source stringlengths 31 168 | text stringlengths 51 3k |
|---|---|
https://en.wikipedia.org/wiki/Basque%20Statistics%20Office | The Basque Statistics Office (Eustat) is an autonomous agency of the Basque Government answering to the Department for the Economy and the Treasury, founded on 25 November 1986 pursuant to the Basque Government Decree 251/1986. Its mission is to collect, analyse and disseminate official statistical information on the relevant aspects of Basque society and the economy. It also carries out research and development, training and methodological support activities and works in partnership with universities, states institutions and Eurostat.
Its work is subject to the Spanish Personal Data Protection Act 15/1999, which guarantees that Eustat does not facilitate any information that can identify or individualise the suppliers of the information, who are protected by statistical confidentiality. It coordinates two advisory bodies, the Basque Statistics Commission, where the Departments of the Basque Government, the provincial councils and the local councils as producers of statistics are represented, and the Basque Statistics Council, where all social agents and representatives of society also participate.
Its statistics operations are established in the four-year plans approved by the Basque Parliament and are implemented through the Annual Statistics Programmes. The 2005-2008 Basque Statistics Plan envisaged 180 operations.
Services
Information Service
Customized statistical information requests
Statistical information on the website
Databank
Sale of publications
Methodology Advice Service
See also
Spanish Personal Data Protection Act 15/1999
References
External links
Official website of Eustat
Department for the Economy and the Treasury of Basque Government
Basque Government
National statistical services |
https://en.wikipedia.org/wiki/Orbit%20trap | In mathematics, an orbit trap is a method of colouring fractal images based upon how close an iterative function, used to create the fractal, approaches a geometric shape, called a "trap". Typical traps are points, lines, circles, flower shapes and even raster images. Orbit traps are typically used to colour two dimensional fractals representing the complex plane.
Examples
Point based
A point-based orbit trap colours a point based upon how close a function's orbit comes to a single point, typically the origin.
Line based
A line-based orbit trap colours a point based upon how close a function's orbit comes to one or more lines, typically vertical or horizontal (x=a or y=a lines). Pickover stalks are an example of a line based orbit trap which use two lines.
Algorithm
Orbit traps are typically used with the class of two-dimensional fractals based on an iterative function. A program that creates such a fractal colours each pixel, which represent discrete points in the complex plane, based upon the behaviour of those points when they pass through a function a set number of times.
The best known example of this kind of fractal is the Mandelbrot set, which is based upon the function zn+1 = zn2 + c. The most common way of colouring Mandelbrot images is by taking the number of iterations required to reach a certain bailout value and then assigning that value a colour. This is called the escape time algorithm.
A program that colours the Mandelbrot set using a point-based orbit trap will assign each pixel with a “distance” variable, that will typically be very high when first assigned:
double distance = 10e5
As the program passes the complex value through the iterative function it will check the distance between each point in the orbit and the trap point. The value of the distance variable will be the shortest distance found during the iteration:
private double getDistance(Complex c,
Complex point,
int maxIteration)
{
double distance = 1e20;
Complex z = new Complex(0, 0);
for (int i=0; i<maxIteration; i++)
{
// Perform Mandelbrot iteration
z = z.multiply(z);
z = z.add(c);
// Set new distance dist = min( dist, |z-point| )
Complex zMinusPoint = new Complex(z);
zMinusPoint = zMinusPoint.subtract(point);
double zMinusPointModulus = zMinusPoint.magnitude();
if (zMinusPointModulus < distance)
distance = zMinusPointModulus;
}
return distance;
}
References
.
.
Fractals |
https://en.wikipedia.org/wiki/Frucht%27s%20theorem | Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite undirected graph. More strongly, for any finite group G there exist infinitely many non-isomorphic simple connected graphs such that the automorphism group of each of them is isomorphic to G.
Proof idea
The main idea of the proof is to observe that the Cayley graph of G, with the addition of colors and orientations on its edges to distinguish the generators of G from each other, has the desired automorphism group. Therefore, if each of these edges is replaced by an appropriate subgraph, such that each replacement subgraph is itself asymmetric and two replacements are isomorphic if and only if they replace edges of the same color, then the undirected graph created by performing these replacements will also have G as its symmetry group.
Graph size
With three exceptions – the cyclic groups of orders 3, 4, and 5 – every group can be represented as the symmetries of a graph whose vertices have only two orbits. Therefore, the number of vertices in the graph is at most twice the order of the group. With a larger set of exceptions, most finite groups can be represented as the symmetries of a vertex-transitive graph, with a number of vertices equal to the order of the group.
Special families of graphs
There are stronger versions of Frucht's theorem that show that certain restricted families of graphs still contain enough graphs to realize any symmetry group. Frucht proved that in fact countably many 3-regular graphs with the desired property exist; for instance, the Frucht graph, a 3-regular graph with 12 vertices and 18 edges, has no nontrivial symmetries, providing a realization of this type for the trivial group. Gert Sabidussi showed that any group can be realized as the symmetry groups of countably many distinct k-regular graphs, k-vertex-connected graphs, or k-chromatic graphs, for all positive integer values k (with for regular graphs and for k-chromatic graphs). From the facts that every graph can be reconstructed from the containment partial order of its edges and vertices, that every finite partial order is equivalent by Birkhoff's representation theorem to a finite distributive lattice, it follows that every finite group can be realized as the symmetries of a distributive lattice, and of the graph of the lattice, a median graph. It is possible to realize every finite group as the group of symmetries of a strongly regular graph. Every finite group can also be realized as the symmetries of a graph with distinguishing number two: one can (improperly) color the graph with two colors so that none of the symmetries of the graph preserve the coloring.
However, some important classes of graphs are incapable of realizing all groups as their symmetries. Camille Jordan characterized the symmetry groups of trees as being the smallest set of finite gro |
https://en.wikipedia.org/wiki/Beta%20skeleton | In computational geometry and geometric graph theory, a β-skeleton or beta skeleton is an undirected graph defined from a set of points in the Euclidean plane. Two points p and q are connected by an edge whenever all the angles prq are sharper than a threshold determined from the numerical parameter β.
Circle-based definition
Let β be a positive real number, and calculate an angle θ using the formulas
For any two points p and q in the plane, let Rpq be the set of points for which angle prq is greater than θ. Then Rpq takes the form of a union of two open disks with diameter βd(p,q) for β ≥ 1 and θ ≤ π/2, and it takes the form of the intersection of two open disks with diameter d(p,q)/β for β ≤ 1 and θ ≥ π/2. When β = 1 the two formulas give the same value θ = π/2, and Rpq takes the form of a single open disk with pq as its diameter.
The β-skeleton of a discrete set S of points in the plane is the undirected graph that connects two points p and q with an edge pq whenever Rpq contains no points of S. That is, the β-skeleton is the empty region graph defined by the regions Rpq. When S contains a point r for which angle prq is greater than θ, then pq is not an edge of the β-skeleton; the β-skeleton consists of those pairs pq for which no such point r exists.
Lune-based definition
Some authors use an alternative definition in which the empty regions Rpq for β > 1 are not unions of two disks but rather lenses (more often called in this context "lunes"), intersections of two congruent disks with diameter βd(pq), such that line segment pq lies on a radius of both disks and such that the points p and q both lie on the boundary of the intersection. As with the circle-based β-skeleton, the lune-based β-skeleton has an edge pq whenever region Rpq is empty of other input points. For this alternative definition, the relative neighborhood graph is a special case of a β-skeleton with β = 2. The two definitions coincide for β ≤ 1, and for larger values of β the circle-based skeleton is a subgraph of the lune-based skeleton.
One important difference between the circle-based and lune-based β-skeletons is that, for any point set that does not lie on a single line, there always exists a sufficiently large value of β such that the circle-based β-skeleton is the empty graph. In contrast, if a pair of points p and q has the property that, for all other points r, one of the two angles pqr and qpr is obtuse, then the lune-based β-skeleton will contain edge pq no matter how large β is.
History
β-skeletons were first defined by as a scale-invariant variation of the alpha shapes of . The name, "β-skeleton", reflects the fact that in some sense the β-skeleton describes the shape of a set of points in the same way that a topological skeleton describes the shape of a two-dimensional region. Several generalizations of the β-skeleton to graphs defined by other empty regions have also been considered.
Properties
If β varies continuously from 0 to ∞, the circle-based β-ske |
https://en.wikipedia.org/wiki/Minimal%20logic | Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson. It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded middle as well as the principle of explosion (ex falso quodlibet), and therefore holding neither of the following two derivations as valid:
where and are any propositions. Most constructive logics only reject the former, the law of excluded middle. In classical logic, the ex falso laws
as well as their variants with and switched, are equivalent to each other and valid. Minimal logic also rejects those principles.
Axiomatization
Minimal logic is axiomatized over the positive fragment of intuitionistic logic. Both of these logics may be formulated in the language using the same axioms for
implication ,
conjunction and
disjunction as the basic connectives, but Minimal Logic adds
falsum or absurdity as part of the language.
Alternatively, direct axioms for negation are discussed below.
Theorems
Here only theorems not already provable in the positive calculus are covered.
Negation introduction
A quick analysis of implication and negation laws gives a good indication of what this logic, lacking full explosion, can and cannot prove.
A natural statement in a language with negation, such as Minimal logic, is, for example, the principle of negation introduction, whereby the negation of a statement is proven by assuming it and deriving a contradiction. Formally, this may be expressed as, for any two propositions,
.
For taken as the contradiction itself, this establishes the law of non-contradiction
.
Assuming any , the introduction rule of the material conditional gives , also when and are not relevantly related. With this and implication elimination, the above introduction principle implies
,
i.e. assuming any contradiction, every proposition can be negated. Negation introduction is possible in minimal logic, so here a contradiction moreover proves every double negation . Explosion would permit to remove the latter double negation, but this principle is not adopted.
Moreover, using the above
This implication is to be compared with the full disjunctive syllogism.
Axiomatization via absurdity
One possible scheme of extending the positive calculus to minimal logic is to treat as an implication, in which case the theorems from the implication calculus of a logic carry over to negation statements. To this end, is introduced as a proposition, not provable unless the system is inconsistent, and negation is then treated as an abbreviation for .
Constructively, represents a proposition for which there can be no reason to believe it.
The already discussed principles can then be from theorems over the positive fragment.
Negation introduction, spelled out in the previous section, is implied as a mere special case of
when . In this way, minimal logic can be characterized as a constructive logic just without negation elimination (a.k.a |
https://en.wikipedia.org/wiki/Five%20points%20determine%20a%20conic | In Euclidean and projective geometry, five points determine a conic (a degree-2 plane curve), just as two (distinct) points determine a line (a degree-1 plane curve). There are additional subtleties for conics that do not exist for lines, and thus the statement and its proof for conics are both more technical than for lines.
Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true over both the Euclidean plane and any pappian projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion.
Proofs
This result can be proven numerous different ways; the dimension counting argument is most direct, and generalizes to higher degree, while other proofs are special to conics.
Dimension counting
Intuitively, passing through five points in general linear position specifies five independent linear constraints on the (projective) linear space of conics, and hence specifies a unique conic, though this brief statement ignores subtleties.
More precisely, this is seen as follows:
conics correspond to points in the five-dimensional projective space
requiring a conic to pass through a point imposes a linear condition on the coordinates: for a fixed the equation is a linear equation in
by dimension counting, five constraints (that the curve passes through five points) are necessary to specify a conic, as each constraint cuts the dimension of possibilities by 1, and one starts with 5 dimensions;
in 5 dimensions, the intersection of 5 (independent) hyperplanes is a single point (formally, by Bézout's theorem);
general linear position of the points means that the constraints are independent, and thus do specify a unique conic;
the resulting conic is non-degenerate because it is a curve (since it has more than 1 point), and does not contain a line (else it would split as two lines, at least one of which must contain 3 of the 5 points, by the pigeonhole principle), so it is irreducible.
The two subtleties in the above analysis are that the resulting point is a quadratic equation (not a linear equation), and that the constraints are independent. The first is simple: if A, B, and C all vanish, then the equation defines a line, and any 3 points on this (indeed any number of points) lie on a line – thus general linear position ensures a conic. The second, that the constraints are independent, is significantly subtler: it corresponds to the fact that given five points in general linear position in the plane, their images in under the Veronese map are in general linear position, which is true because the Veronese map is biregular: i.e., if the image of five points satisfy a relation, then the relation can be pulled back and the original points must |
https://en.wikipedia.org/wiki/Giuseppe%20Antonio%20Borgnis | Giuseppe Antonio Borgnis (Frenchified as Joseph Antoine Borgnis) 1781–1863, was an Italian engineer, professor of Applied mathematics and then of Architecture at the University of Pavia. His book Théorie de la mécanique usuelle is considered by historian Donald Cardwell to be one of the great engineering textbooks of the early 19th century.
Biography
Works
Traité complet de mécanique appliquée aux art, 1818
Théorie de la mécanique usuelle, ou Introduction à l'étude de la mécanique appliquée aux arts, 1821.
Traité élémentaire de construction appliquée à l'architecture civile, 1823; 1838.
Dictionnaire de mécanique appliquée aux arts, 1823.
Elementi di statica architettonica, 1842.
References
French engineers
1863 deaths
1781 births
Italian engineers
People from Craveggia |
https://en.wikipedia.org/wiki/Braikenridge%E2%80%93Maclaurin%20theorem | In geometry, the , named for 18th century British mathematicians William Braikenridge and Colin Maclaurin, is the converse to Pascal's theorem. It states that if the three intersection points of the three pairs of lines through opposite sides of a hexagon lie on a line L, then the six vertices of the hexagon lie on a conic C; the conic may be degenerate, as in Pappus's theorem.
The Braikenridge–Maclaurin theorem may be applied in the Braikenridge–Maclaurin construction, which is a synthetic construction of the conic defined by five points, by varying the sixth point. Namely, Pascal's theorem states that given six points on a conic (the vertices of a hexagon), the lines defined by opposite sides intersect in three collinear points. This can be reversed to construct the possible locations for a sixth point, given five existing ones.
References
Theorems about polygons
Conic sections |
https://en.wikipedia.org/wiki/Inclusion%20%28logic%29 | In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within a second object.
For example, if m and n are two logical matrices, then
The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being 'contained' by another, using the backward letter 'C' to express this. Peirce articulated this clearly in 1870, arguing also that inclusion was a wider concept than equality, and hence a logically simpler one. Schröder (also Frege) calls the same concept 'subordination'.
References
1816 introductions
History of logic
Logic |
https://en.wikipedia.org/wiki/Cantor%20%28software%29 | Cantor is a free software mathematics application for scientific statistics and analysis. It is part of the KDE Software Compilation 4, and was introduced with the 4.4 release as part of the KDE Education Project's kdeedu package.
Features
Cantor is a graphical user interface that delegates its mathematical operations to one of several backends. Its plugin-based structure allows adding different backends. It can make use of Julia, KAlgebra, Lua, Maxima, Octave, Python, Qalculate!, R, SageMath, and Scilab.
Cantor provides a consistent interface to these backends; its project page lists the following features:
Nice Worksheet view for evaluating expressions
View of plotting results inside the worksheet or in a separate window
Typesetting of mathematical formulas using LaTeX
Backend-aware syntax highlighting
Plugin-based assistant dialogs for common tasks (like integrating a function or entering a matrix)
Cantor was the first KDE project to implement upload to the GetHotNewStuff addon service, which is used to download or upload example worksheets. It provided impetus for improvement of this feature for KDE SC 4.4.
References
External links
Notebook interface
Educational math software
Free educational software
KDE Education Project
Science software that uses Qt |
https://en.wikipedia.org/wiki/Recurrent%20sequence | In mathematics, recurrent sequence may refer to:
A sequence satisfying a recurrence relation
Recurrent word, a sequence such that any factor (consecutive subsequence) that appears does so infinitely often, such as the Thue–Morse sequence or a Sturmian word |
https://en.wikipedia.org/wiki/Valentyna%20Semerenko | Valentyna Oleksandrivna Semerenko (), known also as Valya Semerenko () and featuring in statistics as Valj Semerenko (born 18 January 1986) is a Ukrainian biathlete. She is Olympic and World champion, multiple World championships medalist, and one of the most successful Ukrainian winter athletes.
Career
When she was in the fourth grade of elementary school, she took up cross-country skiing together with her twin sister Vita. But later, they decided to switch to biathlon. She began competing internationally a year earlier than Vita. In 2004 she took part at Junior World Championships in Maurienne, France. In 2005 she won two medals at Junior World and one at Junior European Championships. On 17 December 2005, she debuted in Biathlon World Cup. She finished her first World Cup sprint competition in Osrblie, Slovakia, ranking 47th. Valja managed to qualify for 2006 Winter Olympics, where she took part only in individual race, finishing 46th.
For the first time, she participated at the World Championships in 2007 in Antholz-Anterselva, Italy. Valja missed almost half of the 2006-07 Biathlon World Cup. The next season was more successful. She won the silver medal in the 4×6 km relay event at the Biathlon World Championships 2008. First World Cup relay victory came on 7 January 2009, in Oberhof, Germany.
She represented Ukraine at the 2010 Winter Olympics in Vancouver, British Columbia, Canada. In all races she placed between 13 and 23.
First individual podium Valja celebrated on 4 February 2011, in Presque Isle, United States, in sprint. At the 2013 World Championships, Valja won bronze in individual race and silver in relay. On 15 December 2013, Valja won a pursuit race in Annecy, France, which was her first individual World Cup victory.
Together with Juliya Dzhyma, Vita Semerenko and Olena Pidhrushna she won the gold medal in the Women's relay at the 2014 Winter Olympics, in Sochi, Russia.
2014–15 season was the most successful as of January 2018. That season she had four individual podiums and finished third in World Cup classification, being the second-ever Ukrainian biathlete to finish a season in Top-3 after Olena Zubrilova. After Christmas, Valja together with Serhiy Semenov, she won World Team Challenge. The most memorable achievement was victory in mass start at 2015 World Championships in Kontiolahti, Finland. The next two seasons weren't very successful due to illnesses and weak physical conditions. Valja was forced to miss a lot of races.
