source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Selection%20theorem | In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given set-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.
Preliminaries
Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently, is a function from X to the power set of Y.
A function is said to be a selection of F if
In other words, given an input x for which the original function F returns multiple values, the new function f returns a single value. This is a special case of a choice function.
The axiom of choice implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as continuity or measurability. This is where the selection theorems come into action: they guarantee that, if F satisfies certain properties, then it has a selection f that is continuous or has other desirable properties.
Selection theorems for set-valued functions
The approximate selection theorem says that the following conditions are sufficient for the existence of a continuous selection:
: compact metric space
: nonempty compact, convex subset of a normed linear space
a set-valued function, all values nonempty, compact, convex.
has closed graph.
For every there exists a continuous function with , where is the -dilation of , that is, the union of radius- open balls centered on points in .
The Michael selection theorem says that the following conditions are sufficient for the existence of a continuous selection:
X is a paracompact space;
Y is a Banach space;
F is lower hemicontinuous;
for all x in X, the set F(x) is nonempty, convex and closed.
The Deutsch–Kenderov theorem generalizes Michael's theorem as follows:
X is a paracompact space;
Y is a normed vector space;
F is almost lower hemicontinuous, that is, at each for each neighborhood of there exists a neighborhood of such that
for all x in X, the set F(x) is nonempty and convex.
These conditions guarantee that has a continuous approximate selection, that is, for each neighborhood of in there is a continuous function such that for each
In a later note, Xu proved that the Deutsch–Kenderov theorem is also valid if is a locally convex topological vector space.
The Yannelis-Prabhakar selection theorem says that the following conditions are sufficient for the existence of a continuous selection:
X is a paracompact Hausdorff space;
Y is a linear topological space;
for all x in X, the set F(x) is nonempty and convex;
for all y in Y, the inverse set F−1(y) is an open set in X.
The Kuratowski and Ryll-Nardzewski measurable selection theorem says that if X is a Polish space and its Borel σ-algebra, is the set of nonempty closed subsets of X, is a measurable space, and is an measurable map (that |
https://en.wikipedia.org/wiki/Ghale | Ghale () is an indigenous group of Nepal. The Ghale speak Ghale language.
Geographic distribution
The Central Bureau of Statistics of Nepal classifies the Ghale within the broader social group of Mountain/Hill Janajati. At the time of the 2011 Nepal census, 22,881 people (0.1% of the population of Nepal) were Ghale. The frequency of Ghale by province was as follows:
Bagmati Province (0.3%)
Gandaki Province (0.3%)
Koshi Province (0.0%)
Lumbini Province (0.0%)
Madhesh Province (0.0%)
Sudurpashchim Province (0.0%)
Karnali Province (0.0%)
The frequency of Ghale was higher than national average (0.1%) in the following districts:
Manang (7.1%)
Rasuwa (2.4%)
Dhading (2.0%)
Gorkha (1.9%)
Nuwakot (0.4%)
Bhojpur (0.3%)
Chitwan (0.2%)
Sindhupalchowk (0.2%)
References
External links
http://www.ghale.org
http://www.nepal.com/languages/
http://globalrecordings.net/language/4100
http://www.language-archives.org/item/oai:ethnologue.com:ghe
http://globalrecordings.net/language/10195
Indigenous peoples of Nepal |
https://en.wikipedia.org/wiki/Cluster%20algebra | Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions.
Definitions
Suppose that F is an integral domain, such as the field Q(x1,...,xn) of rational functions in n variables over the rational numbers Q.
A cluster of rank n consists of a set of n elements {x, y, ...} of F, usually assumed to be an algebraically independent set of generators of a field extension F.
A seed consists of a cluster {x, y, ...} of F, together with an exchange matrix B with integer entries bx,y indexed by pairs of elements x, y of the cluster. The matrix is sometimes assumed to be skew-symmetric, so that bx,y = –by,x for all x and y. More generally the matrix might be skew-symmetrizable, meaning there are positive integers dx associated with the elements of the cluster such that dxbx,y = –dyby,x for all x and y. It is common to picture a seed as a quiver whose vertices are the generating set, by drawing bx,y arrows from x to y if this number is positive. When bx,y is skew symmetrizable the quiver has no loops or 2-cycles.
A mutation of a seed, depending on a choice of vertex y of the cluster, is a new seed given by a generalization of tilting as follows. Exchange the values of bx,y and by,x for all x in the cluster. If bx,y > 0 and by,z > 0 then replace bx,z by bx,yby,z + bx,z. If bx,y < 0 and by,z < 0 then replace bx,z by -bx,yby,z + bx,z. If bx,y by,z ≤ 0 then do not change bx,z. Finally replace y by a new generator w, where
where the products run through the elements t in the cluster of the seed such that bt,y is positive or negative respectively. The inverse of a mutation is also a mutation, i.e. if A is a mutation of B then B is a mutation of A.
A cluster algebra is constructed from an initial seed as follows. If we repeatedly mutate the seed in all possible ways, we get a finite or infinite graph of seeds, where two seeds are joined by an edge if one can be obtained by mutating the other. The underlying algebra of the cluster algebra is the algebra generated by all the clusters of all the seeds in this graph. The cluster algebra also comes with the extra structure of the seeds of this graph.
A cluster algebra is said to be of finite type if it has only a finite number of seeds. showed that the cluster algebras of finite type can be classified in terms of the Dynkin diagrams of finite-dimensional simple Lie algebras.
Examples
Cluster algebras of rank 1
If {x} is the cluster of a seed of rank 1, then the only mutation takes this to {2x−1}. So a cluster algebra of rank 1 is just a ring k[x,x−1] of Laurent polynomials, and it has just two clusters, {x} and {2x−1}. In particular it is of finite type and is associated with the Dynkin diagram A1.
Cluster algebras of rank 2
Suppose that we start with the cluster {x1, x2} and take the exchange matrix with b12 = – |
https://en.wikipedia.org/wiki/Naomi%20Drake | Naomi Ruth (née Mason Drake; 12 February 1907 – 22 February 1987) was an American who became notable in mid-20th century Louisiana as the Registrar of the Bureau of Vital Statistics for the City of New Orleans (1949–1965), where she imposed strict racial classifications on people under a binary system that recognized only "white" and "black" (or all other). She unilaterally changed records to classify mixed-race individuals as black if she found they had any black (or African) ancestry, an application of hypodescent rules, and did not notify people of her actions.
In other cases, if people would not accept her racial classification, she refused to release the requested birth or death certificate. Her insistence on changing records to classify persons of any suspected African descent was similar to the racial zealotry demonstrated by Dr. Walter Plecker, state registrar of Virginia's Vital Statistics, and a major lobbyist for its Racial Integrity Act of 1924.
Early life and education
Career
In 1938, in Sunseri v, Cassagne (191 La. 209, 185 So. 1 - affirmed on rehearing in 1940, 195 La. 19, 196 So. 7) - the Louisiana Supreme Court proclaimed traceability of African ancestry to be the only requirement for definition of colored.
Drake began with the office as deputy and eventually became director. She directed race-flagging: she would check birth certificates that bore surnames common to blacks. If the baby was listed as white, she directed workers to check the certificate against a "race list" maintained by the Vital Records Office. If the name appeared on the "race list,"" the Vital Records Office conducted a further study of its genealogical records to reach its own assessment of the race of the individual and family. pp. 37–38
If the bureau determined through study of its genealogical records that the person in question had any African ancestors, the applicant was then informed that a certificate would be issued only if it declared the person to be colored. If the applicant refused to accept such a certificate, the bureau in turn refused to issue a certificate. There is evidence that between 1960 and 1965 a minimum of 4,700 applications for certificated copies of birth certificates and a minimum of 1,100 applications for death certificates were held in abeyance by the bureau under the supervision of Naomi Drake (188 So. 2nd 94) ...
Among the practices Drake directed was having her workers check obituaries. They were to assess whether the obituary of a person identified as white provided clues that might help show the individual was "really" black, such as having black relatives, services at a traditionally black funeral home, or burial at a traditionally black cemetery - evidence which she would use to ensure the death certificate classified the person as black.
Not everyone accepted Drake's actions, and people filed thousands of court cases against the office to have racial classifications changed and to protest her withholding leg |
https://en.wikipedia.org/wiki/Heisler%20chart | Heisler charts are a graphical analysis tool for the evaluation of one-dimensional transient conductive heat transfer in thermal engineering. They are a set of two charts per included geometry introduced in 1947 by M. P. Heisler which were supplemented by a third chart per geometry in 1961 by H. Gröber. Heisler charts permit evaluation of the central temperature for transient heat conduction through an infinitely long plane wall of thickness 2L, an infinitely long cylinder of radius ro, and a sphere of radius ro. Each aforementioned geometry can be analyzed by three charts which show the midplane temperature, temperature distribution, and heat transfer.
Although Heisler–Gröber charts are a faster and simpler alternative to the exact solutions of these problems, there are some limitations. First, the body must be at uniform temperature initially. Second, the Fourier's number of the analyzed object should be bigger than 0.2. Additionally, the temperature of the surroundings and the convective heat transfer coefficient must remain constant and uniform. Also, there must be no heat generation from the body itself.
Infinitely long plane wall
These first Heisler–Gröber charts were based upon the first term of the exact Fourier series solution for an infinite plane wall:
where Ti is the initial uniform temperature of the slab, T∞ is the constant environmental temperature imposed at the boundary, x is the location in the plane wall, λ is the root of λ * tan λ = Bi, and α is thermal diffusivity. The position x = 0 represents the center of the slab.
The first chart for the plane wall is plotted using three different variables. Plotted along the vertical axis of the chart is dimensionless temperature at the midplane, Plotted along the horizontal axis is the Fourier number, Fo = αt/L2. The curves within the graph are a selection of values for the inverse of the Biot number, where Bi = hL/k. k is the thermal conductivity of the material and h is the heat transfer coefficient.
The second chart is used to determine the variation of temperature within the plane wall at other location in the x-direction at the same time of for different Biot numbers. The vertical axis is the ratio of a given temperature to that at the centerline where the x/L curve is the position at which T is taken. The horizontal axis is the value of Bi−1.
The third chart in each set was supplemented by Gröber in 1961 and this particular one shows the dimensionless heat transferred from the wall as a function of a dimensionless time variable. The vertical axis is a plot of Q/Qo, the ratio of actual heat transfer to the amount of total possible heat transfer before T = T∞. On the horizontal axis is the plot of (Bi2)(Fo), a dimensionless time variable.
Infinitely long cylinder
For the infinitely long cylinder, the Heisler chart is based on the first term in an exact solution to a Bessel function.
Each chart plots similar curves to the previous examples, and on each axis is pl |
https://en.wikipedia.org/wiki/Doob%E2%80%93Dynkin%20lemma | In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other.
The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the -algebra that is generated by the random variable.
Notations and introductory remarks
In the lemma below, is the -algebra of Borel sets on If and is a measurable space, then
is the smallest -algebra on such that is -measurable.
Statement of the lemma
Let be a function, and a measurable space. A function is -measurable if and only if for some -measurable
Remark. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below.
Remark. The lemma remains valid if the space is replaced with where is bijective with and the bijection is measurable in both directions.
By definition, the measurability of means that for every Borel set Therefore and the lemma may be restated as follows.
Lemma. Let and is a measurable space. Then for some -measurable if and only if .
See also
Conditional expectation
References
A. Bobrowski: Functional analysis for probability and stochastic processes: an introduction, Cambridge University Press (2005),
M. M. Rao, R. J. Swift : Probability Theory with Applications, Mathematics and Its Applications, vol. 582, Springer-Verlag (2006),
Probability theorems
Theorems in measure theory |
https://en.wikipedia.org/wiki/Lax%E2%80%93Wendroff%20theorem | In computational mathematics, the Lax–Wendroff theorem, named after Peter Lax and Burton Wendroff, states that if a conservative numerical scheme for a hyperbolic system of conservation laws converges, then it converges towards a weak solution.
See also
Lax–Wendroff method
Godunov's scheme
References
Randall J. LeVeque, Numerical methods for conservation laws, Birkhäuser, 1992
Numerical differential equations
Computational fluid dynamics
Theorems in analysis |
https://en.wikipedia.org/wiki/Bh%C4%81skara%20I%27s%20sine%20approximation%20formula | In mathematics, Bhāskara I's sine approximation formula is a rational expression in one variable for the computation of the approximate values of the trigonometric sines discovered by Bhāskara I (c. 600 – c. 680), a seventh-century Indian mathematician.
This formula is given in his treatise titled Mahabhaskariya. It is not known how Bhāskara I arrived at his approximation formula. However, several historians of mathematics have put forward different hypotheses as to the method Bhāskara might have used to arrive at his formula. The formula is elegant and simple, and it enables the computation of reasonably accurate values of trigonometric sines without the use of geometry.
Approximation formula
The formula is given in verses 17–19, chapter VII, Mahabhaskariya of Bhāskara I. A translation of the verses is given below:
(Now) I briefly state the rule (for finding the bhujaphala and the kotiphala, etc.) without making use of the Rsine-differences 225, etc. Subtract the degrees of a bhuja (or koti) from the degrees of a half circle (that is, 180 degrees). Then multiply the remainder by the degrees of the bhuja or koti and put down the result at two places. At one place subtract the result from 40500. By one-fourth of the remainder (thus obtained), divide the result at the other place as multiplied by the anthyaphala (that is, the epicyclic radius). Thus is obtained the entire bahuphala (or, kotiphala) for the sun, moon or the star-planets. So also are obtained the direct and inverse Rsines.
(The reference "Rsine-differences 225" is an allusion to Aryabhata's sine table.)
In modern mathematical notations, for an angle x in degrees, this formula gives
Equivalent forms of the formula
Bhāskara I's sine approximation formula can be expressed using the radian measure of angles as follows:
For a positive integer n this takes the following form:
The formula acquires an even simpler form when expressed in terms of the cosine rather than the sine. Using radian measure for angles from to and putting , one gets
To express the previous formula with the constant one can use
Equivalent forms of Bhāskara I's formula have been given by almost all subsequent astronomers and mathematicians of India. For example, Brahmagupta's (598–668 CE)
Brhma-Sphuta-Siddhanta (verses 23–24, chapter XIV) gives the formula in the following form:
Also, Bhāskara II (1114–1185 CE) has given this formula in his Lilavati (Kshetra-vyavahara, Soka No. 48) in the following form:
Accuracy of the formula
The formula is applicable for values of x° in the range from 0° to 180°. The formula is remarkably accurate in this range. The graphs of sin x and the approximation formula are visually indistinguishable and are nearly identical. One of the accompanying figures gives the graph of the error function, namely, the function
in using the formula. It shows that the maximum absolute error in using the formula is around 0.0016. From a plot of the percentage value of the absolu |
https://en.wikipedia.org/wiki/Henry%20Wynn | Henry Philip Wynn (born 19 February 1945) is a British statistician who has been a President of the Royal Statistical Society.
He gained a Bachelor of Arts in mathematics from Oxford and a PhD in Mathematical Statistics from Imperial College, London. He was appointed a Lecturer and then Reader at Imperial College before moving to City University London in 1985 as Professor of Mathematical Statistics (and Dean of Mathematics from 1987 to 1995). At City he co-founded the Engineering Design Centre.
He moved again, in 1995, to the University of Warwick as founding Director of the Risk Initiative and Statistical Consultancy Unit. He is currently, from 2003, Professor of Statistics at the Department of Statistics, London School of Economics where he leads the Decision Support and Risk Group.
He was a founding president of the European Network for Business and Industrial Statistics (ENBIS) and a Co-Investigator on the Research Councils UK funded project Managing Uncertainty in Complex Models (MUCM). He is author of around 140 published papers and three books/monographs.
He holds the Guy Medal in Silver from the Royal Statistical Society and the George Box Medal from the European Network for Business and Industrial Statistics (ENBIS), is an Honorary Fellow of the Institute of Actuaries and a Fellow of the Institute of Mathematical Statistics.
He was the elected President of the Royal Statistical Society in 1977, the first president to be elected by a contested vote. From 1834 to 1978, RSS Presidents had always been nominated and returned unopposed. In 1978 however there had been a lot of opposition when Council arranged for Sir Campbell Adamson to stand for Council, on the understanding that he would stand for and become president the following year. However, for the first time in living memory there was an election for Council, and Campbell Adamson came last out of a 25 candidates. (There were 25 candidates and 24 place on Council.) Despite this, Campbell Adamson was put up for president, and Wynn was nominated as an alternative candidate. Although Wynn was relatively unknown at the time, he won the election and completed his presidency.
