve X defined by
.
The polynomial is a determinant used in the construction of the Dickson invariant of the general linear group, and is an invariant of the special linear group.
The construction of Deligne and Lusztig is a generalization of this fundamental example to other groups. The affine curve X is generalized to a bundle over a "Deligne–Lusztig variety" where T is a maximal torus of G, and instead of using just the first cohomology group they use an alternating sum of ℓ-adic cohomology groups with compact support to construct virtual representations.
The Deligne-Lusztig construction is formally similar to Hermann Weyl's construction of the representations of a compact group from the characters of a maximal torus. The case of compact groups is easier partly because there is only one conjugacy class of maximal tori. The Borel–Weil–Bott construction of representations of algebraic groups using coherent sheaf cohomology is also similar.
For real semisimple groups there is an analogue of the construction of Deligne and Lusztig, using Zuckerman functors to construct representations.
Deligne–Lusztig varieties
The construction of Deligne-Lusztig characters uses a family of auxiliary algebraic varieties XT called Deligne–Lusztig varieties, constructed from a reductive linear algebraic group G defined over a finite field Fq |
https://en.wikipedia.org/wiki/65%2C536 | 65536 is the natural number following 65535 and preceding 65537.
65536 is a power of two: (2 to the 16th power).
65536 is the smallest number with exactly 17 divisors.
In mathematics
65536 is , so in tetration notation 65536 is 42.
When expressed using Knuth's up-arrow notation, 65536 is
,
which is equal to
,
which is equivalent to
or
.
65536 is a superperfect number – a number such that σ(σ(n)) = 2n.
A 16-bit number can distinguish 65536 different possibilities. For example, unsigned binary notation exhausts all possible 16-bit codes in uniquely identifying the numbers 0 to 65535. In this scheme, 65536 is the least natural number that can not be represented with 16 bits. Conversely, it is the "first" or smallest positive integer that requires 17 bits.
65536 is the only power of 2 less than 231000 that does not contain the digits 1, 2, 4, or 8 in its decimal representation.
The sum of the unitary divisors of 65536 is prime (1 + 65536 = 65537, which is prime).
65536 is an untouchable number.
In computing
65536 (216) is the number of different values representable in a number of 16 binary digits (or bits), also known as an unsigned short integer in many computer programming systems.
A 65,536-bit integer can represent up to 265536 (2.00352993...) values.
65,536 is the number of characters in the original Unicode, and currently in a Unicode plane.
This number is a limit in many common hardware and software implementations, some examples of which are:
The Motorola 68000 family, x86 architecture, and other computing platforms have a word size of 16 bits, representing 65536 possible values. (32- or 64-bit operations are supported equally or better in modern microprocessors.)
Many modern CPUs allow a memory page size of 64KiB (65536 bytes) to be configured in their memory-management hardware.
The default buffer size of a pipeline (Unix) is 64KiB (65536 bytes).
65536 is the maximum number of spreadsheet rows supported by Excel 97, Excel 2000, Excel 2002 and Excel 2003. Text files that are larger than 65536 rows cannot be imported to these versions of Excel. (Excel 2007, 2010 and 2013 support 1,048,576 rows (220)).
A 16-bit microprocessor chip can directly access 65536 memory addresses, and the 16-bit highcolor graphics standard supports a color palette of 65536 different colors.
The maximum number of methods allowed in a single dex file Android application is 65536.
The limit for the amount of code in bytes for a non-native, non-abstract method in Java.
The number of available ports to combine with a network address to create a network socket.
The maximum character limit for one message in WhatsApp is 65536.
In popular culture
There are 65536 different charts in Western geomancy.
References
Integers |
https://en.wikipedia.org/wiki/Joseph%20Mazur | Joseph C. Mazur (born in the Bronx in 1942) is Professor Emeritus of Mathematics at Marlboro College, in Marlboro, Vermont.
He holds a B.S. from Pratt Institute, where he first studied architecture. He spent his junior year in Paris, studying mathematics in classes with Claude Chevalley and Roger Godement and returned to Pratt to earn a B.S. in mathematics. From there he went directly to M.I.T to receive his Ph.D. in mathematics (algebraic geometry) in 1972. He has held a visiting scholar position at M.I.T and several visiting professor positions at The Mathematics Institute of the University of Warwick.
In 2006 he was awarded a Guggenheim Fellowship for work on mathematical narrative. In 2008 he was awarded a Bellagio Fellowship from the Rockefeller Foundation, and in 2009 was elected to Fellow of the Vermont Academy of Arts and Sciences. In 2011, 2013, and 2019 he was awarded Bogliasco Fellowships.
Since 1972 he has taught all areas of mathematics, its history and philosophy. He has authored many educational software programs, including Explorations in Calculus, the first interactive, multimedia CD package of simulations for calculus. He is the author of several mathematics books that have been translated into more than a dozen languages. He is also interested in history of science.
Bibliography
Euclid In the Rainforest: Discovering Universal Truth in Logic and Math, Plume, 2005 (Finalist for the PEN/Martha Albrand Award for First Nonfiction).
The Motion Paradox: The 2,500-Year-Old Puzzle Behind All The Mysteries of Time and Space, Dutton, 2007.
Number: The Language of Science (Ed.) Plume, 2005.
What's Luck Got to Do with It? The History, Mathematics and Psychology of the Gambler's Illusion, Princeton, 2010.
Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, Princeton, 2014.
Fluke: The Maths and Myths of Coincidences, London: Oneworld Publications. 2016.
The Clock Mirage: Our Myth of Measured Time, Yale University Press, 2020.
References
External links
Homepage of Joseph Mazur
Joseph Mazur at Marlboro College
American non-fiction writers
1942 births
Living people
20th-century American mathematicians
21st-century American mathematicians
20th-century American Jews
American science journalists
21st-century American Jews |
https://en.wikipedia.org/wiki/Harting%20Old%20Club | The Harting Old Club is a British friendly society, originating in the village of South Harting, West Sussex, and dating back to at least 1800, but in probability at least another 75 years before that. Every Whit Monday the members parade outside St Gabriel's church at 11 o'clock where the secretary calls the roll. The club members then march up and down the high street to the accompaniment of a brass band. In their hand they carry a hazel wand, and on their lapel they wear a red, blue and white rosette. Following a short service the (all male) members retire to enjoy a feast.
References
Footnotes
Notes
Bibliography
External links
Festivities Web-Site
Clubs and societies in West Sussex
Friendly societies of the United Kingdom |
https://en.wikipedia.org/wiki/Triple%20product%20property | In abstract algebra, the triple product property is an identity satisfied in some groups.
Let be a non-trivial group. Three nonempty subsets are said to have the triple product property in if for all elements , , it is the case that
where is the identity of .
It plays a role in research of fast matrix multiplication algorithms.
References
Henry Cohn, Chris Umans. A Group-theoretic Approach to Fast Matrix Multiplication. . Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 11–14 October 2003, Cambridge, MA, IEEE Computer Society, pp. 438–449.
Properties of groups |
https://en.wikipedia.org/wiki/Reda%20Shehata | Reda Shehata (born January 24, 1981 in Egypt) is an Egyptian football midfielder. He is currently the head coach of Ghazl El-Mahalla.
Managerial statistics
References
External links
1981 births
Living people
Egyptian men's footballers
Egypt men's international footballers
Al Ahly SC players
Men's association football midfielders
Al Ittihad Alexandria Club players
Egyptian Premier League players |
https://en.wikipedia.org/wiki/Michael%20Starbird | Michael P. Starbird (born 1948) is a Professor of Mathematics and a University of Texas Distinguished Teaching Professor in the Department of Mathematics at the University of Texas at Austin. He received his B.A from Pomona College and his Ph.D. in mathematics from the University of Wisconsin–Madison.
Starbird's mathematical specialty is topology. He joined the University of Texas at Austin as a faculty member in 1974, and served as an associate dean in Natural Sciences from 1989 to 1997. He serves on the national education committees of the Mathematical Association of America and the American Mathematical Society.
He directs UT's Inquiry Based Learning Project and works to promote the use of Inquiry Based Learning methods of instruction nationally.
Awards
He has received over fifteen teaching awards including the Mathematical Association of America's 2007 national teaching award; the Minnie Stevens Piper Professor award, which is a Texas statewide award given to professors in any subject in any college in the state of Texas; the UT System Regents’ Outstanding Teaching Award in its inaugural year; membership in the UT System Academy of Distinguished Teachers in its inaugural year; member and chair of UT Austin's Academy of Distinguished Teachers; and has received most of the UT-wide teaching awards. He is an inaugural year Fellow of the American Mathematical Society. He received an honorary Doctor of Science degree from Pomona College in 2014.
Administrative work and Service
Starbird served as Associate Dean for Academic and Student Affairs and as Associate Dean for Undergraduate Education in the College of Natural Sciences from 1989 to 1997. He has served on the Steering Committee of the Academy of Distinguished Teachers since 2000 and is currently chair.
He has accepted visiting positions at the Institute for Advanced Study in Princeton, The University of California at San Diego, and the Jet Propulsion Laboratory. In 2012 he became a fellow of the American Mathematical Society, in his inaugural year.
Starbird has served on the national education committees of both the American Mathematical Society and the Mathematical Association of America. He currently serves on the MAA's Committee on the Undergraduate Program and is a member of the Steering Committee for the next CUPM Curriculum Guide.
Publications
He has produced DVD courses for The Teaching Company in the Great Courses Series on calculus, statistics, probability, geometry, and the joy of thinking, which have reached hundreds of thousands of people worldwide. Since 2000, he has given over 200 invited lectures and presented more than 35 workshops on effective teaching to faculty members. He has co-authored two Inquiry Based Learning textbooks published by the MAA: (with David Marshall and Edward Odell) Number Theory Through Inquiry and (with Brian Katz) Distilling Ideas: An Introduction to Mathematical Thinking in the new Mathematics Through Inquiry subseries of the MAA Textbo |
https://en.wikipedia.org/wiki/Positive%20form | In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).
(1,1)-forms
Real (p,p)-forms on a complex manifold M are forms which are of type (p,p) and real, that is, lie in the intersection A real (1,1)-form is called semi-positive (sometimes just positive), respectively, positive (or positive definite) if any of the following equivalent conditions holds:
is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form.
For some basis in the space of (1,0)-forms, can be written diagonally, as with real and non-negative (respectively, positive).
For any (1,0)-tangent vector , (respectively, ).
For any real tangent vector , (respectively, ), where is the complex structure operator.
Positive line bundles
In algebraic geometry, positive definite (1,1)-forms arise as curvature forms of ample line bundles (also known as positive line bundles). Let L be a holomorphic Hermitian line bundle on a complex manifold,
its complex structure operator. Then L is equipped with a unique connection preserving the Hermitian structure and satisfying
.
This connection is called the Chern connection.
The curvature of the Chern connection is always a
purely imaginary (1,1)-form. A line bundle L is called positive if is a positive (1,1)-form. (Note that the de Rham cohomology class of is times the first Chern class of L.) The Kodaira embedding theorem claims that a positive line bundle is ample, and conversely, any ample line bundle admits a Hermitian metric with positive.
Positivity for (p, p)-forms
Semi-positive (1,1)-forms on M form a convex cone. When M is a compact complex surface, , this cone is self-dual, with respect to the Poincaré pairing :
For (p, p)-forms, where , there are two different notions of positivity. A form is called
strongly positive if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real (p, p)-form on an n-dimensional complex manifold M is called weakly positive if for all strongly positive (n-p, n-p)-forms ζ with compact support, we have .
Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are dual with respect to the Poincaré pairing.
Notes
References
P. Griffiths and J. Harris (1978), Principles of Algebraic Geometry, Wiley.
J.-P. Demailly, L2 vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994).
