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https://en.wikipedia.org/wiki/Douglas%20Ravenel | Douglas Conner Ravenel (born February 17, 1947) is an American mathematician known for work in algebraic topology.
Life
Ravenel received his PhD from Brandeis University in 1972 under the direction of Edgar H. Brown, Jr. with a thesis on exotic characteristic classes of spherical fibrations. From 1971 to 1973 he was a C. L. E. Moore instructor at the Massachusetts Institute of Technology, and in 1974/75 he visited the Institute for Advanced Study. He became an assistant professor at Columbia University in 1973 and at the University of Washington in Seattle in 1976, where he was promoted to associate professor in 1978 and professor in 1981. From 1977 to 1979 he was a Sloan Fellow. Since 1988 he has been a professor at the University of Rochester. He was an invited speaker at the International Congress of Mathematicians in Helsinki, 1978, and is an editor of The New York Journal of Mathematics since 1994.
In 2012 he became a fellow of the American Mathematical Society. In 2022 he received the Oswald Veblen Prize in Geometry.
Work
Ravenel's main area of work is stable homotopy theory. Two of his most famous papers are Periodic phenomena in the Adams–Novikov spectral sequence, which he wrote together with Haynes R. Miller and W. Stephen Wilson (Annals of Mathematics 106 (1977), 469–516) and Localization with respect to certain periodic homology theories (American Journal of Mathematics 106 (1984), 351–414).
In the first of these two papers, the authors explore the stable homotopy groups of spheres by analyzing the -term of the Adams–Novikov spectral sequence. The authors set up the so-called chromatic spectral sequence relating this -term to the cohomology of the Morava stabilizer group, which exhibits certain periodic phenomena in the Adams–Novikov spectral sequence and can be seen as the beginning of chromatic homotopy theory. Applying this, the authors compute the second line of the Adams–Novikov spectral sequence and establish the non-triviality of a certain family in the stable homotopy groups of spheres. In all of this, the authors use work by Jack Morava and themselves on Brown–Peterson cohomology and Morava K-theory.
In the second paper, Ravenel expands these phenomena to a global picture of stable homotopy theory leading to the Ravenel conjectures. In this picture, complex cobordism and Morava K-theory control many qualitative phenomena, which were understood before only in special cases. Here Ravenel uses localization in the sense of Aldridge K. Bousfield in a crucial way. All but one of the Ravenel conjectures were proved by Ethan Devinatz, Michael J. Hopkins and Jeff Smith not long after the article got published. Frank Adams said on that occasion:
In June 2023, Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank announced a disproof of the last remaining conjecture.
In further work, Ravenel calculates the Morava K-theories of several spaces and proves important theorems in chromatic homotopy theory together with Hopkins. H |
https://en.wikipedia.org/wiki/Proprism | In geometry of 4 dimensions or higher, a proprism is a polytope resulting from the Cartesian product of two or more polytopes, each of two dimensions or higher. The term was coined by John Horton Conway for product prism. The dimension of the space of a proprism equals the sum of the dimensions of all its product elements. Proprisms are often seen as k-face elements of uniform polytopes.
Properties
The number of vertices in a proprism is equal to the product of the number of vertices in all the polytopes in the product.
The minimum symmetry order of a proprism is the product of the symmetry orders of all the polytopes. A higher symmetry order is possible if polytopes in the product are identical.
A proprism is convex if all its product polytopes are convex.
f-vectors
An f-vector is a number of k-face elements in a polytope from k=0 (points) to k=n-1 (facets). An extended f-vector can also include k=-1 (nullitope), or k=n (body). Prism products include the body element. (The dual to prism products includes the nullitope, while pyramid products include both.)
The f-vector of prism product, A×B, can be computed as (fA,1)*(fB,1), like polynomial multiplication polynomial coefficients.
For example for product of a triangle, f=(3,3), and dion, f=(2) makes a triangular prism with 6 vertices, 9 edges, and 5 faces:
fA(x) = (3,3,1) = 3 + 3x + x2 (triangle)
fB(x) = (2,1) = 2 + x (dion)
fA∨B(x) = fA(x) * fB(x)
= (3 + 3x + x2) * (2 + x)
= 6 + 9x + 5x2 + x3
= (6,9,5,1)
Hypercube f-vectors can be computed as Cartesian products of n dions, { }n. Each { } has f=(2), extended to f=(2,1).
For example, an 8-cube will have extended f-vector power product: f=(2,1)8 = (4,4,1)4 = (16,32,24,8,1)2 = (256,1024,1792,1792,1120,448,112,16,1). If equal lengths, this doubling represents { }8, a square tetra-prism {4}4, a tesseract duo-prism {4,3,3}2, and regular 8-cube {4,3,3,3,3,3,3}.
Double products or duoprisms
In geometry of 4 dimensions or higher, duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an a-polytope, a b-polytope is an (a+b)-polytope, where a and b are 2-polytopes (polygon) or higher.
Most commonly this refers to the product of two polygons in 4-dimensions. In the context of a product of polygons, Henry P. Manning's 1910 work explaining the fourth dimension called these double prisms.
The Cartesian product of two polygons is the set of points:
where P1 and P2 are the sets of the points contained in the respective polygons.
The smallest is a 3-3 duoprism, made as the product of 2 triangles. If the triangles are regular it can be written as a product of Schläfli symbols, {3} × {3}, and is composed of 9 vertices.
The tesseract, can be constructed as the duoprism {4} × {4}, the product of two equal-size orthogonal squares, composed of 16 vertices. The 5-cube can be constructed as a duoprism {4} × {4,3}, the product of a square and cube, while the 6-cube can be con |
https://en.wikipedia.org/wiki/David%20J.%20Thomson | David J. Thomson is a professor in the Department of Mathematics and Statistics at Queen's University in Ontario and a Canada Research Chair in statistics and signal processing, formerly a member of the technical staff at Bell Labs. He is a professional engineer in the province of Ontario, a fellow of the IEEE and a chartered statistician. He holds memberships of the Royal Statistical Society, the American Statistical Association, the Statistical Society of Canada and the American Geophysical Union and, in 2009, received a Killam Research Fellowship (administered through the Canada Council for the Arts). In 2010, he was made a fellow of the Royal Society of Canada. In 2013, he was awarded the Statistical Society of Canada impact award.
He is best known for creation of the multitaper method of spectral estimation, first published in complete form in 1982 in a special issue of Proceedings of the IEEE.
Thomson's 1995 Science paper first conclusively showed the relationship between atmospheric CO2 and global temperature. Thomson and Bell Labs colleagues Carol G. Maclennan and Louis J. Lanzerotti authored a 1995 Nature paper in which they showed evidence that the magnetic signatures of the Sun's normal modes permeate the interplanetary magnetic field as far as Jupiter. He has written over 100 other peer-reviewed journal articles in the fields of statistics, space physics, climatology and paleoclimatology, and seismology.
Career
Thomson joined the Technical Staff at Bell Labs in 1965, where he was assigned to work on the WT4 Millimeter Waveguide System and the Advanced Mobile Phone Service project. In 1983, he was reassigned to the Communications Analysis Research Department where he remained as a Distinguished Member until his retirement in 2001. During this time, he was
a Member of the Panel on Sensors and Electron Devices of the Army Research Laboratory Technical Assessment Board
chairman of Commission C of USNC-URSI
associate editor for Radio Science
associate editor for Communications Theory and for Detection and Estimation of the IEEE Transactions on Information Theory
adjunct professor in the Graduate Department of Scripps Institution of Oceanography
consulted at the Neurological Institute of Columbia University
visiting professor at Princeton University (statistical inference)
visiting professor at Stanford University (time series)
guest lecturer at Massachusetts Institute of Technology (the Houghton lectures)
participant at the Isaac Newton Institute at the University of Cambridge
On retirement from Bell Labs, Thomson took a Canada Research Chair at Queen's University at Kingston, where he has remained to this date.
References
External links
Personal academic web page
Spotlight
Canadian statisticians
Canadian mathematicians
Academic staff of Queen's University at Kingston
Living people
Canada Research Chairs
Polytechnic Institute of New York University alumni
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Tracking%20signal | In statistics and management science, a tracking signal monitors any forecasts that have been made in comparison with actuals, and warns when there are unexpected departures of the outcomes from the forecasts. Forecasts can relate to sales, inventory, or anything pertaining to an organization's future demand.
The tracking signal is a simple indicator that forecast bias is present in the forecast model. It is most often used when the validity of the forecasting model might be in doubt.
Definition
One form of tracking signal is the ratio of the cumulative sum of forecast errors (the deviations between the estimated forecasts and the actual values) to the mean absolute deviation. The formula for this tracking signal is:
where at is the actual value of the quantity being forecast, and ft is the forecast. MAD is the mean absolute deviation. The formula for the MAD is:
where n is the number of periods. Plugging this in, the entire formula for tracking signal is:
Another proposed tracking signal was developed by Trigg (1964). In this model, et is the observed error in period t and |et| is the absolute value of the observed error. The smoothed values of the error and the absolute error are given by:
Then the tracking signal is the ratio:
If no significant bias is present in the forecast, then the smoothed error Et should be small compared to the smoothed absolute error Mt. Therefore, a large tracking signal value indicates a bias in the forecast. For example, with a β of 0.1, a value of Tt greater than .51 indicates nonrandom errors. The tracking signal also can be used directly as a variable smoothing constant.
There have also been proposed methods for adjusting the smoothing constants used in forecasting methods based on some measure of prior performance of the forecasting model. One such approach is suggested by Trigg and Leach (1967), which requires the calculation of the tracking signal. The tracking signal is then used as the value of the smoothing constant for the next forecast. The idea is that when the tracking signal is large, it suggests that the time series has undergone a shift; a larger value of the smoothing constant should be more responsive to a sudden shift in the underlying signal.
See also
Calculating demand forecast accuracy
Demand forecasting
Notes
References
Alstrom, P., Madsen, P. (1996) "Tracking signals in inventory control systems: A simulation study", International Journal of Production Economics, 45 (1-3), 293–302,
Nahmias, Steven (2005) Production & Operations Analysis, Fifth Edition, McGraw-Hill.
Trigg, D.W. (1964) "Monitoring a forecasting system". Operational Research Quarterly, 15, 271–274.
Trigg, D.W. and Leach, A.G. (1967). "Exponential smoothing with an adaptive response rate". Operational Research Quarterly, 18 (1), 53–59
Mita Montero, J David (1973). "Análise de Sistemas de Previsão - Amortecimento Exponencial". Tese de Mestrado de Engenharia Industrial PUC-RJ,Brasil. Aplicação Industria |
https://en.wikipedia.org/wiki/Egon%20Schulte | Egon Schulte (born January 7, 1955 in Heggen (Kreis Olpe), Germany) is a mathematician and a professor of Mathematics at Northeastern University in Boston. He received his Ph.D. in 1980 from the Technical University of Dortmund; his doctoral dissertation was on Regular Incidence Complexes (abstract regular polytopes).
Selected publications
External links
Egon Schulte, Professor, Northeastern University, Department of Mathematics
Egon Schulte Chair and Professor Mathematics
Schulte Publications 1984 - 2010
Living people
1955 births
Northeastern University faculty
Combinatorialists
Technical University of Dortmund alumni
People from Olpe (district) |
https://en.wikipedia.org/wiki/Copa%20Am%C3%A9rica%20records%20and%20statistics | This is a list of records and statistics of the Copa América, including everything from when it was called the South American Football Championship (1916–1975).
Debut of national teams
Overall team records
In this ranking 3 points are awarded for a win, 1 for a draw and 0 for a loss. As per statistical convention in football, matches decided in extra time are counted as wins and losses, while matches decided by penalty shoot-outs are counted as draws. Teams are ranked by total points, then by goal difference, then by goals scored.
Medal table
No third place match was played in 1975, 1979 and 1983.
General statistics by tournament
Note: Carlos Valderrama (1987) was the first player to officially win the best player of the tournament award.
Hosts
Coaches with most games
Teams
Overall
Most Copa América appearances: 45,
For a detailed list, see Copa América participations
Most championships: 15, ,
Most appearances in a Copa América final: 29,
Most appearances in Copa América semi-finals: 36,
For a detailed list of top four appearances, see Copa América results
Most matches played: 206,
Fewest matches played: 3, , ,
Most wins: 124,
Most losses: 87,
Most draws: 41,
Team with the most goals scored in a single match: 12-0
Most goals scored: 465,
Most goals conceded: 323,
Fewest goals scored: 0,
Fewest goals conceded: 5, ,
Highest average of goals scored per match: 2.35,
Lowest average of goals conceded per match: 0.83,
In one tournament
Most wins: 7, (1949)
Most goals scored: 46, (1949)
Fewest goals conceded: 0, (2001)
Most goals conceded: 34
Most minutes without conceding a goal: 1,009
Streaks
Most consecutive championships: 3, 1945, 1946, 1947
Most consecutive final matches: 7, 1923–1937
Most consecutive runners-up:
2, (4 times)
2, (2 times)
2,
2,
Individual
Goals scored
Matches played
Titles by player
Individual records
Most goals scored in a single tournament: 9 – Jair (1949), Humberto Maschio (1957) and Javier Ambrois (1957)
Most goals scored in a single match by a player: 5 – Héctor Scarone (1926), Juan Marvezzi (1941), José Manuel Moreno (1942) and Evaristo (1957)
Most overall assists provided: 17 – Lionel Messi (2007–2021)
Most assists provided in a single tournament: 5 – Lionel Messi (2021)
Fastest goal scored: after 50 seconds – Darío Franco v. Brazil (1991)
Fastest hat-trick: after 10 minutes – José Manuel Moreno (1942)
Most overall matches played: 34 – Sergio Livingstone (1941–1949), Lionel Messi (2007–2021)
List of penalty shoot-outs
Most shoot-outs won: 5 – (1995, 2001, 2004, 2007, 2019)
Most shoot-outs lost: 6
(1993, 2001, 2004, 2007, 2019, 2021)
Most shoot-outs played: 10
(1993, 1995, 1999, 1999, 2001, 2004, 2007, 2011, 2019, 2021)
Championship year in bold
By chronological order
References and footnotes
References
Footnotes
Records |
https://en.wikipedia.org/wiki/Maximum-entropy%20Markov%20model | In statistics, a maximum-entropy Markov model (MEMM), or conditional Markov model (CMM), is a graphical model for sequence labeling that combines features of hidden Markov models (HMMs) and maximum entropy (MaxEnt) models. An MEMM is a discriminative model that extends a standard maximum entropy classifier by assuming that the unknown values to be learnt are connected in a Markov chain rather than being conditionally independent of each other. MEMMs find applications in natural language processing, specifically in part-of-speech tagging and information extraction.
Model
Suppose we have a sequence of observations that we seek to tag with the labels that maximize the conditional probability . In a MEMM, this probability is factored into Markov transition probabilities, where the probability of transitioning to a particular label depends only on the observation at that position and the previous position's label:
Each of these transition probabilities comes from the same general distribution . For each possible label value of the previous label , the probability of a certain label is modeled in the same way as a maximum entropy classifier:
Here, the are real-valued or categorical feature-functions, and is a normalization term ensuring that the distribution sums to one. This form for the distribution corresponds to the maximum entropy probability distribution satisfying the constraint that the empirical expectation for the feature is equal to the expectation given the model:
The parameters can be estimated using generalized iterative scaling. Furthermore, a variant of the Baum–Welch algorithm, which is used for training HMMs, can be used to estimate parameters when training data has incomplete or missing labels.
The optimal state sequence can be found using a very similar Viterbi algorithm to the one used for HMMs. The dynamic program uses the forward probability:
Strengths and weaknesses
An advantage of MEMMs rather than HMMs for sequence tagging is that they offer increased freedom in choosing features to represent observations. In sequence tagging situations, it is useful to use domain knowledge to design special-purpose features. In the original paper introducing MEMMs, the authors write that "when trying to extract previously unseen company names from a newswire article, the identity of a word alone is not very predictive; however, knowing that the word is capitalized, that is a noun, that it is used in an appositive, and that it appears near the top of the article would all be quite predictive (in conjunction with the context provided by the state-transition structure)." Useful sequence tagging features, such as these, are often non-independent. Maximum entropy models do not assume independence between features, but generative observation models used in HMMs do. Therefore, MEMMs allow the user to specify many correlated, but informative features.
Another advantage of MEMMs versus HMMs and conditional random fields (CRFs) is that |
https://en.wikipedia.org/wiki/Peter%20Koellner | Peter Koellner is Professor of Philosophy at Harvard University. He received his Ph.D from MIT in 2003. His main areas of research are mathematical logic, specifically set theory, and philosophy of mathematics, philosophy of physics, analytic philosophy, and philosophy of language.
In 2008 Koellner was awarded a Kurt Gödel Centenary Research Prize Fellowship. Currently, Koellner serves on the American Philosophical Association's Advisory Committee to the Eastern Division Program Committee in the area of Logic.
According to a review by Pierre Matet on Zentralblatt MATH, his joint paper with Hugh Woodin Incompatible Ω-Complete Theories contains an illuminating discussion of the issues involved, which makes it recommended reading for anyone interested in modern set theory.
Papers
On the Question of Absolute Undecidability, Philosophia Mathematica (III) 14 (2006)
On Reflection Principles, Annals of Pure and Applied Logic, Volume 157, Issues 2-3, February 2009, Pages 206-219, Kurt Gödel Centenary Research Prize Fellowships
Incompatible Ω-Complete Theories (with Hugh Woodin), Journal of Symbolic Logic, Volume 74, Issue 4 (2009), 1155-1170..