He has undertaken a wide range of research in theoretical and applied statistics, focusing principally on model building. Projects with a biological focus include work in dynamic modelling in biology (multi-strain models).
Publications
Monographs since 2000
"Dynamical Search" (H.P.Wynn, L Pronzato and A Zhigljavsky), Chapman & Hall/CRC, 2000
"Algebraic Statistics" (H.P.Wynn, E Riccomagno and G Pistone), Chapman and Hall/CRC, 2001.
Selected papers
"The Sequential Generation of D-Optimum Experimental Designs" (Henry P. Wynn), The Annals of Mathematical Statistics (1970) jstor
"Design and Analysis of Computer Experiments" (Jerome Sacks, William J. Welch, Toby J. Mitchell and Henry P. Wynn). Statistical Science (1989). jstor
References
External links
Google scholar page
Personal home page
Academics of C |
https://en.wikipedia.org/wiki/SHAZAM%20%28software%29 | SHAZAM is a comprehensive econometrics and statistics package for estimating, testing, simulating and forecasting many types of econometrics and statistical models. SHAZAM was originally created in 1977 by Kenneth White.
Compatibility
SHAZAM Version 11 is available for all Windows platforms (server, workstation or desktop) from Windows XP or later.
Data management
All SHAZAM editions read and write both fixed and free format text formats using the READ and FORMAT statements. Data can be stored by observation (row) or by variable (column) with or without variable names. Through the supplied Windows Environment formats such as comma-separated values (CSV), Microsoft Excel (both XLS and XLSX) may also be read and written.
SHAZAM Professional Edition contains comprehensive data import capabilities through its Data Connector and SQL editor allowing the import of machine data source such as tab, space or comma separated text formats, other file-based proprietary binary formats (including various Microsoft Excel formats) as well as Microsoft Access or any other LAN, WAN or internet data source for which the user has a driver on their system. In addition SHAZAM can import data from Database Management Systems (DBMS) and SHAZAM ships with drivers for most common data Database Management Systems.
Extensibility
SHAZAM is extensible through the development of SHAZAM Procedures that can be included or reused in SHAZAM command files. Numerous SHAZAM Procedures have been written and are available from the SHAZAM Website or from many sites throughout the internet.
Editions
SHAZAM is currently available in four editions. SHAZAM Command Line Edition (SHAZAMD and SHAZAMQ) are text user interface versions that allow batch processing. They operate in double and quad precisions. SHAZAM Standard Edition (SHAZAMW) adds a native Windows application that incorporates a multiple document interface (MDI) interface including command, data, matrix, graph editors and a Workspace viewer along with integrated online help, samples and examples. SHAZAM Professional Edition (SHAZAMP) adds menu and wizard driven facilities for executing SHAZAM techniques, a data connector with SQL editor as well as an integrated debugger.
SHAZAM user's reference manual
The main guide available for using SHAZAM and offers detailed descriptions of all SHAZAM features, commands and options as well as providing guidance on statistical techniques performed and the algorithms used in the construction of SHAZAM techniques. It contains numerous theoretical explanations, practical examples and sample code.
See also
Comparison of statistical packages
List of statistical packages
References
Further reading
External links
List of Software Reviews .
Research in published academic journals citing SHAZAM .
Official SHAZAM Website
Student Guide to SHAZAM by Diana Whistler
SHAZAM Command Reference
C++ software
Simulation programming languages
Econometrics software
Regression and curve fitting sof |
https://en.wikipedia.org/wiki/Unit%20circle | In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as because it is a one-dimensional unit -sphere.
If is a point on the unit circle's circumference, then and are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, and satisfy the equation
Since for all , and since the reflection of any point on the unit circle about the - or -axis is also on the unit circle, the above equation holds for all points on the unit circle, not only those in the first quadrant.
The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.
One may also use other notions of "distance" to define other "unit circles", such as the Riemannian circle; see the article on mathematical norms for additional examples.
In the complex plane
In the complex plane, numbers of unit magnitude are called the unit complex numbers. This is the set of complex numbers such that When broken into real and imaginary components this condition is
The complex unit circle can be parametrized by angle measure from the positive real axis using the complex exponential function, (See Euler's formula.)
Under the complex multiplication operation, the unit complex numbers are group called the circle group, usually denoted In quantum mechanics, a unit complex number is called a phase factor.
Trigonometric functions on the unit circle
The trigonometric functions cosine and sine of angle may be defined on the unit circle as follows: If is a point on the unit circle, and if the ray from the origin to makes an angle from the positive -axis, (where counterclockwise turning is positive), then
The equation gives the relation
The unit circle also demonstrates that sine and cosine are periodic functions, with the identities
for any integer .
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius from the origin to a point on the unit circle such that an angle with is formed with the positive arm of the -axis. Now consider a point and line segments . The result is a right triangle with . Because has length , length , and has length 1 as a radius on the unit circle, and . Having established these equivalences, take another radius from the origin to a point on the circle such that the same angle is formed with the negative arm of the -axis. Now consider a point and line segments . The result is a right triangle with . It can hence be seen that, because , is at in the same way that P is at . The conclusion is that, since is the same as and is the same as , it is true that and . It |
https://en.wikipedia.org/wiki/List%20of%20statistical%20offices%20in%20Germany | The statistical offices of the German states (German: Statistische Landesämter) carry out the task of collecting official statistics in Germany together and in cooperation with the Federal Statistical Office.
The implementation of statistics according to Article 83 of the constitution is executed at state level. The federal government has, under Article 73 (1) 11. of the constitution, the exclusive legislation for the "statistics for federal purposes."
There are 14 statistical offices for the 16 states:
See also
Federal Statistical Office of Germany
References
Germany
Statistical offices
Germany |
https://en.wikipedia.org/wiki/Somchai%20Chuayboonchum | Somchai Chuayboonchum (Thai สมชาย ชวยบุญชุม) is a Thai football manager and former player.
Playing career
He played as midfielder for Thailand national team.
Managerial statistics
Honours
Manager
Samut Songkhram
Thai Division 1 League runner-up: 2007
Thailand U-19
AFF U-19 Youth Championship: 2011
Nongbua Pitchaya
Thai League 2: 2020–21
Individual
Thai League 1 Coach of the Month: July 2016
References
See also
Wikipedia in Thai
Somchai Chuayboonchum
Somchai Chuayboonchum
Somchai Chuayboonchum
Living people
1954 births
Somchai Chuayboonchum
Men's association football midfielders
Somchai Chuayboonchum
Somchai Chuayboonchum
Somchai Chuayboonchum
Somchai Chuayboonchum
SEA Games medalists in football
Competitors at the 1979 SEA Games |
https://en.wikipedia.org/wiki/Cornacchia%27s%20algorithm | In computational number theory, Cornacchia's algorithm is an algorithm for solving the Diophantine equation , where and d and m are coprime. The algorithm was described in 1908 by Giuseppe Cornacchia.
The algorithm
First, find any solution to (perhaps by using an algorithm listed here); if no such exist, there can be no primitive solution to the original equation. Without loss of generality, we can assume that (if not, then replace with , which will still be a root of ). Then use the Euclidean algorithm to find , and so on; stop when . If is an integer, then the solution is ; otherwise try another root of until either a solution is found or all roots have been exhausted. In this case there is no primitive solution.
To find non-primitive solutions where , note that the existence of such a solution implies that divides (and equivalently, that if is square-free, then all solutions are primitive). Thus the above algorithm can be used to search for a primitive solution to . If such a solution is found, then will be a solution to the original equation.
Example
Solve the equation . A square root of −6 (mod 103) is 32, and 103 ≡ 7 (mod 32); since and , there is a solution x = 7, y = 3.
References
External links
Number theoretic algorithms |
https://en.wikipedia.org/wiki/Gerhard%20Geise | Gerhard Paul Geise (2 January 1930, Stendal – 11 April 2010, Dresden) was a German mathematician and professor of pure mathematics.
He died after a long serious illness in Dresden.
Works (selection)
1961: Über ähnlich-veränderliche ebene Systeme
1976: Senkrechte Projektion
1977: Kegelschnitte, Kugel und Kartenentwürfe
1979: Grundkurs lineare Algebra
1980: Analytische Geometrie für Kristallgitter
1991: Berührungskegelschnitte in Bézierdarstellung
1994: Darstellende Geometrie
1995: Analytische Geometrie
Literature
Geise, Gerhard. In Dorit Petschel (Bearb.): Die Professoren der TU Dresden 1828–2003. Böhlau Verlag, Köln / Weimar / Vienna 2003, .
References
20th-century German mathematicians
Academic staff of TU Dresden
1930 births
2010 deaths
People from Stendal |
https://en.wikipedia.org/wiki/Primitive%20abundant%20number | In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.
For example, 20 is a primitive abundant number because:
The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.
The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number.
The first few primitive abundant numbers are:
20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ...
The smallest odd primitive abundant number is 945.
A variant definition is abundant numbers having no abundant proper divisor . It starts:
12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114
Properties
Every multiple of a primitive abundant number is an abundant number.
Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number.
Every primitive abundant number is either a primitive semiperfect number or a weird number.
There are an infinite number of primitive abundant numbers.
The number of primitive abundant numbers less than or equal to n is
References
Divisor function
Integer sequences |
https://en.wikipedia.org/wiki/Sunao%20Hozaki | is a Japanese footballer for SC Sagamihara.
Career statistics
Updated to 23 February 2018.
References
External links
Profile at SC Sagamihara
1987 births
Living people
Ryutsu Keizai University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
J3 League players
Mito HollyHock players
Thespakusatsu Gunma players
Zweigen Kanazawa players
SC Sagamihara players
Men's association football defenders |
https://en.wikipedia.org/wiki/Hiroki%20Kato | is a former Japanese football player.
Club statistics
References
External links
1986 births
Living people
Ryutsu Keizai University alumni
Association football people from Kanagawa Prefecture
People from Hiratsuka, Kanagawa
Japanese men's footballers
J1 League players
J2 League players
Yokohama F. Marinos players
Mito HollyHock players
Men's association football defenders |
https://en.wikipedia.org/wiki/Method%20of%20steepest%20descent | In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-point approximation is used with integrals in the complex plane, whereas Laplace’s method is used with real integrals.
The integral to be estimated is often of the form
where C is a contour, and λ is large. One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that the following conditions hold:
C′ passes through one or more zeros of the derivative g′(z),
the imaginary part of g(z) is constant on C′.
The method of steepest descent was first published by , who used it to estimate Bessel functions and pointed out that it occurred in the unpublished note by about hypergeometric functions. The contour of steepest descent has a minimax property, see . described some other unpublished notes of Riemann, where he used this method to derive the Riemann–Siegel formula.
Basic idea
The method of steepest descent is a method to approximate a complex integral of the formfor large , where and are analytic functions of . Because the integrand is analytic, the contour can be deformed into a new contour without changing the integral. In particular, one seeks a new contour on which the imaginary part of is constant. Then and the remaining integral can be approximated with other methods like Laplace's method.
Etymology
The method is called the method of steepest descent because for analytic , constant phase contours are equivalent to steepest descent contours.
If is an analytic function of , it satisfies the Cauchy–Riemann equationsThen so contours of constant phase are also contours of steepest descent.
A simple estimate
Let and . If
where denotes the real part, and there exists a positive real number such that
then the following estimate holds:
Proof of the simple estimate:
The case of a single non-degenerate saddle point
Basic notions and notation
Let be a complex -dimensional vector, and
denote the Hessian matrix for a function . If
is a vector function, then its Jacobian matrix is defined as
A non-degenerate saddle point, , of a holomorphic function is a critical point of the function (i.e., ) where the function's Hessian matrix has a non-vanishing determinant (i.e., ).
The following is the main tool for constructing the asymptotics of integrals in the case of a non-degenerate saddle point:
Complex Morse lemma
The Morse lemma for real-valued functions generalizes as follows for holomorphic functions: near a non-degenerate saddle point of a holomorphic function , there exist coordinates in terms of which is exactly quadratic. To make this precise, let be a holomorphic function with domain , and let in be a non-degenerate saddle point of , that is |
https://en.wikipedia.org/wiki/Lagrange%27s%20identity%20%28disambiguation%29 | Lagrange's identity may refer to:
Lagrange's identity, an algebraic identity
Lagrange's identity (boundary value problem), an identity in calculus
Lagrange's trigonometric identities, two trigonometric identities
Lagrange's four-square theorem, a theorem from number theory
Lagrange polynomial for theorems relating to numerical interpolation
Euler–Lagrange equation of variational mechanics |
https://en.wikipedia.org/wiki/Bryan%20Wells%20%28ice%20hockey%29 | Bryan Wells (born February 24, 1966) is a Canadian former professional ice hockey player and ice hockey coach. He coached the Wichita Thunder from 1996–2001.
Career statistics
References
External links
1966 births
Living people
Brandon Wheat Kings players
Canadian expatriate ice hockey players in the United States
Canadian ice hockey coaches
Canadian ice hockey centres
Carolina Thunderbirds players
Regina Pats players
Wichita Thunder coaches
Wichita Thunder players |
https://en.wikipedia.org/wiki/Yuki%20Shimada | is a Japanese football player. He plays for Vonds Ichihara.
Club statistics
References
External links
1986 births
Living people
Komazawa University alumni
Association football people from Saitama Prefecture
Japanese men's footballers
J2 League players
Mito HollyHock players
ReinMeer Aomori players
Vonds Ichihara players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Taiki%20Tsuruno | is a Japanese football player. He plays for Saurcos Fukui.
Club statistics
References
External links
1990 births
Living people
Japanese men's footballers
J2 League players
Japan Football League players
Mito HollyHock players
Zweigen Kanazawa players
Vanraure Hachinohe players
Men's association football midfielders
Association football people from Sapporo |
https://en.wikipedia.org/wiki/HP-17B | HP-17B is an algebraic entry financial and business calculator manufactured by Hewlett-Packard, introduced on 4 January 1988 along with the HP-19B, HP-27S and the HP-28S. It was a simplified business model, like the 19B. There were two versions, the US one working in English only, and the international one with a choice of six languages (English, German, Spanish, French, Italian, and Portuguese).
HP-17B
HP-17B code name was Trader and it belonged to the Pioneer series of Hewlett-Packard calculators. It had a 131×16 LCD dot matrix, 22×2 characters, menu-driven display, used a Saturn processor and had a memory of 8000 bytes, of which 6750 bytes were available to the user for variable and equation storage. The HP-17B had a clock with alarm that allowed for basic agenda capabilities, as well an infrared port for printing to some Hewlett-Packard infrared printers.
HP 17BII
The 17B was replaced by the HP 17BII (F1638A) (code name Trader II) in January 1990, which added RPN entry.
HP 17bII+
The 17BII was replaced by the HP 17bII+ in 2003. Two significantly different case variants of the 17bII+ exist. The newer 17bII+ (F2234A), introduced in 2007, with Sunplus Technology SPLB31A CPU was developed and is manufactured by Kinpo Electronics.
References
External links
HP page of Christoph Gießelink scripts for the HP-17B and more
17B
17B
Sunplus |
https://en.wikipedia.org/wiki/Kunihiro%20Shibazaki | is a former Japanese football player.
Club statistics
References
External links
1981 births
Living people
Shizuoka Sangyo University alumni
Association football people from Tokyo
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Omiya Ardija players
Tochigi SC players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Yuki%20Okada%20%28footballer%2C%20born%201983%29 | is a former Japanese football player.
Club statistics
References
External links
1983 births
Living people
Association football people from Shizuoka Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Hokkaido Consadole Sapporo players
Tochigi SC players
Mito HollyHock players
Fujieda MYFC players
Men's association football defenders
People from Fujieda, Shizuoka |
https://en.wikipedia.org/wiki/Toshikazu%20Irie | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Sakushin Gakuin University alumni
Association football people from Tochigi Prefecture
Japanese men's footballers
J2 League players
Japan Football League players
Tochigi SC players
FB Gulbene players
GKS Górnik Łęczna players
Japanese expatriate men's footballers
Expatriate men's footballers in Latvia
Japanese expatriate sportspeople in Latvia
Expatriate men's footballers in Poland
Men's association football defenders |
https://en.wikipedia.org/wiki/Vladimir%20Andrunakievich | Vladimir Aleksandrovich Andrunakievich (Russian: Владимир Александрович Андрунакиевич; 3 April 1917 – 22 July 1997) was a Soviet and Moldovan mathematician, known for his work in abstract algebra. He was a doctor of physical and mathematical sciences (1958), academician (1961) and vice-president (1964—1969, 1979—1990) of the Moldavian Soviet Academy of Sciences. Laureate of the State Prize of the Moldavian SSR (1972).
