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https://en.wikipedia.org/wiki/Dirk%20van%20Dalen | Dirk van Dalen (born 20 December 1932, Amsterdam) is a Dutch mathematician and historian of science.
Van Dalen studied mathematics and physics and astronomy at the University of Amsterdam. Inspired by the work of Brouwer and Heyting, he received his Ph.D. in 1963 from the University of Amsterdam for the thesis Extension problems in intuitionistic plane Projective geometry. From 1964 to 1966 Van Dalen taught logic and mathematics at MIT, and later Oxford. From 1967 he was professor at the University of Utrecht. In 2003 Dirk van Dalen was awarded the Academy Medal 2003 of the Royal Dutch Academy of Sciences for bringing the works of Brouwer to international attention.
Works
1958: (with Yehoshua Bar-Hillel and Azriel Levy) Foundations of Set Theory, North Holland Publishing
1963: Extension problems in intuitionistic plane projective geometry
1978: (with H.C. Doets and H. De Swart) Sets: Naive, Axiomatic and Applied, Pergamon Press
1980: Logic and Structure, Springer Universitext
1981: (editor) Brouwer's Cambridge Lectures on Intuitionism Cambridge University Press
1988:
2000: (with Heinz-Dieter Ebbinghaus) "Zermelo and the Skolem Paradox", Bulletin of Symbolic Logic 6(2)
2001: "Intuitionistic Logic", in: The Blackwell Guide to Philosophical Logic, Lou Goble (editor), Blackwell
2013: L.E.J. Brouwer - Topologist, Intuitionist, Philosopher: How mathematics is rooted in life, Springer-Verlag
References
The article was originally created as a translation (Google) of the corresponding article in Dutch Wikipedia.
Further reading
Dirk van Dalen Festschrift, Henk Barendregt et al. (editors), University of Utrecht, Department of Philosophy, 1993
Special issue: a tribute to Dirk van Dalen, Yuri Gurevich (editor), North-Holland, Amsterdam, 1995.
External links
Koninklijke bibliotheek over Dirk van Dalen
Homepage at the University of Utrecht
1932 births
Living people
Dutch mathematicians
20th-century Dutch historians
Dutch logicians
Historians of mathematics
Historians of science
Intuitionism
Massachusetts Institute of Technology School of Science faculty
Scientists from Amsterdam
University of Amsterdam alumni
Academic staff of Utrecht University |
https://en.wikipedia.org/wiki/Superfunction | In mathematics, superfunction is a nonstandard name for an iterated function for complexified continuous iteration index. Roughly, for some function f and for some variable x, the superfunction could be defined by the expression
Then, S(z; x) can be interpreted as the superfunction of the function f(x).
Such a definition is valid only for a positive integer index z. The variable x is often omitted.
Much study and many applications of superfunctions employ various extensions of these superfunctions to complex and continuous indices; and the analysis of existence, uniqueness and their evaluation. The Ackermann functions and tetration can be interpreted in terms of superfunctions.
History
Analysis of superfunctions arose from applications of the evaluation of fractional iterations of functions. Superfunctions and their inverses allow evaluation of not only the first negative power of a function (inverse function), but also of any real and even complex iterate of that function. Historically, an early function of this kind considered was ; the function has then been used as the logo of the physics department of the Moscow State University.
At that time, these investigators did not have computational access for the evaluation of such functions, but the function was luckier than : at the very least, the existence of the holomorphic function
such that had been demonstrated in 1950 by Hellmuth Kneser.
Relying on the elegant functional conjugacy theory of Schröder's equation, for his proof, Kneser had constructed the "superfunction" of the exponential map through the corresponding Abel function , satisfying the related Abel equation
so that . The inverse function Kneser found,
is an entire super-exponential, although it is not real on the real axis; it cannot be interpreted as tetrational, because the condition cannot be realized for the entire super-exponential. The real can be constructed with the tetrational (which is also a superexponential); while the real can be constructed with the superfactorial.
There is a book dedicated to superfunctions
Extensions
The recurrence formula of the above preamble can be written as
Instead of the last equation, one could write the identity function,
and extend the range of definition of the superfunction S to the non-negative integers. Then, one may posit
and extend the range of validity to the integer values larger than −2.
The following extension, for example,
is not trivial, because the inverse function may happen to be not defined for some values of .
In particular, tetration can be interpreted as superfunction of exponentiation for some real base ; in this case,
Then, at x = 1,
but
is not defined.
For extension to non-integer values of the argument, the superfunction should be defined in a different way.
For complex numbers and such that belongs to some connected domain ,
the superfunction (from to ) of a holomorphic function f on the domain is
a function , holomorphic on |
https://en.wikipedia.org/wiki/Piero%20Borgi | Piero Borgi (Venice, 1424–1484) was a versatile Italian mathematician. Borgi is the author of several of the best Italian books on mathematics written in the 15th century. Borgi's books include , written in 1483; Arithmetica, written in 1484, a book on arithmetic; and Il Libro de Abacho de Arithmetica e De Arte Mathematiche. The later book was so successful that it went through seventeen editions.
Notes
External links
St Andrews College
D E Smith, The First Great Commercial Arithmetic, Isis 8 (1) (1926), 41–49.
1424 births
1484 deaths
15th-century Italian mathematicians |
https://en.wikipedia.org/wiki/De%20Bruijn%E2%80%93Erd%C5%91s%20theorem | The De Bruijn–Erdős theorem may refer to:
De Bruijn–Erdős theorem (incidence geometry)
De Bruijn–Erdős theorem (graph theory) |
https://en.wikipedia.org/wiki/Finite%20difference%20coefficient | In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward.
Central finite difference
This table contains the coefficients of the central differences, for several orders of accuracy and with uniform grid spacing:
For example, the third derivative with a second-order accuracy is
where represents a uniform grid spacing between each finite difference interval, and .
For the -th derivative with accuracy , there are central coefficients . These are given by the solution of the linear equation system
where the only non-zero value on the right hand side is in the -th row.
An open source implementation for calculating finite difference coefficients of arbitrary derivates and accuracy order in one dimension is available.
The theory of Lagrange polynomials provides explicit formulas for the finite difference coefficients. For the first six derivatives we have the following:
where are generalized harmonic numbers.
Forward finite difference
This table contains the coefficients of the forward differences, for several orders of accuracy and with uniform grid spacing:
For example, the first derivative with a third-order accuracy and the second derivative with a second-order accuracy are
while the corresponding backward approximations are given by
Backward finite difference
To get the coefficients of the backward approximations from those of the forward ones, give all odd derivatives listed in the table in the previous section the opposite sign, whereas for even derivatives the signs stay the same.
The following table illustrates this:
Arbitrary stencil points
For a given arbitrary stencil points of length with the order of derivatives , the finite difference coefficients can be obtained by solving the linear equations
where is the Kronecker delta, equal to one if , and zero otherwise.
Example, for , order of differentiation :
The order of accuracy of the approximation takes the usual form .
See also
Finite difference method
Finite difference
Five-point stencil
Numerical differentiation
References
Finite differences
Numerical differential equations |
https://en.wikipedia.org/wiki/Mean%20signed%20deviation | In statistics, the mean signed difference (MSD), also known as mean signed deviation and mean signed error, is a sample statistic that summarises how well a set of estimates match the quantities that they are supposed to estimate. It is one of a number of statistics that can be used to assess an estimation procedure, and it would often be used in conjunction with a sample version of the mean square error.
For example, suppose a linear regression model has been estimated over a sample of data, and is then used to extrapolate predictions of the dependent variable out of sample after the out-of-sample data points have become available. Then would be the i-th out-of-sample value of the dependent variable, and would be its predicted value. The mean signed deviation is the average value of
Definition
The mean signed difference is derived from a set of n pairs, , where is an estimate of the parameter in a case where it is known that . In many applications, all the quantities will share a common value. When applied to forecasting in a time series analysis context, a forecasting procedure might be evaluated using the mean signed difference, with being the predicted value of a series at a given lead time and being the value of the series eventually observed for that time-point. The mean signed difference is defined to be
Use Cases
The mean signed difference is often useful when the estimations are biased from the true values in a certain direction. If the estimator that produces the values is unbiased, then . However, if the estimations are produced by a biased estimator, then the mean signed difference is a useful tool to understand the direction of the estimator's bias.
See also
Bias of an estimator
Deviation (statistics)
Mean absolute difference
Mean absolute error
Summary statistics
Means
Distance |
https://en.wikipedia.org/wiki/University%20of%20Chicago%20School%20Mathematics%20Project | The University of Chicago School Mathematics Project (UCSMP) is a multi-faceted project of the University of Chicago in the United States, intended to improve competency in mathematics in the United States by elevating educational standards for children in elementary and secondary schools.
Overview
The UCSMP supports educators by supplying training materials to them and offering a comprehensive mathematics curriculum at all levels of primary and secondary education. It seeks to bring international strengths into the United States, translating non-English math textbooks for English students and sponsoring international conferences on the subject of math education. Launched in 1983 with the aid of a six-year grant from Amoco, the UCSMP is used throughout the United States.
UCSMP developed Everyday Mathematics, a pre-K and elementary school mathematics curriculum.
UCSMP publishers
Wright Group-McGraw-Hill (K-6 Materials)
Wright Group-McGraw-Hill (6-12 Materials)
American Mathematical Society (Translations of Foreign Texts)
See also
Zalman Usiskin
References
External links
Official Website
Elementary Component
Secondary Component
1983 establishments in Michigan
Projects established in 1983
Mathematical projects
Mathematics education in the United States
University of Chicago |
https://en.wikipedia.org/wiki/Compacta | Compacta, a Latin adjective for compact, may refer to:
Compacta (genus), a genus of moths
Compacta (typeface), a typeface
and also
In mathematics, the plural of compactum, meaning a compact set
Pars compacta, a portion of the substantia nigra in anatomy
See also
Compactum |
https://en.wikipedia.org/wiki/5%2021%20honeycomb | {{DISPLAYTITLE:5 21 honeycomb}}
In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.
By putting spheres at its vertices one obtains the densest-possible packing of spheres in 8 dimensions. This was proven by Maryna Viazovska in 2016 using the theory of modular forms. Viazovska was awarded the Fields Medal for this work in 2022.
This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure (Gosset regarded honeycombs in n dimensions as degenerate n+1 polytopes).
Each vertex of the 521 honeycomb is surrounded by 2160 8-orthoplexes and 17280 8-simplicies.
The vertex figure of Gosset's honeycomb is the semiregular 421 polytope. It is the final figure in the k21 family.
This honeycomb is highly regular in the sense that its symmetry group (the affine Weyl group) acts transitively on the k-faces for k ≤ 6. All of the k-faces for k ≤ 7 are simplices.
Construction
It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the end of the 2-length branch leaves the 8-orthoplex, 611.
Removing the node on the end of the 1-length branch leaves the 8-simplex.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 421 polytope.
The edge figure is determined from the vertex figure by removing the ringed node and ringing the neighboring node. This makes the 321 polytope.
The face figure is determined from the edge figure by removing the ringed node and ringing the neighboring node. This makes the 221 polytope.
The cell figure is determined from the face figure by removing the ringed node and ringing the neighboring node. This makes the 121 polytope.
Kissing number
Each vertex of this tessellation is the center of a 7-sphere in the densest packing in 8 dimensions; its kissing number is 240, represented by the vertices of its vertex figure 421.
E8 lattice
contains as a subgroup of index 5760. Both and can be seen as affine extensions of from different nodes:
contains as a subgroup of index 270. Both and can be seen as affine extensions of from different nodes:
The vertex arrangement of 521 is called the E8 lattice.
The E8 lattice can also be constructed as a union of the vertices of two 8-demicube honeycombs (called a D82 or D8+ lattice), as well as the union of the vertices of three 8-simplex honeycombs (called an A83 lattice):
= ∪ = ∪ ∪
Regular complex honeycomb
Using a complex number coordinate system, it can also be constructed as a regular complex polytope, given the symbol 3{3}3{3}3{3}3{3}3, and Coxeter diagram . Its elements are in relative proportion as 1 vertex, 80 3-edges, 270 3{3}3 faces, 80 3{3}3{3}3 cells and 1 3{3}3{3}3{3}3 Witting polytope cells.
Relat |
https://en.wikipedia.org/wiki/David%20Hawkins%20%28philosopher%29 | David Hawkins (February 28, 1913 – February 24, 2002) was an American scientist whose interests included the philosophy of science, mathematics, economics, childhood science education, and ethics. He was also an administrative assistant at the Manhattan Project's Los Alamos Laboratory and later one of its official historians. Together with Herbert A. Simon, he discovered and proved the Hawkins–Simon theorem.
Early life
David Hawkins was born in El Paso, Texas, the youngest of seven children of William Ashton Hawkins, and his wife Clara Gardiner. His father was a prominent lawyer noted for his work on water law, who worked for the El Paso and Northeastern Railway, and was one of the founders of the city of Alamogordo, New Mexico. He grew up in La Luz, New Mexico.
Hawkins attended Hotchkiss School in Lakeville, Connecticut, but left after his junior year to enter Stanford University. He initially studied chemistry, but then switched to physics before finally majoring in philosophy. He was awarded his B.A. in 1934 and M.A. in 1936. While he was there, he met Frances Pockman, a teacher and writer. They got married in San Francisco in 1937. They had a daughter, Julie.
In 1936, Hawkins went to the University of California, Berkeley, to work on his doctorate. He became friends with Robert Oppenheimer, with whom he liked to discuss Hindu philosophy and issues in the philosophy of science such as the uncertainty principle and Niels Bohr's complementarity. In 1938, Hawkins and his wife, Frances, joined the Berkeley campus branch of the Communist Party of America. He earned his Ph.D. in 1940, writing his thesis on "A Causal Interpretation of Probability".
Manhattan Project
After graduating, Hawkins worked at Berkeley until May 1943, when Oppenheimer recruited him to work at the Manhattan Project's Los Alamos Laboratory, as his administrative assistant. "I was intrigued by the thought of being part of this extraordinary development," he later explained, "And it was still of course in those days entirely focused on the terrible thought that the Germans might get this weapon and win World War II."
Hawkins saw his role as that of a go-between, mediating between the civilian scientists and the military leadership at Los Alamos, but he also found a kindred spirit in the Polish mathematician Stan Ulam, who was working in Edward Teller's "Super" Group. They investigated the problem of branching a neutron multiplication in a nuclear chain reaction. Stan Frankel and Richard Feynman had tackled the problem using classical physics, but Ulam and Hawkins approached it using probability theory, creating a new sub-field now known as branching process theory. They investigated branching chains using a characteristic function. After the war, Ulam would extend and generalise this work. He described Hawkins as "the most talented amateur mathematician I know".
Hawkins is credited with the selection of the Alamogordo area for the Trinity nuclear test, but he declined to w |
https://en.wikipedia.org/wiki/Least-squares%20support%20vector%20machine | Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Johan Suykens and Joos Vandewalle. LS-SVMs are a class of kernel-based learning methods.
From support-vector machine to least-squares support-vector machine
Given a training set with input data and corresponding binary class labels , the SVM classifier, according to Vapnik's original formulation, satisfies the following conditions:
which is equivalent to
where is the nonlinear map from original space to the high- or infinite-dimensional space.
Inseparable data
In case such a separating hyperplane does not exist, we introduce so-called slack variables such that
According to the structural risk minimization principle, the risk bound is minimized by the following minimization problem:
To solve this problem, we could construct the Lagrangian function:
where are the Lagrangian multipliers. The optimal point will be in the saddle point of the Lagrangian function, and then we obtain
By substituting by its expression in the Lagrangian formed from the appropriate objective and constraints, we will get the following quadratic programming problem:
where is called the kernel function. Solving this QP problem subject to constraints in (), we will get the hyperplane in the high-dimensional space and hence the classifier in the original space.
Least-squares SVM formulation
The least-squares version of the SVM classifier is obtained by reformulating the minimization problem as
subject to the equality constraints
The least-squares SVM (LS-SVM) classifier formulation above implicitly corresponds to a regression interpretation with binary targets .
Using , we have
with Notice, that this error would also make sense for least-squares data fitting, so that the same end results holds for the regression case.