Large Cardinals from Determinacy (with Hugh Woodin), to appear in Handbook of Set Theory
Strong Logics of First and Second Order, to appear in Bulletin of Symbolic Logic
Truth in Mathematics: The Question of Pluralism (to appear in New Waves in Philosophy of Mathematics)
Notes
External links
website at Harvard
Philosophers of mathematics
Harvard University faculty
Harvard University Department of Philosophy faculty
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Zariski%20ring | In commutative algebra, a Zariski ring is a commutative Noetherian topological ring A whose topology is defined by an ideal contained in the Jacobson radical, the intersection of all maximal ideals. They were introduced by under the name "semi-local ring" which now means something different, and named "Zariski rings" by . Examples of Zariski rings are noetherian local rings with the topology induced by the maximal ideal, and -adic completions of Noetherian rings.
Let A be a Noetherian topological ring with the topology defined by an ideal . Then the following are equivalent.
A is a Zariski ring.
The completion is faithfully flat over A (in general, it is only flat over A).
Every maximal ideal is closed.
References
Commutative algebra |
https://en.wikipedia.org/wiki/Zariski%E2%80%93Riemann%20space | In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve.
Zariski–Riemann spaces were introduced by who (rather confusingly) called them Riemann manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by who used them to show that algebraic varieties can be embedded in complete ones.
Local uniformization (proved in characteristic 0 by Zariski) can be interpreted as saying that the Zariski–Riemann space of a variety is nonsingular in some sense, so is a sort of rather weak resolution of singularities. This does not solve the problem of resolution of singularities because in dimensions greater than 1 the Zariski–Riemann space is not locally affine and in particular is not a scheme.
Definition
The Zariski–Riemann space of a field K over a base field k is a locally ringed space whose points are the valuation rings containing k and contained in K. Sometimes the valuation ring K itself is excluded, and sometimes the points are restricted to the zero-dimensional valuation rings (those whose residue field has transcendence degree zero over k).
If S is the Zariski–Riemann space of a subring k of a field K, it has a topology defined by taking a basis of open sets to be the valuation rings containing a given finite subset of K. The space S is quasi-compact. It is made into a locally ringed space by assigning to any open subset the intersection of the valuation rings of the points of the subset. The local ring at any point is the corresponding valuation ring.
The Zariski–Riemann space of a function field can also be constructed as the inverse limit of all complete (or projective) models of the function field.
Examples
The Riemann–Zariski space of a curve
The Riemann–Zariski space of a curve over an algebraically closed field k with function field K is the same as the nonsingular projective model of it. It has one generic non-closed point corresponding to the trivial valuation with valuation ring K, and its other points are the rank 1 valuation rings in K containing k. Unlike the higher-dimensional cases, the Zariski–Riemann space of a curve is a scheme.
The Riemann–Zariski space of a surface
The valuation rings of a surface S over k with function field K can be classified by the dimension (the transcendence degree of the residue field) and the rank (the number of nonzero convex subgroups of the valuation group). gave the following classification:
Dimension 2. The only possibility is the trivial valuation with rank 0, valuation group 0 and valuation ring K.
Dimension 1, rank 1. These correspond to divisors on some blowup of S, or in other words to divisors and infinitely near points of S. They are all discrete. The center in S can be either a point or a curve. The valuation group is Z.
Dimension 0, rank 2. Th |
https://en.wikipedia.org/wiki/Riemann%20manifold | Riemann manifold may refer to:
Riemann surface in complex analysis
Riemannian manifold in Riemannian geometry
Zariski–Riemann space consisting of valuations |
https://en.wikipedia.org/wiki/Zariski%20space | In algebraic geometry, a Zariski space, named for Oscar Zariski, has several different meanings:
A topological space that is Noetherian (every open set is quasicompact)
A topological space that is Noetherian and also sober (every nonempty closed irreducible subset is the closure of a unique point). The spectrum of any commutative Noetherian ring is a Zariski space in this sense
A Zariski–Riemann space of valuations of a field
Algebraic geometry |
https://en.wikipedia.org/wiki/Modular%20lambda%20function | In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2). Over any point τ, its value can be described as a cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve , where the map is defined as the quotient by the [−1] involution.
The q-expansion, where is the nome, is given by:
.
By symmetrizing the lambda function under the canonical action of the symmetric group S3 on X(2), and then normalizing suitably, one obtains a function on the upper half-plane that is invariant under the full modular group , and it is in fact Klein's modular j-invariant.
Modular properties
The function is invariant under the group generated by
The generators of the modular group act by
Consequently, the action of the modular group on is that of the anharmonic group, giving the six values of the cross-ratio:
Relations to other functions
It is the square of the elliptic modulus, that is, . In terms of the Dedekind eta function and theta functions,
and,
where
In terms of the half-periods of Weierstrass's elliptic functions, let be a fundamental pair of periods with .
we have
Since the three half-period values are distinct, this shows that does not take the value 0 or 1.
The relation to the j-invariant is
which is the j-invariant of the elliptic curve of Legendre form
Given , let
where is the complete elliptic integral of the first kind with parameter .
Then
Modular equations
The modular equation of degree (where is a prime number) is an algebraic equation in and . If and , the modular equations of degrees are, respectively,
The quantity (and hence ) can be thought of as a holomorphic function on the upper half-plane :
Since , the modular equations can be used to give algebraic values of for any prime . The algebraic values of are also given by
where is the lemniscate sine and is the lemniscate constant.
Lambda-star
Definition and computation of lambda-star
The function (where ) gives the value of the elliptic modulus , for which the complete elliptic integral of the first kind and its complementary counterpart are related by following expression:
The values of can be computed as follows:
The functions and are related to each other in this way:
Properties of lambda-star
Every value of a positive rational number is a positive algebraic number:
and (the complete elliptic integral of the second kind) can be expressed in closed form in terms of the gamma function for any , as Selberg and Chowla proved in 1949.
The following expression is valid for all :
where is the Jacobi elliptic function delta amplitudinis with modulus .
By knowing one value, this formula can be used to compute related values:
where |
https://en.wikipedia.org/wiki/Luk%C3%A1%C5%A1%20Hejda | Lukáš Hejda (born 9 March 1990) is a Czech professional footballer who plays as a defender for FC Viktoria Plzeň.
Career statistics
As of 1 June 2015
References
External links
1990 births
Living people
Czech men's footballers
Czech Republic men's youth international footballers
Czech Republic men's under-21 international footballers
Czech First League players
AC Sparta Prague players
FK Jablonec players
FK Příbram players
FC Viktoria Plzeň players
People from Bílovec
Czech Republic men's international footballers
Men's association football defenders
Footballers from the Moravian-Silesian Region
FC Baník Ostrava players |
https://en.wikipedia.org/wiki/Equivalent%20latitude | In differential geometry, the equivalent latitude is a Lagrangian coordinate .
It is often used in atmospheric science,
particularly in the study of stratospheric dynamics.
Each isoline in a map of equivalent latitude follows the flow velocity and encloses the same area as the latitude line of equivalent value, hence "equivalent latitude."
Equivalent latitude is calculated from potential vorticity, from passive tracer simulations and from actual measurements of atmospheric tracers such as ozone.
Calculation of equivalent latitude
The calculation of equivalent latitude involves creating a monotonic mapping between the values of equivalent latitude and
the tracer it is based upon: higher values of the tracer map to higher
values of equivalent latitude.
A precise method is to assign a
representative area to each of the tracer measurements, filling the entire globe.
Thus, for a tracer field regularly gridded in longitude and latitude,
grid points closer to the pole will take up a smaller area,
in proportion to the cosine of the latitude.
Now, rank all the tracer values then form the cumulative sum.
The equivalent latitude from the area is given as:
where A is the area enclosed to the South (A = 0 corresponds to the equivalent South Pole) and R is the radius of the Earth.
This method generates a mapping that is as continuous as the data allows
as opposed to binning which produces a coarse-grained mapping.
References
Climatology
Differential geometry
Equivalence (mathematics) |
https://en.wikipedia.org/wiki/Bochner%20measurable%20function | In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,
where the functions each have a countable range and for which the pre-image is measurable for each element x. The concept is named after Salomon Bochner.
Bochner-measurable functions are sometimes called strongly measurable, -measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).
Properties
The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.
Function f is almost surely separably valued (or essentially separably valued) if there exists a subset N ⊆ X with μ(N) = 0 such that f(X \ N) ⊆ B is separable.
A function f : X → B defined on a measure space (X, Σ, μ) and taking values in a Banach space B is (strongly) measurable (with respect to Σ and the Borel algebra on B) if and only if it is both weakly measurable and almost surely separably valued.
In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.
See also
References
.
Functional analysis
Measure theory
Types of functions |
https://en.wikipedia.org/wiki/Motz%27s%20problem | In mathematics, Motz's problem is a problem which is widely employed as a benchmark for singularity problems to compare the effectiveness of numerical methods. The problem was first presented in 1947 by H. Motz in the paper "The treatment of singularities of partial differential equations by relaxation methods".
Partial differential equations
Finite element method |
https://en.wikipedia.org/wiki/SUPRE%20%28suicide%29 | SUPRE is a World Health Organization suicide prevention program. SUPRE gathers statistics and has launched a five-continent study of suicide. SUPRE also publishes advice for media organizations to follow in order to prevent copycat suicides.
References
Suicide prevention
World Health Organization |
https://en.wikipedia.org/wiki/Jovan%20Radivojevi%C4%87 | Jovan Radivojević (Serbian Cyrillic: Јован Радивојевић; born 29 October 1982) is a Serbian footballer who plays for Proleter Novi Sad in the Serbian First League.
Career statistics
External links
Official website
Utakmica profile
1982 births
Living people
Footballers from Novi Sad
Serbian men's footballers
Men's association football midfielders
RFK Novi Sad 1921 players
FK Hajduk Kula players
FK Rad players
OFK Beograd players
FK Borac Čačak players
FK Banat Zrenjanin players
FK Leotar players
NK Zvijezda Gradačac players
FK Proleter Novi Sad players
Serbian SuperLiga players
Serbian expatriate men's footballers
Expatriate men's footballers in Bosnia and Herzegovina |
https://en.wikipedia.org/wiki/Football%20at%20the%201934%20Far%20Eastern%20Championship%20Games | Football at the 1934 Far Eastern Games, held in Manila, Philippines was won by China while no other medals were awarded for second or third place.
Teams
Results
Winner
Statistics
Goalscorers
References
External links
Stokkermans, Karel. "Tenth Far Eastern Games 1934 (Manila)". RSSSF. Retrieved 2010-07-03.
Industrial and Commercial Daily Press, 1934-05-12, section 1 page 3 (total page no. 3)
1934 in Philippine sport
Football at the Far Eastern Championship Games
International association football competitions hosted by the Philippines
1934 in Asian football |
https://en.wikipedia.org/wiki/Landon%20T.%20Clay | Landon T. Clay (born Landon Thomas Clay, 1926 – July 29, 2017) was an American businessman and founder of the Clay Mathematics Institute. He died on July 29, 2017, at his home in Peterborough, New Hampshire.
Biography
Clay graduated from Harvard in 1950 with a B.A. in English. Clay was the chairman of Eaton Vance Corporation from 1971 to 1997 and had been a director of ADE Corporation since 1970 – a mutual fund management and distribution company. Clay served as chairman of the East Hill Management LLC, which is an investment advisory firm that he had founded in 1997. Clay had been a director for the Dakota Mining corporation since 1990 and was a former director for the Golden Queen Mining Company Ltd. from 2006 to 2009. In addition from 1971 to 1997, Clay had served on the board of Museum of Fine Arts. Clay had also served as a director for Plasso Technology Limited.
Clay was also a supporter of astronomy, with one of his gifts through the Harvard College Observatory leading the Magellan Telescopes consortium to name one of the two 6.5-m components of the Magellan Telescope to be named for him, as well as a provider of Clay Postdoctoral Fellowships at the Smithsonian Astrophysical Observatory, part of the Center for Astrophysics Harvard & Smithsonian. He also supported the Clay Center Observatory with its 24-inch telescope at the Dexter-Southfield School in Brookline, Massachusetts.
References
American businesspeople
Harvard College alumni
1926 births
2017 deaths
People from Peterborough, New Hampshire
Harvard College Observatory people |
https://en.wikipedia.org/wiki/Multiple-barrel%20firearm | A multiple-barrel firearm is any type of firearm with more than one gun barrel, usually to increase the rate of fire or hit probability and to reduce barrel erosion or overheating.
History
Volley gun
Multiple-barrel firearms date back to the 14th century, when the first primitive volley guns were developed. They are made with several single-shot barrels assembled together for firing a large number of shots, either simultaneously or in quick succession. These firearms were limited in firepower by the number of barrels bundled, and needed to be manually prepared, ignited, and reloaded after each firing.
In practice the large volley guns were not particularly more useful than a cannon firing canister shot or grapeshot. Since they were still mounted on a carriage, they could be as hard to aim and move around as a heavy cannon, and the many barrels took as long (if not longer) to reload. They also tended to be relatively expensive since they were structurally more complex than a cannon, due to all the barrels and ignition fuses, and each barrel had to be individually maintained and cleaned.
Pepperbox
A pepper-box gun or "pepperbox revolver" has three or more barrels revolving around a central axis, and gets the name from its resemblance to the household pepper shakers. It has existed in all ammunition systems: matchlock, wheellock, flintlock, snaplock, caplock, pinfire, rimfire, and centerfire. They were popular firearms in North America from the 1830s until the 1860s, during the American Civil War, but the concept was introduced much earlier. After each shot, the user manually rotates a next barrel into alignment with the hammer mechanism, and each barrel needs to be reloaded and maintained individually.
In the 15th century, there were design attempts to have several single-shot barrels attached to a stock, being fired individually by means of a match. Around 1790, pepperboxes were built on the basis of flintlock systems, notably by Nock in England and "Segallas" in Belgium. These weapons were built on the success of the earlier two-barrel turnover pistols, which were fitted with three to seven barrels. These early pepperboxes had to be manually rotated by hand.
The invention of the percussion cap building on the innovations of the Rev. Alexander Forsyth's patent of 1807 (which ran until 1821), and the Industrial Revolution allowed pepperbox revolvers to be mass-produced, making them more affordable than the early handmade guns previously only seen in possessions of the rich. Examples of these early weapons are the American three-barrel Manhattan pistol, the English Budding (probably the first English percussion pepperbox) and the Swedish Engholm. Most percussion pepperboxes have a circular flange around the rear of the cylinder to prevent the capped nipples being accidentally fired if the gun were to be knocked while in a pocket, or dropped and to protect the eyes from cap fragments.
Samuel Colt owned a revolving three-barrel matchlo |
https://en.wikipedia.org/wiki/Indian%20influence%20on%20Islamic%20science | The Golden Age of Islam, which saw a flourishing of science,
notably mathematics and astronomy, especially during the 9th and 10th centuries, had a notable Indian influence.
History
For the best part of a millennium, from the Seleucid era and through to the Sassanid period, there had been an exchange of scholarship between the Greek, Persian and Indian cultural spheres. The origin of the number zero and the place-value system notably falls into this period; its early use originates in Indian mathematics of the 5th century (Lokavibhaga), influencing Sassanid era Persian scholars during the 6th century.
The sudden Islamic conquest of Persia in the 640s drove a wedge between the Mediterranean and Indian traditions, but scholarly transfer soon resumed, with translations of both Greek and Sanskrit works into Arabic during the 8th century. This triggered the flourishing of Abbasid-era scholarship centered in Baghdad in the 9th century, and the eventual resumption of transmission to the west via Muslim Spain and Sicily by the 10th century.
There was continuing contact between Indian and Perso-Arabic scholarship during the 9th to 11th centuries while the Muslim conquest of India was temporarily halted.
Al Biruni in the early 11th century traveled widely in India and became an important source of knowledge about India in the Islamic world during that time.
With the establishment of the Delhi Sultanate in the 13th century, northern India fell under Perso-Arabic dominance and the native Sanskrit tradition fell into decline, while at about the same time the "Golden Age of Islam" of the Arab caliphates gave way to Turko-Mongol dominance, leading to the flourishing of a secondary "Golden Age" of Turko-Persian literary tradition during the 13th to 16th centuries, exemplified on either side of Timurid Persia by the Ottoman Empire in the west and the Mughal Empire in the east.
Astronomy
The mathematical astronomy text Brahmasiddhanta of Brahmagupta (598-668) was received in the court of Al-Mansur (753–774). It was translated by Alfazari into Arabic as Az-Zīj ‛alā Sinī al-‛Arab, popularly called Sindhind. This translation was the means by which the Hindu numerals were transmitted from the Sanskrit to the Arabic tradition. According to Al-Biruni,
Alberuni's translator and editor Edward Sachau wrote: "It is Brahmagupta who taught Arabs mathematics before they got acquainted with Greek science."
Al-Fazari also translated the Khandakhadyaka (Arakand) of Brahmagupta.
Through the resulting Arabic translations of Sindhind and Arakand, the use of Indian numerals became established in the Islamic world.
Mathematics
The etymology of the word "sine" comes from the Latin mistranslation of the word jiba, which is an Arabic transliteration of the Sanskrit word for half the chord, jya-ardha.
The sin and cos functions of trigonometry, were important mathematical concepts, imported from the Gupta period of Indian astronomy namely the jyā and koṭi-jyā functions via tr |
https://en.wikipedia.org/wiki/Theodor%20Reye | Karl Theodor Reye (born 20 June 1838 in Ritzebüttel, Germany and died 2 July 1919 in Würzburg, Germany) was a German mathematician. He contributed to geometry, particularly projective geometry and synthetic geometry. He is best known for his introduction of configurations in the second edition of his book, Geometrie der Lage (Geometry of Position, 1876). The Reye configuration of 12 points, 12 planes, and 16 lines is named after him.
Reye also developed a novel solution to the following three-dimensional extension of the problem of Apollonius: Construct all possible spheres that are simultaneously tangent to four given spheres.
Life
Reye obtained his Ph.D. from the University of Göttingen in 1861. His dissertation was entitled "Die mechanische Wärme-Theorie und das Spannungsgesetz der Gase" (The mechanical theory of heat and the potential law of gases).
