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https://en.wikipedia.org/wiki/Rule%20of%20three%20%28statistics%29 | In statistical analysis, the rule of three states that if a certain event did not occur in a sample with subjects, the interval from 0 to 3/ is a 95% confidence interval for the rate of occurrences in the population. When is greater than 30, this is a good approximation of results from more sensitive tests. For example, a pain-relief drug is tested on 1500 human subjects, and no adverse event is recorded. From the rule of three, it can be concluded with 95% confidence that fewer than 1 person in 500 (or 3/1500) will experience an adverse event. By symmetry, for only successes, the 95% confidence interval is .
The rule is useful in the interpretation of clinical trials generally, particularly in phase II and phase III where often there are limitations in duration or statistical power. The rule of three applies well beyond medical research, to any trial done times. If 300 parachutes are randomly tested and all open successfully, then it is concluded with 95% confidence that fewer than 1 in 100 parachutes with the same characteristics (3/300) will fail.
Derivation
A 95% confidence interval is sought for the probability p of an event occurring for any randomly selected single individual in a population, given that it has not been observed to occur in n Bernoulli trials. Denoting the number of events by X, we therefore wish to find the values of the parameter p of a binomial distribution that give Pr(X = 0) ≤ 0.05. The rule can then be derived either from the Poisson approximation to the binomial distribution, or from the formula (1−p)n for the probability of zero events in the binomial distribution. In the latter case, the edge of the confidence interval is given by Pr(X = 0) = 0.05 and hence (1−p)n = .05 so n ln(1–p) = ln .05 ≈ −2.996. Rounding the latter to −3 and using the approximation, for p close to 0, that ln(1−p) ≈ −p (Taylor's formula), we obtain the interval's boundary 3/n.
By a similar argument, the numerator values of 3.51, 4.61, and 5.3 may be used for the 97%, 99%, and 99.5% confidence intervals, respectively, and in general the upper end of the confidence interval can be given as , where is the desired confidence level.
Extension
The Vysochanskij–Petunin inequality shows that the rule of three holds for unimodal distributions with finite variance beyond just the binomial distribution, and gives a way to change the factor 3 if a different confidence is desired. Chebyshev's inequality removes the assumption of unimodality at the price of a higher multiplier (about 4.5 for 95% confidence). Cantelli's inequality is the one-tailed version of Chebyshev's inequality.
See also
Binomial proportion confidence interval
Rule of succession
Notes
References
Ziliak, S. T.; D. N. McCloskey (2008). The cult of statistical significance: How the standard error costs us jobs, justice, and lives. University of Michigan Press.
Clinical trials
Statistical approximations
Medical statistics
Nursing research |
https://en.wikipedia.org/wiki/Boris%20Aronov | Boris Aronov (born March 13, 1963) is a computer scientist, currently a professor at the Tandon School of Engineering, New York University. His main area of research is computational geometry. He is a Sloan Research Fellow.
Aronov earned his B.A. in computer science and mathematics in 1984 from Queens College, City University of New York. He went on to graduate studies at the Courant Institute of Mathematical Sciences of New York University, where he received his M.S. in 1986 and Ph.D. in 1989, under the supervision of Micha Sharir.
References
External links
Aronov's Poly faculty page
1963 births
American computer scientists
Living people
Researchers in geometric algorithms
Queens College, City University of New York alumni
Courant Institute of Mathematical Sciences alumni
Polytechnic Institute of New York University faculty |
https://en.wikipedia.org/wiki/Katsuya%20Eda | is a mathematician, currently a professor at Waseda University. His research centers on set theory and its applications, particularly in algebraic topology. He has done a great deal of work on the fundamental group of the Hawaiian earring and related subjects.
External links
Eda's home page at Waseda University
Living people
21st-century Japanese mathematicians
Set theorists
Topologists
Academic staff of Waseda University
Year of birth missing (living people)
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Essential%20dimension | In mathematics, essential dimension is an invariant defined for certain algebraic structures such as algebraic groups and quadratic forms. It was introduced by J. Buhler and Z. Reichstein
and in its most generality defined by A. Merkurjev.
Basically, essential dimension measures the complexity of algebraic structures via their fields of definition. For example, a quadratic form q : V → K over a field K, where V is a K-vector space, is said to be defined over a subfield L of K if there exists a K-basis e1,...,en of V such that q can be expressed in the form with all coefficients aij belonging to L. If K has characteristic different from 2, every quadratic form is diagonalizable. Therefore, q has a field of definition generated by n elements. Technically, one always works over a (fixed) base field k and the fields K and L in consideration are supposed to contain k. The essential dimension of q is then defined as the least transcendence degree over k of a subfield L of K over which q is defined.
Formal definition
Fix an arbitrary field k and let /k denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : /k → .
For a field extension K/k and an element a of F(K/k) a field of definition of a is an intermediate field K/L/k such that a is contained in the image of the map F(L/k) → F(K/k) induced by the inclusion of L in K.
The essential dimension of a, denoted by ed(a), is the least transcendence degree (over k) of a field of definition for a. The essential dimension of the functor F, denoted by ed(F), is the supremum of ed(a) taken over all elements a of F(K/k) and objects K/k of /k.
Examples
Essential dimension of quadratic forms: For a natural number n consider the functor Qn : /k → taking a field extension K/k to the set of isomorphism classes of non-degenerate n-dimensional quadratic forms over K and taking a morphism L/k → K/k (given by the inclusion of L in K) to the map sending the isomorphism class of a quadratic form q : V → L to the isomorphism class of the quadratic form .
Essential dimension of algebraic groups: For an algebraic group G over k denote by H1(−,G) : /k → the functor taking a field extension K/k to the set of isomorphism classes of G-torsors over K (in the fppf-topology). The essential dimension of this functor is called the essential dimension of the algebraic group G, denoted by ed(G).
Essential dimension of a fibered category: Let be a category fibered over the category of affine k-schemes, given by a functor For example, may be the moduli stack of genus g curves or the classifying stack of an algebraic group. Assume that for each the isomorphism classes of objects in the fiber p−1(A) form a set. Then we get a functor Fp : /k → taking a field extension K/k to the set of isomorphism classes in the fiber . The essential dimension of the fibered category is defined as the essential dimension of the corresponding functor Fp. In case of the |
https://en.wikipedia.org/wiki/Rotating%20calipers | In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points.
The method is so named because the idea is analogous to rotating a spring-loaded vernier caliper around the outside of a convex polygon. Every time one blade of the caliper lies flat against an edge of the polygon, it forms an antipodal pair with the point or edge touching the opposite blade. The complete "rotation" of the caliper around the polygon detects all antipodal pairs; the set of all pairs, viewed as a graph, forms a thrackle. The method of rotating calipers can be interpreted as the projective dual of a sweep line algorithm in which the sweep is across slopes of lines rather than across - or -coordinates of points.
History
The rotating calipers method was first used in the dissertation of Michael Shamos in 1978. Shamos used this method to generate all antipodal pairs of points on a convex polygon and to compute the diameter of a convex polygon in time. Godfried Toussaint coined the phrase "rotating calipers" and demonstrated that the method was applicable in solving many other computational geometry problems.
Shamos's algorithm
Shamos gave the following algorithm in his dissertation (pp. 77–82) for the rotating calipers method, which generated all antipodal pairs of vertices on a convex polygon:
/* p[] is in standard form, ie, counter clockwise order,
distinct vertices, no collinear vertices.
ANGLE(m, n) is a procedure that returns the clockwise angle
swept out by a ray as it rotates from a position parallel
to the directed segment Pm,Pm+1 to a position parallel to Pn, Pn+1
We assume all indices are reduced to mod N (so that N+1 = 1).
*/
GetAllAntiPodalPairs(p[1..n])
// Find first anti-podal pair by locating vertex opposite P1
i = 1
j = 2
while angle(i, j) < pi
j++
yield i, j
/* Now proceed around the polygon taking account of
possibly parallel edges. Line L passes through
Pi, Pi+1 and M passes through Pj, Pj+1
*/
// Loop on j until all of P has been scanned
current = i
while j != n
if angle(current, i + 1) <= angle(current, j + 1)
j++
current = j
else
i++
current = i
yield i, j
// Now take care of parallel edges
if angle(current, i + 1) = angle(current, j + 1)
yield i + 1, j
yield i, j + 1
yield i + 1, j + 1
if current = i
j++
else
i++
Another version of this algorithm appeared in the text by Preparata and Shamos in 1985 that avoided calculation of angles:
GetAllAntiPodalPairs(p[1..n])
i0 = n
i = 1
j = i + 1
while (Area(i, i + 1, j + 1) > Area(i, i + 1, j))
j = j + 1
j0 = j
while (i != j0)
i = i + 1
yield i |
https://en.wikipedia.org/wiki/1968%E2%80%9369%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1968/1969.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1969%E2%80%9370%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1969/1970.
Scottish League Division One
Scottish League Division Two
See also
1969–70 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1970%E2%80%9371%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1970/1971.
Scottish League Division One
Aberdeen, with 15 straight wins of which the last 12 were without conceding, led the league from December until the last week of the season. Aberdeen faced Celtic in their penultimate game needing a win to almost certainly clinch the title, but could only draw 1-1: and then they lost their last game, at Falkirk, allowing Celtic
to take the championship by 2 points.
Scottish League Division Two
See also
1970–71 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1971%E2%80%9372%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1971/1972.
Scottish League Division One
Scottish League Division Two
See also
1971–72 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1972%E2%80%9373%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1972/1973.
Scottish League Division One
Scottish League Division Two
See also
1972–73 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1973%E2%80%9374%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1973–74.
Scottish League Division One
Scottish League Division Two
See also
1973–74 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1974%E2%80%9375%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1974–75. At the end of this season, the leagues were reconstructed into three divisions of 10, 14 and 14. This meant that the top ten teams in Division One entered the new Premier Division, while the rest of the Division One clubs entered the new First Division.
Scottish League Division One
Scottish League Division Two
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1975%E2%80%9376%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1975–76.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1975–76 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1976%E2%80%9377%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1976/1977.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1976–77 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Belgian%20First%20Division | Statistics of Belgian League in season 1988–89.
Overview
It was contested by 18 teams, and KV Mechelen won the championship, while R.W.D. Molenbeek & K.R.C. Genk were relegated.
League standings
Results
Topscorers
References
Belgian Pro League seasons
Belgian
1 |
https://en.wikipedia.org/wiki/1990%E2%80%9391%20Belgian%20First%20Division | Statistics of Belgian League in season 1990–91.
Overview
18 teams participated, and R.S.C. Anderlecht won the championship, while K. Sint-Truidense V.V. and K. Beerschot V.A.C. were relegated.
League standings
Results
Topscorers
References
Belgian Pro League seasons
Belgian
1990–91 in Belgian football |
https://en.wikipedia.org/wiki/Applicative%20computing%20systems | Applicative computing systems, or ACS are the systems of object calculi founded on combinatory logic and lambda calculus.
The only essential notion which is under consideration in these systems is the representation of object. In combinatory logic the only metaoperator is application in a sense of applying one object to other. In lambda calculus two metaoperators are used: application – the same as in combinatory logic, and functional abstraction which binds the only variable in one object.
Features
The objects generated in these systems are the functional entities with the following features:
the number of argument places, or object arity is not fixed but is enabling step by step in interoperations with other objects;
in a process of generating the compound object one of its counterparts—function—is applied to other one—argument—but in other contexts they can change their roles, i.e. functions and arguments are considered on the equal rights;
the self-applying of functions is allowed, i.e. any object can be applied to itself.
ACS give a sound ground for applicative approach to programming.
Research challenge
Applicative computing systems' lack of storage and history sensitivity is the basic reason they have not provided a foundation for computer design. Moreover, most applicative systems employ the substitution operation of the lambda calculus as their basic operation. This operation is one of virtually unlimited power, but its complete and efficient realization presents great difficulties to the machine designer.
See also
Applicative programming language
Categorical abstract machine
Combinatory logic
Functional programming
Lambda calculus
References
Further reading
[This volume reflects the research program and philosophy of H. Curry, one of the founders of computational models and the deductive framework for reasoning in terms of objects.]
Models of computation
Combinatory logic
Lambda calculus |
https://en.wikipedia.org/wiki/Invariant%20estimator | In statistics, the concept of being an invariant estimator is a criterion that can be used to compare the properties of different estimators for the same quantity. It is a way of formalising the idea that an estimator should have certain intuitively appealing qualities. Strictly speaking, "invariant" would mean that the estimates themselves are unchanged when both the measurements and the parameters are transformed in a compatible way, but the meaning has been extended to allow the estimates to change in appropriate ways with such transformations. The term equivariant estimator is used in formal mathematical contexts that include a precise description of the relation of the way the estimator changes in response to changes to the dataset and parameterisation: this corresponds to the use of "equivariance" in more general mathematics.
General setting
Background
In statistical inference, there are several approaches to estimation theory that can be used to decide immediately what estimators should be used according to those approaches. For example, ideas from Bayesian inference would lead directly to Bayesian estimators. Similarly, the theory of classical statistical inference can sometimes lead to strong conclusions about what estimator should be used. However, the usefulness of these theories depends on having a fully prescribed statistical model and may also depend on having a relevant loss function to determine the estimator. Thus a Bayesian analysis might be undertaken, leading to a posterior distribution for relevant parameters, but the use of a specific utility or loss function may be unclear. Ideas of invariance can then be applied to the task of summarising the posterior distribution. In other cases, statistical analyses are undertaken without a fully defined statistical model or the classical theory of statistical inference cannot be readily applied because the family of models being considered are not amenable to such treatment. In addition to these cases where general theory does not prescribe an estimator, the concept of invariance of an estimator can be applied when seeking estimators of alternative forms, either for the sake of simplicity of application of the estimator or so that the estimator is robust.
The concept of invariance is sometimes used on its own as a way of choosing between estimators, but this is not necessarily definitive. For example, a requirement of invariance may be incompatible with the requirement that the estimator be mean-unbiased; on the other hand, the criterion of median-unbiasedness is defined in terms of the estimator's sampling distribution and so is invariant under many transformations.
One use of the concept of invariance is where a class or family of estimators is proposed and a particular formulation must be selected amongst these. One procedure is to impose relevant invariance properties and then to find the formulation within this class that has the best properties, leading to what is called the |
https://en.wikipedia.org/wiki/Principal%20component%20regression | In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model.
In PCR, instead of regressing the dependent variable on the explanatory variables directly, the principal components of the explanatory variables are used as regressors. One typically uses only a subset of all the principal components for regression, making PCR a kind of regularized procedure and also a type of shrinkage estimator.
Often the principal components with higher variances (the ones based on eigenvectors corresponding to the higher eigenvalues of the sample variance-covariance matrix of the explanatory variables) are selected as regressors. However, for the purpose of predicting the outcome, the principal components with low variances may also be important, in some cases even more important.
One major use of PCR lies in overcoming the multicollinearity problem which arises when two or more of the explanatory variables are close to being collinear. PCR can aptly deal with such situations by excluding some of the low-variance principal components in the regression step. In addition, by usually regressing on only a subset of all the principal components, PCR can result in dimension reduction through substantially lowering the effective number of parameters characterizing the underlying model. This can be particularly useful in settings with high-dimensional covariates. Also, through appropriate selection of the principal components to be used for regression, PCR can lead to efficient prediction of the outcome based on the assumed model.
The principle
The PCR method may be broadly divided into three major steps:
1. Perform PCA on the observed data matrix for the explanatory variables to obtain the principal components, and then (usually) select a subset, based on some appropriate criteria, of the principal components so obtained for further use.
2. Now regress the observed vector of outcomes on the selected principal components as covariates, using ordinary least squares regression (linear regression) to get a vector of estimated regression coefficients (with dimension equal to the number of selected principal components).
3. Now transform this vector back to the scale of the actual covariates, using the selected PCA loadings (the eigenvectors corresponding to the selected principal components) to get the final PCR estimator (with dimension equal to the total number of covariates) for estimating the regression coefficients characterizing the original model.
Details of the method
Data representation: Let denote the vector of observed outcomes and denote the corresponding data matrix of observed covariates where, and denote the size of the observed sample and the number of covariates respectively, with . Each of the rows of denotes one set of observations for the |
https://en.wikipedia.org/wiki/Clifford%20Hugh%20Dowker | Clifford Hugh Dowker (; March 2, 1912 – October 14, 1982) was a topologist known for his work in point-set topology and also for his contributions in category theory, sheaf theory and knot theory.