Hence the LS-SVM classifier formulation is equivalent to
with and
Both and should be considered as hyperparameters to tune the amount of regularization versus the sum squared error. The solution does only depend on the ratio , therefore the original formulation uses only as tuning parameter. We use both and as parameters in order to provide a Bayesian interpretation to LS-SVM.
The solution of LS-SVM regressor will be obtained after we construct the Lagrangian function:
where are the Lagrange multipliers. The conditions for optimality are
Elimination of and will yield a linear system instead of a quadratic programming problem:
with , and . Here, is an identity matrix, and is the ke |
https://en.wikipedia.org/wiki/Lo%C3%AFc%20Merel | Loïc Merel (born 13 August 1965) is a French mathematician. His research interests include modular forms and number theory.
Career
Born in Carhaix-Plouguer, Brittany, Merel became a student at the École Normale Supérieure. He finished his doctorate at Pierre and Marie Curie University under supervision of Joseph Oesterlé in 1993. His thesis on modular symbols took inspiration from the work of Yuri Manin and Barry Mazur from the 1970s. In 1996, Merel proved the torsion conjecture for elliptic curves over any number field (which was only known for number fields of degree up to 8 at the time). In recognition of his achievement, in 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin.
Awards
Merel has received numerous awards, including the EMS Prize (1996), the Blumenthal Award (1997) for the advancement of research in pure mathematics, and the (1998) of the French Academy of Sciences.
References
External links
Website at Paris Diderot University
1965 births
Living people
École Normale Supérieure alumni
People from Finistère
21st-century French mathematicians
University of Paris alumni
Scientists from Brittany |
https://en.wikipedia.org/wiki/Sporting%20CP%20in%20European%20football | Sporting Clube de Portugal history and statistics in the UEFA competitions.
1963–64 UEFA Cup Winners' Cup
The 1963–64 European Cup Winners' Cup was won by Sporting Clube de Portugal of Portugal, who defeated MTK Budapest of Hungary in the final. It was the first and only time a Portuguese team side has won a UEFA Cup Winners' Cup trophy.
Sporting CP entered the competition to defeat Atalanta in the qualifying round, past APOEL was to the history of biggest win in UEFA competitions 16–1, Manchester United, Olympique Lyonnais and in the end defeated MTK Budapest, the same final that was played over two legs on neutral ground. Sporting Clube de Portugal win their first European title.
Honours
European Cup Winners' Cup
Winners (1): 1963−64
UEFA Cup
Runners-up (1): 2004–05
Matches
From 1955–56 to 1979–80
Note: Sporting CP score always listed first.
From 1980–81 to 1999–2000
Note: Sporting CP score always listed first.
From 2000–01 to present
Last updated: 26 October 2023Note: Sporting CP score always listed first.
Overall record
By competition
Finals
Notes
References
Sporting CP
Portuguese football clubs in international competitions |
https://en.wikipedia.org/wiki/Wilbur%20Knorr | Wilbur Richard Knorr (August 29, 1945 – March 18, 1997) was an American historian of mathematics and a professor in the departments of philosophy and classics at Stanford University. He has been called "one of the most profound and certainly the most provocative historian of Greek mathematics" of the 20th century.
Biography
Knorr was born August 29, 1945, in Richmond Hill, Queens. He did his undergraduate studies at Harvard University from 1963 to 1966 and stayed there for his Ph.D., which he received in 1973 under the supervision of John Emery Murdoch and G. E. L. Owen. After postdoctoral studies at Cambridge University, he taught at Brooklyn College, but lost his position when the college's Downtown Brooklyn campus was closed as part of New York's mid-1970s fiscal crisis. After taking a temporary position at the Institute for Advanced Study, he joined the Stanford faculty as an assistant professor in 1979, was tenured there in 1983, and was promoted to full professor in 1990.
He died March 18, 1997, in Palo Alto, California, of melanoma.
Knorr was a talented violinist, and played first violin in the Harvard Orchestra, but he gave up his music when he came to Stanford, as the pressures of the tenure process did not allow him adequate practice time.
Books
The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry.
This work incorporates Knorr's Ph.D. thesis. It traces the early history of irrational numbers from their first discovery (in Thebes between 430 and 410 BC, Knorr speculates), through the work of Theodorus of Cyrene, who showed the irrationality of the square roots of the integers up to 17, and Theodorus' student Theaetetus, who showed that all non-square integers have irrational square roots. Knorr reconstructs an argument based on Pythagorean triples and parity that matches the story in Plato's Theaetetus of Theodorus' difficulties with the number 17, and shows that switching from parity to a different dichotomy in terms of whether a number is square or not was the key to Theaetetus' success. Theaetetus classified the known irrational numbers into three types, based on analogies to the geometric mean, arithmetic mean, and harmonic mean, and this classification was then greatly extended by Eudoxus of Cnidus; Knorr speculates that this extension stemmed out of Eudoxus' studies of the golden section.
Along with this history of irrational numbers, Knorr reaches several conclusions about the history of Euclid's Elements and of other related mathematical documents; in particular, he ascribes the origin of the material in Books 1, 3, and 6 of the Elements to the time of Hippocrates of Chios, and of the material in books 2, 4, 10, and 13 to the later period of Theodorus, Theaetetus, and Eudoxos. However, this suggested history has been criticized by van der Waerden, who believed that books 1 through 4 were largely due to the much earlier Pythagorean school.
Anc |
https://en.wikipedia.org/wiki/Black%E2%80%93Karasinski%20model | In financial mathematics, the Black–Karasinski model is a mathematical model of the term structure of interest rates; see short-rate model. It is a one-factor model as it describes interest rate movements as driven by a single source of randomness. It belongs to the class of no-arbitrage models, i.e. it can fit today's zero-coupon bond prices, and in its most general form, today's prices for a set of caps, floors or European swaptions. The model was introduced by Fischer Black and Piotr Karasinski in 1991.
Model
The main state variable of the model is the short rate, which is assumed to follow the stochastic differential equation (under the risk-neutral measure):
where dWt is a standard Brownian motion. The model implies a log-normal distribution for the short rate and therefore the expected value of the money-market account is infinite for any maturity.
In the original article by Fischer Black and Piotr Karasinski the model was implemented using a binomial tree with variable spacing, but a trinomial tree implementation is more common in practice, typically a log-normal application of the Hull–White lattice.
Applications
The model is used mainly for the pricing of exotic interest rate derivatives such as American and Bermudan bond options and swaptions, once its parameters have been calibrated to the current term structure of interest rates and to the prices or implied volatilities of caps, floors or European swaptions. Numerical methods (usually trees) are used in the calibration stage as well as for pricing. It can also be used in modeling credit default risk, where the Black–Karasinski short rate expresses the (stochastic) intensity of default events driven by a Cox process; the guaranteed positive rates are an important feature of the model here. Recent work on Perturbation Methods in Credit Derivatives has shown how analytic prices can be conveniently deduced in many such circumstances, as well as for interest rate options.
References
External links
Simon Benninga and Zvi Wiener (1998). Binomial Term Structure Models, Mathematica in Education and Research, Vol. 7 No. 3 1998
Blanka Horvath, Antoine Jacquier and Colin Turfus (2017). Analytic Option Prices for the Black–Karasinski Short Rate Model
Colin Turfus (2018). Analytic Swaption Pricing in the Black–Karasinski Model
Colin Turfus (2018). Exact Arrow-Debreu Pricing for the Black–Karasinski Short Rate Model
Colin Turfus (2019). Perturbation Expansion for Arrow–Debreu Pricing with Hull-White Interest Rates and Black–Karasinski Credit Intensity
Colin Turfus and Piotr Karasinski (2021). The Black-Karasinski Model: Thirty Years On
Short-rate models
Financial models |
https://en.wikipedia.org/wiki/Trope%20%28mathematics%29 | In geometry, trope is an archaic term for a singular (meaning special) tangent space of a variety, often a quartic surface. The term may have been introduced by , who defined it as "the reciprocal term to node". It is not easy to give a precise definition, because the term is used mainly in older books and papers on algebraic geometry, whose definitions are vague and different, and use archaic terminology. The term trope is used in the theory of quartic surfaces in projective space, where it is sometimes defined as a tangent space meeting the quartic surface in a conic; for example Kummer's surface has 16 tropes.
, describes a trope as a tangent plane where the envelope of nearby tangent planes forms a conic, rather than a plane pencil which we would expect for a generic point. The tangent plane would be tangent to the quartic along the conic, implying that the Gauss map would have a singular point.
See also
Glossary of classical algebraic geometry
References
See page 202 for an early use of the term "trope".
Algebraic geometry |
https://en.wikipedia.org/wiki/Fernando%20%28footballer%2C%20born%201978%29 | Fernando Almeida de Oliveira (born 18 June 1978), known as just Fernando, is a Brazilian football player who currently plays for Vitória.
Club statistics
References
External links
http://awx.jp/cgi/prof/prof.cgi?player=fernando
1978 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Esporte Clube Vitória players
Cruzeiro Esporte Clube players
Club Athletico Paranaense players
Al Ahli SC (Doha) players
Campeonato Brasileiro Série A players
J1 League players
Kashima Antlers players
Expatriate men's footballers in Japan
Expatriate men's footballers in Qatar
Expatriate men's footballers in Saudi Arabia
Men's association football midfielders |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20FK%20Partizan%20season | The 2009–10 season was FK Partizan's 4th season in Serbian SuperLiga. This article shows player statistics and all matches (official and friendly) that the club played during the 2009–10 season.
Tournaments
Players
Squad information
Top scorers
Includes all competitive matches. The list is sorted by shirt number when total goals are equal.
Squad statistics
Competitions
Overview
Serbian SuperLiga
League table
Matches
Serbian Cup
UEFA Champions League
Qualifying phase
UEFA Europa League
Play-off round
Group stage
Friendlies
Transfers
In
Out
Sponsors
See also
List of unbeaten football club seasons
External links
Official website
Partizanopedia 2009-2010 (in Serbian)
FK Partizan seasons
Partizan
Serbian football championship-winning seasons |
https://en.wikipedia.org/wiki/Hot%20game |
In combinatorial game theory, a branch of mathematics, a hot game is one in which each player can improve their position by making the next move.
By contrast, a cold game is one where each player can only worsen their position by making the next move. Cold games have values in the surreal numbers and so can be ordered by value, while hot games can have other values.
Example
For example, consider a game in which players alternately remove tokens of their own color from a table, the Blue player removing only blue tokens and the Red player removing only red tokens, with the winner being the last player to remove a token. Obviously, victory will go to the player who starts off with more tokens, or to the second player if the number of red and blue tokens are equal. Removing a token of one's own color leaves the position slightly worse for the player who made the move, since that player now has fewer tokens on the table. Thus each token represents a "cold" component of the game.
Now consider a special purple token bearing the number "100", which may be removed by either player, who then replaces the purple token with 100 tokens of their own color. (In the notation of Conway, the purple token is the game {100|−100}.) The purple token is a "hot" component, because it is highly advantageous to be the player who removes the purple token. Indeed, if there are any purple tokens on the table, players will prefer to remove them first, leaving the red or blue tokens for last. In general, a player will always prefer to move in a hot game rather than a cold game, because moving in a hot game improves their position, while moving in a cold game injures their position.
Temperature
The temperature of a game is a measure of its value to the two players. A purple "100" token has a temperature of 100 because its value to each player is 100 moves. In general, players will prefer to move in the hottest component available. For example, suppose there is a purple "100" token and also a purple "1,000" token which allows the player who takes it to dump 1,000 tokens of their own color on the table. Each player will prefer to remove the "1,000" token, with temperature 1,000 before the "100" token, with temperature 100.
To take a slightly more complicated example, consider the game {10|2} + {5|−5}. {5|−5} is a token which either player may replace with 5 tokens of their own color, and {10|2} is a token which the Blue player may replace with 10 blue tokens or the Red player may replace with 2 blue tokens.
The temperature of the {10|2} component is ½(10 − 2) = 4, while the temperature of the {5|−5} component is 5. This suggests that each player should prefer to play in the {5|−5} component. Indeed, the best first move for the Red player is to replace {5|−5} with −5, whereupon the Blue player replaces {10|2} with 10, leaving a total of 5; had the Red player moved in the cooler {10|2} component instead, the final position would have been 2 + 5 = 7, which is |
https://en.wikipedia.org/wiki/Aner%20Shalev | Aner Shalev (born 24 January 1958) is a professor at the Einstein Institute of Mathematics at the Hebrew University of Jerusalem, and a writer.
Biography
Shalev was born in Kibbutz Kinneret and grew up in Beit Berl. He moved to Jerusalem at 18 to study mathematics and philosophy at the Hebrew University, and since then, excluding some years abroad, he has been living mainly in Jerusalem.
Shalev received his Ph.D. in mathematics at the Hebrew University in 1989, summa cum laude. His doctoral thesis was written under the supervision of Professors Amitsur and Mann and dealt with group rings, an area combining group theory and ring theory.
Shalev spent his post-doctoral period at Oxford University and at the University of London, returned to Israel in 1992, when he was hired as a senior lecturer at the Hebrew University. Shalev was appointed full professor in 1996, and spent sabbaticals at the Universities of Chicago, Oxford (All Souls College), and London (Imperial College). He was also a visiting scholar at other institutes, such as the Australian National University, MSRI Berkeley, the Institute for Advanced Studies at the Hebrew University, and the Institute for Advanced Study at Princeton.
Shalev is joint editor of the Israel Journal of Mathematics, the Journal of Group Theory, and the Journal of Algebra. He gave an invited talk at the International Congress of Mathematicians (ICM) in Berlin in 1998 and at numerous other mathematical conferences all over the world. Shalev was awarded many grants from various sources, including the ERC Advanced Grant from the European Community (2010–2014).
Aner Shalev is married to Donna Shalev, a senior lecturer at the Classics Department of the Hebrew University of Jerusalem, and they have two daughters.
Research
Shalev's main area of research over the years has been Group Theory, and he often uses methods from other disciplines, such as Lie Algebras and Probability. He has also worked on Ring Theory, Lie Algebras, and other areas. Shalev has published around 120 mathematical articles in various international journals.
The first fruits of Shalev's research solve various problems in Group Rings using a unified method based on dimension subgroups. Subsequently, he worked extensively in p-groups and pro-p groups and was among those who solved the coclass conjectures on the structure of such groups. Likewise Shalev used Lie methods to solve problems on fixed points of automorphisms of p-groups, and studied subgroup growth of profinite and discrete groups.
Probabilistic methods in group theory
From 1995 Shalev developed and applied probabilistic methods to finite groups and (nonabelian) finite simple groups in particular. A formative result in this area shows that almost every pair of elements in a finite simple group generate the group. This result, like many others in the field, were proved by Shalev in collaboration with Martin Liebeck of Imperial College at the University of London. The probabilisti |
https://en.wikipedia.org/wiki/C.%20F.%20Jeff%20Wu | Chien-Fu Jeff Wu (born 1949) is the Coca-Cola Chair in Engineering Statistics and Professor in the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology. He is known for his work on the convergence of the EM algorithm, resampling methods such as the bootstrap and jackknife, and industrial statistics, including design of experiments, and robust parameter design (Taguchi methods).
Born in Taiwan, Wu earned a B.Sc. in Mathematics from National Taiwan University in 1971, and a Ph.D. in Statistics from University of California, Berkeley in 1976. He has been a faculty member at the University of Wisconsin, Madison (1977–1988), the University of Waterloo (1988–1993; GM-NSERC chair in quality and productivity), the University of Michigan (1995–2003; chair of Department of Statistics 1995–98; H.C. Carver professor of statistics, 1997–2003) and currently the Georgia Institute of Technology. He has supervised 50 Ph.D. students and published around 185 peer-reviewed articles and two books.
He has received several awards, including the COPSS Presidents' Award in 1987,
the Shewhart Medal in 2008,
the COPSS R. A. Fisher Lectureship in 2011,
and the Deming Lecturer Award in 2012. He gave the inaugural Akaike Memorial Lecture in 2016. He has been elected as a fellow of the American Statistical Association, the Institute of Mathematical Statistics, the American Society for Quality and the Institute for Operations Research and the Management Sciences. In 2000 he was elected as a member of Academia Sinica. In 2004, he was elected as a member of the National Academy of Engineering. He received the Shewhart Medal of the American Society for Quality and an honorary degree from the University of Waterloo in 2008.