Mathematical work
Reye worked on conic sections, quadrics and projective geometry.
Reye's work on linear manifolds of projective plane pencils and of bundles on spheres influenced later work by Corrado Segre on manifolds. He introduced Reye congruences, the earliest examples of Enriques surfaces.
References
Further reading
(NB. Theodor Reye was a polytechnician in Zürich in 1860, but later became a professor in Straßburg. This paper established and laid the foundation to what is known as Reye–Archard–Khrushchov wear law today.
External links
Theodor Reye (1892) Die Geometrie der Lage from archive.org.
1838 births
1919 deaths
Geometers
19th-century German mathematicians
University of Göttingen alumni
Academic staff of ETH Zurich
Academic staff of RWTH Aachen University
Academic staff of the University of Strasbourg
Heads of universities in Germany
20th-century German mathematicians |
https://en.wikipedia.org/wiki/Forte%20number | In musical set theory, a Forte number is the pair of numbers Allen Forte assigned to the prime form of each pitch class set of three or more members in The Structure of Atonal Music (1973, ). The first number indicates the number of pitch classes in the pitch class set and the second number indicates the set's sequence in Forte's ordering of all pitch class sets containing that number of pitches.
In the 12-TET tuning system (or in any other system of tuning that splits the octave into twelve semitones), each pitch class may be denoted by an integer in the range from 0 to 11 (inclusive), and a pitch class set may be denoted by a set of these integers.
The prime form of a pitch class set is the most compact (i.e., leftwards packed or smallest in lexicographic order) of either the normal form of a set or of its inversion. The normal form of a set is that which is transposed so as to be most compact. For example, a second inversion major chord contains the pitch classes 7, 0, and 4. The normal form would then be 0, 4 and 7. Its (transposed) inversion, which happens to be the minor chord, contains the pitch classes 0, 3, and 7; and is the prime form.
The major and minor chords are both given Forte number 3-11, indicating that it is the eleventh in Forte's ordering of pitch class sets with three pitches. In contrast, the Viennese trichord, with pitch classes 0, 1, and 6, is given Forte number 3-5, indicating that it is the fifth in Forte's ordering of pitch class sets with three pitches. The normal form of the diatonic scale, such as C major; 0, 2, 4, 5, 7, 9, and 11; is 11, 0, 2, 4, 5, 7, and 9; while its prime form is 0, 1, 3, 5, 6, 8, and 10; and its Forte number is 7-35, indicating that it is the thirty-fifth of the seven-member pitch class sets.
Sets of pitches which share the same Forte number have identical interval vectors. Those that have different Forte numbers have different interval vectors with the exception of z-related sets (for example 6-Z44 and 6-Z19).
Calculation
There are two prevailing methods of computing prime form. The first was described by Forte, and the second was introduced in John Rahn's Basic Atonal Theory and used in Joseph N. Straus's Introduction to Post-Tonal Theory. For example, the Forte prime for 6-31 is {0,1,3,5,8,9} whereas the Rahn algorithm chooses {0,1,4,5,7,9}.
In the language of combinatorics, the Forte numbers correspond to the binary bracelets of length 12: that is, equivalence classes of binary sequences of length 12 under the operations of cyclic permutation and reversal. In this correspondence, a one in a binary sequence corresponds to a pitch that is present in a pitch class set, and a zero in a binary sequence corresponds to a pitch that is absent. The rotation of binary sequences corresponds to transposition of chords, and the reversal of binary sequences corresponds to inversion of chords. The most compact form of a pitch class set is the lexicographically maximal sequence within the correspond |
https://en.wikipedia.org/wiki/Exact%20quantum%20polynomial%20time | In computational complexity theory, exact quantum polynomial time (EQP or sometimes QP) is the class of decision problems that can be solved by a quantum computer with zero error probability and in guaranteed worst-case polynomial time. It is the quantum analogue of the complexity class P. This is in contrast to bounded-error quantum computing, where quantum algorithms are expected to run in polynomial time, but may not always do so.
In the original definition of EQP, each language was computed by a single quantum Turing machine (QTM), using a finite gate set whose amplitudes could be computed in polynomial time. However, some results have required the use of an infinite gate set. The amplitudes in the gate set are typically algebraic numbers.
References
Quantum complexity theory |
https://en.wikipedia.org/wiki/Nikolai%20Andreev | Nikolai Andreev (born 5 February 1975 in Saratov, Russia) is a Russian mathematician and popularizer of mathematics. He was awarded with the Leelavati Award in 2022.
Biography
Nikolai is the Head of the Laboratory for Popularization and Promotion of Mathematics at the Steklov Mathematical Institute of the Russian Academy of Sciences (Moscow). He received a Ph.D. in mathematics from Moscow State University in 2000. Among his many highly valued projects by the Russian mathematical community is the creation of the online resource Mathematical Etudes.
Awards and honours
Prize of the President of the Russian Federation in the Area of Sciences and Innovations for Young Scientists (2010)
Gold Medal of the Russian Academy of Sciences (2017) for outstanding achievements in science popularization
The Leelavati Award in 2022 for his contribution to the art of mathematical animation and mathematical model building, in a style that inspires young and old alike, and that mathematicians around the world can adapt to its many uses, as well as for his tireless efforts to popularize genuine mathematics among the public through videos, lectures, and an award-winning book
References
Russian mathematicians
Living people
1975 births |
https://en.wikipedia.org/wiki/Olli%20Sipil%C3%A4inen | Olli Sipiläinen (born 24 July 1977) is a Finnish ice hockey player currently playing for Jukurit of the Finnish Liiga. Between 2002 and 2006 he played for Mikkelin Jukurit.
Career statistics
References
External links
1977 births
Living people
Finnish ice hockey left wingers
Imatran Ketterä players
JYP-Akatemia players
JYP Jyväskylä players
Mikkelin Jukurit players
Ice hockey people from Lappeenranta
SaiPa players
Lukko players |
https://en.wikipedia.org/wiki/AES%20Algiers | The National Higher School of Statistics and Applied Economics in Algiers (in French: École nationale supérieure de statistique et d'économie appliquée, ENSSEA d'Alger, in Arabic: المدرسة الوطنية العليا للاحصاء و الاقتصاد التطبيقي), called as well AES Algiers, is a public institution of higher education in Algeria. It was founded as αν institute of planning technics in 1970. The school is among the few institutions in the field of statistics and applied economics in Africa. It ensures the best programs of Statistics, Finance and Economy in Algeria.
History
In 1970, governmental decree N° 7-109 gave the green light to establish the Institute of Planning Technics (IPT) in order to form trained economists to the development of national plans for development in the field of economic analysis and technics. The institute was renamed to Institute of Planning Technics and Applied Economy (IPTAE) in 1972 and to National Institute of Planning and Statistics (NIPS) in 1983.
By 2008, the National Institute of Planning and Statistics (NIPS) became National Higher School of Statistics and Applied Economy.
Admissions
Students take a competitive entrance examination, known as Le Concours Nationale.
Programs
AES Algiers offers undergraduate and postgraduate programs.
Undergraduate
There are two cycles, first and second.
First cycle The students study for three years in order to make the best choice for the second cycle. In the first year, students study political economy, law, economic geography, mathematical analysis and descriptive statistics.
Students are given a grounding in microeconomic and macroeconomic theory during the second and third of study, when they study probabilities, accounting, international and national economy, operational research, mathematical statistics and computer science.
During the two first years, the students are giving courses in Business English and Business French.
Education in Algiers
Universities and colleges in Algeria
1970 establishments in Algeria |
https://en.wikipedia.org/wiki/Unrooted%20binary%20tree | In mathematics and computer science, an unrooted binary tree is an unrooted tree in which each vertex has either one or three neighbors.
Definitions
A free tree or unrooted tree is a connected undirected graph with no cycles. The vertices with one neighbor are the leaves of the tree, and the remaining vertices are the internal nodes of the tree. The degree of a vertex is its number of neighbors; in a tree with more than one node, the leaves are the vertices of degree one. An unrooted binary tree is a free tree in which all internal nodes have degree exactly three.
In some applications it may make sense to distinguish subtypes of unrooted binary trees: a planar embedding of the tree may be fixed by specifying a cyclic ordering for the edges at each vertex, making it into a plane tree. In computer science, binary trees are often rooted and ordered when they are used as data structures, but in the applications of unrooted binary trees in hierarchical clustering and evolutionary tree reconstruction, unordered trees are more common.
Additionally, one may distinguish between trees in which all vertices have distinct labels, trees in which the leaves only are labeled, and trees in which the nodes are not labeled. In an unrooted binary tree with n leaves, there will be n − 2 internal nodes, so the labels may be taken from the set of integers from 1 to 2n − 1 when all nodes are to be labeled, or from the set of integers from 1 to n when only the leaves are to be labeled.
Related structures
Rooted binary trees
An unrooted binary tree T may be transformed into a full rooted binary tree (that is, a rooted tree in which each non-leaf node has exactly two children) by choosing a root edge e of T, placing a new root node in the middle of e, and directing every edge of the resulting subdivided tree away from the root node. Conversely, any full rooted binary tree may be transformed into an unrooted binary tree by removing the root node, replacing the path between its two children by a single undirected edge, and suppressing the orientation of the remaining edges in the graph. For this reason, there are exactly 2n −3 times as many full rooted binary trees with n leaves as there are unrooted binary trees with n leaves.
Hierarchical clustering
A hierarchical clustering of a collection of objects may be formalized as a maximal family of sets of the objects in which no two sets cross. That is, for every two sets S and T in the family, either S and T are disjoint or one is a subset of the other, and no more sets can be added to the family while preserving this property. If T is an unrooted binary tree, it defines a hierarchical clustering of its leaves: for each edge (u,v) in T there is a cluster consisting of the leaves that are closer to u than to v, and these sets together with the empty set and the set of all leaves form a maximal non-crossing family. Conversely, from any maximal non-crossing family of sets over a set of n elements, one can form a unique unr |
https://en.wikipedia.org/wiki/Division%20of%20international%20labor%20comparisons | The International Labor Comparisons Program (ILC) of the U.S. Bureau of Labor Statistics (BLS) adjusts economic statistics (with an emphasis on labor statistics) to a common conceptual framework in order to make data comparable across countries. Its data can be used to evaluate the economic performance of one country relative to that of other countries and to assess international competitiveness.
Since 2014, the Bureau of Labor Statistics has discontinued this program, but The Conference Board continues to publish the majority of the data series.
History
Precursors of the International Labor Comparisons Program
The first commissioner of the Bureau of Labor Statistics, Carroll Wright, began the BLS tradition of international comparisons. He sent members of his staff to Europe to collect information on foreign labor force trends. In 1898, BLS published a report that compared wages in the United States to those in Europe and in 1902 it published a report that described labor conditions in Mexico.
In 1915, the first issue of the Monthly Labor Review, the Bureau’s research journal, contained articles on employment and various other economic indicators in foreign countries. In the late 1940s, BLS assisted in the implementation of the Marshall Plan by developing international comparisons of labor productivity and providing technical assistance to European governments for developing their own productivity statistics.
Founding of the current program
BLS formed the current international comparisons program in the 1960s as the importance of foreign trade and interest in international competition grew. The first study published by the program was an evaluation of the comparability of unemployment rates undertaken in response to a 1961 request by the Committee to Appraise Employment and Unemployment Statistics. In 1963, the program began to publish trends of labor productivity and unit labor costs for the manufacturing sector. In the mid-1970s, the program published level comparisons of Gross Domestic Product per Capita and by 1980 levels of hourly compensation (wages and benefits) in the manufacturing sector.
Recent developments
Over time, the program expanded its coverage of labor indicators and countries. In addition to the aforementioned labor indicators, the program began to publish a number of related indicators, such as average annual hours, exchange rates, and consumer price indexes. Further, the program originally covered only selected developed countries. As developing countries became more important to U.S. trade, the program expanded its coverage to include selected emerging economies in Asia, Eastern Europe, and Latin America.
In addition, the program produced a number of special international studies on topics, such as compensation and employment in China, youth labor markets, and family structures. The current program has also shown commitment to international cooperation. ILC aided the International Labour Organization (ILO |
https://en.wikipedia.org/wiki/Abd%20as-Salam%20al-Alami | Abd as-Salam ibn Mohammed ibn Ahmed al-Hasani al-Alami al-Fasi () (1834-1895) was a scientist from Fes. He was an expert in the field of astronomy, mathematics and medicine. Al-Alami was the author of several books in these fields and the designer of solar instruments.
References
External links
Clifford Edmund Bosworth, The Encyclopaedia of Islam: Supplement, Volume 12, p. 10 (Retrieved August 2, 2010)
Moroccan scientists
Moroccan writers
Moroccan astronomers
19th-century Moroccan physicians
1834 births
1895 deaths
19th-century Arab people |
https://en.wikipedia.org/wiki/Minkowski%20problem | In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface S whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly positive real function ƒ defined on a sphere, and the surface that is to be constructed should have Gaussian curvature ƒ(n(x)) at the point x, where n(x) denotes the normal to S at x. Eugenio Calabi stated: "From the geometric view point it [the Minkowski problem] is the Rosetta Stone, from which several related problems can be solved."
In full generality, the Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere Sn-1 to be the surface area measure of a convex body in . Here the surface area measure SK of a convex body K is the pushforward of the (n-1)-dimensional Hausdorff measure restricted to the boundary of K via the Gauss map. The Minkowski problem was solved by Hermann Minkowski, Aleksandr Danilovich Aleksandrov, Werner Fenchel and Børge Jessen: a Borel measure μ on the unit sphere is the surface area measure of a convex body if and only if μ has centroid at the origin and is not concentrated on a great subsphere. The convex body is then uniquely determined by μ up to translations.
The Minkowski problem, despite its clear geometric origin, is found to have its appearance in many places. The problem of radiolocation is easily reduced to the Minkowski problem in Euclidean 3-space: restoration of convex shape over the given Gauss surface curvature. The inverse problem of the short-wave diffraction is reduced to the Minkowski problem. The Minkowski problem is the basis of the mathematical theory of diffraction as well as for the physical theory of diffraction.
In 1953 Louis Nirenberg published the solutions of two long standing open problems, the Weyl problem and the Minkowski problem in Euclidean 3-space. L. Nirenberg's solution of the Minkowski problem was a milestone in global geometry. He has been selected to be the first recipient of the Chern Medal (in 2010) for his role in the formulation of the modern theory of non-linear elliptic partial differential equations, particularly for solving the Weyl problem and the Minkowski problems in Euclidean 3-space.
A. V. Pogorelov received Ukraine State Prize (1973) for resolving the multidimensional Minkowski problem in Euclidean spaces. Pogorelov resolved the Weyl problem in Riemannian space in 1969.
Shing-Tung Yau's joint work with Shiu-Yuen Cheng gives a complete proof of the higher-dimensional Minkowski problem in Euclidean spaces. Shing-Tung Yau received the Fields Medal at the International Congress of Mathematicians in Warsaw in 1982 for his work in global differential geometry and elliptic partial differential equations, particularly for solving such difficult problems as the Calabi conjecture of 1954, and a problem of Hermann Minkowski in Euclidean spaces concerning the Dirichlet problem fo |
https://en.wikipedia.org/wiki/Uniform%20matroid | In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry.
Definition
The uniform matroid is defined over a set of elements. A subset of the elements is independent if and only if it contains at most elements. A subset is a basis if it has exactly elements, and it is a circuit if it has exactly elements. The rank of a subset is and the rank of the matroid is .
A matroid of rank is uniform if and only if all of its circuits have exactly elements.
The matroid is called the -point line.
Duality and minors
The dual matroid of the uniform matroid is another uniform matroid . A uniform matroid is self-dual if and only if .
Every minor of a uniform matroid is uniform. Restricting a uniform matroid by one element (as long as ) produces the matroid
and contracting it by one element (as long as ) produces the matroid .
Realization
The uniform matroid may be represented as the matroid of affinely independent subsets of points in general position in -dimensional Euclidean space, or as the matroid of linearly independent subsets of vectors in general position in an -dimensional real vector space.
Every uniform matroid may also be realized in projective spaces and vector spaces over all sufficiently large finite fields. However, the field must be large enough to include enough independent vectors. For instance, the -point line can be realized only over finite fields of or more elements (because otherwise the projective line over that field would have fewer than points): is not a binary matroid, is not a ternary matroid, etc. For this reason, uniform matroids play an important role in Rota's conjecture concerning the forbidden minor characterization of the matroids that can be realized over finite fields.
Algorithms
The problem of finding the minimum-weight basis of a weighted uniform matroid is well-studied in computer science as the selection problem. It may be solved in linear time.
Any algorithm that tests whether a given matroid is uniform, given access to the matroid via an independence oracle, must perform an exponential number of oracle queries, and therefore cannot take polynomial time.
Related matroids
Unless , a uniform matroid is connected: it is not the direct sum of two smaller matroids.
The direct sum of a family of uniform matroids (not necessarily all with the same parameters) is called a partition matroid.
Every uniform matroid is a paving matroid, a transversal matroid and a strict gammoid.
Not every uniform matroid is graphic, and the uniform matroids provide the smallest example of a non-graphic matroid, . The uniform matroid is the graphic matroid of an -edge dipole graph, and the dual uniform matroid is the graphic matroid of its dual graph, the -edge cycle graph. is the graphic matroid of a graph with self-loops, and is the |
https://en.wikipedia.org/wiki/Pr%C3%BCfer%20theorems | In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.
Statement
Let A be an abelian group. If A is finitely generated then by the fundamental theorem of finitely generated abelian groups, A is decomposable into a direct sum of cyclic subgroups, which leads to the classification of finitely generated abelian groups up to isomorphism. The structure of general infinite abelian groups can be considerably more complicated and the conclusion needs not to hold, but Prüfer proved that it remains true for periodic groups in two special cases.