Biography
Clifford Hugh Dowker grew up on a small farm in Western Ontario, Canada. He excelled in mathematics and was paid to teach his math teacher math at his secondary school. He was awarded a scholarship at Western Ontario University, where he got his B.S. in 1933. He wanted to pursue a career as a teacher, but he was persuaded to continue with his education because of his extraordinary mathematical talent. He earned his M.A. from the University of Toronto in 1936 and his Ph.D. from Princeton University in 1938. His dissertation Mapping theorems in non-compact spaces was written under the supervision of Solomon Lefschetz and was published (with additions) in 1947 in the American Journal of Mathematics. After earning his doctorate, Dowker became an instructor at the Western Ontario University for a year. The next year, he worked as an assistant back at Princeton under John von Neumann. During World War II, he worked for the U.S. Air Force, calculating the trajectories of projectiles. He married Yael Naim in 1944. After the war, he was appointed associate professor at the Tufts University. Because of Senator Joseph McCarthy's red scare, he decided to take his family to England shortly thereafter, where he was appointed Reader in applied mathematics at Birkbeck College in 1951. In 1962 he was granted a personal chair, until he retired in 1979. The last years of his life were marked by a long illness, yet he continued working, developing Dowker notation in the weeks before his death.
Work
Dowker showed that Čech and Vietoris homology groups coincide, as do the Čech cohomology and Alexander cohomology groups. Along with Morwen Thistlethwaite, he developed Dowker notation, a simple way of describing knots, suitable for computers.
His most highly cited article is his 1951 paper in which he introduced the concept of countably paracompact spaces.
Dowker conjectured that so-called Dowker spaces could not exist, a conjecture ultimately proven false in a famous 1971 paper by Mary Ellen Rudin.
See also
Dowker notation
Dowker space
Čech cohomology
Morwen Thistlethwaite
Sheaf theory
References
External links
Topologists
1982 deaths
1912 births
University of Western Ontario alumni
20th-century Canadian mathematicians
Canadian emigrants to the United States |
https://en.wikipedia.org/wiki/Jorge%20Delgado%20%28footballer%2C%20born%201975%29 | Jorge Luis Delgado Rueda (born September 30, 1975 in Montevideo) is a retired Uruguayan football striker.
References
External links
Profile at BoliviaGol.com
Profile & Statistics at LFP.es
Living people
1975 births
Uruguayan men's footballers
Men's association football forwards
Liverpool F.C. (Montevideo) players
CD Numancia players
Elche CF players
Racing de Ferrol footballers
Club Nacional de Football players
Montevideo Wanderers F.C. players
Everton de Viña del Mar footballers
C.D. Cuenca footballers
C.A. Cerro players
Club Aurora players
Expatriate men's footballers in Ecuador
Expatriate men's footballers in Bolivia
Expatriate men's footballers in Spain
Expatriate men's footballers in Chile |
https://en.wikipedia.org/wiki/CONCACAF%20Champions%20Cup%20and%20Champions%20League%20records%20and%20statistics | This page details statistics of the CONCACAF Champions Cup and Champions League. Unless notified, these statistics concern all seasons since inception of the Champions' Cup in the 1962 season.
General performances
Finals performances
†Title shared.
*Including one title shared.
Overall team records
In this ranking 3 points are awarded for a win, 1 for a draw and 0 for a loss. As per statistical convention in football, matches decided in extra time are counted as wins and losses, while matches decided by penalty shoot-outs are counted as draws. Teams are ranked by total points, then by goal difference, then by goals scored.
This table includes only Final Rounds as listed by CONCACAF.
CONCACAF Champions League era records
By club
Year in Bold: Club was finalist in that year
By country
Clubs
These records are only from the Champions League era.
Biggest wins
Home:
Herediano 8–0 Alpha United (2011–12)
UNAM 8–0 Isidro Metapán (2011–12)
Querétaro 8–0 Verdes FC (2015–16)
San Francisco 8–0 Verdes FC (2015–16)
Cruz Azul 8–0 Arcahaie (2021)
Away:
Police United 0–11 Pachuca (2016–17)
Highest scoring
Police United 0–11 Pachuca (2016–17)
Puerto Rico Bayamón 1–10 América (2014–15)
Most minutes without conceding a goal
Real Salt Lake – 587
Players
These records are only from the Champions League era.
All-time top scorers
Preliminary round goals included.
Top scorers by season
Most goals in a single game
5 goals: Emanuel Villa (Querétaro), 8–0 against Verdes, group stage, 2015–16
Most goals in a single season
Javier Orozco – 11 goals (2010–11)
Most hat-tricks
Javier Orozco – 5
List of hat-tricks
4 Player scored 4 goals
5 Player scored 5 goals
Managers
By year
Managers with multiple titles
By nationality
This table lists the total number of titles won by managers of each country. Accurate as of the 2023 final.
References
External links
CONCACAF official website
CONCACAF Cup at RSSSF.com
records |
https://en.wikipedia.org/wiki/Ordinal%20collapsing%20function | In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals.
The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordinal is brought "from above" to give a name to that critical point. An example of how this works will be detailed below, for an ordinal collapsing function defining the Bachmann–Howard ordinal (i.e., defining a system of notations up to the Bachmann–Howard ordinal).
The use and definition of ordinal collapsing functions is inextricably intertwined with the theory of ordinal analysis, since the large countable ordinals defined and denoted by a given collapse are used to describe the ordinal-theoretic strength of certain formal systems, typically subsystems of analysis (such as those seen in the light of reverse mathematics), extensions of Kripke–Platek set theory, Bishop-style systems of constructive mathematics or Martin-Löf-style systems of intuitionistic type theory.
Ordinal collapsing functions are typically denoted using some variation of either the Greek letter (psi) or (theta).
An example leading up to the Bachmann–Howard ordinal
The choice of the ordinal collapsing function given as example below imitates greatly the system introduced by Buchholz but is limited to collapsing one cardinal for clarity of exposition. More on the relation between this example and Buchholz's system will be said below.
Definition
Let stand for the first uncountable ordinal , or, in fact, any ordinal which is an -number and guaranteed to be greater than all the countable ordinals which will be constructed (for example, the Church–Kleene ordinal is adequate for our purposes; but we will work with because it allows the convenient use of the word countable in the definitions).
We define a function (which will be non-decreasing and continuous), taking an arbitrary ordinal to a countable ordinal , recursively on , as follows:
Assume has been defined for all , and we wish to define .
Let be the set of ordinals generated starting from , , and by recursively applying the following functions: ordinal addition, multiplication and exponentiation and the function , i.e., the restriction of to ordinals . (Formally, we define and inductively for all natural numbers and we let be the union of the for al |
https://en.wikipedia.org/wiki/Kafr%20Ein | Kafr Ein () is a Palestinian village in the Ramallah and al-Bireh Governorate, located northwest of Ramallah in the central West Bank. According to the Palestinian Central Bureau of Statistics (PCBS), Kafr Ein had a population of 1,958 inhabitants in 2017. Most of the village's population comes from the Barghouti, Rifa' and Rafati clans.
Location
Kafr 'Ein is located 17.7 km northwest of Ramallah. It is bordered by Qarawat Bani Zeid, Bani Zeid ash Sharqiya and Deir as Sudan to the east, Bruqin to the north, Bani Zeid to the west, and An Nabi Salih to the south.
History
Kafr Ein is transliterated as "spring village". The village contains ten springs and ten reservoirs, one of which was recently damaged.
It is believed that there is an ancient site at the top of a local mountain known as Haraek, which contains a church and a mosque. According to local legend, the site was destroyed during the Crusades and the single villager who survived its destruction came down to found Kafr Ein.
Ottoman era
Potsherds from the early Ottoman era have been found. It is noted in the Ottoman tax records of the 16th century as being located in the Sanjak of Al-Quds.
Kafr Ein was ruled by the Barghouti family throughout the later half of the Ottoman rule of Palestine, located within the sheikhdom of Bani Zeid. It produced 52 qintars of olive oil annually, exporting it to Jerusalem or Nablus mainly for traditional soap-making.
In 1838, it was noted under the name of Kefr Iyan as a Muslim village in the District of Beni Zaid, north of Jerusalem.
The French explorer Victor Guérin passed by the village in 1870, and noted that it "did not seem very considerable," while an Ottoman village list of about the same year showed a population of 260, in 69 houses, though the population count only included men.
In 1882, the PEF's Survey of Western Palestine described "Kefr Ain" as a "small hamlet on a hill-slope, supplied by the Ain Mathrun."
In 1896 the population of Kefr ‘ain was estimated to be about 516 persons.
British Mandate era
In the 1922 census of Palestine conducted by the British Mandate authorities, the village, called Kufr 'Ain, had a population of 376, all Muslims, increasing in the 1931 census where In Kafr had a population of 494, still all Muslims, in a total of 133 houses.
In the 1945 statistics, the population was 550 Muslims, while the total land area was 7,145 dunams, according to an official land and population survey. Of this, 4,928 were allocated for plantations and irrigable land, 724 for cereals, while 19 dunams were classified as built-up areas.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Kafr Ein came under Jordanian rule.
The Jordanian census of 1961 found 1,095 inhabitants in Kafr 'Ain.
1967-present
Since the Six-Day War in 1967, Kafr Ein has been held under Israeli occupation.
After the 1995 accords, 97.4% of Kafr Ein land was classified as Area A land, 1.4% as Area B, |
https://en.wikipedia.org/wiki/Median%20graph | In graph theory, a division of mathematics, a median graph is an undirected graph in which every three vertices a, b, and c have a unique median: a vertex m(a,b,c) that belongs to shortest paths between each pair of a, b, and c.
The concept of median graphs has long been studied, for instance by or (more explicitly) by , but the first paper to call them "median graphs" appears to be . As Chung, Graham, and Saks write, "median graphs arise naturally in the study of ordered sets and discrete distributive lattices, and have an extensive literature". In phylogenetics, the Buneman graph representing all maximum parsimony evolutionary trees is a median graph. Median graphs also arise in social choice theory: if a set of alternatives has the structure of a median graph, it is possible to derive in an unambiguous way a majority preference among them.
Additional surveys of median graphs are given by , , and .
Examples
Every tree is a median graph. To see this, observe that in a tree, the union of the three shortest paths between pairs of the three vertices a, b, and c is either itself a path, or a subtree formed by three paths meeting at a single central node with degree three. If the union of the three paths is itself a path, the median m(a,b,c) is equal to one of a, b, or c, whichever of these three vertices is between the other two in the path. If the subtree formed by the union of the three paths is not a path, the median of the three vertices is the central degree-three node of the subtree.
Additional examples of median graphs are provided by the grid graphs. In a grid graph, the coordinates of the median m(a,b,c) can be found as the median of the coordinates of a, b, and c. Conversely, it turns out that, in every median graph, one may label the vertices by points in an integer lattice in such a way that medians can be calculated coordinatewise in this way.
Squaregraphs, planar graphs in which all interior faces are quadrilaterals and all interior vertices have four or more incident edges, are another subclass of the median graphs. A polyomino is a special case of a squaregraph and therefore also forms a median graph.
The simplex graph κ(G) of an arbitrary undirected graph G has a vertex for every clique (complete subgraph) of G; two vertices of κ(G) are linked by an edge if the corresponding cliques differ by one vertex of G . The simplex graph is always a median graph, in which the median of a given triple of cliques may be formed by using the majority rule to determine which vertices of the cliques to include.
No cycle graph of length other than four can be a median graph. Every such cycle has three vertices a, b, and c such that the three shortest paths wrap all the way around the cycle without having a common intersection. For such a triple of vertices, there can be no median.
Equivalent definitions
In an arbitrary graph, for each two vertices a and b, the minimal number of edges between them is called their distance, denot |
https://en.wikipedia.org/wiki/1925%E2%80%9326%20Football%20League | The 1925–26 season was the 34th season of The Football League.
Final league tables
The tables below are reproduced here in the exact form that they can be found at The Rec.Sport.Soccer Statistics Foundation website and in Rothmans Book of Football League Records 1888–89 to 1978–79, with home and away statistics separated.
Match results are drawn from Rothmans for all divisions.
Beginning with the season 1894–95, clubs finishing level on points were separated according to goal average (goals scored divided by goals conceded), or more properly put, goal ratio. In case one or more teams had the same goal difference, this system favoured those teams who had scored fewer goals. The goal average system was eventually scrapped beginning with the 1976–77 season. From the 1922–23 season on, re-election was required of the bottom two teams of both Third Division North and Third Division South.
First Division
Results
Maps
Second Division
Results
Maps
Third Division North
Results
Maps
Third Division South
Results
Maps
See also
1925–26 in English football
1925 in association football
1926 in association football
References
Ian Laschke: Rothmans Book of Football League Records 1888–89 to 1978–79. Macdonald and Jane's, London & Sydney, 1980.
English Football League seasons |
https://en.wikipedia.org/wiki/1928%E2%80%9329%20Football%20League | The 1928–29 season was the 37th season of The Football League.
Final league tables
The tables and results below are reproduced here with home and away statistics separated, as per RSSSF and Rothmans Book of Football League Records 1888–89 to 1978–79.
Beginning with the season 1894–95, clubs finishing level on points were separated according to goal average (goals scored divided by goals conceded), or more properly put, goal ratio. When two teams had the same goal difference, this system favoured those teams who had scored fewer goals. The goal average system was eventually scrapped beginning with the 1976–77 season.
From the 1922–23 season, re-election was required of the bottom two teams of both Third Division North and Third Division South.
First Division
Results
Maps
Second Division
Results
Maps
Third Division North
Results
Maps
Third Division South
Results
Maps
References
English Football League seasons
Eng
1928–29 in English football leagues |
https://en.wikipedia.org/wiki/Bon-dong | Bon-dong is a dong, neighbourhood of Dongjak-gu in Seoul, South Korea.
See also
Administrative divisions of South Korea
References
External links
Map and statistics of Dongjak-gu
Neighbourhoods of Dongjak District |
https://en.wikipedia.org/wiki/Daebang-dong | Daebang-dong is a dong, neighbourhood of Dongjak-gu in Seoul, South Korea.
History
See also
Administrative divisions of South Korea
References
External links
Map and statistics of Dongjak-gu
Neighbourhoods of Dongjak District |
https://en.wikipedia.org/wiki/Noryangjin-dong | Noryangjin-dong is a dong, neighbourhood of Dongjak-gu in Seoul, South Korea.
See also
Administrative divisions of South Korea
References
External links
Map and statistics of Dongjak-gu
Neighbourhoods of Dongjak District |
https://en.wikipedia.org/wiki/Sangdo-dong | Sangdo-dong is a dong, neighbourhood of Dongjak-gu in Seoul, South Korea.
See also
Administrative divisions of South Korea
References
External links
Map and statistics of Dongjak-gu
Neighbourhoods of Dongjak District |
https://en.wikipedia.org/wiki/Sindaebang-dong | Sindaebang-dong is a dong, neighbourhood of Dongjak-gu in Seoul, South Korea.
See also
Administrative divisions of South Korea
References
External links
Map and statistics of Dongjak-gu
Neighbourhoods of Dongjak District |
https://en.wikipedia.org/wiki/Anger%20function | In mathematics, the Anger function, introduced by , is a function defined as
with complex parameter v and complex variable x. It is closely related to the Bessel functions.
The Weber function (also known as Lommel–Weber function), introduced by , is a closely related function defined by
and is closely related to Bessel functions of the second kind.
Relation between Weber and Anger functions
The Anger and Weber functions are related by
so in particular if ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions Jν are the same as Bessel functions Jν, and Weber functions can be expressed as finite linear combinations of Struve functions.
Power series expansion
The Anger function has the power series expansion
While the Weber function has the power series expansion
Differential equations
The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation
More precisely, the Anger functions satisfy the equation
and the Weber functions satisfy the equation
Recurrence relations
The Anger function satisfies this inhomogeneous form of recurrence relation
While the Weber function satisfies this inhomogeneous form of recurrence relation
Delay differential equations
The Anger and Weber functions satisfy these homogeneous forms of delay differential equations
The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations
References
C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952)
H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76
Special functions |
https://en.wikipedia.org/wiki/Guaraque%20Municipality | The Guaraque Municipality is one of the 23 municipalities (municipios) that makes up the Venezuelan state of Mérida and, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, the municipality has a population of 9,965. The town of Guaraque is the shire town of the Guaraque Municipality.