She qualified to represent Ukraine at the 2018 Winter Olympics. Her best personal result at the Games was 19th place in the mass start. She expected to run in the women's relay but wasn't included in the roster. Later she accused coaches of ignoring her results and including not so successful at the Games Iryna Varvynets and Anastasiya Merkushyna in the relay team. Her sister Vita Semerenko later told in an interview that Valja cried for hours after discovering that she wouldn't participate in the |
https://en.wikipedia.org/wiki/2010%20LKL%20All-Star%20Game | The 2010 LKL All-Star Game, was held in Cido Arena, on March 20, 2010.
Roster
Replaced by Tadas Klimavičius.
Statistics
Coaches
The coach of Team Vilkai was Rimas Kurtinaitis, of Lietuvos Rytas, who received 5,490 votes. The coach of Team Ereliai was Antanas Sireika, of Šiauliai, with 3,005 votes.
Other events
All-Star 3-point shootout
The leading scorer in 3-point percentage of the LKL was Donatas Zavackas. Besides him, one player, usually with a good 3-point percentage, was chosen from every team in the league. Below is the list of participating players. The winner received the trophy, a prize and a 4,000 Litas check.
All-Star dunk contest
External links
LKL.com
Lietuvos krepšinio lyga All-Star Game
2009–10 LKL season |
https://en.wikipedia.org/wiki/Krichevsky%E2%80%93Trofimov%20estimator | In information theory, given an unknown stationary source with alphabet A and a sample w from , the Krichevsky–Trofimov (KT) estimator produces an estimate pi(w) of the probability of each symbol i ∈ A. This estimator is optimal in the sense that it minimizes the worst-case regret asymptotically.
For a binary alphabet and a string w with m zeroes and n ones, the KT estimator pi(w) is defined as:
This corresponds to the posterior mean of a Beta-Bernoulli posterior distribution with prior .
For the general case the estimate is made using a Dirichlet-Categorical distribution.
See also
Rule of succession
Bayesian inference using conjugate priors for the categorical distribution
References
Information theory
Data compression |
https://en.wikipedia.org/wiki/Lenhard%20Ng | Lenhard Ng (born 1976, United States) is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University.
Background and education
Lenhard Ng is an American of Chinese descent. His father, Jack Ng, is a professor of physics at University of North Carolina Chapel Hill.
Lenhard earned his B.A. summa cum laude in Mathematics and Physics from Harvard University in three years and his Ph.D. in Mathematics from the Massachusetts Institute of Technology in 2001.
He is married to Astrid Giugni.
Child prodigy
Ng was a child prodigy who was once thought to be the "smartest kid in America". At age 10, he earned a perfect score of 800 on the math portion of what is now called the SAT-I, a feat considered to be a “remarkable achievement” when a high school junior or senior did it. He was the youngest person to have achieved this feat on his first try. At the age of 11, he earned a perfect score on the College Board Test of Standard Written English. He earned a perfect score on the American High School Mathematics Examination in all 4 years of high school at Chapel Hill High School (Chapel Hill, North Carolina). He attended the Johns Hopkins Center for Talented Youth and was one of the gifted children included in the Study of Mathematically Precocious Youth longitudinal cohort. He was estimated to be 1 in approximately 30 million of his age-mates.
At the age of 12, he began taking courses (on a part-time basis) at the University of North Carolina, Chapel Hill. He was not yet 13 when he won the Written Round of the MATHCOUNTS competition. At the age of 14, he participated in the International Mathematical Olympiad and earned a Silver medal. He participated in this competition for the next two years and earned Gold medals. He entered college (Harvard University) full-time at the age of 16 and majored in Mathematics and Physics, graduating summa cum laude in three years. He competed in the William Lowell Putnam Mathematical Competition while at Harvard University and was a three-time fellow (in 1993, 1994, and 1995), one of only 18 people to have achieved this feat since 1938. The first time he became a Putnam Fellow was at the age of 16, making him one of only 6 people (the 5 others being Arthur Rubin, Noam Elkies, Don Zagier, David Ash and John Tillinghast) in the history of the competition to have achieved this feat.
Mathematical work
Ng works in contact and symplectic geometry. His Ph.D. thesis and several other papers concern Legendrian knots, and his best-known work applies symplectic field theory to derive invariants of (topological) knots. More precisely, the conormal bundle of a knot embedded in the three-sphere is a Legendrian torus inside the three-sphere's unit cosphere bundle (a contact five-manifold). Relative contact homology produces symplectic invariants of this pair, which give topological invariants of the knot. Ng computed the linearized contact homology in this case, prov |
https://en.wikipedia.org/wiki/Multivariate | Multivariate is the quality of having multiple variables.
It may also refer to:
In mathematics
Multivariable calculus
Multivariate function
Multivariate polynomial
Multivariate interpolation
Multivariate optimization
In computing
Multivariate cryptography
Multivariate division algorithm
Multivariate optical computing
In statistics
Multivariate analysis
Multivariate random variable
Multivariate regression
Multivariate statistics
See also
Univariate
Bivariate (disambiguation) |
https://en.wikipedia.org/wiki/Calculus%20bovis | Calculus bovis, niu-huang () or ox bezoars are dried gallstones of cattle used in Chinese herbology. In China and Japan it has been long used to treat various diseases, including high fever, convulsion and stroke.
In Asian countries, calculus bovis are sometimes harvested when steers (Bos taurus domesticus Gmelin) are slaughtered. Their gall bladders are taken out, the bile is filtered, and the stones are cleaned and dried. The bezoars may also be surgically removed by veterinarians when working cattle become ill. In western countries, they are usually discarded. Its equivalent in Vedic culture is Gorochana.
Calculus bovis have a color varying from golden yellow to brownish yellow. The shape of a stone is variable and depends on how it was formed, becoming spherical, oval, triangular, tubular, or irregular.
Since natural calculus bovis are scarce, they can be very expensive. There are artificial calculus bovis or bovis calculus artifactus used as substitutes. In China, these are manufactured from cholic acid derived from bovine bile combined with dry bovine bile powder, porcine ursodeoxycholic acid, taurine, bilirubin, cholesterol, etc. Chinese regulations forbid the use of artificial calculus bovis in 42 drugs.
Calculus bovis can cultivated (induced) in vivo by implanting a foreign object and microbials into bovine biles. It can also be cultivated (precipitated) in vitro from bovine bile by adding ursodeoxycholic acid, cholic acid, and calcium bilirubinate, producing calculus bovis sativus. Unlike artificial calculus bovis, either type of cultivated calculus bovis can fully replace calculus bovis under Chinese regulations.
References
External links
Niuhuang
Pharmacopoeia of the People's Republic of China 2015 entries for calculus bovis:
Bovis calculus. ≥ 5% cholic acid, ≥ 25% bilirubin by dry wright.
Bovis calculus artifacus, artificial replacement produced from dry bovine bile. ≥ 13% cholic acid, ≥ 0.63% bilirubin by dry wright.
Bovis calculus sativus, in vitro cultivated replacement produced from bovine bile. ≥ 6% cholic acid, ≥ 35% bilirubin by dry wright.
Traditional Chinese medicine
Animal glandular products
Cattle |
https://en.wikipedia.org/wiki/Owen%27s%20T%20function | In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by
The function was first introduced by Owen in 1956.
Applications
The function T(h, a) gives the probability of the event (X > h and 0 < Y < aX) where X and Y are independent standard normal random variables.
This function can be used to calculate bivariate normal distribution probabilities and, from there, in the calculation of multivariate normal distribution probabilities.
It also frequently appears in various integrals involving Gaussian functions.
Computer algorithms for the accurate calculation of this function are available; quadrature having been employed since the 1970s.
Properties
Here Φ(x) is the standard normal cumulative distribution function
More properties can be found in the literature.
References
Software
Owen's T function (user web site) - offers C++, FORTRAN77, FORTRAN90, and MATLAB libraries released under the LGPL license LGPL
Owen's T-function is implemented in Mathematica since version 8, as OwenT.
External links
Why You Should Care about the Obscure (Wolfram blog post)
Normal distribution
Computational statistics
Functions related to probability distributions |
https://en.wikipedia.org/wiki/Lady%20tasting%20tea | In the design of experiments in statistics, the lady tasting tea is a randomized experiment devised by Ronald Fisher and reported in his book The Design of Experiments (1935). The experiment is the original exposition of Fisher's notion of a null hypothesis, which is "never proved or established, but is possibly disproved, in the course of experimentation".
The example is loosely based on an event in Fisher's life. The woman in question, phycologist Muriel Bristol, claimed to be able to tell whether the tea or the milk was added first to a cup. Her future husband, William Roach, suggested that Fisher give her eight cups, four of each variety, in random order. One could then ask what the probability was for her getting the specific number of cups she identified correct (in fact all eight), but just by chance.
Fisher's description is less than 10 pages in length and is notable for its simplicity and completeness regarding terminology, calculations and design of the experiment. The test used was Fisher's exact test.
The experiment
The experiment provides a subject with eight randomly ordered cups of tea – four prepared by pouring milk and then tea, four by pouring tea and then milk. The subject attempts to select the four cups prepared by one method or the other, and may compare cups directly against each other as desired. The method employed in the experiment is fully disclosed to the subject.
The null hypothesis is that the subject has no ability to distinguish the teas. In Fisher's approach, there was no alternative hypothesis, unlike in the Neyman–Pearson approach.
The test statistic is a simple count of the number of successful attempts to select the four cups prepared by a given method. The distribution of possible numbers of successes, assuming the null hypothesis is true, can be computed using the number of combinations. Using the combination formula, with total cups and cups chosen, there are
possible combinations.
The frequencies of the possible numbers of successes, given in the final column of this table, are derived as follows. For 0 successes, there is clearly only one set of four choices (namely, choosing all four incorrect cups) giving this result. For one success and three failures, there are four correct cups of which one is selected, which by the combination formula can occur in different ways (as shown in column 2, with x denoting a correct cup that is chosen and o denoting a correct cup that is not chosen); and independently of that, there are four incorrect cups of which three are selected, which can occur in ways (as shown in the second column, this time with x interpreted as an incorrect cup which is not chosen, and o indicating an incorrect cup which is chosen). Thus a selection of any one correct cup and any three incorrect cups can occur in any of 4×4 = 16 ways. The frequencies of the other possible numbers of successes are calculated correspondingly. Thus the number of successes is distributed according to th |
https://en.wikipedia.org/wiki/Hoeffding%27s%20lemma | In probability theory, Hoeffding's lemma is an inequality that bounds the moment-generating function of any bounded random variable. It is named after the Finnish–American mathematical statistician Wassily Hoeffding.
The proof of Hoeffding's lemma uses Taylor's theorem and Jensen's inequality. Hoeffding's lemma is itself used in the proof of McDiarmid's inequality.
Statement of the lemma
Let X be any real-valued random variable such that almost surely, i.e. with probability one. Then, for all ,
or equivalently,
Proof
Without loss of generality, by replacing by , we can assume , so that .
Since is a convex function of , we have that for all ,
So,
where . By computing derivatives, we find
and .
From the AMGM inequality we thus see that for all , and thus, from Taylor's theorem, there is some such that
Thus, .
See also
Hoeffding's inequality
Bennett's inequality
Notes
Probabilistic inequalities |
https://en.wikipedia.org/wiki/Coordinating%20Working%20Party%20on%20Fishery%20Statistics | The Coordinating Working Party on Fishery Statistics (CWP) provides a mechanism to coordinate fishery statistical programmes of regional fishery bodies and other inter-governmental organizations with a remit for fishery statistics.
Main function
Functional since 1960, the CWP's purpose is to:
continually review fishery statistics requirements for research, policy-making and management;
agree on standard concepts, definitions, classifications and methodologies for the collection and collation of fishery statistics;
make proposals for the coordination and streamlining of statistical activities among relevant intergovernmental organizations.
Legal Framework
Established by resolution 23/59 of the FAO Conference under Article VI-2 of the Organization's Constitution at its Tenth Session in 1959. The Statutes of CWP were amended and approved by the FAO Council at its Hundred and Eighth Session in June 1995.
Participating Organizations
The CWP is composed of experts nominated by intergovernmental organizations which have a competence in fishery statistics. There are currently 17 participating organizations in the CWP:
Commission for the Conservation of Antarctic Marine Living Resources (CCAMLR)
Commission for the Conservation of Southern Bluefin Tuna (CCSBT)
Food and Agriculture Organization of the United Nations (FAO)
General Fisheries Commission for the Mediterranean (GFCM)
Indian Ocean Tuna Commission (IOTC)
Inter-American Tropical Tuna Commission
International Commission for the Conservation of Atlantic Tunas (ICCAT)
International Council for the Exploration of the Sea (ICES)
International Whaling Commission (IWC)
North Atlantic Salmon Conservation Organization (NASCO)
North East Atlantic Fisheries Commission (NEAFC)
Northwest Atlantic Fisheries Organization (NAFO)
Organisation for Economic Cooperation and Development (OECD)
Secretariat of the Pacific Community (SPC)
Southeast Asian Fisheries Development Center (SEAFDEC)
Statistical Office of the European Communities (Commission of the EU/Eurostat)
Western and Central Pacific Fisheries Commission (WCPFC)
Secretariat
The FAO serves as the CWP Secretariat. The CWP meets in full session approximately every two years and carries out intersessional and ad hoc meetings as required.
Geographical Coverage
Although first mandated to cover North Atlantic fisheries, since 1995 the CWP has extended its remit to all marine water bodies.
Languages
English
References
Bibliography
The Coordinating Working Party on Fishery Statistics: Its Origin, Role and Structure FAO Fisheries Circular No.903. Rome, December 1995.
External links
CWP website
CWP Handbook of Fishery Statistical Standards
FAO Fisheries&Aquaculture department website
Fisheries agencies
Fisheries law
Statistical data agreements
Statistical organizations |
https://en.wikipedia.org/wiki/Gy%C3%B6rgy%20Haj%C3%B3s | György Hajós (February 21, 1912, Budapest – March 17, 1972, Budapest) was a Hungarian mathematician who worked in group theory, graph theory, and geometry.
Biography
Hajós was born February 21, 1912, in Budapest; his great-grandfather, Adam Clark, was the famous Scottish engineer who built the Chain Bridge in Budapest. He earned a teaching degree from the University of Budapest in 1935. He then took a position at the Technical University of Budapest, where he stayed from 1935 to 1949. While at the Technical University of Budapest, he earned a doctorate in 1938. He became a professor at the Eötvös Loránd University in 1949 and remained there until his death in 1972. Additionally he was president of the János Bolyai Mathematical Society from 1963 to 1972.
Research
Hajós's theorem is named after Hajós, and concerns factorizations of Abelian groups into Cartesian products of subsets of their elements. This result in group theory has consequences also in geometry: Hajós used it to prove a conjecture of Hermann Minkowski that, if a Euclidean space of any dimension is tiled by hypercubes whose positions form a lattice, then some pair of hypercubes must meet face-to-face. Hajós used similar group-theoretic methods to attack Keller's conjecture on whether cube tilings (without the lattice constraint) must have pairs of cubes that meet face to face; his work formed an important step in the eventual disproof of this conjecture.
Hajós's conjecture is a conjecture made by Hajós that every graph with chromatic number contains a subdivision of a complete graph . However, it is now known to be false: in 1979, Paul A. Catlin found a counterexample for , and Paul Erdős and Siemion Fajtlowicz later observed that it fails badly for random graphs. The Hajós construction is a general method for constructing graphs with a given chromatic number, also due to Hajós.
Awards and honors
Hajós was a member of the Hungarian Academy of Sciences, first as a corresponding member beginning in 1948 and then as a full member in 1958. In 1965 he was elected to the Romanian Academy of Sciences, and in 1967 to the German Academy of Sciences Leopoldina. He won the Gyula König Prize in 1942, and the Kossuth Prize in 1951 and again in 1962.
References
Hungarian people of Scottish descent
Members of the Hungarian Academy of Sciences
20th-century Hungarian mathematicians
Geometers
Algebraists
1912 births
1972 deaths
Mathematicians from Austria-Hungary |
https://en.wikipedia.org/wiki/Short-term%20business%20statistics | Short-term business statistics (STS) are economic statistics published by Eurostat, the statistical office of the European Commission. They cover all member states of the European Union and - in some areas - also third countries, and provide aggregated results for the euro area and the EU.
Short-term business statistics provide information on the economic development of four major domains: industry, construction, retail trade, and other services. These domains are defined in relation to a classification of economic activities, the NACE Rev.2 (Statistical Classification of Economic Activities in the European Community, second revision).
For the four domains the following variables are reported (most on a monthly, some on a quarterly basis):
Industry: Production, turnover (domestic and non-domestic), number of persons employed, hours worked, gross wages and salaries, output prices (domestic and non-domestic), import prices.
Construction: Production (total, of buildings and of civil engineering), number of persons employed, hours worked, gross wages and salaries, construction costs (material costs, wage costs), construction permits.
Retail trade and repair: Turnover, retail trade volume, number of persons employed, hours worked, gross wages and salaries.
Other services: Turnover, output prices (service producer prices), number of persons employed, hours worked, gross wages and salaries.
The legal authority for the European short-term business statistics is Regulation EC No 1165/98, as amended by Regulation EC No 1158/2005, and further implementing and amending regulations.
References
External links
Eurostat - Short-term business statistics
Eurostat - Statistics Explained: all articles on short-term business statistics
European Commission
Economic data
Economy of the European Union
Year of establishment missing |
https://en.wikipedia.org/wiki/John%20Fitzgerald%20career%20statistics | This is a list of the main career statistics of former professional Australian tennis player John Fitzgerald.
Career statistics
Grand Slam finals
Doubles (7 titles, 4 runners-up)
Mixed Doubles (2 titles, 4 runners-up)
Career finals
Singles: 11 (6-5)
Doubles: 61 (30-31)
External links
Fitzgerald, John |
https://en.wikipedia.org/wiki/Shmuel%20Gal | Shmuel Gal (, born 1940) is a mathematician and professor of statistics at the University of Haifa in Israel.