Andrunakievich was born in Petrograd. He received his Ph.D. from the Moscow State University in 1947 under the supervision of Aleksandr Gennadievich Kurosh and Otto Schmidt.
Monographs
Radicals of algebras and structure theory (with Iu. M. Ryabukhin). Moscow: Nauka, 1979.
Numbers and ideals (with I. D. Chirtoaga). Kishinev: Lumina, 1980.
Applied problems of solid mechanics. Kishinev: Ştiinţa, 1985.
Modules, algebras and topologies. Kishinev: Ştiinţa, 1988.
Constructions of topological rings and modules (with V. I. Arnautov). Kishinev: Ştiinţa, 1988.
Articles
References
1917 births
1997 deaths
People from Saint Petersburg
People from Sankt-Peterburgsky Uyezd
20th-century Moldovan mathematicians
Algebraists
Soviet mathematicians
Alexandru Ioan Cuza University alumni
Academic staff of the D. Mendeleev University of Chemical Technology of Russia
Academic staff of Moldova State University
Moscow State University alumni
Recipients of the Moldavian SSR State Prize
Recipients of the Order of Lenin
Recipients of the Order of the October Revolution
Recipients of the Order of the Red Banner of Labour
Recipients of the Order of the Badge of Honour
Recipients of the Order of the Republic (Moldova) |
https://en.wikipedia.org/wiki/Robert%20Breusch | Robert Hermann Breusch (April 2, 1907 – March 29, 1995) was a German-American number theorist, the William J. Walker Professor of Mathematics at Amherst College.
Breusch was born in Freiburg, Germany, and studied mathematics both at the University of Freiburg and the University of Berlin. Unable to secure a university position after receiving his doctorate, Breusch became a schoolteacher near Freiburg, where he met his future wife, Kate Dreyfuss; Breusch was Protestant, but Dreyfuss was Jewish, and the two of them left Nazi Germany for Chile in the mid-1930s. They married there, and Breusch found a faculty position at Federico Santa María Technical University in Valparaiso. In 1939, they left Chile for the United States, inviting Robert Frucht to take Breusch's place at Santa María; after some years working again as a schoolteacher, Breusch found a position at Amherst College in 1943. He became the Walker professor in 1970, and retired to become an emeritus professor in 1973. The Robert H. Breusch Prize in Mathematics, for the best senior thesis from an Amherst student, was endowed in his memory.
As a mathematician, Breusch was known for his new proof of the prime number theorem and for the many solutions he provided to problems posed in the American Mathematical Monthly. His thesis work combined Bertrand's postulate with Dirichlet's theorem on arithmetic progressions by showing that each of the progressions 3i + 1, 3i + 2, 4i + 1, and 4i + 3 (for i = 0, 1, 2, ...) contains a prime number between x and 2x for every x ≥ 7. For instance, he proved that for n > 47 there is at least one prime between n and (9/8)n. He also wrote a calculus textbook, Calculus and Analytic Geometry with Applications (Prindle, Weber & Schmidt, 1969) with C. Stanley Ogilvy.
See also
Fermat's Last Theorem
Odd greedy expansion
References
1907 births
1995 deaths
Scientists from Freiburg im Breisgau
20th-century German mathematicians
20th-century American mathematicians
Number theorists
Amherst College faculty
Emigrants from Nazi Germany
Immigrants to the United States |
https://en.wikipedia.org/wiki/Lobb%20number | In combinatorial mathematics, the Lobb number Lm,n counts the number of ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses.
Lobb numbers form a natural generalization of the Catalan numbers, which count the number of complete strings of balanced parentheses of a given length. Thus, the nth Catalan number equals the Lobb number L0,n. They are named after Andrew Lobb, who used them to give a simple inductive proof of the formula for the nth Catalan number.
The Lobb numbers are parameterized by two non-negative integers m and n with n ≥ m ≥ 0. The (m, n)th Lobb number Lm,n is given in terms of binomial coefficients by the formula
An alternative expression for Lobb number Lm,n is:
The triangle of these numbers starts as
where the diagonal is
and the left column are the Catalan Numbers
As well as counting sequences of parentheses, the Lobb numbers also count the number of ways in which n + m copies of the value +1 and n − m copies of the value −1 may be arranged into a sequence such that all of the partial sums of the sequence are non-negative.
Ballot counting
The combinatorics of parentheses is replaced with counting ballots in an election with two candidates in Bertrand's ballot theorem, first published by William Allen Whitworth in 1878. The theorem states the probability that winning candidate is ahead in the count, given known final tallies for each candidate.
References
Integer sequences
Factorial and binomial topics
Enumerative combinatorics |
https://en.wikipedia.org/wiki/Brownian%20excursion | In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems.
Definition
A Brownian excursion process, , is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive.
Another representation of a Brownian excursion in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyosi Itô and Henry P. McKean, Jr.)
is in terms of the last time that W hits zero before time 1 and the first time that Brownian motion hits zero after time 1:
Let be the time that a
Brownian bridge process achieves its minimum on [0, 1]. Vervaat (1979) shows that
Properties
Vervaat's representation of a Brownian excursion has several consequences for various functions of . In particular:
(this can also be derived by explicit calculations) and
The following result holds:
and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:
Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of . A formula for a certain double transform of the distribution of this area integral is given by Louchard (1984).
Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of .
For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII.
Connections and applications
The Brownian excursion area
arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g. and the limit distribution of the Betti numbers of certain varieties in cohomology theory. Takacs (1991a) shows that has density
where are the zeros of the Airy function and is the confluent hypergeometric function.
Janson and Louchard (2007) show that
and
They also give higher-order expansions in both cases.
Janson (2007) gives moments of and many other area functionals. In particular,
Brownian excursions also arise in connection with
queuing problems,
railway traffic, and the heights of random rooted binary trees.
Related processes
Brownian bridge
Brownian meander
reflected Brownian motion
skew Brownian motion
Notes
References
Wiener process |
https://en.wikipedia.org/wiki/Approximate%20limit | In mathematics, the approximate limit is a generalization of the ordinary limit for real-valued functions of several real variables.
A function f on has an approximate limit y at a point x if there exists a set F that has density 1 at the point such that if xn is a sequence in F that converges towards x then f(xn) converges towards y.
Properties
The approximate limit of a function, if it exists, is unique. If f has an ordinary limit at x then it also has an approximate limit with the same value.
We denote the approximate limit of f at x0 by
Many of the properties of the ordinary limit are also true for the approximate limit.
In particular, if a is a scalar and f and g are functions, the following equations are true if values on the right-hand side are well-defined (that is the approximate limits exist and in the last equation the approximate limit of g is non-zero.)
Approximate continuity and differentiability
If
then f is said to be approximately continuous at x0. If f is function of only one real variable and the difference quotient
has an approximate limit as h approaches zero we say that f has an approximate derivative at x0. It turns out that approximate differentiability implies approximate continuity, in perfect analogy with ordinary continuity and differentiability.
It also turns out that the usual rules for the derivative of a sum, difference, product and quotient have straightforward generalizations to the approximate derivative. There is no generalization of the chain rule that is true in general however.
External links
Approximate continuity at Encyclopedia of Mathematics
Approximate derivative at Encyclopedia of Mathematics
Approximate differentiability at Encyclopedia of Mathematics
References
Real analysis
Limits (mathematics) |
https://en.wikipedia.org/wiki/Friedrich%20Wilhelm%20Sch%C3%A4fke | Friedrich Wilhelm Heinrich Schäfke (21 July 1922 in Berlin, Weimar Germany – 4 April 2010) was a German mathematician and professor of geometry.
Writings
Mathieusche Funktionen und Sphäroidfunktionen mit Anwendungen auf physikalische und technische Probleme, Springer 1954, with Josef Meixner
Einführung in die Theorie der speziellen Funktionen der mathematischen Physik, Springer 1963
Differenzierbare Abbildungen, Köln 1967, with Dietrich Krekel und Dieter Schmdit
Quasimetrische Räume und quasinormierte Gruppen, Birlinghoven St. Augustin 1971
Gewöhnliche Differentialgleichungen. Die Grundlagen die Theorie im Reellen und Komplexen, Springer 1973, , with Dieter Schmdit
Integrale, 1992, , with Dieter Hoffmann
References
Obituary
External links
Friedrich Wilhelm Schäfke in the catalog of the German National Library
Friedrich Wilhelm Schäfke in the Mathematics Genealogy Project
20th-century German mathematicians
1922 births
2010 deaths |
https://en.wikipedia.org/wiki/Miracle%20Octad%20Generator | In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for studying the Mathieu groups, binary Golay code and Leech lattice.
Description
The Miracle Octad Generator is a 4x6 array of combinations describing any point in 24-dimensional space. It preserves all of the symmetries and maximal subgroups of the Mathieu group M24, namely the monad group, duad group, triad group, octad group, octern group, sextet group, trio group and duum group. It can therefore be used to study all of these symmetries.
Golay code
Another use for the Miracle Octad Generator is to quickly verify codewords of the binary Golay code. Each element of the Miracle Octad Generator can store either a '1' or a '0', usually displayed as an asterisk and blank space, respectively. Each column and the top row have a property known as the count, which is the number of asterisks in that particular line. One of the criteria for a set of 24 coordinates to be a codeword in the binary Golay code is for all seven counts to be of the same parity. The other restriction is that the scores of each column form a word in the hexacode. The score of a column can be either 0, 1, ω, or ω-bar, depending on its contents. The score of a column is evaluated by the following rules:
If a column contains exactly one asterisk, it has a score of 0 if it resides in the top row, 1 if it is in the second row, ω for the third row, and ω-bar for the bottom row.
Simultaneously complementing every bit in a column does not affect its score.
Complementing the bit in the top row does not affect its score, either.
A codeword can be derived from just its top row and score, which proves that there are exactly 4096 codewords in the binary Golay code.
MiniMOG
John Horton Conway developed a 4 × 3 array known as the MiniMOG. The MiniMOG provides the same function for the Mathieu group M12 and ternary Golay code as the Miracle Octad Generator does for M24 and binary Golay code, respectively. Instead of using a quaternary hexacode, the MiniMOG uses a ternary tetracode.
Notes
References
External links
The Miracle Octad Generator
Sporadic groups |
https://en.wikipedia.org/wiki/Fox%20H-function | In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by .
It is defined by a Mellin–Barnes integral
where L is a certain contour separating the poles of the two factors in the numerator.
Relation to other Functions
Lambert W-function
A relation of the Fox H-Function to the -1 branch of the Lambert W-function is given by
where is the complex conjugate of .
Meijer G-function
Compare to the Meijer G-function
The special case for which the Fox H reduces to the Meijer G is Aj = Bk = C, C > 0 for j = 1...p and k = 1...q :
A generalization of the Fox H-function is given by Ram Kishore Saxena. For a further generalization of this function, useful in physics and statistics was given by A.M.Mathai and Ram Kishore Saxena,
References
.
External links
hypergeom on GitLab
Use in solving on MathOverflow
Hypergeometric functions
Special functions |
https://en.wikipedia.org/wiki/Einstein%20function | In mathematics, Einstein function is a name occasionally used for one of the functions
References
E W Lemmon, R Span, 2006, Short Fundamental Equations of State for 20 Industrial Fluids, J. Chem. Eng. Data 51 (3), 785–850 .
Wolfram MathWorld: http://mathworld.wolfram.com/EinsteinFunctions.html
Special functions |
https://en.wikipedia.org/wiki/Co-cultural%20communication%20theory | Co-cultural communication theory was built upon the frameworks of muted group theory and standpoint theory. The cornerstone of co-cultural communication theory is muted group theory as proposed in the mid 1970s by Shirley and Edwin Ardener. The Ardeners were cultural anthropologists who made the observation that most other cultural anthropologists practicing ethnography in the field were talking only to the leaders of the cultures, who were by and large adult males. The researchers would then use this data to represent the culture as a whole, leaving out the perspectives of women, children and other groups made voiceless by the cultural hierarchy (S. Ardener, 1975). The Ardeners maintained that groups which function at the top of the society hierarchy determine to a great extent the dominant communication system of the entire society (E. Ardener, 1978). Ardener's 1975 muted group theory also posited that dominant group members formulate a "communication system that support their perception of the world and conceptualized it as the appropriate language for the rest of society".
Communication faculty Stanback and Pearce (1981) referred to these non-dominant groups as "subordinate social groups". They noted 4 ways in which the non-dominant groups tend to communicate with the dominant groups. They also asserted that "From the perspective of the dominant group, the behaviors in each form of communication are appropriate. However, the meaning of these behaviors to the members of the lower-statused group are quite different, making them different forms of communication with different implications for the relations among the groups".
In the study of communication, Stanback and Pearce as well as Kramarae used muted group theory to help explain communication patterns and social representation of non-dominant cultural groups Kramarae (1981) believed that "those experiences unique to subordinate group members often cannot be effectively expressed within the confinements of the dominant communication system". She suggested that people within these groups create alternative forms of communication to articulate their experiences. Although, Kramarae used muted group theory to communications strategies of women she suggested that the framework can be applied with equal validity to a number of dominant/non-dominant relationships (Orbe, 1996).
Kramarae (1981) presented three assumptions of muted group theory as applied to communication between men and women concluding that women traditionally have been muted by a male-dominated communications system. Additionally, Kramarae proposed seven hypotheses originating in muted group theory. Standpoint theory was mainly used as a feminist theoretical framework to explore experiences of women as they participate in and oppose their own subordination, however, (Smith, 1987) suggested that the theory had applications for other subordinate groups. A basic tenet of standpoint theory is that it "seeks to include the experience |
https://en.wikipedia.org/wiki/Shawn%20Carlson | Shawn Carlson (born 1960) is an American physicist, science writer, and a STEM educator.
Education
Carlson graduated from U.C. Berkeley with Bachelor of Science degrees in both Applied Mathematics and Physics in 1981. He graduated from UCLA with a master's degree in physics in 1983, and with a Ph.D. in Nuclear Physics in 1989. As a post doc, Carlson ran the Leuschner Observatory for the Center for Particle Astrophysics at the Lawrence Berkeley National Laboratory and was chief observer for the Berkeley Automated Supernovae Search.
Career
Astrology test
While an undergraduate, Carlson carried out a double-blind test of astrologers' abilities to extract information about their clients from the apparent positions of celestial objects as seen from the places and times of their clients' births.
Carlson's experiment involved twenty-eight astrologers who were held in high esteem by their peers. They agreed to match over 100 natal charts to psychological profiles that were generated by the California Psychological Inventory (CPI), a standard and well accepted personality test, which the astrologers themselves identified as the scientific instrument that was best aligned with type of information they believed they could divine from their art. The astrologers agreed that the experimental protocol provided a "fair test" of astrology prior to taking part in it.
The participating astrologers were nominated by the National Council for Geocosmic Research (NCGR), which acted as the astrological advisors to ensure that the test was fair. The astrologers came from Europe and the United States. The astrologers also identified the central proposition of natal astrology to be tested.
The results were published in Nature on December 5, 1985. The study found that astrologers were unable to match natal charts to their corresponding personality tests better than chance. Moreover, astrologers were no more likely to be correct even when they had high confidence that they had made a match correctly. Carlson concluded that the result "clearly refutes the astrological hypothesis".
Other activities
Carlson left academia in 1994 and founded the Society for Amateur Scientists. He contributed to the columns "Science on Society" on The Humanist from 1990-1992, "The Amateur Scientist" in Scientific American from 1995 to 2001, and "The Citizen Scientist" for Make magazine from 2005 to 2007. In 2010, he launched LabRats Science Education Program, an organization that organizes activities for and encourages amateur scientists. He currently serves as Executive Director for the organization.