The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups. The second Prüfer theorem states that a countable abelian p-group whose non-trivial elements have finite p-height is isomorphic to a direct sum of cyclic groups. Examples show that the assumption that the group be countable cannot be removed.
The two Prüfer theorems follow from a general criterion of decomposability of an abelian group into a direct sum of cyclic subgroups due to L. Ya. Kulikov:
An abelian p-group A is isomorphic to a direct sum of cyclic groups if and only if it is a union of a sequence {Ai} of subgroups with the property that the heights of all elements of Ai are bounded by a constant (possibly depending on i).
References
László Fuchs (1970), Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press
Abelian group theory
Infinite group theory
Theorems in group theory |
https://en.wikipedia.org/wiki/Matching%20preclusion | In graph theory, a branch of mathematics, the matching preclusion number of a graph G (denoted mp(G)) is the minimum number of edges whose deletion results in the elimination of all perfect matchings or near-perfect matchings (matchings that cover all but one vertex in a graph with an odd number of vertices). Matching preclusion measures the robustness of a graph as a communications network topology for distributed algorithms that require each node of the distributed system to be matched with a neighboring partner node.
In many graphs, mp(G) is equal to the minimum degree of any vertex in the graph, because deleting all edges incident to a single vertex prevents that vertex from being matched. This set of edges is called a trivial matching preclusion set. A variant definition, the conditional matching preclusion number, asks for the minimum number of edges the deletion of which results in a graph that has neither a perfect or near-perfect matching nor any isolated vertices.
It is NP-complete to test whether the matching preclusion number of a given graph is below a given threshold.
The strong matching preclusion number (or simply, SMP number) is a generalization of the matching preclusion number; the SMP number of a graph G, smp(G) is the minimum number of vertices and/or edges whose deletion results in a graph that has neither perfect matchings nor almost-perfect matchings.
Other numbers defined in a similar way by edge deletion in an undirected graph include the edge connectivity, the minimum number of edges to delete in order to disconnect the graph, and the cyclomatic number, the minimum number of edges to delete in order to eliminate all cycles.
References
Graph invariants
Matching (graph theory) |
https://en.wikipedia.org/wiki/Height%20%28abelian%20group%29 | In mathematics, the height of an element g of an abelian group A is an invariant that captures its divisibility properties: it is the largest natural number N such that the equation Nx = g has a solution x ∈ A, or the symbol ∞ if there is no such N. The p-height considers only divisibility properties by the powers of a fixed prime number p. The notion of height admits a refinement so that the p-height becomes an ordinal number. Height plays an important role in Prüfer theorems and also in Ulm's theorem, which describes the classification of certain infinite abelian groups in terms of their Ulm factors or Ulm invariants.
Definition of height
Let A be an abelian group and g an element of A. The p-height of g in A, denoted hp(g), is the largest natural number n such that the equation pnx = g has a solution in x ∈ A, or the symbol ∞ if a solution exists for all n. Thus hp(g) = n if and only if g ∈ pnA and g ∉ pn+1A.
This allows one to refine the notion of height.
For any ordinal α, there is a subgroup pαA of A which is the image of the multiplication map by p iterated α times, defined using
transfinite induction:
p0A = A;
pα+1A = p(pαA);
pβA=∩α < β pαA if β is a limit ordinal.
The subgroups pαA form a decreasing filtration of the group A, and their intersection is the subgroup of the p-divisible elements of A, whose elements are assigned height ∞. The modified p-height hp∗(g) = α if g ∈ pαA, but g ∉ pα+1A. The construction of pαA is functorial in A; in particular, subquotients of the filtration are isomorphism invariants of A.
Ulm subgroups
Let p be a fixed prime number. The (first) Ulm subgroup of an abelian group A, denoted U(A) or A1, is pωA = ∩n pnA, where ω is the smallest infinite ordinal. It consists of all elements of A of infinite height. The family {Uσ(A)} of Ulm subgroups indexed by ordinals σ is defined by transfinite induction:
U0(A) = A;
Uσ+1(A) = U(Uσ(A));
Uτ(A) = ∩σ < τ Uσ(A) if τ is a limit ordinal.
Equivalently, Uσ(A) = pωσA, where ωσ is the product of ordinals ω and σ.
Ulm subgroups form a decreasing filtration of A whose quotients Uσ(A) = Uσ(A)/Uσ+1(A) are called the Ulm factors of A. This filtration stabilizes and the smallest ordinal τ such that Uτ(A) = Uτ+1(A) is the Ulm length of A. The smallest Ulm subgroup Uτ(A), also denoted U∞(A) and p∞A, is the largest p-divisible subgroup of A; if A is a p-group, then U∞(A) is divisible, and as such it is a direct summand of A.
For every Ulm factor Uσ(A) the p-heights of its elements are finite and they are unbounded for every Ulm factor except possibly the last one, namely Uτ−1(A) when the Ulm length τ is a successor ordinal.
Ulm's theorem
The second Prüfer theorem provides a straightforward extension of the fundamental theorem of finitely generated abelian groups to countable abelian p-groups without elements of infinite height: each such group is isomorphic to a direct sum of cyclic groups whose orders are powers of p. Moreover, the cardinality of the set o |
https://en.wikipedia.org/wiki/Paul%20Erd%C5%91s%20Prize | The Paul Erdős Prize (formerly Mathematical Prize) is given to Hungarian mathematicians not older than 40 by the Mathematics Department of the Hungarian Academy of Sciences. It was established and originally funded by Paul Erdős.
Awardees
See also
List of mathematics awards
Sources
The list on the homepage of the Hungarian academy
Paul Erdős
Mathematics awards
Hungarian awards
Awards established in 1973 |
https://en.wikipedia.org/wiki/Nadine%20Kraus | Nadine Kraus (born 14 February 1988) is a German footballer who plays as a midfielder for 1. FFC Recklinghausen. She studied at Friedrich Schiller University during her career.
Career
Statistics
References
External links
1988 births
Living people
German women's footballers
Women's association football midfielders
Frauen-Bundesliga players
2. Frauen-Bundesliga players
1. FC Saarbrücken (women) players
SGS Essen players
FF USV Jena players
VfL Bochum (women) players
People from Weiden in der Oberpfalz
Footballers from the Upper Palatinate |
https://en.wikipedia.org/wiki/Basic%20subgroup | In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulikov (for p-groups) and by László Fuchs (in general) in an attempt to formulate classification theory of infinite abelian groups that goes beyond the Prüfer theorems. It helps to reduce the classification problem to classification of possible extensions between two well understood classes of abelian groups: direct sums of cyclic groups and divisible groups.
Definition and properties
A subgroup, , of an abelian group, , is called p-basic, for a fixed prime number, , if the following conditions hold:
is a direct sum of cyclic groups of order and infinite cyclic groups;
is a p-pure subgroup of ;
The quotient group, , is a p-divisible group.
Conditions 1–3 imply that the subgroup, , is Hausdorff in the p-adic topology of , which moreover coincides with the topology induced from , and that is dense in . Picking a generator in each cyclic direct summand of creates a p-basis of B, which is analogous to a basis of a vector space or a free abelian group.
Every abelian group, , contains p-basic subgroups for each , and any 2 p-basic subgroups of are isomorphic. Abelian groups that contain a unique p-basic subgroup have been completely characterized. For the case of p-groups they are either divisible or bounded; i.e., have bounded exponent. In general, the isomorphism class of the quotient, by a basic subgroup, , may depend on .
Generalization to modules
The notion of a p-basic subgroup in an abelian p-group admits a direct generalization to modules over a principal ideal domain. The existence of such a basic submodule and uniqueness of its isomorphism type continue to hold.
References
László Fuchs (1970), Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press
L. Ya. Kulikov, On the theory of abelian groups of arbitrary cardinality (in Russian), Mat. Sb., 16 (1945), 129–162
Abelian group theory
Infinite group theory
Subgroup properties |
https://en.wikipedia.org/wiki/Response%20Prompting%20Procedures | Response Prompting Procedures are systematic strategies used to increase the probability of correct responding and opportunities for positive reinforcement for learners by providing and then systematically removing prompts. Response prompting is sometimes called errorless learning because teaching using these procedures usually results in few errors by the learner. The goal of response prompting is to transfer stimulus control from the prompt to the desired discriminative stimulus. Several response prompting procedures are commonly used in special education research: (a) system of least prompts, (b) most to least prompting, (c) progressive and constant time delay, and (d) simultaneous prompting.
System of least prompts
The SLP prompting procedure uses and removes prompts by moving through a hierarchy from less to more restrictive prompts. If the student emits the correct behavior at any point during this instructional trial (with or without prompts), reinforcement is provided. The system of least prompts gives the learner the opportunity to exhibit the correct response with the least restrictive level of prompting needed. Because teachers are required to use multiple types of prompts (e.g., verbal and physical prompts), the SLP prompting procedure may be complicated for use in typical settings, but may be similar to non-systematic teaching procedures typically used by teachers that involve giving learners an opportunity to exhibit a behavior independently before providing a prompt.
Example: SLP Trial for Teaching Sitting Behavior
SLP has been widely used for a variety of learners and skills. It has been most widely used in a 1:1 format (individual instruction) for chained skills. It has recently been used in conjunction with new technology, like portable DVD players and video iPods, to each self-help skills to young adults with intellectual disabilities and to improve transition skills for elementary school students with autism spectrum disorders.
Most to least prompting
The MTL prompting procedure (Cuvo, Leaf, & Borakove, 1978) removes prompts by moving through a hierarchy from most restrictive to less restrictive. The MTL prompting procedure begins with the most restrictive prompt, usually a physical prompt. After the learner has received reinforcement for completing the task with physical prompts, a less restrictive prompt is given (e.g., a partial physical prompt), and then an even less restrictive prompt (e.g., verbal prompt). Usually, a specific criterion is set for each prompt change (e.g., after three days of correct performance of the behavior with the use of a partial physical prompt, a verbal prompt will be used). If the individual fails to perform the behavior correctly with the less intrusive prompt, the instructor would return to a more intrusive prompt for a specified number of trials. Eventually, the discriminative stimulus for the behavior is the typically occurring stimulus (e.g., when lunch is finished, student independent |
https://en.wikipedia.org/wiki/Icosian | In mathematics, the icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell. The term can be used to refer to two related, but distinct, concepts:
The icosian group: a multiplicative group of 120 quaternions, positioned at the vertices of a 600-cell of unit radius. This group is isomorphic to the binary icosahedral group of order 120.
The icosian ring: all finite sums of the 120 unit icosians.
Unit icosians
The 120 unit icosians, which form the icosian group, are all even permutations of:
8 icosians of the form ½(±2, 0, 0, 0)
16 icosians of the form ½(±1, ±1, ±1, ±1)
96 icosians of the form ½(0, ±1, ±1/φ, ±φ)
In this case, the vector (a, b, c, d) refers to the quaternion a + bi + cj + dk, and φ represents the golden ratio ( + 1)/2. These 120 vectors form the H4 root system, with a Weyl group of order 14400. In addition to the 120 unit icosians forming the vertices of a 600-cell, the 600 icosians of norm 2 form the vertices of a 120-cell. Other subgroups of icosians correspond to the tesseract, 16-cell and 24-cell.
Icosian ring
The icosians lie in the golden field, (a + b) + (c + d)i + (e + f)j + (g + h)k, where the eight variables are rational numbers. This quaternion is only an icosian if the vector (a, b, c, d, e, f, g, h) is a point on a lattice L, which is isomorphic to an E8 lattice.
More precisely, the quaternion norm of the above element is (a + b)2 + (c + d)2 + (e + f)2 + (g + h)2. Its Euclidean norm is defined as u + v if the quaternion norm is u + v. This Euclidean norm defines a quadratic form on L, under which the lattice is isomorphic to the E8 lattice.
This construction shows that the Coxeter group embeds as a subgroup of . Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.
References
John H. Conway, Neil Sloane: Sphere Packings, Lattices and Groups (2nd edition)
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss: The Symmetries of Things (2008)
Frans Marcelis Icosians and ADE
Adam P. Goucher Good fibrations
Quaternions
John Horton Conway |
https://en.wikipedia.org/wiki/UEFA%20Euro%201996%20statistics | The following article outlines statistics for UEFA Euro 1996, which took place in England from 8 to 30 June 1996. Goals scored during penalty shoot-outs are not counted, and matches decided by a penalty shoot-out are counted as draws.
Goalscorers
Discipline
Sanctions against foul play at UEFA Euro 1996 are in the first instance the responsibility of the referee, but when he deems it necessary to give a caution, or dismiss a player, UEFA keeps a record and may enforce a suspension. Referee decisions are generally seen as final. However, UEFA's disciplinary committee may additionally penalise players for offences unpunished by the referee.
Overview
Red cards
A player receiving a red card is automatically suspended for the next match. A longer suspension is possible if the UEFA disciplinary committee judges the offence as warranting it. In keeping with the FIFA Disciplinary Code (FDC) and UEFA Disciplinary Regulations (UDR), UEFA does not allow for appeals of red cards except in the case of mistaken identity. The FDC further stipulates that if a player is sent off during his team's final Euro 1996 match, the suspension carries over to his team's next competitive international(s). For Euro 1996 these were the qualification matches for the 1998 FIFA World Cup.
Any player who was suspended due to a red card that was earned in Euro 1996 qualifying was required to serve the balance of any suspension unserved by the end of qualifying either in the Euro 1996 finals (for any player on a team that qualified, whether he had been selected to the final squad or not) or in World Cup qualifying (for players on teams that did not qualify).
Yellow cards
Any player receiving a single yellow card during two of the three group stage matches plus the quarter-final match was suspended for the next match. A single yellow card does not carry over to the semi-finals. This means that no player will be suspended for final unless he gets sent off in semi-final or he is serving a longer suspension for an earlier incident. Suspensions due to yellow cards will not carry over to the World Cup qualifiers. Yellow cards and any related suspensions earned in the Euro 1996 qualifiers are neither counted nor enforced in the final tournament.
In the event a player is sent off for two bookable offences, only the red card is counted for disciplinary purposes. However, in the event a player receives a direct red card after being booked in the same match, then both cards are counted. If the player was already facing a suspension for two tournament bookings when he was sent off, this would result in separate suspensions that would be served consecutively. The one match ban for the yellow cards would be served first unless the player's team is eliminated in the match in which he was sent off. If the player's team is eliminated in the match in which he was serving his ban for the yellow cards, then the ban for the sending off would be carried over to the World Cup qualifiers.
Add |
https://en.wikipedia.org/wiki/1999%20Djurg%C3%A5rdens%20IF%20season |
Player statistics
Appearances for competitive matches only
|}
Topscorers
Allsvenskan
Svenska Cupen
Friendlies
Competitions
Overall
Allsvenskan
League table
1999–00 Svenska Cupen
Friendlies
References
1999
Swedish football clubs 1999 season |
https://en.wikipedia.org/wiki/List%20of%20Coventry%20City%20F.C.%20records%20and%20statistics | This is a list of records and statistics for Coventry City F.C., an English professional association football club based in Coventry. The club was founded as Singers F.C. in 1883 and turned professional in 1893, before joining the Football League in 1920. In 1898 the club was renamed Coventry City. Coventry City currently play in the EFL Championship, the second tier of English football. They were relegated out of the top tier for the first time in 34 years in 2001.
This list encompasses the major honours won by Coventry City and records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Coventry City players on the international stage, and the highest transfer fees paid and received by the club. The club's attendance records, both at Ricoh Arena, their home since 2005, and Highfield Road, their home from 1899 to 2005, are also included in the list.
Team records
Record wins and defeats
League sequences
Wins, draws and defeats
* Season concluded with 10 games remaining due to the COVID-19 pandemic.
Points
Attendances
Club honours
Correct as of June 2020.
All-time FA Premier League table
Correct as of the end of the 2019–20 Premier League season.
Teams in bold are part of the 2020–21 Premier League. 47 teams have played at least one season in the Premier League, since it formed for the 1992–93 season. Coventry City were in this league from 1992 until their relegation in 2001.
Player records
Appearances
Most appearances (all competitions)
Correct as of match on 7 October 2023.
Goals
Most goals (all competitions)
Correct as of match on 7 October 2023.
Internationals
Transfers in
Correct as of 2 September 2023.
Highest transfer fees paid
The club's record signing came in August 2023, when they signed Haji Wright from Antalyaspor for £7.7 million, which surpassed the previous record in August 2000 for Craig Bellamy, who signed from Norwich City for £6.5 million. Previously, the record was £6 million for Robbie Keane in 1999 from Wolverhampton Wanderers, which made Keane the most expensive teenager in British football.
Progression of record fee paid
Transfers out
Correct as of 2 September 2023.
Highest transfer fees received
The club's record sale came in July 2023, when they sold Viktor Gyökeres to Sporting CP for £20.5 million, which eclipsed the previous record in August 2000 of Robbie Keane to Inter Milan for £13 million. The sale of Chris Kirkland to Liverpool for £6 million in 2001 set a British record transfer fee for a goalkeeper and the sale of Phil Babb also to Liverpool in 1994 set a British record transfer fee for a defender.
Progression of record fee received
Managerial records
References
External links
Coventry City on the box - links to various match videos
English football club statistics
Records and Statistics
Coventry-relate |
https://en.wikipedia.org/wiki/Rankin%E2%80%93Selberg%20method | In mathematics, the Rankin–Selberg method, introduced by and , also known as the theory of integral representations of L-functions, is a technique for directly constructing and analytically continuing several important examples of automorphic L-functions. Some authors reserve the term for a special type of integral representation, namely those that involve an Eisenstein series. It has been one of the most powerful techniques for studying the Langlands program.