He devised the Gal's accurate tables method for the computer evaluation of elementary functions. With Zvi Yehudai he developed in 1993 a new algorithm for sorting which is used by IBM.
Gal has solved the Princess and monster game and made several significant contributions to the area of search games.
He has been working on rendezvous problems with his collaborative colleagues Steve Alpern, Vic Baston, and John Howard.
Gal received a Ph.D. in mathematics from the Hebrew University of Jerusalem. His thesis advisor was Aryeh Dvoretzky.
References
External links
{https://sites.google.com/edu.haifa.ac.il/sgal/home }
Game theorists
Hebrew University of Jerusalem alumni
Israeli mathematicians
Israeli operations researchers
Academic staff of the University of Haifa
Jewish systems scientists
Living people
1940 births |
https://en.wikipedia.org/wiki/Placidus%20de%20Titis | Placidus de Titis (also de Titus, Latinization of Placido de Titi, pseudonym Didacus Prittus Pelusiensis; 1603–1668) was an Olivetan monk and professor of mathematics, physics and astronomy at the University of Pavia from 1657 until his death. Placidus popularized the system of astrological houses now known as the "Placidian system", current in modern astrology. He did not invent the method; it is acknowledged by the 12th century Hebrew astrologer Abraham Ibn Ezra as the system employed by Ptolemy, an attribution that was accepted by Placidus.
Biography
Placidus was born in Perugia, into the Titi noble family. His father died early, and he was looked after by his mother Cecilia. He studied at the University of Padua where his uncle Girolamo de Titi was professor of theology. One of his teachers was the astronomer Andrea Argoli. The Duchy of Milan at the time was owned by Habsburg Spain, administered by Archduke Leopold Wilhelm of Austria. The Archduke showed strong interest in science, especially occult sciences of alchemy and astrology, and Placidus dedicated his astrological house tables to him. In 1657 he was appointed professor of mathematics at the University of Pavia, a position he held for the rest of his life. Like his contemporary Jean-Baptiste Morin, Placidus opposed the copernican theory and retained a geocentric perspective, although there have been suggestions that he might have been a closet Copernican.
He died in Pavia in 1668.
English translations of Placidus' Primum Mobile were published by Manoah Sibly (1789) and John Cooper (1814).
Works
De motibus directionum coelestium mobilium (1641).
Physiomathematica sive coelestis philosophia (1650), Placidus' magnum opus, first published as Quaestionum physiomathematicarum libri tres, under the pseudonym of Didacus Prittus Pelusiensis, second edition by C. Francobacci und A. Scirota (pseudonyms of two of Placidus' students, F. Brunacci and F. M. Onorati).
Nuncius astronomicus (1654).
Il corriere astronomico (1656).
Tabulae primi mobilis cum thesibus et canonibus (1657).
Commentaria in Ptolemaeum de siderum judiciis (1658).
De siderum judiciis, 2 vols. (1660, 1665).
De diebus decretoriis et aegrorum decubitu, 2 vols. (1661, 1665).
Ephemerides coelestium motuum (1661-1665).
Tocco di paragone onde evidentemente appare che l’astrologia nelle parti concesse da S. Chiesa è vera scienza, naturale, nobile, et utile quanto la filosofia (1666), in defense of astrology as a natural science.
Notes
Further reading
Gansten, Martin. 2011. “Placidean Teachings in Early Nineteenth-Century Britain: John Worsdale and Thomas Oxley.” In Astrologies: Plurality and Diversity: The Proceedings of the Eighth Annual Conference of the Sophia Centre for the Study of Cosmology in Culture, University of Wales, Trinity Saint David, 24–25 July 2010, ed. Nicholas Campion and Liz Green. Ceredigion: Sophia Centre Press.
17th-century Italian astronomers
17th-century Italian Christian monk |
https://en.wikipedia.org/wiki/Annual%20business%20survey%20%28UK%29 | The annual business survey, formerly the annual business inquiry, is a census of production in the United Kingdom, produced by the Office for National Statistics. It was introduced in 1988 and consolidated earlier surveys. Results were first published in 2000.
External links
Office for National Statistics: Annual Business Survey: Survey at a glance
Office for National Statistics: Annual Business Survey: ABS News
Economy of the United Kingdom
Business in the United Kingdom
1988 introductions
Censuses in the United Kingdom |
https://en.wikipedia.org/wiki/Robert%20Kozma%20%28professor%29 | Robert Kozma is First Tennessee University Professor of Mathematics at the University of Memphis.
Biography
Kozma received his MS in Power Engineering from the Moscow Power Engineering Institute in 1982, his MS in mathematics from the Eötvös Loránd University in 1988, and his PhD in Applied Physics from Delft University of Technology in 1992
Kozma has been associate professor at the Department of Quantum Science and Engineering, Tohoku University, Sendai, Japan since 1993. He became Assistant Professor/Lecturer at the Department of Information Sciences, Otago University, Dunedin, New Zealand in 1996. In 1998 in the USA he became a joint appointment at the Division of Neurobiology and the EECS Department, University of California, Berkeley. In 2009 he was appointed Professor of Computer Science, University of Memphis, Memphis, Tennessee, and since 2009 he is Professor of Mathematical Sciences, University of Memphis, Memphis, Tennessee. Since 2001 he is also director of Computational Neurodynamics Laboratory, presently CLION, FedEx Institute of Technology of the University of Memphis, Memphis.
He serves on the AdCom of IEEE Computational Intelligence Society CIS (2009–2012) and on the Governing Board of the International Neural Network Society INNS (2004–2012). He is Chair of the Distinguished Lecturer Program, IEEE CIS. He has been Technical Committee Member of IEEE Computational Intelligence Society since 1996, and IEEE Senior Member. He also served in leading positions at over 20 international conferences, including General Chair of IEEE/INNS International Joint Conference on Neural Networks IJCNN09 in Atlanta; Program Co-chair of International Joint Conference on Neural Networks IJCNN08/WCCI08 in Hong Kong; Program Co-chair of IJCNN04, Budapest, Hungary; chair for Finances of IEEE WCCI06, Vancouver, Canada.
He is Associate Editor of ‘Neural Networks (Elsevier),’ ‘IEEE Transactions on Neural Networks,’ ‘Neurocomputing’ (Elsevier), ‘Journal of Cognitive Neurodynamics’ (Springer), Area Editor of ‘New Mathematics and Natural Computation’ (World Scientific), and ‘Cognitive Systems Research.’
Work
Kozma's current research interests include spatio-temporal dynamics of neural processes, random graph approaches to large-scale networks, such as neural networks, computational intelligence methods for knowledge acquisition and autonomous decision making in biological and artificial systems.
Publications
He has published 100+ papers and 3+ books in the several fields including signal processing; and design, analysis, and control of intelligent systems.
References
External links
Kozma's homepage
CLION
Robert Kozma; Modeling Cortical Phase Transitions (2007)
September 2009 Newsletter for University of Memphis
Living people
American computer scientists
20th-century American mathematicians
21st-century American mathematicians
University of Memphis faculty
Delft University of Technology alumni
Year of birth missing (living people)
Network scientis |
https://en.wikipedia.org/wiki/SEARCH%2C%20The%20National%20Consortium%20for%20Justice%20Information%20and%20Statistics | SEARCH, The National Consortium for Justice Information and Statistics (also called SEARCH), is a nonprofit criminal justice support organization created by and for the states. Its headquarters are in Sacramento, California.
SEARCH was established in 1969. It provides support to the information sharing, interoperability, information technology, cybercrime and criminal records needs of justice and public safety agencies and practitioners nationwide at the state, local and tribal levels.
Mission
SEARCH has a mission to improve "the quality of justice and public safety through the use, management, and exchange of information; application of new technologies; and responsible law and policy, while safeguarding security and privacy."
House Resolution
On November 17, 2009, the U.S. House of Representatives passed House Resolution 851 to recognize the organization's 40th anniversary. The resolution was sponsored by Congresswoman Doris O. Matsui (D-CA 5).
Programs
SEARCH programs offer assistance to justice and public safety agencies to develop, operate, secure, and improve information sharing and identification systems. Its programs focus on: information sharing initiatives, criminal history records systems, information technology planning, acquisition, and management, cybercrime investigation and systems security, information law and policy, public safety and communications interoperability.
Membership
SEARCH is governed by a Membership Group composed of one gubernatorial appointee from the 50 states, the District of Columbia, Puerto Rico, and the U.S. Virgin Islands, and eight at-large appointees selected by the SEARCH Chair. Members are responsible for operational decisions and policymaking concerning the management of criminal justice and criminal history information. The current Chairman of SEARCH is Mr. Brad Truitt, Director of Information Systems, Tennessee Bureau of Investigation.
Partnerships
SEARCH partners with other agencies, organizations and associations to support IT initiatives and information sharing, such as rap sheet standards, Global Justice Information Sharing Initiative, National Information Exchange Model (NIEM), and the FBI's National Data Exchange Project (N-DEx).
References
External links
SEARCH Website
National Criminal Justice Reference Service Website
The Library of Congress Website
U.S. Representative Doris Matsui Website
Legal organizations based in the United States |
https://en.wikipedia.org/wiki/Julian%20Pottage | Julian Y. Pottage (born 1962) is a British contract bridge player, writer, and teacher, who studied mathematics at Trinity College, Cambridge. He is also well known as a collector of bridge problems, and writes a monthly problem column in Britain's Bridge Magazine.
He is the Bridge Correspondent for The Daily Telegraph, a regular contributor to English Bridge and, prior to Mr Bridge’s retirement, was Associate Editor of BRIDGE.
He has written or co-authored 26 books on bridge, including Bridge Problems for a New Millennium and The Extra Edge In Play with Terence Reese (1913–1996). He also co-edited the 2010 second edition of Clyde E. Love's Bridge Squeezes Complete.
His book Play or Defend? won the International Bridge Press Association's 2004 Book of the Year Award, as did A Great Deal of Bridge Problems, in 2014.
Pottage is from Hampshire and went to school at St John's College in Southsea and Millfield school in Somerset. After reading mathematics at Trinity College Cambridge, Julian had a career in the occupational pensions industry. More recently his work has concentrated on bridge writing and breeding golden retriever dogs, his kennel name being Yorkbeach.
As a player, Pottage has participated in several national and international events, notably winning the Pachabo and Tollemache double in 1999. In recent years he has won several simultaneous pairs, including the inaugural Peter Jordan trophy. Julian was in the winning England team in the 1984 Junior Camrose and, having moved to Wales in 2005, has represented Wales in a number of international competitions. Since 2023, his caps for Wales include the most recent 3 European Championships and at least 1 Camrose weekend for each of the preceding 6 years.
He lives in Wales. He and his wife have 4 now adult children.
Bibliography
References
External links
Tollemache Cup History
Living people
1962 births
People from Basingstoke
Contract bridge writers
British and Irish contract bridge players
Welsh contract bridge players
Alumni of Trinity College, Cambridge
Date of birth missing (living people)
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Bangkok%20United%20F.C.%20in%20Asia | Statistics of Bangkok United in Asian competition.
Results
References
Bangkok United F.C. |
https://en.wikipedia.org/wiki/BEC%20Tero%20Sasana%20F.C.%20in%20Asia | Statistics of BEC Tero Sasana in Asian competition.
AFC Champions League
Results
Asian Club Championship
Results
References
Asia |
https://en.wikipedia.org/wiki/Osotspa%20Saraburi%20F.C.%20in%20Asia | Statistics of Osotspa Saraburi in Asian competition.
AFC Champions League
Results
AFC Cup
Results
References
Asia |
https://en.wikipedia.org/wiki/Sergey%20Zhukov%20%28ice%20hockey%29 | Sergei Petrovich Zhukov (born 23 November 1975) is a Russian former professional ice hockey player who last played for Lokomotiv Yaroslavl of the Kontinental Hockey League (KHL).
Career statistics
Regular season and playoffs
International
External links
1975 births
Boston Bruins draft picks
HC Sibir Novosibirsk players
Ice hockey players at the 2006 Winter Olympics
Living people
Lokomotiv Yaroslavl players
Olympic ice hockey players for Russia
Sportspeople from Novosibirsk
Russian ice hockey defencemen |
https://en.wikipedia.org/wiki/Rudhmuli | Rudhmuli is a village in Bah Tehsil of Agra District of Uttar Pradesh in India.
Demographics
India census, Rudhmuli follows the same statistics as of Bah [Nearest Town having 2011 census statistics]. Rudhmuli had a population of 1,445. The Male population is 734 and female population is 711.15% of the population is under 6 years of age. Pincode for Rudhmuli is 283104 and STD code is 05614.
Overview
It is around 71.8 km from Agra city and around 59 km from Etawah district. It was founded in early 19th century by Shri Angad singh Bhadoria of Bhadawar state.
Rudhmuli is near the famous Bateshwarnath Temple (Lord Shiva). November local festival at Bateshwarnath attracts thousands of devotees from nearby areas and states ( Rajasthan & Madhya Pradesh).
It is situated on the Bank of Yamuna River and most of the land in village is not Plain. In old times, the Village was controlled by the Bhadoria Rulers nominated by the King of Bhadawar state.
References
External links
http://duckduckgo.com/Rudhmuli
http://agra.travelhawk.ca/news/Rudhmuli.html
http://agra.nic.in/smsMaster/GramPanchayat.pdf
http://www.indiamapped.com/uttar-pradesh/agra/bah/rudmuli/
Villages in Agra district
bpy:বাহ
it:Bah (India)
new:बाह
pt:Bah
vi:Bah |
https://en.wikipedia.org/wiki/Jonas%20Solberg%20Andersen | Jonas Solberg Andersen (born March 8, 1981 in Sarpsborg, Norway) is a Norwegian former professional ice hockey player.
Career statistics
Regular season and playoffs
International
External links
1981 births
Living people
Ice hockey players at the 2010 Winter Olympics
Leksands IF players
Norwegian expatriate ice hockey people
Norwegian ice hockey forwards
Olympic ice hockey players for Norway
Sportspeople from Sarpsborg
Sparta Sarpsborg players |
https://en.wikipedia.org/wiki/Nikolai%20Nikolaevich%20Yanenko | Nikolai Nikolaevich Yanenko (; 22 May 1921 – 16 January 1984) was a Soviet mathematician and academician. He was known for his contributions to computational mathematics and fluid mechanics. He served as Director of the Institute of Theoretical and Applied Mechanics of the Siberian Division of the USSR Academy of Sciences.
Education and research
Yanenko received his Ph.D. degree from Lomonosov Moscow State University in 1954 under the supervision of Petr Rashevskii. The topic of his doctoral thesis was a problem in multidimensional differential geometry. His scientific interests span several areas of fundamental research, including gas dynamics, the theory of difference schemes, and computational fluid dynamics.
In recognition of Yanenko's scientific contributions the International Conference on "Mathematical Models and Numerical Methods of Continuum Mechanics" was held on May 26, 1996 at the Novosibirsk Scientific Center, Novosibirsk.
References
External links
N.N. Yanenko's portraits
1921 births
1984 deaths
Computational fluid dynamicists
Soviet mathematicians
20th-century Russian mathematicians
Full Members of the USSR Academy of Sciences
Recipients of the Stalin Prize
Recipients of the USSR State Prize
Heroes of Socialist Labour
Recipients of the Order of Lenin |
https://en.wikipedia.org/wiki/Schur%20algebra | In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. Green first published in 1980. The name "Schur algebra" is due to Green. In the modular case (over infinite fields of positive characteristic) Schur algebras were used by Gordon James and Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent. Schur algebras were used by Friedlander and Suslin to prove finite generation of cohomology of finite group schemes.
Construction
The Schur algebra can be defined for any commutative ring and integers . Consider the algebra of polynomials (with coefficients in ) in commuting variables , 1 ≤ i, j ≤ . Denote by the homogeneous polynomials of degree . Elements of are k-linear combinations of monomials formed by multiplying together of the generators (allowing repetition). Thus
Now, has a natural coalgebra structure with comultiplication and counit the algebra homomorphisms given on generators by
(Kronecker's delta).
Since comultiplication is an algebra homomorphism, is a bialgebra. One easily
checks that is a subcoalgebra of the bialgebra , for every r ≥ 0.
Definition. The Schur algebra (in degree ) is the algebra . That is, is the linear dual of .
It is a general fact that the linear dual of a coalgebra is an algebra in a natural way, where the multiplication in the algebra is induced by dualizing the comultiplication in the coalgebra. To see this, let
and, given linear functionals , on , define their product to be the linear functional given by
The identity element for this multiplication of functionals is the counit in .
Main properties
One of the most basic properties expresses as a centralizer algebra. Let be the space of rank column vectors over , and form the tensor power
Then the symmetric group on letters acts naturally on the tensor space by place permutation, and one has an isomorphism
In other words, may be viewed as the algebra of endomorphisms of tensor space commuting with the action of the symmetric group.
is free over of rank given by the binomial coefficient .
Various bases of are known, many of which are indexed by pairs of semistandard Young tableaux of shape , as varies over the set of partitions of into no more than parts.
In case k is an infinite field, may also be identified with the enveloping algebra (in the sense of H. Weyl) for the action of the general linear group acting on (via the diagonal action on tensors, induced from the natural action of on given by matrix multiplication).
Schur algebras are "defined over the integers". This means that they satisfy the following change of scalars property: |
https://en.wikipedia.org/wiki/Savilian%20Professor%20of%20Geometry | The position of Savilian Professor of Geometry was established at the University of Oxford in 1619. It was founded (at the same time as the Savilian Professorship of Astronomy) by Sir Henry Savile, a mathematician and classical scholar who was Warden of Merton College, Oxford, and Provost of Eton College, reacting to what has been described by one 20th-century mathematician as "the wretched state of mathematical studies in England" at that time. He appointed Henry Briggs as the first professor. Edward Titchmarsh (professor 1931–63) said when applying that he was not prepared to lecture on geometry, and the requirement was removed from the duties of the post to enable his appointment, although the title of the chair was not changed. The two Savilian chairs have been linked with professorial fellowships at New College, Oxford, since the late 19th century. Before then, for over 175 years until the middle of the 19th century, the geometry professors had an official residence adjoining the college in New College Lane.