Awards
1999 MacArthur Fellows Program
Selected works
References
External links
"The Amateur Scientist Column, Scientific American magazine"
"LabRats Science Education project"
"TEDx Talk, 2012 "Connecting the Dots to Your Future."
"Introduction to Engagement Education."
1960 births
21st-century American physicists
UC Berkeley College of Letters and Science alumni
University of Cal |
https://en.wikipedia.org/wiki/List%20of%20Melbourne%20City%20FC%20records%20and%20statistics | Melbourne City Football Club is an Australian professional soccer club based in Bundoora, Melbourne. The club was formed in 2009 as Melbourne Heart before being renamed as Melbourne City. They became the second Victorian member admitted into the A-League Men in 2010 after Melbourne Victory.
The list encompasses the honours won by Melbourne City. The player records section itemises the club's leading goalscorers and those who have made the most appearances in first-team competitions. It also records notable achievements by Melbourne City players on the international stage, and the highest transfer fees paid and received by the club.
Melbourne City has won one A-League Men Premiership in 2020–21, one A-League Men Championship in 2021 and an Australia Cup in 2016. The club's record appearance maker is Scott Jamieson, who made 161 appearances, between 2017 and 2023. Jamie Maclaren is Melbourne City's record goalscorer, scoring 105 goals in total.
All figures are correct as of the match played on 29 October 2023.
Honours and achievements
Domestic
A-League Men Premiership
Winners (3): 2020–21, 2021–22, 2022–23
Runners-up (1): 2019–20
A-League Men Championship
Winners (1): 2021
Runners-up (3): 2020, 2022, 2023
Australia Cup
Winners (1): 2016
Runners-up (1): 2019
Player records
Appearances
Most A-League Men appearances: Scott Jamieson, 140
Most Australia Cup appearances: Scott Jamieson, 15
Most AFC Champions League appearances: Jamie Maclaren, 9
Youngest first-team player: Idrus Abdulahi, 15 years, 216 days (against Central Coast Mariners, A-League, 26 April 2019)
Oldest first-team player: Thomas Sørensen, 40 years, 285 days (against Western Sydney Wanderers, A-League, 24 March 2017)
Most consecutive appearances: Bruno Fornaroli, 55 (from 26 August 2015 to 4 February 2017)
Most appearances
Competitive matches only, includes appearances as substitute. Numbers in brackets indicate goals scored.
Goalscorers
Most goals in a season: Jamie Maclaren, 29 goals (in the 2019–20 season)
Most A-League Men goals in a season: Jamie Maclaren, 25 goals in the A-League, 2020–21
Youngest goalscorer: Max Caputo, 17 years, 228 days (against Newcastle Jets, A-League Men, 2 April 2023)
Oldest goalscorer: Tim Cahill, 37 years, 131 days (against Perth Glory, A-League, 16 April 2017)
Top goalscorers
Competitive matches only. Numbers in brackets indicate appearances made.
International
This section refers to caps won while a Melbourne City player.
First capped player: Michael Beauchamp for Australia against New Zealand on 24 May 2010.
Most capped player: Jamie Maclaren with 16 caps.
First player to play in the World Cup finals: Daniel Arzani, for Australia against France on 16 June 2018
Managerial records
First full-time manager: John van 't Schip managed Melbourne City from October 2009 to April 2012
Longest-serving manager: John van 't Schip, (30 December 2013 to 3 January 2017)
Shortest tenure as manager: Michael Valkanis, 5 months, 17 days |
https://en.wikipedia.org/wiki/Isoclinism%20of%20groups | In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated aspects of character theory, as described in and . The word "isoclinism" comes from the Greek ισοκλινης meaning equal slope.
Some textbooks discussing isoclinism include and and .
Definition
The isoclinism class of a group G is determined by the groups G/Z(G) (the inner automorphism group) and (the commutator subgroup) and the commutator map from G/Z(G) × G/Z(G) to (taking a, b to aba−1b−1).
In other words, two groups G1 and G2 are isoclinic if there are isomorphisms from G1/Z(G1) to G2/Z(G2) and from G1 to G2 commuting with the commutator map.
Examples
All Abelian groups are isoclinic since they are equal to their centers and their commutator subgroups are always the identity subgroup. Indeed, a group is isoclinic to an abelian group if and only if it is itself abelian, and G is isoclinic with G×A if and only if A is abelian. The dihedral, quasidihedral, and quaternion groups of order 2n are isoclinic for n≥3, in more detail.
Isoclinism divides p-groups into families, and the smallest members of each family are called stem groups. A group is a stem group if and only if Z(G) ≤ [G,G], that is, if and only if every element of the center of the group is contained in the derived subgroup (also called the commutator subgroup), . Some enumeration results on isoclinism families are given in .
Isoclinism is used in theory of projective representations of finite groups, as all Schur covering groups of a group are isoclinic, a fact already hinted at by Hall according to . This is used in describing the character tables of the finite simple groups .
References
Finite groups |
https://en.wikipedia.org/wiki/1946%E2%80%9347%20Wolverhampton%20Wanderers%20F.C.%20season | Season:1946-1947
1st Division
Final league position: 3
Statistics
External links
WolvesStats for 46/47 season
Wolverhampton Wanderers F.C. seasons
Wolverhampton Wanderers |
https://en.wikipedia.org/wiki/Anabelian%20geometry | Anabelian geometry is a theory in number theory which describes the way in which the algebraic fundamental group G of a certain arithmetic variety X, or some related geometric object, can help to restore X. The first results for number fields and their absolute Galois groups were obtained by Jürgen Neukirch, Masatoshi Gündüz Ikeda, Kenkichi Iwasawa, and Kôji Uchida (Neukirch–Uchida theorem, 1969) prior to conjectures made about hyperbolic curves over number fields by Alexander Grothendieck. As introduced in Esquisse d'un Programme the latter were about how topological homomorphisms between two arithmetic fundamental groups of two hyperbolic curves over number fields correspond to maps between the curves. These Grothendieck conjectures were partially solved by Hiroaki Nakamura and Akio Tamagawa, while complete proofs were given by Shinichi Mochizuki.
Anabelian geometry can be viewed as one of the three generalizations of class field theory. Unlike two other generalizations — abelian higher class field theory and representation theoretic Langlands program — anabelian geometry is non-abelian and highly non-linear.
Formulation of a conjecture of Grothendieck on curves
The "anabelian question" has been formulated as
A concrete example is the case of curves, which may be affine as well as projective. Suppose given a hyperbolic curve C, i.e., the complement of n points in a projective algebraic curve of genus g, taken to be smooth and irreducible, defined over a field K that is finitely generated (over its prime field), such that
.
Grothendieck conjectured that the algebraic fundamental group G of C, a profinite group, determines C itself (i.e., the isomorphism class of G determines that of C). This was proved by Mochizuki. An example is for the case of (the projective line) and , when the isomorphism class of C is determined by the cross-ratio in K of the four points removed (almost, there being an order to the four points in a cross-ratio, but not in the points removed). There are also results for the case of K a local field.
Mono-anabelian geometry
Shinichi Mochizuki introduced and developed the mono-anabelian geometry, an approach which restores, for a certain class of hyperbolic curves over number fields or some other fields, the curve from its algebraic fundamental group. Key results of mono-anabelian geometry were published in Mochizuki's "Topics in Absolute Anabelian Geometry" I (2012), II (2013), and III (2015).
The opposite approach of mono-anabelian geometry is bi-anabelian geometry, a term coined by Mochizuki in "Topics in Absolute Anabelian Geometry III" to indicate the classical approach.
Mono-anabelian geometry deals with certain types (strictly Belyi type) of hyperbolic curves over number fields and local fields. This theory considerably extends anabelian geometry. Its main aim to construct algorithms which produce the curve, up to an isomorphism, from the étale fundamental group of such a curve. In particular, for the first |
https://en.wikipedia.org/wiki/Digital%20manifold | In mathematics, a digital manifold is a special kind of combinatorial manifold which is defined in digital space i.e. grid cell space. A combinatorial manifold is a kind of manifold which is a discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes.
Concepts
Parallel-move is used to extend an i-cell to (i+1)-cell. In other words, if A and B are two i-cells
and A is a parallel-move of B, then {A,B} is an (i+1)-cell.
Therefore, k-cells can be defined recursively.
Basically, a connected set of grid points M can be viewed as a digital k-manifold if:
(1) any two k-cells are (k-1)-connected, (2) every (k-1)-cell has
only one or two parallel-moves, and (3) M does not contain any (k+1)-cells.
See also
Digital geometry
Digital topology
Topological data analysis
Topology
Discrete mathematics
References
Digital topology
Digital geometry |
https://en.wikipedia.org/wiki/W-curve | In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge alone. Many well-known curves are W-curves, among them conics, logarithmic spirals, powers (like y = x3), logarithms and the helix, but not e.g. the sine. W-curves occur widely in the realm of plants.
Name
The 'W' stands for the German 'Wurf' – a throw – which in this context refers to a series of four points on a line. A 1-dimensional W-curve (read: the motion of a point on a projective line) is determined by such a series.
The German "W-Kurve" sounds almost exactly like "Weg-Kurve" and the last can be translated by "path curve". That is why in the English literature one often finds "path curve" or "pathcurve".
See also
Homography
Further reading
Felix Klein and Sophus Lie: Ueber diejenigen ebenen Curven... in Mathematische Annalen, Band 4, 1871; online available at the University of Goettingen
For an introduction on W-curves and how to draw them, see Lawrence Edwards Projective Geometry, Floris Books 2003,
On the occurrence of W-curves in nature see Lawrence Edwards The vortex of life, Floris Books 1993,
For an algebraic classification of 2- and 3-dimensional W-curves see Classification of pathcurves
Georg Scheffers (1903) "Besondere transzendente Kurven", Klein's encyclopedia Band 3–3.
Curves
Projective geometry |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20Stevenage%20F.C.%20season | The 2010–11 season was Stevenage F.C.'s first season in the Football League, where the club competed in League Two. This article shows statistics of the club's players in the season, and also lists all matches that the club played during the season. Their first-place finish and subsequent promotion means it will be their first ever season of playing in the Football League, having featured in the Conference National for the past sixteen years. The season also marked the first season that the club played under its new name – Stevenage Football Club, dropping 'Borough' from its title as of 1 June 2010. It was the third year in charge for manager Graham Westley during his second spell at the club; having previously managed the Hertfordshire side from 2003 to 2006.
Ahead of the club's first season in the Football League, Westley stated a desire to give the majority of the squad that had won promotion from the Conference National a chance to play in League Two. This subsequently meant there was a similar level of transfer activity to the club's 2009–10 campaign. Three players left the club ahead of the 2010–11 season. Andy Drury decided to leave Stevenage in favour of a move back into non-league, joining Luton Town on a free transfer, while both Mark Albrighton and Eddie Odhiambo were released in June 2010. Five players joined the club during the close season. Darius Charles was the first signing of the season, joining Stevenage from Ebbsfleet United for a fee set by a tribunal. Wingers Peter Winn and Rob Sinclair signed on free transfers from Scunthorpe United and Salisbury City respectively, while defender Luke Foster joined the club after being released by Mansfield Town. Stevenage also announced the signing of midfielder John Mousinho in late June 2010, who rejected a contract extension at Wycombe Wanderers in order to sign for the club. In terms of transfers during the 2010–11 campaign, strikers Lee Boylan, Tim Sills and Peter Vincenti were all allowed to find new clubs in January 2011, while both Marvin Williams and Taiwo Atieno had brief spells with the club. Three strikers joined the club during the season, Ben May and Byron Harrison signed for the club on free transfers, whilst Craig Reid joined from Newport County for what was a "club record fee" in January 2011.
Stevenage recorded just one league win from their first seven league fixtures, and suffered from inconsistency for the first half of the league campaign. Following four defeats in six games in December 2010 and January 2011, Stevenage found themselves in 18th position, just four points above the relegation zone. However, during a congested period throughout February and March 2011, Stevenage won nine games out of eleven, propelling the club up the league table and into the play-off positions. This included winning six games on the trot, a sequence only matched by Bury. A 3–3 draw on the last day of the season against Bury confirmed Stevenage's place in the play-offs, finishing sixt |
https://en.wikipedia.org/wiki/Norm%20residue%20isomorphism%20theorem | In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime and any natural number . John Milnor speculated that this theorem might be true for and all , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions. The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.
Statement
For any integer ℓ invertible in a field there is a map
where denotes the Galois module of ℓ-th roots of unity in some separable closure of k. It induces an isomorphism . The first hint that this is related to K-theory is that is the group K1(k). Taking the tensor products and applying the multiplicativity of étale cohomology yields an extension of the map to maps:
These maps have the property that, for every element a in , vanishes. This is the defining relation of Milnor K-theory. Specifically, Milnor K-theory is defined to be the graded parts of the ring:
where is the tensor algebra of the multiplicative group and the quotient is by the two-sided ideal generated by all elements of the form . Therefore the map factors through a map:
This map is called the Galois symbol or norm residue map. Because étale cohomology with mod-ℓ coefficients is an ℓ-torsion group, this map additionally factors through .
The norm residue isomorphism theorem (or Bloch–Kato conjecture) states that for a field k and an integer ℓ that is invertible in k, the norm residue map
from Milnor K-theory mod-ℓ to étale cohomology is an isomorphism. The case is the Milnor conjecture, and the case is the Merkurjev–Suslin theorem.
History
The étale cohomology of a field is identical to Galois cohomology, so the conjecture equates the ℓth cotorsion (the quotient by the subgroup of ℓ-divisible elements) of the Milnor K-group of a field k with the Galois cohomology of k with coefficients in the Galois module of ℓth roots of unity. The point of the conjecture is that there are properties that are easily seen for Milnor K-groups but not for Galois cohomology, and vice versa; the norm residue isomorphism theorem makes it possible to apply techniques applicable to the object on one side of the isomorphism to the object on the other side of the isomorphism.
The case when n is 0 is trivial, and the case when follows easily from Hilbert's Theorem 90. The case and was proved by . An important advance was the case an |
https://en.wikipedia.org/wiki/Dieudonn%C3%A9%20module | In mathematics, a Dieudonné module introduced by , is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms and called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.
Finite flat commutative group schemes over a perfect field of positive characteristic can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring
,
which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of . The endomorphisms and are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Dieudonné and Pierre Cartier constructed an antiequivalence of categories between finite commutative group schemes over of order a power of and modules over with finite -length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps , and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected -group schemes correspond to -modules for which is nilpotent, and étale group schemes correspond to modules for which is an isomorphism.
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Tadao Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Alexander Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze -divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Andrew Wiles's work on the Shimura–Taniyama conjecture.
Dieudonné rings
If is a field of characteristic , its ring of Witt vectors consists of sequences of elements of , and has an endomorphism induced by the Frobenius endomorphism of , so . The Dieudonné ring, often denoted by or , is the non-commutative ring over generated by 2 elements and subject to the relations
.
It is a -graded ring, where the piece of degree is a 1-dimensional free module over , spanned by if and by if .
Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by and .
Dieudonné modules and groups
Special sorts of modules over the Dieudonné ring correspond to certain algebraic group schemes. For example, finite length modules over the Dieudonné ring form an abelian category equivalent to the opposite of the category of finite commuta |
https://en.wikipedia.org/wiki/Carlo%20Minnaja | Carlo Minnaja (born 19 March 1940) is a retired professor of Mathematics, a native Esperanto speaker, and Esperanto translator, author and researcher. He authored many books about Esperanto, in Esperanto and Italian, including a vocabulary of Esperanto, and is a member of the Akademio de Esperanto.
Academic career
Minnaja graduated with a bachelor's degree in mathematics from the University of Pisa in 1963. He earned his doctorate in Mathematics at the Scuola Normale Superiore di Pisa, and began his academic career as a professor at the Accademia Navale di Livorno in 1965.
In 1973, Minnaja became a mathematics professor in the University of Padua's Engineering Department. He taught the courses "Mathematics for Electrical Engineering," "History of Mathematics," and "Linear Algebra and Geometry for Information Engineering."
Between 1979 and 1994, he was a member of the Scientific Council of Institute of Computational Linguistics of Pisa. On a Fulbright scholarship, in 1983 and 1984, he was a visiting professor at Virginia Tech and UCLA.
Awards and appointments
Minnaja is a member of the International Academy of Sciences San Marino and a former vice president. He has also lectured in several European universities, and was author or coauthor of fifty scientific papers. Many of which are his books and articles on the international language Esperanto.