History
The theory in some sense dates back to Bernhard Riemann, who constructed his zeta function as the Mellin transform of Jacobi's theta function. Riemann used asymptotics of the theta function to obtain the analytic continuation, and the automorphy of the theta function to prove the functional equation. Erich Hecke, and later Hans Maass, applied the same Mellin transform method to modular forms on the upper half-plane, after which Riemann's example can be seen as a special case.
Robert Alexander Rankin and Atle Selberg independently constructed their convolution L-functions, now thought of as the Langlands L-function associated to the tensor product of standard representation of GL(2) with itself. Like Riemann, they used an integral of modular forms, but one of a different type: they integrated the product of two weight k modular forms f, g with a real analytic Eisenstein series E(τ,s) over a fundamental domain D of the modular group SL2(Z) acting on the upper half plane
.
The integral converges absolutely if one of the two forms is cuspidal; otherwise the asymptotics must be used to get a meromorphic continuation like Riemann did. The analytic continuation and functional equation then boil down to those of the Eisenstein series. The integral was identified with the convolution L-function by a technique called "unfolding", in which the definition of the Eisenstein series and the range of integration are converted into a simpler expression that more readily exhibits the L-function as a Dirichlet series. The simultaneous combination of an unfolding together with global control over the analytic properties, is special and what makes the technique successful.
Modern adelic theory
Hervé Jacquet and Robert Langlands later gave adelic integral representations for the standard, and tensor product L-functions that had been earlier obtained by Riemann, Hecke, Maass, Rankin, and Selberg. They gave a very complete theory, in that they elucidated formulas for all local factors, stated the functional equation in a precise form, and gave sharp analytic continuations.
Generalizations and limitations
Nowadays one has integral representations for a large constellation of automorphic L-functions, however with two frustrating caveats. The first is that it is not at all clear which L-functions possibly have integral representations, or how they may be found; it is feared that the method is near exhaustion, though time and again new examples are found via clever arguments. The second is th |
https://en.wikipedia.org/wiki/Octagonal%20tiling | In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.
Uniform colorings
Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.
Regular maps
The regular map {8,3}2,0 can be seen as a 6-coloring of the {8,3} hyperbolic tiling. Within the regular map, octagons of the same color are considered the same face shown in multiple locations. The 2,0 subscripts show the same color will repeat by moving 2 steps in a straight direction following opposite edges. This regular map also has a representation as a double covering of a cube, represented by Schläfli symbol {8/2,3}, with 6 octagonal faces, double wrapped {8/2}, with 24 edges, and 16 vertices. It was described by Branko Grünbaum in his 2003 paper Are Your Polyhedra
the Same as My Polyhedra?
Related polyhedra and tilings
This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.
And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.
From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.
See also
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Regular tilings |
https://en.wikipedia.org/wiki/1998%20Djurg%C3%A5rdens%20IF%20season |
Player statistics
Appearances for competitive matches only
|}
Goals
Division 1 Norra
Svenska Cupen
Friendlies
Competitions
Overall
Division 1 Norra
League table
Svenska Cupen
1997–98
Source: https://www.rsssf.org/tablesz/zwed98.html
1998–99
Friendlies
References
1998
Swedish football clubs 1998 season
1998 in Swedish football |
https://en.wikipedia.org/wiki/Paul%20Erd%C5%91s%20Award | The Paul Erdős Award, named after Paul Erdős, is given by the
World Federation of National Mathematics Competitions for those who "have played a significant role in the development of mathematical challenges at the national or international level and which have been a stimulus for the enrichment of mathematics learning". The awards have been given in two-year periods since 1992.
Awardees
1992:
Luis Davidson, Cuba
Nikolay Konstantinov, Russia
John Webb, South Africa
1994:
Ronald Dunkley, Canada
Walter Mientka, USA
Urgengtserengiin Sanjmyatav, Mongolia
Jordan Tabov, Bulgaria
Peter Taylor, Australia
Qiu Zonghu, People's Republic of China
1996:
George Berzsenyi, USA
Tony Gardiner, United Kingdom
Derek Holton, New Zealand
1998:
Agnis Andzans, Latvia
, Germany
Mark Saul, USA
2000:
Francisco Bellot Rosado, Spain
, Hungary
, Hungary
2002:
Bogoljub Marinkovic, Yugoslavia
Harold Braun Reiter, United States of America
Wen-Hsien Sun, Taiwan
2004:
Warren Atkins, Australia
, France
Patricia Fauring, Argentina
2006:
Simon Chua, Philippines
Ali Rejali, Iran
Alexander Soifer, USA
2008:
, Germany
Bruce Henry, Australia
Leou Shian, Taiwan
2010:
Rafael Sanchez-Lamoneda, Venezuela
Yahya Tabesh, Iran
2012:
Cecil C. Rousseau, USA
Paul Vaderlind, Sweden
2014:
Petar Kenderov, Bulgaria
, Hungary
Richard Rusczyk, USA
2016:
Luis Caceres, Puerto Rico
David Christopher Hunt, Australia
Kar-Ping Shum, Hong Kong, China
2018:
Bin Xiong, China
David Monk, United Kingdom
Carlos Gustavo Tamm de Araujo Moreira, Brazil
See also
List of mathematics awards
Sources
Homepage of the award.
References
Paul Erdős
Mathematics awards |
https://en.wikipedia.org/wiki/Blee | Blee may refer to:
Chromatics
Color
Colorfulness
Hue
Complexion
Form
Visual Appearance
Shape
Configuration (geometry)
A surname
Francis J. Blee (a.k.a. Francis J. "Frank" Blee), an American Republican Party politician
Kathleen M. Blee, a professor of sociology at the University of Pittsburgh
Robert Blee, the mayor of Cleveland, Ohio from 1893 to 1894
Owen Blee, Dispensing Optician, Ireland
Place
Blee, a former village, now part of Monheim am Rhein
Color |
https://en.wikipedia.org/wiki/Harry%20Coonce | Harry Bernard Coonce (born 1939) is an American mathematician notable for being the originator of the now-popular Mathematics Genealogy Project, launched in 1996, a web-based catalog of mathematics doctoral advisors and students.
Coonce conceived of the idea while reading the unsigned thesis of his academic advisor Malcolm Robertson, in the Princeton University library, and wondering who his advisor's advisor was. The amount of time it took Coonce, without the existence of a central database of such information, to find out that Robertson's advisor was C. Einar Hille, gave him the idea for the project. In a 2000 interview, Coonce estimated that the project would top out at about 80,000 entries. In June 2016, the number of entries surpassed 200,000.
Education
Coonce completed his PhD in 1969 at the University of Delaware with a dissertation on A Variational Method for Functions of Bounded Boundary Rotation. Coonce presently is a retired mathematics professor of Minnesota State University, Mankato.
Private life
He married Susan Schilling, a computer scientist who died in 2016.
References
20th-century American mathematicians
21st-century American mathematicians
Living people
1939 births
University of Delaware alumni
Minnesota State University, Mankato alumni |
https://en.wikipedia.org/wiki/Horvitz%E2%80%93Thompson%20estimator | In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample. Inverse probability weighting is applied to account for different proportions of observations within strata in a target population. The Horvitz–Thompson estimator is frequently applied in survey analyses and can be used to account for missing data, as well as many sources of unequal selection probabilities.
The method
Formally, let be an independent sample from n of N ≥ n distinct strata with a common mean μ. Suppose further that is the inclusion probability that a randomly sampled individual in a superpopulation belongs to the ith stratum. The Horvitz–Thompson estimator of the total is given by:
and the Horvitz–Thompson estimate of the mean is given by:
In a Bayesian probabilistic framework is considered the proportion of individuals in a target population belonging to the ith stratum. Hence, could be thought of as an estimate of the complete sample of persons within the ith stratum. The Horvitz–Thompson estimator can also be expressed as the limit of a weighted bootstrap resampling estimate of the mean. It can also be viewed as a special case of multiple imputation approaches.
For post-stratified study designs, estimation of and are done in distinct steps. In such cases, computating the variance of is not straightforward. Resampling techniques such as the bootstrap or the jackknife can be applied to gain consistent estimates of the variance of the Horvitz–Thompson estimator. The "survey" package for R conducts analyses for post-stratified data using the Horvitz–Thompson estimator.
Proof of Horvitz-Thompson Unbiased Estimation of the Mean
The Horvitz–Thompson estimator can be shown to be unbiased when evaluating the expectation of the Horvitz–Thompson estimator, , as follows:
The Hansen-Hurwitz (1943) is known to be inferior to the Horvitz–Thompson (1952) strategy, associated with a number of Inclusion Probabilities Proportional to Size (IPPS) sampling procedures.
References
External links
Survey Package Website for R
Sampling (statistics)
Survey methodology
Missing data |
https://en.wikipedia.org/wiki/Gerhard%20Hochschild | Gerhard Paul Hochschild (April 29, 1915 in Berlin – July 8, 2010 in El Cerrito, California) was a German-born American mathematician who worked on Lie groups, algebraic groups, homological algebra and algebraic number theory.
Early life
On April 29, 1915, Hochschild was born to a middle-class Jewish family in Berlin, Germany, the son of Lilli and Heinrich Hochschild. Hochschild had an older brother. His father was a patent attorney who had an engineering degree. After the rise of the National Socialist German Workers' Party in 1933, his father sent him to South Africa where he was able to enroll in school with funding from the Hochschild Family Foundation established by Berthold Hochschild, a cousin of his grandfather.
Education
In 1936, Hochschild earned a BS degree in mathematics from University of Cape Town in Union of South Africa. In 1937, Hochschild earned a MS degree in mathematics from University of Cape Town.
In 1941, Hochschild earned his PhD in mathematics from Princeton University. Hochschild completed his thesis in 1941 at Princeton University with Claude Chevalley on Semisimple Algebras and Generalized Derivations.
Career
In 1956–7 Hochschild was at the Institute for Advanced Study. Hochschild was a professor at University of Illinois at Urbana-Champaign. In the late 1950s Hochschild was a professor at University of California, Berkeley.
introduced Hochschild cohomology, a cohomology theory for algebras, which classifies deformations of algebras. introduced cohomology into class field theory. Along with Bertram Kostant and Alex F. T. W. Rosenberg, the Hochschild–Kostant–Rosenberg theorem is named after him.
Among his students were Andrzej Białynicki-Birula and James Ax.
In 1955, Hochschild was a Guggenheim Fellow. In 1979 Hochschild was elected to the National Academy of Sciences, and in 1980 he was awarded the Leroy P. Steele Prize of the AMS.
In 1982, Hochschild retired but continued teaching part-time until 1985.
Personal life
In July 1950, Hochschild married Ruth Heinsheimer. Ruth was born in Germany and fled with her mother in 1939; the couple met at the University of Illinois where she was earning her M.S. in mathematics and Gerhard was working as an assistant professor. Hochschild's children are Ann Hochschild (b. 1955) and Peter Hochschild (b. 1957).
On July 8, 2010, Hochschild died at his home. Hochschild was 95.
See also
Hochschild–Mostow group
Publications
References
External links
Pictures of Gerhard Hochschild from Oberwolfach
Finding Aid to the Gerhard P. Hochschild papers, 1941-2004, The Bancroft Library
Hochschild Family Foundation at foundation center.org
20th-century American mathematicians
21st-century American mathematicians
Algebraists
1915 births
2010 deaths
Princeton University alumni
University of Cape Town alumni
University of Illinois Urbana-Champaign faculty
University of California, Berkeley College of Letters and Science faculty
Members of the United States National Academy o |
https://en.wikipedia.org/wiki/Abel%27s%20irreducibility%20theorem | In mathematics, Abel's irreducibility theorem, a field theory result described in 1829 by Niels Henrik Abel, asserts that if ƒ(x) is a polynomial over a field F that shares a root with a polynomial g(x) that is irreducible over F, then every root of g(x) is a root of ƒ(x). Equivalently, if ƒ(x) shares at least one root with g(x) then ƒ is divisible evenly by g(x), meaning that ƒ(x) can be factored as g(x)h(x) with h(x) also having coefficients in F.
Corollaries of the theorem include:
If ƒ(x) is irreducible, there is no lower-degree polynomial (other than the zero polynomial) that shares any root with it. For example, x2 − 2 is irreducible over the rational numbers and has as a root; hence there is no linear or constant polynomial over the rationals having as a root. Furthermore, there is no same-degree polynomial that shares any roots with ƒ(x), other than constant multiples of ƒ(x).
If ƒ(x) ≠ g(x) are two different irreducible monic polynomials, then they share no roots.
References
External links
Larry Freeman. Fermat's Last Theorem blog: Abel's Lemmas on Irreducibility. September 4, 2008.
Field (mathematics) |
https://en.wikipedia.org/wiki/Netherlands%20national%20football%20team%20records%20and%20statistics | This page details Netherlands men's national football team records; the most capped players, the players with the most goals, Netherlands' match record by opponent and decade.
Honours
Major tournaments
FIFA World Cup:
Runners-up (3): 1974, 1978, 2010
Third place (1): 2014
Fourth place (1): 1998
UEFA European Championship:
Winners (1): 1988
Third place (1): 1976
Semi-finals (3): 1992, 2000, 2004
UEFA Nations League:
Runners-up (1): 2019
Fourth place (1): 2023
Olympic football tournament:
Bronze medal/Third place (3): 1908, 1912, 1920
Fourth place (1): 1924
Friendly tournaments
Olympic Football Consolation Tournament
Winners: 1928
75th Anniversary FIFA Cup
Runners-up: 1979
World Champions' Gold Cup
Fourth Place: 1980
Copa Confraternidad
Runners-up: 2011
Individual records
Player records
Most-capped players
Last updated: 18 June 2023 Source: voetbalstats.nl
Top goalscorers
Last updated: 29 November 2022 Source: voetbalstats.nl
Hat-tricks
Robin van Persie scored four goals for the Dutch national side during their 11–0 win over minnows San Marino on 2 September 2011. He scored three goals in the 2014 FIFA World Cup qualifier against Hungary on 11 October 2013.
Last updated: 19 November 2021 Source: voetbalstats.nl
Age records
Oldest player to make debut: Sander Boschker, aged 39 years and 256 days vs , 1 June 2010
Youngest player to make debut: Jan van Breda Kolff, aged 17 years and 74 days vs , 2 April 1911 RSSSF
Oldest player to score: Abe Lenstra, aged 38 years and 143 days vs , 19 April 1959
Youngest player to score: Jan van Breda Kolff, aged 17 years and 74 days vs , 2 April 1911
Progression of the Netherlands association football goalscoring record
This is a progressive list of association footballers who have held or co-held the record for international goals for the Netherlands national football team.
For the early decades, records of players appearances and goals were often considered unreliable. RSSSF and IFFHS have spent much effort trying to produce definitive lists of full international matches, and corresponding data on players' international caps and goals. Using this data, the following records can be retrospectively produced. Note that, at the time, these records may not have been recognised.
Manager records
Team records
Tournament records
The Netherlands used all 23 players during the 2014 World Cup, making it the first team in World Cup history to ever use all of its squad players.
World Cup appearances: 11 (most recent in 2022)
Most successful World Cup appearance: Runners-up (1974, 1978, 2010)
European Championship appearances: 10 (most recent in 2020)
Most successful European Championship appearance: Winners (1988)
Summer Olympics appearances: 8 (most recent in 2008)
Most successful Olympics appearance: Third place (1908, 1912, 1920)
Match records
Firsts
First match: 1–4 (Antwerp, Belgium; 30 April 1905)
First World Cup finals match: 3–2 (Milan, Italy; 27 May |
https://en.wikipedia.org/wiki/Rolled%20throughput%20yield | Rolled throughput yield (RTY) in production economics is the probability that a process with more than one step will produce a defect free unit. It is the product of yields for each process step of the entire process.
For any process, it is ideal for that process to produce its product without defects and without rework. Rolled throughput yield quantifies the cumulative effects of inefficiencies found throughout the process. Rolled throughput yield and rolled throughput yield loss (RTYL) are often used in Six Sigma.
See also
Defects per million opportunities
Business process
Process capability
Total quality management
Total productive maintenance
References
Business terms
Production economics
Six Sigma |
https://en.wikipedia.org/wiki/University%20Gardens%20High%20School | University Gardens High School (Spanish: Escuela Superior University Gardens, generally abbreviated as UGHS), formally University Gardens Community School Specialized in Science and Mathematics, is a secondary magnet school located in Hato Rey Sur, San Juan, Puerto Rico. University Gardens is run by the Puerto Rico Department of Education and is overseen by its Specialized Schools Unit (UnEE, for its initials in Spanish).
University Gardens is known for its high standards, its "hard work" culture coming into the national culture. Recognized for featuring some of the highest standardized test scores in Puerto Rico, it has been criticized for having 27.8% of its student population come from living below the poverty line.
History
Since at least 1972 a high and middle school already existed in the gated eponymous community, located in the Río Piedras district. The University Gardens neighborhood is located right along PR-52, and close to the University of Puerto Rico, Río Piedras campus. It has historically been composed mostly of middle class and upper lower class families, and serves as "serves as a buffering zone between the rich and lower class areas of San Juan". Currently, there has been an influx of upper class individuals, who have transformed some residence into mansions. In 1976, it was converted into a magnet school.
Roxánna I. Duntley-Matos described the school's appearance as it was in 1981 as:
It used to offer grades 10th-12th, however this was changed by a Department of Education reorganization.
Hurricane María, earthquakes and COVID-19
After the 2019–20 Puerto Rico earthquakes UGHS was evaluated and, while the field inspector declared the school "suitable for immediate operation and occupancy," they did note that there were fissures on the walls, as well as the façade of a wall that was close to breakings off.
During February 2021, it was one of 172 public schools identified as "suitable to open" that formed part of an initial phase proposed by the Department to resume in-person teaching starting in March. Nevertheless, this plan was abandoned, and in-person classes did not occur until 13 May.