There have been 20 professors; Frances Kirwan, the current () and first female holder of the chair, was appointed in 2017. The post has been held by a number of distinguished mathematicians. Briggs helped to develop the common logarithm, described as "one of the most useful systems for mathematics". The third professor, John Wallis, introduced the use of for infinity, and was regarded as "one of the leading mathematicians of his time". Both Edmond Halley, who successfully predicted the return of the comet named in his honour, and his successor Nathaniel Bliss held the post of Astronomer Royal in addition to the professorship. Stephen Rigaud (professor 1810–27) has been called "the foremost historian of astronomy and mathematics in his generation". The life and work of James Sylvester (professor 1883–94) was commemorated by the Royal Society by the inauguration of the Sylvester Medal; this was won by a later professor, Edward Titchmarsh. Two professors, Sylvester and Michael Atiyah (professor 1963–69), have been awarded the Copley Medal of the Royal Society; Atiyah also won the Fields Medal while he was professor.
Foundation and duties
Sir Henry Savile, the Warden of Merton College, Oxford, and Provost of Eton College, was deeply saddened by what the 20th-century mathematician Ida Busbridge has termed "the wretched state of mathematical studies in England", and so founded professorships in geometry and astronomy at the University of Oxford in 1619; both chairs were named after him. He also donated his books to the university's Bodleian Library "for the use chiefly of mathematical readers". He required the professors to be men of good character, at least 26 years old, and to have "imbibed the purer philosophy from the springs of Aristotle and Plato" before acquiring a thorough knowledge of science. The professors could come from any Christian country, but he specified that a professor from England should have a Master of Arts degree as |
https://en.wikipedia.org/wiki/P-derivation | In mathematics, more specifically differential algebra, a p-derivation (for p a prime number) on a ring R, is a mapping from R to R that satisfies certain conditions outlined directly below. The notion of a p-derivation is related to that of a derivation in differential algebra.
Definition
Let p be a prime number. A p-derivation or Buium derivative on a ring is a map that satisfies the following "product rule":
and "sum rule":
as well as
Note that in the "sum rule" we are not really dividing by p, since all the relevant binomial coefficients in the numerator are divisible by p, so this definition applies in the case when has p-torsion.
Relation to Frobenius endomorphisms
A map is a lift of the Frobenius endomorphism provided . An example of such a lift could come from the Artin map.
If is a ring with a p-derivation, then the map defines a ring endomorphism which is a lift of the Frobenius endomorphism. When the ring R is free the correspondence is a bijection.
Examples
For the unique p-derivation is the map
The quotient is well-defined because of Fermat's little theorem.
If R is any p-torsion free ring and is a lift of the Frobenius endomorphism then
defines a p-derivation.
See also
Witt vector
Arithmetic derivative
Derivation
Fermat quotient
References
.
External links
Project Euclid
Differential algebra
Generalizations of the derivative |
https://en.wikipedia.org/wiki/Chern%20Medal | The Chern Medal is an international award recognizing outstanding lifelong achievement of the highest level in the field of mathematics. The prize is given at the International Congress of Mathematicians (ICM), which is held every four years.
Introduction
It is named in honor of the late Chinese mathematician Shiing-Shen Chern. The award is a joint effort of the International Mathematical Union (IMU) and the Chern Medal Foundation (CMF) to be bestowed in the same fashion as the IMU's other three awards (the Fields Medal, the Abacus Medal, and the Gauss Prize), i.e. at the opening ceremony of the International Congress of Mathematicians (ICM), which is held every four years. The first such occasion was at the 2010 ICM in Hyderabad, India.
Each recipient receives a medal decorated with Chern's likeness, a cash prize of $250,000 (USD), and the opportunity to direct $250,000 of charitable donations to one or more organizations for the purpose of supporting research, education, or outreach in mathematics.
Laureates
See also
Fields Medal
Gauss Prize
International Congress of Mathematicians (ICM)
International Mathematical Union (IMU)
Nevanlinna Prize
List of mathematics awards
Notes
References
, International Mathematical Union
Press release, International Mathematical Union
Program Guidelines, International Mathematical Union
Awards of the International Mathematical Union |
https://en.wikipedia.org/wiki/Van%20Houtum%20distribution | In probability theory and statistics, the Van Houtum distribution is a discrete probability distribution named after prof. Geert-Jan van Houtum. It can be characterized by saying that all values of a finite set of possible values are equally probable, except for the smallest and largest element of this set. Since the Van Houtum distribution is a generalization of the discrete uniform distribution, i.e. it is uniform except possibly at its boundaries, it is sometimes also referred to as quasi-uniform.
It is regularly the case that the only available information concerning some discrete random variable are its first two moments. The Van Houtum distribution can be used to fit a distribution with finite support on these moments.
A simple example of the Van Houtum distribution arises when throwing a loaded dice which has been tampered with to land on a 6 twice as often as on a 1. The possible values of the sample space are 1, 2, 3, 4, 5 and 6. Each time the die is thrown, the probability of throwing a 2, 3, 4 or 5 is 1/6; the probability of a 1 is 1/9 and the probability of throwing a 6 is 2/9.
Probability mass function
A random variable U has a Van Houtum (a, b, pa, pb) distribution if its probability mass function is
Fitting procedure
Suppose a random variable has mean and squared coefficient of variation . Let be a Van Houtum distributed random variable. Then the first two moments of match the first two moments of if , , and are chosen such that:
There does not exist a Van Houtum distribution for every combination of and . By using the fact that for any real mean the discrete distribution on the integers that has minimal variance is concentrated on the integers and , it is easy to verify that a Van Houtum distribution (or indeed any discrete distribution on the integers) can only be fitted on the first two moments if
References
Discrete distributions |
https://en.wikipedia.org/wiki/Bridge%20Squeezes%20Complete | Bridge Squeezes Complete is a book on contract bridge written by Ann Arbor, Michigan-based mathematics professor Clyde E. Love, originally published in 1959. Written in a "dry, mathematical way", it is still considered one of the most important bridge books ever written and the squeeze vocabulary Love invented remains the basis for all discussions of squeezes.
Reprints of the original text have been published in 1961 by Mayflower (London), in 1968 by Dover (New York) and in 1969 by Constable (London). A revised edition, edited by Linda Lee and Julian Pottage, was published by Master Point Press in 2010.
An interactive software presentation of the original text was developed by Lorne Russell in 2006 and translated into French by Theo VanDommelen in 2008.
See also
Squeeze play bibliography
References
Contract bridge books
1959 non-fiction books |
https://en.wikipedia.org/wiki/Clyde%20E.%20Love | Clyde Elton Love (December 12, 1882 – January 31, 1960) was an American contract bridge author and mathematics professor at the University of Michigan, Ann Arbor. He was a native of Bancroft, Michigan and graduated from the University of Michigan in 1905.
Love is well known in bridge circles for his 1959 book Bridge Squeezes Complete, one of the earliest efforts to codify then-existing squeeze play. Love established rules for recognizing bridge squeezes, and for executing them when they occurred. His system of classifying squeezes has been used by most bridge writers since. He was also the author of many magazine articles.
He died at his home in Ann Arbor, aged 77. He was survived by his wife and daughter.
Bibliography
Differential and Integral Calculus, 1916-1970
Analytic Geometry, 1923-1955
Elements of Analytic Geometry, 1931-1956
On the Asymptotic Solutions of Linear Differential Equations, 1913-1914
See also
Squeeze plays in contract bridge
References
External links
(including 3 "from old catalog")
1882 births
1960 deaths
20th-century American mathematicians
University of Michigan alumni
University of Michigan faculty
Contract bridge writers
People from Shiawassee County, Michigan |
https://en.wikipedia.org/wiki/Carl%20King-Millward | Carl Edward King-Millward (17 July 1935 in Bedfordshire, United Kingdom – August 2000) was a British mathematician. He became head of applied mathematics at the Institute of Historical Research in London in 1965, thus becoming the youngest non-literary scholar to do so in the post-war era.
King-Millward's parents were of Slavonic extraction, moving to Britain in 1933.
References
1935 births
2000 deaths
20th-century English mathematicians
People from Bedfordshire |
https://en.wikipedia.org/wiki/Conductor%20%28class%20field%20theory%29 | In algebraic number theory, the conductor of a finite abelian extension of local or global fields provides a quantitative measure of the ramification in the extension. The definition of the conductor is related to the Artin map.
Local conductor
Let L/K be a finite abelian extension of non-archimedean local fields. The conductor of L/K, denoted , is the smallest non-negative integer n such that the higher unit group
is contained in NL/K(L×), where NL/K is field norm map and is the maximal ideal of K. Equivalently, n is the smallest integer such that the local Artin map is trivial on . Sometimes, the conductor is defined as where n is as above.
The conductor of an extension measures the ramification. Qualitatively, the extension is unramified if, and only if, the conductor is zero, and it is tamely ramified if, and only if, the conductor is 1. More precisely, the conductor computes the non-triviality of higher ramification groups: if s is the largest integer for which the "lower numbering" higher ramification group Gs is non-trivial, then , where ηL/K is the function that translates from "lower numbering" to "upper numbering" of higher ramification groups.
The conductor of L/K is also related to the Artin conductors of characters of the Galois group Gal(L/K). Specifically,
where χ varies over all multiplicative complex characters of Gal(L/K), is the Artin conductor of χ, and lcm is the least common multiple.
More general fields
The conductor can be defined in the same way for L/K a not necessarily abelian finite Galois extension of local fields. However, it only depends on Lab/K, the maximal abelian extension of K in L, because of the "norm limitation theorem", which states that, in this situation,
Additionally, the conductor can be defined when L and K are allowed to be slightly more general than local, namely if they are complete valued fields with quasi-finite residue field.
Archimedean fields
Mostly for the sake of global conductors, the conductor of the trivial extension R/R is defined to be 0, and the conductor of the extension C/R is defined to be 1.
Global conductor
Algebraic number fields
The conductor of an abelian extension L/K of number fields can be defined, similarly to the local case, using the Artin map. Specifically, let θ : Im → Gal(L/K) be the global Artin map where the modulus m is a defining modulus for L/K; we say that Artin reciprocity holds for m if θ factors through the ray class group modulo m. We define the conductor of L/K, denoted , to be the highest common factor of all moduli for which reciprocity holds; in fact reciprocity holds for , so it is the smallest such modulus.
Example
Taking as base the field of rational numbers, the Kronecker–Weber theorem states that an algebraic number field K is abelian over Q if and only if it is a subfield of a cyclotomic field , where denotes a primitive nth root of unity. If n is the smallest integer for which this holds, the conductor of K is then n if K is fixe |
https://en.wikipedia.org/wiki/Huber%20loss | In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used.
Definition
The Huber loss function describes the penalty incurred by an estimation procedure . Huber (1964) defines the loss function piecewise by
This function is quadratic for small values of , and linear for large values, with equal values and slopes of the different sections at the two points where . The variable often refers to the residuals, that is to the difference between the observed and predicted values , so the former can be expanded to
The Huber loss is the convolution of the absolute value function with the rectangular function, scaled and translated. Thus it "smoothens out" the former's corner at the origin.
Motivation
Two very commonly used loss functions are the squared loss, , and the absolute loss, . The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case). The squared loss has the disadvantage that it has the tendency to be dominated by outliers—when summing over a set of 's (as in ), the sample mean is influenced too much by a few particularly large -values when the distribution is heavy tailed: in terms of estimation theory, the asymptotic relative efficiency of the mean is poor for heavy-tailed distributions.
As defined above, the Huber loss function is strongly convex in a uniform neighborhood of its minimum ; at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points and . These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimator (using the absolute value function).
Pseudo-Huber loss function
The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function. It combines the best properties of L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. The scale at which the Pseudo-Huber loss function transitions from L2 loss for values close to the minimum to L1 loss for extreme values and the steepness at extreme values can be controlled by the value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as
As such, this function approximates for small values of , and approximates a straight line with slope for large values of .
While the above is the most common form, other smooth approximations of the Huber loss function also exist.
Variant for classification
For classification purposes, a variant of the Huber loss called modified Huber is so |
https://en.wikipedia.org/wiki/Generic%20flatness | In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due to Alexander Grothendieck.
Generic flatness states that if Y is an integral locally noetherian scheme, is a finite type morphism of schemes, and F is a coherent OX-module, then there is a non-empty open subset U of Y such that the restriction of F to u−1(U) is flat over U.
Because Y is integral, U is a dense open subset of Y. This can be applied to deduce a variant of generic flatness which is true when the base is not integral. Suppose that S is a noetherian scheme, is a finite type morphism, and F is a coherent OX module. Then there exists a partition of S into locally closed subsets S1, ..., Sn with the following property: Give each Si its reduced scheme structure, denote by Xi the fiber product , and denote by Fi the restriction ; then each Fi is flat.
Generic freeness
Generic flatness is a consequence of the generic freeness lemma. Generic freeness states that if A is a noetherian integral domain, B is a finite type A-algebra, and M is a finite type B-module, then there exists a non-zero element f of A such that Mf is a free Af-module. Generic freeness can be extended to the graded situation: If B is graded by the natural numbers, A acts in degree zero, and M is a graded B-module, then f may be chosen such that each graded component of Mf is free.
Generic freeness is proved using Grothendieck's technique of dévissage. Another version of generic freeness can be proved using Noether's normalization lemma.
References
Bibliography
Algebraic geometry
Commutative algebra
Theorems in abstract algebra |
https://en.wikipedia.org/wiki/D%C3%A9vissage | In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent.
Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme. They did this by defining an object called a relative dévissage which is well-suited to certain kinds of inductive arguments. They used this technique to give a new criterion for a module to be flat. As a consequence, they were able to simplify and generalize the results of EGA IV 11 on descent of flatness.
The word dévissage is French for unscrewing.
Grothendieck's dévissage theorem
Let X be a noetherian scheme. Let C be a subset of the objects of the category of coherent OX-modules which contains the zero sheaf and which has the property that, for any short exact sequence of coherent sheaves, if two of A, A′, and A′′ are in C, then so is the third. Let X′ be a closed subspace of the underlying topological space of X. Suppose that for every irreducible closed subset Y of X′, there exists a coherent sheaf G in C whose fiber at the generic point y of Y is a one-dimensional vector space over the residue field k(y). Then every coherent OX-module whose support is contained in X′ is contained in C.
In the particular case that , the theorem says that C is the category of coherent OX-modules. This is the setting in which the theorem is most often applied, but the statement above makes it possible to prove the theorem by noetherian induction.
A variation on the theorem is that if every direct factor of an object in C is again in C, then the condition that the fiber of G at x be one-dimensional can be replaced by the condition that the fiber is non-zero.
Gruson and Raynaud's relative dévissages
Suppose that is a finitely presented morphism of affine schemes, s is a point of S, and M is a finite type OX-module. If n is a natural number, then Gruson and Raynaud define an S-dévissage in dimension n to consist of:
A closed finitely presented subscheme X′ of X containing the closed subscheme defined by the annihilator of M and such that the dimension of is less than or equal to n.
A scheme T and a factorization of the restriction of f to X′ such that is a finite morphism and is a smooth affine morphism with geometrically integral fibers of dimension n. Denote the generic point of by τ and the pushforward of M to T by N.
A free finite type OT-module L and a homomorphism such that is bijective.
If n1, n2, ..., nr is a strictly decreasing sequence of natural numbers, then an S-dévissage in dimensions n1, n2, ..., nr is de |
https://en.wikipedia.org/wiki/Michael%20Feinberg | Michael Jai Feinberg is a photographer and software designer best known for creating the computer games Endorfun and Ishido. He is also the creator of lightSource Sacred Geometry software, and more recently Pypeline, a rich-media software platform.
References
Living people
American photographers
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Infinite%20product | In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product
is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to converge when the limit exists and is not zero. Otherwise the product is said to diverge. A limit of zero is treated specially in order to obtain results analogous to those for infinite sums. Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence an as n increases without bound must be 1, while the converse is in general not true.
The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète (Viète's formula, the first published infinite product in mathematics) and John Wallis (Wallis product):
Convergence criteria
The product of positive real numbers
converges to a nonzero real number if and only if the sum
converges. This allows the translation of convergence criteria for infinite sums into convergence criteria for infinite products. The same criterion applies to products of arbitrary complex numbers (including negative reals) if the logarithm is understood as a fixed branch of logarithm which satisfies ln(1) = 0, with the proviso that the infinite product diverges when infinitely many an fall outside the domain of ln, whereas finitely many such an can be ignored in the sum.
For products of reals in which each , written as, for instance, , where , the bounds
show that the infinite product converges if the infinite sum of the pn converges. This relies on the Monotone convergence theorem. We can show the converse by observing that, if , then
and by the limit comparison test it follows that the two series
are equivalent meaning that either they both converge or they both diverge.
The same proof also shows that if for some , then converges to a non-zero number if and only if converges.
If the series diverges to , then the sequence of partial products of the an converges to zero. The infinite product is said to diverge to zero.
For the case where the have arbitrary signs, the convergence of the sum does not guarantee the convergence of the product . For example, if , then converges, but diverges to zero. However, if is convergent, then the product converges absolutely–that is, the factors may be rearranged in any order without altering either the convergence, or the limiting value, of the infinite product. Also, if is convergent, then the sum and the product are either both convergent, or both divergent.
Product representations of functions
One important result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire function |
https://en.wikipedia.org/wiki/Amida%20%28Mesopotamia%29 | {
"type": "FeatureCollection",
"features": [
{
"type": "Feature",
"properties": {},
"geometry": {
"type": "Point",
"coordinates": [ 40.253906, 37.951385 ]
}
}
]
}Amida (, , ) was an ancient city in Mesopotamia located where modern Diyarbakır, Turkey now stands.
The city was located on the right bank of the Tigris. The walls are lofty and substantial, and constructed of the recycled stones from older buildings.