History
The Guaraque Municipality became a part of the former Rivas Davila district (today Rivas Davila Municipality) in 1904. Later, in 1984, this area became the Autonomous Municipality of Guaraque, formed by the Mesa de Quintero and Río Negro Municipalities. In 1992, as a result of the modification of the territorial law, the current municipal parishes were created. The most popular celebration of the municipality are that of Santa Barbara that take place from November 23 until January 4.
Geography
This is a mountainous region located in the Venezuelan Andes and part of the municipality is protected by the Páramos Batallón and La Negra national parks. The Negro, Guaraque, and El Molino rivers are the municipality's main bodies of water which have formed a valley at 1,500 meters above sea level, although most of the municipality is above 2,000 meters above sea level, with a maximum altitude of 3,532 meters above sea level.
Demographics
The Guaraque Municipality, according to a 2007 population estimate by the National Institute of Statistics of Venezuela, has a population of 9,965 (up from 8,641 in 2000). This amounts to 1.2% of the state's population. The municipality's population density is .
Government
The mayor of the Guaraque Municipality is Carlos Alí Guerrero, elected on October 31, 2004, with 61% of the vote. He replaced Rosmel Javier Sanchez shortly after the elections. The municipality is divided into three parishes; Capital Guaraque, Mesa de Quintero, and Río Negro.
References
External links
alcaldiadeguaraquemerida.gob.ve
Municipalities of Mérida (state) |
https://en.wikipedia.org/wiki/Dol%C3%A9ans-Dade%20exponential | In stochastic calculus, the Doléans-Dade exponential or stochastic exponential of a semimartingale X is the unique strong solution of the stochastic differential equation where denotes the process of left limits, i.e., .
The concept is named after Catherine Doléans-Dade. Stochastic exponential plays an important role in the formulation of Girsanov's theorem and arises naturally in all applications where relative changes are important since measures the cumulative percentage change in .
Notation and terminology
Process obtained above is commonly denoted by . The terminology "stochastic exponential" arises from the similarity of to the natural exponential of : If X is absolutely continuous with respect to time, then Y solves, path-by-path, the differential equation , whose solution is .
General formula and special cases
Without any assumptions on the semimartingale , one has where is the continuous part of quadratic variation of and the product extends over the (countably many) jumps of X up to time t.
If is continuous, then In particular, if is a Brownian motion, then the Doléans-Dade exponential is a geometric Brownian motion.
If is continuous and of finite variation, then Here need not be differentiable with respect to time; for example, can be the Cantor function.
Properties
Stochastic exponential cannot go to zero continuously, it can only jump to zero. Hence, the stochastic exponential of a continuous semimartingale is always strictly positive.
Once has jumped to zero, it is absorbed in zero. The first time it jumps to zero is precisely the first time when .
Unlike the natural exponential , which depends only of the value of at time , the stochastic exponential depends not only on but on the whole history of in the time interval . For this reason one must write and not .
Natural exponential of a semimartingale can always be written as a stochastic exponential of another semimartingale but not the other way around.
Stochastic exponential of a local martingale is again a local martingale.
All the formulae and properties above apply also to stochastic exponential of a complex-valued . This has application in the theory of conformal martingales and in the calculation of characteristic functions.
Useful identities
Yor's formula: for any two semimartingales and one has
Applications
Stochastic exponential of a local martingale appears in the statement of Girsanov theorem. Criteria to ensure that the stochastic exponential of a continuous local martingale is a martingale are given by Kazamaki's condition, Novikov's condition, and Beneš's condition.
Derivation of the explicit formula for continuous semimartingales
For any continuous semimartingale X, take for granted that is continuous and strictly positive. Then applying Itō's formula with gives
Exponentiating with gives the solution
This differs from what might be expected by comparison with the case where X has finite variation due to the exi |
https://en.wikipedia.org/wiki/Isomonodromic%20deformation | In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems.
Isomonodromic deformations were first studied by Richard Fuchs, with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, René Garnier, and Ludwig Schlesinger. Inspired by results in statistical mechanics, a seminal contribution to the theory was made by Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, who studied cases involving irregular singularities.
Fuchsian systems and Schlesinger's equations
Fuchsian system
A Fuchsian system is the system of linear differential equations
where x takes values in the complex projective line , the y takes values in and the Ai are constant n×n matrices. Solutions to this equation have polynomial growth in the limit x = λi.
By placing n independent column solutions into a fundamental matrix
then and
one can regard as taking values in . For simplicity, assume that there is no further pole at infinity, which amounts to the condition that
Monodromy data
Now, fix a basepoint b on the Riemann sphere away from the poles. Analytic continuation of a fundamental solution around any pole λi and back to the basepoint will produce a new solution defined near b. The new and old solutions are linked by the monodromy matrix Mi as follows:
One therefore has the Riemann–Hilbert homomorphism from the fundamental group of the punctured sphere to the monodromy representation:
A change of basepoint merely results in a (simultaneous) conjugation of all the monodromy matrices. The monodromy matrices modulo conjugation define the monodromy data of the Fuchsian system.
Hilbert's twenty-first problem
Now, with given monodromy data, can a Fuchsian system be found which exhibits this monodromy? This is one form of Hilbert's twenty-first problem. One does not distinguish between coordinates x and which are related by Möbius transformations, and also do not distinguish between gauge equivalent Fuchsian systems - this means that A and
are regarded as being equivalent for any holomorphic gauge transformation g(x). (It is thus most natural to regard a Fuchsian system geometrically, as a connection with simple poles on a trivial rank n vector bundle over the Riemann sphere).
For generic monodromy data, the answer to Hilbert's twenty-first problem is 'yes'. The first proof was given by Josip Plemelj. However, the proof only holds for generic data, and it was shown in 1989 by Andrei Bolibrukh that there are certain 'degenerate' cases when the answer is 'no'. Here, the generic case is focused upon entirely.
Schlesinger's equations
There are generically many Fuchsian systems with the same monodromy data. Thus, given any such Fuchsian system with specified monodromy data, |
https://en.wikipedia.org/wiki/Klaus%20Wagner | Klaus Wagner (March 31, 1910 – February 6, 2000) was a German mathematician known for his contributions to graph theory.
Education and career
Wagner studied topology at the University of Cologne under the supervision of who had been a student of Issai Schur. Wagner received his Ph.D. in 1937, with a dissertation concerning the Jordan curve theorem and four color theorem, and taught at Cologne for many years himself. In 1970, he moved to the University of Duisburg, where he remained until his retirement in 1978.
Graph minors
Wagner is known for his contributions to graph theory and particularly the theory of graph minors, graphs that can be formed from a larger graph by contracting and removing edges.
Wagner's theorem characterizes the planar graphs as exactly those graphs that do not have as a minor either a complete graph K5 on five vertices or a complete bipartite graph K3,3 with three vertices on each side of its bipartition. That is, these two graphs are the only minor-minimal non-planar graphs. It is closely related to, but should be distinguished from, Kuratowski's theorem, which states that the planar graphs are exactly those graphs that do not contain as a subgraph a subdivision of K5 or K3,3.
Another result of his, also known as Wagner's theorem, is that a four-connected graph is planar if and only if it has no K5 minor. This implies a characterization of the graphs with no K5 minor as being constructed from planar graphs and Wagner graph (an eight-vertex Möbius ladder) by clique-sums, operations that glue together subgraphs at cliques of up to three vertices and then possibly remove edges from those cliques. This characterization was used by Wagner to show that the case k = 5 of the Hadwiger conjecture on the chromatic number of Kk-minor-free graphs is equivalent to the four color theorem. Analogous characterizations of other families of graphs in terms of the summands of their clique-sum decompositions have since become standard in graph minor theory.
Wagner conjectured in the 1930s (although this conjecture was not published until later) that in any infinite set of graphs, one graph is isomorphic to a minor of another. The truth of this conjecture implies that any family of graphs closed under the operation of taking minors (as planar graphs are) can automatically be characterized by finitely many forbidden minors analogously to Wagner's theorem characterizing the planar graphs. Neil Robertson and Paul Seymour finally published a proof of Wagner's conjecture in 2004 and it is now known as the Robertson–Seymour theorem.
Recognition
Wagner was honored in 1990 by a festschrift on graph theory, and in June 2000, following Wagner's death, the University of Cologne hosted a Festkolloquium in his memory.
Selected publications
.
References
1910 births
2000 deaths
20th-century German mathematicians
Topologists
Graph theorists |
https://en.wikipedia.org/wiki/Spectral%20theory%20of%20ordinary%20differential%20equations | In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
Introduction
Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm–Liouville theory. In modern language, it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl extended this theory to second order ordinary differential equations with singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced boundary conditions in terms of his celebrated dichotomy between limit points and limit circles.
In the 1920s, John von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira used to streamline Weyl's method. Kodaira also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the spectral measure. The same formula had also been obtained independently by E. C. Titchmarsh in 1946 (scientific communication between Japan and the United Kingdom had been interrupted by World War II). Titchmarsh had followed the method of the German mathematician Emil Hilb, who derived the eigenfunction expansions using complex function theory instead of operator theory. Other methods avoiding the spectral theorem were later developed independently by Levitan, Levinson and Yoshida, who used the fact that the resolvent of the singular differential operator could be approximated by compact resolvents corresponding to Sturm–Liouville problems for proper subintervals. Another method was found by Mark Grigoryevich Krein; his use of direction functionals was subsequently generalised by Izrail Glazman to arbitrary ordinary differential equations of even order.
Weyl app |
https://en.wikipedia.org/wiki/Thomas%20A.%20Scott%20Professorship%20of%20Mathematics | The Thomas A. Scott Professorship of Mathematics is an academic grant made to the University of Pennsylvania. It was established in 1881 by the railroad executive and financier Thomas Alexander Scott.
Recipients
Ezra Otis Kendall, 1881–1899
Edwin Schofield Crawley, 1899–1933
George Hervey Hallett, 1933–1941
John Robert Kline, 1941–1955
Hans A. Rademacher, 1956–1962
Eugenio Calabi, 1967–1993
Shmuel Weinberger, 1994–1996
Herbert S. Wilf, 1998–2006
Charles Epstein, 2008–present
See also
Thomas A. Scott Fellowship in Hygiene
External links
University of Pennsylvania Mathematics Department page about the Professorship
References
Professorships in mathematics
Awards established in 1881
University of Pennsylvania
Mathematics education in the United States
1881 establishments in Pennsylvania |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Belgian%20First%20Division | Statistics of Belgian League in season 1983–84.
Overview
It was performed by 18 teams, and K.S.K. Beveren won the championship.
League standings
Results
Top scorers
References
Belgian Pro League seasons
Belgian
1 |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Belgian%20First%20Division | Statistics of Belgian League in season 1984–85.
Overview
It was performed by 18 teams, and R.S.C. Anderlecht won the championship, while K. Sint-Niklase S.K.E. and Racing Jet de Bruxelles were relegated.
League standings
Results
Topscorers
References
Belgian Pro League seasons
Belgian
1 |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Belgian%20First%20Division | Statistics of Belgian League in season 1985–86.
Overview
It was performed by 18 teams, and R.S.C. Anderlecht won the championship, while K. Waterschei S.V. Thor Genk & Lierse S.K. were relegated.
League standings
Results
Topscorers
References
Belgian Pro League seasons
Belgian
1 |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Belgian%20First%20Division | Statistics of Belgian League in season 1987–88.
Overview
It was contested by 18 teams, and Club Brugge K.V. won the championship, while K.A.A. Gent & Racing Jet de Bruxelles were relegated.
League standings
Results
Topscorers
References
Belgian Pro League seasons
Belgian
1 |
https://en.wikipedia.org/wiki/1932%E2%80%9333%20Football%20League | The 1932–33 season was the 41st season of The Football League.
Final league tables
Match results are drawn from The Rec.Sport.Soccer Statistics Foundation website and Rothmans for the First Division and from Rothmans for the Second Division and for the two Third Divisions.
From the 1922–23 season onwards, re-election was required of the bottom two teams of both the Third Division North and Third Division South leagues.
First Division
Results
Maps
Second Division
Results
Maps
Third Division North
Results
Maps
Third Division South
Results
Maps
See also
1932–33 in English football
1932 in association football
1933 in association football
References
Ian Laschke: Rothmans Book of Football League Records 1888–89 to 1978–79. Macdonald and Jane’s, London & Sydney, 1980.
English Football League seasons
Eng
1932–33 in English football leagues |
https://en.wikipedia.org/wiki/1938%E2%80%9339%20Football%20League | The 1938–39 season was the 47th season of the Football League.
Final league tables
The tables below are reproduced here in the exact form that they can be found at The Rec.Sport.Soccer Statistics Foundation website and in Rothmans Book of Football League Records 1888–89 to 1978–79, with home and away statistics separated.
Match results are drawn from The Rec.Sport.Soccer Statistics Foundation website and Rothmans for the First Division and from Rothmans for the Second Division and for the two Third Divisions.
Beginning with the season 1894–95, clubs finishing level on points were separated according to goal average (goals scored divided by goals conceded), or more properly put, goal ratio. In case one or more teams had the same goal difference, this system favoured those teams who had scored fewer goals, if the teams had a positive goal difference. The goal average system was eventually scrapped beginning with the 1976–77 season.
From the 1922–23 season, the bottom two teams of both Third Division North and Third Division South were required to apply for re-election.
First Division
Results
Maps
Second Division
Results
Maps
Third Division North
Results
Maps
Third Division South
Results
Maps
See also
1938-39 in English football
1938 in association football
1939 in association football
References
Ian Laschke: Rothmans Book of Football League Records 1888–89 to 1978–79. Macdonald and Jane’s, London & Sydney, 1980.
English Football League seasons
Eng
1938–39 in English football leagues |
https://en.wikipedia.org/wiki/Reduced%20derivative | In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have derivatives in the sense of Radon measures, it is desirable to have a derivative that takes values in the same space as the functions themselves. Although the precise definition of the reduced derivative is quite involved, its key properties are quite easy to remember:
it is a multiple of the usual derivative wherever it exists;
at jump points, it is a multiple of the jump vector.
The notion of reduced derivative appears to have been introduced by Alexander Mielke and Florian Theil in 2004.
Definition
Let X be a separable, reflexive Banach space with norm || || and fix T > 0. Let BV−([0, T]; X) denote the space of all left-continuous functions z : [0, T] → X with bounded variation on [0, T].
For any function of time f, use subscripts +/− to denote the right/left continuous versions of f, i.e.
For any sub-interval [a, b] of [0, T], let Var(z, [a, b]) denote the variation of z over [a, b], i.e., the supremum
The first step in the construction of the reduced derivative is the "stretch" time so that z can be linearly interpolated at its jump points. To this end, define
The "stretched time" function τ̂ is left-continuous (i.e. τ̂ = τ̂−); moreover, τ̂− and τ̂+ are strictly increasing and agree except at the (at most countable) jump points of z. Setting T̂ = τ̂(T), this "stretch" can be inverted by
Using this, the stretched version of z is defined by
where θ ∈ [0, 1] and
The effect of this definition is to create a new function ẑ which "stretches out" the jumps of z by linear interpolation. A quick calculation shows that ẑ is not just continuous, but also lies in a Sobolev space:
The derivative of ẑ(τ) with respect to τ is defined almost everywhere with respect to Lebesgue measure. The reduced derivative of z is the pull-back of this derivative by the stretching function τ̂ : [0, T] → [0, T̂]. In other words,
Associated with this pull-back of the derivative is the pull-back of Lebesgue measure on [0, T̂], which defines the differential measure μz:
Properties
The reduced derivative rd(z) is defined only μz-almost everywhere on [0, T].
If t is a jump point of z, then
If z is differentiable on (t1, t2), then
and, for t ∈ (t1, t2),
,
For 0 ≤ s < t ≤ T,
References
Differential calculus
Mathematical analysis |
https://en.wikipedia.org/wiki/219%20%28number%29 | 219 (two hundred [and] nineteen) is the natural number following 218 and preceding 220.
In mathematics
219 is a happy number.
Mertens function(219) = 4, a record high.
There are 219 partially ordered sets on four labeled elements.
219 is the smallest number that can be represented as a sum of four positive cubes in two different ways.
There are 219 different space groups, discrete and full-dimensional sets of symmetries of three-dimensional space or of crystal structures.