On the first day of class seniors came in a caravan on the first day of school, as part of their traditional celebration for the beginning of their final school year. Due to the COVID-19 pandemic, from August 2021 onwards, classes will start at 8:00 a.m. and end at 1:00 p.m., after students receive their take-out only lunch.
School community
Accreditation
Accreditation is obtained from the Puerto Rico Education Council for six years. However, in 2014, it was approved, but not accredited, by the Council.
University Gardens is part of the Puerto Rico Department of Education's Specialized Schools Unit (UnEE, for its initials in Spanish), which at times has been under the Assistant Secretary for Academic Services, Educational Transformation Projects, Curriculum and Pedagogical Innovation Division, and the Undersecretariat for Academic |
https://en.wikipedia.org/wiki/Fitting%20ideal | In commutative algebra, the Fitting ideals of a finitely generated module over a commutative ring describe the obstructions to generating the module by a given number of elements. They were introduced by .
Definition
If M is a finitely generated module over a commutative ring R generated by elements m1,...,mn
with relations
then the ith Fitting ideal of M is generated by the minors (determinants of submatrices) of order of the matrix .
The Fitting ideals do not depend on the choice of generators and relations of M.
Some authors defined the Fitting ideal to be the first nonzero Fitting ideal .
Properties
The Fitting ideals are increasing
If M can be generated by n elements then Fittn(M) = R, and if R is local the converse holds. We have Fitt0(M) ⊆ Ann(M) (the annihilator of M), and Ann(M)Fitti(M) ⊆ Fitti−1(M), so in particular if M can be generated by n elements then Ann(M)n ⊆ Fitt0(M).
Examples
If M is free of rank n then the Fitting ideals are zero for i<n and R for i ≥ n.
If M is a finite abelian group of order (considered as a module over the integers) then the Fitting ideal is the ideal .
The Alexander polynomial of a knot is a generator of the Fitting ideal of the first homology of the infinite abelian cover of the knot complement.
Fitting image
The zeroth Fitting ideal can be used also to give a variant of the notion of scheme-theoretic image of a morphism, a variant that behaves well in families. Specifically, given a finite morphism of noetherian schemes , the -module is coherent, so we may define as a coherent sheaf of -ideals; the corresponding closed subscheme of is called the Fitting image of f.
References
Commutative algebra |
https://en.wikipedia.org/wiki/Joseph%20S.%20B.%20Mitchell | Joseph S. B. Mitchell is an American computer scientist and mathematician. He is Distinguished Professor and Department Chair of Applied Mathematics and Statistics and Research Professor of Computer Science at Stony Brook University.
Biography
Mitchell received a BS (1981, Physics and Applied Mathematics), and an MS (1981, Mathematics) from Carnegie Mellon University, and Ph.D. (1986, Operations Research) from Stanford University (under advisership of Christos Papadimitriou). He was with Hughes Research Laboratories (1981–86) and then on the faculty of Cornell University (1986–1991). He now serves as Distinguished Professor of Applied Mathematics and Statistics and Research Professor of Computer Science at Stony Brook University. He serves as Chair of the Department of Applied Mathematics and Statistics (since 2014).
Mitchell has served for several years on the Computational Geometry Steering Committee, often as Chair. He is on the editorial board of the journals Discrete and Computational Geometry, Computational Geometry: Theory and Applications, Journal of Computational Geometry, and the Journal of Graph Algorithms and Applications, and is an editor-in-chief of the International Journal of Computational Geometry and Applications. He has served on numerous program committees and was co-chair of the PC for the 21st ACM Symposium on Computational Geometry (2005).
Research
Mitchell's primary research area is computational geometry, applied to problems in computer graphics, visualization, air traffic management, manufacturing, and geographic information systems.
Awards and honors
Mitchell has been an NSF Presidential Young Investigator, Fulbright Scholar, and a recipient of the President's Award for Excellence in Scholarship and Creative Activities. He shared the 2010 Gödel Prize with Sanjeev Arora for devising a polynomial-time approximation scheme for the Euclidean travelling salesman problem.
In 2011 the Association for Computing Machinery listed him as an ACM Fellow for his research in computational geometry and approximation algorithms.
He has also won numerous teaching awards.
References
External links
Joseph S. B. Mitchell's Homepage
Living people
Gödel Prize laureates
Stanford University alumni
Researchers in geometric algorithms
American computer scientists
Stony Brook University faculty
Fellows of the Association for Computing Machinery
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Suicide%20in%20Australia | According to the Australian Bureau of Statistics, the age standardised death rate for suicide in Australia, for the year 2019, was 13.1 deaths per 100,000 people; preliminary estimates for years 2020 and 2021 are respectively 12.1 and 12.0. In 2020, 3,139 deaths were due to suicide (2,384 males and 755 females); in 2021, 3,144 deaths were due to suicide (2,358 males and 786 females).
The World Health Organization reported the 2019 age standardised suicide rate in Australia at 11.3 per 100,000 people per year.
Deaths from suicide occur among males at a rate three times greater than that for females: in 2019, the standardised suicide rate for males was 20.1 deaths per 100,000 people, while for females it was 6.3 deaths per 100,000 people, according to the Australian Bureau of Statistics. The Australian Institute of Health and Welfare reports similar data.
Background information
In Australia, 48% of all suicides in 2000 were by 35 to 64-year-olds; an additional 13% were by 65 year olds and over. The suicide rates for children younger than 15 years is estimated to have increased by 92% between the 1960s to 1990s. Suicide rates are generally higher amongst males, rural and regional dwellers, Aboriginal and Torres Strait Islander people. Suicide prevention researcher, Gerry Georgatos has found that suicide rates among Aboriginal and Torres Strait Islander people, particularly in the Kimberley, Northern Territory and far north Queensland regions, are among the highest in the world. He describes the high rates as "a humanitarian crisis."
For a death to be considered a suicide and counted as such in Australian statistics, three criteria need to be met:
The death must be due to unnatural causes, such as injury, poisoning or suffocation rather than an illness
The actions which result in death must be self-inflicted
The person who injures himself or herself must have had the intention to die
(ResponseAbility, 2012)
A study suggest that 1-2% of the NSW population has contemplated suicide in 1992. This was estimated to be approximately 90 000 people out of a population of about 6 million.
Helplines
Beyond Blue
Lifeline Australia
Menslink
Suicide Prevention Australia
Risk factors
Gender
In every state and territory of Australia, suicide is much more common among males than females, with the ratio standing at 3:1 in 2012.
According to hospital data, females are more likely to deliberately injure themselves than males. In the 2008–2009 financial year, 62% of those who were hospitalised due to self-harm were female.
Researchers have attributed the difference between attempted and completed suicides among the sexes to males using more lethal means to end their lives.
Suicide rates for both males and females have generally decreased since the mid-90s with the overall suicide rate decreasing by 23% between 1999 and 2009. Suicide rates for males peaked in 1997 at 23.6 per 100,000 but have steadily decreased since then and stood at 14.9 per 100,00 |
https://en.wikipedia.org/wiki/Cannonball%20problem | In the mathematics of figurate numbers, the cannonball problem asks which numbers are both square and square pyramidal. The problem can be stated as: given a square arrangement of cannonballs, for what size squares can these cannonballs also be arranged into a square pyramid.
Equivalently, which squares can be represented as the sum of consecutive squares, starting from 1.
Formulation as a Diophantine equation
When cannonballs are stacked within a square frame, the number of balls is a square pyramidal number; Thomas Harriot gave a formula for this number around 1587, answering a question posed to him by Sir Walter Raleigh on their expedition to America.
Édouard Lucas formulated the cannonball problem as a Diophantine equation
or
Solution
Lucas conjectured that the only solutions are N = 1, M = 1, and N = 24, M = 70, using either 1 or 4900 cannon balls. It was not until 1918 that G. N. Watson found a proof for this fact, using elliptic functions. More recently, elementary proofs have been published.
Applications
The solution N = 24, M = 70 can be used for constructing the Leech lattice. The result has relevance to the bosonic string theory in 26 dimensions.
Although it is possible to tile a geometric square with unequal squares, it is not possible to do so with a solution to the cannonball problem. The squares with side lengths from 1 to 24 have areas equal to the square with side length 70, but they cannot be arranged to tile it.
Related problems
A triangular-pyramid version of the Cannon Ball Problem, which is to yield a perfect square from the Nth Tetrahedral number, would have N = 48. That means that the (24 × 2 = ) 48th tetrahedral number equals to (702 × 22 = 1402 = ) 19600. This is comparable with the 24th square pyramid having a total of 702 cannon balls.
Similarly, a pentagonal-pyramid version of the Cannon Ball problem to produce a perfect square, would have N = 8, yielding a total of (14 × 14 = ) 196 cannon balls.
The only numbers that are simultaneously triangular and square pyramidal, are 1, 55, 91, and 208335.
There are no numbers (other than the trivial solution 1) that are both tetrahedral and square pyramidal.
See also
Square triangular number, the numbers that are simultaneously square and triangular
Close-packing of equal spheres
References
External links
Diophantine equations
Figurate numbers |
https://en.wikipedia.org/wiki/G%C3%A1bor%20Hal%C3%A1sz | Gábor Halász (born 25 December 1941, in Budapest) is a Hungarian mathematician. He specialised in number theory and mathematical analysis, especially in analytic number theory. He is a member of the Hungarian Academy of Sciences. Since 1985, he is professor at the Faculty of Sciences of the Eötvös Loránd University, Budapest.
With Pál Turán, Halász proved zero density results on the roots of the Riemann zeta function. He co-invented the Halász–Montgomery inequality with Hugh Montgomery.
Awards, prizes
Alfréd Rényi Prize (1972)
Paul Erdős Prize (1976)
Tibor Szele Commemorative Medal of the János Bolyai Mathematical Society (1985)
References
Sources
Ki Kicsoda, 2006, MTI, Budapest.
Halász's homepage at the Hungarian Academy of Sciences.
A Magyar Tudományos Akadémai Almanachja, Budapest, 2006.
1941 births
Living people
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Number theorists
Members of the Hungarian Academy of Sciences
Mathematicians from Budapest
Academic staff of Eötvös Loránd University |
https://en.wikipedia.org/wiki/Journal%20of%20Statistics%20Education | The Journal of Statistics and Data Science Education is a triannual open access peer-reviewed< academic journal. It was established in 1992 at North Carolina State University by E. Jacquelin Dietz as the Journal of Statistics Education, obtaining its current title in 2020. It is published by Taylor & Francis on behalf of the American Statistical Association of which it became an official publication in 1999. The journal covers subjects related to statistical literacy and statistics education at all levels of education.
See also
Comparison of statistics journals.
References
External links
English-language journals
Academic journals established in 1993
Statistics journals
Education journals
Triannual journals
Open access journals
Taylor & Francis academic journals |
https://en.wikipedia.org/wiki/Cantor%27s%20intersection%20theorem | Cantor's intersection theorem refers to two closely related theorems in general topology and real analysis, named after Georg Cantor, about intersections of decreasing nested sequences of non-empty compact sets.
Topological statement
Theorem. Let be a topological space. A decreasing nested sequence of non-empty compact, closed subsets of has a non-empty intersection. In other words, supposing is a sequence of non-empty compact, closed subsets of S satisfying
it follows that
The closedness condition may be omitted in situations where every compact subset of is closed, for example when is Hausdorff.
Proof. Assume, by way of contradiction, that . For each , let . Since and , we have . Since the are closed relative to and therefore, also closed relative to , the , their set complements in , are open relative to .
Since is compact and is an open cover (on ) of , a finite cover can be extracted. Let . Then because , by the nesting hypothesis for the collection . Consequently, . But then , a contradiction. ∎
Statement for real numbers
The theorem in real analysis draws the same conclusion for closed and bounded subsets of the set of real numbers . It states that a decreasing nested sequence of non-empty, closed and bounded subsets of has a non-empty intersection.
This version follows from the general topological statement in light of the Heine–Borel theorem, which states that sets of real numbers are compact if and only if they are closed and bounded. However, it is typically used as a lemma in proving said theorem, and therefore warrants a separate proof.
As an example, if , the intersection over is . On the other hand, both the sequence of open bounded sets and the sequence of unbounded closed sets have empty intersection. All these sequences are properly nested.
This version of the theorem generalizes to , the set of -element vectors of real numbers, but does not generalize to arbitrary metric spaces. For example, in the space of rational numbers, the sets
are closed and bounded, but their intersection is empty.
Note that this contradicts neither the topological statement, as the sets are not compact, nor the variant below, as the rational numbers are not complete with respect to the usual metric.
A simple corollary of the theorem is that the Cantor set is nonempty, since it is defined as the intersection of a decreasing nested sequence of sets, each of which is defined as the union of a finite number of closed intervals; hence each of these sets is non-empty, closed, and bounded. In fact, the Cantor set contains uncountably many points.
Theorem. Let be a sequence of non-empty, closed, and bounded subsets of satisfying
Then,
Proof. Each nonempty, closed, and bounded subset admits a minimal element . Since for each , we have
,
it follows that
,
so is an increasing sequence contained in the bounded set . The monotone convergence theorem for bounded sequences of real numbers now guarantees the existence of a l |
https://en.wikipedia.org/wiki/John%20C.%20Gittins | John Charles Gittins (born 1938) is a researcher in applied probability and operations research, who is a professor and Emeritus Fellow at Keble College, Oxford University.
He is renowned as the developer of the "Gittins index", which is used for sequential decision-making, especially in research and development in the pharmaceutical industry. He has research interests in applied probability, decision analysis and optimal decisions, including optimal stopping and stochastic optimization.
Gittins was an Assistant Director of Research at the Department of Engineering, Cambridge University from 1967 to 1974. Then he was a lecturer at Oxford University from 1975 to 2005 and head of the Department of Statistics there for 6 years. In 1992, Oxford University awarded him the degree Doctor of Science (D. Sci.). In 1996 he became a Professor of Statistics at Oxford University.
He has been awarded the Rollo Davidson Prize (1982) for early-career probabilists, and the Guy Medal in Silver (1984).
Selected publications
(1989) Multi-Armed Bandit Allocation Indices, Wiley.
(1985) (with Bergman, S.W.) Statistical Methods for Pharmaceutical Research Planning, CRC Press.
(2000) (with H. Pezeshk) "How Large Should a Clinical Trial Be?", The Statistician, 49 (2), 177–187 )
(2001) (with G. Harper) "Bounds on the Performance of a Greedy Algorithm for Probabilities". Mathematics of Operations Research, 26, 313–323
(2003) "Stochastic Models for the Planning of Pharmaceutical Research", Journal of Statistical Theory and Applications, 2 (2), 198–214.
(2011) (with K. D. Glazebrook and R. R. Weber) Multi-Armed Bandit Allocation Indices, second edition, Wiley,
References
External links
Home page for John Gittins
1938 births
British statisticians
Probability theorists
British operations researchers
Fellows of Keble College, Oxford
Living people
Alumni of the University of Wales
Academics of the University of Cambridge
Academics of the University of Oxford
Alumni of the University of Oxford |
https://en.wikipedia.org/wiki/Republic%20of%20Korea%20Armed%20Forces%20statistics%20in%20the%20Vietnam%20War |
See also
Vietnam War
External links
Republic of Korea Ministry of National Defense Institute for Military History Compilation (Korean)
Military history of South Korea during the Vietnam War |
https://en.wikipedia.org/wiki/Dwork%20family | In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer n, studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-functions, such families have been shown to have relationships with mirror symmetry and extensions of the modularity theorem.
Definition
The Dwork family is given by the equations
for all .
References
Algebraic geometry |
https://en.wikipedia.org/wiki/Mikkel%20Cramer | Mikkel Engvang Cramer (born 25 January 1992) is a retired Danish footballer who played as a left back.
References
External links
Mikkel Cramer official Danish Superliga statistics at danskfodbold.com
1992 births
Living people
Danish men's footballers
Randers FC players
Akademisk Boldklub players
AC Horsens players
Danish Superliga players
Men's association football defenders |
https://en.wikipedia.org/wiki/Complete%20sequence | In mathematics, a sequence of natural numbers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
For example, the sequence of powers of two (1, 2, 4, 8, ...), the basis of the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 1001012 = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include the even numbers, since adding even numbers produces only even numbers—no odd number can be formed.
Conditions for completeness
Without loss of generality, assume the sequence an is in non-decreasing order, and define the partial sums of an as:
.
Then the conditions
are both necessary and sufficient for an to be a complete sequence.
A corollary to the above states that
are sufficient for an to be a complete sequence.
However there are complete sequences that do not satisfy this corollary, for example , consisting of the number 1 and the first prime after each power of 2.
Other complete sequences
The complete sequences include:
The sequence of the number 1 followed by the prime numbers (studied by S. S. Pillai and others); this follows from Bertrand's postulate.
The sequence of practical numbers which has 1 as the first term and contains all other powers of 2 as a subset.
The Fibonacci numbers, as well as the Fibonacci numbers with any one number removed. This follows from the identity that the sum of the first n Fibonacci numbers is the (n + 2)nd Fibonacci number minus 1.
Applications
Just as the powers of two form a complete sequence due to the binary numeral system, in fact any complete sequence can be used to encode integers as bit strings. The rightmost bit position is assigned to the first, smallest member of the sequence; the next rightmost to the next member; and so on. Bits set to 1 are included in the sum. These representations may not be unique.