In 1980, he received the Culture Award from the prime minister of Italy.
Activities in the Esperanto movement
In 1960, Minnaja became a member of the Board of the World Esperanto Youth Organization, addressing specialized sections and services. He is still very active in the fields of esperantology and Esperanto literature, and is a member of the literary circle Patrol.
In 1981, he translated Carlo Goldoni's The Mistress of the Inn into Esperanto, and in 1982, he published his translations of the poems of Cesare Pavese.
In 1996, he published his Italian-Esperanto dictionary, which, at more than 1400 pages, is the most comprehensive Italian-Esperanto dictionary.
Minnaja is a member of the Scientific Committee of the Centro Italiano di Interlinguistica, works regularly as a reviewer for the esperanto literary magazine Monato and was editor-in-chief of the magazine Literatura Foiro in 2014.
Books
Following a brief list of Carlo Minnaja's books, some others can be found in his Padua University account page.
Eugenio Montale kay aliaj liguriaj aŭtoroj (Esperanto: "Eugenio Montale and other ligurian authors") pub. Edizioni Eva by C. Minnaja, 2013, .
I matematici nell'Università di Padova. Dal suo nascere al XX secolo (Italian: "Mathematicians at the University of Padua. From its birth to the twentieth century") by C. Minnaja, E. Giusti, F. Baldassarri, 2007, .
Historio de la esperanta literaturo (Esperanto: "History of Esperanto Literature") by C. Minnaja, G. Silfer, 2016, .
L'esperanto in Italia. Alla ricerca della democrazia linguistica (Italian: "Esperanto in Italy. In search of Linguis |
https://en.wikipedia.org/wiki/Thomas%20Murray%20MacRobert | Thomas Murray MacRobert (4 April 1884, in Dreghorn, Ayrshire – 1 November 1962, in Glasgow) was a Scottish mathematician. He became professor of mathematics at the University of Glasgow and introduced the MacRobert E function, a generalisation of the generalised hypergeometric series.
Life
He was born on 4 April 1884 in the manse at Dreghorn, Ayrshire in south-west Scotland, the son of Rev Thomas MacRobert and his wife, Isabella Edgely Fisher. He was educated at Irvine Royal Academy with his identical twin brother, Alexander, then studied divinity at Glasgow University but transferred to study mathematics and natural philosophy (physics), graduating in 1905. He then took a second degree at Trinity College, Cambridge.
In 1910 he joined the staff of Glasgow University as an assistant to Professor Gibson, lecturing in mathematics.
In the First World War he served in the Royal Garrison Artillery and saw active service in France.
In 1921 he was elected a fellow of the Royal Society of Edinburgh. His proposers were Andrew Gray, George Alexander Gibson, James Gordon Gray and Robert Alexander Houston. He was President of the Edinburgh Mathematical Society 1921/22. He resigned from the RSE in 1940.
Glasgow University granted him an honorary doctorate (LLD) in 1955.
He retired in 1954 and died in Glasgow on 1 November 1962.
Family
In 1914, before going to war, he married Violet McIlwraith; they initially lived in a flat in North Kelvinside in Glasgow. They had three children: Violet, Tom and Alexander. He was a member of the Glasgow temperance movement and enjoyed hill-walking.
Artistic recognition
His portrait by Norman Hepple is held by the Hunterian Art Gallery in Glasgow.
Publications
Functions of a Complex Variable (1917)
Spherical Harmonics (1927)
Trigonometry (1938)
Higher Trigonometry (1943)
Spherical Trigonometry (1946)
A Short Introduction to Fine Typography (1957)
References
Biography of Thomas Murray MacRobert
1884 births
1962 deaths
Academics of the University of Glasgow
British identical twins
People from Dreghorn
Alumni of the University of Glasgow
Alumni of Trinity College, Cambridge
Royal Garrison Artillery officers
Fellows of the Royal Society of Edinburgh
20th-century Scottish mathematicians
British Army personnel of World War I |
https://en.wikipedia.org/wiki/B%C3%A1lint%20Vir%C3%A1g | Bálint Virág (born 1973) is a Hungarian mathematician working in Canada, known for his work in probability theory, particularly determinantal processes, random matrix theory, and random walks and other probabilistic questions on groups. He received his Ph.D. from U.C. Berkeley in 2000, under the direction of Yuval Peres, and was a post-doc at MIT. Since 2003 he has been a Canada research chair at the University of Toronto.
Virág was awarded a Sloan Fellowship (2004), the Rollo Davidson Prize (2008), the Coxeter–James Prize (2010), and the John L. Synge Award (2014). He was an invited speaker at the International Congress of Mathematicians in 2014.
References
External links
Bálint Virág at the University of Toronto
Probability theorists
Living people
Canada Research Chairs
1973 births
21st-century Hungarian mathematicians
21st-century Canadian mathematicians
Academic staff of the University of Toronto
Massachusetts Institute of Technology alumni
University of California, Berkeley alumni
Scientists from Toronto |
https://en.wikipedia.org/wiki/Mehdi%20Rahimzadeh | Mehdi Rahimzadeh (born March 1, 1989) is an Iranian footballer.
Club career
Rahimzadeh started his career with Malavan F.C.
Club career statistics
References
1989 births
Living people
Malavan F.C. players
Gostaresh Foulad F.C. players
S.C. Damash Gilan players
Iranian men's footballers
Men's association football goalkeepers
Footballers from Gilan province |
https://en.wikipedia.org/wiki/Mohammad%20Rostami | Mohammad Rostami (born September 15, 1985) is an Iranian footballer who plays for Malavan F.C. in the IPL.
Club career
Rostami has played his entire career with Malavan F.C.
Club career statistics
Last Update: 22 August 2011
References
1985 births
Living people
Malavan F.C. players
S.C. Damash Gilan players
Sepidrood Rasht S.C. players
Iranian men's footballers
Men's association football defenders
Footballers from Gilan province |
https://en.wikipedia.org/wiki/Sign%20sequence | In mathematics, a sign sequence, or ±1–sequence or bipolar sequence, is a sequence of numbers, each of which is either 1 or −1. One example is the sequence (1, −1, 1, −1, ...).
Such sequences are commonly studied in discrepancy theory.
Erdős discrepancy problem
Around 1932, mathematician Paul Erdős conjectured that for any infinite ±1-sequence and any integer C, there exist integers k and d such that
The Erdős discrepancy problem asks for a proof or disproof of this conjecture.
In February 2014, Alexei Lisitsa and Boris Konev of the University of Liverpool showed that every sequence of 1161 or more elements satisfies the conjecture in the special case C = 2, which proves the conjecture for C ≤ 2. This was the best such bound available at the time. Their proof relied on a SAT-solver computer algorithm whose output takes up 13 gigabytes of data, more than the entire text of Wikipedia at that time, so it cannot be independently verified by human mathematicians without further use of a computer.
In September 2015, Terence Tao announced a proof of the conjecture, building on work done in 2010 during Polymath5 (a form of crowdsourcing applied to mathematics) and a suggestion made by German mathematician Uwe Stroinski on Tao's blog. His proof was published in 2016, as the first paper in the new journal Discrete Analysis.
Erdős discrepancy of finite sequences has been proposed as a measure of local randomness in DNA sequences. This is based on the fact that in the case of finite-length sequences discrepancy is bounded, and therefore one can determine the finite sequences with discrepancy less than a certain value. Those sequences will also be those that "avoid" certain periodicities. By comparing the expected versus observed distribution in the DNA or using other correlation measures, one can make conclusions related to the local behavior of DNA sequences.
Barker codes
A Barker code is a sequence of N values of +1 and −1,
such that
for all .
Barker codes of lengths 11 and 13 are used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties.
See also
Binary sequence
Discrepancy of hypergraphs
Rudin–Shapiro sequence
Notes
References
External links
The Erdős discrepancy problem – Polymath Project
Computer cracks Erdős puzzle – but no human brain can check the answer—The Independent (Friday, 21 February 2014)
Binary sequences
Computer-assisted proofs |
https://en.wikipedia.org/wiki/Larry%20Wos | Lawrence T. Wos (1930–2020) was an American mathematician, a researcher in the Mathematics and Computer Science Division of Argonne National Laboratory.
Biography
Wos studied at the University of Chicago, receiving a bachelor's degree in 1950 and a master's in mathematics in 1954, and went on for doctoral studies at the University of Illinois at Urbana-Champaign where he received PhD in 1957 supervised by Reinhold Baer. He joined the Argonne in 1957, and began using computers to prove mathematical theorems in 1963.
Wos was congenitally blind. He was an avid bowler, the best male blind bowler in the US.
Awards and honors
In 1982, Wos and his colleague Steve Winker were the first to win the Automated Theorem Proving Prize, given by the American Mathematical Society.
In 1992, Wos was the first to win the Herbrand Award for his contributions to the field of automated deduction. A festschrift in his honor, Automated reasoning and its applications: essays in honor of Larry Wos (Robert Veroff, ed.) was published by the MIT Press in 1997 ().
Books
Wos and Gail W. Pieper are the coauthors of the books A Fascinating Country in the World of Computing: Your Guide to Automated Reasoning (World Scientific, 1999, ) and Automated Reasoning and the Discovery of Missing and Elegant Proofs (Rinton Press, 2003, ). Wos's collected works were published by World Scientific in 2000, in two volumes ().
References
External links
Publication list at DBLP
Maria Paola Bonacina (with Franz Baader, Alan Bundy, Ulrich Furbach, Frank Pfenning, John Slaney, and Christoph Weidenbach), "In Memoriam: Larry Wos"
American scientists with disabilities
Academics from Chicago
20th-century American mathematicians
21st-century American mathematicians
University of Chicago alumni
University of Illinois Urbana-Champaign alumni
1930 births
2020 deaths |
https://en.wikipedia.org/wiki/Mumford%E2%80%93Tate%20group | In algebraic geometry, the Mumford–Tate group (or Hodge group) MT(F) constructed from a Hodge structure F is a certain algebraic group G. When F is given by a rational representation of an algebraic torus, the definition of G is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the p-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of on p-divisible groups, and named them Mumford–Tate groups.
Formulation
The algebraic torus T used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of a+bi on the basis {1,i} of the complex numbers C over R:
The circle group inside this group of matrices is the unitary group U(1).
Hodge structures arising in geometry, for example on the cohomology groups of Kähler manifolds, have a lattice consisting of the integral cohomology classes. Not quite so much is needed for the definition of the Mumford–Tate group, but it does assume that the vector space V underlying the Hodge structure has a given rational structure, i.e. is given over the rational numbers Q. For the purposes of the theory the complex vector space VC, obtained by extending the scalars of V from Q to C, is used.
The weight k of the Hodge structure describes the action of the diagonal matrices of T, and V is supposed therefore to be homogeneous of weight k, under that action. Under the action of the full group VC breaks up into subspaces Vpq, complex conjugate in pairs under switching p and q. Thinking of the matrix in terms of the complex number λ it represents, Vpq has the action of λ by the pth power and of the complex conjugate of λ by the qth power. Here necessarily
p + q = k.
In more abstract terms, the torus T underlying the matrix group is the Weil restriction of the multiplicative group GL(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to GL(1), interchanged by complex conjugation.
Once formulated in this fashion, the rational representation ρ of T on V setting up the Hodge structure F determines the image ρ(U(1)) in GL(VC); and MT(F) is by definition the smallest algebraic group defined over Q containing this image.
Mumford–Tate conjecture
The original context for the formulation of the group in question was the question of the Galois representation on the Tate module of an abelian variety A. Conjecturally, the image of such a Galois representation, which is an l-adic Lie group for a given prime number l, is determined by the corresponding Mumford–Tate group G (coming from the Hodge structure on H1(A)), to the extent that knowledge of G determines the Lie algebra of the Galois image. This conjecture is known only in particular cases. Through generalisations of this conjecture, the Mumford–Tat |
https://en.wikipedia.org/wiki/Paul%20Olum | Paul Olum (August 16, 1918 – January 19, 2001) was an American mathematician (algebraic topology), professor of mathematics, and university administrator.
Early years
Born in Binghamton, New York to a father who was a Russian Jew who immigrated at age of nine to escape persecution, Olum took an interest in mathematics at an early age. He graduated summa cum laude from Harvard University in 1940. In 1942 he married Vivian Goldstein, completed an MA in physics at Princeton University, and joined the scientific staff of the Manhattan Project. During his time at Los Alamos, Olum was among the Los Alamos scientists who questioned the implications of the atomic bomb, and after its use against Japan, he became a lifelong advocate for world peace and for nuclear arms control.
Reportedly, one reason he switched from physics to mathematics as his field was that compared to his office mate, future Nobel laureate Richard Feynman, Olum did not think he was good at physics. He returned to Harvard after the war to complete his Ph.D. in mathematics in 1947 under Hassler Whitney as his thesis advisor.
Among his close friends was Feynman, who wrote in his autobiography of Paul's intelligence. In one anecdote, Feynman told of an experience at Los Alamos when he had claimed to be able to take any problem that could be stated in ten seconds and find an answer to within ten percent in no more than sixty seconds. When Feynman made this challenge to Olum, he quickly responded, “Find the tangent of 10 to the 100th.”
Cornell
Following a postdoctoral year at the Institute for Advanced Study, Olum joined the Cornell University faculty in 1949. Over the next 25 years at Cornell, Olum rose to the rank of professor, served in various administrative roles, and spent time as a visiting faculty member at the University of Paris, Hebrew University, Stanford University, and the University of Washington, and also returned as a member of the Institute for Advanced Study.
As a mathematician, Olum was widely respected for his research in algebraic topology. He made significant contributions in the area of obstruction theory. His Ph.D. students include Martin Arkowitz, Robert Lewis, Jean-Pierre Meyer, and Norman Stein.
In 1962 Olum initiated the Cornell Topology Festival, an annual regional mathematics conference. From 1963 to 1966, Olum served as Mathematics Department chair, and recruited a number of talented faculty.
Olum advocated the abolition of the House Committee on Unamerican Activities, was an early critic of the Vietnam War, and sought to remove the Reserve Officer Training Corps from the Cornell campus. Olum assisted in the establishment of Cornell's Women's Studies Program in 1972. Following the Willard Straight Hall Takeover in 1969, Olum chaired a committee to propose a major overhaul of Cornell's governance, including its Board of Trustees. Olum's relationship with his fellow Los Alamos physicist Dale Corson, who had just become Cornell's President, assisted in |
https://en.wikipedia.org/wiki/Doomsday%20conjecture | In algebraic topology, the doomsday conjecture was a conjecture about Ext groups over the Steenrod algebra made by Joel Cohen, named by Michael Barratt, published by and disproved by . stated a modified version called the new doomsday conjecture.
The original doomsday conjecture was that for any prime p and positive integer s there are only a finite number of permanent cycles in
found an infinite number of permanent cycles for p = s = 2, disproving the conjecture. Minami's new doomsday conjecture is a weaker form stating (in the case p = 2) that there are no nontrivial permanent cycles in the image of (Sq0)n for n sufficiently large depending on s.
References
Algebraic topology |
https://en.wikipedia.org/wiki/Visionlearning | Visionlearning is a free, web-based resource for students and educators in the science, technology, engineering and mathematics (STEM) disciplines. Geared toward those studying at high school and undergraduate levels, Visionlearning takes advantage of recent advances in new media to provide students and educators with learning and teaching materials. Research by project personnel has shown that this peer-reviewed and bilingual content improves student understanding of science and facilitates multidisciplinary teaching. The project also strives to build community around improving STEM education.
Visionlearning is supported by the National Science Foundation and the U.S. Department of Education.
History
In 1998, Dr. Anthony Carpi, an environmental chemist and then-assistant professor at John Jay College of the City University of New York, designed and launched a prototype Internet learning resource for science students, called The Natural Science Pages. This was in part a response to the growing evidence that poor textbook content and deficient teaching materials contribute to inadequate science education. The purpose of the prototype was to see if presenting important course information in a new form would improve what students understood, what information they remembered, and how well they performed in their class overall.
What this research showed, through standardized exam scores and student retention data, was that using the prototype significantly improved science comprehension and performance in the targeted natural science course. Further evaluations also showed that the prototype helped improve communication skills and student engagement with the course.