Complex manifolds
Algebraic geometry
Differential forms |
https://en.wikipedia.org/wiki/Information%20inequality | Information inequality may mean
in statistics, the Cramér–Rao bound, an inequality for the variance of an estimator based on the information in a sample
in information theory, inequalities in information theory describes various inequalities specific to that context.
in sociology, Information Inequality and Social Barriers
also in sociology, information inequity |
https://en.wikipedia.org/wiki/Non-binding | Non-binding or nonbinding may refer to
Nonbinding allocation of responsibility (NBAR) in a superfund
Non-binding authority in law
Non-binding arbitration
Non-binding constraint, mathematics
Non-binding opinion in patent law:
International preliminary report on patentability objective
Non-binding opinion (United Kingdom patent law)
Non-binding resolution
Non-binding referendum
See also
Binding (disambiguation) |
https://en.wikipedia.org/wiki/Bankruptcy%20costs%20of%20debt | Within the theory of corporate finance, bankruptcy costs of debt are the increased costs of financing with debt instead of equity that result from a higher probability of bankruptcy. The fact that bankruptcy is generally a costly process in itself and not only a transfer of ownership implies that these costs negatively affect the total value of the firm. These costs can be thought of as a financial cost, in the sense that the cost of financing increases because the probability of bankruptcy increases. One way to understand this is to realize that when a firm goes bankrupt investors holding its debt are likely to lose part or all of their investment, and therefore investors require a higher rate of return when investing in bonds of a firm that can easily go bankrupt. This implies that an increase in debt which ends up increasing a firm's bankruptcy probability causes an increase in these bankruptcy costs of debt.
In the trade-off theory of capital structure, firms are supposedly choosing their level of debt financing by trading off these bankruptcy costs of debt against tax benefits of debt. In particular, a firm that is trying to maximize the value for its shareholders will equalize the marginal cost of debt that results from these bankruptcy costs with the marginal benefit of debt that results from tax benefits.
In the personal bankruptcy there is a cost associated with filling the paperwork. For Chapter 13 Bankruptcy there is a fee of $281 and for Chapter 7 Bankruptcy it is $306. Additionally there can be other payments required, like Lawyer's fee, Conversion fee, Credit counselling and debtor education fee.
See also
Corporate finance
Trade-Off Theory
Capital structure
Tax benefits of debt
Financial distress
Financial risk management
References
Corporate finance
Debt |
https://en.wikipedia.org/wiki/United%20Nations%20peacekeeping%20missions%20involving%20Pakistan | Pakistan has served in 46 United Nations peacekeeping missions in 29 countries around the world. As of 2023, United Nations (UN) statistics show that 168 Pakistani UN peacekeepers have been killed since 1948. The biggest Pakistani loss occurred on 5 June 1993 in Mogadishu. Pakistan joined the United Nations on 30 September 1947, despite opposition from Afghanistan because of the Durand Line issue. The Pakistan Armed Forces are the sixth largest contributor of troops towards UN peacekeeping efforts, behind Ethiopia and Rwanda.
Peacekeeping, as defined by the United Nations, is the practice of helping countries torn by conflict create conditions for sustainable peace. UN peacekeepers — usually military officers and regular troops alongside civilian personnel from many countries — monitor and observe peace processes that emerge in regions post-war and assist ex-combatants in implementing the peace agreements they have signed. Such assistance comes in many forms, including confidence-building measures, power-sharing arrangements, electoral support, strengthening the rule of law, and economic and social development. Pakistan's contributions have consisted mainly of regular military personnel, but also include paramilitary troops and civilian police officers as peacekeepers. All operations must include the resolution of conflicts through the use of force to be considered valid under the charter of the United Nations.
Foundation
{{Blockquote|text=Our foreign policy is one of friendliness and goodwill towards all the nations of the world. We believe in the principle of honesty and fair play in national and international dealings and are prepared to make our utmost contribution to the promotion of peace and prosperity among the nations of the world. Pakistan will never be found lacking in extending its material and moral support to the oppressed and suppressed peoples of the world and in upholding the principles of the United Nations Charter."|author=Quaid-e-Azam Muhammad Ali Jinnah, founder and 1st Governor-General of Pakistan|title=|source=}}
Completed missions
Congo (August 1960 to May 1964)Contribution: 400 Troops, Ordnance, Transport units and Staff PersonnelCasualties: None.
United Nations Operation in the Congo was a United Nations peacekeeping force in Congo that was established after United Nations Security Council Resolution 143 of July 14, 1960.It was active during the Congo Crisis. During the operation Pakistan provided logistic support under Lt Col Naseer, the first ever Pakistani officer commanding an Ordnance Company in United Nation, during movement of troops to and from Congo and inland movement of the United Nation troops. Pakistan Army Ordnance Corps and Pakistan Army Supply Corps (ASC) organized the whole operation. It continued uninterrupted from 1960 to 1964 with four Independent Army Supply and Ordnance Corps companies, consisting of about 100 personnel each.The movement control entailed through sea, air, rail, river and road |
https://en.wikipedia.org/wiki/Pseudotriangle | In Euclidean plane geometry, a pseudotriangle (pseudo-triangle) is the simply connected subset of the plane that lies between any three mutually tangent convex sets. A pseudotriangulation (pseudo-triangulations) is a partition of a region of the plane into pseudotriangles, and a pointed pseudotriangulation is a pseudotriangulation in which at each vertex the incident edges span an angle of less than π.
Although the words "pseudotriangle" and "pseudotriangulation" have been used with various meanings in mathematics for much longer, the terms as used here were introduced in 1993 by Michel Pocchiola and Gert Vegter in connection with the computation of visibility relations and bitangents among convex obstacles in the plane. Pointed pseudotriangulations were first considered by Ileana Streinu (2000, 2005) as part of her solution to the carpenter's ruler problem, a proof that any simple polygonal path in the plane can be straightened out by a sequence of continuous motions. Pseudotriangulations have also been used for collision detection among moving objects and for dynamic graph drawing and shape morphing. Pointed pseudotriangulations arise in rigidity theory as examples of minimally rigid planar graphs, and in methods for placing guards in connection with the art gallery theorem. The shelling antimatroid of a planar point set gives rise to pointed pseudotriangulations, although not all pointed pseudotriangulations can arise in this way.
For a detailed survey of much of the material discussed here, see Rote, Santos, and Streinu (2008).
Pseudotriangles
Pocchiola and Vegter (1996a,b,c) originally defined a pseudotriangle to be a simply-connected region of the plane bounded by three smooth convex curves that are tangent at their endpoints. However, subsequent work has settled on a broader definition that applies more generally to polygons as well as to regions bounded by smooth curves, and that allows nonzero angles at the three vertices. In this broader definition, a pseudotriangle is a simply-connected region of the plane, having three convex vertices. The three boundary curves connecting these three vertices must be convex, in the sense that any line segment connecting two points on the same boundary curve must lie entirely outside or on the boundary of the pseudotriangle. Thus, the pseudotriangle is the region between the convex hulls of these three curves, and more generally any three mutually tangent convex sets form a pseudotriangle that lies between them.
For algorithmic applications it is of particular interest to characterize pseudotriangles that are polygons. In a polygon, a vertex is convex if it spans an interior angle of less than π, and concave otherwise (in particular, we consider an angle of exactly π to be concave). Any polygon must have at least three convex angles, because the total exterior angle of a polygon is 2π, the convex angles contribute less than π each to this total, and the concave angles contribute zero or negative |
https://en.wikipedia.org/wiki/Quasi-polynomial | In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.
A quasi-polynomial can be written as , where is a periodic function with integral period. If is not identically zero, then the degree of is . Equivalently, a function is a quasi-polynomial if there exist polynomials such that when . The polynomials are called the constituents of .
Examples
Given a -dimensional polytope with rational vertices , define to be the convex hull of . The function is a quasi-polynomial in of degree . In this case, is a function . This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
Given two quasi-polynomials and , the convolution of and is
which is a quasi-polynomial with degree
See also
Ehrhart polynomial
References
Stanley, Richard P. (1997). Enumerative Combinatorics, Volume 1. Cambridge University Press. , 0-521-56069-1.
Polynomials
Algebraic combinatorics |
https://en.wikipedia.org/wiki/Mittag-Leffler%20star | In complex analysis, a branch of mathematics, the Mittag-Leffler star of a complex-analytic function is a set in the complex plane obtained by attempting to extend that function along rays emanating from a given point. This concept is named after Gösta Mittag-Leffler.
Definition and elementary properties
Formally, the Mittag-Leffler star of a complex-analytic function ƒ defined on an open disk U in the complex plane centered at a point a is the set of all points z in the complex plane such that ƒ can be continued analytically along the line segment joining a and z (see analytic continuation along a curve).
It follows from the definition that the Mittag-Leffler star is an open star-convex set (with respect to the point a) and that it contains the disk U. Moreover, ƒ admits a single-valued analytic continuation to the Mittag-Leffler star.
Examples
The Mittag-Leffler star of the complex exponential function defined in a neighborhood of a = 0 is the entire complex plane.
The Mittag-Leffler star of the complex logarithm defined in the neighborhood of point a = 1 is the entire complex plane without the origin and the negative real axis. In general, given the complex logarithm defined in the neighborhood of a point a ≠ 0 in the complex plane, this function can be extended all the way to infinity on any ray starting at a, except on the ray which goes from a to the origin, one cannot extend the complex logarithm beyond the origin along that ray.
Any open star-convex set is the Mittag-Leffler star of some complex-analytic function, since any open set in the complex plane is a domain of holomorphy.
Uses
Any complex-analytic function ƒ defined around a point a in the complex plane can be expanded in a series of polynomials which is convergent in the entire Mittag-Leffler star of ƒ at a. Each polynomial in this series is a linear combination of the first several terms in the Taylor series expansion of ƒ around a.
Such a series expansion of ƒ, called the Mittag-Leffler expansion, is convergent in a larger set than the Taylor series expansion of ƒ at a. Indeed, the largest open set on which the latter series is convergent is a disk centered at a and contained within the Mittag-Leffler star of ƒ at a
References
External links
Analytic functions |
https://en.wikipedia.org/wiki/Anderson%27s%20theorem | In mathematics, Anderson's theorem is a result in real analysis and geometry which says that the integral of an integrable, symmetric, unimodal, non-negative function f over an n-dimensional convex body K does not decrease if K is translated inwards towards the origin. This is a natural statement, since the graph of f can be thought of as a hill with a single peak over the origin; however, for n ≥ 2, the proof is not entirely obvious, as there may be points x of the body K where the value f(x) is larger than at the corresponding translate of x.
Anderson's theorem, named after Theodore Wilbur Anderson, also has an interesting application to probability theory.
Statement of the theorem
Let K be a convex body in n-dimensional Euclidean space Rn that is symmetric with respect to reflection in the origin, i.e. K = −K. Let f : Rn → R be a non-negative, symmetric, globally integrable function; i.e.
f(x) ≥ 0 for all x ∈ Rn;
f(x) = f(−x) for all x ∈ Rn;
Suppose also that the super-level sets L(f, t) of f, defined by
are convex subsets of Rn for every t ≥ 0. (This property is sometimes referred to as being unimodal.) Then, for any 0 ≤ c ≤ 1 and y ∈ Rn,
Application to probability theory
Given a probability space (Ω, Σ, Pr), suppose that X : Ω → Rn is an Rn-valued random variable with probability density function f : Rn → [0, +∞) and that Y : Ω → Rn is an independent random variable. The probability density functions of many well-known probability distributions are p-concave for some p, and hence unimodal. If they are also symmetric (e.g. the Laplace and normal distributions), then Anderson's theorem applies, in which case
for any origin-symmetric convex body K ⊆ Rn.