Fibonacci coding
For example, in the Fibonacci arithmetic system, based on the Fibonacci sequence, the number 17 can be encoded in six different ways:
110111 (F6 + F5 + F3 + F2 + F1 = 8 + 5 + 2 + 1 + 1 = 17, maximal form)
111001 (F6 + F5 + F4 + F1 = 8 + 5 + 3 + 1 = 17)
111010 (F6 + F5 + F4 + F2 = 8 + 5 + 3 + 1 = 17)
1000111 (F7 + F3 + F2 + F1 = 13 + 2 + 1 + 1 = 17)
1001001 (F7 + F4 + F1 = 13 + 3 + 1 = 17)
1001010 (F7 + F4 + F2 = 13 + 3 + 1 = 17, minimal form, as used in Fibonacci coding)
The maximal form above will always use F1 and will always have a trailing one. The full coding without the trailing one can be found at . By dropping the trailing one, the coding for 17 above occurs as the 16th term of A104326. The minimal form will never use F1 and will always have a trailing zero. The full coding without the t |
https://en.wikipedia.org/wiki/Bender%27s%20method | In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups , and Gorenstein and Walter's classification of groups with dihedral Sylow 2-subgroups. Bender's method involves studying a maximal subgroup M containing the centralizer of an involution, and its generalized Fitting subgroup F*(M).
One succinct version of Bender's method is the result that if M, N are two distinct maximal subgroups of a simple group with F*(M) ≤ N and F*(N) ≤ M, then there is a prime p such that both F*(M) and F*(N) are p-groups. This situation occurs whenever M and N are distinct maximal parabolic subgroups of a simple group of Lie type, and in this case p is the characteristic, but this has only been used to help identify groups of low Lie rank. These ideas are described in textbook form in ,
, , and .
References
Group theory |
https://en.wikipedia.org/wiki/Barnette%27s%20conjecture | Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.
Definitions
A planar graph is an undirected graph that can be embedded into the Euclidean plane without any crossings. A planar graph is called polyhedral if and only if it is 3-vertex-connected, that is, if there do not exist two vertices the removal of which would disconnect the rest of the graph. A graph is bipartite if its vertices can be colored with two different colors such that each edge has one endpoint of each color. A graph is cubic (or 3-regular) if each vertex is the endpoint of exactly three edges. Finally, a graph is Hamiltonian if there exists a cycle that passes through each of its vertices exactly once. Barnette's conjecture states that every cubic bipartite polyhedral graph is Hamiltonian.
By Steinitz's theorem, a planar graph represents the edges and vertices of a convex polyhedron if and only if it is polyhedral. A three-dimensional polyhedron has a cubic graph if and only if it is a simple polyhedron.
And, a planar graph is bipartite if and only if, in a planar embedding of the graph, all face cycles have even length. Therefore, Barnette's conjecture may be stated in an equivalent form: suppose that a three-dimensional simple convex polyhedron has an even number of edges on each of its faces. Then, according to the conjecture, the graph of the polyhedron has a Hamiltonian cycle.
History
conjectured that every cubic polyhedral graph is Hamiltonian; this came to be known as Tait's conjecture. It was disproven by , who constructed a counterexample with 46 vertices; other researchers later found even smaller counterexamples. However, none of these known counterexamples is bipartite. Tutte himself conjectured that every cubic 3-connected bipartite graph is Hamiltonian, but this was shown to be false by the discovery of a counterexample, the Horton graph. proposed a weakened combination of Tait's and Tutte's conjectures, stating that every bipartite cubic polyhedron is Hamiltonian, or, equivalently, that every counterexample to Tait's conjecture is non-bipartite.
Equivalent forms
showed that Barnette's conjecture is equivalent to a superficially stronger statement, that for every two edges e and f on the same face of a bipartite cubic polyhedron, there exists a Hamiltonian cycle that contains e but does not contain f. Clearly, if this statement is true, then every bipartite cubic polyhedron contains a Hamiltonian cycle: just choose e and f arbitrarily. In the other directions, Kelmans showed that a counterexample could be transformed into a counterexample to the original Barnette conjecture.
Barnette's conjecture is also equivalent to the statement that the vertices of the dual of every cubic bipartite |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20Wycombe%20Wanderers%20F.C.%20season | The 2010–11 Football League Two is Wycombe Wanderers F.C.'s seventeenth season of League football. This article shows statistics of the club's players in the season, and also lists all matches that the club has played during the season.
League table
Match results
Legend
Friendlies
Football League Two
FA Cup
League Cup
Football League Trophy
Squad statistics
Appearances for competitive matches only
See also
2010–11 in English football
Wycombe Wanderers F.C.
References
External links
Wycombe Wanderers official website
Wycombe Wanderers F.C. seasons
Wycombe Wanderers |
https://en.wikipedia.org/wiki/Quantum%20topology | Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.
Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associated with topological spaces and the related embedding of one space within another such as knots and links in three-dimensional space. This bra–ket notation of kets and bras can be generalised, becoming maps of vector spaces associated with topological spaces that allow tensor products.
Topological entanglement involving linking and braiding can be intuitively related to quantum entanglement.
See also
Topological quantum field theory
Reshetikhin–Turaev invariant
References
External links
Quantum Topology, a journal published by EMS Publishing House
Quantum mechanics
Topology |
https://en.wikipedia.org/wiki/Gradually%20varied%20surface | In mathematics, a gradually varied surface is a special type of digital surfaces. It is a function from a 2D digital space (see digital geometry) to an ordered set or a chain.
A gradually varied function is a function from a digital space to where and are real numbers. This function possesses the following property: If x and y are two adjacent points in , assume , then , , or .
The concept of the continuous function in digital space (can be called digitally continuous functions) was proposed by Azriel Rosenfeld in 1986. It is a function in which the value (an integer) at a digital point is the same or almost the same as its neighbors. In other words, if x and y are two adjacent points in a digital space, |f(x) − f(y)| ≤ 1.
So we can see that the gradually varied function is defined to be more general than the digitally continuous function. The gradually varied function was defined by L. Chen in 1989.
An extension theorem related to above functions was mentioned by Rosenfeld (1986) and completed by Chen (1989). This theorem states: Let and . The necessary and sufficient condition for the existence of the gradually varied extension of is : for each pair of points and in , assume and , we have , where is the (digital) distance between and .
The gradually varied surface has direct relationship to graph homomorphism.
References
L. Chen, The necessary and sufficient condition and the efficient algorithms for gradually varied fill, Chinese Sci. Bull. 35 (10), pp 870–873, 1990.
A Rosenfeld, `Continuous' functions on digital pictures, Pattern Recognition Letters, v.4 n.3, p. 177-184, 1986.
G. Agnarsson and L. Chen, On the extension of vertex maps to graph homomorphisms, Discrete Mathematics, Vol 306, No 17, pp. 2021–2030, 2006.
L. Boxer, Digitally continuous functions, Pattern Recognition Letters, Vol 15, No 8, pp 833–839, 1994.
L.M. Chen, Digital Functions and Data Reconstruction, Springer, 2013
Digital geometry |
https://en.wikipedia.org/wiki/Multiple%20representations%20%28mathematics%20education%29 | In mathematics education, a representation is a way of encoding an idea or a relationship, and can be both internal (e.g., mental construct) and external (e.g., graph). Thus multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulatives, pictures, and sounds. Representations are thinking tools for doing mathematics.
Higher-order thinking
The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities. Estimation, another complex task, can strongly benefit from multiple representations
Curricula that support starting from conceptual understanding, then developing procedural fluency, for example, AIMS Foundation Activities, frequently use multiple representations.
Supporting student use of multiple representations may lead to more open-ended problems, or at least accepting multiple methods of solutions and forms of answers. Project-based learning units, such as WebQuests, typically call for several representations.
Motivation
Some representations, such as pictures, videos and manipulatives, can motivate because of their richness, possibilities of play, use of technologies, or connections with interesting areas of life. Tasks that involve multiple representations can sustain intrinsic motivation in mathematics, by supporting higher-order thinking and problem solving.
Multiple representations may also remove some of the gender biases that exist in math classrooms. For example, explaining probability solely through baseball statistics may potentially alienate students who have no interest in sports. When showing a tie to real-life applications, teachers should choose representations that are varied and of interest to all genders and cultures.
Assessment
Tasks that involve construction, use, and interpretation of multiple representations can lend themselves to rubric assessment, and other assessment types suitable for open-ended activities. For example, tapping into visualization for math problem solving manifests multiple representations. These multiple representations arise when each student uses their knowledge base and experience—to create a visualization of the problem domain on the way toward a solution. Since visualization can be categorized into two main areas, schematic or pictorial, most students will make use of one method, or sometimes |
https://en.wikipedia.org/wiki/2005%E2%80%9306%20Saudi%20Premier%20League | Statistics of the 2005–06 Saudi Premier League.
Stadia and locations
Final league table
Championship playoffs
Match against fourth place
Match against third place
Final
Season statistics
Top scorers
References
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
2005–06 in Saudi Arabian football |
https://en.wikipedia.org/wiki/2004%E2%80%9305%20Saudi%20Premier%20League | Statistics of the 2004–05 Saudi Premier League, officially known as The Custodian of The Two Holy Mosques League Cup.
Stadia and locations
Final league table
Championship playoffs
Match against fourth place
Match against third place
Final
Season statistics
Top scorers
References
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20Saudi%20Premier%20League | Statistics of the 2003–04 Saudi Premier League.
Stadia and locations
Final league table
Championship playoff
Match against fourth place
Match Against third place
Final
Season statistics
Top scorers
References
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20Saudi%20Premier%20League | Statistics of the 2002–03 Saudi Premier League.
Stadia and locations
Final league table
Championship playoff
Match against fourth place
Match against third place
Final
Season statistics
Top scorers
References
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20Saudi%20Premier%20League | Statistics of the 2001–02 Saudi Premier League.
Stadia and locations
Final league table
Championship playoff
Fourth place game
Third place game
Final
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/2000%E2%80%9301%20Saudi%20Premier%20League | Statistics of the 2000–01 Saudi Premier League.
Stadia and locations
Regular season
Final four
Semifinals
First legs
Second legs
Final
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
1 |
https://en.wikipedia.org/wiki/1999%E2%80%932000%20Saudi%20Premier%20League | Statistics of the 1999–2000 Saudi Premier League.
Stadia and locations
Final league table
Championship playoffs
Semifinals
First legs
Second legs
Championship final
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/1998%E2%80%9399%20Saudi%20Premier%20League | Statistics of the 1998-1999 Saudi Premier League.
Stadia and locations
Final league table
Playoffs
Semifinals
Final
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/1997%E2%80%9398%20Saudi%20Premier%20League | Statistics of the 1997–98 Saudi Premier League.
Stadia and locations
Final League table
Playoffs
Semifinals
Third place match
Final
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
goalzz
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Saudi%20Premier%20League | Statistics of the 1987–88 Saudi Premier League.
Stadia and locations
League table
Promoted: Al Hajr and Al Rouda
Full records are not known at this time
External links
RSSSF Stats
Saudi Arabia Football Federation
Saudi League Statistics
Saudi Premier League seasons
Saudi Professional League
Professional League |
https://en.wikipedia.org/wiki/Romanian%20Master%20of%20Mathematics%20and%20Sciences | The Romanian Master of Mathematics and Sciences (formerly known as the Romanian Masters in Mathematics) is an annual competition for students at the pre-university level, held in Bucharest, Romania. The contestants compete individually, in four different sections: mathematics, physics, chemistry and computer science. The participating teams (national and local teams) can have up to four students for each section (plus two coaches: a leader and a deputy leader). The contest follows the same structure as IMO and IPhO and is usually held at the end of February.
History
The first Romanian Master in Mathematics was held in 2008 and has been initiated by Prof. Severius Moldoveanu and Prof. Radu Gologan. In 2010 Physics was also added as a section, therefore the name changed to RMMS. At the beginning, the competition structure had been 4 problems in 5 hours, but also in 2010, it was changed to 6 problems over 2 days, with 4.5 hours of exam each day. The first country that won the competition was the United Kingdom. The 4th edition was held between 23–28 of February 2011 and included also Chemistry and Computer Science. The 5th edition, held in 2012 was only for Physics and Mathematics.
The current champion team in Mathematics is the United States of America.
Teams reaching the top three in mathematics
* = teams finished equal points
Organizers
The contest is organised at the Tudor Vianu National College of Computer Science in collaboration with the Sector 1 town council. As a host, Tudor Vianu has the right to have its own team entering the contest in each section, thus participating against countries.
References
Mathematics competitions
Physics competitions
Science competitions
Science events in Romania
Annual events in Romania
2008 establishments in Romania
Recurring events established in 2008 |
https://en.wikipedia.org/wiki/Dynamic%20risk%20measure | In financial mathematics, a conditional risk measure is a random variable of the financial risk (particularly the downside risk) as if measured at some point in the future. A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra.
A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures.
A different approach to dynamic risk measurement has been suggested by Novak.
Conditional risk measure
Consider a portfolio's returns at some terminal time as a random variable that is uniformly bounded, i.e., denotes the payoff of a portfolio. A mapping is a conditional risk measure if it has the following properties for random portfolio returns :
Conditional cash invariance
Monotonicity
Normalization
If it is a conditional convex risk measure then it will also have the property:
Conditional convexity
A conditional coherent risk measure is a conditional convex risk measure that additionally satisfies:
Conditional positive homogeneity
Acceptance set
The acceptance set at time associated with a conditional risk measure is
.
If you are given an acceptance set at time then the corresponding conditional risk measure is
where is the essential infimum.
Regular property
A conditional risk measure is said to be regular if for any and then where is the indicator function on . Any normalized conditional convex risk measure is regular.
The financial interpretation of this states that the conditional risk at some future node (i.e. ) only depends on the possible states from that node. In a binomial model this would be akin to calculating the risk on the subtree branching off from the point in question.
Time consistent property
A dynamic risk measure is time consistent if and only if .
Example: dynamic superhedging price
The dynamic superhedging price involves conditional risk measures of the form
.
It is shown that this is a time consistent risk measure.
References
Financial risk modeling |
https://en.wikipedia.org/wiki/Enrico%20Arbarello | Enrico Arbarello is an Italian mathematician who is a leading expert in algebraic geometry.
He earned a Ph.D. at Columbia University in New York in 1973. He was a visiting scholar at the Institute for Advanced Study from 1993-94. He is now a Mathematics Professor at Sapienza University of Rome.
In 2012 he became a fellow of the American Mathematical Society.
References
External links
University site
Living people
Columbia University alumni
Institute for Advanced Study visiting scholars
20th-century Italian mathematicians
21st-century Italian mathematicians
Fellows of the American Mathematical Society
1945 births
Algebraic geometers
Academic staff of the Sapienza University of Rome
Academic staff of the Scuola Normale Superiore di Pisa |
https://en.wikipedia.org/wiki/Gabrielle%20A.%20Brenner | Gabrielle A. Brenner is an Associate Professor of Economics. She holds a Bachelor of Science in Mathematics and master's degree in economics from the University of Jerusalem as well as a Ph.D. in economics from the University of Chicago. She is particularly interested by the areas of entrepreneurship, game theory, and the operation of competitive and regulated markets. Brenner specializes in decision making in situations involving risk.
Career
Brenner is an associate professor in Economics at the Hautes Études Commerciales (HEC Montréal). She is also an Associate member of the Chaire d’entrepreneuriat Rogers – J.-A.-Bombardier.
Brenner has participated in several studies on the development of entrepreneurship in North America and West Africa. She also contributed to studies on entrepreneurship in ethnic communities and the role it plays in the social integration of immigrants.
Brenner has worked on numerous programs to develop entrepreneurship. She served as a consultant for the World Bank, UNIDO, and CIDA.
Publications
Books
Brenner, R., Brenner, G.A., and Brown, A., A World of Chance, Cambridge University Press, 2008.
Brenner, R. and Brenner, G., Gambling and Speculation, Cambridge, Cambridge University Press, 1990, pp. vii-286, published in French in Paris by Presses Universitaires de France, 1993.
Articles
(and Brenner, R.) "Gambling: Shaping an opinion", Journal of Gambling Studies, v. 6, no. 4 (1990), pp. 297–311.
(and Brenner, R.) "Les Innovations et La Loi Sur la Concurrence", Actualité Economique, v. 65, no.1 (1989), pp. 146–63.
(and Brenner, R.), "Intrepreneurship - Le nouveau nom d'un vieux phénomène", Revue internationale de gestion, September 1988.
External links
Page on HEC Montréal: https://web.archive.org/web/20100601163331/http://www.hec.ca/en/profs/gabrielle.brenner.html
Page on Chaire d’entrepreneuriat Rogers – J.-A.-Bombardier: http://expertise.hec.ca/chaire_entrepreneuriat/2009/09/10/gabrielle-brenner/
Living people
Canadian economists
Canadian women economists
University of Chicago alumni
Hebrew University of Jerusalem Faculty of Social Sciences alumni
Academic staff of HEC Montréal
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Julian%20Besag | Julian Ernst Besag FRS (26 March 1945 – 6 August 2010) was a British statistician known chiefly for his work in spatial statistics (including its applications to epidemiology, image analysis and agricultural science), and Bayesian inference (including Markov chain Monte Carlo algorithms).
Early life and education
Besag was born in Loughborough and was educated at Loughborough Grammar School. He began studying engineering at the University of Cambridge but moved to the University of Birmingham to study statistics, obtaining his BSc in 1968.
Career
He then spent a year as a research assistant to Maurice Bartlett at the University of Oxford before obtaining a lectureship at the University of Liverpool. Inspired by John Tukey, he visited Princeton for a year.
He moved to the University of Durham in 1975, where he became a professor in 1986. He was a visiting professor at the University of Washington in Seattle during 1989–90 and, after a year at Newcastle University, returned to Seattle long-term. He officially retired in 2007 but remained an emeritus professor. At his death in 2010 he was also a visiting professor at the Universities of Bath and Bristol.
Besag was an ISI highly cited researcher; his 1986 paper "On the Statistical Analysis of Dirty Pictures" was the most cited paper by a UK mathematical scientist in the 1980s. The Royal Statistical Society awarded him its Guy Medal in Silver in 1983 for his contributions to spatial statistics, and he was elected a Fellow of the Royal Society in 2004.
Notable contributions
Spatial statistics
For an array of random variables Yij, stochastic dependence was known to be important. Julian initially researched a model to for the correlation between Yij pairs as a function of the distance between the corresponding lattice point pairs. However, this proved to be difficult due to ambiguous conditions for self-consistency.