In response to the success of The Natural Science Pages prototype and with support of the NSF, Carpi developed the Visionlearning project, which brought on a group of scientists and educators to create content and became a website open to students and educators worldwide. In the last decade, Visionlearning has evolved into a more comprehensive website that provides free educational materials to support the science, technology, engineering, and mathematics (STEM) disciplines, with translations into Spanish. These materials include independent learning modules that focus on distinct topics within STEM disciplines, teaching resources, a MyClassroom tool (similar to Moodle or WebCT) that allows educators to customize and maintain a virtual learning environment for their students, and new media elements to benefit multiple learning styles. A key feature of all of Visionlearning’s material is an emphasis on the process of science and discovery.
The Process of Science: A Philosophy for Teaching and Learning
Throughout the development of the learning modules and multimedia tools, Visionlearning has emphasized an importance on teaching science as a process, rather than “a simple set of facts and terms to be memorized.” This comes from an organizational ideology that understanding how scientif |
https://en.wikipedia.org/wiki/2010%E2%80%9311%20PFC%20CSKA%20Sofia%20season | The 2010–11 season was PFC CSKA Sofia's 63rd consecutive season in A Group. This article shows player statistics and all matches (official and friendly) that the club will play during the 2010–11 season.
Players
Squad statistics
Appearances for competitive matches only
|-
|colspan="14"|Players sold or loaned out after the start of the season:
|}
As of 29 May 2011
Players in/out
Summer transfers
In:
Out:
Winter transfers
In:
Out:
Player seasonal records
Competitive matches only. Updated to games played 29 May 2011.
Goalscorers
Start formations
Overall
{|class="wikitable" style="text-align: center;"
|-
!
!Total
! Home
! Away
|-
|align=left| Games played || 30 || 15 || 15
|-
|align=left| Games won || 18 || 10 || 8
|-
|align=left| Games drawn || 7 || 3 || 4
|-
|align=left| Games lost || 4 || 2 || 3
|-
|align=left| Biggest win || 4–0 vs Chernomorets || 3–1 vs Lokomotiv Sofia3–1 vs Akademik Sofia || 4–0 vs Chernomorets
|-
|align=left| Biggest loss || 1–4 vs Lokomotiv Plovdiv || 1–2 vs Chernomorets || 1–4 vs Lokomotiv Plovdiv
|-
|align=left| Clean sheets || 13 || 8 || 5
|-
|align=left| Goals scored || 53 || 23 || 30
|-
|align=left| Goals conceded || 26 || 10 || 16
|-
|align=left| Goal difference || +27 || +11 || +14
|-
|align=left| Top scorer || Delev – 13 || Delev – 7 || Delev – 6
|-
|align=left| Winning rate || % || % || %
|-
Source: Soccerway
Pre-season and friendlies
Pre-season
On-season (autumn)
Mid-season
On-season (spring)
Competitions
A Group
Table
Results summary
Results by round
Fixtures and results
Bulgarian Cup
Europa League
Third qualifying round
Play-off round
Group stage
UEFA Club Rankings
This is the current UEFA Club Rankings, including season 2009–10.
See also
PFC CSKA Sofia
References
External links
CSKA Official Site
Bulgarian A Professional Football Group
UEFA Profile
PFC CSKA Sofia seasons
Cska Sofia |
https://en.wikipedia.org/wiki/2001%E2%80%9302%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb It also lists all matches that Dinamo Zagreb played in the 2001–02 season.
Players
Squad
(Correct as of September 2001)
Goalscorers
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: 2001–02 Prva HNL article
Matches
Competitive
Last updated 4 May 2002Sources:
External links
Dinamo Zagreb official website
GNK Dinamo Zagreb seasons
Dinamo Zagreb |
https://en.wikipedia.org/wiki/Graph%20algebra%20%28social%20sciences%29 | Graph algebra is systems-centric modeling tool for the social sciences. It was first developed by Sprague, Pzeworski, and Cortes as a hybridized version of engineering plots to describe social phenomena.
Notes and references
Algebra
Social science methodology |
https://en.wikipedia.org/wiki/Carl-Gustav%20Esseen | Carl-Gustav Esseen (13 September 1918 in Linköping – 10 November 2001) was a Swedish mathematician. His work was in the theory of probability. The Berry–Esseen theorem is named after him.
Life
Carl-Gustav Esseen attended school in Linköping. Starting in 1936, he studied mathematics, astronomy, physics and chemistry at the University of Uppsala. Inspired by the harmonic-analytic research of Harald Cramér and Arne Beurling, Esseen examined the accuracy of the approximation to the normal distribution in the central limit theorem in the case of independent and identically distributed summands. Esseen's bound is now called "the Berry-Esseen theorem", because it was independently proved by Andrew C. Berry, also.
In 1944 Esseen received his doctorate with a thesis on the Fourier analysis of probability distributions. In 1949 he was appointed full professor of applied mathematics at the Royal Institute of Technology in Stockholm. In 1962 his professorship moved to the field of mathematical statistics and in 1967, he became first holder of the chair of mathematical statistics at the University of Uppsala. He retired in 1984.
Scientific work
Although Esseen worked mostly on the central limit theorem and related topics, he also worked in other areas. Some industrial applications were considered in his writings, for example, his studies on control theory and in telecommunications. After retirement, Esseen worked on topics from number theory, especially factorization, a topic of importance in cryptology.
Esseen supervised several doctoral students. His lectures and writings were meticulously prepared and delivered.
Honours
1963: Elected Member of the Royal Swedish Academy of Engineering Sciences.
Selected works
"On the Liapounoff limit of error in the theory of probability", Arkiv för Mathematik, Astronomi och Fysik 28A, #9 (1942), 19 pp., .
"Determination of the maximum deviation from the Gaussian law", Arkiv för Mathematik, Astronomi och Fysik 29A, #20 (1943), 10 pp., .
"Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law", Acta Mathematica 77, #1 (December 1945), pp. 1–125, , . (Dissertation)
"On mean central limit theorems", Kungl. Tekn. Högsk. Handl. Stockholm #121 (1958), 31 pp., .
"Bounds for the absolute third moment", Scandinavian Journal of Statistics () 2, #3 (1975), pp. 149–152, , .
(with Svante Janson) "On moment conditions for normed sums of independent variables and martingale differences", Stochastic Processes and their Applications 19, #1 (1985), pp. 173–182, , .
"A stochastic model for primitive roots", Revue Roumaine de Mathématiques Pures et Appliquées 38, #6 (1993), pp. 481–501, .
References
External links
Members of the Royal Swedish Academy of Engineering Sciences
Probability theorists
20th-century Swedish mathematicians
Swedish statisticians
Academic staff of Uppsala University
Uppsala University alumni
Academic staff of the KTH Royal Institute of Technology
2001 deaths
1918 |
https://en.wikipedia.org/wiki/2002%E2%80%9303%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb It also lists all matches that Dinamo Zagreb played in the 2002–03 season.
Squad
(Correct as of November 2002)
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: 2002–03 Prva HNL article
Matches
Competitive
Last updated 4 May 2002Sources:
External links
2002–03 in Croatian Football at Rec.Sport.Soccer Statistics Foundation
2002–03 season at Prva HNL official website
Dinamo Zagreb Official website
GNK Dinamo Zagreb seasons
Dinamo Zagreb
Croatian football championship-winning seasons |
https://en.wikipedia.org/wiki/Srirangam%20Kannan | Srirangam Kannan (born 5 May 1952) is an Indian musician and artist, known for playing the morsing. He has a degree in mathematics.
Profile
Vidwan Srirangam S. Kannan was born on 5 May 1952 in Srirangam to K Sathyamurthy and Kamalam. Growing up, he had little experience with Carnatic music.
When he was 19 years old he heard a concert where Sri Pudukkotai S. Mahadevan played the morsing. Shortly afterwards, he became Mahadevan's disciple. He also learned more about laya (tempo) from Kanadukathan Rajaraman, a kanjeera and mridangam artist and a friend of Mahadevan. By age 23, Srirangam Kannan had started his career as a full-fledged morsing artist.
After graduating from university with a degree in mathematics, he joined Indian Bank, where he worked for 30 years before retiring in 2000 after having become manager.
He continues to play in concerts across India. He also performs regularly for AIR Chennai.
Awards and honours
Srirangam Kannan has been the recipient of many awards and recognitions, listed here.
Awarded Mannargudi Natesa Pillai Award, instituted by Sri Raagam Fine Arts, Chennai, presented by Dr. M Balamuralikrishna in 1996.
Kalaimamani Award by the Government of Tamil Nadu in 1998
Best Upapakkavadhyam Award from the Music Academy, instituted by Dr. Ramamurthy, in 1998 & 2001.
Honoured as the Asthana Vidwan of Sri Kanchi Kamakoti Peetam in the year 2000.
Best Upapakkavadhyam Award from Narada Gana Sabha, instituted by Obul Reddy, in 2003.
Lifetime Achievement Award in the field of Carnatic Music from the Kanchi Kamakoti Peetam in 2003.
A Top Graded artiste in All India Radio
Meritorious Award for achievement in Carnatic Music, instituted by the Maharajapuram Santhanam Foundation, Chennai in 2005.
Vani Kala Sudhakara award for the most proficient morsing vidwan, instituted by Sri Thyaga Bhrama Gana Sabha, Chennai in 2005.
Lifetime Achievement Award in the field of Carnatic Music from Sri Sachidananda Swamy of Datta Peetam, Mysore in 2006
Nada Vidya Bhupathi, instituted by Nada Dweepam Trust, Chennai in 2009.
Tours and concerts
Listed here are Srirangam Kannan's tours and concerts.
1988: Festival of India in USSR with Shri. Karaikudi R Mani and Dr. N Ramani
1990: Tala Vaadya Concerts in France, Italy, Belgium and the UK.
1990: Participated in the Collegium Instrumentale Hale, Chamber Orchestra in Germany, presented by Dr. L Subramaniam
1991: Tala Vaadya ensemble conducted by Zakir Hussain in Malaysia.
1992: Tala Vaadya ensemble organized by Indian Council for Cultural Relations in Hungary, Germany, and the UK.
1997: Indian Independence Golden Jubilee Celebration at New Delhi, Tala Vaadya ensemble with Umayalpuram Sivaraman and Pt Kishan Maharaj.
1998: Participated in the International Music Festival, held at Helsinki, Finland
1998: Participated in the Telstra Adelaide Music Festival, Australia.
2000: Participated in the Jazz Festival, Copenhagen, Denmark.
2000: Participated in the World Expo, Hannover, Germany.
|
https://en.wikipedia.org/wiki/%C5%81o%C5%9B%E2%80%93Tarski%20preservation%20theorem | The Łoś–Tarski theorem is a theorem in model theory, a branch of mathematics, that states that the set of formulas preserved under taking substructures is exactly the set of universal formulas. The theorem was discovered by Jerzy Łoś and Alfred Tarski.
Statement
Let be a theory in a first-order logic language and
a set of formulas of .
(The sequence of variables need not be
finite.) Then the following are equivalent:
If and are models of , , is a sequence of elements of . If , then .( is preserved in substructures for models of )
is equivalent modulo to a set of formulas of .
A formula is if and only if it is of the form where is quantifier-free.
In more common terms, this states that
every first-order formula is preserved under induced substructures if and only if it is , i.e. logically equivalent to a first-order universal formula.
As substructures and embeddings are dual notions, this theorem is sometimes stated in its dual form:
every first-order formula is preserved under embeddings on all structures if and only if it is , i.e. logically equivalent to a first-order existential formula.
Note that this property fails for finite models.
Citations
References
Model theory
Metalogic |
https://en.wikipedia.org/wiki/Variable%20kernel%20density%20estimation | In statistics, adaptive or "variable-bandwidth" kernel density estimation is a form of kernel density estimation in which the size of the kernels used in the estimate are varied
depending upon either the location of the samples or the location of the test point.
It is a particularly effective technique when the sample space is multi-dimensional.
Rationale
Given a set of samples, , we wish to estimate the
density, , at a test point, :
where n is the number of samples, K is the
"kernel", h is its width and D is the number of dimensions in .
The kernel can be thought of as a simple, linear filter.
Using a fixed filter width may mean that in regions of low density, all samples
will fall in the tails of the filter with very low weighting, while regions of high
density will find an excessive number of samples in the central region with weighting
close to unity. To fix this problem, we vary the width of the kernel in different
regions of the sample space.
There are two methods of doing this: balloon and pointwise estimation.
In a balloon estimator, the kernel width is varied depending on the location
of the test point. In a pointwise estimator, the kernel width is varied depending
on the location of the sample.
For multivariate estimators, the parameter, h, can be generalized to
vary not just the size, but also the shape of the kernel. This more complicated approach
will not be covered here.
Balloon estimators
A common method of varying the kernel width is to make it inversely proportional to the density at the test point:
where k is a constant.
If we back-substitute the estimated PDF, and assuming a Gaussian kernel function,
we can show that W is a constant:
A similar derivation holds for any kernel whose normalising function is of the order , although with a different constant factor in place of the term. This produces a generalization of the k-nearest neighbour algorithm.
That is, a uniform kernel function will return the
KNN technique.
There are two components to the error: a variance term and a bias term. The variance term is given as:
.
The bias term is found by evaluating the approximated function in the limit as the kernel
width becomes much larger than the sample spacing. By using a Taylor expansion for the real function, the bias term drops out:
An optimal kernel width that minimizes the error of each estimate can thus be derived.
Use for statistical classification
The method is particularly effective when applied to statistical classification.
There are two ways we can proceed: the first is to compute the PDFs of
each class separately, using different bandwidth parameters,
and then compare them as in Taylor.
Alternatively, we can divide up the sum based on the class of each sample:
where ci is the class of the ith sample.
The class of the test point may be estimated through maximum likelihood.
External links
akde1d.m - Matlab m-file for one-dimensional adaptive kernel density estimation.
|
https://en.wikipedia.org/wiki/Siegel%20G-function | In mathematics, the Siegel G-functions are a class of functions in transcendental number theory introduced by C. L. Siegel. They satisfy a linear differential equation with polynomial coefficients, and the coefficients of their power series expansion lie in a fixed algebraic number field and have heights of at most exponential growth.
Definition
A Siegel G-function is a function given by an infinite power series
where the coefficients an all belong to the same algebraic number field, K, and with the following two properties.
f is the solution to a linear differential equation with coefficients that are polynomials in z;
the projective height of the first n coefficients is O(cn) for some fixed constant c > 0.
The second condition means the coefficients of f grow no faster than a geometric series. Indeed, the functions can be considered as generalisations of geometric series, whence the name G-function, just as E-functions are generalisations of the exponential function.
References
C. L. Siegel, "Über einige Anwendungen diophantischer Approximationen", Ges. Abhandlungen, I, Springer (1966)
Analytic number theory |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb It also lists all matches that Dinamo Zagreb played in the 2003–04 season.
Squad
(Correct as of November 2003)
Transfers
Summer
Incoming
Outgoing
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Source: 2003–04 Prva HNL article
Matches
Competitive
External links
2003–04 in Croatian Football at Rec.Sport.Soccer Statistics Foundation
2003–04 season at Prva HNL official website
Dinamo Zagreb Official website
GNK Dinamo Zagreb seasons
Dinamo Zagreb |
https://en.wikipedia.org/wiki/Ground%20field | In mathematics, a ground field is a field K fixed at the beginning of the discussion.
Use
It is used in various areas of algebra:
In linear algebra
In linear algebra, the concept of a vector space may be developed over any field.
In algebraic geometry
In algebraic geometry, in the foundational developments of André Weil the use of fields other than the complex numbers was essential to expand the definitions to include the idea of abstract algebraic variety over K, and generic point relative to K.
In Lie theory
Reference to a ground field may be common in the theory of Lie algebras (qua vector spaces) and algebraic groups (qua algebraic varieties).
In Galois theory
In Galois theory, given a field extension L/K, the field K that is being extended may be considered the ground field for an argument or discussion. Within algebraic geometry, from the point of view of scheme theory, the spectrum Spec(K) of the ground field K plays the role of final object in the category of K-schemes, and its structure and symmetry may be richer than the fact that the space of the scheme is a point might suggest.
In Diophantine geometry
In diophantine geometry the characteristic problems of the subject are those caused by the fact that the ground field K is not taken to be algebraically closed. The field of definition of a variety given abstractly may be smaller than the ground field, and two varieties may become isomorphic when the ground field is enlarged, a major topic in Galois cohomology.