In 1985, in a lecture given to the Chinese Academy of Sciences in Beijing, he used the term Data Science for the first time as an alternative name for statistics. Later, in November 1997, he gave the inaugural lecture entitled "Statistics = Data Science?" for his appointment to the H. C. Carver Professorship at the University of Michigan.
He popularized the term "data science" and advocated that statistics be renamed data science and statisticians data scientists.
He also presented his lecture entitled "Statistics = Data Science?" as the first of his 1998 P.C. Mahalanobis Memorial Lectures. These lectures honor Prasanta Chandra Mahalanobis, an Indian scientist and statistician and founder of the Indian Statistical Institute.
In Mile, Yunnan, China, a conference was held in July 2014 celebrating Professor Wu's 65th birthday. In 2014 he gave the Bradley Lecture at the University of Georgia. In 2016 he was the inaugural recipient of the Akaike Memorial Lecture Award. In 2017 Jeff Wu received the George Box Medal from ENBIS. In 2020, Jeff Wu received Georgia Institute of Technology’s highest award given to a faculty member: the Class of 1934 Distinguished Professor Award. In the same year, he also got t |
https://en.wikipedia.org/wiki/List%20of%20SANU%20members | List of Serbian Academy of Sciences and Arts members:
Department of Mathematics, Physics and Geo Sciences
Bogoljub Stanković
Stevan Karamata
Zoran Maksimović - Secretary of the Department of Mathematics, Physics and Geo Sciences
Stevan Koički - Vice-President
Zvonko Marić - Representative of the Department
Milosav Marjanović
Mileva Prvanović
Olga Hadžić
Dragoš Cvetković
Fedor Mesinger
Vojislav Marić
Aleksandar Ivić
Božidar Vujanović
Fedor Herbut
Nikola Konjević
Marko Ercegovac
Stevo Todorčević
Corresponding members
Nikola Konjević
Marko Ercegovac
Zaviša Janjić
Stevan Pilipović
Đorde Šijački
Vidojko Jović
Milan Damnjanović
Gradimir Milovanović
Nonresident members
Nemanja Kaloper
Foreign members
Bogdan Maglić
Sergei Novikov
Vilen Andreyevich Zharikov
Tihomir Novakov
Vasiliy Sergeyevich Vladimirov
Yuriy Tsolakovich Oganesian
Pantó György
Blagovest Sendov
William N. Everitt
Dietrich H. Welte
Julius Erich Wess
André Berger
Dionigi Galletto
Igor Shafarevich
Anton Zeilinger
Endre Süli
Milan Herak - resigned
Vladimir Majer - resigned
Department of Chemical and Biological Sciences
Dušan Kanazir
Milutin Stefanović
Vojislav Petrović - Representative of the Department
Dragomir Vitorović
Paula Putanov - Representative in Novi Sad
Dušan Čamprag
Slobodan Ribnikar
Miroslav Gašić - Secretary of the Department of Chemical and Biological Sciences
Iván Gutman
Dragoslav Marinković
Živorad Čeković
Corresponding members
Radoslav Adžić
Živorad Čeković
Dragan Škorić
Marko Anđelković
Miljenko Perić
Nikola Tucić
Vladimir Stevanović
Bogdan Šolaja
Nonresident members
Radomir Crkvenjakov
Slobodan Macura
Stanko Stojilković
Nenad Kostić
Dražen Zimonjić
Foreign members
Seymour Cohen
Paul M. Doty
Dušan Hadžić
Miha Tišler
Drago Grdenić
Chintamani Nagesa Ramachandra Rao
Richard M. Spriggs
John O'Mara Bockris
Guy Ourisson
Michael Simic
Branislav Vidić
Emil Špaldon
Constantin E. Sekeris
Igor Vladimirovich Torgov
Velibor Krsmanović
Frank E. Karas
Francisco J. Ayala
Stanley Prusiner
Borislav Bogdanović
Paul Greengard
Department of Technical Sciences
Ilija Obradović
Nikola Hajdin - President
Petar Miljanić - Secretary of the Department of Technical Sciences
Jovan Surutka
Momčilo Ristić
Dragutin Dražić - Representative of the Department
Dušan Milović
Đorđe Zloković
Miomir Vukobratović
Vladan Đorđević
Aleksandar Marinčić
Ilija Stojanović
Pantelija Nikolić
Đorđe Đukić
Boško Petrović
Antonije Đorđević
Corresponding members
Boško Petrović
Dragutin Zelenović
Antonije Đorđević
Zoran Lj. Petrović
Teodor Atanacković
Ninoslav Stojadinović
Zoran Đurić
Zoran Popović
Foreign members
Dragoslav D. Šiljak
Bruno Thürlimann
Konstantin Vasilyevich Frolov
Vukan R. Vuchic
Valeriy Vladimirovich Skorohod
Anthony N. Kounadis
Valentin Vitalyevich Rumyancev
Felix Leonidovich Chernoushko
Dmitriy Yevgeniyevich Ohotsimski
Miloš Ercegovac
Zoran D. Popović
Zoja Popović
Ingo Müller
Miroslav Krstić
Department of Medical Sciences
Ljubisav Rakić - Representative of the Department
Zlatibor Petrović - Secretar |
https://en.wikipedia.org/wiki/Chinese%20multiplication%20table | The Chinese multiplication table is the first requisite for using the Rod calculus for carrying out multiplication, division, the extraction of square roots, and the solving of equations based on place value decimal notation. It was known in China as early as the Spring and Autumn period, and survived through the age of the abacus; pupils in elementary school today still must memorise it.
The Chinese multiplication table consists of eighty-one terms. It was often called the nine-nine table, or simply nine-nine, because in ancient times, the nine nine table started with 9×9: nine nines beget eighty-one, eight nines beget seventy-two ... seven nines beget sixty three, etc. two ones beget one. In the opinion of Wang Guowei, a noted scholar, the nine-nine table probably started with nine because of the "worship of nine" in ancient China; the emperor was considered the "nine five supremacy" in the Book of Change. See also .
It is also known as nine-nine song (or poem), as the table consists of eighty-one lines with four or five Chinese characters per lines; this thus created a constant metre and render the multiplication table as a poem. For example, 9x9=81 would be rendered as "九九八十一", or "nine nine eighty one", with the world for "begets" "得" implied. This makes it easy to learn by heart. A shorter version of the table consists of only forty-five sentences, as terms such as "nine eights beget seventy-two" are identical to "eight nines beget seventy-two" so there is no need to learn them twice. When the abacus replaced the counting rods in the Ming dynasty, many authors on the abacus advocated the use of the full table instead of the shorter one. They claimed that memorising it without needing a moment of thinking makes abacus calculation much faster.
The existence of the Chinese multiplication table is evidence of an early positional decimal system: otherwise a much larger multiplication table would be needed with terms beyond 9×9.
The Nine-nine song text in Chinese
It can be read in either row-major or column-major order.
The Nine-nine table in Chinese literature
Many Chinese classics make reference to the nine-nine table:
Zhoubi Suanjing: "nine nine eighty one"
Guan Zi has sentences of the form "three eights beget twenty four, three sevens beget twenty-one"
The Nine Chapters on the Mathematical Art: "Fu Xi invented the art of nine-nine".
In Huainanzi, there were eight sentences: "nine nines beget eighty one", "eight nines beget seventy two", all the way to "two nines beget eighteen".
A nine-nine table manuscript was discovered in Dun Huang.
Xia Houyang's Computational Canons: "To learn the art of multiplication and division,one must understand nine-nine".
The Song dynasty author Hong Zhai's Notebooks said: "three threes as nine, three fours as twelve, two eights as sixteen, four fours as sixteen, three nines as twenty seven, four nines as thirty six, six sixes as thirty six, five eights as forty, five nines as forty five, seven n |
https://en.wikipedia.org/wiki/Discrete%20Chebyshev%20polynomials | In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev and rediscovered by Gram. They were later found to be applicable to various algebraic properties of spin angular momentum.
Elementary Definition
The discrete Chebyshev polynomial is a polynomial of degree n in x, for , constructed such that two polynomials of unequal degree are orthogonal with respect to the weight function
with being the Dirac delta function. That is,
The integral on the left is actually a sum because of the delta function, and we have,
Thus, even though is a polynomial in , only its values at a discrete set of points,
are of any significance. Nevertheless, because these polynomials can be defined in terms of orthogonality with respect to a nonnegative weight function, the entire theory of orthogonal polynomials is applicable. In particular, the polynomials are complete in the sense that
Chebyshev chose the normalization so that
This fixes the polynomials completely along with the sign convention, .
If the independent variable is linearly scaled and shifted so that the end points assume the values and , then as , times a constant, where is the Legendre polynomial.
Advanced Definition
Let be a smooth function defined on the closed interval [−1, 1], whose values are known explicitly only at points , where k and m are integers and . The task is to approximate f as a polynomial of degree n < m. Consider a positive semi-definite bilinear form
where and are continuous on [−1, 1] and let
be a discrete semi-norm. Let be a family of polynomials orthogonal to each other
whenever is not equal to . Assume all the polynomials have a positive leading coefficient and they are normalized in such a way that
The are called discrete Chebyshev (or Gram) polynomials.
Connection with Spin Algebra
The discrete Chebyshev polynomials have surprising connections to various algebraic properties of spin: spin transition probabilities,
the probabilities for observations of the spin in Bohm's spin-s version of the Einstein-Podolsky-Rosen experiment,
and Wigner functions for various spin states.
Specifically, the polynomials turn out to be the eigenvectors of the absolute square of the rotation matrix (the Wigner D-matrix). The associated eigenvalue is the Legendre polynomial , where is the rotation angle. In other words, if
where are the usual angular momentum or spin eigenstates,
and
then
The eigenvectors are scaled and shifted versions of the Chebyshev polynomials. They are shifted so as to have support on the points instead of for with corresponding to , and corresponding to . In addition, the can be scaled so as to obey other normalization conditions. For example, one could demand that they satisfy
along with .
References
Orthogonal polynomials
Approximation theory |
https://en.wikipedia.org/wiki/Beulah%20College | Beulah College is a coeducational Christian secondary school in Tongatapu, Tonga, established in 1938. It was formally opened by Sālote Tupou III in February 1939. The SDA Annual Statistics first report on Beulah College in 1941. It lists 109 students and five teachers for only grades 1–8. Four students graduated. The 2009 report lists 202 students, 97 of which were Seventh-day Adventists. The school provided a complete secondary school education. There were 16 graduates.
In October 2015 boarders at the school were sent home after a girl in the village was diagnosed with typhoid.
See also
List of Seventh-day Adventist secondary schools
References
Further reading
Page 150 relates a brief account of Adventists on Tonga.
External links
Beulah College - Adventist Yearbook
This site gives a brief, yet comprehensive, history of Tongan SDA education
Educational institutions established in 1938
Schools in Tonga
1938 establishments in Tonga
Tongatapu
Secondary schools affiliated with the Seventh-day Adventist Church |
https://en.wikipedia.org/wiki/Welch%20bounds | In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory. The bounds were originally published in a 1974 paper by L. R. Welch.
Mathematical statement
If are unit vectors in , define , where is the usual inner product on . Then the following inequalities hold for :Welch bounds are also sometimes stated in terms of the averaged squared overlap between the set of vectors. In this case, one has the inequality
Applicability
If , then the vectors can form an orthonormal set in . In this case, and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if . This will be assumed throughout the remainder of this article.
Proof for k = 1
The "first Welch bound," corresponding to , is by far the most commonly used in applications. Its proof proceeds in two steps, each of which depends on a more basic mathematical inequality. The first step invokes the Cauchy–Schwarz inequality and begins by considering the Gram matrix of the vectors ; i.e.,
The trace of is equal to the sum of its eigenvalues. Because the rank of is at most , and it is a positive semidefinite matrix, has at most positive eigenvalues with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of as with and applying the Cauchy-Schwarz inequality to the inner product of an -vector of ones with a vector whose components are these eigenvalues yields
The square of the Frobenius norm (Hilbert–Schmidt norm) of satisfies
Taking this together with the preceding inequality gives
Because each has unit length, the elements on the main diagonal of are ones, and hence its trace is . So,
or
The second part of the proof uses an inequality encompassing the simple observation that the average of a set of non-negative numbers can be no greater than the largest number in the set. In mathematical notation, if for , then
The previous expression has non-negative terms in the sum, the largest of which is . So,
or
which is precisely the inequality given by Welch in the case that .
Achieving the Welch bounds
In certain telecommunications applications, it is desirable to construct sets of vectors that meet the Welch bounds with equality. Several techniques have been introduced to obtain so-called Welch Bound Equality (WBE) sets of vectors for the bound.
The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when . The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix are equal, which happens precisely when the vectors constitute a tight frame for .
The other inequality in |
https://en.wikipedia.org/wiki/Edson%20%28footballer%2C%20born%201962%29 | Edson Aparecido de Souza (born 29 November 1962), known as just Edson, is a former Brazilian football player.
Club statistics
References
External links
1962 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Japan Soccer League players
J1 League players
Japan Football League (1992–1998) players
Japan Football League players
Tokyo Verdy players
Shonan Bellmare players
Tokyo Musashino United FC players
Expatriate men's footballers in Japan
Men's association football midfielders
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Saskatoon%20metropolitan%20area | The Saskatoon region is the greater metropolitan area of Saskatoon, Saskatchewan. As of 2021 the Statistics Canada estimates the region's population to be 317,480 people.
The area is served by the Saskatoon John G. Diefenbaker International Airport, the 22nd busiest airport in the country.
Unlike many major North American urbanized areas, yet similarly to other prairie centers in Canada, Saskatoon has absorbed numerous neighbouring communities in its history. In the past, when the city limits reached the borders of neighbouring municipalities, such as Sutherland and Nutana, they were simply annexed into Saskatoon's jurisdiction. The vast majority of the region's inhabitants reside within the City of Saskatoon, which now has a population in excess of 270,000.
Geography
Census metropolitan area (CMA) is the term Statistics Canada uses to determine the demographics of greater Saskatoon (as well as other large Canadian cities). The Saskatoon CMA includes the City of Saskatoon, Rural Municipality of Corman Park No. 344, the cities of Martensville and Warman, and other smaller communities within the region.
According to Canada's 2021 census, the Saskatoon CMA has surpassed a quarter of a million people and is the 17th largest metropolitan area in the country with an estimated population of 317,480. It is also the largest CMA in Saskatchewan and has a land area of .
List of municipalities
Demographics
Ethnicity
Language
The question on knowledge of languages allows for multiple responses. The following figures are from the 2021 Canadian Census, and lists languages that were selected by at least 500 respondents.
Religion
Notes
References
External links
Statistics Canada
The City of Saskatoon
Saskatoon Regional Economic Development Authority
R.M. Of Corman Park
City Of Martensville
Metropolitan areas of Saskatchewan |
https://en.wikipedia.org/wiki/Number%20Theory%20Library | NTL is a C++ library for doing number theory. NTL supports arbitrary length integer and arbitrary precision floating point arithmetic, finite fields, vectors, matrices, polynomials, lattice basis reduction and basic linear algebra. NTL is free software released under the GNU Lesser General Public License v2.1.
References
External links
Official NTL website
C++ libraries
Free mathematics software
2015 software |
https://en.wikipedia.org/wiki/JsMath | jsMath was a JavaScript library for displaying mathematics in browsers in a cross-platform way. jsMath is free software released under the Apache License.
jsMath was succeeded by MathJax.
See also
MathJax
TeX and LaTeX, from which jsMath inherits its syntax and layout algorithms
MathML, a W3C standard enabling direct math rendering in the browser, using an XML syntax
ASCIIMathML, a client-side library for writing MathML in a subset of LaTeX math syntax
Google Chart API
References
External links
Free mathematics software
Free TeX software
JavaScript libraries |
https://en.wikipedia.org/wiki/V%C3%A1lber%20%28footballer%2C%20born%201971%29 | Válber da Silva Costa (born 6 December 1971), better known as just Válber, is a former Brazilian football player.