References
Integers |
https://en.wikipedia.org/wiki/Sebasti%C3%A1n%20Penco | Sebastián Ariel Penco (born 22 September 1983 in Morón, Buenos Aires) is an Argentine footballer who currently plays as a forward for San Martín SJ.
External links
Profile & Statistics at Guardian's Stats Centre
Apertura 2004 Statistics at Terra.com.ar
Ascenso MX Profile
1983 births
Living people
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football forwards
Footballers from Buenos Aires Province
Everton de Viña del Mar footballers
Xanthi F.C. players
Club Atlético Nueva Chicago footballers
Club Almirante Brown footballers
Club Atlético Independiente footballers
Racing Club de Avellaneda footballers
Deportivo Español footballers
Club Atlético Aldosivi footballers
Club Almagro players
Once Caldas footballers
San Martín de San Juan footballers
Atlético San Luis footballers
Murciélagos F.C. footballers
Club Atlético Sarmiento footballers
Sport Boys footballers
Chilean Primera División players
Argentine Primera División players
Primera Nacional players
Super League Greece players
Categoría Primera A players
Ascenso MX players
Peruvian Primera División players
Argentine expatriate sportspeople in Chile
Argentine expatriate sportspeople in Greece
Argentine expatriate sportspeople in Colombia
Argentine expatriate sportspeople in Mexico
Argentine expatriate sportspeople in Peru
Expatriate men's footballers in Chile
Expatriate men's footballers in Greece
Expatriate men's footballers in Colombia
Expatriate men's footballers in Mexico
Expatriate men's footballers in Peru |
https://en.wikipedia.org/wiki/Polynomial%20SOS | In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms of degree m such that
Every form that is SOS is also a positive polynomial, and although the converse is not always true, Hilbert proved that for n = 2, 2m = 2, or n = 3 and 2m = 4 a form is SOS if and only if it is positive. The same is also valid for the analog problem on positive symmetric forms.
Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found. Moreover, every real nonnegative form can be approximated as closely as desired (in the -norm of its coefficient vector) by a sequence of forms that are SOS.
Square matricial representation (SMR)
To establish whether a form is SOS amounts to solving a convex optimization problem. Indeed, any can be written as
where is a vector containing a base for the forms of degree m in x (such as all monomials of degree m in x), the prime ′ denotes the transpose, H is any symmetric matrix satisfying
and is a linear parameterization of the linear space
The dimension of the vector is given by
whereas the dimension of the vector is given by
Then, is SOS if and only if there exists a vector such that
meaning that the matrix is positive-semidefinite. This is a linear matrix inequality (LMI) feasibility test, which is a convex optimization problem. The expression was introduced in with the name square matricial representation (SMR) in order to establish whether a form is SOS via an LMI. This representation is also known as Gram matrix.
Examples
Consider the form of degree 4 in two variables . We have Since there exists α such that , namely , it follows that h(x) is SOS.
Consider the form of degree 4 in three variables . We have Since for , it follows that is SOS.
Generalizations
Matrix SOS
A matrix form F(x) (i.e., a matrix whose entries are forms) of dimension r and degree 2m in the real n-dimensional vector x is SOS if and only if there exist matrix forms of degree m such that
Matrix SMR
To establish whether a matrix form F(x) is SOS amounts to solving a convex optimization problem. Indeed, similarly to the scalar case any F(x) can be written according to the SMR as
where is the Kronecker product of matrices, H is any symmetric matrix satisfying
and is a linear parameterization of the linear space
The dimension of the vector is given by
Then, is SOS if and only if there exists a vector such that the following LMI holds:
The expression was introduced in in order to establish whether a matrix form is SOS via an LMI.
Noncommutative polynomial SOS
Consider the free algebra R⟨X⟩ generated by the n noncommuting letters X = (X1, ..., Xn) and equipped with the involution T, such that T fixes R and X1, ..., Xn and reverses words formed by X1, ..., Xn.
By analogy with the commutative case, the noncommutative symmetric polynomials f are th |
https://en.wikipedia.org/wiki/Gabriel%20Roth%20%28footballer%29 | Gabriel Fernando Roth (born 5 May 1979 in Venado Tuerto, Santa Fe) is an Argentine footballer playing for Rangers.
External links
Argentine Primera statistics
Apertura 2007 Statistics at Terra.com.ar
1979 births
Living people
Argentine men's footballers
Argentine expatriate men's footballers
Men's association football midfielders
Gimnasia y Esgrima de Jujuy footballers
San Martín de San Juan footballers
Independiente Rivadavia footballers
Talleres de Córdoba footballers
Córdoba CF players
Rangers de Talca footballers
Club Atlético Patronato footballers
Atlético Bucaramanga footballers
Categoría Primera A players
Primera B de Chile players
Expatriate men's footballers in Chile
Expatriate men's footballers in Spain
Expatriate men's footballers in Colombia
Argentine people of German descent
People from General López Department
Footballers from Santa Fe Province |
https://en.wikipedia.org/wiki/1975%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1975. This was the 8th season of the NASL.
Overview
The league comprised 20 teams with the Tampa Bay Rowdies winning the championship.
Pelé joined the New York Cosmos in 1975.
1975 was the first year the league used the term Soccer Bowl for their championship game.
Changes from the previous season
Rules changes
The 1975 season saw the removal of tie games. Matches that were level after 90 minutes would go to 15 minutes of sudden death overtime, and then onto penalty kicks if needed. It would not be until 2000 that a top-tier American soccer league would again allow matches to end in a draw.
New teams
Chicago Sting
Hartford Bicentennials
Portland Timbers
San Antonio Thunder
Tampa Bay Rowdies
Teams folding
None
Teams moving
None
Name changes
Toronto Metros to Toronto Metros-Croatia*
*after merger with Toronto Croatia of National Soccer League
Regular season
W = Wins, L = Losses, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win,
1 point for a shootout win,
0 points for a loss,
1 point for each regulation goal scored up to three per game.
-Premiers (most points). -Other playoff teams.
NASL League Leaders
Scoring
*(2 points per goal, 1 per assist)
Goalkeeping
*(1,260 minutes minimum)
NASL All-Stars
Playoffs
All playoff games in all rounds including Soccer Bowl '75 were single game elimination match ups.
Bracket
Quarterfinals
Semifinals
Soccer Bowl '75
1975 NASL Champions: Tampa Bay Rowdies
Post season awards
Most Valuable Player: Steve David, Miami
Coach of the year: John Sewell, St. Louis
Rookie of the year: Chris Bahr, Philadelphia
References
External links
Complete Results and Standings
North American Soccer League (1968–1984) seasons
1975 in American soccer leagues
1975 in Canadian soccer |
https://en.wikipedia.org/wiki/1970%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1970. This was the 3rd season of the NASL.
Overview
Six teams competed with the Rochester Lancers winning the championship. Santos FC of Brazil beat the NASL All-Stars 4–3 at Soldier Field in Chicago to finish the season. In 1970, NASL teams rounded out their schedules by playing an assortment of foreign clubs including Hapoel Petah Tikva, Varzim, Hertha Berlin and Coventry City. These games weren't just for attendance but also counted in the standings. The Washington Darts went 2-2-0 versus the international teams earning the "International Cup".
Changes from the previous season
New teams
Rochester Lancers*
Washington Darts*
*joined from American Soccer League
Teams folding
Baltimore Bays
Teams moving
None
Name changes
None
Regular season
W = Wins, L = Losses, T= Ties, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win,
3 points for a tie,
0 points for a loss,
1 point for each goal scored up to three per game.
-Premiers (most points). -Other playoff team.
NASL All-Stars
NASL Final 1970
First leg
Second leg
1970 NASL Champions: Rochester Lancers
Post season awards
Most Valuable Player: Carlos Metidieri, Rochester
Coach of the year: Sal DeRosa, Rochester
Rookie of the year: Jim Leeker, St. Louis
References
North American Soccer League (1968–1984) seasons
1970 in American soccer leagues |
https://en.wikipedia.org/wiki/1971%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1971. This was the 4th season of the NASL.
Overview
Eight teams competed in the 1971 season. The Dallas Tornado won the league championship after playing several playoff matches under the new sudden death rule, which replaced the penalty shootout. In Game 1 of the best-of-three semifinal against the Rochester Lancers, league scoring champion Carlos Metidieri of Rochester scored to end the match at 2–1 in the 176th minute after six periods of overtime—shortly before midnight. Three days later, Dallas tied the series with a 3–1 regulation win. In the deciding match, the two teams ended regulation with a 1–1 tie and played four overtime periods before Bobby Moffat scored in the 148 minute for a 2–1 victory. Four days later, Dallas lost Game 1 of the NASL Championship Series, 2–1, in the 3rd overtime to the Atlanta Chiefs after 123 minutes. All totaled, Dallas had played 537 minutes of soccer (3 minutes short of six games) in 13 days. The Tornado won 4–1 in Game 2 and 2–0 in Game 3 to clinch the league championship.
Changes From the previous season
Rule changes
Playoffs series switched from a two-game aggregate score to a best-two-out-of-three match format. Any playoff games tied after 90 minutes would now be settled by golden goal (or sudden death) overtime periods lasting 15 minutes each.
New teams
Montreal Olympique
New York Cosmos
Toronto Metros
Teams folding
Kansas City Spurs
Teams moving
None
Name changes
None
Regular season
W = Wins, L = Losses, T= Ties, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win, 3 points for a tie, 0 points for a loss, 1 point for each goal scored up to three per game.
-Premiers (most points). -Other playoff teams.
1971 NASL All-Stars
Playoffs
Bracket
Semifinals
NASL Final 1971
Game one
Game two
Game three
1971 NASL Champions: Dallas Tornado
Post season awards
Most Valuable Player: Carlos Metidieri, Rochester
Coach of the year: Ron Newman, Dallas
Rookie of the year: Randy Horton, New York
References
North American Soccer League (1968–1984) seasons
1971 in American soccer leagues
1971 in Canadian soccer |
https://en.wikipedia.org/wiki/1972%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1972. This was the 5th season of the NASL.
Overview
Eight teams took part in the league with the New York Cosmos winning the championship.
Changes from previous season
Rules changes
The league changed its offside rule during the season on June 26. They created a "Blue Line" which was an offside line across the field, 35 yards from the goal line. Thereafter, no player could be offside unless he had crossed the 35-yard line. This made the NASL unique in the soccer world; the league received temporary approval for the change from FIFA on an experimental basis only. The league also switched the playoff format to single-match elimination contests rather than series.
New teams
None
Teams folding
None
Teams moving
Washington Darts to Miami Gatos
Name changes
None
Regular season
W = Wins, L = Losses, T= Ties, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win,
3 points for a tie,
0 points for a loss,
1 point for each goal scored up to three per game.
-Premiers (most points). -Other playoff teams.
NASL All-Stars
Playoffs
All playoff games in all rounds including the NASL Final were single game elimination match ups.
Bracket
Semifinals
NASL Final 1972
1972 NASL Champions: New York Cosmos
Post season awards
Most Valuable Player: Randy Horton, New York
Coach of the year: Casey Frankiewicz, St. Louis
Rookie of the year: Mike Winter, St. Louis
References
External links
Complete Results and Standings
North American Soccer League (1968–1984) seasons
1972
1972 in Canadian soccer |
https://en.wikipedia.org/wiki/1973%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1973. This was the 6th season of the NASL.
Overview
Nine teams took part in the league with the Philadelphia Atoms winning the championship.
During the season, Tiburones Rojos de Veracruz from Vera Cruz, Mexico, played each of the nine NASL clubs in exhibition games that counted in the league's final standings. The 1973 season would be the last season in which games from non-league clubs counted in league standings.
A week before the NASL Final 1973, commissioner Phil Woosnam announced that no team in the league made a profit during the season.
In a unique twist, the team with home field for the NASL Championship Game determined the date and time the game was to be played. When the Dallas Tornado won their semi-final, setting up the final with Philadelphia, they chose August 25 as the date of the game. They did this because the NASL loan agreements with players from the English First Division (the precursor to today's Premier League) expired before that date.
Because of this, Philadelphia's two leading scorers, Andy "The Flea" Provan and Jim Fryatt, were on their way back to England when the championship match was played on the 25th. Despite this, Philadelphia coach, Al Miller, put Bill Straub, a defender who had not played a minute for the club prior to the championship game, into the lineup at forward. The move paid off as Straub headed home the second goal in a 2–0 win with under five minutes remaining in the final.
Changes from the previous season
New teams
Philadelphia Atoms
Teams folding
None
Teams moving
None
Name changes
Atlanta Chiefs to Atlanta Apollos
Miami Gatos to Miami Toros
Regular season
W = Wins, L = Losses, T= Ties, GF = Goals For, GA = Goals Against, BP = Bonus Points, PTS= Total Points
POINT SYSTEM
6 points for a win, 3 points for a tie, 0 points for a loss, 1 bonus point for each goal scored up to three per game.
-Premiers (most points). -Other playoff teams.
NASL All-Stars
Playoffs
All playoff games in all rounds including the NASL Final were single game elimination match ups.
Bracket
Semifinals
NASL Final 1973
1973 NASL Champions: Philadelphia Atoms
Post season awards
Most Valuable Player: Warren Archibald, Miami
Coach of the year: Al Miller, Philadelphia
Rookie of the year: Kyle Rote, Jr., Dallas
References
External links
Video highlights of NASL in 1973
Complete Results and Standings
North American Soccer League (1968–1984) seasons
1973
1973 in Canadian soccer |
https://en.wikipedia.org/wiki/1974%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1974. This was the 7th season of the NASL.
Overview
Fifteen teams comprised the league with the Los Angeles Aztecs winning the championship in a penalty kick shootout over the Miami Toros.
Changes from the previous season
Rules changes
The league decided to do away with tie games. If a match was tied after 90 minutes, the teams would go directly to a standard penalty shootout with no extra time played. The outcome would appear in the standings as a 'tie-win'. The tie-winner would gain three points, plus goals in regulation, while the loser of the tie-breaker received no points, except for regulation goals. Including the 1974 NASL Final, 33 matches were decided using this method.
New teams
Baltimore Comets
Boston Minutemen
Denver Dynamos
Los Angeles Aztecs
San Jose Earthquakes
Seattle Sounders
Vancouver Whitecaps
Washington Diplomats
Teams folding
Atlanta Apollos
Montreal Olympique
Teams moving
None
Name changes
None
Regular season
W = Wins, L = Losses, T= PK Shootout Wins, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win,
3 points for a PK shootout win,
0 points for a loss,
1 point for each goal scored up to three per game.
-Premiers (most points). -Other playoff teams.
NASL All-Stars
Playoffs
All playoff games in all rounds including the NASL Final were single game elimination match ups.
Bracket
Quarterfinals
Semifinals
NASL Final 1974
1974 NASL Champions: Los Angeles Aztecs
Post season awards
Most Valuable Player: Peter Silvester, Baltimore
Coach of the year: John Young, Miami
Rookie of the year: Douglas McMillan, Los Angeles
References
External links
Complete Results and Standings
North American Soccer League (1968–1984) seasons
1974 in American soccer leagues
1974 in Canadian soccer |
https://en.wikipedia.org/wiki/1982%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1982. This was the 15th season of the NASL.
Overview
The league comprised 14 teams. The New York Cosmos won the championship. The NASL no longer used the 35-yard line for offside, but retained its presence for use in tie-breaker shootouts.
Changes from the previous season
New teams
None
Teams folding
Atlanta Chiefs
Calgary Boomers
California Surf
Dallas Tornado
Los Angeles Aztecs
Minnesota Kicks
Washington Diplomats
Atlanta, Calgary, California, Dallas and Washington folded in September 1981, while Los Angeles and Minnesota folded in November–December 1981.
Teams moving
None
Name changes
None
Regular season
W = Wins, L = Losses, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win in regulation and overtime, 4 point for a shootout win,
0 points for a loss,
1 bonus point for each regulation goal scored, up to three per game.
Premiers (most points). Other playoff teams.
NASL All-Stars
Playoffs
Bracket
Quarterfinals
† Higher seed hosts Games 1 and 3
* Montreal Manic hosted Game 1 (instead of Game 2) due to stadium conflicts with the Expos baseball club.