History
Amid(a) was established as an Aramean settlement, circa the 3rd millennium BC, later as the capital of Bit-Zamani. The oldest artefact from Amida is the famous stele of king Naram-Sin also believed to be from third millennia BC. The name Amida first appears in the writings of Assyrian King Adad-nirari I (C. 1310 -1281 BC) who ruled the city as a part of the Assyrian homeland. Amida remained an important region of the Assyrian homeland throughout the reign of king Tiglath-Pileser I (1114–1076 BC) and the name Amida appeared in the annals of Assyrian rulers until 705 BC, and also appears in the archives of Armenian king Tiridates II in 305 AD, and the Roman historian Ammianus Marcellinus (325–391 AD).
It was enlarged and strengthened by Constantius II, in whose reign it was besieged and taken after seventy-three days by the Sassanid king Shapur II (359). The Roman soldiers and a large part of the population of the town were massacred by the Persians. The historian Ammianus Marcellinus, who took part in the defence of the town, has given a minute account of the siege. The Persians did not attempt to garrison the city after the siege.
Amida was besieged by the Sassanid king Kavadh I during the Anastasian War through the autumn and winter (502-503). The siege of the city proved to be a far more difficult enterprise than Kavadh expected; the defenders, although unsupported by troops, repelled the Sassanid assaults for three months before they were finally beaten. Part of the prisoners of Amida were deported to Arrajan, a city refounded by Kavad I, who then named it "Weh-az-Amid-Kawad" (literally, "better than Amida, Kavad [built this]". During that same war, the Romans attempted an ultimately unsuccessful siege of the Persian-held Amida, led by generals Patricius and Hypatius. In 504, however, the Byzantines reconquered the city, and Justinian I repaired its walls and fortifications.
The Sassanids captured the city for a third time in 602 and held it for more than twenty years. In 628 the Roman emperor Heraclius recovered Amida.
Finally, in 639 the city was captured by the Arab armies of Islam and it remained in Arab hands until the Kurdish dynasty of the Marwanids ruled the area during the 10th and 11th centuries.
In 1085, the Seljuq Turks captured the region from the Marwanids, and they settled many Turcomans in the region. However, the Ayyubids received the city from their vassal State the Artiquids in 1232, and the city ruled by them until the Mongolian Ilkhanate captured t |
https://en.wikipedia.org/wiki/Arcsine%20laws%20%28Wiener%20process%29 | In probability theory, the arcsine laws are a collection of results for one-dimensional random walks and Brownian motion (the Wiener process). The best known of these is attributed to .
All three laws relate path properties of the Wiener process to the arcsine distribution. A random variable X on [0,1] is arcsine-distributed if
Statement of the laws
Throughout we suppose that (Wt)0 ≤ t ≤ 1 ∈ R is the one-dimensional Wiener process on [0,1]. Scale invariance ensures that the results can be generalised to Wiener processes run for t ∈[0,∞).
First (Lévy's) arcsine law
The first arcsine law states that the proportion of time that the one-dimensional Wiener process is positive follows an arcsine distribution. Let
be the measure of the set of times in [0,1] at which the Wiener process is positive. Then is arcsine distributed.
Second arcsine law
The second arcsine law describes the distribution of the last time the Wiener process changes sign. Let
be the time of the last zero. Then L is arcsine distributed.
Third arcsine law
The third arcsine law states that the time at which a Wiener process achieves its maximum is arcsine distributed.
The statement of the law relies on the fact that the Wiener process has an almost surely unique maxima, and so we can define the random variable M which is the time at which the maxima is achieved. i.e. the unique M such that
Then M is arcsine distributed.
Equivalence of the second and third laws
Defining the running maximum process Mt of the Wiener process
then the law of Xt = Mt − Wt has the same law as a reflected Wiener process |Bt| (where Bt is a Wiener process independent of Wt).
Since the zeros of B and |B| coincide, the last zero of X has the same distribution as L, the last zero of the Wiener process. The last zero of X occurs exactly when W achieves its maximum. It follows that the second and third laws are equivalent.
Notes
References
Wiener process
Statistical mechanics |
https://en.wikipedia.org/wiki/Erd%C5%91s%20arcsine%20law | In number theory, the Erdős arcsine law, named after Paul Erdős in 1969, states that the prime divisors of a number have a distribution related to the arcsine distribution.
Specifically, say that the jth prime factor p of a given number n (in the sorted sequence of distinct prime factors) is "small" when .
Then, for any fixed parameter u, in the limit as x goes to infinity, the proportion of the integers n less than x that have fewer than small prime factors converges to
References
Number theory |
https://en.wikipedia.org/wiki/Arcsine%20distribution | In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root:
for 0 ≤ x ≤ 1, and whose probability density function is
on (0, 1). The standard arcsine distribution is a special case of the beta distribution with α = β = 1/2. That is, if is an arcsine-distributed random variable, then . By extension, the arcsine distribution is a special case of the Pearson type I distribution.
The arcsine distribution appears in the Lévy arcsine law, in the Erdős arcsine law, and as the Jeffreys prior for the probability of success of a Bernoulli trial.
Generalization
Arbitrary bounded support
The distribution can be expanded to include any bounded support from a ≤ x ≤ b by a simple transformation
for a ≤ x ≤ b, and whose probability density function is
on (a, b).
Shape factor
The generalized standard arcsine distribution on (0,1) with probability density function
is also a special case of the beta distribution with parameters .
Note that when the general arcsine distribution reduces to the standard distribution listed above.
Properties
Arcsine distribution is closed under translation and scaling by a positive factor
If
The square of an arcsine distribution over (-1, 1) has arcsine distribution over (0, 1)
If
The coordinates of points uniformly selected on a circle of radius centered at the origin (0, 0), have an distribution
For example, if we select a point uniformly on the circumference, , we have that the point's x coordinate distribution is , and its y coordinate distribution is
Characteristic function
The characteristic function of the arcsine distribution is a confluent hypergeometric function and given as .
Related distributions
If U and V are i.i.d uniform (−π,π) random variables, then , , , and all have an distribution.
If is the generalized arcsine distribution with shape parameter supported on the finite interval [a,b] then
If X ~ Cauchy(0, 1) then has a standard arcsine distribution
References
Further reading
Continuous distributions |
https://en.wikipedia.org/wiki/Foundations%20of%20Algebraic%20Geometry | Foundations of Algebraic Geometry is a book by that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two subvarieties.
Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over a finite field.
Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.
In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by Grothendieck's development of schemes.
References
External links
Extracts from the preface of Foundations of Algebraic Geometry
1946 non-fiction books
Algebraic geometry
Mathematics books
History of mathematics |
https://en.wikipedia.org/wiki/Three%20subgroups%20lemma | In mathematics, more specifically group theory, the three subgroups lemma is a result concerning commutators. It is a consequence of Philip Hall and Ernst Witt's eponymous identity.
Notation
In what follows, the following notation will be employed:
If H and K are subgroups of a group G, the commutator of H and K, denoted by [H, K], is defined as the subgroup of G generated by commutators between elements in the two subgroups. If L is a third subgroup, the convention that [H,K,L] = [[H,K],L] will be followed.
If x and y are elements of a group G, the conjugate of x by y will be denoted by .
If H is a subgroup of a group G, then the centralizer of H in G will be denoted by CG(H).
Statement
Let X, Y and Z be subgroups of a group G, and assume
and
Then .
More generally, for a normal subgroup of , if and , then .
Proof and the Hall–Witt identity
Hall–Witt identity
If , then
Proof of the three subgroups lemma
Let , , and . Then , and by the Hall–Witt identity above, it follows that and so . Therefore, for all and . Since these elements generate , we conclude that and hence .
See also
Commutator
Lower central series
Grün's lemma
Jacobi identity
Notes
References
Lemmas in group theory
Articles containing proofs |
https://en.wikipedia.org/wiki/Argentina%20national%20football%20team%20records%20and%20statistics | This is a list of statistical records for the Argentina national football team.
Individual records
Players in bold are still active with Argentina.
Most capped players
Top goalscorers
World Cup winning captains
Goals
Most goals scored: 106, Lionel Messi, 2005–
Oldest goalscorer: Martín Palermo, 36 years and 7 months old in 2010 against Greece
Youngest goalscorer: Diego Maradona, 18 years, 284 days old in 1979 against Scotland
Youngest player to score in a FIFA World Cup match: Lionel Messi 18 years and 357 days, against Serbia and Montenegro in 2006
Most goals scored in a single match: 5 – Manuel Seoane in 1925, Juan Marvezzi in 1941, Lionel Messi in 2022
Most goals scored in a calendar year: 18 – Lionel Messi (2022)
Manager records
Most appearances
Guillermo Stábile: 127 Guillermo coached Argentina in 123 matches which made him among the few coaches who were in charge of more than 100 international matches. While still with the national team, he led them to victories in the South American Championship in 1941, 1945, 1946, 1947, 1955, and 1957.
Competition records
FIFA World Cup
*Draws include knockout matches decided via penalty shoot-out.
Copa América
FIFA Confederations Cup
Olympic Games
Pan American Games
Head-to-head record
This is a list of the official games played by the Argentina national team. The team has played a number of countries around the world, some repeatedly, although it has played the most games (201) against neighboring Uruguay.
Updated to 17 October 2023 after the match against .
See also
Argentina–Brazil football rivalry
Argentina–Uruguay football rivalry
Argentina–England football rivalry
Notes
References
Argentina national football team records and statistics
National association football team records and statistics |
https://en.wikipedia.org/wiki/Circular%20uniform%20distribution | In probability theory and directional statistics, a circular uniform distribution is a probability distribution on the unit circle whose density is uniform for all angles.
Description
Definition
The probability density function (pdf) of the circular uniform distribution, e.g. with , is:
Moments with respect to a parametrization
We consider the circular variable with at base angle . In these terms, the circular moments of the circular uniform distribution are all zero, except for :
where is the Kronecker delta symbol.
Descriptive statistics
Here the mean angle is undefined, and the length of the mean resultant is zero.
Distribution of the mean
The sample mean of a set of N measurements drawn from a circular uniform distribution is defined as:
where the average sine and cosine are:
and the average resultant length is:
and the mean angle is:
The sample mean for the circular uniform distribution will be concentrated about zero, becoming more concentrated as N increases. The distribution of the sample mean for the uniform distribution is given by:
where consists of intervals of in the variables, subject to the constraint that and are constant, or, alternatively, that and are constant. The distribution of the angle is uniform
and the distribution of is given by:
where is the Bessel function of order zero. There is no known general analytic solution for the above integral, and it is difficult to evaluate due to the large number of oscillations in the integrand. A 10,000 point Monte Carlo simulation of the distribution of the mean for N=3 is shown in the figure.
For certain special cases, the above integral can be evaluated:
For large N, the distribution of the mean can be determined from the central limit theorem for directional statistics. Since the angles are uniformly distributed, the individual sines and cosines of the angles will be distributed as:
where or . It follows that they will have zero mean and a variance of 1/2. By the central limit theorem, in the limit of large N, and , being the sum of a large number of i.i.d's, will be normally distributed with mean zero and variance . The mean resultant length , being the square root of the sum of squares of two normally distributed independent variables, will be Chi-distributed with two degrees of freedom (i.e.Rayleigh-distributed) and variance :
Entropy
The differential information entropy of the uniform distribution is simply
where is any interval of length . This is the maximum entropy any circular distribution may have.
See also
Wrapped distribution
References
Continuous distributions
Directional statistics |
https://en.wikipedia.org/wiki/Wrapped%20L%C3%A9vy%20distribution | In probability theory and directional statistics, a wrapped Lévy distribution is a wrapped probability distribution that results from the "wrapping" of the Lévy distribution around the unit circle.
Description
The pdf of the wrapped Lévy distribution is
where the value of the summand is taken to be zero when , is the scale factor and is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:
In terms of the circular variable the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:
where is some interval of length . The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:
The mean angle is
and the length of the mean resultant is
See also
Wrapped distribution
Directional statistics
References
Continuous distributions
Directional statistics |
https://en.wikipedia.org/wiki/Korn%C3%A9l%20Kulcs%C3%A1r | Kornél Kulcsár (born 11 November 1991) is a Hungarian football player who plays for Romanian club Târgu Mureș.
Club statistics
Updated to games played as of 15 May 2021.
References
Player profile at HLSZ
1991 births
People from Nagyatád
Footballers from Somogy County
Living people
Hungarian men's footballers
Men's association football midfielders
Kaposvári Rákóczi FC players
Szombathelyi Haladás footballers
Zalaegerszegi TE players
Pápai FC footballers
Mezőkövesdi SE footballers
Kisvárda FC players
Győri ETO FC players
Soproni VSE players
Mosonmagyaróvári TE footballers
Budafoki MTE footballers
Kozármisleny SE footballers
ASA Târgu Mureș (1962) players
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
Hungarian expatriate men's footballers
Expatriate men's footballers in Romania
Hungarian expatriate sportspeople in Romania |
https://en.wikipedia.org/wiki/Graphlets | Graphlets in mathematics are induced subgraph isomorphism classes in a graph, i.e. two graphlet occurrences are isomorphic, whereas two graphlets are non-isomorphic. Graphlets differ from network motifs in a statistical sense, network motifs are defined as over- or under-represented graphlets with respect to some random graph null model.
Graphlet-based network properties
Relative graphlet frequency distance
RGF-distance compares the frequencies of the appearance of all 3-5-node graphlets in two networks. Let Ni(G) be the number of graphlets of type () in network G, and let be the total number of graphlets of G. The "similarity" between two graphs should be independent of the total number of nodes or edges, and should depend only upon the differences between relative frequencies of graphlets. Thus, relative graphlet frequency distance D(G,H) between two graphs G and H is defined as:
,
where . The logarithm of the graphlet frequency is used because frequencies of different graphlets can differ by several orders of magnitude and the distance measure should not be entirely dominated by the most frequent graphlets.
Graphlet degree distribution agreement
GDD-agreement generalizes the notion of the degree distribution to the spectrum of graphlet degree distributions (GDDs) in the following way. The degree distribution measures the number of nodes of degree k in graph G, i.e., the number of nodes "touching" k edges, for each value of k. Note that an edge is the only graphlet with two nodes. GDDs generalize the degree distribution to other graphlets: they measure for each 2-5-node graphlet Gi, , such as a triangle or a square, the number of nodes "touching" k graphlets Gi at a particular node. A node at which a graphlet is "touched" is topologically relevant, since it allows us to distinguish between nodes "touching", for example, a three node path at an end node or at the middle node. This is summarized by automorphism orbits (or just orbits, for brevity): by taking into account the "symmetries" between nodes of a graphlet, there are 73 different orbits across all 2-5-node graphlets (see [Pržulj, 2007] for details).
For each orbit j, one needs to measure the jth GDD, dGj(k), i.e., the distribution of the number of nodes in G "touching" the corresponding graphlet at orbit j k times. Clearly, the degree distribution is the 0th GDD. dGj(k) is scaled as
to decrease the contribution of larger degrees in a GDD and then normalized with respect to its total area
giving the "normalized distribution"
.
The jth GDD-agreement compares the jth GDDs of two networks.
For two networks G and H and a particular orbit j, the "distance" Dj(G,H) between their normalized jth GDDs is:
.
The distance is between 0 and 1, where 0 means that G and H have identical jth GDDs, and 1 means that their jth GDDs
are far away. Next, Dj(G,H) is reversed to obtain the jth GDD-agreement:
, for .
The total GDD-agreement between two networks G and H is the arithmetic or t |
https://en.wikipedia.org/wiki/Graph%20Aligner | Graph Aligner (GRAAL) is an algorithm for global network alignment that is based solely on network topology. It aligns two networks and by producing an alignment that consists of a set of ordered pairs , where is a node in and is a node in . GRAAL matches pairs of nodes originating in different networks based on their graphlet degree signature similarities, where a higher similarity between two nodes corresponds to a higher topological similarity between their extended neighborhoods (out to distance 4). GRAAL produces global alignments, i.e., it aligns each node in the smaller network to exactly one node in the larger network. The matching proceeds using a technique analogous to the "seed and extend" approach of the popular BLAST algorithm for sequence alignment: it first chooses a single "seed" pair of nodes (one node from each network) with high graphlet degree signature similarity. It then expands the alignment radially outward around the seed as far as practical using a greedy algorithm (see [Kuchaiev et al., 2010] for details).
Method
When aligning two graphs and , GRAAL first computes costs of aligning each node in G with each node in . The cost of aligning two nodes takes into account the graphlet degree signature similarity between them, modified to reduce the cost as the degrees of both nodes increase, since higher degree nodes with similar signatures provide a tighter constraint than correspondingly similar low degree nodes. In this way, GRAAL align the densest parts of the networks first. Let be the degree of a node in network , let be the maximum degree of nodes in , let be the graphlet degree signature similarity of nodes and , and let be a parameter in [0, 1] that controls the contribution of the node signature similarity to the cost function (that is, is the parameter that controls the contribution of node degrees to the cost function), then the cost of aligning nodes and is computed as:
.
A cost of corresponds to a pair of topologically identical nodes and , while a cost close to corresponds to a pair of topologically different nodes.
GRAAL chooses as the initial seed a pair of nodes , and , that have the smallest cost. Ties are broken randomly. Once the seed is found, GRAAL builds "spheres" of all possible radii around nodes and . A sphere of radius around node is the set of nodes that are at distance from , where the distance is the length of the shortest path from to . Spheres of the same radius in two networks are then greedily aligned together by searching for the pairs and that are not already aligned and that can be aligned with the minimal cost. When all spheres around the seed have been aligned, some nodes in both networks may remain unaligned. For this reason, GRAAL repeats the same algorithm on a pair of networks for and , and searches for the new seed again, if necessary. Network is defined as a new network having the same set of nodes as and having if and only if the dista |
https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20Horv%C3%A1th%20%28footballer%2C%20born%201988%29 | László Horváth (born 23 February 1988) is a Hungarian football player who currently plays for Kazincbarcikai SC.
Club statistics
Updated to games played as of 19 May 2019.