Club statistics
References
External links
1971 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Mogi Mirim Esporte Clube players
Sport Club Corinthians Paulista players
Sociedade Esportiva Palmeiras players
Sport Club Internacional players
CR Vasco da Gama players
Goiás Esporte Clube players
Associação Atlética Ponte Preta players
Club Athletico Paranaense players
Santa Cruz Futebol Clube players
Expatriate men's footballers in Japan
J1 League players
Yokohama Flügels players
Yokohama F. Marinos players
Men's association football forwards |
https://en.wikipedia.org/wiki/Cho%20Kwi-jae | Cho Kwi-jae (born 16 January 1969) is a former South Korean football player and manager He is the current manager J1 League club of Kyoto Sanga.
Club statistics
Managerial statistics
Honours
Managerial
Shonan Bellmare
J2 League: 2014, 2017
J.League Cup: 2018
References
External links
1969 births
Living people
Waseda University alumni
Association football people from Kyoto Prefecture
South Korean men's footballers
South Korean expatriate men's footballers
Japan Soccer League players
J1 League players
Japan Football League (1992–1998) players
Kashiwa Reysol players
Urawa Red Diamonds players
Vissel Kobe players
J1 League managers
J2 League managers
Shonan Bellmare managers
Kyoto Sanga FC managers
South Korean expatriate sportspeople in Japan
Expatriate men's footballers in Japan
Men's association football defenders
South Korean football managers
South Korean expatriate football managers |
https://en.wikipedia.org/wiki/Jeff%20Kahn | Jeffry Ned Kahn is a professor of mathematics at Rutgers University notable for his work in combinatorics.
Education
Kahn received his Ph.D. from Ohio State University in 1979 after completing his dissertation under his advisor Dijen K. Ray-Chaudhuri.
Research
In 1980 he showed the importance of the bundle theorem for ovoidal Möbius planes. In 1993, together with Gil Kalai, he disproved Borsuk's conjecture. In 1996 he was awarded the Pólya Prize (SIAM).
Awards and honors
He was an invited speaker at the 1994 International Congress of Mathematicians in Zurich.
In 2012, he was awarded the Fulkerson Prize (jointly with Anders Johansson and Van H. Vu) for determining the threshold of edge density above which a random graph can be covered by disjoint copies of a given smaller graph. Also in 2012, he became a fellow of the American Mathematical Society.
References
Living people
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Rutgers University faculty
Fellows of the American Mathematical Society
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Causal%20analysis | Causal analysis is the field of experimental design and statistics pertaining to establishing cause and effect. Typically it involves establishing four elements: correlation, sequence in time (that is, causes must occur before their proposed effect), a plausible physical or information-theoretical mechanism for an observed effect to follow from a possible cause, and eliminating the possibility of common and alternative ("special") causes. Such analysis usually involves one or more artificial or natural experiments.
Motivation
Data analysis is primarily concerned with causal questions. For example, did the fertilizer cause the crops to grow? Or, can a given sickness be prevented? Or, why is my friend depressed? The potential outcomes and regression analysis techniques handle such queries when data is collected using designed experiments. Data collected in observational studies require different techniques for causal inference (because, for example, of issues such as confounding). Causal inference techniques used with experimental data require additional assumptions to produce reasonable inferences with observation data. The difficulty of causal inference under such circumstances is often summed up as "correlation does not imply causation".
In philosophy and physics
The nature of causality is systematically investigated in several academic disciplines, including philosophy and physics.
In academia, there are a significant number of theories on causality; The Oxford Handbook of Causation encompasses 770 pages. Among the more influential theories within philosophy are Aristotle's Four causes and Al-Ghazali's occasionalism. David Hume argued that beliefs about causality are based on experience, and experience similarly based on the assumption that the future models the past, which in turn can only be based on experience – leading to circular logic. In conclusion, he asserted that causality is not based on actual reasoning: only correlation can actually be perceived. Immanuel Kant, according to , held that "a causal principle according to which every event has a cause, or follows according to a causal law, cannot be established through induction as a purely empirical claim, since it would then lack strict universality, or necessity".
Outside the field of philosophy, theories of causation can be identified in classical mechanics, statistical mechanics, quantum mechanics, spacetime theories, biology, social sciences, and law. To establish a correlation as causal within physics, it is normally understood that the cause and the effect must connect through a local mechanism (cf. for instance the concept of impact) or a nonlocal mechanism (cf. the concept of field), in accordance with known laws of nature.
From the point of view of thermodynamics, universal properties of causes as compared to effects have been identified through the Second law of thermodynamics, confirming the ancient, medieval and Cartesian view that "the cause is greater than the e |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20Alberta | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
At the 2021 Census of Population, Alberta had 311 designated places, an increase from 304 in 2011. Designated place types in Alberta include 18 dissolved municipalities, 10 Métis settlements, and 283 unincorporated places. In 2021, the 311 designated places had a cumulative population of 78,571 and an average population of 253. Alberta's largest designated place is Langdon with a population of 5,497.
List
Retired designated places
T & E Trailer Park, located within the City of Grande Prairie, was last recognized as a designated place in the 2006 Census of Canada.
See also
List of census agglomerations in Alberta
List of census divisions of Alberta
List of communities in Alberta
List of hamlets in Alberta
List of localities in Alberta
List of municipalities in Alberta
List of population centres in Alberta
Notes
References
Designated places |
https://en.wikipedia.org/wiki/Lam%20Lay%20Yong | Lam Lay Yong (maiden name Oon Lay Yong, ; born 1936) is a retired Professor of Mathematics.
Academic career
From 1988 to 1996 she was Professor at the Department of Mathematics, National University of Singapore (NUS). She graduated from the University of Malaya (later becoming University of Singapore) in 1957 and pursued graduate study in Cambridge University, obtaining her Ph.D. degree from University of Singapore in 1966, and becoming a lecturer at the University of Singapore. She was promoted to full professor in 1988, taught in NUS for 35 years, and retired in 1996.
From 1974 to 1990, Lam Lay Yong was the associate editor of Historia Mathematica. Lam was a member of Académie Internationale d'Histoire des Sciences.
In 2001, Lam Lay Yong was awarded the Kenneth O. May Prize jointly with Ubiratan D'Ambrosio. Lam was the first Asian and first woman to receive this award. Her reception speech was Ancient Chinese Mathematics and its influence on World Mathematics.
Lam Lay Yong also won the 2005 Outstanding Science Alumni Award from NUS. She is the granddaughter of Tan Kah Kee and niece of Lee Kong Chian.
Chinese origins of Hindu-Arabic Numerals Hypothesis
Lam Lay Yong has hypothesised that Hindu–Arabic numeral system originated in China based on her comparative studies on Chinese counting rods system. She states that the rod numerals and the hindu numerals have a few in common, that they're nine signs, concept of zero, a place value system, and decimal base. She claims that, "While no one knows how the Hindu-Arabic system originates in India, on the other hand, there is strong evidence of a transmission of the concept of the rod system to India." She even claims that there is no unquestionable evidence that the system originated in India, and that she claims that there are two factors concerning this. One was from mathematician's mention, for example a critique of Severus Sebokht on Indian ingenuity, and Al-Khwarizmi's book on Hindu Calculation. The other factor is the presence of Brahmi numerals.
However Michel Danino criticised this by saying that Lam Lay Yong's evidence for this was not at all evidence-based nor rigorous, and that she is ill-qualified for crosscultural studies. According to Michael Danino Her thesis has not been accepted, thus, the Chinese origin of Hindu-Arabic numerals remains to be hypothetical, and not widely accepted at all. All of this seems to contradict Yong's claims that there is strong evidence of rod numerals in India.
Publication
Jiu Zhang Suanshu (1994) "(Nine Chapters on the Mathematical Art): An Overview, Archive for History of Exact Sciences, vol. 47: pp. 1–51.
Zhang Qiujian Suanjing (1997) "(The Mathematical Classic of Zhang Qiujian): An Overview", Archive for History of Exact Sciences, vol. 50: pp. 201–240.
Lam Lay Yong, Ang Tian Se (2004) Fleeting Footsteps. Tracing the Conception of Arithmetic and Algebra in Ancient China, Revised Edition, World Scientific, Singapore.
Lam Lay Yong (1977) A Critical |
https://en.wikipedia.org/wiki/George%20David%20Birkhoff%20Prize | The George David Birkhoff Prize in applied mathematics is awarded – jointly by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM) – in honour of George David Birkhoff (1884–1944). It is currently awarded every three years for an outstanding contribution to: "applied mathematics in the highest and broadest sense". The recipient of the prize has to be a member of one of the awarding societies, as well as a resident of the United States of America, Canada or Mexico. The prize was endowed in 1967, first issued in 1968, and currently (2020) amounts to US$5,000.
Recipients
See also
List of mathematics awards
Prizes named after people
Notes
Awards established in 1968
Awards of the American Mathematical Society
Awards of the Society for Industrial and Applied Mathematics
Triennial events
Applied mathematics
North American awards
1968 establishments in North America |
https://en.wikipedia.org/wiki/Method%20of%20conditional%20probabilities | In mathematics and computer science, the method of conditional probabilities is a systematic method for converting non-constructive probabilistic existence proofs into efficient deterministic algorithms that explicitly construct the desired object.
Often, the probabilistic method is used to prove the existence of mathematical objects with some desired combinatorial properties. The proofs in that method work by showing that a random object, chosen from some probability distribution, has the desired properties with positive probability. Consequently, they are nonconstructive — they don't explicitly describe an efficient method for computing the desired objects.
The method fo conditional probabilities converts such a proof, in a "very precise sense", into an efficient deterministic algorithm, one that is guaranteed to compute an object with the desired properties. That is, the method derandomizes the proof. The basic idea is to replace each random choice in a random experiment by a deterministic choice, so as to keep the conditional probability of failure, given the choices so far, below 1.
The method is particularly relevant in the context of randomized rounding (which uses the probabilistic method to design approximation algorithms).
When applying the method of conditional probabilities, the technical term pessimistic estimator refers to a quantity used in place of the true conditional probability (or conditional expectation) underlying the proof.
Overview
Raghavan gives this description:
We first show the existence of a provably good approximate solution using the probabilistic method... [We then] show that the probabilistic existence proof can be converted, in a very precise sense, into a deterministic approximation algorithm.
Raghavan is discussing the method in the context of randomized rounding, but it works with the probabilistic method in general.
To apply the method to a probabilistic proof, the randomly chosen object in the proof must be choosable by a random experiment that consists of a sequence of "small" random choices.
Here is a trivial example to illustrate the principle.
Lemma: It is possible to flip three coins so that the number of tails is at least 2.
Probabilistic proof. If the three coins are flipped randomly, the expected number of tails is 1.5. Thus, there must be some outcome (way of flipping the coins) so that the number of tails is at least 1.5. Since the number of tails is an integer, in such an outcome there are at least 2 tails. QED
In this example the random experiment consists of flipping three fair coins. The experiment is illustrated by the rooted tree in the adjacent diagram. There are eight outcomes, each corresponding to a leaf in the tree. A trial of the random experiment corresponds to taking a random walk from the root (the top node in the tree, where no coins have been flipped) to a leaf. The successful outcomes are those in which at least two coins came up tails. The interior nodes in the |
https://en.wikipedia.org/wiki/Uncertainty%20coefficient | In statistics, the uncertainty coefficient, also called proficiency, entropy coefficient or Theil's U, is a measure of nominal association. It was first introduced by Henri Theil and is based on the concept of information entropy.
Definition
Suppose we have samples of two discrete random variables, X and Y. By constructing the joint distribution, , from which we can calculate the conditional distributions, and , and calculating the various entropies, we can determine the degree of association between the two variables.
The entropy of a single distribution is given as:
while the conditional entropy is given as:
The uncertainty coefficient or proficiency is defined as:
and tells us: given Y, what fraction of the bits of X can we predict? In this case we can think of X as containing the total information, and of Y as allowing one to predict part of such information.
The above expression makes clear that the uncertainty coefficient is a normalised mutual information I(X;Y). In particular, the uncertainty coefficient ranges in [0, 1] as I(X;Y) < H(X) and both I(X,Y) and H(X) are positive or null.
Note that the value of U (but not H!) is independent of the base of the log since all logarithms are proportional.
The uncertainty coefficient is useful for measuring the validity of a statistical classification algorithm and has the advantage over simpler accuracy measures such as precision and recall in that it is not affected by the relative fractions of the different classes, i.e., P(x).
It also has the unique property that it won't penalize an algorithm for predicting the wrong classes, so long as it does so consistently (i.e., it simply rearranges the classes). This is useful in evaluating clustering algorithms since cluster labels typically have no particular ordering.
Variations
The uncertainty coefficient is not symmetric with respect to the roles of X and Y. The roles can be reversed and a symmetrical measure thus defined as a weighted average between the two:
Although normally applied to discrete variables, the uncertainty coefficient can be extended to continuous variables using density estimation.
See also
Mutual information
Rand index
F-score
Binary classification
References
External links
libagf Includes software for calculating uncertainty coefficients.
Statistical ratios
Summary statistics for contingency tables
Information theory
Statistics articles needing expert attention |
https://en.wikipedia.org/wiki/Steve%20Awodey | Steven M. Awodey (; born 1959) is an American mathematician and logician. He is a Professor of Philosophy and Mathematics at Carnegie Mellon University.
Biography
Awodey studied mathematics and philosophy at the University of Marburg and the University of Chicago. He earned his Ph.D. from Chicago under Saunders Mac Lane in 1997. He is an active researcher in the areas of category theory and logic, and has also written on the philosophy of mathematics. He is one of the originators of the field of homotopy type theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2012–13.
Bibliography
References
External links
American logicians
20th-century American mathematicians
21st-century American mathematicians
Philosophers of mathematics
Living people
University of Marburg alumni
University of Chicago alumni
Institute for Advanced Study visiting scholars
Carnegie Mellon University faculty
1959 births |
https://en.wikipedia.org/wiki/Andrei%20Knyazev%20%28mathematician%29 | Andrew Knyazev is an American mathematician. He graduated from the Faculty of Computational Mathematics and Cybernetics of Moscow State University under the supervision of Evgenii Georgievich D'yakonov () in 1981 and obtained his PhD in Numerical Mathematics at the Russian Academy of Sciences under the supervision of Vyacheslav Ivanovich Lebedev () in 1985. He worked at the Kurchatov Institute between 1981–1983, and then to 1992 at the Marchuk Institute of Numerical Mathematics () of the Russian Academy of Sciences, headed by Gury Marchuk ().
From 1993–1994, Knyazev held a visiting position at the Courant Institute of Mathematical Sciences of New York University, collaborating with Olof B. Widlund. From 1994 until retirement in 2014, he was a Professor of Mathematics at the University of Colorado Denver, supported by the National Science Foundation and United States Department of Energy grants. He was a recipient of the 2008 Excellence in Research Award, the 2000 college Teaching Excellence Award, and a finalist of the CU President's Faculty Excellence Award for Advancing Teaching and Learning through Technology in 1999.
He was awarded the title of Professor Emeritus at the University of Colorado Denver
and named the SIAM Fellow Class of 2016
and
AMS Fellow Class of 2019.
From 2012–2018, Knyazev worked at the Mitsubishi Electric Research Laboratories on algorithms for image and video processing, data sciences, optimal control, and material sciences, resulting in dozens of publications and 13 patent applications. Since 2018, he contributed to numerical techniques in quantum computing at Zapata Computing, real-time embedded anomaly detection in automotive data, and algorithms for silicon photonics-based hardware.
Knyazev is mostly known for his work in numerical solution of large sparse eigenvalue problems, particularly preconditioning and the iterative method LOBPCG. Knyazev's implementation of LOBPCG is available in many open source software packages, e.g., BLOPEX, SciPy, and ABINIT.
Knyazev collaborated with John Osborn
on the theory of the Ritz method in the finite element method context
and with Nikolai Sergeevich Bakhvalov () (Erdős number 3 via Leonid Kantorovich) on numerical solution of elliptic partial differential equations with large jumps in the main coefficients.
Jointly with his Ph.D. students, Knyazev pioneered using majorization for bounds in the Rayleigh–Ritz method
(see and references there) and contributed to the theory of angles between flats.