Semifinals
† Higher seed hosts Games 1 and 3
Soccer Bowl '82
1982 NASL Champions: New York Cosmos
Post season awards
Most Valuable Player: Peter Ward, Seattle
Coach of the year: Johnny Giles, Vancouver
Rookie of the year: Pedro DeBrito, Tampa Bay
North American Player of the Year: Mark Peterson, Seattle
Soccer Bowl MVP: Giorgio Chinaglia, New York
References
External links
Video of 1982 NASL goals of the year
Complete Results and Standings
North American Soccer League (1968–1984) seasons
1982 in American soccer leagues
1982 in Canadian soccer |
https://en.wikipedia.org/wiki/1983%20North%20American%20Soccer%20League%20season | Statistics of North American Soccer League in season 1983. This was the 16th and penultimate season of the NASL.
Overview
There were 12 teams in the league. The Tulsa Roughnecks won the championship. Though Vancouver won two more games than any other club, for the fourth time in league history, the team with the most wins did not win the regular season due to the NASL's system of awarding points.
Changes from the previous season
New teams
Team America
Teams folding
Edmonton Drillers
Jacksonville Tea Men
Portland Timbers
Teams moving
None
Name changes
San Jose to Golden Bay
Regular season
W = Wins, L = Losses, GF = Goals For, GA = Goals Against, PT= point system
6 points for a win in regulation and overtime, 4 point for a shootout win,
0 points for a loss,
1 bonus point for each regulation goal scored, up to three per game.
-Premiers (most points). -Best record. -Other playoff teams.
NASL All-Stars
Playoffs
Bracket
Quarterfinals
Semifinals
Soccer Bowl '83
1983 NASL Champions: Tulsa Roughnecks
Post season awards
Most Valuable Player: Roberto Cabanas, New York
Coach of the year: Don Popovic, Golden Bay
Rookie of the year: Gregg Thompson, Tampa Bay
North American Player of the Year: Tino Lettieri, Vancouver
Soccer Bowl MVP: Njego Pesa, Tulsa
References
External links
Video highlights of 1983 season
Complete Results and Standings
North American Soccer League (1968–1984) seasons
1983 in American soccer leagues
1983 in Canadian soccer |
https://en.wikipedia.org/wiki/1979%E2%80%9380%20Scottish%20Football%20League | Statistics of Scottish Football League in season 1979/1980.
Scottish Premier Division
Scottish First Division
Scottish Second Division
See also
1979–80 in Scottish football
References
Scottish Football League seasons |
https://en.wikipedia.org/wiki/Hilbert%E2%80%93Poincar%C3%A9%20series | In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say , where the coefficient of gives the dimension (or rank) of the sub-structure of elements homogeneous of degree . It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of its argument .
Definition
Let K be a field, and let be an -graded vector space over K, where each subspace of vectors of degree i is finite-dimensional. Then the Hilbert–Poincaré series of V is the formal power series
A similar definition can be given for an -graded R-module over any commutative ring R in which each submodule of elements homogeneous of a fixed degree n is free of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the Hilbert–Poincaré series is considered has additional structure, for instance, that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure.
Example: Since there are monomials of degree k in variables (by induction, say), one can deduce that the sum of the Hilbert–Poincaré series of is the rational function .
Hilbert–Serre theorem
Suppose M is a finitely generated graded module over with an Artinian ring (e.g., a field) A. Then the Poincaré series of M is a polynomial with integral coefficients divided by . The standard proof today is an induction on n. Hilbert's original proof made a use of Hilbert's syzygy theorem (a projective resolution of M), which gives more homological information.
Here is a proof by induction on the number n of indeterminates. If , then, since M has finite length, if k is large enough. Next, suppose the theorem is true for and consider the exact sequence of graded modules (exact degree-wise), with the notation ,
.
Since the length is additive, Poincaré series are also additive. Hence, we have:
.
We can write . Since K is killed by , we can regard it as a graded module over ; the same is true for C. The theorem thus now follows from the inductive hypothesis.
Chain complex
An example of graded vector space is associated to a chain complex, or cochain complex C of vector spaces; the latter takes the form
The Hilbert–Poincaré series (here often called the Poincaré polynomial) of the graded vector space for this comple |
https://en.wikipedia.org/wiki/Varignon%27s%20theorem | In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was published posthumously in 1731.
Theorem
The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex), then the area of the parallelogram is half the area of the quadrilateral.
If one introduces the concept of oriented areas for n-gons, then this area equality also holds for complex quadrilaterals.
The Varignon parallelogram exists even for a skew quadrilateral, and is planar whether the quadrilateral is planar or not. The theorem can be generalized to the midpoint polygon of an arbitrary polygon.
Proof
Referring to the diagram above, triangles ADC and HDG are similar by the side-angle-side criterion, so angles DAC and DHG are equal, making HG parallel to AC. In the same way EF is parallel to AC, so HG and EF are parallel to each other; the same holds for HE and GF.
Varignon's theorem can also be proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or barycentric coordinates. The proof applies even to skew quadrilaterals in spaces of any dimension.
Any three points E, F, G are completed to a parallelogram (lying in the plane containing E, F, and G) by taking its fourth vertex to be E − F + G. In the construction of the Varignon parallelogram this is the point (A + B)/2 − (B + C)/2 + (C + D)/2 = (A + D)/2. But this is the point H in the figure, whence EFGH forms a parallelogram.
In short, the centroid of the four points A, B, C, D is the midpoint of each of the two diagonals EG and FH of EFGH, showing that the midpoints coincide.
From the first proof, one can see that the sum of the diagonals is equal to the perimeter of the parallelogram formed. Also, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram.
The Varignon parallelogram
Properties
A planar Varignon parallelogram also has the following properties:
Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.
A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.
The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.
The perimeter of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.
The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.
The two bimedians |
https://en.wikipedia.org/wiki/D%20notation | D notation or D-notation may refer to:
D notation (computing), scientific notation for double precision numbers in some versions of FORTRAN and BASIC
Dice notation, dice algebra in gaming |
https://en.wikipedia.org/wiki/Induced%20homomorphism | In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y.
More generally, in category theory, any functor by definition provides an induced morphism in the target category for each morphism in the source category.
For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the category of topological spaces to (e.g.) the category of groups or rings. This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism.
A homomorphism induced from a map is often denoted .
Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other.
A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred from relations between the homomorphisms they induce. The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in.
In fundamental groups
Let X and Y be topological spaces with points x0 in X and y0 in Y.
Let h : X→Y be a continuous map such that .
Then we can define a map from the fundamental group to the fundamental group as follows:
any element of , represented by a loop f in X based at x0, is mapped to the loop in obtained by composing with h:
Here [f] denotes the equivalence class of f under homotopy, as in the definition of the fundamental group.
It is easily checked from the definitions that is a well-defined function → : loops in the same equivalence class, i.e. homotopic loops in X, are mapped to homotopic loops in Y, because a homotopy can be composed with h as well.
It also follows from the definition of the group operation in fundamental groups (namely by concatenation of loops) that is a group homomorphism:
(where + denotes concatenation of loops, with the first + in X and the second + in Y).
The resulting homomorphism is the homomorphism induced from h.
It may also be denoted as (h).
Indeed, gives a functor from the category of pointed spaces to the category of groups: it associates the fundamental group to each pointed space and it associates the induced homomorphism to each preserving continuous map h: → .
To prove it satis |
https://en.wikipedia.org/wiki/Substitution%20principle | Substitution principle can refer to several things:
Substitution principle (mathematics)
Substitution principle (sustainability)
Liskov substitution principle (computer science) |
https://en.wikipedia.org/wiki/1979%E2%80%9380%20Ekstraklasa | Statistics for the 1979–80 season of the Ekstraklasa (the top tier of association football in Poland).
Overview
It was contested by 16 teams, and Szombierki Bytom won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1979–80 in Polish football
Pol |
https://en.wikipedia.org/wiki/1980%E2%80%9381%20Ekstraklasa | Statistics of Ekstraklasa in the 1980–81 season.
Overview
It was contested by 16 teams, and Widzew Łódź won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1980–81 in Polish football
Pol |
https://en.wikipedia.org/wiki/1981%E2%80%9382%20Ekstraklasa | Statistics of Ekstraklasa in the 1981–82 season.
Overview
16 teams competed in the 1981–82 season. Widzew Łódź won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1981–82 in Polish football
Pol |
https://en.wikipedia.org/wiki/Asira%20ash-Shamaliya | Asira ash-Shamaliya () is a Palestinian town in the Nablus Governorate, located 3.5 kilometers north of Nablus in the northern West Bank. According to the Palestinian Central Bureau of Statistics, the town had a population of approximately 8,813 inhabitants in 2017.
Location
‘Asira ash Shamaliya is located 3.5 km north of Nablus. It is bordered by Talluza, Al Badhan, and 'Azmut to the east, Nablus to the south, Zawata, Ijnisinya, and Nisf Jubeil to the west, and Beit Imrin and Yasid to the north.
Etymology
According to Palmer, the old name Asiret el Hatab means The difficult place of timber.
According to the local municipality, in Arabic, the word Asira means "firewood" and refers to the town's (and nearby Asira al-Qibliya's) abundance of forests which was used by residents to sell firewood.
History
Pottery sherds from the Iron Age I, Iron Age II, late Roman, Byzantine, early Muslim and Medieval eras have been found here.
South east of the village centre (at grid no. 175/183) is a site where a quantity of pottery from Iron Age I has been found.
In 1166, a Crusader estate called Asine was located here.
Ottoman era
In 1517, the village was included in the Ottoman empire with the rest of Palestine, and in the 1596 tax-records it appeared as Asirah, located in the Nahiya of Jabal Sami, part of Nablus Sanjak. The population was 19 households and 5 bachelors, all Muslim. They paid a fixed tax rate of 33.3% on agricultural products, such as wheat, barley, summer crops, olive trees, goats and beehives, in addition to occasional revenues and a fixed tax for people of Nablus area; a total of 3,335 akçe.
In 1838 Robinson placed '''Asiret el Hatab in the Wady esh-Sha'ir district, west of Nablus.
In 1870, Victor Guérin noted about the village, which he called A'sireh, that it was: "a considerable village, whose inhabitants are considered industrious. Their houses are better built than in many other places in Palestine. Around the village, there are some gardens planted with figs, olive trees and vegetables."
In 1882, the PEF's Survey of Western Palestine (SWP) described Asira ash-Shamaliya, which they called Asiret el Hatab as: "a large village on a round knoll, with olive groves on every side."
British Mandate era
In the 1922 census of Palestine, conducted by the British Mandate authorities, '''Asira Shamaliyeh had a population of 1,179; 1,178 Muslims and 1 Orthodox Christian, increasing in the 1931 census to 1,544, all Muslim, in 329 houses.
In the 1945 statistics the population was 2,060, all Muslims, with 30,496 dunams of land, according to an official land and population survey. Of this, 4,850 dunams were used for plantations and irrigable land, 11,765 were for cereals, while 101 dunams were built-up (urban) land.
Jordanian era
In the wake of the 1948 Arab–Israeli War, and after the 1949 Armistice Agreements, Asira ash-Shamaliya came under Jordanian rule.
In 1961, the population of Asira Shamaliya was 3,232.
Post-1967
Since the Six-D |
https://en.wikipedia.org/wiki/Centerpoint%20%28geometry%29 | In statistics and computational geometry, the notion of centerpoint is a generalization of the median to data in higher-dimensional Euclidean space. Given a set of points in d-dimensional space, a centerpoint of the set is a point such that any hyperplane that goes through that point divides the set of points in two roughly equal subsets: the smaller part should have at least a 1/(d + 1) fraction of the points. Like the median, a centerpoint need not be one of the data points. Every non-empty set of points (with no duplicates) has at least one centerpoint.
Related concepts
Closely related concepts are the Tukey depth of a point (the minimum number of sample points on one side of a hyperplane through the point) and a Tukey median of a point set (a point maximizing the Tukey depth). A centerpoint is a point of depth at least n/(d + 1), and a Tukey median must be a centerpoint, but not every centerpoint is a Tukey median. Both terms are named after John Tukey.
For a different generalization of the median to higher dimensions, see geometric median.
Existence
A simple proof of the existence of a centerpoint may be obtained using Helly's theorem. Suppose there are n points, and consider the family of closed half-spaces that contain more than dn/(d + 1) of the points. Fewer than n/(d + 1) points are excluded from any one of these halfspaces, so the intersection of any subset of d + 1 of these halfspaces must be nonempty. By Helly's theorem, it follows that the intersection of all of these halfspaces must also be nonempty. Any point in this intersection is necessarily a centerpoint.
Algorithms
For points in the Euclidean plane, a centerpoint may be constructed in linear time. In any dimension d, a Tukey median (and therefore also a centerpoint) may be constructed in time O(nd − 1 + n log n).
A randomized algorithm that repeatedly replaces sets of d + 2 points by their Radon point can be used to compute an approximation to a centerpoint of any point set, in the sense that its Tukey depth is linear in the sample set size, in an amount of time that is polynomial in both the number of points and the dimension.
References
Citations
Sources
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.
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Euclidean geometry
Multi-dimensional geometry
Means |
https://en.wikipedia.org/wiki/Slowly%20varying%20function | In real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata, and have found several important applications, for example in probability theory.
Basic definitions
. A measurable function is called slowly varying (at infinity) if for all ,
. Let . Then is a regularly varying function if and only if . In particular, the limit must be finite.
These definitions are due to Jovan Karamata.
Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function.
Basic properties
Regularly varying functions have some important properties: a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by .
Uniformity of the limiting behaviour
. The limit in and is uniform if is restricted to a compact interval.
Karamata's characterization theorem
. Every regularly varying function is of the form
where
is a real number,
is a slowly varying function.
Note. This implies that the function in has necessarily to be of the following form
where the real number is called the index of regular variation.
Karamata representation theorem
. A function is slowly varying if and only if there exists such that for all the function can be written in the form
where
is a bounded measurable function of a real variable converging to a finite number as goes to infinity
is a bounded measurable function of a real variable converging to zero as goes to infinity.
Examples
If is a measurable function and has a limit
then is a slowly varying function.
For any , the function is slowly varying.
The function is not slowly varying, nor is for any real . However, these functions are regularly varying.
See also
Analytic number theory
Hardy–Littlewood tauberian theorem and its treatment by Karamata
Notes
References
.
Real analysis
Tauberian theorems
Types of functions |
https://en.wikipedia.org/wiki/Words%20of%20estimative%20probability | Words of estimative probability (WEP or WEPs) are terms used by intelligence analysts in the production of analytic reports to convey the likelihood of a future event occurring. A well-chosen WEP gives a decision maker a clear and unambiguous estimate upon which to base a decision. Ineffective WEPs are vague or misleading about the likelihood of an event. An ineffective WEP places the decision maker in the role of the analyst, increasing the likelihood of poor or snap decision making. Some intelligence and policy failures appear to be related to the imprecise use of estimative words.
History
Intelligence
In 1964 Sherman Kent, one of the first contributors to a formal discipline of intelligence analysis addressed the problem of misleading expressions of odds in National Intelligence Estimates (NIE). In Words of Estimative Probability, Kent distinguished between ‘poets’ (those preferring wordy probabilistic statements) from ‘mathematicians’ (those preferring quantitative odds). To bridge the gap between them and decision makers, Kent developed a paradigm relating estimative terms to odds. His goal was to "... set forth the community's findings in such a way as to make clear to the reader what is certain knowledge and what is reasoned judgment, and within this large realm of judgment what varying degrees of certitude lie behind each key judgment."
Kent's initiative was not adopted although the idea was well received and remains compelling today.
Policy and intelligence failures related to WEPs
An example of the damage that missing or vague WEPs can do is to be found in the President's Daily Brief (PDB), entitled Bin Laden Determined to Strike in US. The President's Daily Brief is arguably the pinnacle of concise, relevant, actionable analytic writing in the U.S. Intelligence Community (IC). The PDB is intended to keep the President informed on a wide range of issues, the best analysts write it and senior leaders review it. This, the “August 6 PDB,” is at the center of much controversy for the USIC. The August 6 PDB began with not only a vague warning in the title, but also continued with vague warnings:“Bin Ladin since 1997 has wanted to conduct terrorist attacks in the US” (CIA, 2001, para. 1);
“Bin Ladin implied...that his followers would ‘bring the fighting to America’” (CIA, 2001, para. 1);
Bin Ladin’s “attacks against...US embassies...in 1998 demonstrate that he prepares operations years in advance and is not deterred by setbacks” (CIA, 2001, para. 6);
“FBI information...indicates patterns of suspicious activity in this country consistent with preparations for hijackings or other types of attacks” (CIA, 2001, para. 10);
“a call to [the US] Embassy in the UAE in May [said] that a group of Bin Ladin supporters was in the US planning attacks with explosives” (CIA, 2001, para. 11).