References
Theorems in geometry
Probability theorems
Theorems in real analysis |
https://en.wikipedia.org/wiki/Shephard%27s%20problem | In mathematics, Shephard's problem, is the following geometrical question asked by Geoffrey Colin Shephard in 1964: if K and L are centrally symmetric convex bodies in n-dimensional Euclidean space such that whenever K and L are projected onto a hyperplane, the volume of the projection of K is smaller than the volume of the projection of L, then does it follow that the volume of K is smaller than that of L?
In this case, "centrally symmetric" means that the reflection of K in the origin, −K, is a translate of K, and similarly for L. If k : Rn → Πk is a projection of Rn onto some k-dimensional hyperplane Πk (not necessarily a coordinate hyperplane) and Vk denotes k-dimensional volume, Shephard's problem is to determine the truth or falsity of the implication
Vk(k(K)) is sometimes known as the brightness of K and the function Vk o k as a (k-dimensional) brightness function.
In dimensions n = 1 and 2, the answer to Shephard's problem is "yes". In 1967, however, Petty and Schneider showed that the answer is "no" for every n ≥ 3. The solution of Shephard's problem requires Minkowski's first inequality for convex bodies and the notion of projection bodies of convex bodies.
See also
Busemann–Petty problem
Notes
References
Convex geometry
Convex analysis |
https://en.wikipedia.org/wiki/Minkowski%27s%20first%20inequality%20for%20convex%20bodies | In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality.
Statement of the inequality
Let K and L be two n-dimensional convex bodies in n-dimensional Euclidean space Rn. Define a quantity V1(K, L) by
where V denotes the n-dimensional Lebesgue measure and + denotes the Minkowski sum. Then
with equality if and only if K and L are homothetic, i.e. are equal up to translation and dilation.
Remarks
V1 is just one example of a class of quantities known as mixed volumes.
If L is the n-dimensional unit ball B, then n V1(K, B) is the (n − 1)-dimensional surface measure of K, denoted S(K).
Connection to other inequalities
The Brunn–Minkowski inequality
One can show that the Brunn–Minkowski inequality for convex bodies in Rn implies Minkowski's first inequality for convex bodies in Rn, and that equality in the Brunn–Minkowski inequality implies equality in Minkowski's first inequality.
The isoperimetric inequality
By taking L = B, the n-dimensional unit ball, in Minkowski's first inequality for convex bodies, one obtains the isoperimetric inequality for convex bodies in Rn: if K is a convex body in Rn, then
with equality if and only if K is a ball of some radius.
References
Calculus of variations
Geometric inequalities
Normed spaces |
https://en.wikipedia.org/wiki/Principal%20series%20representation | In mathematics, the principal series representations of certain kinds of topological group G occur in the case where G is not a compact group. There, by analogy with spectral theory, one expects that the regular representation of G will decompose according to some kind of continuous spectrum, of representations involving a continuous parameter, as well as a discrete spectrum. The principal series representations are some induced representations constructed in a uniform way, in order to fill out the continuous part of the spectrum.
In more detail, the unitary dual is the space of all representations relevant to decomposing the regular representation. The discrete series consists of 'atoms' of the unitary dual (points carrying a Plancherel measure > 0). In the earliest examples studied, the rest (or most) of the unitary dual could be parametrised by starting with a subgroup H of G, simpler but not compact, and building up induced representations using representations of H which were accessible, in the sense of being easy to write down, and involving a parameter. (Such an induction process may produce representations that are not unitary.)
For the case of a semisimple Lie group G, the subgroup H is constructed starting from the Iwasawa decomposition
G = KAN
with K a maximal compact subgroup. Then H is chosen to contain AN (which is a non-compact solvable Lie group), being taken as
H := MAN
with M the centralizer in K of A. Representations ρ of H are considered that are irreducible, and unitary, and are the trivial representation on the subgroup N. (Assuming the case M a trivial group, such ρ are analogues of the representations of the group of diagonal matrices inside the special linear group.) The induced representations of such ρ make up the principal series. The spherical principal series consists of representations induced from 1-dimensional representations of MAN obtained by extending characters of A
using the homomorphism of MAN onto A.
There may be other continuous series of representations relevant to the unitary dual: as their name implies, the principal series are the 'main' contribution.
This type of construction has been found to have application to groups G that are not Lie groups (for example, finite groups of Lie type, groups over p-adic fields).
Examples
For examples, see the representation theory of SL2(R). For the general linear group GL2 over a local field, the dimension of the Jacquet module of a principal series representation is two.
References
External links
Computing the unitary dual (PDF)
Unitary representation theory
Representation theory of Lie groups |
https://en.wikipedia.org/wiki/Principal%20series | Principal series may refer to:
Principal series (spectroscopy), series of spectral lines
Principal series representation , topological group theory,
Science disambiguation pages |
https://en.wikipedia.org/wiki/Palestinian%20citizens%20of%20Israel | Palestinian citizens of Israel, also known as 48-Palestinians () are Arab citizens of Israel that self-identify as Palestinian. According to Israel's Central Bureau of Statistics, the Arab population in 2019 was estimated at 1,890,000, representing 20.95% of the country's population. The majority of these identify themselves as Arab or Palestinian by nationality and Israeli by citizenship. Many Arabs have family ties to Palestinians in the West Bank and Gaza Strip as well as to Palestinian refugees in Jordan, Syria and Lebanon.
Identification as Palestinian
Historical
Between 1948 and 1967, very few Arab citizens of Israel identified openly as "Palestinian". An "Israeli-Arab" identity, the preferred phrase of the Israeli establishment and public, was predominant. Public expressions of Palestinian identity, such as displays of the Palestinian flag or the singing and reciting of nationalist songs or poetry were illegal. Ever since the Nakba, the Palestinians that have remained within Israel's 1948 borders have been colloquially known as "48 Arabs" (). With the end of military administrative rule in 1966 and following the 1967 war, national consciousness and its expression among Israel's Arab citizens spread. A majority then self-identified as Palestinian, preferring this descriptor to Israeli Arab in numerous surveys over the years.
Terminology
How to refer to the Arab citizenry of Israel is a highly politicized issue, and there are a number of self-identification labels used by members of this community. Generally speaking, supporters of Israel tend to use Israeli Arab or Arab Israeli to refer to this population without mentioning Palestine, while critics of Israel (or supporters of Palestinians) tend to use Palestinian or Palestinian Arab without referencing Israel. According to The New York Times, most preferred to identify themselves as Palestinian citizens of Israel rather than as Israeli Arabs, as of 2012. The New York Times uses both 'Palestinian Israelis' and 'Israeli Arabs' to refer to the same population.
Common practice in contemporary academic literature is to identify this community as Palestinian as it is how the majority self-identify. Terms preferred by most Arab citizens to identify themselves include Palestinians, Palestinians in Israel, Israeli Palestinians, the Palestinians of 1948, Palestinian Arabs, Palestinian Arab citizens of Israel or Palestinian citizens of Israel. There are, however, individuals from among the Arab citizenry who reject the term Palestinian altogether. A minority of Israel's Arab citizens include "Israeli" in some way in their self-identifying label; the majority identify as Palestinian by nationality and Israeli by citizenship.
The Israeli establishment prefers Israeli Arabs or Arabs in Israel, and also uses the terms the minorities, the Arab sector, Arabs of Israel and Arab citizens of Israel. These labels have been criticized for denying this population a political or national identification, obscu |
https://en.wikipedia.org/wiki/Section%20%28category%20theory%29 | In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism.
In other words, if and are morphisms whose composition is the identity morphism on , then is a section of , and is a retraction of .
Every section is a monomorphism (every morphism with a left inverse is left-cancellative), and every retraction is an epimorphism (every morphism with a right inverse is right-cancellative).
In algebra, sections are also called split monomorphisms and retractions are also called split epimorphisms. In an abelian category, if is a split epimorphism with split monomorphism , then is isomorphic to the direct sum of and the kernel of . The synonym coretraction for section is sometimes seen in the literature, although rarely in recent work.
Properties
A section that is also an epimorphism is an isomorphism. Dually a retraction that is also a monomorphism is an isomorphism.
Terminology
The concept of a retraction in category theory comes from the essentially similar notion of a retraction in topology: where is a subspace of is a retraction in the topological sense, if it's a retraction of the inclusion map in the category theory sense. The concept in topology was defined by Karol Borsuk in 1931.
Borsuk's student, Samuel Eilenberg, was with Saunders Mac Lane the founder of category theory, and (as the earliest publications on category theory concerned various topological spaces) one might have expected this term to have initially be used. In fact, their earlier publications, up to, e.g., Mac Lane (1963)'s Homology, used the term right inverse. It was not until 1965 when Eilenberg and John Coleman Moore coined the dual term 'coretraction' that Borsuk's term was lifted to category theory in general. The term coretraction gave way to the term section by the end of the 1960s.
Both use of left/right inverse and section/retraction are commonly seen in the literature: the former use has the advantage that it is familiar from the theory of semigroups and monoids; the latter is considered less confusing by some because one does not have to think about 'which way around' composition goes, an issue that has become greater with the increasing popularity of the synonym f;g for g∘f.
Examples
In the category of sets, every monomorphism (injective function) with a non-empty domain is a section, and every epimorphism (surjective function) is a retraction; the latter statement is equivalent to the axiom of choice.
In the category of vector spaces over a field K, every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.
In the category of abelian groups, the epimorphism Z → Z/2Z which sends every integer to its remainder modulo 2 does not split; in fact the only morphism Z/2Z → Z is the zero map. Similarly, the natural monomorphism Z/2Z → Z/4Z doesn't split even though there |
https://en.wikipedia.org/wiki/Closed | Closed may refer to:
Mathematics
Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
Closed set, a set which contains all its limit points
Closed interval, an interval which includes its endpoints
Closed line segment, a line segment which includes its endpoints
Closed manifold, a compact manifold which has no boundary
Other uses
Closed (poker), a betting round where no player will have the right to raise
Closed (album), a 2010 album by Bomb Factory
Closed GmbH, a German fashion brand
Closed class, in linguistics, a class of words or other entities which rarely changes
See also
Close (disambiguation)
Closed loop (disambiguation)
Closing (disambiguation)
Closure (disambiguation)
Open (disambiguation) |
https://en.wikipedia.org/wiki/Gelfand%E2%80%93Graev%20representation | In representation theory, a branch of mathematics, the Gelfand–Graev representation is a representation of a reductive group over a finite field introduced by , induced from a non-degenerate character of a Sylow subgroup.
The Gelfand–Graev representation is reducible and decomposes as the sum of irreducible representations, each of multiplicity at most 1. The irreducible representations occurring in the Gelfand–Graev representation are called regular representations. These are the analogues for finite groups of representations with a Whittaker model.
References
English translation in volume 2 of Gelfand's collected works.
Representation theory |
https://en.wikipedia.org/wiki/Paul%20de%20Nourquer%20du%20Camper | Paul de Nourquer du Camper was Governor General for Inde française in the Second French Colonial Empire during the July Monarchy. During his period an annual statistics manual was written by Pierre Constant Sicé in 1842, which describes and narrates various situations in Inde française.
Titles Held
See also
European and American voyages of scientific exploration
French colonial governors and administrators
Governors of French India
Governors of French Guiana
People of the July Monarchy
Year of birth missing
Year of death missing |
https://en.wikipedia.org/wiki/Patrick%20du%20Val | Patrick du Val (March 26, 1903 – January 22, 1987) was a British mathematician, known for his work on algebraic geometry, differential geometry, and general relativity. The concept of Du Val singularity of an algebraic surface is named after him.