References
External links
Patents granted to Andrei Kniazev and patent applications filed by Andrei Kniazev at USPTO and world-wide
MathSciNet (subscription required) reviews for Andrew Knyazev
Zentralblatt MATH (subscription required) reviews
arXiv Reports
SIGPORT Contributions
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) at GitHub
Knyazev's software in MATLAB
Andrew Knyazev on ResearchGate
20th-century American mathematicians
21st-century American |
https://en.wikipedia.org/wiki/Pochhammer%20contour | In mathematics, the Pochhammer contour, introduced by and , is a contour in the complex plane with two points removed, used for contour integration. If A and B are loops around the two points, both starting at some fixed point P, then the Pochhammer contour is the commutator ABA−1B−1, where the superscript −1 denotes a path taken in the opposite direction. With the two points taken as 0 and 1, the fixed basepoint P being on the real axis between them, an example is the path that starts at P, encircles the point 1 in the counter-clockwise direction and returns to P, then encircles 0 counter-clockwise and returns to P, after that circling 1 and then 0 clockwise, before coming back to P. The class of the contour is an actual commutator when it is considered in the fundamental group with basepoint P of the complement in the complex plane (or Riemann sphere) of the two points looped. When it comes to taking contour integrals, moving basepoint from P to another choice Q makes no difference to the result, since there will be cancellation of integrals from P to Q and back.
Homologous to zero but not homotopic to zero
Within the doubly punctured plane this curve is homologous to zero but not homotopic to zero. Its winding number about any point is 0 despite the fact that within the doubly punctured plane it cannot be shrunk to a single point.
Applications
The beta function is given by Euler's integral
provided that the real parts of α and β are positive, which may be converted into an integral over the Pochhammer contour C as
The contour integral converges for all values of α and β and so gives the analytic continuation of the beta function. A similar method can be applied to Euler's integral for the hypergeometric function to give its analytic continuation.
Notes
References
Special functions |
https://en.wikipedia.org/wiki/Cl%C3%A1udio%20%28footballer%2C%20born%201972%29 | Cláudio Luís Assunção de Freitas (born 31 March 1972) is a former Brazilian football player.
Club statistics
References
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Cerezo Osaka
1972 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
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CR Flamengo footballers
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Expatriate men's footballers in Japan
J1 League players
Shonan Bellmare players
Cerezo Osaka players
Men's association football defenders
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Pericles%20%28footballer%2C%20born%201975%29 | Pericles de Oliveira Ramos (born 2 January 1975) is a former Brazilian football player.
Club statistics
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1975 births
Living people
Brazilian men's footballers
J1 League players
J2 League players
Japan Football League players
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Sagan Tosu players
Gainare Tottori players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Gar%C3%A7a%20%28footballer%29 | Edson Rodrigues (born March 13, 1967) is a former Brazilian football player. He has played for Nagoya Grampus Eight.
Club statistics
References
External links
1967 births
Living people
Brazilian men's footballers
J1 League players
Nagoya Grampus players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Fernando%20%28footballer%2C%20born%201967%29 | Fernando Henrique Mariano (born 3 April 1967) is a Brazilian football player.
Club statistics
References
External links
1967 births
Living people
Brazilian men's footballers
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Esporte Clube Santo André players
Marília Atlético Clube players
Botafogo de Futebol e Regatas players
Esporte Clube Juventude players
Sociedade Esportiva Palmeiras players
Sport Club Internacional players
Guarani FC players
Associação Portuguesa de Desportos players
Mogi Mirim Esporte Clube players
Uberlândia Esporte Clube players
Avispa Fukuoka players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Luiz%20Carlos%20%28footballer%2C%20born%201971%29 | Luiz Carlos Guarnieri (born August 13, 1971) is a former Brazilian football player.
Club statistics
References
External links
kyotosangadc
1971 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J1 League players
Japan Football League (1992–1998) players
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Clube Atlético Mineiro players
Coritiba Foot Ball Club players
Santa Cruz Futebol Clube players
Kyoto Sanga FC players
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Jos%C3%A9%20Reginaldo%20Vital | Jose Reginaldo Vital (born 29 February 1976) is a former Brazilian football player.
Club statistics
2012
Atualmente joga futebol amador no Inter de Campo Largo
References
External links
1976 births
Living people
Brazilian men's footballers
Campeonato Brasileiro Série A players
J1 League players
J2 League players
Gamba Osaka players
Hokkaido Consadole Sapporo players
Club Athletico Paranaense players
Paraná Clube players
Associação Atlética Ponte Preta players
Coritiba Foot Ball Club players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football midfielders |
https://en.wikipedia.org/wiki/Carlos%20Alberto%20Dias | Carlos Alberto Costa Dias (born 5 May 1967) is a former Brazilian football player. He has played for Brazil's national team.
Club statistics
National team statistics
References
External links
1967 births
Living people
Brazilian men's footballers
Brazilian football managers
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Men's association football forwards
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CR Vasco da Gama players
Grêmio Foot-Ball Porto Alegrense players
Paraná Clube players
Japan Soccer League players
J1 League players
Shonan Bellmare players
Shimizu S-Pulse players
Tokyo Verdy players
Expatriate men's footballers in Japan
Sociedade Esportiva do Gama managers
Brazil men's international footballers
Footballers from Brasília |
https://en.wikipedia.org/wiki/Projective%20polyhedron | In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.
Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, a synonym for "spherical polyhedron". However, the term elliptic geometry applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.
As cellular decompositions of the projective plane, they have Euler characteristic 1, while spherical polyhedra have Euler characteristic 2. The qualifier "globally" is to contrast with locally projective polyhedra, which are defined in the theory of abstract polyhedra.
Non-overlapping projective polyhedra (density 1) correspond to spherical polyhedra (equivalently, convex polyhedra) with central symmetry. This is elaborated and extended below in relation with spherical polyhedra and relation with traditional polyhedra.
Examples
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra:
Hemi-cube, {4,3}/2
Hemi-octahedron, {3,4}/2
Hemi-dodecahedron, {5,3}/2
Hemi-icosahedron, {3,5}/2
Hemi-dihedron, {2p,2}/2, p>=1
Hemi-hosohedron, {2,2p}/2, p>=1
These can be obtained by taking the quotient of the associated spherical polyhedron by the antipodal map (identifying opposite points on the sphere).
On the other hand, the tetrahedron does not have central symmetry, so there is no "hemi-tetrahedron". See relation with spherical polyhedra below on how the tetrahedron is treated.
Hemipolyhedra
Note that the prefix "hemi-" is also used to refer to hemipolyhedra, which are uniform polyhedra having some faces that pass through the center of symmetry. As these do not define spherical polyhedra (because they pass through the center, which does not map to a defined point on the sphere), they do not define projective polyhedra by the quotient map from 3-space (minus the origin) to the projective plane.
Of these uniform hemipolyhedra, only the tetrahemihexahedron is topologically a projective polyhedron, as can be verified by its Euler characteristic and visually obvious connection to the Roman surface. It is 2-covered by the cuboctahedron, and can be realized as the quotient of the spherical cuboctahedron by the antipodal map. It is the only uniform (traditional) polyhedron that is projective – that is, the only uniform projective polyhedron that immerses in Euclidean three-space as a uniform traditional polyhedron.
Relation with spherical polyhedra
There is a 2-to-1 covering map of the sphere to the projective plane, and under this map, projective polyhedra correspond to spherical polyhedra with central sym |
https://en.wikipedia.org/wiki/Gammoid | In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph.
The concept of a gammoid was introduced and shown to be a matroid by , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths. Gammoids were given their name by and studied in more detail by .
Definition
Let be a directed graph, be a set of starting vertices, and be a set of destination vertices (not necessarily disjoint from ). The gammoid derived from this data has as its set of elements. A subset of is independent in if there exists a set of vertex-disjoint paths whose starting points all belong to and whose ending points are exactly .
A strict gammoid is a gammoid in which the set of destination vertices consists of every vertex in . Thus, a gammoid is a restriction of a strict gammoid, to a subset of its elements.
Example
Consider the uniform matroid on a set of elements, in which every set of or fewer elements is independent. One way to represent this matroid as a gammoid would be to form a complete bipartite graph with a set of vertices on one side of the bipartition, with a set of vertices on the other side of the bipartition, and with every edge directed from to In this graph, a subset of is the set of endpoints of a set of disjoint paths if and only if it has or fewer vertices, for otherwise there aren't enough vertices in to start the paths. The special structure of this graph shows that the uniform matroid is a transversal matroid as well as being a gammoid.
Alternatively, the same uniform matroid may be represented as a gammoid on a smaller graph, with only vertices, by choosing a subset of vertices and connecting each of the chosen vertices to every other vertex in the graph. Again, a subset of the vertices of the graph can be endpoints of disjoint paths if and only if it has or fewer vertices, because otherwise there are not enough vertices that can be starts of paths. In this graph, every vertex corresponds to an element of the matroid, showing that the uniform matroid is a strict gammoid.
Menger's theorem and gammoid rank
The rank of a set in a gammoid defined from a graph and vertex subsets and is, by definition, the maximum number of vertex-disjoint paths from to . By Menger's theorem, it also equals the minimum cardinality of a set that intersects every path from to .
Relation to transversal matroids
A transversal matroid is defined from a family of sets: its elements are the elements of the sets, and a set of these elements is independent whenever there exists a one-to-one matching of the elements of to disjoint sets containing them, called a system of distinct representatives. Equivalently, a transversal matroid may be represented by a special kind of gammoid, defined from a directed bipartite graph that has a vertex in for each s |
https://en.wikipedia.org/wiki/Bas%C3%ADlio%20%28footballer%29 | Valdeci Basílio da Silva (born 14 July 1972) is a Brazilian former football player.
Club statistics
References
External links
1972 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Coritiba Foot Ball Club players
Sociedade Esportiva Palmeiras players
Associação Atlética Ponte Preta players
Grêmio Foot-Ball Porto Alegrense players
Santos FC players
Itumbiara Esporte Clube players
Kashiwa Reysol players
Tokyo Verdy players
Expatriate men's footballers in Japan
J1 League players
J2 League players
Associação Atlética Internacional (Bebedouro) players
Men's association football forwards
People from Andradina
Footballers from São Paulo (state) |
https://en.wikipedia.org/wiki/Marcelo%20Trivisonno | Marcelo Trivisonno (born 8 June 1966) is a former Argentine football player.
Club statistics
References
External links
1966 births
Living people
Argentine men's footballers
Argentine expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
Rosario Central footballers
San Martín de San Juan footballers
Club Atlético Douglas Haig players
Atlético Tucumán footballers
Tiro Federal footballers
Club Olimpo footballers
Urawa Red Diamonds players
Men's association football defenders |
https://en.wikipedia.org/wiki/Moacir%20%28footballer%2C%20born%201970%29 | Moacir Rodrigues Santos (born 21 March 1970), known as Moacir, is a former Brazilian football player. He has played for Brazil national team.
Club statistics
National team statistics
References
External links
1970 births
Living people
Brazilian men's footballers
Brazil men's international footballers
Brazil men's under-20 international footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
La Liga players
J1 League players
Clube Atlético Mineiro players
Sport Club Corinthians Paulista players
Atlético Madrid footballers
Sevilla FC players
Associação Portuguesa de Desportos players
Ituano FC players
Uberaba Sport Club players
Tokyo Verdy players
Expatriate men's footballers in Spain
Expatriate men's footballers in Japan
Men's association football midfielders
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Edinho%20Baiano | Édson Manoel do Nascimento (born 27 July 1967), better known as Edinho Baiano, is a former Brazilian footballer.
Club statistics
References
External links
kyotosangadc
1967 births
Living people
Brazilian men's footballers
J1 League players
Kyoto Sanga FC players
Esporte Clube Vitória players
Joinville Esporte Clube players
Sociedade Esportiva Palmeiras players
Paraná Clube players
Club Athletico Paranaense players
Londrina Esporte Clube players
Coritiba Foot Ball Club players
Avaí FC players
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Alair%20%28footballer%29 | Alair de Souza Camargo Júnior, or simply Alair (born 27 January 1982) is a Brazilian football player. He recently plays for Ehime FC in the J2 League.
Club statistics
References
External links
1982 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
J2 League players
Shimizu S-Pulse players
Ventforet Kofu players
Ehime FC players
Kyoto Sanga FC players
Men's association football defenders |
https://en.wikipedia.org/wiki/Alessandro%20Cambalhota | Alessandro Andrade de Oliveira (born 27 May 1973), known as Alessandro Cambalhota, is a former Brazilian football player. He has played for the Brazil national team once.
Club statistics
National team statistics
Honours
Santos
Copa CONMEBOL: 1998
Porto
Taça de Portugal: 1999–2000
Supertaça Cândido de Oliveira: 1999
References
External links
1973 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
Primeira Liga players
Süper Lig players
J1 League players
CR Vasco da Gama players
Santos FC players
Fluminense FC players
Cruzeiro Esporte Clube players
Clube Atlético Mineiro players
Sport Club Corinthians Paulista players
Figueirense FC players
Esporte Clube Noroeste players
Clube Atlético Linense players
Júbilo Iwata players
Kayseri Erciyesspor footballers
Denizlispor footballers
FC Porto players
Al-Ahli Saudi FC players
Expatriate men's footballers in Japan
Expatriate men's footballers in Turkey
Expatriate men's footballers in Portugal
Men's association football forwards
Brazil men's international footballers
People from Teixeira de Freitas
Footballers from Bahia |
https://en.wikipedia.org/wiki/Rodrigo%20Batata | Rodrigo Pinheiro da Silva (born 10 September 1977), known as Rodrigo Batata, is a former Brazilian football player.
Club statistics
References
External links
(in Japanese)
1977 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J1 League players
Paraná Clube players
Yokohama Flügels players
Portimonense S.C. players
Coritiba Foot Ball Club players
Paulista Futebol Clube players
J. Malucelli Futebol players
Club Puebla players
Expatriate men's footballers in France
Expatriate men's footballers in Portugal
Expatriate men's footballers in Japan
Expatriate men's footballers in Mexico
Men's association football midfielders
Footballers from Curitiba |
https://en.wikipedia.org/wiki/Kenta%20Suzuki | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Japan Football League players
Ventforet Kofu players
SC Sagamihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Schwarz%27s%20list | In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles.
The wider importance of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that cases of finite monodromy for such equations and regular singularities could be attributed to changes of variable (complex analytic mappings of the Riemann sphere to itself) that reduce the equation to hypergeometric form. In fact more is true: Schwarz's list underlies all second-order equations with regular singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation's data.
The numbers are (up to permutations, sign changes and addition of with even) the differences of the exponents of the hypergeometric differential equation at the three singular points . They are rational numbers if and only if and are, a point that matters in arithmetic rather than geometric approaches to the theory.
Further work
An extension of Schwarz's results was given by T. Kimura, who dealt with cases where the identity component of the differential Galois group of the hypergeometric equation is a solvable group. A general result connecting the differential Galois group G and the monodromy group Γ states that G is the Zariski closure of Γ — this theorem is attributed in the book of Matsuda to Michio Kuga. By general differential Galois theory, the resulting Kimura-Schwarz table classifies cases of integrability of the equation by algebraic functions and quadratures.
Another relevant list is that of K. Takeuchi, who classified the (hyperbolic) triangle groups that are arithmetic groups (85 examples).
Émile Picard sought to extend the work of Schwarz in complex geometry, by means of a generalized hypergeometric function, to construct cases of equations where the monodromy was a discrete group in the projective unitary group PU(1, n). Pierre Deligne and George Mostow used his ideas to construct lattices in the projective unitary group. This work recovers in the classical case the finiteness of Takeuchi's list, and by means of a characterisation of the lattices they construct that are arithmetic gr |
https://en.wikipedia.org/wiki/Derivation%20of%20the%20conjugate%20gradient%20method | In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system
where is symmetric positive-definite. The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction method for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems.
The intent of this article is to document the important steps in these derivations.
Derivation from the conjugate direction method
The conjugate gradient method can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function
The conjugate direction method
In the conjugate direction method for minimizing
one starts with an initial guess and the corresponding residual , and computes the iterate and residual by the formulae
where are a series of mutually conjugate directions, i.e.,
for any .
The conjugate direction method is imprecise in the sense that no formulae are given for selection of the directions . Specific choices lead to various methods including the conjugate gradient method and Gaussian elimination.