The PDB described Bin Laden's previous activities. It did not present the President with a critically important clear estimate of Bin Laden's likely activities in t |
https://en.wikipedia.org/wiki/Affine%20Grassmannian%20%28manifold%29 | In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn (described on this page), while in the other the affine Grassmannian is a quotient of a group-ring based on formal Laurent series.
Formal definition
Given a finite-dimensional vector space V and a non-negative integer k, then Graffk(V) is the topological space of all affine k-dimensional subspaces of V.
It has a natural projection p:Graffk(V) → Grk(V), the Grassmannian of all linear k-dimensional subspaces of V by defining p(U) to be the translation of U to a subspace through the origin. This projection is a fibration, and if V is given an inner product, the fibre containing U can be identified with , the orthogonal complement to p(U).
The fibres are therefore vector spaces, and the projection p is a vector bundle over the Grassmannian, which defines the manifold structure on Graffk(V).
As a homogeneous space, the affine Grassmannian of an n-dimensional vector space V can be identified with
where E(n) is the Euclidean group of Rn and O(m) is the orthogonal group on Rm. It follows that the dimension is given by
(This relation is easier to deduce from the identification of next section, as the difference between the number of coefficients, (n−k)(n+1) and the dimension of the linear group acting on the equations, (n−k)2.)
Relationship with ordinary Grassmannian
Let be the usual linear coordinates on Rn. Then Rn is embedded into Rn+1 as the affine hyperplane xn+1 = 1. The k-dimensional affine subspaces of Rn are in one-to-one correspondence with the (k+1)-dimensional linear subspaces of Rn+1 that are in general position with respect to the plane xn+1 = 1. Indeed, a k-dimensional affine subspace of Rn is the locus of solutions of a rank n − k system of affine equations
These determine a rank n−k system of linear equations on Rn+1
whose solution is a (k + 1)-plane that, when intersected with xn+1 = 1, is the original k-plane.
Because of this identification, Graff(k,n) is a Zariski open set in Gr(k + 1, n + 1).
References
Differential geometry
Projective geometry
Algebraic homogeneous spaces
Algebraic geometry |
https://en.wikipedia.org/wiki/Line-intercept%20sampling | In statistics, more specifically in biostatistics, line-intercept sampling (LIS) is a method of sampling elements in a region whereby an element is sampled if a chosen line segment, called a “transect”, intersects the element.
Line intercept sampling has proven to be a reliable, versatile, and easy to implement method to analyze an area containing various objects of interest. It has recently also been applied to estimating variances during particulate material sampling.
References
See also
Sampling (statistics)
Sampling techniques
Environmental statistics
sl:Metoda linijskega transekta |
https://en.wikipedia.org/wiki/1289%20%28number%29 | The number 1289 (twelve hundred eighty-nine) is the natural number following 1288 and preceding 1290.
In mathematics
The number 1289 is an odd prime number, following 1283 and preceding 1291. It is classified as an apocalyptic power, a deficient number, and an evil number:
The number 1289 is called an apocalyptic power because 21289 contains the consecutive digits 666 (in decimal), as the Number of the beast: in the sequence ..., 1281, 1285, 1286, 1289, 1290, 1298, 1301, etc.
The number 1289 is a deficient number because the sum of all its positive divisors (except itself) totals less than 1289. Compare with perfect and abundant numbers.
The number 1289 is an evil number because it has an even number of 1's contained in its binary expansion.
In technology
1289 is a 4.5 volt battery with 3 "B Size" cells in flat side-by-side format with brass-strip connections, also known as MN1203, 3LR12 and many other IEC "R12" variations. Used in Rear cycle lamps, hand torches and in pairs in many Continental European Radio models.
Historical years
1289 AD, 1289 BC.
Notes
Integers |
https://en.wikipedia.org/wiki/Jim%20Berger%20%28statistician%29 | James Orvis Berger (born April 6, 1950 in Minneapolis, Minnesota) is an American statistician best known for his work on Bayesian statistics and decision theory. He won the COPSS Presidents' Award, one of the two highest awards in statistics, in 1985 at the age of 35. He received a Ph.D. in mathematics from Cornell University in 1974. He was a faculty member in the Department of Statistics at Purdue University until 1997, at which time he moved to the Institute of Statistics and Decision Sciences (now the Department of Statistical Science) at Duke University, where he is currently the Arts and Sciences Professor of Statistics. He was also director of the Statistical and Applied Mathematical Sciences Institute from 2002-2010, and has been a visiting professor at the University of Chicago since 2011.
Contributions to science
Berger has worked on the decision theoretic bases of Bayesian inference, including advances on the Stein phenomenon during and after his thesis. He has also greatly contributed to advances in the so-called objective Bayes approach where prior distributions are constructed from the structure of the sampling distributions and/or of frequentist properties. He is also recognized for his analysis of the opposition between Bayesian and frequentist visions on testing statistical hypotheses, with criticisms of the use of p-values and critical levels.
Awards and honors
Berger has received numerous awards for his work: Guggenheim Fellowship, the COPSS Presidents' Award and the R. A. Fisher Lectureship. He was elected as a Fellow of the American Statistical Association and to the National Academy of Sciences in 2003. In 2004, he was awarded an honorary Doctor of Science degree by Purdue University.
Bibliography
References
External links
Duke University faculty
American statisticians
Fellows of the American Statistical Association
Members of the United States National Academy of Sciences
Presidents of the Institute of Mathematical Statistics
Cornell University alumni
Living people
1950 births
Scientists from Minneapolis
Annals of Statistics editors
Mathematical statisticians |
https://en.wikipedia.org/wiki/Nothing%20Could%20Come%20Between%20Us | "Nothing Could Come Between Us" is a song by Canadian hard rock group Theory of a Deadman. It was released in June 2002 as the lead single from their eponymous debut album.
It represented the first major success of the band and helped propel it into the music scene, reaching #2 in Canada and #8 on the US Billboard Mainstream Rock Tracks.
Content
The song lyrics deal with the disillusionment of a man that, despite his feelings, he does not feel like he can spend the rest of his life with someone. The song also involves him reminiscing about the good times they had and some of his favorite mannerisms of her ("Nothing could come between us, no nothing, nothing / One the favorite things she used to say").
Track listing
The Single Track Listing
"Nothing Could Come Between Us" (3:27)
"Above This" (2:14)
"Invisible Man" (2:42)
The Single CD included bonus track "Above This" and the song "Invisible Man". "Invisible Man" was included on the Theory of a Deadman album while "Above This" became a downloadable track.
Charts
Year-end charts
References
2002 debut singles
Theory of a Deadman songs
604 Records singles
2002 songs |
https://en.wikipedia.org/wiki/Shelly%20Chaiken | Rochelle Lynne "Shelly" Chaiken (born 1949) is an American social psychologist. She first received her BS from the University of Maryland, College Park in 1971 for mathematics. She later earned her MS (in 1975) and her PhD (in 1978) at the University of Massachusetts Amherst in social psychology. She was a professor of psychology at New York University, but is now retired.
Chaiken is a member of many psychological organizations including the Society of Experimental Social Psychology, the American Psychological Association (Fellow, Div. 8), and the American Psychological Society.
Research
She completed work involving attitude, persuasion, and social cognition and is most well-known for the developing the heuristic-systematic model of information processing.
Chaiken completed a study researching interracial contact. The study found that participant who were exposed to more white faces in a positive way, had a more negative view or increased prejudice toward black faces.
Chaiken edited many psychological books including Attitude Research in the 21st Century: The Current State of Knowledge, and Dual-Process Theories in Social Psychology.
Dual-Process Theories in Social Psychology conglomerates the theories of informational processing in an organized way, along with reviews and research of these theories.
Much of her work involving persuasion has been helpful to conflict resolution centers and negotiations with their patients.
For her work on dual process theories of attitudes, on October 17, 2009 Chaiken was a co-recipient of the Society of Experimental Social Psychology's Scientific Impact Award, which "[h]onors the author(s) of a specific article or chapter that has proven highly influential over the last 25 years."
References
American social psychologists
Scientists from New York City
Living people
1949 births |
https://en.wikipedia.org/wiki/Poincar%C3%A9%20space | In algebraic topology, a Poincaré space is an n-dimensional topological space with a distinguished element µ of its nth homology group such that taking the cap product with an element of the kth cohomology group yields an isomorphism to the (n − k)th homology group. The space is essentially one for which Poincaré duality is valid; more precisely, one whose singular chain complex forms a Poincaré complex with respect to the distinguished element µ.
For example, any closed, orientable, connected manifold M is a Poincaré space, where the distinguished element is the fundamental class
Poincaré spaces are used in surgery theory to analyze and classify manifolds. Not every Poincaré space is a manifold, but the difference can be studied, first by having a normal map from a manifold, and then via obstruction theory.
Other uses
Sometimes, Poincaré space means a homology sphere with non-trivial fundamental group—for instance, the Poincaré dodecahedral space in 3 dimensions.
See also
Stable normal bundle
References
Algebraic topology
Abstract algebra |
https://en.wikipedia.org/wiki/United%20Kingdom%20agency%20worker%20law | United Kingdom agency worker law refers to the law which regulates people's work through employment agencies in the United Kingdom. Though statistics are disputed, there are currently between half a million and one and a half million agency workers in the UK, and probably over 17,000 agencies. As a result of judge made law and absence of statutory protection, agency workers have more flexible pay and working conditions than permanent staff covered under the Employment Rights Act 1996.
For most of the 20th century, employment agencies were quasi-legal entities in international law. The International Labour Organization in many Conventions called on member states to abolish them. However, the UK never signed up. The major piece of legislation which regulates agency practices is the Employment Agencies Act 1973, though it was slimmed considerably by the Deregulation and Contracting Out Act 1994. This abolished licences, so agencies operate without governmental oversight, except for a small inspectorate and occasional court cases. After the 2004 Morecambe Bay cockling disaster, Parliament enacted the Gangmasters (Licensing) Act 2004, requiring agencies (gangmasters) in the agricultural, shellfish and food packing sectors to be licensed.
In January 2010, the Government passed The Agency Workers Regulations 2010 (SI 2010/93) which require, at least, equal pay and working time rights when compared with what a direct worker would be paid. This is designed to implement the EU Agency Workers Directive, which is the first transnational legal measure to ensure agency workers are treated equally. The Directive was the culmination of initial resistance by the Government under Tony Blair, and a final surge of Parliamentary support for a Temporary and Agency Workers (Equal Treatment) Bill. The Regulations and the Directive are the third pillar of law, along with the Part-time Workers (Prevention of Less Favourable Treatment) Regulations 2000 and Fixed Term Employees (Prevention of Less Favourable Treatment) Regulations 2002 to regulate atypical workers.
Employment agency regulation
The Employment Agencies Act 1973 regulates the conduct of the 17,000 odd agencies operating in the UK. It prohibits most agencies charging upfront fees, makes it an offence to put out misleading advertising for jobs which do not exist, sets standards for assessing an employee's experience, and more. It was introduced after similar (though stronger) legislation was passed in France and Germany regulating agencies (for Germany, see Arbeitnehmerüberlassungsgesetz). The 1973 Act was amended by the Conservative government through the Deregulation and Contracting Out Act 1994, ostensibly to increase efficiency. It abolished the system of agency licensing, so that agencies can operate freely, unless inspectors find violations and close them down.
Supporting the Act are The Conduct of Employment Agencies and Employment Businesses Regulations 2003. These regulations restrict agencies from |
https://en.wikipedia.org/wiki/Shia%20Islam%20in%20Nigeria | Although the majority of the Nigerian Muslim population is Sunni, there is a small Shia minority, particularly in the northern states of Kano and Sokoto. However, there are no actual statistics that reflect a Shia population in Nigeria, and a figure of even 5% of the total Nigerian Muslim population is thought to be too high “because of the routine conflation of Shi’a with Sunnis who express solidarity with the Iranian revolutionary program, such as those of Zakzaky’s Ikhwani.”
Introduction of Shia in Nigeria
Shia faith was "almost unknown" in Nigeria until the 1980s, when Ibraheem Zakzaky introduced Shia Islam. Zakzaky's gained a following among those disenchanted with the political and religious establishment.
Persecution
Members of the Nigerian Shia community have been persecuted in some cases, but in other cases have united with Nigerian Sunni in the Islamic Movement in Nigeria. Cleric Sheikh Ibraheem Zakzaky is a primary figure in the movement.
Saudi Arabia’s linked Sunni politicians, organizations and Nigerian security apparatus are behind the persecution of Shia Muslims in Nigeria. The Salafist movement Izala Society, is close to both Riyadh and Abuja and its satellite television channel Manara often broadcasts anti-Shiite sectarian propaganda.
The state government of Sokoto has reacted to the rise of Shia Islam in the state by taking such measures as demolishing the Islamic Center in 2007. Furthermore, clashes between Sunni and Shia residents followed the assassination of Salafi Imam Umaru Danmaishiyya, who was known for his fiery anti-Shia preaching.
In 2014, the Zaria Quds Day massacres took place, leaving 35 dead. In 2015, the Zaria massacre during which 348 Shia Muslims were killed by the Nigerian Army.
In April 2018, clashes broke out as Nigerian police fired teargas Shia protesters who were demanding the release of Sheikh Ibrahim Zakzaky, who had been detained for two years with no trial. The clashes left at least one protester dead and several others injured. Further, Nigerian police detained at least 115 protesters.
In October 2018, Nigerian military killed at least 45 peaceful Shia protesters. After soldiers began to fire, they targeted protesters fleeing the chaos. Many of the injured were shot in the back or legs.
In July 2021, Shaikh Zakzaky has been acquitted of all charges and has been freed.
See also
Islamic Movement (Nigeria)
Zaria Quds Day massacres
Shia in Bahrain
References
Further reading
Muhammad Mansur Nigeria’s Zaria city crammed with mourners of Imam Hussain Jafariya News, January 29, 2007.
Mohamed Ali Mosques vandalised in (Abuja) Nigeria clashes Jafariya News, June 10, 2004.
Elise Aymer Nigeria: Clash of Religions The Yale International Forum, Winter 1996.
Biography of Zakzaky
External links
Islamic Movement in Nigeria –Nigeria's foremost Shi'a organization. |
https://en.wikipedia.org/wiki/Breusch%E2%80%93Godfrey%20test | In statistics, the Breusch–Godfrey test is used to assess the validity of some of the modelling assumptions inherent in applying regression-like models to observed data series. In particular, it tests for the presence of serial correlation that has not been included in a proposed model structure and which, if present, would mean that incorrect conclusions would be drawn from other tests or that sub-optimal estimates of model parameters would be obtained.
The regression models to which the test can be applied include cases where lagged values of the dependent variables are used as independent variables in the model's representation for later observations. This type of structure is common in econometric models.
The test is named after Trevor S. Breusch and Leslie G. Godfrey.
Background
The Breusch–Godfrey test is a test for autocorrelation in the errors in a regression model. It makes use of the residuals from the model being considered in a regression analysis, and a test statistic is derived from these. The null hypothesis is that there is no serial correlation of any order up to p.
Because the test is based on the idea of Lagrange multiplier testing, it is sometimes referred to as an LM test for serial correlation.
A similar assessment can be also carried out with the Durbin–Watson test and the Ljung–Box test. However, the test is more general than that using the Durbin–Watson statistic (or Durbin's h statistic), which is only valid for nonstochastic regressors and for testing the possibility of a first-order autoregressive model (e.g. AR(1)) for the regression errors. The BG test has none of these restrictions, and is statistically more powerful than Durbin's h statistic.
The BG test is considered to be more general than the Ljung-Box test because the latter requires the assumption of strict exogeneity, but the BG test does not. However, the BG test requires the assumptions of stronger forms of predeterminedness and conditional homoskedastictiy.
Procedure
Consider a linear regression of any form, for example
where the errors might follow an AR(p) autoregressive scheme, as follows:
The simple regression model is first fitted by ordinary least squares to obtain a set of sample residuals .