References
Player profile at HLSZ
1988 births
Living people
Footballers from Kaposvár
Hungarian men's footballers
Men's association football goalkeepers
Kaposvári Rákóczi FC players
Kaposvölgye VSC footballers
Szigetszentmiklósi TK footballers
Balmazújvárosi FC players
MTK Budapest FC players
Kazincbarcikai SC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players
21st-century Hungarian people |
https://en.wikipedia.org/wiki/Colin%20McLarty | Colin McLarty (born July 12, 1951) is an American logician whose publications have ranged widely in philosophy and the foundations of mathematics, as well as in the history of science and of mathematics.
Research
Category theory
He has written papers about Saunders Mac Lane, one of the founders of category theory.
McLarty's Elementary Categories and Elementary Toposes describes category theory and topos theory at an elementary level.
McLarty worked on establishing that Fermat's Last Theorem can be proven in a setting with much weaker assumptions than the ones used in Wiles' proof, which makes use of involved category theoretical constructions.
History of Mathematics
He is a member of the Grothendieck Circle, which provides on-line and open access to many writings about the mathematician Alexandre Grothendieck, who revolutionized Banach-space theory and algebraic geometry and whose life has fascinated many biographers and mathematical scientists.
McLarty has also written about the German algebraist Emmy Noether and her involvement with German political history.
Positions
At Case Western Reserve University (CWRU), Colin McLarty is the Truman P. Handy Professor of Philosophy and the current Chair of the Philosophy Department.
At CWRU, he is also a professor of mathematics.
Bibliography
Colin McLarty, The Uses and Abuses of the History of Topos Theory, Br. J. Philos. Sci, 41 (1990) p 355.
Colin McLarty, 'Emmy Noether’s ‘Set Theoretic’ Topology: From Dedekind to the rise of functors' in The Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) p. 187–208.
McLarty, Colin, 2005, "Saunders Mac Lane (1909–2005): His Mathematical Life and Philosophical Works," Philosophia Mathematica 13: 237–51. With selected bibliography emphasizing Mac Lane's philosophical writings.
--------, 2007, "The Last Mathematician from Hilbert's Göttingen: Saunders Mac Lane as Philosopher of Mathematics,", British Journal for the Philosophy of Science 58(1): 77–112.
See also
"Abstract nonsense", a (usually ironic) term for category theory and category-theoretic arguments in homological algebra
"Generalized abstract nonsense", an ironic and affectionate term for topos theory
Other topos theorists (besides Saunders Mac Lane):
Martin Hyland
Peter Johnstone
William Lawvere
References
External links
Personal website of Colin McLarty
CWRU website of Colin McLarty
Grothendieck Circle, which lists McLarty as a member. (A collection of mathematical and biographical information, photos, links about Alexander Grothendieck.)
Category theorists
Mathematical logicians
American logicians
Philosophers of mathematics
American historians of mathematics
Philosophers of science
Historians of science
Intellectual historians
Algebraists
Algebraic geometers
20th-century American mathematicians
21st-century American mathematicians
20th-century American philosophers
Living people
|
https://en.wikipedia.org/wiki/F-test%20of%20equality%20of%20variances | In statistics, an F-test of equality of variances is a test for the null hypothesis that two normal populations have the same variance.
Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the test statistic used is the ratio of two sample variances. This particular situation is of importance in mathematical statistics since it provides a basic exemplar case in which the F-distribution can be derived. For application in applied statistics, there is concern that the test is so sensitive to the assumption of normality that it would be inadvisable to use it as a routine test for the equality of variances. In other words, this is a case where "approximate normality" (which in similar contexts would often be justified using the central limit theorem), is not good enough to make the test procedure approximately valid to an acceptable degree.
The test
Let X1, ..., Xn and Y1, ..., Ym be independent and identically distributed samples from two populations which each has a normal distribution. The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let
be the sample means. Let
be the sample variances. Then the test statistic
has an F-distribution with n − 1 and m − 1 degrees of freedom if the null hypothesis of equality of variances is true. Otherwise it follows an F-distribution scaled by the ratio of true variances. The null hypothesis is rejected if F is either too large or too small based on the desired alpha level (i.e., statistical significance).
Properties
This F-test is known to be extremely sensitive to non-normality, so Levene's test, Bartlett's test, or the Brown–Forsythe test are better tests for testing the equality of two variances. (However, all of these tests create experiment-wise type I error inflations when conducted as a test of the assumption of homoscedasticity prior to a test of effects.) F-tests for the equality of variances can be used in practice, with care, particularly where a quick check is required, and subject to associated diagnostic checking: practical text-books suggest both graphical and formal checks of the assumption.
F-tests are used for other statistical tests of hypotheses, such as testing for differences in means in three or more groups, or in factorial layouts. These F-tests are generally not robust when there are violations of the assumption that each population follows the normal distribution, particularly for small alpha levels and unbalanced layouts. However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the F-test is relatively robust, although (if the normality assumption does not hold) it suffers from a loss in comparative statistical power as compared with non-parametric counterparts.
Generalization
The immediate generalization of the problem outlined above is to situations where there |
https://en.wikipedia.org/wiki/ImaGem%20Inc. | ImaGem Inc. is a gem information company based on patented technology for grading and identifying gems. The parent company, ImageStatistics, offers applications in imaging and image processing for the medical, pharmaceutical, manufacturing, chemical and quality assurance industries. ImaGem was created after 20 years of research done on human perception and ImageStatistics, a statistical tool developed by Dr. Lalit K. Aggarwal. ImaGem’s Systems employ advanced optics, imaging technology and proprietary software programs to automate and integrate diamond grading. In addition to evaluating the 4C’s of diamond grading, ImaGem also analyzes Light behavior information based on precise and repeatable measures; three complementary dimensions of brilliance, intensity and sparkle. ImaGem has created a system of uniquely identifying and registering (fingerprinting) a stone without any need for laser inscription. By offering all this information, ImaGem has promoted decision-making and efficiency in the gem industry supply chain. In 1998, ImaGem Inc. was incorporated in the state of Pennsylvania, USA after funding from a major retailer, Ben Franklin Technology Partnership, and private funding. The company has developed an integrated technology to collect image data for diamonds and gemstones, analyze it using direct measurement methodology and grade for precise and repeatable measurements.
Breakthroughs and technological advances
Color
Color grading of diamonds (see Diamond color) and gemstones is subjective - an individual compares a subject stone under a light source to a master stone and assigns a color grade. This approach is well-known to provide a color grade that is not repeatable because, in the past, there has been confusion among different methods of grading used in various labs. Because of this, ImaGem solved several problems which contribute to a lack of precision and repeatability in color grading.
The human perceptual system is not capable of precise color memory. Perception of color by an individual can vary during different times of the day, by environmental factors, and by eye color.
Light sources are not standardized either for their color temperature or their UV component.
Master Stones used in grading are generally of a particular weight at a grading lab. Different labs may use master stones of different weights.
All these factors therefore, in varying degrees, contribute to inconsistent color grading as it has been practiced in previous years.
ImaGem’s color grading system was designed based on measurements with a system error of 1/400 of 1% to achieve high levels of precision and repeatability. The system assigns grades within 1/3 of the color grading system used by major diamond grading labs. The color grading system corrects for weight and fluorescence and uses CIE Chromaticity System (see CIE 1931 color space).
Fluorescence
The most frequent color of fluorescence in diamonds is blue, but it occurs in many other colors. Flu |
https://en.wikipedia.org/wiki/Uncertainty%20principle%20%28disambiguation%29 | Heisenberg's uncertainty principle is a fundamental concept in quantum physics.
Uncertainty principle may also refer to:
Fourier uncertainty principle, a concept in mathematics akin to Heisenberg's uncertainty principle
Küpfmüller's uncertainty principle, a concept in electronic engineering formulated by Karl Küpfmüller
The Uncertainty Principle (film), a 2002 Portuguese drama
The Uncertainty Principle (audio drama), a production based on the British television series Doctor Who
"The Uncertainty Principle", a season 1 episode of the TV series Joan of Arcadia
"Uncertainty Principle" (Numbers), a 2005 episode of the TV series Numbers
"The Uncertainty Principle" (The Spectacular Spider-Man), a 2008 episode of the TV series The Spectacular Spider-Man
"The Uncertainty Principle", a 2009 episode of the TV series Holby City
The Uncertainty Principle, a 1978 novel by Dmitri Bilenkin
Uncertainty Principle, a 2003 novel by Gregorio Morales
Uncertainty Principle, a British drone/funeral doom metal one-man band
See also
Uncertainty, a situation involving ambiguous or unknown information |
https://en.wikipedia.org/wiki/Gilles%20Pisier | Gilles I. Pisier (born 18 November 1950) is a professor of mathematics at the Pierre and Marie Curie University and a distinguished professor and A.G. and M.E. Owen Chair of Mathematics at the Texas A&M University. He is known for his contributions to several fields of mathematics, including functional analysis, probability theory, harmonic analysis, and operator theory. He has also made fundamental contributions to the theory of C*-algebras. Gilles is the younger brother of French actress Marie-France Pisier.
Research
Pisier has obtained many fundamental results in various parts of mathematical analysis.
Geometry of Banach spaces
In the "local theory of Banach spaces", Pisier and Bernard Maurey developed the theory of Rademacher type, following its use in probability theory by J. Hoffman–Jorgensen and in the characterization of Hilbert spaces among Banach spaces by S. Kwapień. Using probability in vector spaces, Pisier proved that super-reflexive Banach spaces can be renormed with the modulus of uniform convexity having "power type". His work (with Per Enflo and Joram Lindenstrauss) on the "three–space problem" influenced the work on quasi–normed spaces by Nigel Kalton.
Operator theory
Pisier transformed the area of operator spaces. In the 1990s, he solved two long-standing open problems. In the theory of C*-algebras, he solved, jointly with Marius Junge, the problem of the uniqueness of C* -norms on the tensor product of two copies of B(H), the bounded linear operators on a Hilbert space H. He and Junge were able to produce two such tensor norms that are nonequivalent. In 1997, he constructed an operator that was polynomially bounded but not similar to a contraction, answering a famous question of Paul Halmos.
Awards
He was an invited speaker at the 1983 ICM and
a plenary speaker at the 1998 ICM. In 1997, Pisier received the Ostrowski Prize for this work. He is also a recipient of the Grands Prix de l'Académie des Sciences de Paris in 1992 and the Salem Prize in 1979. In 2012 he became a fellow of the American Mathematical Society.
Books
Pisier has authored several books and monographs in the fields of functional analysis, harmonic analysis, and operator theory. Among them are:
See also
Pisier–Ringrose inequality
References
External links
1950 births
Living people
20th-century French mathematicians
21st-century French mathematicians
Academic staff of the University of Paris
École Polytechnique alumni
Members of the French Academy of Sciences
Texas A&M University faculty
Fellows of the American Mathematical Society
Foreign Fellows of the Indian National Science Academy |
https://en.wikipedia.org/wiki/Kristin%20Lauter | Kristin Estella Lauter (born 1969) is an American mathematician and cryptographer whose research interest is broadly in application of number theory and algebraic geometry in cryptography. She is particularly known for her work in the area of elliptic curve cryptography. She was a researcher at Microsoft Research in Redmond, Washington, from 1999–2021 and the head of the Cryptography Group from 2008–2021; her group developed Microsoft SEAL. In April 2021, Lauter joined Facebook AI Research (FAIR) as the West Coast Head of Research Science. She became the President-Elect of the Association for Women in Mathematics in February 2014 and served as President February 1, 2015 - January 31, 2017.
Education and career
Lauter received her Bachelor of Arts, Master of Science, and Ph.D, all in mathematics, from the University of Chicago, in 1990, 1991, and 1996, respectively.
Prior to joining Microsoft, she held positions as a visiting scholar at Max Planck Institut fur Mathematik in Bonn, Germany (1997), T.H. Hildebrandt research assistant professor at the University of Michigan (1996-1999), and a visiting researcher at Institut de Mathematiques Luminy in France (1999).
In 1999, Lauter joined Microsoft to work on her research in cryptography. She has worked on developing new cryptographic systems, research on post quantum systems, and researching to find faults in current cryptographic systems.
In 2005, she worked with coworkers at Microsoft to develop a cryptographic algorithm from supersingular isogeny graphs. She created a HASH function from it and presented it at the NIST hash function competition.
Dr. Kristin Lauter is also known for her work in homomorphic encryption, which has been commonly used in machine-learning, building mathematical models, private AI, and the collection of genomic data. She has also worked on encryption with the Cloud. She has given many tutorials on homomorphic encryption for broad audiences. A specific tutorial on homomorphic encryption allowed her to meet some iDASH organizers to whom she explained her encryption techniques, such as edit distance and chi-squared statistics.
Beginning in April 2021, Lauter joined Facebook AI Research as West Coast Head of Research Science, leading the Seattle and Menlo Park Labs with groups in Core Machine Learning, Computer Vision, Robotics, Natural Language Processing, and other areas.
Service
She is a co-founder of the Women in Numbers Network, a research collaboration community for women in number theory. The first conference was held in 2008 and it was called the WIN Conference. Groups of number theorists work on research and have published about 50 papers. She served on the Advisory Board of the Banff International Research Station and on the Council of the American Mathematical Society. She served as president of the Association for Women in Mathematics 2015–2017. From 2014 to 2016, Lauter served as a Council Member at Large for the American Mathematical Society. She cu |
https://en.wikipedia.org/wiki/Adi%20Nimni | Adi Nimni (born 27 August 1991) is an Israeli footballer.
Honours
Liga Leumit:
Runner-up (1): 2012-13
Toto Cup (Leumit):
Runner-up (1): 2012–13
Club career statistics
(correct as of April 2013)
External links
Profile at One
1991 births
Living people
Israeli Jews
Israeli men's footballers
Maccabi Netanya F.C. players
Hapoel Ra'anana A.F.C. players
Hapoel Ramat Gan Givatayim F.C. players
Hapoel Rishon LeZion F.C. players
F.C. Tira players
Maccabi Yavne F.C. players
Israeli Premier League players
Liga Leumit players
Israeli people of Libyan-Jewish descent
Footballers from Central District (Israel)
Men's association football defenders |
https://en.wikipedia.org/wiki/Oddness | Oddness may refer to:
Eccentricity (behavior)
Oddness of numbers, for which see parity (mathematics) |
https://en.wikipedia.org/wiki/Andrei%20Scobioal%C4%83 | Andrei Scobioală (1884–1971) was a Bessarabian politician, a professor of mathematics and a deputy in the Country Council between 1917-1919 and in the Parliament of Romania between 1928 - 1930.
Biography
He studied at the Normal School in Bairancea and the Odessa Physics Faculty, taking his degree in 1909. It was mobilized for weapons during the First World War, more precisely in 1917, in a Russian unit on the Romanian front. Besides working on the front, he sent a letter to Gubernial Zemstvo at the beginning of the school year 1917–1918 in which he proposed to bring Transylvanian and Bucovina teachers from the prisoners and the refugees of war because he was worried about the situation of the native language education in Bessarabia, given the small number of Bessarabian teachers. On 18 February 1918 he participated in the opening of the Popular University of Chișinău, where he gave a speech.
Besides the political activity carried out in some Bessarabian and Romanian bodies and institutions, during the interwar period he was also a mathematics professor at the Normal School and the Military High School in Chișinău, as well as at the Lazăr High School in Bucharest (in 1940). In 1941 he returned to Chișinău, where he worked until 1944 as a teacher at the Normal School, after which he moved definitively to Bucharest. He died in 1971 because of a heart attack. He was buried at the Cărămidari cemetery in Bucharest (South Square).
Gallery
Bibliography
Gheorghe E. Cojocaru, Sfatul Țării: itinerar, Civitas, Chişinău, 1998,
Mihai Taşcă, Sfatul Țării şi actualele autorităţi locale, "Timpul de dimineaţă", no. 114 (849), June 27, 2008 (page 16)
External links
Arhiva pentru Sfatul Tarii
Deputaţii Sfatului Ţării şi Lavrenti Beria
References
1884 births
1971 deaths
Moldovan MPs 1917–1918
People from Ungheni District |
https://en.wikipedia.org/wiki/Future%20of%20mathematics | The progression of both the nature of mathematics and individual mathematical problems into the future is a widely debated topic; many past predictions about modern mathematics have been misplaced or completely false, so there is reason to believe that many predictions today will follow a similar path. However, the subject still carries an important weight and has been written about by many notable mathematicians. Typically, they are motivated by a desire to set a research agenda to direct efforts to specific problems, or a wish to clarify, update and extrapolate the way that subdisciplines relate to the general discipline of mathematics and its possibilities. Examples of agendas pushing for progress in specific areas in the future, historical and recent, include Felix Klein's Erlangen program, Hilbert's problems, Langlands program, and the Millennium Prize Problems. In the Mathematics Subject Classification section 01Axx History of mathematics and mathematicians, subsection 01A67 is titled Future prospectives.
The accuracy of predictions about mathematics has varied widely and has proceeded very closely to that of technology. As such, it is important to keep in mind that many of the predictions by researchers below may be misguided or turn out to be untrue.
Motivations and methodology for speculation
According to Henri Poincaré writing in 1908 (English translation), "The true method of forecasting the future of mathematics lies in the study of its history and its present state".
The historical approach can consist of the study of earlier predictions, and comparing them to the present state of the art to see how the predictions have fared, e.g. monitoring the progress of Hilbert's problems. A subject survey of mathematics itself however is now problematic: the sheer expansion of the subject gives rise to issues of mathematical knowledge management.
The development of technology has also significantly impacted the outcomes of many predictions; because of the uncertain nature of the future of technology, this leads to quite a bit of uncertainty in the future of mathematics. Also entailed by this is that successful predictions about future technology may also result in successful mathematical predictions.