Early life
Du Val was born in Cheadle Hulme, Cheshire. He was the son of a cabinet maker, but his parents' marriage broke up. As a child, he suffered ill-health, in particular asthma, and was educated mostly by his mother. He was awarded a first class honours degree from the University of London External Programme in 1926, which he took by correspondence course.
He was a talented linguist, for example teaching himself Norwegian so that he might read Peer Gynt. He also had a strong interest in history but his love of mathematics led him to pursue that as a career. His earliest publications show a leaning towards applied mathematics.
His mother moved to a village near Cambridge and he became acquainted with Henry Baker, Lowndean Professor of Astronomy and Geometry. Baker turned his interest towards algebraic geometry, and he entered Trinity College, Cambridge in 1927.
Research in geometry
Du Val's early work before becoming a research student was on relativity, including a paper on the De Sitter model of the universe and Grassmann's tensor calculus. His doctorate was on algebraic geometry and in his thesis he generalised a result of Schoute. He worked on algebraic surfaces and later in his career became interested in elliptic functions.
He received his Ph.D. with a thesis entitled 'On Certain Configurations of Algebraic Geometry Having Groups of Self-Transformations Representable by Symmetry Groups of Certain Polygons' under Baker's supervision in 1930. While a research student he had many famous geometers including Hodge as fellow research students, and he formed a particular friendship with Coxeter and Semple. He was elected a fellow of Trinity in 1930 for four years. During that time he travelled extensively, visiting Rome and working with Federigo Enriques, then in 1934 Princeton University, where he attended lectures by James W. Alexander, Luther P. Eisenhart, Solomon Lefschetz, Oswald Veblen, Joseph Wedderburn, and Hermann Weyl.
In 1936, Du Val took up an assistant lectureship in the Mathematics Department at Manchester, where he stayed for five years. He was then funded by a British Council scheme to go to Istanbul University as a professor of pure mathematics. There he learnt Turkish and even wrote a book on coordinate geometry in that language.
After a spell in the United States at the University of Georgia, he returned to the United Kingdom, first taking up a post in Bristol, then at the University College London in 1954, where he remained until he retired in 1970. Together with Semple he led the London Geometry Seminar during the time he spent in London.
Du Val had three children.
Later life
After retirement, Du Val returned to Istanbul. For three years he held the same post as be |
https://en.wikipedia.org/wiki/Prime%20zeta%20function | In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by . It is defined as the following infinite series, which converges for :
Properties
The Euler product for the Riemann zeta function ζ(s) implies that
which by Möbius inversion gives
When s goes to 1, we have .
This is used in the definition of Dirichlet density.
This gives the continuation of P(s) to , with an infinite number of logarithmic singularities at points s where ns is a pole (only ns = 1 when n is a squarefree number greater than or equal to 1), or zero of the Riemann zeta function ζ(.). The line is a natural boundary as the singularities cluster near all points of this line.
If one defines a sequence
then
(Exponentiation shows that this is equivalent to Lemma 2.7 by Li.)
The prime zeta function is related to Artin's constant by
where Ln is the nth Lucas number.
Specific values are:
Analysis
Integral
The integral over the prime zeta function is usually anchored at infinity,
because the pole at prohibits defining a nice lower bound
at some finite integer without entering a discussion on branch cuts in the complex plane:
The noteworthy values are again those where the sums converge slowly:
Derivative
The first derivative is
The interesting values are again those where the sums converge slowly:
Generalizations
Almost-prime zeta functions
As the Riemann zeta function is a sum of inverse powers over the integers
and the prime zeta function a sum of inverse powers of the prime numbers,
the k-primes (the integers which are a product of not
necessarily distinct primes) define a sort of intermediate sums:
where is the total number of prime factors.
Each integer in the denominator of the Riemann zeta function
may be classified by its value of the index , which decomposes the Riemann zeta
function into an infinite sum of the :
Since we know that the Dirichlet series (in some formal parameter u) satisfies
we can use formulas for the symmetric polynomial variants with a generating function of the right-hand-side type. Namely, we have the coefficient-wise identity that when the sequences correspond to where denotes the characteristic function of the primes. Using Newton's identities, we have a general formula for these sums given by
Special cases include the following explicit expansions:
Prime modulo zeta functions
Constructing the sum not over all primes but only over primes which are in the same modulo class introduces further types of infinite series that are a reduction of the Dirichlet L-function.
See also
Divergence of the sum of the reciprocals of the primes
References
External links
Zeta and L-functions |
https://en.wikipedia.org/wiki/Positive-definite%20function%20on%20a%20group | In mathematics, and specifically in operator theory, a positive-definite function on a group relates the notions of positivity, in the context of Hilbert spaces, and algebraic groups. It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.
Definition
Let G be a group, H be a complex Hilbert space, and L(H) be the bounded operators on H.
A positive-definite function on G is a function that satisfies
for every function h: G → H with finite support (h takes non-zero values for only finitely many s).
In other words, a function F: G → L(H) is said to be a positive-definite function if the kernel K: G × G → L(H) defined by K(s, t) = F(s−1t) is a positive-definite kernel.
Unitary representations
A unitary representation is a unital homomorphism Φ: G → L(H) where Φ(s) is a unitary operator for all s. For such Φ, Φ(s−1) = Φ(s)*.
Positive-definite functions on G are intimately related to unitary representations of G. Every unitary representation of G gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of G in a natural way.
Let Φ: G → L(H) be a unitary representation of G. If P ∈ L(H) is the projection onto a closed subspace H` of H. Then F(s) = P Φ(s) is a positive-definite function on G with values in L(H`). This can be shown readily:
for every h: G → H` with finite support. If G has a topology and Φ is weakly(resp. strongly) continuous, then clearly so is F.
On the other hand, consider now a positive-definite function F on G. A unitary representation of G can be obtained as follows. Let C00(G, H) be the family of functions h: G → H with finite support. The corresponding positive kernel K(s, t) = F(s−1t) defines a (possibly degenerate) inner product on C00(G, H). Let the resulting Hilbert space be denoted by V.
We notice that the "matrix elements" K(s, t) = K(a−1s, a−1t) for all a, s, t in G. So Uah(s) = h(a−1s) preserves the inner product on V, i.e. it is unitary in L(V). It is clear that the map Φ(a) = Ua is a representation of G on V.
The unitary representation is unique, up to Hilbert space isomorphism, provided the following minimality condition holds:
where denotes the closure of the linear span.
Identify H as elements (possibly equivalence classes) in V, whose support consists of the identity element e ∈ G, and let P be the projection onto this subspace. Then we have PUaP = F(a) for all a ∈ G.
Toeplitz kernels
Let G be the additive group of integers Z. The kernel K(n, m) = F(m − n) is called a kernel of Toeplitz type, by analogy with Toeplitz matrices. If F is of the form F(n) = Tn where T is a bounded operator acting on some Hilbert space. One can show that the kernel K(n, m) is positive if and only if T is a contraction. By the discussion from the previous section, we have a unitary representation of Z, Φ(n) = Un for a unitary operator U. Moreover, the property |
https://en.wikipedia.org/wiki/Boca%20Juniors%20top%20scorers | This article includes statistics of Boca Juniors all-time top goal scorers.
Martín Palermo is Boca Juniors all time goal scorer with 236 goals, 193 of those goals were scored in Argentine Primera División tournaments and the other 43 in International tournaments.
Palermo is also the club's top international scorer with 43 goals, followed by Rodrigo Palacio with 28.
All time top scorers
All official tournaments
Note: Only goals in official competitions are included.
Last updated on: 22 September 2023 – Top 20 scorers of all time (all competitions) at historiadeboca.com.ar
League goals
Last updated on: 20 September 2023 – Top 20 league scorers at historiadeboca.com.ar
International goals
Last updated on: 20 September 2023 – Top 20 international scorers at historiadeboca.com.ar
Top scorers per season
Those players that are bolded were also the Top Scorers of that championship.
Note:League goals only.
Most frequent Boca top scorers
Note:League goals only.
References
External links
Boca Players - Historia de Boca
Boca Tournaments - Historia de Boca
Club's Top scorers per season - Agrupación Nuevo Boca
Historical Top scorers - Agrupación Nuevo Boca
Boca Juniors footballers
Association football player non-biographical articles
Boca |
https://en.wikipedia.org/wiki/Preimage%20theorem | In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.
Statement of Theorem
Definition. Let be a smooth map between manifolds. We say that a point is a regular value of if for all the map is surjective. Here, and are the tangent spaces of and at the points and
Theorem. Let be a smooth map, and let be a regular value of Then is a submanifold of If then the codimension of is equal to the dimension of Also, the tangent space of at is equal to
There is also a complex version of this theorem:
Theorem. Let and be two complex manifolds of complex dimensions Let be a holomorphic map and let be such that for all Then is a complex submanifold of of complex dimension
See also
References
Theorems in differential topology |
https://en.wikipedia.org/wiki/J.%20A.%20Green | J. A. Green may refer to:
Sandy Green (mathematician) (James Alexander Green, 1926–2014), professor of mathematics
J. A. Green (photographer) (1873–1905), Nigerian photographer |
https://en.wikipedia.org/wiki/Sandy%20Green%20%28mathematician%29 | James Alexander "Sandy" Green FRS (26 February 1926 – 7 April 2014) was a mathematician and Professor at the Mathematics Institute at the University of Warwick, who worked in the field of representation theory.
Early life
Sandy Green was born in February 1926 in Rochester, New York, but moved to Toronto with his emigrant Scottish parents later that year. The family returned to Britain in May 1935 when his father, Frederick C. Green, took up the Drapers Professorship of French at the University of Cambridge.
Education
Green was educated at the Perse School, Cambridge. He won a scholarship to the University of St Andrews and matriculated aged 16 in 1942. He took an ordinary BSc in 1944, and then, after scientific service in the war, was awarded a BSc Honours in 1947. He gained his PhD at St John's College, Cambridge in 1951, under the supervision of Philip Hall and David Rees.
Career
World War II
In the summer of 1944, he was conscripted for national scientific service at the age of eighteen, and was he was assigned to work at Bletchley Park, where he acted as a human "computer" carrying out calculations in Hut F, the "Newmanry", a department led by Max Newman, which used special-purpose Colossus computers to assist in breaking German naval codes.
Academic work
His first lecturing post (1950) was at the University of Manchester, where Newman was his Head of department. In 1964 he became a Reader at the University of Sussex, and then in 1965 was appointed as a professor at the newly formed Mathematics Institute at Warwick University, where he led the algebra group. He spent several periods as a visiting academic in the United States, beginning with a year at the Institute for Advanced Study in Princeton, New Jersey in 1960–61, as well as similar visits to universities in France, Germany and Portugal. After retiring from Warwick he became a member of the faculty and Professor Emeritus at the Mathematics Institute of the University of Oxford, in whose meetings he participated actively. His final publication was produced at the age of eighty.
Work in mathematics
Green found all the characters of general linear groups over finite fields (Green 1955) and invented the Green correspondence in modular representation theory. Both Green functions in the representation theory of groups of Lie type and Green's relations in the area of semigroups are named after him. His final publication (2007) was a revised and augmented edition of his 1980 work, Polynomial Representations of GL(n).
Personal life
Green met his wife, Margaret Lord, at Bletchley Park, where she worked as a Colossus operator, also in the Newmanry section (Hut F). The couple married in August 1950, and have two daughters and a son. Up to his death, he lived in Oxford.
Honours
He was elected to the Royal Society of Edinburgh in 1968 and the Royal Society in 1987 and was awarded two London Mathematical Society prizes: the Senior Berwick Prize in 1984 and the de Morgan Medal in 2001.