Breusch and Godfrey proved that, if the following auxiliary regression model is fitted
and if the usual Coefficient of determination ( statistic) is calculated for this model:
,
where stands for the arithmetic mean over the last samples. With number of data-points available for the second regression , where is the total number of observations. Note that the value of n depends on the number of lags of the error term ().
Then the following asymptotic approximation can be used for the distribution of the test statistic
when the null hypothesis holds (that is, there is no serial correlation of any order up to p). Here n is
Software
In R, this test is performed by function bgtest, available in package lmtest.
In Stata, this test is pe |
https://en.wikipedia.org/wiki/Fuzzy%20math | Fuzzy Math may refer to:
In mathematics, Fuzzy mathematics.
In education, a derogatory term for Reform mathematics.
A derogatory political term, Fuzzy math (politics) |
https://en.wikipedia.org/wiki/Polar%20sine | In geometry, the polar sine generalizes the sine function of angle to the vertex angle of a polytope. It is denoted by psin.
Definition
n vectors in n-dimensional space
Let v1, ..., vn (n ≥ 1) be non-zero Euclidean vectors in n-dimensional space (Rn) that are directed from a vertex of a parallelotope, forming the edges of the parallelotope. The polar sine of the vertex angle is:
where the numerator is the determinant
which equals the signed hypervolume of the parallelotope with vector edges
and where the denominator is the n-fold product
of the magnitudes of the vectors, which equals the hypervolume of the n-dimensional hyperrectangle with edges equal to the magnitudes of the vectors ||v1||, ||v2||, ... ||vn|| rather than the vectors themselves. Also see Ericksson.
The parallelotope is like a "squashed hyperrectangle", so it has less hypervolume than the hyperrectangle, meaning (see image for the 3d case):
as for the ordinary sine, with either bound being reached only in the case that all vectors are mutually orthogonal.
In the case n = 2, the polar sine is the ordinary sine of the angle between the two vectors.
In higher dimensions
A non-negative version of the polar sine that works in any -dimensional space can be defined using the Gram determinant. The numerator is given as
where the superscript T indicates matrix transposition. This can be nonzero only if . In the case m = n, this is equivalent to the absolute value of the definition given previously. In the degenerate case , the determinant will be of a singular matrix, giving , because it is not possible to have linearly independent vectors in -dimensional space.
Properties
Interchange of vectors
The polar sine changes sign whenever two vectors are interchanged, due to the antisymmetry of row-exchanging in the determinant; however, its absolute value will remain unchanged.
Invariance under scalar multiplication of vectors
The polar sine does not change if all of the vectors v1, ..., vn are scalar-multiplied by positive constants ci, due to factorization
If an odd number of these constants are instead negative, then the sign of the polar sine will change; however, its absolute value will remain unchanged.
Vanishes with linear dependencies
If the vectors are not linearly independent, the polar sine will be zero. This will always be so in the degenerate case that the number of dimensions is strictly less than the number of vectors .
Relationship to pairwise cosines
The cosine of the angle between two non-zero vectors is given by
using the dot product. Comparison of this expression to the definition of the absolute value of the polar sine as given above gives:
In particular, for , this is equivalent to
which is the Pythagorean theorem.
History
Polar sines were investigated by Euler in the 18th century.
See also
Trigonometric functions
List of trigonometric identities
Solid angle
Simplex
Law of sines
Cross product and Seven-dimensional cross product
Grad |
https://en.wikipedia.org/wiki/List%20of%20Collingwood%20Football%20Club%20coaches | The following is a list of coaches who have coached the Collingwood Football Club at a game of Australian rules football in the Australian Football League (AFL), formerly the VFL.
Statistics are correct as of the end of round 17 of the 2023 AFL season.
Key:
C = Coached
W = Won
L = Lost
D = Drew
W% = Win percentage
Notes
1: Bob Rush stood in to perform the match day coaching duties in the 1930 Grand Final, including delivering the half time address, because regular coach Jock McHale was absent on the day of the game, having fallen ill with influenza days before the game. For many years, Rush was credited with having coached the game; but after a decision in 2014 by the AFL's historians, McHale is now credited as Collingwood's sole coach in the game for the purposes of coaching statistics.
2: Although Neil Mann's coaching span is listed as being from 1972 to 1974 he coached the Magpies once in 1960 and again for a game in 1967 as caretaker coach.
References
AFL Tables List of Collingwood coaches
Collingwood Football Club
Collingwood Football Club coaches |
https://en.wikipedia.org/wiki/Additive%20smoothing | In statistics, additive smoothing, also called Laplace smoothing or Lidstone smoothing, is a technique used to smooth categorical data. Given a set of observation counts from a -dimensional multinomial distribution with trials, a "smoothed" version of the counts gives the estimator:
where the smoothed count and the "pseudocount" α > 0 is a smoothing parameter. α = 0 corresponds to no smoothing. (This parameter is explained in below.) Additive smoothing is a type of shrinkage estimator, as the resulting estimate will be between the empirical probability (relative frequency) , and the uniform probability . Invoking Laplace's rule of succession, some authors have argued that α should be 1 (in which case the term add-one smoothing is also used), though in practice a smaller value is typically chosen.
From a Bayesian point of view, this corresponds to the expected value of the posterior distribution, using a symmetric Dirichlet distribution with parameter α as a prior distribution. In the special case where the number of categories is 2, this is equivalent to using a beta distribution as the conjugate prior for the parameters of the binomial distribution.
History
Laplace came up with this smoothing technique when he tried to estimate the chance that the sun will rise tomorrow. His rationale was that even given a large sample of days with the rising sun, we still can not be completely sure that the sun will still rise tomorrow (known as the sunrise problem).
Pseudocount
A pseudocount is an amount (not generally an integer, despite its name) added to the number of observed cases in order to change the expected probability in a model of those data, when not known to be zero. It is so named because, roughly speaking, a pseudo-count of value weighs into the posterior distribution similarly to each category having an additional count of . If the frequency of each item is out of samples, the empirical probability of event is
but the posterior probability when additively smoothed is
as if to increase each count by a priori.
Depending on the prior knowledge, which is sometimes a subjective value, a pseudocount may have any non-negative finite value. It may only be zero (or the possibility ignored) if impossible by definition, such as the possibility of a decimal digit of pi being a letter, or a physical possibility that would be rejected and so not counted, such as a computer printing a letter when a valid program for pi is run, or excluded and not counted because of no interest, such as if only interested in the zeros and ones. Generally, there is also a possibility that no value may be computable or observable in a finite time (see the halting problem). But at least one possibility must have a non-zero pseudocount, otherwise no prediction could be computed before the first observation. The relative values of pseudocounts represent the relative prior expected probabilities of their possibilities. The sum of the pseudocounts, which may be v |
https://en.wikipedia.org/wiki/Generalized%20Verma%20module | In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. The motivation for their study is that their homomorphisms correspond to invariant differential operators over generalized flag manifolds. The study of these operators is an important part of the theory of parabolic geometries.
Definition
Let be a semisimple Lie algebra and a parabolic subalgebra of . For any irreducible finite-dimensional representation of we define the generalized Verma module to be the relative tensor product
.
The action of is left multiplication in .
If λ is the highest weight of V, we sometimes denote the Verma module by .
Note that makes sense only for -dominant and -integral weights (see weight) .
It is well known that a parabolic subalgebra of determines a unique grading so that .
Let .
It follows from the Poincaré–Birkhoff–Witt theorem that, as a vector space (and even as a -module and as a -module),
.
In further text, we will denote a generalized Verma module simply by GVM.
Properties of GVMs
GVM's are highest weight modules and their highest weight λ is the highest weight of the representation V. If is the highest weight vector in V, then is the highest weight vector in .
GVM's are weight modules, i.e. they are direct sum of its weight spaces and these weight spaces are finite-dimensional.
As all highest weight modules, GVM's are quotients of Verma modules. The kernel of the projection is
where is the set of those simple roots α such that the negative root spaces of root are in (the set S determines uniquely the subalgebra ), is the root reflection with respect to the root α and
is the affine action of on λ. It follows from the theory of (true) Verma modules that is isomorphic to a unique submodule of . In (1), we identified . The sum in (1) is not direct.
In the special case when , the parabolic subalgebra is the Borel subalgebra and the GVM coincides with (true) Verma module. In the other extremal case when , and the GVM is isomorphic to the inducing representation V.
The GVM is called regular, if its highest weight λ is on the affine Weyl orbit of a dominant weight . In other word, there exist an element w of the Weyl group W such that
where is the affine action of the Weyl group.
The Verma module is called singular, if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight so that is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).
Homomorphisms of GVMs
By a homomorphism of GVMs we mean -homomorphism.
For any two weights a homomorphism
may exist only if and are linked with an affine action of the Weyl group of the Lie algebra . This follows easily from the Harish-Chandra theorem on infinitesimal central characters.
Unlike in the case of (true) Verma modules, the homomorphisms of GVM |
https://en.wikipedia.org/wiki/Menachem%20Magidor | Menachem Magidor (Hebrew: מנחם מגידור; born January 24, 1946) is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of the Association for Symbolic Logic from 1996 to 1998 and as president of the
Division for Logic, Methodology and Philosophy of Science and Technology of the International Union for History and Philosophy of Science (DLMPST/IUHPS) from 2016 to 2019. In 2016 he was elected an honorary foreign member of the American Academy of Arts and Sciences. In 2018 he received the Solomon Bublick Award.
Biography
Menachem Magidor was born in Petah Tikva, Israel. He received his Ph.D. in 1973 from the Hebrew University of Jerusalem. His thesis, On Super Compact Cardinals, was written under the supervision of Azriel Lévy. He served as president of the Hebrew University of Jerusalem from 1997 to 2009, following Hanoch Gutfreund and succeeded by Menachem Ben-Sasson. The Oxford philosopher Ofra Magidor is his daughter.
Mathematical theories
Magidor obtained several important consistency results on powers of singular cardinals substantially developing the method of forcing. He generalized the Prikry forcing in order to change the cofinality of a large cardinal to a predetermined regular cardinal. He proved that the least strongly compact cardinal can be equal to the least measurable cardinal or to the least supercompact cardinal (but not at the same time). Assuming consistency of huge cardinals he constructed models (1977) of set theory with first examples of nonregular ultrafilters over very small cardinals (related to the famous Guilmann–Keisler problem concerning existence of nonregular ultrafilters), even with the example of jumping cardinality of ultrapowers. He proved consistent that is strong limit, but . He even strengthened the condition that is strong limit to that generalised continuum hypothesis holds below . This constituted a negative solution to the singular cardinals hypothesis. Both proofs used the consistency of very large cardinals. Magidor, Matthew Foreman, and Saharon Shelah formulated and proved the consistency of Martin's maximum, a provably maximal form of Martin's axiom. Magidor also gave a simple proof of the Jensen and the Dodd-Jensen covering lemmas. He proved that if 0# does not exist then every primitive recursive closed set of ordinals is the union of countably many sets in .
Selected published works
References
1946 births
Living people
Academic staff of the Hebrew University of Jerusalem
20th-century Israeli mathematicians
21st-century Israeli mathematicians
Set theorists
Presidents of universities in Israel |
https://en.wikipedia.org/wiki/Geometry%20Festival | The Geometry Festival is an annual mathematics conference held in the United States.
The festival has been held since 1985 at the University of Pennsylvania, the University of Maryland, the University of North Carolina, the State University of New York at Stony Brook, Duke University and New York University's Courant Institute of Mathematical Sciences. It is a three day conference that focuses on the major recent results in geometry and related fields.
Previous Geometry Festival speakers
1985 at Penn
Marcel Berger
Pat Eberlein
Jost Eschenburg
Friedrich Hirzebruch
Blaine Lawson
Leon Simon
Scott Wolpert
Deane Yang
1986 at Maryland
Uwe Abresch, Explicit constant mean curvature tori
Zhi-yong Gao, The existence of negatively Ricci curved metrics
David Hoffman, New results in the global theory of minimal surfaces
Jack Lee, Conformal geometry and the Yamabe problem
Ngaiming Mok, Compact Kähler manifolds of non-negative curvature
John Morgan, Self dual connections and the topology of 4-manifolds
Chuu-Lian Terng, Submanifolds with flat normal bundle
1987 at Penn
Robert Bryant, The construction of metrics with exceptional holonomy
Francis Bonahon, Hyperbolic 3-manifolds with arbitrarily short geodesics
Keith Burns, Geodesic flows on the 2-sphere
Andreas Floer, Instantons and Casson's invariant
Hermann Karcher, Embedded minimal surfaces in the 3-sphere
Jürgen Moser, Minimal foliations of tori
Edward Witten. Applications of quantum field theory to topology
1988 at North Carolina
Detlef Gromoll, On complete spaces of non-negative Ricci curvature
Nicolas Kapouleas, Constant mean curvature surfaces in E3
Robert Osserman, Gauss map of complete minimal surfaces
Pierre Pansu, Lp-cohomology of negatively curved manifolds
Peter Petersen, Bounding homotopy types by geometry
Gang Tian, Kähler-Einstein metrics on quasiprojective manifolds
DaGang Yang, Some new examples of manifolds of positive Ricci curvature
Wolfgang Ziller, Recent results on Einstein metrics
1989 at Stony Brook
Eugenio Calabi, Extremal singular metrics on surfaces
Harold Donnelly, Nodal sets of eigenfunctions on Riemannian manifolds
Yakov Eliashberg, Symplectic geometric methods in several complex variables
F. Thomas Farrell, A topological analogue of Mostow's rigidity theorem
Lesley Sibner, Solutions to Yang-Mills equations which are not self-dual
Carlos Simpson, Moduli spaces of representations of fundamental groups
1990 at Maryland
Michael T. Anderson, Behavior of metrics under Ricci curvature bounds
Kevin Corlette, Harmonic maps and geometric superrigidity
Kenji Fukaya, Fundamental groups of almost non-negatively curved manifolds
Mikhail Gromov, Recent progress in symplectic geometry
Werner Müller, On spectral theory for locally symmetric manifolds with finite volume
Rick Schoen, Least area problems for Lagrangian submanifolds
Gudlaugur Thorbergsson, Isoparametric submanifolds and their Tits buildings
Shing-Tung Yau, Some theorems in Kähler ge |
https://en.wikipedia.org/wiki/Network%20probability%20matrix | The network probability matrix describes the probability structure of a network based on the historical presence or absence of edges in a network. For example, individuals in a social network are not connected to other individuals with uniform random probability. The probability structure is much more complex. Intuitively, there are some people whom a person will communicate with or be connected more closely than others. For this reason, real-world networks tend to have clusters or cliques of nodes that are more closely related than others (Albert and Barabasi, 2002, Carley [year], Newmann 2003). This can be simulated by varying the probabilities that certain nodes will communicate. The network probability matrix was originally proposed by Ian McCulloh.
References
McCulloh, I., Lospinoso, J. & Carley, K.M. (2007). Probability Mechanics in Communications Networks. In Proceedings of the 12th International Conference on Applied Mathematics of the World Science Engineering Academy and Society, Cairo, Egypt. 30–31 December 2007.
"Understanding Network Science," (Archived article) https://wayback-beta.archive.org/web/20080830045705/http://zangani.com/blog/2007-1030-networkingscience
Linked: The New Science of Networks, A.-L. Barabási (Perseus Publishing, Cambridge (2002).
Network Science, The National Academies Press (2005)
External links
Center for Computational Analysis of Social and Organizational Systems (CASOS) at Carnegie Mellon University
U.S. Military Academy Network Science Center
The Center for Interdisciplinary Research on Complex Systems at Northeastern University
Social statistics |
https://en.wikipedia.org/wiki/Confusion%20of%20the%20inverse | Confusion of the inverse, also called the conditional probability fallacy or the inverse fallacy, is a logical fallacy whereupon a conditional probability is equated with its inverse; that is, given two events A and B, the probability of A happening given that B has happened is assumed to be about the same as the probability of B given A, when there is actually no evidence for this assumption. More formally, P(A|B) is assumed to be approximately equal to P(B|A).
Examples
Example 1
In one study, physicians were asked to give the chances of malignancy with a 1% prior probability of occurring. A test can detect 80% of malignancies and has a 10% false positive rate. What is the probability of malignancy given a positive test result? Approximately 95 out of 100 physicians responded the probability of malignancy would be about 75%, apparently because the physicians believed that the chances of malignancy given a positive test result were approximately the same as the chances of a positive test result given malignancy.