Given the support of research by governments and other funding bodies, concerns about the future form part of the rationale of the distribution of funding. Mathematical education must also consider changes that are happening in the mathematical requirements of the workplace; course design will be influenced both by current and by possible future areas of application of mathematics. László Lovász, in Trends in Mathematics: How they could Change Education? describes how the mathematics community and mathematical research activity is growing and states that this will mean changes in the way things are done: larger organisations mean more resources are spent on overheads (coordination and communication); in mathematics this would equate to mor |
https://en.wikipedia.org/wiki/Impossibility%20of%20a%20gambling%20system | The principle of the impossibility of a gambling system is a concept in probability. It states that in a random sequence, the methodical selection of subsequences does not change the probability of specific elements. The first mathematical demonstration is attributed to Richard von Mises (who used the term collective rather than sequence).
The principle states that no method for forming a subsequence of a random sequence (the gambling system) improves the odds for a specific event. For instance, a sequence of fair coin tosses produces equal and independent 50/50 chances for heads and tails. A simple system of betting on heads every 3rd, 7th, or 21st toss, etc., does not change the odds of winning in the long run. As a mathematical consequence of computability theory, more complicated betting strategies (such as a martingale) also cannot alter the odds in the long run.
Von Mises' mathematical demonstration defines an infinite sequence of zeros and ones as a random sequence if it is not biased by having the frequency stability property. With this property, the frequency of zeroes in the sequence stabilizes at 1/2, and every possible subsequence selected by any systematic method is likewise not biased.
The subsequence selection criterion is important, because although the sequence 0101010101... is not biased, selecting the odd positions results in 000000... which is not random. Von Mises did not fully define what constituted a "proper" selection rule for subsequences, but in 1940 Alonzo Church defined it as any recursive function which having read the first N elements of the sequence decides if it wants to select element number N+1. Church was a pioneer in the field of computable functions, and the definition he made relied on the Church Turing Thesis for computability.
In the mid-1960s, A. N. Kolmogorov and D. W. Loveland independently proposed a more permissive selection rule. In their view Church's recursive function definition was too restrictive in that it read the elements in order. Instead they proposed a rule based on a partially computable process which having read any N elements of the sequence, decides if it wants to select another element which has not been read yet.
The principle influenced modern concepts in randomness, e.g. the work by A. N. Kolmogorov in considering a finite sequence random (with respect to a class of computing systems) if any program that can generate the sequence is at least as long as the sequence itself.
See also
Gambler's ruin
History of randomness
No free lunch theorem
References
Probability theory |
https://en.wikipedia.org/wiki/Charles%20Whitworth%20%28MP%29 | Sir Charles Whitworth (c. 1721 – 22 August 1778) was a British politician who sat in the House of Commons for 31 years from 1747 to 1778. He was known for his expertise in statistics and finance.
Whitworth was the son of Francis Whitworth, Member of Parliament for Minehead from 1723 to 1742 and was educated at Westminster School and at Lincoln's Inn.
Whitworth represented the constituencies of Minehead from 1747 until 1761, and Bletchingley until 1768, in which year he was knighted. He then represented Minehead until 1774, East Looe until 1775 and Saltash until his death in 1778. He served in the army for a short time and was made lieutenant-governor of Gravesend and Tilbury for life in 1758. Whitworth was Chairman of Ways and Means from 1768 until his death.
The major contribution made by Sir Charles to the statistics of Great Britain consisted of the production of the first complete Balance of Trade (Visible trade) for the country together with individual commercial accounts of Great Britain and all known countries during the period 1697–1773, a major task he completed two years before his death.
Whitworth was an active early member of the Society for the Encouragement of Arts, Manufactures and Commerce in 1754.
He married Martha, the daughter of Richard Shelley, a commissioner of the stamp office. He had three sons and four daughters, and his heir was his eldest son, Charles Whitworth, 1st Earl Whitworth.
References
1720s births
1778 deaths
People educated at Westminster School, London
Members of Lincoln's Inn
Knights Bachelor
Members of the Parliament of Great Britain for constituencies in Cornwall
Members of the Parliament of Great Britain for English constituencies
British MPs 1747–1754
British MPs 1754–1761
British MPs 1761–1768
British MPs 1768–1774
British MPs 1774–1780 |
https://en.wikipedia.org/wiki/Novikov%27s%20compact%20leaf%20theorem | In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that
A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have a compact leaf.
Novikov's compact leaf theorem for S3
Theorem: A smooth codimension-one foliation of the 3-sphere S3 has a compact leaf. The leaf is a torus T2 bounding a solid torus with the Reeb foliation.
The theorem was proved by Sergei Novikov in 1964. Earlier Charles Ehresmann had conjectured that every smooth codimension-one foliation on S3 had a compact leaf, which was known to be true for all known examples; in particular, the Reeb foliation has a compact leaf that is T2.
Novikov's compact leaf theorem for any M3
In 1965, Novikov proved the compact leaf theorem for any M3:
Theorem: Let M3 be a closed 3-manifold with a smooth codimension-one foliation F. Suppose any of the following conditions is satisfied:
the fundamental group is finite,
the second homotopy group ,
there exists a leaf such that the map induced by inclusion has a non-trivial kernel.
Then F has a compact leaf of genus g ≤ 1.
In terms of covering spaces:
A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have a compact leaf.
References
S. Novikov. The topology of foliations//Trudy Moskov. Mat. Obshch, 1965, v. 14, p. 248–278.
I. Tamura. Topology of foliations — AMS, v.97, 2006.
D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds, Invent. Math., 36 (1976), p. 225–255.
Foliations
Theorems in topology |
https://en.wikipedia.org/wiki/Latin%20rectangle | In combinatorial mathematics, a Latin rectangle is an matrix (where ), using symbols, usually the numbers or as its entries, with no number occurring more than once in any row or column.
An Latin rectangle is called a Latin square. Latin rectangles and Latin squares may also be described as the optimal colorings of rook's graphs, or as optimal edge colorings of complete bipartite graphs.
An example of a 3 × 5 Latin rectangle is:
{| class="wikitable"
|-
|0|| 1|| 2|| 3|| 4
|-
|3|| 4 ||0 ||1|| 2
|-
|4|| 0|| 3|| 2|| 1
|}
Normalization
A Latin rectangle is called normalized (or reduced) if its first row is in natural order and so is its first column.
The example above is not normalized.
Enumeration
Let () denote the number of normalized × Latin rectangles. Then the total number of × Latin rectangles is
A 2 × Latin rectangle corresponds to a permutation with no fixed points. Such permutations have been called discordant permutations. An enumeration of permutations discordant with a given permutation is the famous problème des rencontres. The enumeration of permutations discordant with two permutations, one of which is a simple cyclic shift of the other, is known as the reduced problème des ménages.
The number of normalized Latin rectangles, , of small sizes is given by
{| class="wikitable" style="text-align:center;"
|-
! k\n !!1!!2!!3!!4!!5!!6!!7!!8
|-
! 1
|1||1||1||1||1||1||1||1
|-
! 2
| ||1||1||3||11||53||309||2119
|-
! 3
| || ||1||4||46||1064||35792||1673792
|-
! 4
| || || ||4||56||6552||1293216||420909504
|-
! 5
| || || || ||56||9408||11270400||27206658048
|-
! 6
| || || || || ||9408||16942080||335390189568
|-
! 7
| || || || || || ||16942080||535281401856
|-
! 8
| || || || || || || ||535281401856
|}
When = 1, that is, there is only one row, since the Latin rectangles are normalized there is no choice for what this row can be. The table also shows that , which follows since if only one row is missing, the missing entry in each column can be determined from the Latin square property and the rectangle can be uniquely extended to a Latin square.
Extendability
The property of being able to extend a Latin rectangle missing one row to a Latin square mentioned above, can be significantly strengthened. Namely, if , then it is possible to append rows to an Latin rectangle to form a Latin square, using Hall's marriage theorem.
Semi-Latin squares
A semi-Latin square is another type of incomplete Latin square. A semi-Latin square is an × array, , in which some positions are unoccupied and other positions are occupied by one of the integers }, such that, if an integer occurs in , then it occurs times and no two 's belong to the same row or column. If different integers occur in , then has index .
For example, a semi-Latin square of order 5 and index 3 is:
{| class="wikitable"
|-
| 1 || || 0 || || 2
|-
| || 2 || 1 || || 0
|-
| 0 || 1 || || 2 ||
|-
| 2 || 0 || || 1 ||
|-
| || || 2 || 0 || 1
|}
A semi-Lat |
https://en.wikipedia.org/wiki/Alfr%C3%A9d%20R%C3%A9nyi%20Prize | The Alfréd Rényi Prize is awarded biennially by the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Science in honor of founder Alfréd Rényi. By the current rules it is given to one or two fellows of the Institute in recognition of their outstanding performance in mathematics research of the previous five-year period. Members of the Hungarian Academy of Sciences and the director are not eligible.
Laureates
See also
List of mathematics awards
List of prizes named after people
References
Awards of the Hungarian Academy of Sciences
Awards established in 1972
External links
Official Website |
https://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3%20Pyber | László Pyber (born 8 May 1960 in Budapest) is a Hungarian mathematician. He is a researcher at the Alfréd Rényi Institute of Mathematics, Budapest. He works in combinatorics and group theory.
Biography
Pyber received his Ph.D. from the Hungarian Academy of Sciences in 1989 under the direction of László Lovász and Gyula O.H. Katona with the thesis Extremal Structures and Covering Problems.
In 2007, he was awarded the Academics Prize by the Hungarian Academy of Sciences.
In 2017, he was the recipient of an ERC Advanced Grant.
Mathematical contributions
Pyber has solved a number of conjectures in graph theory. In 1985, he proved the conjecture of Paul Erdős and Tibor Gallai that edges of a simple graph with n vertices can be covered with at most n-1 circuits and edges. In 1986, he proved the conjecture of Paul Erdős that a graph with n vertices and its complement can be covered with n2/4+2 cliques.
He has also contributed to the study of permutation groups. In 1993, he provided an upper bound for the order of a 2-transitive group of degree n not containing An avoiding the use of the classification of finite simple groups. Together with Tomasz Łuczak, Pyber proved the conjecture of McKay that for every ε>0, there is a constant C such that C randomly chosen elements invariably generate the symmetric group Sn with probability greater than 1-ε.
Pyber has made fundamental contributions in enumerating finite groups of a given order n. In 1993, he proved that if the prime power decomposition of n is n=p1g1 ⋯ pkgk and μ=max(g1,...,gk), then the number of groups of order n is at mostIn 2004, Pyber settled several questions in subgroup growth by completing the investigation of the spectrum of possible subgroup growth types.
In 2011, Pyber and Andrei Jaikin-Zapirain obtained a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability. They also explored related questions for profinite groups and settled several open problems.
In 2016, Pyber and Endre Szabó proved that in a finite simple group L of Lie type, a generating set A of L either grows, i.e., |A3| ≥ |A|1+ε for some ε depending only on the Lie rank of L, or A3=L. This implies that diameters of Cayley graphs of finite simple groups of bounded rank are polylogarithmic in the size of the group, partially resolving a well-known conjecture of László Babai.
References
External links
Pyber's home page.
Pyber's nomination for Hungarian Academy of Sciences membership
Combinatorialists
Group theorists
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Living people
1960 births |
https://en.wikipedia.org/wiki/KKC | KKC may refer to:
Kahn-Kalai conjecture, a mathematical problem in the field of probability
Karnataka Knowledge Commission, a Karnatakan governmental body
Kennedy–King College, a public two-year community college in Chicago, Illinois, United States
Khon Kaen Airport (IATA code: KKC), a Thai military/public airport
The Kingkiller Chronicle, a fantasy trilogy by Patrick Rothfuss
Klynveld Kraayenhof & Co., a Dutch accountancy firm
Kohn, Kohn & Colapinto, a Washington, D.C. law firm
Krispy Kreme Challenge, an annual charity event
Krispy Krunchy Chicken, an American chain of fried chicken fast food restaurants
Kulture Kiari Cephus, daughter of Cardi B and Offset (rapper) |
https://en.wikipedia.org/wiki/Hypercyclic%20operator | In mathematics, especially functional analysis, a hypercyclic operator on a Banach space X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense in the whole space X. In other words, the smallest closed invariant subset containing x is the whole space. Such an x is then called hypercyclic vector.
There is no hypercyclic operator in finite-dimensional spaces, but the property of hypercyclicity in spaces of infinite dimension is not a rare phenomenon: many operators are hypercyclic.
The hypercyclicity is a special case of broader notions of topological transitivity (see topological mixing), and universality. Universality in general involves a set of mappings from one topological space to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning to hypercyclicity. Examples of universal objects were discovered already in 1914 by Julius Pál, in 1935 by Józef Marcinkiewicz, or MacLane in 1952. However, it was not until the 1980s when hypercyclic operators started to be more intensively studied.
Examples
An example of a hypercyclic operator is two times the backward shift operator on the ℓ2 sequence space, that is the operator, which takes a sequence
(a1, a2, a3, …) ∈ ℓ2
to a sequence
(2a2, 2a3, 2a4, …) ∈ ℓ2.
This was proved in 1969 by Rolewicz.
Known results
On every infinite-dimensional separable Banach space there is a hypercyclic operator. On the other hand, there is no hypercyclic operator on a finite-dimensional space, nor on a non-separable Banach space.
If x is a hypercyclic vector, then Tnx is hypercyclic as well, so there is always a dense set of hypercyclic vectors.
Moreover, the set of hypercyclic vectors is a connected Gδ set, and always contains a dense vector space, up to {0}.
constructed an operator on ℓ1, such that all the non-zero vectors are hypercyclic, providing a counterexample to the invariant subspace problem (and even invariant subset problem) in the class of Banach spaces. The problem, whether such an operator (sometimes called hypertransitive, or orbit transitive) exists on a separable Hilbert space, is still open (as of 2014).
References
See also
Topological mixing
Functional analysis
Operator theory
Invariant subspaces |
https://en.wikipedia.org/wiki/Waring%E2%80%93Goldbach%20problem | The Waring–Goldbach problem is a problem in additive number theory, concerning the representation of integers as sums of powers of prime numbers. It is named as a combination of Waring's problem on sums of powers of integers, and the Goldbach conjecture on sums of primes. It was initiated by Hua Luogeng in 1938.
Problem statement
It asks whether large numbers can be expressed as a sum, with at most a constant number of terms, of like powers of primes. That is, for any given natural number, k, is it true that for sufficiently large integer N there necessarily exist a set of primes, {p1, p2, ..., pt}, such that N = p1k + p2k + ... + ptk, where t is at most some constant value?
The case, k=1, is a weaker version of the Goldbach conjecture. Some progress has been made on the cases k=2 to 7.
Heuristic justification
By the prime number theorem, the number of k-th powers of a prime below x is of the order x1/k/log x.
From this, the number of t-term expressions with sums ≤x is roughly xt/k/(log x)t.
It is reasonable to assume that for some sufficiently large number t this is x-c, i.e., all numbers up to x are t-fold sums of k-th powers
of primes. This argument is, of course, a long way from a strict proof.
Relevant results
In his monograph, using and refining the methods of Hardy, Littlewood and Vinogradov, Hua Luogeng obtains a O(k2log k) upper bound for the number of terms required to exhibit all sufficiently large numbers as the sum of k-th powers of primes.
Every sufficiently large odd integer is the sum of 21 fifth powers of primes.
References
Additive number theory
Conjectures about prime numbers
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Statistical%20model%20validation | In statistics, model validation is the task of evaluating whether a chosen statistical model is appropriate or not. Oftentimes in statistical inference, inferences from models that appear to fit their data may be flukes, resulting in a misunderstanding by researchers of the actual relevance of their model. To combat this, model validation is used to test whether a statistical model can hold up to permutations in the data. This topic is not to be confused with the closely related task of model selection, the process of discriminating between multiple candidate models: model validation does not concern so much the conceptual design of models as it tests only the consistency between a chosen model and its stated outputs.
There are many ways to validate a model. Residual plots plot the difference between the actual data and the model's predictions: correlations in the residual plots may indicate a flaw in the model. Cross validation is a method of model validation that iteratively refits the model, each time leaving out just a small sample and comparing whether the samples left out are predicted by the model: there are many kinds of cross validation. Predictive simulation is used to compare simulated data to actual data. External validation involves fitting the model to new data. Akaike information criterion estimates the quality of a model.
Overview
Model validation comes in many forms and the specific method of model validation a researcher uses is often a constraint of their research design. To emphasize, what this means is that there is no one-size-fits-all method to validating a model. For example, if a researcher is operating with a very limited set of data, but data they have strong prior assumptions about, they may consider validating the fit of their model by using a Bayesian framework and testing the fit of their model using various prior distributions. However, if a researcher has a lot of data and is testing multiple nested models, these conditions may lend themselves toward cross validation and possibly a leave one out test. These are two abstract examples and any actual model validation will have to consider far more intricacies than describes here but these example illustrate that model validation methods are always going to be circumstantial.
In general, models can be validated using existing data or with new data, and both methods are discussed more in the following subsections, and a note of caution is provided, too.
Validation with Existing Data
Validation based on existing data involves analyzing the goodness of fit of the model or analyzing whether the residuals seem to be random (i.e. residual diagnostics). This method involves using analyses of the models closeness to the data and trying to understand how well the model predicts its own data. One example of this method is in Figure 1, which shows a polynomial function fit to some data. We see that the polynomial function does not conform well to the data, which appears lin |
https://en.wikipedia.org/wiki/Intra-rater%20reliability | In statistics, intra-rater reliability is the degree of agreement among repeated administrations of a diagnostic test performed by a single rater. Intra-rater reliability and inter-rater reliability are aspects of test validity.
See also
Inter-rater reliability
Rating (pharmaceutical industry)
Reliability (statistics)
Repeatability
Test-retest reliability
References
Comparison of assessments
Statistical reliability |
https://en.wikipedia.org/wiki/Jan%20Mycielski | Jan Mycielski (born February 7, 1932 in Wiśniowa, Podkarpackie Voivodeship, Poland) is a Polish-American mathematician, a professor emeritus of mathematics at the University of Colorado at Boulder.
Academic career
Mycielski received his Ph.D. in mathematics from the University of Wrocław in 1957 under the supervision of . His dissertation was entitled "Applications of Free Groups to Geometrical Constructions". Following positions at the Centre National de la Recherche Scientifique in Paris, the Institute of Mathematics of the Polish Academy of Sciences, the University of California, Berkeley, and Case Western Reserve University, he took a permanent faculty position at Colorado in 1969.