Bibli |
https://en.wikipedia.org/wiki/%CE%A0%20pad | The Π pad (pi pad) is a specific type of attenuator circuit in electronics whereby the topology of the circuit is formed in the shape of the Greek capital letter pi (Π).
Attenuators are used in electronics to reduce the level of a signal. They are also referred to as pads due to their effect of padding down a signal by analogy with acoustics. Attenuators have a flat frequency response attenuating all frequencies equally in the band they are intended to operate. The attenuator has the opposite task of an amplifier. The topology of an attenuator circuit will usually follow one of the simple filter sections. However, there is no need for more complex circuitry, as there is with filters, due to the simplicity of the frequency response required.
Circuits are required to be balanced or unbalanced depending on the geometry of the transmission lines they are to be used with. For radio frequency applications, the format is often unbalanced, such as coaxial. For audio and telecommunications, balanced circuits are usually required, such as with the twisted pair format. The Π pad is intrinsically an unbalanced circuit. However, it can be converted to a balanced circuit by placing half the series resistance in the return path. Such a circuit is called a box section because the circuit is formed in the shape of a box.
Terminology
An attenuator is a form of a two-port network with a generator connected to one port and a load connected to the other. In all of the circuits given below it is assumed that the generator and load impedances are purely resistive (though not necessarily equal) and that the attenuator circuit is required to perfectly match to these. The symbols used for these impedances are;
the impedance of the generator
the impedance of the load
Popular values of impedance are 600Ω in telecommunications and audio, 75Ω for video and dipole antennae, and 50Ω for RF.
The voltage transfer function, A, is,
While the inverse of this is the loss, L, of the attenuator,
The value of attenuation is normally marked on the attenuator as its loss, LdB, in decibels (dB). The relationship with L is;
Popular values of attenuator are 3dB, 6dB, 10dB, 20dB, and 40dB.
However, it is often more convenient to express the loss in nepers,
where is the attenuation in nepers (one neper is approximately 8.7 dB).
Impedance and loss
The values of resistance of the attenuator's elements can be calculated using image parameter theory. The starting point here is the image impedances of the L section in figure 2. The image admittance of the input is,
and the image impedance of the output is,
The loss of the L section when terminated in its image impedances is,
where the image parameter transmission function, γL is given by,
The loss of this L section in the reverse direction is given by,
For an attenuator, Z and Y are simple resistors and γ becomes the image parameter attenuation (that is, the attenuation when terminated with the image impedance |
https://en.wikipedia.org/wiki/Evelyn%20Nelson%20%28mathematician%29 | Evelyn Merle Nelson (November 25, 1943 – August 1, 1987), born Evelyn Merle Roden, was a Canadian mathematician. Nelson made contributions to the area of universal algebra with applications to theoretical computer science. She, along with Cecilia Krieger, is the namesake of the Krieger–Nelson Prize,
awarded by the Canadian Mathematical Society for outstanding research by a female mathematician.
Early life
Nelson was born on November 25, 1943, in Hamilton, Ontario, Canada. Her parents were immigrants from Russia in the 1920s. Nelson went to high school at Westdale Secondary School in Hamilton.
Education
After spending two years at the University of Toronto, Nelson returned to Hamilton to study at McMaster University. She received her B.Sc in mathematics from
McMaster in 1965, followed by an M.Sc in mathematics from McMaster in 1967. She succeeded in having her thesis work published in the Canadian Journal of Mathematics, also in 1967; the article was entitled "Finiteness of semigroups of operators in universal algebra".
Nelson completed her Ph.D in 1970. Her thesis was entitled "The lattice of equational classes of commutative semigroups", and the ideas also formed a journal paper published in the Canadian Journal of Mathematics.
Career
Following completion of her Ph.D., Nelson continued at McMaster. She first worked as a post-doctoral researcher, later as a "research associate", and in 1978 was appointed associate professor. Serving as chair of the Unit of Computer Science at McMaster from 1982 until 1984, Nelson became a full professor in 1983.
Nelson's teaching record was, according to one colleague, "invariably of the highest order". However, before earning a faculty position at McMaster, prejudice against her led to doubts about her teaching ability.
Nelson published over 40 papers during her 20-year career. She died from cancer in 1987.
Recognition
Nelson is the namesake, along with Cecilia Krieger, of the Krieger–Nelson Prize, which is awarded to a female mathematician in recognition
of outstanding achievement. The Department of Mathematics at McMaster University has a lecture series, "The Evelyn
Nelson Lectures", held since 1991.
Selected publications
References
External links
"Evelyn M. Nelson", Biographies of Women Mathematicians, Agnes Scott College
Canadian women mathematicians
1943 births
1987 deaths
People from Hamilton, Ontario
McMaster University alumni
Academic staff of McMaster University
20th-century Canadian mathematicians
20th-century women mathematicians
20th-century Canadian women scientists |
https://en.wikipedia.org/wiki/How%20Students%20Learn | How Students Learn: History, Mathematics, and Science in the Classroom is the title of a 2001 educational psychology book edited by M. Suzanne Donovan and John D. Bransford and published by the United States National Academy of Sciences's National Academies Press.
The book focuses on "three fundamental and well-established principles of learning that are highlighted in How People Learn and are particularly important for teachers to understand and be able to incorporate in their teaching:
"Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information, or they may learn them for purposes of a test but revert to their preconceptions outside the classroom.
"To develop competence in an area of inquiry, students must (a) have a deep foundation of factual knowledge, (b) understand the facts and ideas in the context of a conceptual framework, and (c) organize knowledge in ways that facilitate retrieval and application.
"A 'metacognitive' approach to instruction can help students learn to take control of their own learning by defining learning goals and monitoring their progress in achieving them."
References
Adomanis, James F. (May 2006). "How Students Learn: History in the Classroom, edited by M. Suzanne Donovan and John D. Bransford. Washington, DC: The National Academies Press, 2005. 615 pages. $34.95, paper, with a CD-ROM". The History Teacher. Society for History Education. 39 (3): 410–411. doi:10.2307/30036810. ISSN 0018-2745.
Gilbert, John K. (2005-09-26). "How Students Learn: Science in the Classroom". Science Education. Wiley. 89 (6): 1043–1045. doi:10.1002/sce.20115. ISSN 0036-8326.
Coffey, David (March 2006). "How Students Learn: Mathematics in the Classroom". Mathematics Teaching in the Middle School. National Council of Teachers of Mathematics. 11 (7): 351–352. ISSN 1072-0839. JSTOR 41182323.
each, John T. (2005-12-16). "Book review: How students learn: Science in the classroom". International Journal of Science Education. Routledge. 27 (15): 1883–1886. doi:10.1080/09500690500247576. ISSN 0950-0693.
Carboni, Lisa Wilson (March 2006). "How Students Learn: Mathematics in the Classroom". Teaching Children Mathematics. National Council of Teachers of Mathematics. 12 (7): 384. ISSN 1073-5836. JSTOR 41198776.
Godsell, Sarah (2016-12-03). "What is history? Views from a primary school teacher education programme". South African Journal of Childhood Education. University of Johannesburg. 6 (1). doi:10.4102/sajce.v6i1.485. ISSN 2223-7682. Archived from the original (PDF) on 2020-03-30.
Davis, Seonaid (September 2005). "Teaching Science Through the Use of Modelling [How Students Learn History, Mathematics, and Science in the Classroom]". The Crucible. Science Teachers' Association of Ontario. 371 (1): 16–18. ISSN 0381-8047.
External links
Free online executive summary
Educational psychology books |
https://en.wikipedia.org/wiki/Eric%20van%20Douwen | Eric Karel van Douwen (April 25, 1946 in Voorburg, South Holland, Netherlands – July 28, 1987 in Athens, Ohio, United States) was a Dutch mathematician specializing in set-theoretic topology. He received his Ph.D. in 1975 from Vrije Universiteit under the supervision of Maarten Maurice and Johannes Aarts, both of whom were in turn students of Johannes de Groot.
He began his academic career studying physics, but became dissatisfied partway through the program. His wife helped inspire his choice to switch to mathematics by asking, "Why not mathematics? It's what you work on all the time anyway". He produced the content of his dissertation unsupervised, and seeking better credentials, he transferred to Vrije to defend, a maneuver permitted by the Dutch university rules.
References
External links
Eric van Douwen's papers Includes a short bio. From Scott Williams's pages at SUNY Buffalo
1946 births
1987 deaths
20th-century Dutch mathematicians
Topologists
People from Voorburg
Vrije Universiteit Amsterdam alumni |
https://en.wikipedia.org/wiki/693%20%28number%29 | 693 (six hundred [and] ninety-three) is the natural number following 692 and preceding 694.
In mathematics
693 has twelve divisors: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, and 693. Thus, 693 is tied with 315 for the highest number of divisors for any odd natural number below 900. The smallest positive odd integer with more divisors is 945, which has 16 divisors. Consequently, 945 is also the smallest odd abundant number, having an abundancy index of 1920/945 ≈ 2.03175.
693 appears as the first three digits after the decimal point in the decimal form for the natural logarithm of 2. To 10 digits, this number is 0.6931471805. As a result, if an event has a constant probability of 0.1% of occurring, 693 is the smallest number of trials that must be performed for there to be at least a 50% chance that the event occurs at least once. More generally, for any probability p, the probability that the event occurs at least once in a sample of n items, assuming the items are independent, is given by the following formula:
1 − (1 − p)n
For p = 10−3 = 0.001, plugging in n = 692 gives, to four decimal places, 0.4996, while n = 693 yields 0.5001.
693 is the lowest common multiple of 7, 9, and 11. Multiplying 693 by 5 gives 3465, the smallest positive integer divisible by 3, 5, 7, 9, and 11.
693 is a palindrome in bases 32, 62, 76, 98, 230, and 692. It is also a palindrome in binary: 1010110101.
The reciprocal of 693 has a period of six: = 0..
693 is a triangular matchstick number.
References
Integers |
https://en.wikipedia.org/wiki/New%20Orleans%20Charter%20Science%20and%20Mathematics%20High%20School | New Orleans Charter Science & Math High School is an open enrollment charter school in New Orleans, Louisiana, USA. Students commonly refer to the school as "SciHigh", "Science & Math", or vice versa, "Math and Science".
The organization, Advocates for Science and Mathematics Education, governs the school, which is located in Uptown, in the former Allen Elementary School campus. The school is supported in part by the Foundation for Science and Mathematics Education, a 501(c)(3) nonprofit organization that advocates for "an open-admissions, rigorous, hands-on educational model paired with the belief that any student can succeed when provided with a safe and supportive environment."
History
The school was founded in 1993 by Barbara MacPhee as a half day school focused on the rigorous instruction of mathematics and science. Students from any New Orleans Public School were able to enroll part-time at NOCSMHS and part-time at their "home" school. From its inception until Hurricane Katrina, the school was housed on the campus of Delgado Community College.
After Katrina, the school was chartered under the auspices of the Orleans Parish School Board as a full-day, grades 9-12 high school, offering instruction in all subjects, including English, social studies, and foreign language. Due to the damage, Delgado received from Katrina, NOCSMHS moved to the building which housed the former Henry W. Allen Elementary School in the uptown New Orleans neighborhood.
In October 2010, the school met its performance growth goal by attaining a School Performance Score over 80.0, giving it a two-star rating.
The following year, in October 2011, the school far surpassed its growth goal and earned a "B+" grade—the highest grade given to any open-enrollment school in the New Orleans area.
In April 2018 the board for Sci High selected former Delgado Community College employee Monique Cola as the school's headmaster; she has a PhD from Tulane University.
The groundbreaking for the Mid-City campus was scheduled for fall 2018. The school relocated there in 2020. It has a 400-seat gymnasium on the second floor.