The correct probability of malignancy given a positive test result as stated above is 7.5%, derived via Bayes' theorem:
Other examples of confusion include:
Hard drug users tend to use marijuana; therefore, marijuana users tend to use hard drugs (the first probability is marijuana use given hard drug use, the second is hard drug use given marijuana use).
Most accidents occur within 25 miles from home; therefore, you are safest when you are far from home.
Terrorists tend to have an engineering background; so, engineers have a tendency towards terrorism.
For other errors in conditional probability, see the Monty Hall problem and the base rate fallacy. Compare to illicit conversion.
Example 2
In order to identify individuals having a serious disease in an early curable form, one may consider screening a large group of people. While the benefits are obvious, an argument against such screenings is the disturbance caused by false positive screening results: If a person not having the disease is incorrectly found to have it by the initial test, they will most likely be distressed, and even if they subsequently take a more careful test and are told they are well, their lives may still be affected negatively. If they undertake unnecessary treatment for the disease, they may be harmed by the treatment's side effects and costs.
The magnitude of this problem is best understood in terms of conditional probabilities.
Suppose 1% of the group suffer from the disease, and the rest are well. Choosing an individual at random,
Suppose that when the screening test is applied to a person not having the disease, there is a 1% chance of getting a false positive result (and hence 99% chance of getting a true negative result, a number known as the specificity of the test), i.e.
Finally, suppose that when the test is applied to a person having the disease, there is a 1% chance of a false negative result (and 99% chance of getting a true positive |
https://en.wikipedia.org/wiki/Harold%20J.%20Kushner | Harold Joseph Kushner is an American applied mathematician and a Professor Emeritus of Applied Mathematics at Brown University. He is known for his work on the theory of stochastic stability (based on the concept of supermartingales as Lyapunov functions), the theory of non-linear filtering (based on the Kushner equation), and for the development of numerical methods for stochastic control problems such as the Markov chain approximation method. He is commonly cited as the first person to study Bayesian optimization, based on work he published in 1964.
Harold Kushner received his Ph.D. in Electrical Engineering from the University of Wisconsin in 1958.
Awards and honors
In 1992 the IEEE Control Systems Award
In 1994 the Louis E. Levy Medal from The Franklin Institute
In 2004 the Richard E. Bellman Control Heritage Award from the American Automatic Control Council, for "fundamental contributions to stochastic systems theory and engineering applications, and for inspiring generations of researchers in the field"
Bibliography
References
External links
Brown University profile
AACC profile
Control theorists
Richard E. Bellman Control Heritage Award recipients
Living people
Fellows of the Society for Industrial and Applied Mathematics
Brown University faculty
University of Wisconsin–Madison College of Engineering alumni
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/S.%20N.%20Bose%20National%20Centre%20for%20Basic%20Sciences | S. N. Bose National Centre for Basic Sciences (SNBNCBS) is an autonomous research institute dedicated to basic research in mathematics sciences under the Department of Science and Technology of Government of India. It is located in West Bengal, Salt Lake, Kolkata. This institute was named after the Indian scientist Satyendra Nath Bose and established in 1986. Chanchal Kumar Majumdar was the founder director of this institute.
As this is a research institute, mainly Ph.D. program is done here. From 2001 Integrated Ph.D. (M.Sc.+Ph.D.) program was started. After completion of two years of study, a M.Sc. degree is given by University of Calcutta. Formerly, this degree was given by West Bengal University of Technology. Students of this institute can submit their Ph.D. thesis to Jadavpur University, the University of Calcutta, West Bengal University of Technology or any other university which allows students to do so.
See also
List of colleges in West Bengal
Education in West Bengal
References
External links
Research institutes in West Bengal
Research institutes in Kolkata
Multidisciplinary research institutes
Educational institutions established in 1986
University of Calcutta
Maulana Abul Kalam Azad University of Technology
1986 establishments in West Bengal |
https://en.wikipedia.org/wiki/John%20Zaborszky | John Zaborszky (May 13, 1914 – February 11, 2008) was a noted Hungarian-born applied mathematician and a professor in the Department of systems science and mathematics, Washington University in St. Louis. He received the Richard E. Bellman Control Heritage Award in 1986. He was elected to the National Academy of Engineering in 1984.
Biography
Zaborszky earned a master's degree and PhD in 1937 and 1943, respectively, "under auspices of the Regent of Hungary" from the Technical University of Budapest. He continued as a docent at that institution and was chief engineer of the city's municipal power system before emigrating to the United States in 1947. He was an assistant professor at UMR and in 1954 moved to St. Louis to join Washington University. In 1974, he founded and was first chairman of the Systems Science Department. He was the 1970 President of the IEEE Control Systems Society and he received its Distinguished Member Award in 1983. He was an IEEE Fellow and was elected to Eta Kappa Nu (HKN).
References
External links
Washington University Obituary
Zaborszky Distinguished Lecture Series
National Academy of Engineering: John Zaborszky
1914 births
2008 deaths
Control theorists
Richard E. Bellman Control Heritage Award recipients
Members of the United States National Academy of Engineering
Washington University in St. Louis faculty
Washington University in St. Louis mathematicians
Hungarian emigrants to the United States
Missouri University of Science and Technology faculty |
https://en.wikipedia.org/wiki/Langford%20pairing | In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number k are k units apart. Langford pairings are named after C. Dudley Langford, who posed the problem of constructing them in 1958.
Langford's problem is the task of finding Langford pairings for a given value of n.
The closely related concept of a Skolem sequence is defined in the same way, but instead permutes the sequence 0, 0, 1, 1, ..., n − 1, n − 1.
Example
A Langford pairing for n = 3 is given by the sequence 2, 3, 1, 2, 1, 3.
Properties
Langford pairings exist only when n is congruent to 0 or 3 modulo 4; for instance, there is no Langford pairing when n = 1, 2, or 5.
The numbers of different Langford pairings for n = 1, 2, …, counting any sequence as being the same as its reversal, are
0, 0, 1, 1, 0, 0, 26, 150, 0, 0, 17792, 108144, 0, 0, 39809640, 326721800, 0, 0, 256814891280, 2636337861200, 0, 0, … .
As describes, the problem of listing all Langford pairings for a given n can be solved as an instance of the exact cover problem, but for large n the number of solutions can be calculated more efficiently by algebraic methods.
Applications
used Skolem sequences to construct Steiner triple systems.
In the 1960s, E. J. Groth used Langford pairings to construct circuits for integer multiplication.
See also
Stirling permutation, a different type of permutation of the same multiset
Notes
References
.
.
.
.
.
External links
John E. Miller, Langford's Problem, 2006. (with an extensive bibliography).
Combinatorics
Permutations |
https://en.wikipedia.org/wiki/List%20of%20Clube%20de%20Regatas%20do%20Flamengo%20records%20and%20statistics | This is a list of statistics and records related to Clube de Regatas do Flamengo. Flamengo is a Brazilian professional association football club based in Rio de Janeiro, RJ, that currently plays in the Campeonato Brasileiro Série A.
Football honours
International
Intercontinental Cup: 1
1981
Continental
Copa Libertadores: 3
1981, 2019, 2022
Copa Mercosul: 1 (record)
1999
Copa de Oro: 1 (record)
1996
Recopa Sudamericana: 1
2020
National
League
Campeonato Brasileiro Série A: 7
1980, 1982, 1983, 1992, 2009, 2019, 2020
Copa União: 1 (record)
1987
Cups
Supercopa do Brasil: 2 (record)
2020, 2021
Copa do Brasil: 4
1990, 2006, 2013, 2022
Copa dos Campeões: 1 (record)
2001
Regional
Torneio Rio-São Paulo: 1
1961
Local
Campeonato Carioca: 37 (record)
1914, 1915, 1920, 1921, 1925, 1927, 1939, 1942, 1943, 1944, 1953, 1954, 1955, 1963, 1965, 1972, 1974, 1978, 1979 (C), 1979 (S), 1981, 1986, 1991, 1996, 1999, 2000, 2001, 2004, 2007, 2008, 2009, 2011, 2014, 2017, 2019, 2020, 2021
Friendly
Marlboro Cup : 1
1990
Women's
National
Campeonato Brasileiro de Futebol Feminino: 1
2016
Local
Campeonato Carioca de Futebol Feminino: 6
2015, 2016 , 2017 , 2018 , 2019 , 2021
Players records
Appearances
Youngest player: Lorran – (against Audax Rio, Campeonato Carioca, 12 January 2023)
Youngest first-team player: Lorran – (against Bangu, Campeonato Carioca, 24 January 2023)
All-time records
All matches, including friendlies and non-official matches.
Players in bold currently still play for the club.
Players in italic currently still play professional football.
Foreign players all-time records
All matches, including friendlies and non-official matches.
Players in bold currently still play for the club.
Players in italic currently still play professional football.
1 Includes only official matches.
Brazilian League appearance records
Includes only matches for the Brazilian National League, created only in 1971.
Players in bold currently still play for the club.
Players in italic currently still play professional football.
Foreign players Brazilian League appearance records
Includes only matches for the Brazilian National League, created only in 1971.
Players in bold currently still play for the club.
Players in italic currently still play professional football.
*Dual or multiple citizenship.
Goalscorers
Youngest goalscorer: Lorran – (against Bangu, Campeonato Carioca, 24 January 2023)
Oldest goalscorer:b Diego Ribas – (against Coritiba, Campeonato Brasileiro Série A, 16 July 2022)
Most goals in a season in all competitions:a 43 – Gabriel Barbosa, 2019
Most League goals: 135 – Zico
Most League goals in a season: 25 – Gabriel Barbosa, 2019
Most League goals in a 38-game season: 25 – Gabriel Barbosa, 2019
Most goals scored in a match:b 4
Pedro vs Deportes Tolima, Copa Libertadores, 6 July 2022
Pedro vs Maringá, Copa do Brasil, 26 April 2023
Goals in consecutive League matches:b 7 consecutive matches – Gabriel Barbosa, 17 August 2019 to |
https://en.wikipedia.org/wiki/Ibn%20Shuayb | Abu l-`Abbas Ahmad ibn Muhammad ibn Shuayb al-Kirjani, known as Ibn Shuayb or Ibn Suhayb (; died 1 March 1349) was a Moroccan scholar of medicine, alchemy, botany, astronomy, mathematics, a poet, and the chancellor of the Marinid sultan Abu al Hassan. He was born in Taza, and died in Tunis.
References
Sources
Moroccan scholars
Moroccan writers
1349 deaths
Medieval Moroccan astronomers
Astronomers of the medieval Islamic world
Medieval Moroccan mathematicians
People from Taza
Year of birth unknown
14th-century Moroccan physicians |
https://en.wikipedia.org/wiki/Yusuke%20Murayama | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Kokushikan University alumni
Association football people from Shizuoka Prefecture
Japanese men's footballers
J1 League players
J2 League players
Shonan Bellmare players
Omiya Ardija players
Oita Trinita players
Expatriate men's footballers in Thailand
FISU World University Games gold medalists for Japan
Universiade medalists in football
Men's association football defenders |
https://en.wikipedia.org/wiki/Heidelberg%20University%20Faculty%20of%20Mathematics%20and%20Computer%20Science | The Faculty of Mathematics and Computer Science is one of twelve faculties at the University of Heidelberg. It comprises the Institute of Mathematics, the Institute of Applied Mathematics, the School of Applied Sciences, and the Institute of Computer Science. The faculty maintains close relationships to the Interdisciplinary Center for Scientific Computing (IWR) and the Mathematics Center Heidelberg (MATCH). The first chair of mathematics was entrusted to the physician Jacob Curio in the year 1547.
Institute of Mathematics
In 1547, the first chair of mathematics was entrusted to the physician Jacob Curio. Today, areas of research include:
Complex analysis: automorphic functions and modular forms
Arithmetic: algebraic number theory, algorithmic algebra, and arithmetical geometry
Topology and geometry: geometric partial differential equations, algebraic topology, differential topology, and differential geometry
Institute of Applied Mathematics
In 1957, Gottfried Köthe became the first director of the Institute of Applied Mathematics. Today, areas of research include:
Probability theory and statistics: time-series analysis, nonparametrics, asymptotic statistical procedures, and computer-intensive statistical methods
Applied analysis, numerical analysis and optimization, notably in the field of modelling and scientific computing.
Institute of Applied Sciences
In 1969, the Institute of Applied Sciences was founded. Its areas of research include:
Media Computing, Business Computing and Health Care Computing.
Communication, Robotics and Strategic Management.
Institute of Computer Science
In 2001, the Institute of Computer Science was founded. Today, areas of research include:
Computability and computational complexity theory
Efficient use of high-power computing systems
Development, administration and use of web-based information systems
Knowledge management in software development
Noted mathematicians and computer scientists
Moritz Benedikt Cantor: "History of mathematics"
Immanuel Lazarus Fuchs: "Fuchsian group", "Picard–Fuchs equation"
Emil Julius Gumbel: "Gumbel distribution"
Otto Hesse: "Hessian curve", "Hessian matrix", "Hesse normal form"
Leo Koenigsberger
Sofia Kovalevskaya: "Cauchy–Kowalevski theorem"
Emanuel Lasker: "Lasker–Noether theorem"
Jacob Lüroth
Hans Maaß
Max Noether: "Max Noether's theorem"
Oskar Perron: "Perron–Frobenius theorem", "Perron's formula", "Perron integral"
Hermann Schapira
Friedrich Karl Schmidt
Herbert Seifert: "Seifert fiber space", "Seifert surface", "Seifert–van Kampen theorem", "Seifert conjecture", Seifert–Weber space
Paul Stäckel: "twin prime"
William Threlfall
Heinrich Weber: "Kronecker–Weber theorem", "Weber's theorem"
Notes and references
Heidelberg University |
https://en.wikipedia.org/wiki/Sy%20Friedman | Sy-David Friedman (born May 23, 1953, in Chicago) is an American and Austrian mathematician and a (retired) professor of mathematics at the University of Vienna and the former director of the Kurt Gödel Research Center for Mathematical Logic. His main research interest lies in mathematical logic, in particular in set theory and recursion theory.
Friedman is the brother of Ilene Friedman and the brother of mathematician Harvey Friedman.
Biography
He studied at Northwestern University and, from 1970, at the Massachusetts Institute of Technology. He received his Ph.D. in 1976 from MIT (his thesis Recursion on Inadmissible Ordinals was written under the supervision of Gerald E. Sacks).
In 1979, Sy Friedman accepted a position at MIT, and in 1990 he became a full professor there. Since 1999, he has been a professor of mathematical logic at the University of Vienna (since 2018 retired). He is a Fellow of Collegium Invisibile.
Selected publications and results
He has authored about 70 research articles, including:
He also published a research monograph
References
External links
20th-century American mathematicians
21st-century American mathematicians
Austrian mathematicians
American emigrants to Austria
Set theorists
Fellows of Collegium Invisibile
1953 births
Living people
Massachusetts Institute of Technology alumni |
https://en.wikipedia.org/wiki/Elastic%20pendulum | In physics and mathematics, in the area of dynamical systems, an elastic pendulum (also called spring pendulum or swinging spring) is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The system exhibits chaotic behaviour and is sensitive to initial conditions. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations.
Analysis and interpretation
The system is much more complex than a simple pendulum, as the properties of the spring add an extra dimension of freedom to the system. For example, when the spring compresses, the shorter radius causes the spring to move faster due to the conservation of angular momentum. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum.
Lagrangian
The spring has the rest length and can be stretched by a length . The angle of oscillation of the pendulum is .
The Lagrangian is:
where is the kinetic energy and is the potential energy.
Hooke's law is the potential energy of the spring itself:
where is the spring constant.
The potential energy from gravity, on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is:
where is the gravitational acceleration.
The kinetic energy is given by:
where is the velocity of the mass. To relate to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:
So the Lagrangian becomes:
Equations of motion
With two degrees of freedom, for and , the equations of motion can be found using two Euler-Lagrange equations:
For :
isolated:
And for :
isolated:
The elastic pendulum is now described with two coupled ordinary differential equations. These can be solved numerically. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order in this system.
See also
Double pendulum
Duffing oscillator
Pendulum (mathematics)
Spring-mass system
References
Further reading
External links
Holovatsky V., Holovatska Y. (2019) "Oscillations of an elastic pendulum" (interactive animation), Wolfram Demonstrations Project, published February 19, 2019.
Chaotic maps
Dynamical systems
Mathematical physics
Pendulums |
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