Notes
Field (mathematics) |
https://en.wikipedia.org/wiki/Raikov%27s%20theorem | Raikov’s theorem, named for Russian mathematician Dmitrii Abramovich Raikov, is a result in probability theory. It is well known that if each of two independent random variables ξ1 and ξ2 has a Poisson distribution, then their sum ξ=ξ1+ξ2 has a Poisson distribution as well. It turns out that the converse is also valid.
Statement of the theorem
Suppose that a random variable ξ has Poisson's distribution and admits a decomposition as a sum ξ=ξ1+ξ2 of two independent random variables. Then the distribution of each summand is a shifted Poisson's distribution.
Comment
Raikov's theorem is similar to Cramér’s decomposition theorem. The latter result claims that if a sum of two independent random variables has normal distribution, then each summand is normally distributed as well. It was also proved by Yu.V.Linnik that a convolution of normal distribution and Poisson's distribution possesses a similar property ().
An extension to locally compact Abelian groups
Let be a locally compact Abelian group. Denote by the convolution semigroup of probability distributions on , and by the degenerate distribution concentrated at . Let .
The Poisson distribution generated by the measure is defined as a shifted distribution of the form
One has the following
Raikov's theorem on locally compact Abelian groups
Let be the Poisson distribution generated by the measure . Suppose that , with . If is either an infinite order element, or has order 2, then is also a Poisson's distribution. In the case of being an element of finite order , can fail to be a Poisson's distribution.
References
Characterization of probability distributions
Probability theorems
Theorems in statistics |
https://en.wikipedia.org/wiki/C%C3%A9dric%20Villani | Cédric Patrice Thierry Villani (; born 5 October 1973) is a French politician and mathematician working primarily on partial differential equations, Riemannian geometry and mathematical physics. He was awarded the Fields Medal in 2010, and he was the director of Sorbonne University's Institut Henri Poincaré from 2009 to 2017. As of September 2022, he is a professor at Institut des Hautes Études Scientifiques.
Villani has given two lectures at the Royal Institution, the first titled 'Birth of a Theorem'. The English translation of his book Théorème vivant (Living Theorem) has the same title.
In the book he describes the links between his research on kinetic theory and the one of the mathematician Carlo Cercignani: Villani, in fact, proved the so-called Cercignani's conjecture.
His second lecture at the Royal Institution is titled 'The Extraordinary Theorems of John Nash'.
Villani was elected as the deputy for Essonne's 5th constituency in the National Assembly, the lower house of the French Parliament, during the 2017 legislative election. He was elected as a member of La République En Marche! (LREM), but in May 2020 left the party to form a new party, Ecology, Democracy, Solidarity (EDS). Following the dissolution of EDS, Villani joined Ecology Generation, and ran for re-election under the banner of the NUPES. He was elected Vice President of the French Parliamentary Office for the Evaluation of Scientific and Technological Choices in July 2017.
He lost his seat in the 2022 French legislative election to La Republique En Marche! candidate Paul Midy by 19 votes.
Biography
After attending the Lycée Louis-le-Grand, Villani was admitted at the École Normale Supérieure in Paris and studied there from 1992 to 1996, after which he was appointed an agrégé préparateur at the same school. He received his doctorate at Paris Dauphine University in 1998, under the supervision of Pierre-Louis Lions, and became professor at the École normale supérieure de Lyon in 2000. He is now professor at the University of Lyon. He was director of the Institut Henri Poincaré in Paris from 2009 to 2017.
He has held various visiting positions at Georgia Tech (Fall 1999), the University of California, Berkeley (Spring 2004), and the Institute for Advanced Study, Princeton (Spring 2009).
Mathematical work
Villani has worked on the theory of partial differential equations involved in statistical mechanics, specifically the Boltzmann equation, where, with Laurent Desvillettes, he was the first to prove how quickly convergence occurs for initial values not near equilibrium. He has written with Giuseppe Toscani on this subject. With Clément Mouhot, he has worked on nonlinear Landau damping. He has worked on the theory of optimal transport and its applications to differential geometry, and with John Lott has defined a notion of bounded Ricci curvature for general measured length spaces. He also served on the Mathematical Sciences jury for the Infosys Prize in 2015 and 2016.
|
https://en.wikipedia.org/wiki/Watson%27s%20lemma | In mathematics, Watson's lemma, proved by G. N. Watson (1918, p. 133), has significant application within the theory on the asymptotic behavior of integrals.
Statement of the lemma
Let be fixed. Assume , where has an infinite number of derivatives in the neighborhood of , with , and .
Suppose, in addition, either that
where are independent of , or that
Then, it is true that for all positive that
and that the following asymptotic equivalence holds:
See, for instance, for the original proof or for a more recent development.
Proof
We will prove the version of Watson's lemma which assumes that has at most exponential growth as . The basic idea behind the proof is that we will approximate by finitely many terms of its Taylor series. Since the derivatives of are only assumed to exist in a neighborhood of the origin, we will essentially proceed by removing the tail of the integral, applying Taylor's theorem with remainder in the remaining small interval, then adding the tail back on in the end. At each step we will carefully estimate how much we are throwing away or adding on. This proof is a modification of the one found in .
Let and suppose that is a measurable function of the form , where and has an infinite number of continuous derivatives in the interval for some , and that for all , where the constants and are independent of .
We can show that the integral is finite for large enough by writing
and estimating each term.
For the first term we have
for , where the last integral is finite by the assumptions that is continuous on the interval and that . For the second term we use the assumption that is exponentially bounded to see that, for ,
The finiteness of the original integral then follows from applying the triangle inequality to .
We can deduce from the above calculation that
as .
By appealing to Taylor's theorem with remainder we know that, for each integer ,
for , where . Plugging this in to the first term in we get
To bound the term involving the remainder we use the assumption that is continuous on the interval , and in particular it is bounded there. As such we see that
Here we have used the fact that
if and , where is the gamma function.
From the above calculation we see from that
as .
We will now add the tails on to each integral in . For each we have
and we will show that the remaining integrals are exponentially small. Indeed, if we make the change of variables we get
for , so that
If we substitute this last result into we find that
as . Finally, substituting this into we conclude that
as .
Since this last expression is true for each integer we have thus shown that
as , where the infinite series is interpreted as an asymptotic expansion of the integral in question.
Example
When , the confluent hypergeometric function of the first kind has the integral representation
where is the gamma function. The change of variables puts this into the form
which is |
https://en.wikipedia.org/wiki/Ulf%20Grenander | Ulf Grenander (23 July 1923 – 12 May 2016) was a Swedish statistician and professor of applied mathematics at Brown University.
His early research was in probability theory, stochastic processes, time series analysis, and statistical theory (particularly the order-constrained estimation of cumulative distribution functions using his sieve estimator). In recent decades, Grenander contributed to computational statistics, image processing, pattern recognition, and artificial intelligence. He coined the term pattern theory to distinguish from pattern recognition.
Honors
In 1966 Grenander was elected to the Royal Academy of Sciences of Sweden, and in 1996 to the US National Academy of Sciences. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. He received an honorary doctorate in 1994 from the University of Chicago, and in 2005 from the Royal Institute of Technology of Stockholm, Sweden.
Education
Grenander earned his undergraduate degree at Uppsala University. He earned his Ph.D. at Stockholm University in 1950 under the supervision of Harald Cramér.
Appointments
He was active as a 1950–1951 Associate Professor at Stockholm University, 1951–1952 at University of Chicago, At 1952–1953 University of California–Berkeley, At Stockholm University 1953–1957, at Brown University 1957–1958 and 1958–1966 again at Stockholm University, where he succeeded in 1959 Harald Cramér as the Professor in actuarial science and mathematical statistics. From 1966 until his retirement, Grenander was L. Herbert Ballou University Professor at Brown University. In 1969–1974 he was also professor of Applied Mathematics at The Royal Institute of Technology.
Selected works
References
External links
Homepage of Ulf Grenander at Brown University
Pattern Theory: Grenander's Ideas and Examples – a video lecture by David Mumford
1923 births
2016 deaths
Swedish mathematicians
Swedish statisticians
American statisticians
Mathematical analysts
Probability theorists
20th-century American mathematicians
21st-century American mathematicians
Brown University faculty
Academic staff of the KTH Royal Institute of Technology
Stockholm University alumni
Uppsala University alumni
Swedish emigrants to the United States
People from Västervik Municipality
Members of the United States National Academy of Sciences
Members of the Royal Swedish Academy of Sciences
Mathematical statisticians
Artificial intelligence researchers |
https://en.wikipedia.org/wiki/Vinay%20V.%20Deodhar | Vinay Vithal Deodhar (3 December 1948 – 18 January 2015) was a Professor Emeritus in the Department of Mathematics at Indiana University. He worked in the area of algebraic groups and representation theory.
Early life
Deodhar was born in Mumbai (Bombay), India in 1948.
Career
Deodhar earned his Ph.D. from the University of Mumbai in 1974 for his work On Central Extensions of Rational Points of Algebraic Groups done under the supervision of M. S. Raghunathan.
After his doctorate, he was invited to join the School of Mathematics of the Tata Institute of Fundamental Research. Simultaneously he was a visiting scholar at the Institute for Advanced Study (IAS) in Princeton during 1975-77 and then a visiting professor at the Australian National University in Canberra. In 1981 he was appointed to a professorship at Indiana University, Bloomington, Indiana, where he remained until his death in 2015. He spent a further period as a visiting scholar at the IAS in 1992-93.
References
External links
20th-century Indian mathematicians
Algebraists
University of Mumbai alumni
Scientists from Mumbai
Tata Institute of Fundamental Research alumni
Institute for Advanced Study visiting scholars
1948 births
2015 deaths |
https://en.wikipedia.org/wiki/A%20Passage%20to%20Infinity | A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact is a 2009 book by George Gheverghese Joseph chronicling the social and mathematical origins of the Kerala school of astronomy and mathematics. The book discusses the highlights of the achievements of Kerala school and also analyses the hypotheses and conjectures on the possible transmission of Kerala mathematics to Europe.
An outline of the contents
Introduction
The Social Origins of the Kerala School
The Mathematical Origins of the Kerala School
The Highlights of Kerala Mathematics and Astronomy
Indian Trigonometry: From Ancient Beginnings to Nilakantha
Squaring the Circle: The Kerala Answer
Reaching for the Stars: The Power Series for Sines and Cosines
Changing Perspectives on Indian Mathematics
Exploring Transmissions: A Case Study of Kerala Mathematics
A Final Assessment
See also
Indian astronomy
Indian mathematics
History of mathematics
References
Further references
In association with the Royal Society's 350th anniversary celebrations in 2010, Asia House presented a talk based on A Passage to Infinity. See :
For an audio-visual presentation of George Gheverghese Joseph's views on the ideas presented in the book, see :
The Economic Times talks to George Gheverghese Joseph on The Passage to Infinity. See :
Review of "A PASSAGE TO INFINITY: Medieval Indian Mathematics from Kerala and its impact" by M. Ram Murty in Hardy-Ramanujan Journal, 36 (2013), 43–46.
Hindu astronomy
Indian mathematics
Kerala school of astronomy and mathematics
Astronomy books |
https://en.wikipedia.org/wiki/Guofang%20Wei | Guofang Wei is a mathematician in the field of differential geometry. She is a professor at the University of California, Santa Barbara.
Education
Wei earned a doctorate in mathematics from the State University of New York at Stony Brook in 1989, under the supervision of Detlef Gromoll. Her dissertation produced fundamental new examples of manifolds with positive Ricci curvature and was published in the Bulletin of the American Mathematical Society. These examples were later expanded upon by Burkard Wilking.
Research
In addition to her work on the topology of manifolds with nonnegative Ricci curvature, she has completed work on the isometry groups of manifolds with negative Ricci curvature with coauthors Xianzhe Dai and Zhongmin Shen. She also has major work with Peter Petersen on manifolds with integral Ricci curvature bounds.
Starting in 2000 Wei began working with Christina Sormani on limits of manifolds with lower Ricci curvature bounds using techniques of Jeff Cheeger and Tobias Colding, particularly Kenji Fukaya's metric measure convergence. The limit spaces in this setting are metric measure spaces. Wei was invited to present this work in a series of talks at the Seminaire Borel in Switzerland. Sormani and Wei also developed a notion called the covering spectrum of a Riemannian manifold. Dr. Wei has completed research with her student, Will Wylie, on smooth metric measure spaces and the Bakry–Emery Ricci tensor.
Guofang Wei was twice invited to present her work at the prestigious Geometry Festival both in 1996 and 2009.
Outreach
In addition to conducting research, Guofang Wei has mentored the Dos Pueblos High School Math Team, which won second place in the International Shing-Tung Yau High School Math Awards competition in Beijing in 2008.
Awards and honors
In 2013 she became a fellow of the American Mathematical Society, for "contributions to global Riemannian geometry and its relation with Ricci curvature".
Selected publications
Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups, Bull. Amer. Math. Soc. Vol. 19, no. 1 (1988), 311–313.
with X. Dai and Z. Shen, Negative Ricci curvature and isometry group, Duke Math J. 76 (1994) 59–73.
with X. Dai and R. Ye, Smoothing Riemannian manifolds with Ricci curvature bounds, MANUSCR MATH, vol. 90, no. 1, pp. 49–61, 1996.
with P. Petersen, Relative volume comparison with integral curvature bounds, GAFA 7 (1997) 1031–1045.
with C. Sormani, The covering spectrum of a compact length space, Journal of Diff. Geom. 67 (2004) 35–77.
with X. Dai and X. Wang, On stability of Riemannian manifolds with parallel spinors, Invent Math, vol. 161, no. 1, pp. 151–176, 2005
with W. Wylie Comparison Geometry for the Bakry–Emery Ricci Tensor, Journal of Diff. Geom. 83, no. 2 (2009), 377–405.
References
External links
Guofang Wei at UCSB
Seminaire Borel
Geometry Festival 2009
Dos Pueblos High School Math Team wins Second Place in S T Yau HS Math Awards
196 |
https://en.wikipedia.org/wiki/Luc%20Illusie | Luc Illusie (; born 1940) is a French mathematician, specializing in algebraic geometry. His most important work concerns the theory of the cotangent complex and deformations, crystalline cohomology and the De Rham–Witt complex, and logarithmic geometry. In 2012, he was awarded the Émile Picard Medal of the French Academy of Sciences.
Biography
Luc Illusie entered the École Normale Supérieure in 1959. At first a student of the mathematician Henri Cartan, he participated in the Cartan–Schwartz seminar of 1963–1964. In 1964, following Cartan's advice, he began to work with Alexandre Grothendieck, collaborating with him on two volumes of the latter's Séminaire de Géométrie Algébrique du Bois Marie. In 1970, Illusie introduced the concept of the cotangent complex.
A researcher in the Centre national de la recherche scientifique from 1964 to 1976, Illusie then became a professor at the University of Paris-Sud, retiring as emeritus professor in 2005. Between 1984 and 1995, he was the director of the arithmetic and algebraic geometry group in the department of mathematics of that university. and Gérard Laumon are among his students.
Thesis
In May 1971, Illusie defended a state doctorate ( Thèse d’État) entitled "Cotangent complex; application to the theory of deformations" at the University of Paris-Sud, in front of a jury composed of
Alexander Grothendieck, Michel Demazure and Jean-Pierre Serre and presided by Henri Cartan.
The thesis was published in French by Springer-Verlag as a two-volume book (in 1971 & 1972). The main results of the thesis are summarized in a paper in English (entitled "Cotangent complex and Deformations of torsors and group schemes") presented in Halifax, at Dalhousie University, in January 1971 as part of a colloquium on algebraic geometry. This paper, originally published by Springer-Verlag in 1972, also exists in a slightly extended version.
Illusie's construction of the cotangent complex generalizes that of Michel André and Daniel Quillen to morphisms of ringed topoi. The generality of the framework makes it possible to apply the formalism to
various first-order deformation problems: schemes, morphisms of schemes,
group schemes and torsors under group schemes. Results concerning commutative
group schemes in particular were the key tool in Grothendieck's proof of his
existence and structure theorem for infinitesimal deformations of Barsotti–Tate groups, an ingredient in Gerd Faltings' proof of the Mordell conjecture. In Chapter VIII of the second volume of the thesis, Illusie introduces
and studies derived de Rham complexes.