Contributions
Among the mathematical concepts named after Mycielski are:
The Ehrenfeucht–Mycielski sequence, a sequence of binary digits with pseudorandom properties
The Mycielskian, a construction for embedding any undirected graph into a larger graph with strictly higher chromatic number without creating any additional triangles.
The Mycielski–Grötzsch graph, the Mycielskian of the 5-cycle, an 11-vertex triangle-free graph that is the smallest possible triangle-free graph requiring four colors.
Mycielski's theorem that there exist triangle-free graphs with arbitrarily large chromatic number.
Awards and honors
In 2012 he became a fellow of the American Mathematical Society.
Selected works
1991. A Note on S. M. Ulam's Mathematics.
A note in Adventures of a Mathematician. Stanislaw Ulam. University of California Press, 1991.
See also
List of Poles
References
External links
Mycielski's personal web page at the Univ. of Colorado.
1932 births
Living people
Polish set theorists
Polish logicians
University of Colorado Boulder faculty
Polish emigrants to the United States
University of Wrocław alumni
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
Jan
20th-century Polish philosophers |
https://en.wikipedia.org/wiki/Pythagorean%20theorem | In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation:
The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points.
The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.
Proofs using constructed squares
Rearrangement proofs
In one rearrangement proof, two squares are used whose sides have a measure of and which contain four right triangles whose sides are a, b and c, with the hypotenuse being c. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c. Each outer square has an area of as well as , with representing the total area of the four triangles. Within the big square on the left side, the four triangles are moved to form two similar rectangles with sides of length a and b. These rectangles in their new position have now delineated two new squares, one having side length a is formed in the bottom-left corner, and another square of side length b formed in the top-right corner. In this new position, this left side now has a square of area as well as . Since both squares have the area of it follows that the other measure of the square area also equal each other such that = . With the area of the four triangles removed from both side of the equation what remains is
In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas and which will again lead to a second square |
https://en.wikipedia.org/wiki/Richard%20Lashof | Richard K. Lashof (November 9, 1922 – February 4, 2010) was an American mathematician. He contributed to the field of geometric and differential topology, working with Shiing-Shen Chern, Stephen Smale, among others. Lashof is regarded as "the key figure in sustaining the Chicago Mathematics Department as an international center for research and the training of topologists" by Melvin Rothenberg.
Born in Philadelphia, Pennsylvania, Lashof earned a Bachelor of Science degree in chemical engineering from the University of Pennsylvania in 1943, then served as a communications officer in the United States Navy from 1943 to 1946. He earned his doctoral degree in mathematics from Columbia University in 1954, under supervision of Richard Vincent Kadison. In 1971 he was an Invited Speaker at the International Congress of Mathematicians in Nice.
His wife, Joyce Lashof, was awarded the Sedgwick Memorial Medal in 1995.
References
External links
1922 births
2010 deaths
Mathematicians from Philadelphia
20th-century American mathematicians
21st-century American mathematicians
Topologists
University of Pennsylvania School of Engineering and Applied Science alumni
Columbia Graduate School of Arts and Sciences alumni
University of Chicago faculty
United States Navy personnel of World War II |
https://en.wikipedia.org/wiki/Lie%20point%20symmetry | Lie point symmetry is a concept in advanced mathematics. Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations. This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Élie Cartan.
Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are translations, rotations and scalings.
The Lie symmetry theory is a well-known subject. In it are discussed continuous symmetries opposed to, for example, discrete symmetries. The literature for this theory can be found, among other places, in these notes.
Overview
Types of symmetries
Lie groups and hence their infinitesimal generators can be naturally "extended" to act on the space of independent variables, state variables (dependent variables) and derivatives of the state variables up to any finite order. There are many other kinds of symmetries. For example, contact transformations let coefficients of the transformations infinitesimal generator depend also on first derivatives of the coordinates. Lie-Bäcklund transformations let them involve derivatives up to an arbitrary order. The possibility of the existence of such symmetries was recognized by Noether. For Lie point symmetries, the coefficients of the infinitesimal generators depend only on coordinates, denoted by .
Applications
Lie symmetries were introduced by Lie in order to solve ordinary differential equations. Another application of symmetry methods is to reduce systems of differential equations, finding equivalent systems of differential equations of simpler form. This is called reduction. In the literature, one can find the classical reduction process, and the moving frame-based reduction process. Also symmetry groups can be used for classifying different symmetry classes of solutions.
Geometrical framework
Infinitesimal approach
Lie's fundamental theorems underline that Lie groups can be characterized by elements known as infinitesimal generators. These mathematical objects form a Lie algebra of infinitesimal generators. Deduced "infinitesimal symmetry conditions" (defining equations of the symmetry group) can be explicitly solved in order to find the closed form of symmetry groups, and thus the associated infinitesimal generators.
Let be |
https://en.wikipedia.org/wiki/Sperner%20property%20of%20a%20partially%20ordered%20set | In order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset. Since every rank level is itself an antichain, the Sperner property is equivalently the property that some rank level is a maximum antichain. The Sperner property and Sperner posets are named after Emanuel Sperner, who proved Sperner's theorem stating that the family of all subsets of a finite set (partially ordered by set inclusion) has this property. The lattice of partitions of a finite set typically lacks the Sperner property.
Variations
A k-Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels, or, equivalently, the poset has a maximum k-family consisting of k rank levels.
A strict Sperner poset is a graded poset in which all maximum antichains are rank levels.
A strongly Sperner poset is a graded poset which is k-Sperner for all values of k up to the largest rank value.
References
Order theory |
https://en.wikipedia.org/wiki/Truncated%207-orthoplexes | In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.
There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated 7-orthoplex are located on the triangular faces of the 7-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 7-orthoplex. The final three truncations are best expressed relative to the 7-cube.
Truncated 7-orthoplex
Alternate names
Truncated heptacross
Truncated hecatonicosoctaexon (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 7-orthoplex, centered at the origin, are all 168 vertices are sign (4) and coordinate (42) permutations of
(±2,±1,0,0,0,0,0)
Images
Construction
There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or [4,35] Coxeter group, and a lower symmetry with the D7 or [34,1,1] Coxeter group.
Bitruncated 7-orthoplex
Alternate names
Bitruncated heptacross
Bitruncated hecatonicosoctaexon (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±1,0,0,0,0)
Images
Tritruncated 7-orthoplex
The tritruncated 7-orthoplex can tessellation space in the quadritruncated 7-cubic honeycomb.
Alternate names
Tritruncated heptacross
Tritruncated hecatonicosoctaexon (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-orthoplex, centered at the origin, are all sign and coordinate permutations of
(±2,±2,±2,±1,0,0,0)
Images
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
x3x3o3o3o3o4o - tez, o3x3x3o3o3o4o - botaz, o3o3x3x3o3o4o - totaz
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
7-polytopes |
https://en.wikipedia.org/wiki/Andrew%20Ronald%20Mitchell | Andrew Ronald Mitchell (22 June 1921 – 22 November 2007) was a British applied mathematician and numerical analyst. He was a professor of mathematics at the University of St Andrews, Dundee, Scotland. He was known for his contributions to the field of numerical analysis of partial differential equations in general and finite difference method and finite element method in particular. Mitchell has authored several influential books on numerical solution of partial differential equations, including "The Finite Element Analysis in Partial Differential Equations" with Richard Wait and "The Finite Difference Method in Partial Differential Equations" with David F. Griffiths.
Early life and education
Mitchell was born in Dundee, Scotland, on 22 June 1921. His father was a blacksmith. Mitchell went to school at Morgan Academy, Dundee. He played football at school and was invited to sign for North End Junior Football Club in Dundee. He left Morgan Academy in 1938 after receiving a scholarships through the school to do a mathematics degree in the University College, Dundee. He graduated with First Class Honours in 1942, and was called up and sent to the wartime Ministry of Aircraft Production in London, where he remained until after the end of the war. While he was in London, he continued to play football during the war, turning out a few times for Chelsea. After the war he played for a number of Scottish clubs including St Johnstone, East Fife, Brechin City before ending his playing career at Berwick Rangers in 1955.
References
External links
1921 births
2007 deaths
Mitchell, Andrew Ronald
Alumni of the University of Dundee
Footballers from Dundee
Berwick Rangers F.C. players
Brechin City F.C. players
Chelsea F.C. wartime guest players
East Fife F.C. players
St Johnstone F.C. players
Scottish Football League players
Men's association football wing halves
Scottish men's footballers |
https://en.wikipedia.org/wiki/Ordinal%20number | In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as linearly ordered labels that include the natural numbers and have the property that every set of ordinals has a least element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number (omega) that is greater than every natural number, along with ordinal numbers , , etc., which are even greater than .
A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered, and given two well-ordered sets, one is isomorphic to an initial segment of the other. So ordinal numbers exist and are essentially unique.
Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not always apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.
Ordinals were introduced by Georg Cantor in 1883 in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.
Ordinals extend the natural numbers
A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets, these two concepts coincide, since all linear orders of a finite set are isomorphic.
When dealing with infinite sets, however, one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which leads to the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.
Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered. A well-ordered set is a totally ordered set in which every non-empty subset has a least element (a totally ordered set is an ordered set such that, given two distinct elements, one is less than the other). Equivalently, assuming the axiom of dependent choice, it is a totally ordered set without any infinite decreasing se |
https://en.wikipedia.org/wiki/Nonlinear%20complementarity%20problem | In applied mathematics, a nonlinear complementarity problem (NCP) with respect to a mapping ƒ : Rn → Rn, denoted by NCPƒ, is to find a vector x ∈ Rn such that
where ƒ(x) is a smooth mapping. The case of a discontinuous mapping was discussed by Habetler and Kostreva (1978).
References
Applied mathematics |
https://en.wikipedia.org/wiki/Division%20algebra | In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra D over a field. We call D a division algebra if for any element a in D and any non-zero element b in D there exists precisely one element x in D with a = bx and precisely one element y in D such that .
For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element a has a multiplicative inverse (i.e. an element x with ).
Associative division algebras
The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-dimensional as a vector space over the reals). The Frobenius theorem states that up to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4).
Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field.
Over an algebraically closed field K (for example the complex numbers C), there are no finite-dimensional associative division algebras, except K itself.
Associative division algebras have no nonzero zero divisors. A finite-dimensional unital associative algebra (over any field) is a division algebra if and only if it has no nonzero zero divisors.
Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion.
The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F.
One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).
For infinite-dimensional associative division algebras, the most important cases are those where the space has some reasonable topology. See for example normed division algebras and Banach algebras.
Not necessarily associative division algebras
If the division algebra is not assumed to be associative, usually some weaker condition (such as alternativity or power associativity) is imposed instead. See algebra over a field for a list of such conditions.
Over the reals there are (up to isomorphism) onl |
https://en.wikipedia.org/wiki/Limit%20%28mathematics%29 | In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals.
The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.
In formulas, a limit of a function is usually written as
(although a few authors use "Lt" instead of "lim")
and is read as "the limit of of as approaches equals ". The fact that a function approaches the limit as approaches is sometimes denoted by a right arrow (→ or ), as in
which reads " of tends to as tends to ".
History
Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."
The modern definition of a limit goes back to Bernard Bolzano who, in 1817, developed the basics of the epsilon-delta technique to define continuous functions. However, his work remained unknown to other mathematicians until thirty years after his death.
Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit.
The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, who introduced it in his book A Course of Pure Mathematics in 1908.
Types of limits
In sequences
Real numbers
The expression 0.999... should be interpreted as the limit of the sequence 0.9, 0.99, 0.999, ... and so on. This sequence can be rigorously shown to have the limit 1, and therefore this expression is meaningfully interpreted as having the value 1.
Formally, suppose is a sequence of real numbers. When the limit of the sequence exists, the real number is the limit of this sequence if and only if for every real number , there exists a natural number such that for all , we have .
The common notation
is read as:
"The limit of an as n approaches infinity equals L" or "The limit as n approaches infinity of an equals L".
The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value is the distance between and .
Not every sequence has a limit. A sequence with a limit is called convergent; otherwise it is called divergent. One can show that a convergent sequence has only one limit.
The limit of a sequence and the limit of a function are closely related. On one hand, the limit as approaches infinity of a sequence is simply the limit at infinity of a function —defined on the natural numbers . On the other hand, if X is the domain of a function and if the limit as approaches inf |
https://en.wikipedia.org/wiki/Igor%20Girsanov | Igor Vladimirovich Girsanov (; (10 September 1934 – 16 March 1967) was a Russian mathematician. He made major contributions to probability theory and its applications.
Early life
Igor Girsanov was born on 10 September 1934, in Turkestan (then Kazakh ASSR). He studied in Baku until his family moved to Moscow in 1950. While at school he was an active member of the Moscow State University maths club and won multiple Moscow mathematics olympiads.
Education
Between 1952 and 1960, Girsanov was an undergraduate and graduate student at Moscow State University. After his graduation he joined the faculty. In 1965 he became Head of the newly formed Probability and Statistics Laboratory at MSU.
Academic work
One can distinguish two periods in Girsanov's academic work.
Prior to 1961, he worked as a member of a group of mathematicians united around E. B. Dynkin who were developing the theory of Markov processes. His thesis introduced the concept of a strong Feller process which proved to be particularly useful. In the same thesis he considered the problem of applying Markov processes to the solution of partial differential equations. He was among the first scientists to study elliptic and parabolic equations with discontinuous coefficients. In his papers on stochastic differential equations he determined the conditions when the discontinuity of the coefficients does not prevent the solution from being unique. He also produced important papers on the general theory of Markov processes.
Girsanov was able to learn areas of mathematics that were unfamiliar to him very quickly. At the same time he was considering questions unrelated to stochastic processes. For example, he constructed an example of a dynamical system with a simple spectrum. In collaboration with B. S. Mityagin he worked on quasi-invariant measures on topological linear spaces.
Around 1960 the problems of optimal management in industry and economics came to the fore in USSR. Many mathematicians were working in this area, mostly using familiar mathematical tools. Girsanov also started to work within this framework. But in 1961 he changed his approach. He began to develop a broader understanding of the issues at hand, developing new mathematical techniques.
Girsanov became an advocate of mathematical economics, actively defending it against the opponents of quantitative methods. His research results had applications in industry.
He published fifteen papers on applications to chemistry. His interests included the optimal control of chemical reactors.
He also maintained his interest in theoretical mathematics developing the functional analysis required for optimisation problems.
Death
Girsanov died on 16 March 1967 at the age of 32 as a result of an accident during a hike in the Sayan Mountains.
References
1934 births
1967 deaths
20th-century Russian mathematicians |
https://en.wikipedia.org/wiki/Reeb%20sphere%20theorem | In mathematics, Reeb sphere theorem, named after Georges Reeb, states that
A closed oriented connected manifold M n that admits a singular foliation having only centers is homeomorphic to the sphere Sn and the foliation has exactly two singularities.
Morse foliation
A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are level sets of a Morse function, being the singularity a critical point of the function. The singularity is a center if it is a local extremum of the function; otherwise, the singularity is a saddle.
The number of centers c and the number of saddles , specifically , is tightly connected with the manifold topology.
We denote , the index of a singularity , where k is the index of the corresponding critical point of a Morse function. In particular, a center has index 0, index of a saddle is at least 1.
A Morse foliation F on a manifold M is a singular transversely oriented codimension one foliation of class with isolated singularities such that:
each singularity of F is of Morse type,
each singular leaf L contains a unique singularity p; in addition, if then is not connected.
Reeb sphere theorem
This is the case , the case without saddles.
Theorem: Let be a closed oriented connected manifold of dimension . Assume that admits a -transversely oriented codimension one foliation with a non empty set of singularities all of them centers. Then the singular set of consists of two points and is homeomorphic to the sphere .
It is a consequence of the Reeb stability theorem.
Generalization
More general case is
In 1978, Edward Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles. He showed that the number of centers cannot be too much as compared with the number of saddles, notably, . So there are exactly two cases when :
(1)
(2)
He obtained a description of the manifold admitting a foliation with singularities that satisfy (1).
Theorem: Let be a compact connected manifold admitting a Morse foliation with centers and saddles. Then .
In case ,
is homeomorphic to ,
all saddles have index 1,
each regular leaf is diffeomorphic to .
Finally, in 2008, César Camacho and Bruno Scardua considered the case (2), . This is possible in a small number of low dimensions.
Theorem: Let be a compact connected manifold and a Morse foliation on . If , then
or ,
is an Eells–Kuiper manifold.
References
Foliations
Theorems in topology |
https://en.wikipedia.org/wiki/Pakistanis%20in%20Oman | Pakistanis in Oman are either Pakistani people who live in Oman, Pakistani immigrants to Oman and people born in Oman of Pakistani descent. According to official government statistics published by the Sultanate of Oman, the population of Pakistani expatriates stood at 231,685 in November 2016.
See also
Al Balushi
References
Oman
Ethnic groups in Oman |
https://en.wikipedia.org/wiki/Eells%E2%80%93Kuiper%20manifold | In mathematics, an Eells–Kuiper manifold is a compactification of by a sphere of dimension , where , or . It is named after James Eells and Nicolaas Kuiper.
If , the Eells–Kuiper manifold is diffeomorphic to the real projective plane . For it is simply-connected and has the integral cohomology structure of the complex projective plane (), of the quaternionic projective plane () or of the Cayley projective plane ().
Properties
These manifolds are important in both Morse theory and foliation theory:
Theorem: Let be a connected closed manifold (not necessarily orientable) of dimension . Suppose admits a Morse function of class with exactly three singular points. Then is a Eells–Kuiper manifold.
Theorem: Let be a compact connected manifold and a Morse foliation on . Suppose the number of centers of the foliation is more than the number of saddles . Then there are two possibilities:
, and is homeomorphic to the sphere ,
, and is an Eells–Kuiper manifold, or .
See also
Reeb sphere theorem
References
Foliations
Manifolds |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.