Faculty
In 2018 the school retained 68% of its faculty members, with 13 members starting work and 28 continuing to work from the previous year.
School building format
The first floor is primarily science classes. It also holds the cafeteria, Phys. Ed. room, Art class, and counselors office
The second floor is primarily language arts and social studies classes. It also holds the Computer Lab, Auditorium, and Main Office
The third floor is primarily the math and English classes. It also holds the Library and Special Education classes
Athletics
New Orleans Charter Science and Mathematics High athletics competes in the LHSAA.
References
External links
New Orleans Charter Science & Math High School Official Website.
EducateNow! website
Foundation for Science and Mathematics Education Official Website.
Charter schools in New Orleans
Public high schools |
https://en.wikipedia.org/wiki/Karen%20King | Karen King may refer to:
Karen Leigh King (born 1954), historian of religion
Karen D. King (born 1970), African-American mathematics educator |
https://en.wikipedia.org/wiki/Interval%20order | In mathematics, especially order theory,
the interval order for a collection of intervals on the real line
is the partial order corresponding to their left-to-right precedence relation—one interval, I1, being considered less than another, I2, if I1 is completely to the left of I2.
More formally, a countable poset is an interval order if and only if
there exists a bijection from to a set of real intervals,
so ,
such that for any we have
in exactly when .
Such posets may be equivalently
characterized as those with no induced subposet isomorphic to the
pair of two-element chains, in other words as the -free posets
. Fully written out, this means that for any two pairs of elements and one must have or .
The subclass of interval orders obtained by restricting the intervals to those of unit length, so they all have the form , is precisely the semiorders.
The complement of the comparability graph of an interval order (, ≤)
is the interval graph .
Interval orders should not be confused with the interval-containment orders, which are the inclusion orders on intervals on the real line (equivalently, the orders of dimension ≤ 2).
Interval orders and dimension
An important parameter of partial orders is order dimension: the dimension of a partial order is the least number of linear orders whose intersection is . For interval orders, dimension can be arbitrarily large. And while the problem of determining the dimension of general partial orders is known to be NP-hard, determining the dimension of an interval order remains a problem of unknown computational complexity.
A related parameter is interval dimension, which is defined analogously, but in terms of interval orders instead of linear orders. Thus, the interval dimension of a partially ordered set is the least integer for which there exist interval orders on with exactly when and .
The interval dimension of an order is never greater than its order dimension.
Combinatorics
In addition to being isomorphic to -free posets,
unlabeled interval orders on are also in bijection
with a subset of fixed-point-free involutions
on ordered sets with cardinality
. These are the
involutions with no so-called left- or right-neighbor nestings where, for any involution
on , a left nesting is
an such that and a right nesting is an such that
.
Such involutions, according to semi-length, have ordinary generating function
The coefficient of in the expansion of gives the number of unlabeled interval orders of size . The sequence of these numbers begins
1, 2, 5, 15, 53, 217, 1014, 5335, 31240, 201608, 1422074, 10886503, 89903100, 796713190, 7541889195, 75955177642, …
Notes
References
.
.
.
.
.
Further reading
Order theory
Combinatorics |
https://en.wikipedia.org/wiki/Hossein%20Malek-Afzali | Hossein Malek-Afzali (; born 1939) is an Iranian scientist, physician, and an associate of World Health Organization.
He is currently professor at the Department of Biostatistics and Epidemiology, School of Public Health, Tehran University. He also acted as deputy health minister of Iran.
Malek-Afzali is the author of more than 80 articles in international journals and several books in English and Persian.
In 2007, Malek-Afzali was awarded the United Nations Population Award. He has helped design strategies to improve health procedures, particularly adolescent health, reproductive health and family planning. In the field of reproductive health, he has engaged policymakers and religious leaders in the planning and implementation of reproductive health programmes in Iran.
Awards
United Nations Population Award (2007)
Notes
See also
Intellectual movements in Iran
Contemporary Medicine in Iran
Iranian public health doctors
20th-century Iranian inventors
Iranian Vice Ministers
Academic staff of the University of Tehran
World Health Organization officials
1939 births
Living people
Iranian officials of the United Nations |
https://en.wikipedia.org/wiki/Semiregular%20polytope | In geometry, by Thorold Gosset's definition a semiregular polytope is usually taken to be a polytope that is vertex-transitive and has all its facets being regular polytopes. E.L. Elte compiled a longer list in 1912 as The Semiregular Polytopes of the Hyperspaces which included a wider definition.
Gosset's list
In three-dimensional space and below, the terms semiregular polytope and uniform polytope have identical meanings, because all uniform polygons must be regular. However, since not all uniform polyhedra are regular, the number of semiregular polytopes in dimensions higher than three is much smaller than the number of uniform polytopes in the same number of dimensions.
The three convex semiregular 4-polytopes are the rectified 5-cell, snub 24-cell and rectified 600-cell. The only semiregular polytopes in higher dimensions are the k21 polytopes, where the rectified 5-cell is the special case of k = 0. These were all listed by Gosset, but a proof of the completeness of this list was not published until the work of for four dimensions, and for higher dimensions.
Gosset's 4-polytopes (with his names in parentheses)
Rectified 5-cell (Tetroctahedric),
Rectified 600-cell (Octicosahedric),
Snub 24-cell (Tetricosahedric), , or
Semiregular E-polytopes in higher dimensions
5-demicube (5-ic semi-regular), a 5-polytope, ↔
221 polytope (6-ic semi-regular), a 6-polytope, or
321 polytope (7-ic semi-regular), a 7-polytope,
421 polytope (8-ic semi-regular), an 8-polytope,
Euclidean honeycombs
Semiregular polytopes can be extended to semiregular honeycombs. The semiregular Euclidean honeycombs are the tetrahedral-octahedral honeycomb (3D), gyrated alternated cubic honeycomb (3D) and the 521 honeycomb (8D).
Gosset honeycombs:
Tetrahedral-octahedral honeycomb or alternated cubic honeycomb (Simple tetroctahedric check), ↔ (Also quasiregular polytope)
Gyrated alternated cubic honeycomb (Complex tetroctahedric check),
Semiregular E-honeycomb:
521 honeycomb (9-ic check) (8D Euclidean honeycomb),
additionally allowed Euclidean honeycombs as facets of higher-dimensional Euclidean honeycombs, giving the following additional figures:
Hypercubic honeycomb prism, named by Gosset as the (n – 1)-ic semi-check (analogous to a single rank or file of a chessboard)
Alternated hexagonal slab honeycomb (tetroctahedric semi-check),
Hyperbolic honeycombs
There are also hyperbolic uniform honeycombs composed of only regular cells , including:
Hyperbolic uniform honeycombs, 3D honeycombs:
Alternated order-5 cubic honeycomb, ↔ (Also quasiregular polytope)
Tetrahedral-octahedral honeycomb,
Tetrahedron-icosahedron honeycomb,
Paracompact uniform honeycombs, 3D honeycombs, which include uniform tilings as cells:
Rectified order-6 tetrahedral honeycomb,
Rectified square tiling honeycomb,
Rectified order-4 square tiling honeycomb, ↔
Alternated order-6 cubic honeycomb, ↔ (Also quasiregular)
Alternated hexagonal tiling honeycomb, ↔
Alternated o |
https://en.wikipedia.org/wiki/Nemanja%20Jovanovi%C4%87 | Nemanja Jovanović (; born on 3 March 1984) is a Serbian former football striker.
Career statistics
Honours
Vaslui
UEFA Intertoto Cup (1): 2008
References
External links
1984 births
Living people
Serbian men's footballers
Serbian expatriate men's footballers
Red Star Belgrade footballers
FK Železnik players
FC Universitatea Cluj players
CS Sporting Vaslui players
CSM Unirea Alba Iulia players
FC Unirea Urziceni players
FC Argeș Pitești players
SCM Râmnicu Vâlcea players
CS Pandurii Târgu Jiu players
Liga I players
Liga II players
Men's association football forwards
FC Kairat players
FC Yelimay players
Sandnes Ulf players
FC AGMK players
FC Taraz players
Kazakhstan Premier League players
Uzbekistan Super League players
Eliteserien players
Serbian expatriate sportspeople in Romania
Expatriate men's footballers in Romania
Expatriate men's footballers in Norway
Expatriate men's footballers in Kazakhstan
Expatriate men's footballers in Uzbekistan
People from Negotin
Sportspeople from Bor District |
https://en.wikipedia.org/wiki/D%27Agostino%27s%20K-squared%20test | In statistics, D'Agostino's K2 test, named for Ralph D'Agostino, is a goodness-of-fit measure of departure from normality, that is the test aims to gauge the compatibility of given data with the null hypothesis that the data is a realization of independent, identically distributed Gaussian random variables. The test is based on transformations of the sample kurtosis and skewness, and has power only against the alternatives that the distribution is skewed and/or kurtic.
Skewness and kurtosis
In the following, { xi } denotes a sample of n observations, g1 and g2 are the sample skewness and kurtosis, mj’s are the j-th sample central moments, and is the sample mean. Frequently in the literature related to normality testing, the skewness and kurtosis are denoted as and β2 respectively. Such notation can be inconvenient since, for example, can be a negative quantity.
The sample skewness and kurtosis are defined as
These quantities consistently estimate the theoretical skewness and kurtosis of the distribution, respectively. Moreover, if the sample indeed comes from a normal population, then the exact finite sample distributions of the skewness and kurtosis can themselves be analysed in terms of their means μ1, variances μ2, skewnesses γ1, and kurtosis γ2. This has been done by , who derived the following expressions:
and
For example, a sample with size drawn from a normally distributed population can be expected to have a skewness of and a kurtosis of , where SD indicates the standard deviation.
Transformed sample skewness and kurtosis
The sample skewness g1 and kurtosis g2 are both asymptotically normal. However, the rate of their convergence to the distribution limit is frustratingly slow, especially for g2. For example even with observations the sample kurtosis g2 has both the skewness and the kurtosis of approximately 0.3, which is not negligible. In order to remedy this situation, it has been suggested to transform the quantities g1 and g2 in a way that makes their distribution as close to standard normal as possible.
In particular, suggested the following transformation for sample skewness:
where constants α and δ are computed as
and where μ2 = μ2(g1) is the variance of g1, and γ2 = γ2(g1) is the kurtosis — the expressions given in the previous section.
Similarly, suggested a transformation for g2, which works reasonably well for sample sizes of 20 or greater:
where
and μ1 = μ1(g2), μ2 = μ2(g2), γ1 = γ1(g2) are the quantities computed by Pearson.
Omnibus K2 statistic
Statistics Z1 and Z2 can be combined to produce an omnibus test, able to detect deviations from normality due to either skewness or kurtosis :
If the null hypothesis of normality is true, then K2 is approximately χ2-distributed with 2 degrees of freedom.
Note that the statistics g1, g2 are not independent, only uncorrelated. Therefore, their transforms Z1, Z2 will be dependent also , rendering the validity of χ2 approximation questionable. Simulat |
https://en.wikipedia.org/wiki/Lee%20Jang-kwan | Lee Jang-Kwan (born July 4, 1974) is a South Korean football manager and retired player. He currently manages Jeonnam Dragons of K League 2.
Club career statistics
References
External links
1974 births
Living people
Men's association football midfielders
South Korean men's footballers
Busan IPark players
Incheon United FC players
K League 1 players
Footballers from Seoul |
https://en.wikipedia.org/wiki/Convex%20hull%20algorithms | Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science.
In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexities.
Computing the convex hull means that a non-ambiguous and efficient representation of the required convex shape is constructed. The complexity of the corresponding algorithms is usually estimated in terms of n, the number of input points, and sometimes also in terms of h, the number of points on the convex hull.