Derivation from the Arnoldi/Lanczos iteration
The conjugate gradient method can also be seen as a variant of the Arnoldi/Lanczos iteration applied to solving linear systems.
The general Arnoldi method
In the Arnoldi iteration, one starts with a vector and gradually builds an orthonormal basis of the Krylov subspace
by defining where
In other words, for , is found by Gram-Schmidt orthogonalizing against followed by normalization.
Put in matrix form, the iteration is captured by the equation
where
with
When applying the Arnoldi iteration to solving linear systems, one starts with , the residual corresponding to an initial guess . After each step of iteration, one computes and the new iterate .
The direct Lanczos method
For the rest of discussion, we assume that is symmetric positive-definite. With symmetry of , the upper Hessenberg matrix becomes symmetric and thus tridiagonal. It then can be more clearly denoted by
This enables a short three-term recurrence for in the iteration, and the Arnoldi iteration is reduced to the Lanczos iteration.
Since is symmetric positive-definite, so is . Hence, can be LU factorized without partial pivoting into
with convenient recurrences for and :
Rewrite as
with
It is now important to observe that
In fact, there are short recurrences for and as well:
With this formulation, we arrive at a simple recurrence for :
The relations above straightforwardly lead to the direct Lanczos method, which turns out to be slightly more complex.
The conjugate gradient method from imposing orthogonality and conjugacy
If we allow to scale and compensate for the scaling in the constant factor, we potentially can have simpler recurrences of the form:
As premises for the simplification, we now derive the orthogonality of and conjugacy of , i.e., fo |
https://en.wikipedia.org/wiki/British%20Annals%20of%20Medicine%2C%20Pharmacy%2C%20Vital%20Statistics%2C%20and%20General%20Science | The British Annals of Medicine, Pharmacy, Vital Statistics, and General Science was a weekly publication edited by William Farr that ran from only January to August 1837. Although short-lived, it was succeeded by Farr's other journals and was extremely influential in the development of vital statistics.
References
Weekly journals
Publications established in 1837
Publications disestablished in 1837
English-language journals
General medical journals |
https://en.wikipedia.org/wiki/Caroline%20Wozniacki%20career%20statistics | This is a list of the main career statistics of Danish tennis player Caroline Wozniacki. She won 30 singles titles including a Grand Slam title, a WTA Finals title, three Premier Mandatory titles and three Premier 5 titles. She was the winner of the 2018 Australian Open and the 2017 WTA Finals, and the runner-up at the 2009 US Open, the 2010 WTA Tour Championships, and the 2014 US Open. She also reached another 25 singles finals, and won two doubles titles. Wozniacki was first ranked world No. 1 by the WTA on October 11, 2010.
Performance timelines
Only main-draw results in WTA Tour, Grand Slam tournaments and Olympic Games are included in win–loss records.SinglesCurrent through the 2023 US Open.Doubles
TeamLevels of Fed Cup in which Denmark did not compete in a particular year are marked "not participating" or "NP".Significant finals
Grand Slam finals
Singles: 3 (1 title, 2 runner-ups)
WTA Finals finals
Singles: 2 (1 title, 1 runner-up)
WTA Premier Mandatory & 5 finals
Singles: 12 (6 titles, 6 runner-ups)
WTA career finals
Singles: 55 (30 titles, 25 runner-ups)
Doubles: 4 (2 titles, 2 runners-up)
ITF Circuit finals
Singles: 6 (4 titles, 2 runner–ups)
Junior Grand Slam finals
Singles: 2 (1 title, 1 runner-up)
Fed Cup participation
Singles: 22 (17–5)
Doubles: 9 (3–6)
WTA Tour career earnings
Wozniacki earned more than 35 million dollars during her career.
Career Grand Slam statistics
Career Grand Slam seedings
The tournaments won by Wozniacki are in boldface, and advanced into finals by Wozniacki are in italics.
Best Grand Slam tournament results details
Grand Slam winners are in boldface, and runner–ups are in italics Australian Open2018 Australian Open (2nd seed)RoundOpponentRankScore1R Mihaela Buzărnescu446–2, 6–32R Jana Fett1193–6, 6–2, 7–53R Kiki Bertens (30)326–4, 6–34R Magdaléna Rybáriková (19)216–3, 6–0QF Carla Suárez Navarro396–0, 6–7(3–7), 6–2SF Elise Mertens376–3, 7–6(7–2)W Simona Halep (1)17–6(7–2), 3–6, 6–4French Open2010 French Open (3rd seed)RoundOpponentRankScore1R Alla Kudryavtseva786–0, 6–32R Tathiana Garbin566–3, 6–13R Alexandra Dulgheru (31)326–3, 6–44R Flavia Pennetta (14)157–6(7–5), 6–7(4–7), 6–2QF Francesca Schiavone (17)172–6, 3–62017 French Open (11th seed)RoundOpponentRankScore1R Jaimee Fourlis (WC)3376–4, 3–6, 6–22R Françoise Abanda (Q)1956–0, 6–03R CiCi Bellis 486–2, 2–6, 6–34R Svetlana Kuznetsova (8)96–1, 4–6, 6–2QF Jeļena Ostapenko476–4, 2–6, 2–6 Wimbledon Championships2009 Wimbledon (9th seed)RoundOpponentRankScore1R Kimiko Date (WC)1425–7, 6–3, 6–12R Maria Kirilenko596–0, 6–43R Anabel Medina Garrigues (20)206–2, 6–24R Sabine Lisicki414–6, 4–62010 Wimbledon (3rd seed)RoundOpponentRankScore1R Tathiana Garbin536–1, 6–12R Chang Kai-chen896–4, 6–33R Anastasia Pavlyuchenkova (29)327–5, 6–44R Petra Kvitová622–6, 0–62011 Wimbledon (1st seed)RoundOpponentRankScore1R Arantxa Parra Santonja1056–2, 6–12R Virginie Razzano966–1, 6–33R Jarmila Gajdošová (27)286–3, 6–24R Dominika Cibulková (24)246–1, |
https://en.wikipedia.org/wiki/Heyting%20field | A Heyting field is one of the inequivalent ways in constructive mathematics to capture the classical notion of a field. It is essentially a field with an apartness relation.
Definition
A commutative ring is a Heyting field if ¬, either or is invertible for every , and each noninvertible element is zero. The first two conditions say that the ring is local; the first and third conditions say that it is a field in the classical sense.
The apartness relation is defined by writing # if is invertible. This relation is often now written as with the warning that it is not equivalent to ¬. For example, the assumption ¬ is not generally sufficient to construct the inverse of , but is.
Example
The prototypical Heyting field is the real numbers.
See also
Constructive analysis
Pseudo-order
References
Mines, Richman, Ruitenberg. A Course in Constructive Algebra. Springer, 1987.
Constructivism (mathematics) |
https://en.wikipedia.org/wiki/Jean%20Kuntzmann | Jean Kuntzmann (1 June 1912 – 18 December 1992) was a French mathematician, known for his works in applied mathematics and computer science, pushing and developing both fields at a very early time.
Kuntzmann earned his Ph.D. in mathematics from the University of Paris under supervision of Georges Valiron (thesis: ).
In 1960, he established the École nationale supérieure d'informatique et de mathématiques appliquées de Grenoble.
References
.
External links
Jean Kuntzmann (1912-1992) Un extraordinaire pionnier
1912 births
1992 deaths
20th-century French mathematicians
French computer scientists
University of Paris alumni
Academic staff of Grenoble Alpes University |
https://en.wikipedia.org/wiki/Villebois%2C%20Quebec | Villebois is an unconstituted locality within the municipality of Baie-James in the Nord-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Villebois had a population of 173 living in 77 of its 85 total private dwellings, a change of from its 2016 population of 157. With a land area of , it had a population density of in 2021.
References
Communities in Nord-du-Québec
Designated places in Quebec
Unconstituted localities in Quebec |
https://en.wikipedia.org/wiki/Val-Paradis%2C%20Quebec | Val-Paradis is an unconstituted locality within the municipality of Baie-James in the Nord-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Val-Paradis had a population of 186 living in 79 of its 84 total private dwellings, a change of from its 2016 population of 155. With a land area of , it had a population density of in 2021.
References
Communities in Nord-du-Québec
Designated places in Quebec
Unconstituted localities in Quebec |
https://en.wikipedia.org/wiki/Beaucanton%2C%20Quebec | Beaucanton is an unconstituted locality within the municipality of Baie-James in the Nord-du-Québec region of Quebec, Canada.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Beaucanton had a population of 153 living in 73 of its 79 total private dwellings, a change of from its 2016 population of 152. With a land area of , it had a population density of in 2021.
References
Communities in Nord-du-Québec
Designated places in Quebec
Unconstituted localities in Quebec |
https://en.wikipedia.org/wiki/Auerbach%27s%20lemma | In mathematics, Auerbach's lemma, named after Herman Auerbach, is a theorem in functional analysis which asserts that a certain property of Euclidean spaces holds for general finite-dimensional normed vector spaces.
Statement
Let (V, ||·||) be an n-dimensional normed vector space. Then there exists a basis {e1, ..., en} of V such that
||ei|| = 1 and ||ei|| = 1 for i = 1, ..., n,
where {e1, ..., en} is a basis of V* dual to {e1, ..., en}, i. e. ei(ej) = δij.
A basis with this property is called an Auerbach basis.
If V is an inner product space (or even infinite-dimensional Hilbert space) then this result is obvious as one may take for {ei} any orthonormal basis of V (the dual basis is then {(ei|·)}).
Geometric formulation
An equivalent statement is the following: any centrally symmetric convex body in has a linear image which contains the unit cross-polytope (the unit ball for the norm) and is contained in the unit cube (the unit ball for the norm).
Corollary
The lemma has a corollary with implications to approximation theory.
Let V be an n-dimensional subspace of a normed vector space (X, ||·||). Then there exists a projection P of X onto V such that ||P|| ≤ n.
Proof
Let {e1, ..., en} be an Auerbach basis of V and {e1, ..., en} corresponding dual basis. By Hahn–Banach theorem each ei extends to f i ∈ X* such that
||f i|| = 1.
Now set
P(x) = Σ f i(x) ei.
It's easy to check that P is indeed a projection onto V and that ||P|| ≤ n (this follows from triangle inequality).
References
Joseph Diestel, Hans Jarchow, Andrew Tonge, Absolutely Summing Operators, p. 146.
Joram Lindenstrauss, Lior Tzafriri, Classical Banach Spaces I and II: Sequence Spaces; Function Spaces, Springer 1996, , p. 16.
Reinhold Meise, Dietmar Vogt, Einführung in die Funktionalanalysis, Vieweg, Braunschweig 1992, .
Przemysław Wojtaszczyk, Banach spaces for analysts. Cambridge Studies in Advancod Mathematics, Cambridge University Press, vol. 25, 1991, p. 75.
Banach spaces
Lemmas in analysis |
https://en.wikipedia.org/wiki/Khabibullin%27s%20conjecture%20on%20integral%20inequalities | Khabibullin's conjecture is a conjecture in mathematics related to Paley's problem for plurisubharmonic functions and to various extremal problems in the theory of entire functions of several variables. The conjecture was named after its proposer, B. N. Khabibullin.
There are three versions of the conjecture, one in terms of logarithmically convex functions, one in terms of increasing functions, and one in terms of non-negative functions. The conjecture has implications in the study of complex functions and is related to Euler's Beta function. While the conjecture is known to hold for certain conditions, counterexamples have also been found.
The first statement in terms of logarithmically convex functions
Khabibullin's conjecture (version 1, 1992). Let be a non-negative increasing function on the half-line such that . Assume that is a convex function of . Let , , and . If
then
This statement of the Khabibullin's conjecture completes his survey.
Relation to Euler's Beta function
The product in the right hand side of the inequality () is related to the Euler's Beta function :
Discussion
For each fixed the function
turns the inequalities () and () to equalities.
The Khabibullin's conjecture is valid for without the assumption of convexity of . Meanwhile, one can show that this conjecture is not valid without some convexity conditions for . In 2010, R. A. Sharipov showed that the conjecture fails in the case and for .
The second statement in terms of increasing functions
Khabibullin's conjecture (version 2). Let be a non-negative increasing function on the half-line and . If
then
The third statement in terms of non-negative functions
Khabibullin's conjecture (version 3). Let be a non-negative continuous function on the half-line and . If
then
See also
Logarithmically convex function
References
Conjectures
Inequalities |
https://en.wikipedia.org/wiki/S%C3%A1ndor%20Hajd%C3%BA | Sándor Hajdú (born 21 January 1985 in Budapest) is a Hungarian football player who currently plays for MTK Hungária FC.
Career statistics
References
HLSZ
1985 births
Living people
Footballers from Budapest
Hungarian men's footballers
Men's association football midfielders
Soroksár SC players
MTK Budapest FC players
Aqvital FC Csákvár players
Nemzeti Bajnokság I players
21st-century Hungarian people |
https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley%20algorithm | In mathematics, economics, and computer science, the Gale–Shapley algorithm (also known as the deferred acceptance algorithm or propose-and-reject algorithm) is an algorithm for finding a solution to the stable matching problem, named for David Gale and Lloyd Shapley. It takes polynomial time, and the time is linear in the size of the input to the algorithm. It is a truthful mechanism from the point of view of the proposing participants, for whom the solution will always be optimal.
Background
The stable matching problem, in its most basic form, takes as input equal numbers of two types of participants ( job applicants and employers, for example), and an ordering for each participant giving their preference for whom to be matched to among the participants of the other type. A stable matching always exists, and the algorithmic problem solved by the Gale–Shapley algorithm is to find one. A matching is not stable if:
In other words, a matching is stable when there is no pair (A, B) where both participants prefer each other to their matched partners. If such a pair exists, the matching is not stable, in the sense that the members of this pair would prefer to leave the system and be matched to each other, possibly leaving other participants unmatched.
Solution
In 1962, David Gale and Lloyd Shapley proved that, for any equal number of participants of each type, it is always possible to find a matching in which all pairs are stable. They presented an algorithm to do so. In 1984, Alvin E. Roth observed that essentially the same algorithm had already been in practical use since the early 1950s, in the National Resident Matching Program.
The Gale–Shapley algorithm involves a number of "rounds" (or "iterations"). In terms of job applicants and employers, it can be expressed as follows:
In each round, any subset of the employers with open job positions makes a job offer to the applicant they prefer, among the ones they have not yet already made an offer to.
Each applicant who has received an offer evaluates it against their current position (if they have one). If the applicant is not yet employed, or if they receive an offer from an employer they like better than their current employer, they accept the new offer and become matched to the new employer (possibly leaving a previous employer with an open position). Otherwise, they reject the new offer.
This process is repeated until everyone is employed.
The runtime complexity of this algorithm is where is the number of participants of each type. Since the input preference lists also have size proportional to , the runtime is linear in the input size.
This algorithm guarantees that:
Everyone gets matched At the end, there cannot be an applicant and employer both unmatched. An employer left unmatched at the end of the process must have made an offer to all applicants. But an applicant who receives an offer remains employed for the rest of the process, so there can be no unemployed applicants. Sin |
https://en.wikipedia.org/wiki/Real%20rank | Real rank may refer to:
Real rank of a Lie group, the rank of a maximal split torus in the group.
Real rank (C*-algebras) |
https://en.wikipedia.org/wiki/National%20Center%20for%20Charitable%20Statistics | The National Center for Charitable Statistics (NCCS) is a clearing house for information about the nonprofit sector of the U.S. economy. The National Center for Charitable Statistics builds national, state, and regional databases and develops standards for reporting on the activities of all tax-exempt organizations.
Services
The National Center for Charitable Statistics collects data on charities in the U.S. and shares this data with the public.
The National Center for Charitable Statistics maintains a free online directory of charities, listed by mission and location.
When the Electronic Data Initiative for Nonprofits Coalition was formed in 2002, the National Center for Charitable Statistics advised the group in furtherance of the goal of integrated federal and state electronic reporting and dissemination of data on nonprofit organizations.
GuideStar works with the National Center for Charitable Statistics to get each Form 990 filed by a nonprofit organization online and readily available to the public. The National Center for Charitable Statistics buys scans of each organization's annual Form 990 on CDs from the Internal Revenue Service, and the scans are then posted online in order to help donors decide to which organizations they wish to give their donations.