He therefore suggested using multivariate distributions for such a variable, taking inspiration from statistical physics and unpublished work by Peter Clifford and John Hammersley:
He published his findings to the Royal Statistical Society in March 1974.
Death
Besag died in Bristol, 2010 following a surgery.
Notable publications
Besag, J. (1974) "Spatial Interaction and the Statistical Analysis of Lattice Systems", Journal of the Royal Statistical Society, Series B, 36 (2), 192–236.
Besag, J. (1975) "Statistical Analysis of Non-Lattice Data." Journal of the Royal Statistical Society, Series D, 24(3), 179–195.
Besag, J. (1977) "Comments on Ripley's paper." Journal of the Royal Statistical Society, Series B, 39(2), 193–195
Besag, J.E. (1986) "On the Statistical Analysis of Dirty Pictures," Journal of the Royal Statistical Society, Series B, 48, 259–302.
See also
Ripley's K and Besag's L function
Spatial point process
References
Profile: Julian Besag, FRS. IMS Bulletin Sept/Oct 2004; Vol. 33 No. 5, p13.
Prof Julian Besag at Debrett's People of Today
Julian Besag – Department o |
https://en.wikipedia.org/wiki/Boehmians | In mathematics, Boehmians are objects obtained by an abstract algebraic construction of "quotients of sequences." The original construction was motivated by regular operators introduced by T. K. Boehme. Regular operators are a subclass of Mikusiński operators, that are defined as equivalence classes of convolution quotients of functions on . The original construction of Boehmians gives us a space of generalized functions that includes all regular operators and has the algebraic character of convolution quotients. On the other hand, it includes all distributions eliminating the restriction of regular operators to .
Since the Boehmians were introduced in 1981, the framework of Boehmians has been used to define a variety of spaces of generalized functions on and generalized integral transforms on those spaces. It was also applied to function spaces on other domains, like locally compact groups and manifolds.
The general construction of Boehmians
Let be an arbitrary nonempty set and let be a commutative semigroup acting on . Let be a collection of sequences of elements of such that the following two conditions are satisfied:
(1) If , then ,
(2) If and for some and all , then .
Now we define a set of pairs of sequences:
.
In we introduce an equivalence relation:
~ if .
The space of Boehmians is the space of equivalence classes of , that is ~.
References
J. Mikusiński, Operational Calculus, Pergamon Press (1959).
T. K. Boehme, The support of Mikusiński operators, Trans. Amer. Math. Soc. 176 (1973), 319–334.
J. Mikusiński and P. Mikusiński, Quotients de suites et leurs applications dans l'analyse fonctionnelle (French), [Quotients of sequences and their applications in functional analysis], C. R. Acad. Sci. Paris Sr. I Math. 293 (1981), 463-464.
P. Mikusiński, Convergence of Boehmians, Japan. J. Math. (N.S.) 9 (1983), 159–179.
Generalized functions |
https://en.wikipedia.org/wiki/Wimin | Wimin may refer to:
Womyn, an alternate spelling of the word 'women', sometimes used by feminists.
WIMIN, the Women in Mathematics in New England conference, hosted by the Center for Women in Mathematics |
https://en.wikipedia.org/wiki/San%20Pedro%2C%20Chile | San Pedro () is a commune of the Melipilla Province in central Chile's Santiago Metropolitan Region.
Demographics
According to the 2002 census of the National Statistics Institute, San Pedro spans an area of and has 7,549 inhabitants (4,080 men and 3,469 women), and the commune is an entirely rural area. The population grew by 11.9% (803 persons) between the 1992 and 2002 censuses.
Administration
As a commune, San Pedro is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2012-2016 alcalde is Florentino Flores Armijo (PDC), and his council members are:
Emilio Cerda Sagurie (PPD)
Jeremías Vilches Mondaca (PS)
Samuel Espinoza Vilches (PRSD)
Avelino Farías Piña (UDI)
Juan Ignacio Zúñiga Godoy (RN)
Pedro Ulloa Ulloa (PH)
Within the electoral divisions of Chile, San Pedro is represented in the Chamber of Deputies by Denise Pascal (PS) and Gonzalo Uriarte (UDI) as part of the 31st electoral district, (together with Talagante, Peñaflor, El Monte, Isla de Maipo, Melipilla, María Pinto, Curacaví, Alhué and Padre Hurtado). The commune is represented in the Senate by Guido Girardi Lavín (PPD) and Jovino Novoa Vásquez (UDI) as part of the 7th senatorial constituency (Santiago-West).
References
External links
Official link - Municipality of San Pedro
Communes of Chile
Populated places in Melipilla Province |
https://en.wikipedia.org/wiki/Entropic%20risk%20measure | In financial mathematics (concerned with mathematical modeling of financial markets), the entropic risk measure is a risk measure which depends on the risk aversion of the user through the exponential utility function. It is a possible alternative to other risk measures as value-at-risk or expected shortfall.
It is a theoretically interesting measure because it provides different risk values for different individuals whose attitudes toward risk may differ. However, in practice it would be difficult to use since quantifying the risk aversion for an individual is difficult to do. The entropic risk measure is the prime example of a convex risk measure which is not coherent. Given the connection to utility functions, it can be used in utility maximization problems.
Mathematical definition
The entropic risk measure with the risk aversion parameter is defined as
where is the relative entropy of Q << P.
Acceptance set
The acceptance set for the entropic risk measure is the set of payoffs with positive expected utility. That is
where is the exponential utility function.
Dynamic entropic risk measure
The conditional risk measure associated with dynamic entropic risk with risk aversion parameter is given by
This is a time consistent risk measure if is constant through time,
and can be computed efficiently using forward-backwards differential equations
.
See also
Entropic value at risk
List of financial performance measures
References
Financial risk modeling
Utility |
https://en.wikipedia.org/wiki/List%20of%20lakes%20of%20British%20Columbia | This is an incomplete list of lakes of British Columbia, a province of Canada.
Larger lake statistics
*
List of lakes
1
101 Mile Lake
103 Mile Lake
105 Mile Lake
108 Mile Lake
A
Adams Lake
Alouette Lake
Alta Lake (British Columbia)
Ambrose Lake (British Columbia)
Anderson Lake (British Columbia)
Angora Lake
Angus Horne Lake
Arrow Lakes
Atlin Lake
Azure Lake
Azouzetta Lake
B
Babine Lake
Ball Lake
Battleship Lake
Bear Lake (Bear River)
Bennett Lake
Berg Lake
Bolton Lake (British Columbia)
Bridge Lake (British Columbia)
Brigade Lake
Bughouse Lake
Buntzen Lake
Burnaby Lake
C
Cahilty Lake (British Columbia)
Canim Lake (British Columbia)
Capilano Lake
Carp Lake Provincial Park
Carpenter Lake
Cecil Lake (Peace River Country)
Chadsey Lake
Champion Lakes Provincial Park
Charlie Lake (British Columbia)
Charlotte Lake (British Columbia)
Cheakamus Lake
Chehalis Lake
Cheslatta Lake
Chilcotin Lake
Chilko Lake
Chilliwack Lake
Christina Lake (British Columbia)
Chute Lake
Clearwater Lake (British Columbia)
Clendinning Lake
Clowhom Lake
Columbia Lake
Como Lake (British Columbia)
Consinka Lake
Coquitlam Lake
Costalot Lake
Cowichan Lake
Croteau Lake
Cuisson Lake
Cultus Lake, British Columbia
Cunningham Lake
D
Davis Lake (British Columbia)
Dease Lake (British Columbia)
Decker Lake (British Columbia)
Deer Lake (British Columbia)
Deka Lake
Devick Lake (British Columbia)
Divers Lake
Duncan (Amazay) Lake
Duncan Lake (British Columbia)
E
Eagle Lake (British Columbia)
Elfin Lakes
Elk Lake (British Columbia)
Elk Lakes (British Columbia)
Elsay Lake
Emerald Lake (British Columbia)
Emma Lake, (Powell River Area)
Lake Errock (British Columbia)
Evans Lake (British Columbia)
F
Floe Lake
Fortress Lake
François Lake
Frog Lakes
Frozen Lake (Montana)
G
Garibaldi Lake
Gates Lake
Granite Lake (Kawdy Plateau)
Granite Lake (Powell River)
Green Lake (Cariboo)
Green Lake (Whistler)
Greendrop Lake
Greig Lake (Vancouver Island)
Gun Lake (British Columbia)
Gunanoot Lake
H
Harrison Lake
Hatzic Lake
Hawthorn Lake
Hayward Lake
Hobson Lake
Horsefly Lake
Howard Lake (British Columbia)
I
Ilthpaya Lake
Inga Lake
Inland Lake Provincial Park
J
Jack Shark Lake
John Hart Lake
Junker Lake
K
Kalamalka Lake
Kamloops Lake
Kawkawa Lake
Kennedy Lake (Vancouver Island)
Kinbasket Lake
Kinney Lake
Kitkiata Lake
Kluskus Lakes
Knewstubb Lake
Knouff Lake (British Columbia)
Kokanee Lake
Lake Koocanusa
Kootenay Lake
Kostal Lake
Kotcho Lake
L
Lac La Hache
Lafarge Lake
Lajoie Lake
Lava Lake
Lava Lakes
Lac Le Jeune
Lightning Lake
Lillooet Lake
Lindeman Lake (Chilkoot Trail)
Lindeman Lake (Chilliwack)
Little Lillooet Lake
Little Shuswap Lake
Lizard Lake (Juan de Fuca, Vancouver Island)
Lizard Lake (Vancouver Island)
Lizard Pond
Long Lake (British Columbia)
Long Lake (Smith Inlet)
Long Lake (Vancouver Island)
Loon Lake, British Columbia
Lost Lagoon
Lost Lake (Whistler)
M
N
Nanaimo Lakes
Nation Lakes
Nechako Lakes
Ness Lake
Nicola Lake
Nimpkish Lake
Nimpo Lake
Nitinat Lake
O
Lake O'Hara
Lake Oesa
Okanaga |
https://en.wikipedia.org/wiki/Ian%20R.%20Porteous | Ian Robertson Porteous (9 October 1930 – 30 January 2011) was a Scottish mathematician at the University of Liverpool and an educator on Merseyside. He is best known for three books on geometry and modern algebra. In Liverpool he and Peter Giblin are known for their registered charity Mathematical Education on Merseyside which promotes enthusiasm for mathematics through sponsorship of an annual competition.
Family and early life
Porteous was born on 9 October 1930. He was one of six children of Reverend Norman Walker Porteous (later a theologian and Old Testament academic), from Crossgates, Fife and May Hadwen Robertson of Kirkcaldy, Fife. He attended George Watson's College in Edinburgh, and the University of Edinburgh, obtaining his first mathematical degree in 1952. After a time in national service, he took up study at Trinity College, Cambridge. Porteous wrote his thesis Algebraic Geometry under W.V.D. Hodge and Michael Atiyah at University of Cambridge in 1961.
Early career
Porteous began teaching at the University of Liverpool as a lecturer in 1959, becoming senior lecturer in 1972. During a year (1961–62) at Columbia University in New York, Porteous was influenced by Serge Lang. He continued to do research on manifolds in differential geometry.
In 1971 his article "The normal singularities of a submanifold" was published in Journal of Differential Geometry 5:543–64. It was concerned with the smooth embeddings of an m-manifold in Rn.
In 1969 Porteous published Topological Geometry with Van Nostrand Reinhold and Company. It was reviewed in Mathematical Reviews by J. Eells, who interpreted it as a three-term textbook for a sequence in abstract algebra, geometric algebra, and differential calculus in Euclidean and Banach spaces and on manifolds. Eells says "Surely this book is the product of substantial thought and care, both from the standpoints of consistent mathematical presentation and of student's pedagogical requirements." In 1981 a second edition was published with Cambridge University Press.
Later career and works
In 1995 Ian Porteous published Clifford Algebras and the Classical Groups which was reviewed by Peter R. Law. In praise, Law says "Porteous' presentation of the subject matter sets a standard by which others may be judged." The book has 24 chapters including 8:quaternions, 13:The classical groups, 15:Clifford algebras, 16:Spin groups, 17:Conjugation, 20:Topological spaces, 21:Manifolds, 22:Lie groups. In the preface Porteous acknowledges the contribution of his master's degree student Tony Hampson and anticipatory work by Terry Wall. See references to a link where misprints may be found.
The textbook Geometric Differentiation (1994) is a modern, elementary study of differential geometry. The subtitle, "for the intelligence of curves and surfaces" indicates its extent in the differential geometry of curves and differential geometry of surfaces.
The review by D.R.J. Chillingworth says it is "aimed at advanced undergraduat |
https://en.wikipedia.org/wiki/Solvency%20cone | The solvency cone is a concept used in financial mathematics which models the possible trades in the financial market. This is of particular interest to markets with transaction costs. Specifically, it is the convex cone of portfolios that can be exchanged to portfolios of non-negative components (including paying of any transaction costs).
Mathematical basis
If given a bid-ask matrix for assets such that and is the number of assets which with any non-negative quantity of them can be "discarded" (traditionally ), then the solvency cone is the convex cone spanned by the unit vectors and the vectors .
Definition
A solvency cone is any closed convex cone such that and .
Uses
A process of (random) solvency cones is a model of a financial market. This is sometimes called a market process.
The negative of a solvency cone is the set of portfolios that can be obtained starting from the zero portfolio. This is intimately related to self-financing portfolios.
The dual cone of the solvency cone () are the set of prices which would define a friction-less pricing system for the assets that is consistent with the market. This is also called a consistent pricing system.
Examples
Assume there are 2 assets, A and M with 1 to 1 exchange possible.
Frictionless market
In a frictionless market, we can obviously make (1A,-1M) and (-1A,1M) into non-negative portfolios, therefore . Note that (1,1) is the "price vector."
With transaction costs
Assume further that there is 50% transaction costs for each deal. This means that (1A,-1M) and (-1A,1M) cannot be exchanged into non-negative portfolios. But, (2A,-1M) and (-1A,2M) can be traded into non-negative portfolios. It can be seen that .
The dual cone of prices is thus easiest to see in terms of prices of A in terms of M (and similarly done for price of M in terms of A):
someone offers 1A for tM: therefore there is arbitrage if
someone offers tM for 1A: therefore there is arbitrage if
Properties
If a solvency cone :
contains a line, then there is an exchange possible without transaction costs.
, then there is no possible exchange, i.e. the market is completely illiquid.
References
Financial risk modeling |
https://en.wikipedia.org/wiki/Coarse%20function | In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous. Although continuous functions are usually observed on a small scale, coarse functions are usually observed on a large scale.
See also
Coarse structure
References
Types of functions |
https://en.wikipedia.org/wiki/Distribution%20of%20the%20product%20of%20two%20random%20variables | A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product is a product distribution.
The product distribution is the PDF of the product of sample values. This is not the same as the product of their PDF's yet the concepts are often ambiguously termed as "product of Gaussians".
Algebra of random variables
The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios.
Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables.
Derivation for independent random variables
If and are two independent, continuous random variables, described by probability density functions and then the probability density function of is
Proof
We first write the cumulative distribution function of starting with its definition
We find the desired probability density function by taking the derivative of both sides with respect to . Since on the right hand side, appears only in the integration limits, the derivative is easily performed using the fundamental theorem of calculus and the chain rule. (Note the negative sign that is needed when the variable occurs in the lower limit of the integration.)
where the absolute value is used to conveniently combine the two terms.
Alternate proof
A faster more compact proof begins with the same step of writing the cumulative distribution of starting with its definition:
where is the Heaviside step function and serves to limit the region of integration to values of and satisfying .
We find the desired probability density function by taking the derivative of both sides with respect to .
where we utilize the translation and scaling properties of the Dirac delta function .
A more intuitive description of the procedure is illustrated in the figure below. The joint pdf exists in the - plane and an arc of constant value is shown as the shaded line. To find the marginal probability on this arc, integrate over increments of area on this contour.
Starting with , we have . So the probability increment is . Since implies , we can relate the probability increment to the -increment, namely . Then integration over , yields .
A Bayesian interpretation
Let be a random sample drawn from probability distribution . Scaling by generates a sample from scaled distribution which can be written as a conditional distribution .
Letting be a random variable with pdf , the distribution of the scaled sample becomes and integrating out we get so is drawn from this distributi |
https://en.wikipedia.org/wiki/Alhu%C3%A9 | Alhué () is a Chilean town and commune located in Melipilla Province, Santiago Metropolitan Region.
Demographics
According to the 2002 census of the National Statistics Institute, Alhué spans an area of and has 4,435 inhabitants (2,343 men and 2,092 women). Of these, 2,593 (58.5%) lived in urban areas and 1,842 (41.5%) in rural areas. The population grew by 10.5% (422 persons) between the 1992 and 2002 censuses.
Administration
As a commune, Alhué is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2012-2016 alcalde is Roberto Torres Huerta (Ind.). The communal council has the following members:
Danilo Salazar Morales (PC)
Hugo Lazo Segovia (PC)
Roberto Aravena Miranda (PDC)
Mario Huerta Mora (PH)
Luis Núñez Pérez (PH)
Nancy Cofré Quiroz (RN)
Within the electoral divisions of Chile, Alhué is represented in the Chamber of Deputies by Denise Pascal (PS) and Gonzalo Uriarte (UDI) as part of the 31st electoral district, (together with Talagante, Peñaflor, El Monte, Isla de Maipo, Melipilla, María Pinto, Curacaví, San Pedro and Padre Hurtado). The commune is represented in the Senate by Guido Girardi Lavín (PPD) and Jovino Novoa Vásquez (UDI) as part of the 7th senatorial constituency (Santiago-West).
References
External links
Municipality of Alhué
Communes of Chile
Populated places in Melipilla Province |
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