Planar case
Consider the general case when the input to the algorithm is a finite unordered set of points on a Cartesian plane. An important special case, in which the points are given in the order of traversal of a simple polygon's boundary, is described later in a separate subsection.
If not all points are on the same line, then their convex hull is a convex polygon whose vertices are some of the points in the input set. Its most common representation is the list of its vertices ordered along its boundary clockwise or counterclockwise. In some applications it is convenient to represent a convex polygon as an intersection of a set of half-planes.
Lower bound on computational complexity
For a finite set of points in the plane, the lower bound on the computational complexity of finding the convex hull represented as a convex polygon is easily shown to be the same as for sorting using the following reduction. For the set numbers to sort consider the set of points in the plane. Since they lie on a parabola, which is a convex curve, it is easy to see that the vertices of the convex hull, when traversed along the boundary, produce the sorted order of the numbers . Clearly, linear time is required for the described transformation of numbers into points and then extracting their sorted order. Therefore, in the general case the convex hull of n points cannot be computed more quickly than sorting.
The standard Ω(n log n) lower bound for sorting is proven in the decision tree model of computing, in which only numerical comparisons but not arithmetic operations can be performed; however, in this model, convex hulls cannot be computed at all. Sorting also requires Ω(n log n) time in the algebraic decision tree model of computation, a model that is more suitable for convex hulls, and in this model convex hulls also require Ω(n log n) time. However, in models of computer arithmetic that allow numbers to be sorted more quickly than O(n log n) time, for instance by using integer sorting algorithms, planar convex hulls can also be computed more quickly: the Graham scan algorithm for convex hulls consists of a single sorting step followed by a linear amount of additional work.
Optimal output-sensitive algorithms
As stated above, the complexity of finding a convex hull as a function of the input size n is lower bounded by Ω(n log n). However, the complexity of some |
https://en.wikipedia.org/wiki/Problems%20involving%20arithmetic%20progressions | Problems involving arithmetic progressions are of interest in number theory, combinatorics, and computer science, both from theoretical and applied points of view.
Largest progression-free subsets
Find the cardinality (denoted by Ak(m)) of the largest subset of {1, 2, ..., m} which contains no progression of k distinct terms. The elements of the forbidden progressions are not required to be consecutive. For example, A4(10) = 8, because {1, 2, 3, 5, 6, 8, 9, 10} has no arithmetic progressions of length 4, while all 9-element subsets of {1, 2, ..., 10} have one.
In 1936, Paul Erdős and Pál Turán posed a question related to this number and Erdős set a $1000 prize for an answer to it. The prize was collected by Endre Szemerédi for a solution published in 1975, what has become known as Szemerédi's theorem.
Arithmetic progressions from prime numbers
Szemerédi's theorem states that a set of natural numbers of non-zero upper asymptotic density contains finite arithmetic progressions, of any arbitrary length k.
Erdős made a more general conjecture from which it would follow that
The sequence of primes numbers contains arithmetic progressions of any length.
This result was proven by Ben Green and Terence Tao in 2004 and is now known as the Green–Tao theorem.
See also Dirichlet's theorem on arithmetic progressions.
, the longest known arithmetic progression of primes has length 27:
224584605939537911 + 81292139·23#·n, for n = 0 to 26. (23# = 223092870)
As of 2011, the longest known arithmetic progression of consecutive primes has length 10. It was found in 1998. The progression starts with a 93-digit number
100 99697 24697 14247 63778 66555 87969 84032 95093 24689
19004 18036 03417 75890 43417 03348 88215 90672 29719
and has the common difference 210.
Primes in arithmetic progressions
The prime number theorem for arithmetic progressions deals with the asymptotic distribution of prime numbers in an arithmetic progression.
Covering by and partitioning into arithmetic progressions
Find minimal ln such that any set of n residues modulo p can be covered by an arithmetic progression of the length ln.
For a given set S of integers find the minimal number of arithmetic progressions that cover S
For a given set S of integers find the minimal number of nonoverlapping arithmetic progressions that cover S
Find the number of ways to partition {1, ..., n} into arithmetic progressions.
Find the number of ways to partition {1, ..., n} into arithmetic progressions of length at least 2 with the same period.
See also Covering system
See also
Arithmetic combinatorics
PrimeGrid
Notes
Mathematical series
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Boris%20Rufimovich%20Vainberg | Boris Rufimovich Vainberg () is a professor of mathematics at the University of North Carolina at Charlotte. He was born in 1938. He received his Dr.S. from Moscow State University, under the supervision of Samarii Galpern. He taught mathematics at Moscow State University for nearly 30 years, then held a visiting professor position at the University of Delaware before taking his current position at UNCC.
His research concerns differential equations, scattering theory, and spectral theory. A survey of his research and publications is also presented in an article on his 80th birthday in the Russian Mathematical Surveys (Russian) and (English).
Publications
Books
Asymptotic Methods in Equations of Mathematical Physics, 1982 (in Russian).
Asymptotic Methods in Equations of Mathematical Physics (revised and expanded English version), Gordon and Breach Science Publishers, New York--London, 1989
Linear Water Waves: A Mathematical Approach, Cambridge University Press, 2002 (with N. Kuznetsov and V. Maz'ya)
Book chapters
Large Time Asymptotic Expansion of the Solutions of Exterior Boundary Value Problems for Hyperbolic Equations and Quasiclassical Approximations, Chapter in "Partial Differential Equations, V", 1999, Springer-Verlag, Berlin-Heidelberg-New York, Series: Encyclopaedia of Math. Sciences.
Papers
He has written over 170 published papers.
References
External links
Dr. Vainberg's home page at UNCC
Soviet mathematicians
Living people
1938 births |
https://en.wikipedia.org/wiki/Log-Laplace%20distribution | In probability theory and statistics, the log-Laplace distribution is the probability distribution of a random variable whose logarithm has a Laplace distribution. If X has a Laplace distribution with parameters μ and b, then Y = eX has a log-Laplace distribution. The distributional properties can be derived from the Laplace distribution.
Characterization
A random variable has a log-Laplace(μ, b) distribution if its probability density function is:
The cumulative distribution function for Y when y > 0, is
Generalization
Versions of the log-Laplace distribution based on an asymmetric Laplace distribution also exist. Depending on the parameters, including asymmetry, the log-Laplace may or may not have a finite mean and a finite variance.
References
External links
Continuous distributions
Probability distributions with non-finite variance
Non-Newtonian calculus |
https://en.wikipedia.org/wiki/Hilbert%27s%20theorem%20%28differential%20geometry%29 | In differential geometry, Hilbert's theorem (1901) states that there exists no complete regular surface of constant negative gaussian curvature immersed in . This theorem answers the question for the negative case of which surfaces in can be obtained by isometrically immersing complete manifolds with constant curvature.
History
Hilbert's theorem was first treated by David Hilbert in "Über Flächen von konstanter Krümmung" (Trans. Amer. Math. Soc. 2 (1901), 87–99).
A different proof was given shortly after by E. Holmgren in "Sur les surfaces à courbure constante négative" (1902).
A far-leading generalization was obtained by Nikolai Efimov in 1975.
Proof
The proof of Hilbert's theorem is elaborate and requires several lemmas. The idea is to show the nonexistence of an isometric immersion
of a plane to the real space . This proof is basically the same as in Hilbert's paper, although based in the books of Do Carmo and Spivak.
Observations: In order to have a more manageable treatment, but without loss of generality, the curvature may be considered equal to minus one, . There is no loss of generality, since it is being dealt with constant curvatures, and similarities of multiply by a constant. The exponential map is a local diffeomorphism (in fact a covering map, by Cartan-Hadamard theorem), therefore, it induces an inner product in the tangent space of at : . Furthermore, denotes the geometric surface with this inner product. If is an isometric immersion, the same holds for
.
The first lemma is independent from the other ones, and will be used at the end as the counter statement to reject the results from the other lemmas.
Lemma 1: The area of is infinite.
Proof's Sketch:
The idea of the proof is to create a global isometry between and . Then, since has an infinite area, will have it too.
The fact that the hyperbolic plane has an infinite area comes by computing the surface integral with the corresponding coefficients of the First fundamental form. To obtain these ones, the hyperbolic plane can be defined as the plane with the following inner product around a point with coordinates
Since the hyperbolic plane is unbounded, the limits of the integral are infinite, and the area can be calculated through
Next it is needed to create a map, which will show that the global information from the hyperbolic plane can be transfer to the surface , i.e. a global isometry. will be the map, whose domain is the hyperbolic plane and image the 2-dimensional manifold , which carries the inner product from the surface with negative curvature. will be defined via the exponential map, its inverse, and a linear isometry between their tangent spaces,
.
That is
,
where . That is to say, the starting point goes to the tangent plane from through the inverse of the exponential map. Then travels from one tangent plane to the other through the isometry , and then down to the surface with another exponential map.
The following step inv |
https://en.wikipedia.org/wiki/William%20Skiles | William Skiles may refer to:
William W. Skiles, U.S. Representative from Ohio
William West Skiles, American missionary
William Vernon Skiles, professor of mathematics
Bill Skiles, of American stand-up comedy act Skiles and Henderson |
https://en.wikipedia.org/wiki/Linear%20extension | In order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order.
Definitions
Linear extension of a partial order
A partial order is a reflexive, transitive and antisymmetric relation. Given any partial orders and on a set is a linear extension of exactly when
is a total order, and
For every if then
It is that second property that leads mathematicians to describe as extending
Alternatively, a linear extension may be viewed as an order-preserving bijection from a partially ordered set to a chain on the same ground set.
Linear extension of a preorder
A preorder is a reflexive and transitive relation. The difference between a preorder and a partial-order is that a preorder allows two different items to be considered "equivalent", that is, both and hold, while a partial-order allows this only when . A relation is called a linear extension of a preorder if:
is a total preorder, and
For every if then , and
For every if then . Here, means " and not ".
The difference between these definitions is only in condition 3. When the extension is a partial order, condition 3 need not be stated explicitly, since it follows from condition 2. Proof: suppose that and not . By condition 2, . By reflexivity, "not " implies that . Since is a partial order, and imply "not ". Therefore, .
However, for general preorders, condition 3 is needed to rule out trivial extensions. Without this condition, the preorder by which all elements are equivalent ( and hold for all pairs x,y) would be an extension of every preorder.
Order-extension principle
The statement that every partial order can be extended to a total order is known as the order-extension principle. A proof using the axiom of choice was first published by Edward Marczewski (Szpilrajin) in 1930. Marczewski writes that the theorem had previously been proven by Stefan Banach, Kazimierz Kuratowski, and Alfred Tarski, again using the axiom of choice, but that the proofs had not been published.
There is an analogous statement for preorders: every preorder can be extended to a total preorder. This statement was proved by Hansson.
In modern axiomatic set theory the order-extension principle is itself taken as an axiom, of comparable ontological status to the axiom of choice. The order-extension principle is implied by the Boolean prime ideal theorem or the equivalent compactness theorem, but the reverse implication doesn't hold.
Applying the order-extension principle to a partial order in which every two elements are incomparable shows that (under this principle) every set can be linearly ordered. This assertion that every set can be linearly ordered is known as the ordering principle, OP, and is a weakening of the well-ordering theorem. However, there are mode |
https://en.wikipedia.org/wiki/Spruce%20View | Spruce View is a hamlet in central Alberta, Canada within Red Deer County. It is located on Highway 54, approximately west of Innisfail. Spruce View is also recognized by Statistics Canada as a designated place.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Spruce View had a population of 138 living in 64 of its 73 total private dwellings, a change of from its 2016 population of 175. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Spruce View had a population of 175 living in 73 of its 84 total private dwellings, a change of from its 2011 population of 163. With a land area of , it had a population density of in 2016.