History
The National Center for Charitable Statistics was established on March 15, 1982, as a research division of Independent Sector. Russy Sumariwalla was the first executive director of the National Center for Charitable Statistics. Prior to its establishment, no one knew exactly how many nonprofit organizations existed and how nonprofit organizations were using their donations, and enacting laws and policies related to nonprofit organization was very difficult.
The National Center for Charitable Statistics was transferred to the Center on Nonprofits and Philanthropy (CNP) at the Urban Institute on July 1, 1996.
The National Center for Charitable Statistics, along with several other nonprofit organizations, was instrumental in creating the National Taxonomy of Exempt Entities classification system or NTEE Codes. The National Taxonomy of Exempt Entities classifies organizations into more than 100 different categories based on the mission and program activities of an organization. The Internal Revenue Service uses this system to classify newly registered tax-exempt organizations.
References
External links
Urban Institute
Charities based in the United States
Statistical organizations in the United States
Organizations established in 1982 |
https://en.wikipedia.org/wiki/Mathesis | Mathesis may refer to
454 Mathesis, an asteroid discovered in 1900
Mathesis (journal), a Belgian mathematics journal founded in 1881
Mathesis (philosophy), the science of establishing a systematic order for things according to Michel Foucault
Mathesis (society), an Italian association of mathematics teachers
Mathesis universalis, a hypothetical universal science advocated by Leibniz and Descartes among others
Mathesis universalis, a treatise on integral calculus published by John Wallis in 1657
Mathesis Universalis (journal), a philosophy journal published by the University of Białystok in Poland
Matheseos Libri Oct., commonly referred to as Mathesis, a book on astrology by fourth-century author Julius Firmicus Maternus
Mathesis biceps, vetus et nova, a treatise published by Juan Caramuel y Lobkowitz in 1670
Mad Mathesis, fictional characters in The Dunciad by Alexander Pope and A Tangled Tale by Lewis Carroll
Mathesis Publications, the publisher of Ancient Philosophy (journal) |
https://en.wikipedia.org/wiki/Graphon | In graph theory and statistics, a graphon (also known as a graph limit) is a symmetric measurable function , that is important in the study of dense graphs. Graphons arise both as a natural notion for the limit of a sequence of dense graphs, and as the fundamental defining objects of exchangeable random graph models. Graphons are tied to dense graphs by the following pair of observations: the random graph models defined by graphons give rise to dense graphs almost surely, and, by the regularity lemma, graphons capture the structure of arbitrary large dense graphs.
Statistical formulation
A graphon is a symmetric measurable function . Usually a graphon is understood as defining an exchangeable random graph model according to the following scheme:
Each vertex of the graph is assigned an independent random value
Edge is independently included in the graph with probability .
A random graph model is an exchangeable random graph model if and only if it can be defined in terms of a (possibly random) graphon in this way.
The model based on a fixed graphon is sometimes denoted ,
by analogy with the Erdős–Rényi model of random graphs.
A graph generated from a graphon in this way is called a -random graph.
It follows from this definition and the law of large numbers that, if , exchangeable random graph models are dense almost surely.
Examples
The simplest example of a graphon is for some constant . In this case the associated exchangeable random graph model is the Erdős–Rényi model that includes each edge independently with probability .
If we instead start with a graphon that is piecewise constant by:
dividing the unit square into blocks, and
setting equal to on the block,
the resulting exchangeable random graph model is the
community stochastic block model, a generalization of the Erdős–Rényi model.
We can interpret this as a random graph model consisting of distinct Erdős–Rényi graphs with parameters respectively, with bigraphs between them where each possible edge between blocks and is included independently with probability .
Many other popular random graph models can be understood as exchangeable random graph models defined by some graphon, a detailed survey is included in Orbanz and Roy.
Jointly exchangeable adjacency matrices
A random graph of size can be represented as a random adjacency matrix. In order to impose consistency (in the sense of projectivity) between random graphs of different sizes it is natural to study the sequence of adjacency matrices arising as the upper-left sub-matrices of some infinite array of random variables; this allows us to generate by adding a node to and sampling the edges for . With this perspective, random graphs are defined as random infinite symmetric arrays .
Following the fundamental importance of exchangeable sequences in classical probability, it is natural to look for an analogous notion in the random graph setting. One such notion is given by jointly exchangeable |
https://en.wikipedia.org/wiki/Bateman%20function | In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by
Bateman discovered this function, when Theodore von Kármán asked for the solution of the following differential equation which appeared in the theory of turbulence
and Bateman found this function as one of the solutions. Bateman denoted this function as "k" function in honor of Theodore von Kármán.
This is not to be confused with another function of the same name which is used in Pharmacokinetics.
Properties
for real values of and
for if is a positive integer
, where is the Modified Bessel function of the second kind.
References
Special hypergeometric functions
Special functions |
https://en.wikipedia.org/wiki/Toronto%20function | In mathematics, the Toronto function T(m,n,r) is a modification of the confluent hypergeometric function defined by , Weisstein, as
Later, Heatley (1964) recomputed to 12 decimals the table of the M(R)-function, and gave some corrections of the original tables. The table was also extended from x = 4 to x = 16 (Heatley, 1965). An example of the Toronto function has appeared in a study on the theory of turbulence (Heatley, 1965).
References
Heatley, A. H. (1964), "A short table of the Toronto function", Mathematics of Computation, 18, No.88: 361
Heatley, A. H. (1965), "An extension of the table of the Toronto function", Mathematics of Computation, 19, No.89: 118-123
Weisstein, E. W., "Toronto Function", From Math World - A Wolfram Web Resource
Special hypergeometric functions |
https://en.wikipedia.org/wiki/Norman%20L.%20Biggs | Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics.
Education
Biggs was educated at Harrow County Grammar School and then studied mathematics at Selwyn College, Cambridge. In 1962, Biggs gained first-class honours in his third year of the university's undergraduate degree in mathematics.
1946–1952: Uxendon Manor Primary School, Kenton, Middlesex
1952–1959: Harrow County Grammar School
1959–1963: Selwyn College, Cambridge (Entrance Exhibition 1959, Scholarship 1961)
1960: First Class, Mathematical Tripos Pt. I
1962: Wrangler, Mathematical Tripos Pt. II; B.A. (Cantab.)
1963: Distinction, Mathematical Tripos Pt. III
1988: D.Sc. (London); M.A. (Cantab.)
Career
He was a lecturer at University of Southampton, lecturer then reader at Royal Holloway, University of London, and Professor of Mathematics at the London School of Economics. He has been on the editorial board of a number of journals, including the Journal of Algebraic Combinatorics. He has been a member of the Council of the London Mathematical Society.
He has written 12 books and over 100 papers on mathematical topics, many of them in algebraic combinatorics and its applications. He became Emeritus Professor in 2006 and continues to teach History of Mathematics in Finance and Economics for undergraduates. He is also vice-president of the British Society for the History of Mathematics.
Family
Biggs married Christine Mary Farmer in 1975 and has one daughter Clare Juliet born in 1980.
Interests and Hobbies
Biggs' interests include computational learning theory, the history of mathematics and historical metrology. Since 2006, he has been an emeritus professor at the London School of Economics.
Biggs hobbies consist of writing about the history of weights and scales. He currently holds the position of Chair of the International Society of Antique Scale Collectors (Europe), and a member of the British Numismatic Society.
Work
Mathematics
In 2002, Biggs wrote the second edition of Discrete Mathematics breaking down a wide range of topics into a clear and organised style. Biggs organised the book into four major sections; The Language of Mathematics, Techniques, Algorithms and Graphs, and Algebraic Methods. This book was an accumulation of Discrete Mathematics, first edition, textbook published in 1985 which dealt with calculations involving a finite number of steps rather than limiting processes. The second edition added nine new introductory chapters; Fundamental language of mathematicians, statements and proofs, the logical framework, sets and functions, and number system. This book stresses the significance of simple logical reasoning, shown by the exercises and examples given in the book. Each chapter contains modelled solutions, examples, exercises including hints and answers.
Algebraic Graph Theory
In 1974, Biggs published Algebraic Graph Theory which articulates properties of graphs |
https://en.wikipedia.org/wiki/Kronecker%20sum%20of%20discrete%20Laplacians | In mathematics, the Kronecker sum of discrete Laplacians, named after Leopold Kronecker, is a discrete version of the separation of variables for the continuous Laplacian in a rectangular cuboid domain.
General form of the Kronecker sum of discrete Laplacians
In a general situation of the separation of variables in the discrete case, the multidimensional discrete Laplacian is a Kronecker sum of 1D discrete Laplacians.
Example: 2D discrete Laplacian on a regular grid with the homogeneous Dirichlet boundary condition
Mathematically, using the Kronecker sum:
where and are 1D discrete Laplacians in the x- and y-directions, correspondingly, and are the identities of appropriate sizes. Both and must correspond to the case of the homogeneous Dirichlet boundary condition at end points of the x- and y-intervals, in order to generate the 2D discrete Laplacian L corresponding to the homogeneous Dirichlet boundary condition everywhere on the boundary of the rectangular domain.
Here is a sample OCTAVE/MATLAB code to compute L on the regular 10×15 2D grid:
nx = 10; % number of grid points in the x-direction;
ny = 15; % number of grid points in the y-direction;
ex = ones(nx,1);
Dxx = spdiags([ex -2*ex ex], [-1 0 1], nx, nx); %1D discrete Laplacian in the x-direction ;
ey = ones(ny,1);
Dyy = spdiags([ey, -2*ey ey], [-1 0 1], ny, ny); %1D discrete Laplacian in the y-direction ;
L = kron(Dyy, speye(nx)) + kron(speye(ny), Dxx) ;
Eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid
Knowing all eigenvalues and eigenvectors of the factors, all eigenvalues and eigenvectors of the Kronecker product can be explicitly calculated. Based on this, eigenvalues and eigenvectors of the Kronecker sum
can also be explicitly calculated.
The eigenvalues and eigenvectors of the standard central difference approximation of the second derivative on an interval for traditional combinations of boundary conditions at the interval end points are well known. Combining these expressions with the formulas of eigenvalues and eigenvectors for the Kronecker sum, one can easily obtain the required answer.
Example: 3D discrete Laplacian on a regular grid with the homogeneous Dirichlet boundary condition
where and are 1D discrete Laplacians in every of the 3 directions, and are the identities of appropriate sizes. Each 1D discrete Laplacian must correspond to the case of the homogeneous Dirichlet boundary condition, in order to generate the 3D discrete Laplacian L corresponding to the homogeneous Dirichlet boundary condition everywhere on the boundary. The eigenvalues are
where , and the corresponding eigenvectors are
where the multi-index pairs the eigenvalues and the eigenvectors, while the multi-index
determines the location of the value of every eigenvector at the regular grid. The boundary points, where
the homogeneous Dirichlet boundary condition are imposed, are just outside the grid.
Available software
An OCTAVE/MATLAB code http: |
https://en.wikipedia.org/wiki/Lucien%20Szpiro | Lucien Serge Szpiro (23 December 1941 – 18 April 2020) was a French mathematician known for his work in number theory, arithmetic geometry, and commutative algebra. He formulated Szpiro's conjecture and was a Distinguished Professor at the CUNY Graduate Center and an emeritus at the CNRS.
Early life and education
Lucien Serge Szpiro was born on 23 December 1941 in the 20th arrondissement of Paris, France. Szpiro attended Paris-Sud University where he earned his Ph.D. under Pierre Samuel. His doctoral work was heavily influenced by the seminars of Maurice Auslander, Claude Chevalley, and Alexander Grothendieck. He earned his Doctorat d'État (DrE) in 1971.
Career
From 1963 to 1965, Szpiro worked as an assistant high school teacher in Paris. From 1965 to 1969, he was an assistant professor (maître assistant) at the University of Paris. From 1969 to 1999, Szpiro worked at the CNRS, initially as an attaché at Paris Diderot University before rising to the rank of a distinguished professor (Directeur de Recherche de Classe Exceptionnelle) at Paris-Sud University. In 1999, he became an emeritus professor (Directeur de Recherche émérite) at the CNRS and moved to the CUNY Graduate Center as a Distinguished Professor. He also held visiting positions at several institutions including Columbia University and the Institute for Advanced Study.
Szpiro was the editor-in-chief of Astérisque from 1991 to 1993 and an editor of the Bulletin de la Société Mathématique de France from 1984 to 1990. He was also head of the commission that oversaw the Société mathématique de France libraries.
Szpiro advised 17 doctoral students, including Ahmed Abbes, Emmanuel Ullmo, and Shou-Wu Zhang.
Research
In the 1970s, Szpiro's research in commutative algebra led to his proof of the Auslander zero divisor conjecture. Together with Christian Peskine, he developed the liaison theory of algebraic varieties.
In the 1980s, Szpiro's research interests shifted to Diophantine geometry, first over function fields and then over number fields. The Institut des hautes études scientifiques described Szpiro as being "the first to realise the importance of a paper by Arakelov for questions of Diophantine geometry", which ultimately led to the development of Arakelov theory as a tool of modern Diophantine geometry exemplified by Gerd Faltings's proof of the Mordell conjecture. Szpiro also showed the link between the positivity of the dualising sheaf of a curve and the Bogomolov conjecture.
In 1981, Szpiro formulated a conjecture (now known as Szpiro's conjecture) relating the discriminant of an elliptic curve with its conductor. His conjecture inspired the abc conjecture, which was later shown to be equivalent to a modified form of Szpiro's conjecture in 1988. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theore |
https://en.wikipedia.org/wiki/Fran%C3%A7ois%20Labourie | François Labourie (born 15 December 1960) is a French mathematician who has made various contributions to geometry, including pseudoholomorphic curves, Anosov diffeomorphism, and convex geometry. In a series of papers with Yves Benoist and Patrick Foulon, he solved a conjecture on Anosov's flows in compact contact manifolds.
He was educated at the École Normale Supérieure and Paris Diderot University, where he earned his Ph.D. under supervision of Mikhail Gromov. In 1992 he was awarded one of the inaugural prizes of the European Mathematical Society. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin.
References
External links
Curriculum Vitae
1960 births
Living people
École Normale Supérieure alumni
20th-century French mathematicians
21st-century French mathematicians
University of Paris alumni
French geometers |
https://en.wikipedia.org/wiki/List%20of%20algebraic%20constructions | An algebraic construction is a method by which an algebraic entity is defined or derived from another.
Instances include:
Cayley–Dickson construction
Proj construction
Grothendieck group
Gelfand–Naimark–Segal construction
Ultraproduct
ADHM construction
Burnside ring
Simplicial set
Fox derivative
Mapping cone (homological algebra)
Prym variety
Todd class
Adjunction (field theory)
Vaughan Jones construction
Strähle construction
Coset construction
Plus construction
Algebraic K-theory
Gelfand–Naimark–Segal construction
Stanley–Reisner ring construction
Quotient ring construction
Ward's twistor construction
Hilbert symbol
Hilbert's arithmetic of ends
Colombeau's construction
Vector bundle
Integral monoid ring construction
Integral group ring construction
Category of Eilenberg–Moore algebras
Kleisli category
Adjunction (field theory)
Lindenbaum–Tarski algebra construction
Freudenthal magic square
Stone–Čech compactification
Mathematics-related lists
Algebra |
https://en.wikipedia.org/wiki/Abelson%27s%20paradox | Abelson's paradox is an applied statistics paradox identified by Robert P. Abelson. The paradox pertains to a possible paradoxical relationship between the magnitude of the r2 (i.e., coefficient of determination) effect size and its practical meaning.
Abelson's example was obtained from the analysis of the r2 of batting average in baseball and skill level. Although batting average is considered among the most significant characteristics necessary for success, the effect size was only a tiny 0.003.
See also
List of paradoxes
References
Statistical paradoxes |
https://en.wikipedia.org/wiki/N50%2C%20L50%2C%20and%20related%20statistics | In computational biology, N50 and L50 are statistics of a set of contig or scaffold lengths. The N50 is similar to a mean or median of lengths, but has greater weight given to the longer contigs. It is used widely in genome assembly, especially in reference to contig lengths within a draft assembly. There are also the related U50, UL50, UG50, UG50%, N90, NG50, and D50 statistics.