Awards
Illusie has received the Langevin Prize of the French Academy of Sciences in 1977 and, in 2012, the Émile Picard Medal of the French Academy of Sciences for "his fundamental work on the cotangent complex, the Picard–Lefschetz formula, Hodge theory and logarithmic geometry".
Selected works
Complexe cotangent et déformations, Lecture Notes in Mathematics 239 et 283, Berlin and New York, |
https://en.wikipedia.org/wiki/Michel%20Brou%C3%A9 | Michel Broué (born 28 October 1946) is a French mathematician. He holds a chair at Paris Diderot University. Broué has made contributions to algebraic geometry and representation theory.
In 2012 he became a fellow of the American Mathematical Society.
He is the son of French historian Pierre Broué and the father of French director and screenwriter Isabelle Broué and of French journalist and radio producer Caroline Broué.
References
External links
Website at Paris Diderot University
1946 births
Living people
20th-century French mathematicians
21st-century French mathematicians
University of Paris alumni
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/1946%E2%80%9347%20NK%20Dinamo%20Zagreb%20season | This article shows statistics of individual players for the football club Dinamo Zagreb. It also lists all matches that Dinamo Zagreb played in the 1946–47 season.
Competitions
Overall
Prva HNL
Classification
Results summary
Results by round
Results by opponent
Matches
Competitive
Player details
Player statistics
FW = Forward, MF = Midfielder, GK = Goalkeeper, DF = Defender
References
External links
Dinamo Zagreb official website
1946-47
Yugoslav football clubs 1946–47 season |
https://en.wikipedia.org/wiki/Gilah%20Leder | Gilah Chaja Leder (born 1941) is an adjunct professor at Monash University and a professor emerita at La Trobe University. Her research interests are in mathematics education, gender, affect, and exceptionality. Leder was the 2009 recipient of the Felix Klein Medal.
Early life
She was born in 1941, during the II World War, in Hilversum, North Holland. Being a Jewish child, she was hidden and protected by the catholic Zwanikken family of Laren. The father of the house, Cornelis Zwanikken worked at the municipality department of Social Affairs. Here, she was accepted as one of their own and affectionately called “zusje” (little sister). She learned to read and write in her early childhood. After the war she was reunited with her family. They started to live in Netherlands, where she visited coeducational elementary school. In November 1953 she moved to Adelaide, Australia. She started her 7th grade there at Woodwille High School, a coeducational government school. She got her bachelor's degree with honours in mathematics at University of Adelaide.
Career
She started her career teaching maths at a high school in Melbourne. Later she was offered a position at Melbourne Secondary Teachers College.
After having given birth to her 2 children, she completed her PhD and a doctorate at Monash University. Later she was appointed as a lecturer in the Faculty of Education at Monash University.
In 1990 she edited and published a journal Mathematics and Gender together with Elizabeth Fennema.
In 1993 she was named Monash's University 'Supervisor of the Year' for her talent in supervising postgraduate students.
In 1994 she was appointed a professor of Education at La Trobe University.
In 2010 she was honoured by the International Commission on Mathematical Instruction for her achievements in mathematics education, research and development.
She was elected a Fellow of the Academy of the Social Sciences in Australia in 2001. She is past President and life member of the Mathematics Research Group of Australasia and of the International group of Psychology of Mathematics Education.
Leder was made a Member of the Order of Australia (AM) in the 2019 Queen's Birthday Honours in recognition of her "significant service to higher education, and to the Jewish community of Victoria".
Publications
Leder has almost 200 scholarly publications, including:
Mathematics and gender: Changing perspectives. Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. Leder, Gilah C. In Grouws, Douglas A. (Ed). (1992). Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. (pp. 597–622). New York, NY, England: Macmillan Publishing Co, Inc.
See also
Mathematical anxiety
References
Notable Women in Mathematics, a Biographical Dictionary, edited by Charlene Morrow and Teri Perl, Greenwood Press, 1998. pp 118–123
Mathematics educators
Living people
1 |
https://en.wikipedia.org/wiki/Henrique%20%28footballer%2C%20born%201966%29 | Henrique Arlindo Etges (born 15 March 1966) is a former Brazilian footballer.
Club statistics
References
External links
1966 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Brazil men's under-20 international footballers
Brazil men's international footballers
J1 League players
Grêmio Foot-Ball Porto Alegrense players
Associação Portuguesa de Desportos players
União São João Esporte Clube players
Sport Club Corinthians Paulista players
Tokyo Verdy players
Expatriate men's footballers in Japan
Men's association football defenders
People from Venâncio Aires |
https://en.wikipedia.org/wiki/Rafael%20%28footballer%2C%20born%20August%201982%29 | Rafael dos Santos Silva (born 27 August 1982) is a Brazilian football player.
Club statistics
References
External links
Oita Trinita
1982 births
Living people
Brazilian men's footballers
Super League Greece players
Campeonato Brasileiro Série A players
J1 League players
Guarani FC players
Sport Club Corinthians Paulista players
Grêmio Foot-Ball Porto Alegrense players
Clube Atlético Juventus players
Oita Trinita players
Brazilian expatriate men's footballers
Expatriate men's footballers in Greece
Expatriate men's footballers in Japan
Men's association football forwards
Footballers from São Paulo |
https://en.wikipedia.org/wiki/Paulo%20Henrique%20%28footballer%2C%20born%201972%29 | Paulo Henrique (born 21 February 1972) is a Brazilian footballer.
Club statistics
References
External links
1972 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J1 League players
J2 League players
JEF United Chiba players
Vegalta Sendai players
Eastern Sports Club footballers
Metro Gallery FC players
Hong Kong First Division League players
Expatriate men's footballers in Japan
Expatriate men's footballers in Hong Kong
Men's association football midfielders |
https://en.wikipedia.org/wiki/Ta%C3%ADlson%20%28footballer%2C%20born%201975%29 | José Ilson dos Santos also known as Taílson (born 28 November 1975) is a Brazilian football player.
Club statistics
References
External links
"Tailson" José Ilson Dos Santos
Profile & stats - Lokeren
Profile - Lierse
1975 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in China
Expatriate men's footballers in Japan
Belgian Pro League players
J1 League players
Gamba Osaka players
Changsha Ginde players
Chinese Super League players
Botafogo Futebol Clube (SP) players
S.C. Braga players
América Futebol Clube (MG) players
Club Athletico Paranaense players
Fortaleza Esporte Clube players
Paulista Futebol Clube players
Esporte Clube Juventude players
Sport Club do Recife players
Royal Excel Mouscron players
K.S.C. Lokeren Oost-Vlaanderen players
Lierse S.K. players
Expatriate men's footballers in Belgium
Esporte Clube XV de Novembro (Piracicaba) players
Sociedade Esportiva Matonense players
Brazil men's under-20 international footballers
Men's association football forwards |
https://en.wikipedia.org/wiki/Ari%20%28footballer%2C%20born%201980%29 | Arivaldo Alves dos Santos (born November 19, 1980), known as just Ari, is a Brazilian football player.
Club statistics
References
External links
jsgoal
1980 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
Kashima Antlers players
Esporte Clube Bahia players
Fortaleza Esporte Clube players
Sport Club Internacional players
Atlético Clube Goianiense players
Men's association football midfielders |
https://en.wikipedia.org/wiki/Angelo%20%28footballer%2C%20born%201965%29 | Angelo Carlos Pretti (born 10 August 1965) is a former Brazilian football player.
Club statistics
References
External links
1965 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J1 League players
Japan Football League (1992–1998) players
Yokohama Flügels players
Kyoto Sanga FC players
Montedio Yamagata players
FC Tokyo players
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Ca%C3%ADco | Aírton Graciliano dos Santos (born 15 May 1974), commonly known as Caíco, is a former Brazilian football player.
Club statistics
Honours
Internacional
Campeonato Gaúcho: 1992, 1994
Copa do Brasil: 1992
Santos
Torneio Rio-São Paulo: 1997
Atlético Paranaense
Campeonato Paranaense: 1998
Atlético Mineiro
Campeonato Mineiro: 2000
Itumbiara
Campeonato Goiano: 2008
Brazil
Toulon Tournament: 1993
FIFA World Youth Championship: 1993
References
External links
1974 births
Living people
Footballers from Porto Alegre
Men's association football midfielders
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Switzerland
Expatriate men's footballers in Portugal
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in Japan
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
Primeira Liga players
Sport Club Internacional players
Tokyo Verdy players
CR Flamengo footballers
Santos FC players
Club Athletico Paranaense players
Clube Atlético Mineiro players
FC Lugano players
Associação Atlética Ponte Preta players
Goiás Esporte Clube players
U.D. Leiria players
Esporte Clube Juventude players
C.S. Marítimo players
Coritiba Foot Ball Club players
Itumbiara Esporte Clube players
Vila Nova Futebol Clube players |
https://en.wikipedia.org/wiki/Martin%20M%C3%BCller%20%28footballer%2C%20born%201970%29 | Martin Müller (born 6 November 1970) is a Czech former professional football player.
Club statistics
References
External links
1970 births
Living people
Czech men's footballers
Czech First League players
J1 League players
SK Slavia Prague players
FK Chmel Blšany players
FC Viktoria Plzeň players
1. FK Drnovice players
Vissel Kobe players
Czech expatriate men's footballers
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Alison%20%28footballer%2C%20born%201982%29 | Alison Barros Moraes (born 30 June 1982) is a Brazilian football player.
Club statistics
References
External links
Omiya Ardija
Guardian's Stats Centre
1982 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Brazilian expatriate sportspeople in South Korea
Expatriate men's footballers in South Korea
Brazilian expatriate sportspeople in Japan
Expatriate men's footballers in Japan
Ulsan Hyundai FC players
Daejeon Hana Citizen players
Marília Atlético Clube players
Omiya Ardija players
K League 1 players
J1 League players
Men's association football forwards |
https://en.wikipedia.org/wiki/Paulo%20Isidoro%20%28footballer%2C%20born%201973%29 | Alex Sandro Santana de Oliveira (born 30 October 1973), known as Paulo Isidoro, is a retired Brazilian football player.
Club statistics
References
External links
1973 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Cruzeiro Esporte Clube players
Fluminense FC players
Sociedade Esportiva Palmeiras players
Sport Club Internacional players
América Futebol Clube (RN) players
Fortaleza Esporte Clube players
Esporte Clube Bahia players
Mogi Mirim Esporte Clube players
J1 League players
Kawasaki Frontale players
Men's association football forwards
Footballers from Salvador, Bahia |
https://en.wikipedia.org/wiki/Kim%20Hwang-jung | Kim Hwang-Jung (Hangul: 김황정), or Terumasa Kin (金 晃正) (born 19 November 1975) is a former South Korean football player.
Club statistics
References
External links
1975 births
Living people
Hannan University alumni
Association football people from Osaka Prefecture
South Korean men's footballers
South Korean expatriate men's footballers
J1 League players
J2 League players
K League 1 players
JEF United Chiba players
Ventforet Kofu players
Ulsan Hyundai FC players
South Korean expatriate sportspeople in Japan
Expatriate men's footballers in Japan
Men's association football forwards |
https://en.wikipedia.org/wiki/Yuki%20Tamura | is a former Japanese football player.
Club statistics
References
External links
1985 births
Living people
Japanese men's footballers
J1 League players
J2 League players
Japan Football League players
Sanfrecce Hiroshima players
Ehime FC players
FC Gifu players
Gainare Tottori players
Japanese expatriate men's footballers
Japanese expatriate sportspeople in Paraguay
Expatriate men's footballers in Paraguay
Club Guaraní players
Sportivo Trinidense footballers
Sportivo Luqueño players
Men's association football forwards
Association football people from Hiroshima |
https://en.wikipedia.org/wiki/Marcelo%20Rosa%20%28footballer%2C%20born%201976%29 | Marcelo Rosa da Silva (born 29 January 1976) is a former Brazilian football player.
Club statistics
References
External links
Cerezo Osaka
1976 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
J1 League players
Expatriate men's footballers in Japan
Sport Club Internacional players
CR Flamengo footballers
Cerezo Osaka players
Servette FC players
Men's association football midfielders
Footballers from Porto Alegre |
https://en.wikipedia.org/wiki/W%C3%A1gner%20%28footballer%2C%20born%201973%29 | Wágner Pires de Almeida (born 27 December 1973) is a Brazilian football player.
Club statistics
References
External links
Cerezo Osaka
1973 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
J1 League players
Expatriate men's footballers in Poland
Expatriate men's footballers in Japan
Esporte Clube XV de Novembro (Piracicaba) players
Clube Atlético Mineiro players
Guarani FC players
Mirassol Futebol Clube players
União São João Esporte Clube players
Esporte Clube Santo André players
Figueirense FC players
Botafogo Futebol Clube (SP) players
Uberlândia Esporte Clube players
Adap Galo Maringá Futebol Clube players
América Futebol Clube (RN) players
Pogoń Szczecin players
Cerezo Osaka players
Men's association football forwards
Footballers from Porto Alegre |
https://en.wikipedia.org/wiki/Ken%20Matsumoto | is a former Japanese football player.
Club statistics
References
External links
jsgoal
1987 births
Living people
Association football people from Kanagawa Prefecture
Japanese men's footballers
J1 League players
Singapore Premier League players
JEF United Chiba players
Albirex Niigata Singapore FC players
YSCC Yokohama players
Iwate Grulla Morioka players
Japanese expatriate men's footballers
Men's association football midfielders |
https://en.wikipedia.org/wiki/Emerson%20%28footballer%2C%20born%20March%201973%29 | Emerson Orlando de Melo (born 2 March 1973), known as Emerson, is a former Brazilian football player.
Club statistics
References
External links
1973 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
J1 League players
Esporte Clube Bahia players
Esporte Clube Vitória players
Grêmio Foot-Ball Porto Alegrense players
Guarani FC players
Brasiliense FC players
Ituano FC players
América Futebol Clube (SP) players
Tokyo Verdy players
Expatriate men's footballers in Japan
Men's association football defenders |
https://en.wikipedia.org/wiki/Tatsuro%20Kimura | is a former Japanese football player.
Club statistics
References
External links
1984 births
Living people
Japanese men's footballers
J1 League players
J2 League players
Sanfrecce Hiroshima players
V-Varen Nagasaki players
Zweigen Kanazawa players
Men's association football midfielders
Association football people from Hiroshima |
https://en.wikipedia.org/wiki/Danilo%20Gomes%20%28footballer%2C%20born%201981%29 | Danilo Gustavo Vergne Gomes (born 15 October 1981) is a Brazilian former football player.
Club career statistics
References
External links
jsgoal
1981 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Esporte Clube Bahia players
Sport Club Internacional players
J1 League players
FC Tokyo players
Paulista Futebol Clube players
Red Bull Bragantino II players
Club León footballers
Salgueiro Atlético Clube players
Clube Atlético Bragantino players
Treze Futebol Clube players
Expatriate men's footballers in Japan
Expatriate men's footballers in Mexico
Men's association football midfielders
Footballers from Salvador, Bahia |
https://en.wikipedia.org/wiki/Dedimar | Dedimar Souza Lima (born 27 January 1976) is a Brazilian former professional footballer and manager who is currently a youth scout for Palmeiras.
Club statistics
References
External links
1976 births
Living people
Brazilian men's footballers
Brazilian football managers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
J2 League players
Brazil men's under-20 international footballers
Campeonato Brasileiro Série D managers
Esporte Clube Vitória players
Sociedade Esportiva Palmeiras players
Clube Atlético Mineiro players
Júbilo Iwata players
Coritiba Foot Ball Club players
Paulista Futebol Clube players
Esporte Clube Santo André players
Tokyo Verdy players
Marília Atlético Clube players
Associação Desportiva São Caetano players
Clube Atlético Juventus players
Paulista Futebol Clube managers
Esporte Clube Santo André managers
Men's association football defenders
People from Irecê
Footballers from Bahia |
https://en.wikipedia.org/wiki/Moabe%20Platini | Moabe Platini Dias Ramos (born 22 June 1987) is a Brazilian football player.
Club statistics
References
External links
Oita Trinita
1987 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Expatriate men's footballers in Japan
J1 League players
União São João Esporte Clube players
Oita Trinita players
Reilac Shiga FC players
Men's association football midfielders
Footballers from São Paulo |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.