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Designated places in Alberta
Hamlets in Alberta
Red Deer County |
https://en.wikipedia.org/wiki/Petzval%20lens | The Petzval objective or Petzval lens is the first photographic portrait objective lens (with a 160 mm focal length) in the history of photography. It was developed by the Hungarian mathematics professor Joseph Petzval in 1840 in Vienna, with technical advice provided by . The Voigtländer company went on to build the first Petzval lens in 1840 on behalf of Petzval, whereupon it became known throughout Europe. Later, the optical instruments maker Carl Dietzler in Vienna also produced the Petzval lens.
History
The Voigtländer-Petzval objective lens was revolutionary and attracted the attention of the scientific world because it was the first mathematically calculated precision objective in the history of photography. Petzval's lens established two new features: firstly, it was faster compared to previous lenses, with a maximum aperture of 1:3.6. In comparison to Daguerre's daguerreotype camera lens of 1839, Petzval's design had 22 times the light-gathering capacity, which for the first time enabled portraits under favourable conditions with exposure times of less than a minute.
Additionally, Petzval calculated for the first time the composition of the lenses based on optical laws, whereas optics before had previously been ground and polished according to experience. For the calculations, 8 artillery gunners and 3 corporals were made available to Petzval by Archduke Louis of Austria (commander of the artillery), since the artillery was one of the few professions where mathematical calculations were made.
By 1845 Petzval's collaboration with Voigtländer, who held the license to produce the lenses, had become "mired in disputes'. Voigtländer moved production outside of Austria and therefore beyond Petzval's patent limitations. The Petzval objective was produced by Voigtländer and sold worldwide; by 1862 Voigtländer had produced 60,000 pieces.
One disadvantage of Petzval's design was a sharp drop in sharpness at the edges, which was corrected in the Aplanat lens developed by .
Optical design
The lens consisted of two doublet lenses with an aperture stop in between. The front lens is well corrected for spherical aberrations but introduces coma. The second doublet corrects for this and the position of the stop corrects most of the astigmatism. However, this results in additional field curvature and vignetting. The total field of view is therefore restricted to about 30 degrees. An f-number of 3.6 was achievable, which was considerably faster than other lenses of the time.
Lens revival
In 2013 Lomography successfully launched a crowdfunding campaign at kickstarter.com to produce a new Petzval lens in Russia for film and digital cameras.
Lensbaby offers Petzval lenses for modern cameras under the Burnside and Twist names.
William Optics produces a 51 mm F/4.9 (250 mm focal length) Petzval lens for astrophotography.
Image gallery
References
Photographic lenses
Hungarian inventions
Austrian inventions |
https://en.wikipedia.org/wiki/Cs%C3%A1sz%C3%A1r%20polyhedron | In geometry, the Császár polyhedron () is a nonconvex toroidal polyhedron with 14 triangular faces.
This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph onto a topological torus. Of the 35 possible triangles from vertices of the polyhedron, only 14 are faces.
Complete graph
The tetrahedron and the Császár polyhedron are the only two known polyhedra (having a manifold boundary) without any diagonals: every two vertices of the polygon are connected by an edge, so there is no line segment between two vertices that does not lie on the polyhedron boundary. That is, the vertices and edges of the Császár polyhedron form a complete graph.
The combinatorial description of this polyhedron has been described earlier by Möbius. Three additional different polyhedra of this type can be found in a paper by .
If the boundary of a polyhedron with v vertices forms a surface with h holes, in such a way that every pair of vertices is connected by an edge, it follows by some manipulation of the Euler characteristic that
This equation is satisfied for the tetrahedron with h = 0 and v = 4, and for the Császár polyhedron with h = 1 and v = 7. The next possible solution, h = 6 and v = 12, would correspond to a polyhedron with 44 faces and 66 edges, but it is not realizable as a polyhedron. It is not known whether such a polyhedron exists with a higher genus.
More generally, this equation can be satisfied only when v is congruent to 0, 3, 4, or 7 modulo 12.
History and related polyhedra
The Császár polyhedron is named after Hungarian topologist Ákos Császár, who discovered it in 1949. The dual to the Császár polyhedron, the Szilassi polyhedron, was discovered later, in 1977, by Lajos Szilassi; it has 14 vertices, 21 edges, and seven hexagonal faces, each sharing an edge with every other face. Like the Császár polyhedron, the Szilassi polyhedron has the topology of a torus.
There are other known polyhedra such as the Schönhardt polyhedron for which there are no interior diagonals (that is, all diagonals are outside the polyhedron) as well as non-manifold surfaces with no diagonals.
References
External links
Császár’s polyhedron in virtual reality in NeoTrie VR.
Nonconvex polyhedra
Toroidal polyhedra
Articles containing video clips |
https://en.wikipedia.org/wiki/Acevedo%20Municipality%2C%20Miranda | Acevedo is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 88,289. The town of Caucagua is the municipal seat of the Acevedo Municipality.
Demographics
The Acevedo Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 88,289 (up from 75,868 in 2000). This amounts to 3.1% of the state's population. The municipality's population density is .
Government
The mayor of the Acevedo Municipality is Juan José Aponte Mijares, elected on October 31, 2004, with 47% of the vote. He replaced Vicente Apicella shortly after the elections. The municipality is divided into eight parishes; Caucagua, Aragüita, Arévalo González, Capaya, El Café, Marizapa, Panaquire, and Ribas.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Bri%C3%B3n%20Municipality%2C%20Miranda | Brion is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 56,699. The town of Higuerote is the municipal seat of the Brion Municipality.
Demographics
The Brion Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 56,699 (up from 48,976 in 2000). This amounts to 2% of the state's population. The municipality's population density is 171.8 people per square mile (106.78/km2).
Government
The mayor of the municipality is Liliana Coromoto Gonzalez Guachi chosen on November 23, 2008, triumphing with 48% of the votes and his her opponent Raimundo Teran with 40% and volume posecion on December 1, 2008, defeating Raúl Ceballos.
Mayors of the municipality and political organizations which have governed (1989 onwards)
Manuel González: (1989-1992) COPEI (Committee of Political Electoral Independent Organization)
Domingo Palacios: (1992-2000) (Independent)
Ramón Ramos: (2000-2004) COPEI-Electors of Miranda
Raúl Ceballos: (2004-2008) PPT (Motherland For All)-PSUV (United Socialist Party of Venezuela)
Liliana González: (2008) PSUV (United Socialist party of Venezuela)
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Carrizal%20Municipality | Carrizal is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 52,224. The town of Carrizal is the municipal seat of the Carrizal Municipality.
Demographics
The Carrizal Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 52,224 (up from 44,431 in 2000). This amounts to 1.8% of the state's population. The municipality's population density is .
Government
Since 2017, the mayor of the city is Farith Fraija, after winning the municipal elections widely against José Luis Rodríguez, who ruled the municipality after replacing Orlando Urdaneta in 2002, shortly after a special election.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Buroz%20Municipality | Buroz is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 25,755. The town of Mamporal is the municipal seat of the Buroz Municipality.
Demographics
The Buroz Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 25,755 (up from 21,624 in 2000). This amounts to 0.9% of the state's population. The municipality's population density is .
Government
The mayor of the Buroz Municipality is Ramón Gómez Serrano, re-elected on October 31, 2004, with 68% of the vote. The municipality is divided into one parish (Mamporal).
References
External links
buroz-miranda.gob.ve]
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Andr%C3%A9s%20Bello%20Municipality%2C%20Miranda | Andrés Bello is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 25,208. The town of San José de Barlovento is the shire town of the Andrés Bello Municipality. The municipality is one of a number in Venezuela named "Andrés Bello Municipality", in honour of the writer Andrés Bello.
Demographics
The Andrés Bello Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 25,208 (up from 21,725 in 2000). This amounts to 0.9% of the state's population. The municipality's population density is .
Government
The mayor of the Andrés Bello Municipality is Albaro Ramón Hidalgo Rudas, elected on October 31, 2004, with 50% of the vote. He replaced Ramon Lobo shortly after the elections. The municipality is divided into two parishes; San José de Barlovento and Cumbo.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Crist%C3%B3bal%20Rojas%20Municipality | Cristóbal Rojas is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 96,369. The town of Charallave is the municipal seat of the Cristóbal Rojas Municipality. The municipality is named for Venezuelan painter Cristóbal Rojas.
Demographics
The Cristóbal Rojas Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 96,369 (up from 83,568 in 2000). This amounts to 3.4% of the state's population. The municipality's population density is .
Government
The mayor of the Cristóbal Rojas Municipality is Marisela Mendoza de Brito, re-elected on October 31, 2004, with 49% of the vote. The municipality is divided into two parishes; Charallave and Las Brisas.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Guaicaipuro%20Municipality | Guaicaipuro is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 280,687. The town of Los Teques is the municipal seat of the Guaicaipuro Municipality. The municipality is named for the sixteenth century cacique Guaicaipuro.
History
The city of Los Teques was founded in 1777 and was named after the Aractoeques Carabs, an indigenous tribe that once inhabited the area. On February 13, 1927, the capital of Miranda was moved to this city from Petare (before being in Petare, the capital of Miranda was in Ocumare del Tuy).
Geography
Temperature: Varies from 18 and 26 degrees Celsius.
Demographics
The Guaicaipuro Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 280,687 (up from 240,731 in 2000). This amounts to 9.8% of the state's population. The municipality's population density is .
Government
The mayor of the Guaicaipuro Municipality is Francisco Garcés, elected on December 8, 2013, with 52% of the vote. The municipality is divided into seven parishes; Los Teques, Altagracia de La Montaña, Cecilio Acosta, El Jarillo, Paracotos, San Pedro, and Tácata.
Transportation
On November 3, 2006, President Hugo Chávez inaugurated the Los Teques Metro. This metro system is connected to the Caracas Metro.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Independencia%20Municipality%2C%20Miranda | Independencia is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 160,899. The town of Santa Teresa del Tuy is the municipal seat of the Independencia Municipality.
Demographics
The Independencia Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 160,899 (up from 137,469 in 2000). This amounts to 5.6% of the state's population. The municipality's population density is .
Government
The mayor of the Independencia Municipality is Wilmer Salazar, re-elected on October 31, 2004, with 50% of the vote. The municipality is divided into two parishes; Santa Teresa del Tuy and El Cartanal.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/Lander%20Municipality | Lander is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 135,739. The town of Ocumare del Tuy is the municipal seat of the Lander Municipality.
Demographics
The Lander Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 135,739 (up from 117,819 in 2000). This amounts to 4.7% of the state's population. The municipality's population density is .
Government
The mayor of the Lander Municipality is José Gregorio Arvelo, elected on October 31, 2004, with 53% of the vote. He replaced Manuel Garcia shortly after the elections. The municipality is divided into three parishes; Ocumare del Tuy, La Democracia, and Santa Bárbara.
References
Municipalities of Miranda (state) |
https://en.wikipedia.org/wiki/P%C3%A1ez%20Municipality%2C%20Miranda | Páez is one of the 21 municipalities (municipios) that makes up the Venezuelan state of Miranda and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 39,097. The town of Río Chico is the municipal seat of the Páez Municipality.
Name
The municipality is one of several in Venezuela named "Páez Municipality" for independence hero José Antonio Páez.
Demographics
The Páez Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 39,097 (up from 33,259 in 2000). This amounts to 1.4% of the state's population. The municipality's population density is .
Government
The mayor of the Páez Municipality is Emilio Ruiz, re-elected on October 31, 2004, with 39% of the vote. The municipality is divided into five parishes; Río Chico, El Guapo, Tacarigua de La Laguna, Paparo, and San Fernando del Guapo.
References
External links
paez-miranda.gob.ve
Municipalities of Miranda (state) |
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