To provide a better assessment of assembly output for viral and microbial datasets, a new metric called U50 should be used. The U50 identifies unique, target-specific contigs by using a reference genome as baseline, aiming at circumventing some limitations that are inherent to the N50 metric. The use of the U50 metric allows for a more accurate measure of assembly performance by analyzing only the unique, non-overlapping contigs. Most viral and microbial sequencing have high background noise (i.e., host and other non-targets), which contributes to having a skewed, misrepresented N50 value - this is corrected by U50.
Definition
N50
N50 statistic defines assembly quality in terms of contiguity. Given a set of contigs, the N50 is defined as the sequence length of the shortest contig at 50% of the total assembly length. It can be thought of as the point of half of the mass of the distribution; the number of bases from all contigs longer than the N50 will be close to the number of bases from all contigs shorter than the N50. For example, consider 9 contigs with the lengths 2,3,4,5,6,7,8,9,and 10; their sum is 54, half of the sum is 27, and the size of the genome also happens to be 54. 50% of this assembly would be 10 + 9 + 8 = 27 (half the length of the sequence). Thus the N50=8, which is the size of the contig which, along with the larger contigs, contain half of sequence of a particular genome. Note: When comparing N50 values from different assemblies, the assembly sizes must be the same size in order for N50 to be meaningful.
N50 can be described as a weighted median statistic such that 50% of the entire assembly is contained in contigs or scaffolds equal to or larger than this value.
L50
Given a set of contigs, each with its own length, the L50 is defined as count of smallest number of contigs whose length sum makes up half of genome size. From the example above the L50=3.
N90
The N90 statistic is less than or equal to the N50 statistic; it is the length for which the collection of all contigs of that length or longer contains at least 90% of the sum of the lengths of all contigs.
NG50
Note that N50 is calculated in the context of the assembly size rather than the genome size. Therefore, comparisons of N50 values derived from assemblies of significantly different lengths are usually not informative, even if for the same genome. To address this, the authors of the Assemblathon competition came up with a new measure called NG50. The NG50 statistic is the same as N50 except that it is 50% of the known or estimated genome size that must be of the NG50 length or lo |
https://en.wikipedia.org/wiki/Evolutionary%20multimodal%20optimization | In applied mathematics, multimodal optimization deals with optimization tasks that involve finding all or most of the multiple (at least locally optimal) solutions of a problem, as opposed to a single best solution.
Evolutionary multimodal optimization is a branch of evolutionary computation, which is closely related to machine learning. Wong provides a short survey, wherein the chapter of Shir and the book of Preuss cover the topic in more detail.
Motivation
Knowledge of multiple solutions to an optimization task is especially helpful in engineering, when due to physical (and/or cost) constraints, the best results may not always be realizable. In such a scenario, if multiple solutions (locally and/or globally optimal) are known, the implementation can be quickly switched to another solution and still obtain the best possible system performance. Multiple solutions could also be analyzed to discover hidden properties (or relationships) of the underlying optimization problem, which makes them important for obtaining domain knowledge.
In addition, the algorithms for multimodal optimization usually not only locate multiple optima in a single run, but also preserve their population diversity, resulting in their global optimization ability on multimodal functions. Moreover, the techniques for multimodal optimization are usually borrowed as diversity maintenance techniques to other problems.
Background
Classical techniques of optimization would need multiple restart points and multiple runs in the hope that a different solution may be discovered every run, with no guarantee however. Evolutionary algorithms (EAs) due to their population based approach, provide a natural advantage over classical optimization techniques. They maintain a population of possible solutions, which are processed every generation, and if the multiple solutions can be preserved over all these generations, then at termination of the algorithm we will have multiple good solutions, rather than only the best solution. Note that this is against the natural tendency of classical optimization techniques, which will always converge to the best solution, or a sub-optimal solution (in a rugged, “badly behaving” function). Finding and maintenance of multiple solutions is wherein lies the challenge of using EAs for multi-modal optimization. Niching is a generic term referred to as the technique of finding and preserving multiple stable niches, or favorable parts of the solution space possibly around multiple solutions, so as to prevent convergence to a single solution.
The field of Evolutionary algorithms encompasses genetic algorithms (GAs), evolution strategy (ES), differential evolution (DE), particle swarm optimization (PSO), and other methods. Attempts have been made to solve multi-modal optimization in all these realms and most, if not all the various methods implement niching in some form or the other.
Multimodal optimization using genetic algorithms/evolution strategies |
https://en.wikipedia.org/wiki/1999%20J.League%20Division%202 | Statistics of J.League Division 2 in the 1999 season.
Overview
This was the first season of J2, the professional second tier in Japan. It replaced the JFL, which was moved to the 3rd tier. The league was contested by 10 teams, and Kawasaki Frontale won the championship.
Team changes
Final table
References
J2 League seasons
2
Japan
Japan |
https://en.wikipedia.org/wiki/2000%20J.League%20Division%202 | Statistics of J.League Division 2 in the 2000 season.
Overview
It was contested by 11 teams, and Consadole Sapporo won the championship.
Final table
References
J2 League seasons
2
Japan
Japan |
https://en.wikipedia.org/wiki/2001%20J.League%20Division%202 | Statistics of J.League Division 2 in the 2001 season.
Overview
It was contested by 12 teams, and Kyoto Purple Sanga won the championship.
Final table
References
J2 League seasons
2
Japan
Japan |
https://en.wikipedia.org/wiki/Flat%20function | In mathematics, especially real analysis, a flat function is a smooth function all of whose derivatives vanish at a given point . The flat functions are, in some sense, the antitheses of the analytic functions. An analytic function is given by a convergent power series close to some point :
In the case of a flat function, all derivatives vanish at , i.e. for all . This means that a meaningful Taylor series expansion in a neighbourhood of is impossible. In the language of Taylor's theorem, the non-constant part of the function always lies in the remainder for all .
The function need not be flat at just one point. Trivially, constant functions on are flat everywhere. But there are also other, less trivial, examples.
Example
The function defined by
is flat at . Thus, this is an example of a non-analytic smooth function. The pathological nature of this example is partially illuminated by the fact that its extension to the complex numbers is, in fact, not differentiable.
References
Real analysis
Algebraic geometry
Differential calculus
Smooth functions
Differential structures |
https://en.wikipedia.org/wiki/Finsler%E2%80%93Hadwiger%20theorem | The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality relating the side lengths and area of a triangle.
Statement
To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints of B'D and D'B respectively, and let F and H be the centers of the two squares. Then the theorem states that the quadrilateral EFGH is a square as well.
The square EFGH is called the Finsler–Hadwiger square of the two given squares.
Application
Repeated application of the Finsler–Hadwiger theorem can be used to prove Van Aubel's theorem, on the congruence and perpendicularity of segments through centers of four squares constructed on the sides of an arbitrary quadrilateral. Each pair of consecutive squares forms an instance of the theorem, and the two pairs of opposite Finsler–Hadwiger squares of those instances form another two instances of the theorem, having the same derived square.
References
External links
Theorems about quadrilaterals
Euclidean geometry |
https://en.wikipedia.org/wiki/Bezdek | Bezdek may refer to:
People
Hugo Bezdek, a Czech-American sports figure
Károly Bezdek, a professor of mathematics at the University of Calgary
Pavel Bezdek, a Czech actor and stunt performer
Rudolf Bezděk, a Czech boxer who competed in the 1936 Summer Olympics.
Other
23199 Bezdek a main belt asteroid |
https://en.wikipedia.org/wiki/Elliptical%20distribution | In probability and statistics, an elliptical distribution is any member of a broad family of probability distributions that generalize the multivariate normal distribution. Intuitively, in the simplified two and three dimensional case, the joint distribution forms an ellipse and an ellipsoid, respectively, in iso-density plots.
In statistics, the normal distribution is used in classical multivariate analysis, while elliptical distributions are used in generalized multivariate analysis, for the study of symmetric distributions with tails that are heavy, like the multivariate t-distribution, or light (in comparison with the normal distribution). Some statistical methods that were originally motivated by the study of the normal distribution have good performance for general elliptical distributions (with finite variance), particularly for spherical distributions (which are defined below). Elliptical distributions are also used in robust statistics to evaluate proposed multivariate-statistical procedures.
Definition
Elliptical distributions are defined in terms of the characteristic function of probability theory. A random vector on a Euclidean space has an elliptical distribution if its characteristic function satisfies the following functional equation (for every column-vector )
for some location parameter , some nonnegative-definite matrix and some scalar function . The definition of elliptical distributions for real random-vectors has been extended to accommodate random vectors in Euclidean spaces over the field of complex numbers, so facilitating applications in time-series analysis. Computational methods are available for generating pseudo-random vectors from elliptical distributions, for use in Monte Carlo simulations for example.
Some elliptical distributions are alternatively defined in terms of their density functions. An elliptical distribution with a density function f has the form:
where is the normalizing constant, is an -dimensional random vector with median vector (which is also the mean vector if the latter exists), and is a positive definite matrix which is proportional to the covariance matrix if the latter exists.
Examples
Examples include the following multivariate probability distributions:
Multivariate normal distribution
Multivariate t-distribution
Symmetric multivariate stable distribution
Symmetric multivariate Laplace distribution
Multivariate logistic distribution
Multivariate symmetric general hyperbolic distribution
Properties
In the 2-dimensional case, if the density exists, each iso-density locus (the set of x1,x2 pairs all giving a particular value of ) is an ellipse or a union of ellipses (hence the name elliptical distribution). More generally, for arbitrary n, the iso-density loci are unions of ellipsoids. All these ellipsoids or ellipses have the common center μ and are scaled copies (homothets) of each other.
The multivariate normal distribution is the special case in which . While |
https://en.wikipedia.org/wiki/Gathera | Gathera is a settlement in Kenya's Central Province about 87 kilometers from the Kenyan capital of Nairobi.
Statistics
There is an infant mortality of 44 per 1000 births. Malnutrition is apparent as 1.54% of children below five years old are underweight. Most of the land coverage is open broadleaved deciduous forest. The Cultivation Intensity is about 20% cultivated, and 80% natural vegetation.
Schools
Gathera Secondary School is located at 2- 10205 Chinga-Kairu Road.
References
Populated places in Central Province (Kenya) |
https://en.wikipedia.org/wiki/Philip%20Treisman | Philip Uri Treisman is an American mathematician and mathematics educator. He is the Director of the Charles A. Dana Center, and is a Professor of Mathematics at The University of Texas at Austin. He is credited with pioneering the Emerging Scholars Program (ESP), aimed at helping students from underprivileged backgrounds excel in calculus and other courses in science. The program was first implemented at the University of California, Berkeley and has now disseminated throughout college campuses across the United States. His efforts to improve American education have been recognized by Newsweek, the Harvard Foundation and the MacArthur Foundation, among other publications and societies.
He graduated summa cum laude with a B.S. in Mathematics from the University of California, Los Angeles, and from the University of California, Berkeley with a Ph.D. in 1985.
Awards
1987 Charles A. Dana Award for Pioneering Achievement in American Higher Education.
1992 MacArthur Fellows Program
2006 The Harvard Foundation's Scientist of the Year Award
2019 Yueh-Gin Gung and Dr. Charles Y. Hu Award
References
External links
"Dr. Philip Treisman Meets Isaac Newton in Surreal Space", Digital Writing & Research Lab, Amanda Dulcinea Cuéllar
"Philip Uri Treisman", Mathematics Genealogy Project
, Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College
Living people
20th-century American mathematicians
21st-century American mathematicians
MacArthur Fellows
UC Berkeley College of Letters and Science alumni
University of Texas at Austin faculty
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Cunningham%20function | In statistics, the Cunningham function or Pearson–Cunningham function ωm,n(x) is a generalisation of a special function introduced by and studied in the form here by . It can be defined in terms of the confluent hypergeometric function U, by
The function was studied by Cunningham in the context of a multivariate generalisation of the Edgeworth expansion for approximating a probability density function based on its (joint) moments. In a more general context, the function is related to the solution of the constant-coefficient diffusion equation, in one or more dimensions.
The function ωm,n(x) is a solution of the differential equation for X:
The special function studied by Pearson is given, in his notation by,
Notes
References
See exercise 10, chapter XVI, p. 353
Special hypergeometric functions
Statistical approximations |
https://en.wikipedia.org/wiki/Mazzucato | Mazzucato is a surname. Notable people with the surname include:
Alberto Mazzucato, Italian musician
Anna Mazzucato, professor of mathematics
Augusto Mazzucato, Italian soccer player
Carla Carli Mazzucato, Italian artist
Francesca Mazzucato, Italian writer and translator
Giovanni Mazzucato (1787-1814), Italian botanist
Mariana Mazzucato, Italian economist
Michele Mazzucato, Italian astronomer
Nicola Mazzucato, Italian rugby coach
Roberto Mazzucato, Italian athlete
Valerio Mazzucato, Italian soccer player
Mazzucato may also refer to:
35461 Mazzucato, main belt asteroid
Surnames
Italian-language surnames |
https://en.wikipedia.org/wiki/Pignedoli | Pignedoli may refer to:
22263 Pignedoli, minor aster discovered by Antonio Pignedoli, a mathematics professor at the Military Academy of Modena
People
Sabrina Pignedoli (born 1983), Italian politician and Member of the European Parliament since 2019
Sergio Pignedoli (1910–1980), a prominent Italian Cardinal of the Roman Catholic Church and a top candidate for Pope |
https://en.wikipedia.org/wiki/Unit%20sphere | In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ball is the closed set of points of distance less than or equal to 1 from a fixed central point. Usually the center is at the origin of the space, so one speaks of "the unit ball" or "the unit sphere".
Special cases are the unit circle and the unit disk.
The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of translation and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere.
Unit spheres and balls in Euclidean space
In Euclidean space of n dimensions, the -dimensional unit sphere is the set of all points which satisfy the equation
The n-dimensional open unit ball is the set of all points satisfying the inequality
and the n-dimensional closed unit ball is the set of all points satisfying the inequality
General area and volume formulas
The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the x-, y-, or z- axes:
The volume of the unit ball in n-dimensional Euclidean space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit ball in n dimensions, which we denote Vn, can be expressed by making use of the gamma function. It is
where n!! is the double factorial.
The hypervolume of the (n−1)-dimensional unit sphere (i.e., the "area" of the boundary of the n-dimensional unit ball), which we denote An−1, can be expressed as
where the last equality holds only for . For example, is the "area" of the boundary of the unit ball , which simply counts the two points. Then is the "area" of the boundary of the unit disc, which is the circumference of the unit circle. is the area of the boundary of the unit ball , which is the surface area of the unit sphere .
The surface areas and the volumes for some values of are as follows:
where the decimal expanded values for n ≥ 2 are rounded to the displayed precision.
Recursion
The An values satisfy the recursion:
for .
The Vn values satisfy the recursion:
for .
Non-negative real-valued dimensions
The value at non-negative real values of is sometimes used for normalization of Hausdorff measure.
Other radii
The surface area of an (n−1)-dimensional sphere with radius r is An−1 rn−1 and the volume of an n-dimensional ball with radius r is Vn rn. For instance, the area is for the two-dimensional surface of the three-dimensional ball of radius r. The volume is for the three-dimensional ball of radius r.
Unit balls in normed vector spaces
The open unit ball of a normed vector space with the norm is given by
It is the topological interior of the closed unit ball of (V,||·||):
The latter is the disjoint union of the former and their c |
https://en.wikipedia.org/wiki/420%20%28number%29 | 420 (four hundred [and] twenty) is the natural number following 419 and preceding 421.
In mathematics
420 is:
the sum of four consecutive primes ().
the sum of the first twenty positive even numbers.
a zero of the Mertens function and is sparsely totient.
a pronic number.
the smallest number divisible by the numbers 1 to 7; as a consequence of that, it is a Harshad number in bases 2 to 10, except in base 5.
a 141-gonal number.
a balanced number.
In other fields
420 is a slang term that refers to the consumption of cannabis. April 20th is commonly celebrated as a holiday dedicated to the drug, due to the numerical form of the day being 4/20.
420 is the country calling code for Czech Republic.
References
Integers
Internet memes |
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